From 35167ec1568723a83ecd435a79ab004b5416adf3 Mon Sep 17 00:00:00 2001 From: Adam Getchell Date: Sat, 25 Jul 2026 22:48:37 -0700 Subject: [PATCH 1/3] feat(validation)!: add certified realization intersection fast paths - Add orientation, shared-face, and exact linear-program certificates before the conservative active-set fallback. - Dispatch finite floating-point systems through la-stack's exact Bareiss solver while preserving rational fallbacks and typed witnesses. - Bound the 4D and 5D release regressions and document the broader evidence deferred to #482 and #483. - Align release documentation and tooling with Rust 1.97.1. - Refresh pinned development tools and override Semgrep's vulnerable MCP dependency with the patched release. - Harden Tectonic dependency discovery across supported development environments. BREAKING CHANGE: Rust 1.97.1 is now the minimum supported Rust version. Closes #506 --- .config/nextest.toml | 8 + .github/workflows/semgrep-sarif.yml | 2 +- AGENTS.md | 2 +- Cargo.toml | 2 +- README.md | 2 +- clippy.toml | 2 +- docs/dev/commands.md | 18 +- docs/dev/tooling-alignment.md | 22 +- docs/limitations.md | 11 +- docs/roadmap.md | 21 +- justfile | 106 +- papers/ARTIFACT.md | 6 +- papers/validation.pdf | Bin 1211787 -> 1228570 bytes papers/validation.tex | 64 + pyproject.toml | 7 +- rust-toolchain.toml | 2 +- scripts/tests/test_benchmark_models.py | 4 + src/geometry/matrix.rs | 30 +- src/geometry/realization.rs | 1809 +++++++++++++++++++++++- tests/large_scale_debug.rs | 22 +- uv.lock | 383 ++--- 21 files changed, 2209 insertions(+), 314 deletions(-) diff --git a/.config/nextest.toml b/.config/nextest.toml index a88ba22e..33cce4e7 100644 --- a/.config/nextest.toml +++ b/.config/nextest.toml @@ -69,3 +69,11 @@ success-output = "never" fail-fast = false retries = 0 slow-timeout = { period = "900s", terminate-after = 1 } + +# Full Levels 1-5 validation at the bounded 2D-5D regression scales is +# CPU-intensive. Give each large-scale case the complete worker pool so +# concurrent slow fixtures cannot consume its 600-second internal wall-clock +# budget. Broader Level 4 and Level 5 performance work remains in #482/#483. +[[profile.slow.overrides]] +filter = 'test(debug_large_scale_)' +threads-required = "num-cpus" diff --git a/.github/workflows/semgrep-sarif.yml b/.github/workflows/semgrep-sarif.yml index b8cadcf4..fc08eaf9 100644 --- a/.github/workflows/semgrep-sarif.yml +++ b/.github/workflows/semgrep-sarif.yml @@ -67,7 +67,7 @@ jobs: id: semgrep run: | set +e - uv run semgrep \ + uv run --locked semgrep \ --error \ --strict \ --timeout 30 \ diff --git a/AGENTS.md b/AGENTS.md index efd74a04..2b79940e 100644 --- a/AGENTS.md +++ b/AGENTS.md @@ -79,7 +79,7 @@ workflow references do not need to be loaded preemptively. - **Language**: Rust - **Project**: d-dimensional Delaunay triangulation library -- **MSRV**: 1.97.0 +- **MSRV**: 1.97.1 - **Edition**: 2024 - **Primary architecture hub**: `docs/code_organization.md` diff --git a/Cargo.toml b/Cargo.toml index 2996251c..060bf721 100644 --- a/Cargo.toml +++ b/Cargo.toml @@ -2,7 +2,7 @@ name = "delaunay" version = "0.7.8" edition = "2024" -rust-version = "1.97.0" +rust-version = "1.97.1" autobins = false authors = [ "Adam Getchell " ] homepage = "https://github.com/acgetchell/delaunay" diff --git a/README.md b/README.md index bb424d3c..5ad22670 100644 --- a/README.md +++ b/README.md @@ -123,7 +123,7 @@ cargo add delaunay@0.7.8 Use `cargo add delaunay` instead if you want Cargo to select the newest published release. -- Rust 1.97.0 or newer. The minimum supported version is declared in +- Rust 1.97.1 or newer. The minimum supported version is declared in `Cargo.toml`, while `rust-toolchain.toml` pins the exact repository toolchain. - `f64` coordinates for caller-facing construction, predicate, validation, and generator APIs. diff --git a/clippy.toml b/clippy.toml index ecfe128a..485b6079 100644 --- a/clippy.toml +++ b/clippy.toml @@ -2,7 +2,7 @@ # This ensures consistent configuration across all developers and IDEs # Minimum Supported Rust Version - matches Cargo.toml and rust-toolchain.toml -msrv = "1.97.0" +msrv = "1.97.1" # Lint levels are controlled by Cargo.toml plus the `just clippy` / CI # invocation flags. This file contains Clippy configuration options that are diff --git a/docs/dev/commands.md b/docs/dev/commands.md index 06eab368..4b1f150e 100644 --- a/docs/dev/commands.md +++ b/docs/dev/commands.md @@ -632,13 +632,17 @@ the canonical figures and reviewer PDF through those named artifact owners. Tectonic and `tex-fmt` are pinned Cargo-installed tools. `chktex` comes from a TeX distribution or system package manager. Installing or upgrading Tectonic -from Cargo also requires `pkg-config` and development headers for its native -bridge libraries, including fontconfig, FreeType, Graphite2, HarfBuzz, ICU, -libpng, and zlib. When the pinned Tectonic version is absent, -`just setup-tools` checks ICU locally and auto-detects common Homebrew ICU -pkg-config directories before it asks for a manual `PKG_CONFIG_PATH`; an -already-correct Tectonic installation does not require those native build -prerequisites. Paper CI installs the platform native package set explicitly. +from Cargo also requires a `pkg-config` implementation and development headers +for its externally resolved native bridge libraries, including fontconfig, +FreeType, Graphite2, ICU, libpng, and zlib; the pinned default build vendors +HarfBuzz. When the pinned Tectonic version is absent, `just setup-tools` uses an +existing `pkg-config` or obtains it ephemerally with +`pkgx +freedesktop.org/pkg-config`, then checks the complete external native +dependency set. On macOS it auto-detects common Homebrew metadata directories, +including the active SDK metadata used for system compression libraries, before +it asks for a manual `PKG_CONFIG_PATH`. An already-correct Tectonic installation +does not require those native build prerequisites. Paper CI installs the +platform native package set explicitly. Reviewer-facing validation diagrams under `docs/assets/validation/` use the same deterministic notebook with a separate explicit output switch: diff --git a/docs/dev/tooling-alignment.md b/docs/dev/tooling-alignment.md index bae07f84..aa4fb590 100644 --- a/docs/dev/tooling-alignment.md +++ b/docs/dev/tooling-alignment.md @@ -58,7 +58,7 @@ Both repositories now share the same core Rust and Python support-tooling loop: The useful updates ported in this pass are: -- Rust MSRV metadata uses Rust 1.97.0 for the v0.8.0 release line. +- Rust MSRV metadata uses Rust 1.97.1 for the v0.8.0 release line. `Cargo.toml`, `rust-toolchain.toml`, `clippy.toml`, contributor docs, and agent guidance use the same baseline so upcoming `la-stack` and `markov-chain-monte-carlo` releases have no MSRV conflict. @@ -66,15 +66,17 @@ The useful updates ported in this pass are: `justfile`, with `cargo-nextest` on 0.9.140 and `cargo-llvm-cov` on 0.8.7, which is still shared across the repositories. - CI command-runner, Markdown, YAML formatting, and spelling tool pins now - track the current reviewed versions: `just` 1.56.0, `rumdl` 0.2.34, + track the current reviewed versions: `just` 1.57.0, `rumdl` 0.2.43, `dprint` 0.55.2, and `typos-cli` 1.48.0. The linter and formatter pins keep local setup and CI on the same reviewed releases. - uv-managed Python support-tool pins now use exact reviewed versions for the - shared dev tools: `ruff` 0.15.21, `semgrep` 1.169.0, and `ty` 0.0.59. - Semgrep is intentionally ahead of the older sibling baseline so its transitive - dependency graph stays on the current reviewed toolchain baseline. Delaunay - previously used lower-bound specifiers for those tools, which allowed local - and CI environments to drift away from that reviewed baseline. + shared dev tools: `ruff` 0.15.21, `semgrep` 1.171.0, and `ty` 0.0.59. + Semgrep still pins the vulnerable `mcp` 1.23.3 release, so the uv project + configuration overrides that transitive dependency to patched `mcp` 1.28.1. + Remove the override once Semgrep's published dependency constraint reaches + that version or later. Delaunay previously used lower-bound specifiers for + those tools, which allowed local and CI environments to drift away from that + reviewed baseline. - Local just helpers now version-check the pinned `uv`, `just`, `cargo-nextest`, `cargo-machete`, `taplo-cli`, `dprint`, `git-cliff`, `rumdl`, `samply`, `typos-cli`, and `zizmor` tools instead of accepting any installed @@ -241,7 +243,11 @@ The useful updates ported in this pass are: install the same Cargo-tool pins with the sibling repositories' `taiki-e/cache-cargo-install-action`/`cargo install --locked` pattern before the corresponding recipes run. Pinned Rust CLI tools are installed through - Cargo rather than Homebrew so local setup cannot drift from CI pins. The CI + Cargo rather than Homebrew so local setup cannot drift from CI pins. The + local Tectonic bootstrap likewise avoids a permanent `pkg-config` installation: + when that command is absent, `just setup-tools` obtains it ephemerally through + `pkgx` for the dependency probe and Cargo installation. The external native + bridge libraries and their pkg-config metadata remain platform-managed. The CI build matrix runs `just ci`, including `cargo machete` dependency hygiene, on Linux, macOS, and Windows after syncing the locked uv dev group and installing the pinned Cargo tools. The papers workflow diff --git a/docs/limitations.md b/docs/limitations.md index fd92d465..32f96401 100644 --- a/docs/limitations.md +++ b/docs/limitations.md @@ -110,10 +110,13 @@ release characterization, not a portable performance promise: defaults to 36,000 vertices, 3D defaults to 7,500 vertices, 4D defaults to 800 vertices, and 5D defaults to 140 vertices. Pass `n` explicitly when a run must match a documented scale exactly. -- The raw `slow-tests` harness defaults use slightly heavier defaults for some - dimensions (currently 40,000 vertices in 2D, 900 vertices in 4D, and - 150 vertices in 5D). Prefer the `just` helpers for routine - acceptance/profiling runs. +- The raw `slow-tests` harness defaults are 40,000 vertices in 2D, 7,500 in + 3D, 100 in 4D, and 50 in 5D. The 4D and 5D defaults are deliberately bounded + release-regression probes, not scale or performance evidence. Broader Level 4 + realization and Level 5 Delaunay validation work remains in + [#482](https://github.com/acgetchell/delaunay/issues/482) and + [#483](https://github.com/acgetchell/delaunay/issues/483) for v0.8.1. + Prefer the `just` helpers for explicit acceptance and profiling runs. Current 2D scale envelope: diff --git a/docs/roadmap.md b/docs/roadmap.md index 7175cf38..ab256984 100644 --- a/docs/roadmap.md +++ b/docs/roadmap.md @@ -31,8 +31,8 @@ Key takeaways from v0.7.8: ### v0.8.0 paper-facing API and topology push v0.8.0 is the next feature-bearing release and is expected to carry the larger -work intentionally deferred from v0.7.8 cleanup. It will require Rust 1.97.0; -the final release gate is an explicit audit of the 1.97.0 toolchain surface +work intentionally deferred from v0.7.8 cleanup. It will require Rust 1.97.1; +the final release gate is an explicit audit of the 1.97.1 toolchain surface before shipping. - **Pachner/Edit API shape (#252/#253/#350/#337):** unify the Pachner move API, @@ -51,15 +51,26 @@ before shipping. - **Naming cleanup (#323):** make the breaking `Cell` → `Simplex` rename. - **Iterator cleanup (#353):** prefer iterator-based collection-building paths where that improves clarity and allocation behavior. -- **Rust 1.97.0 release gate (#329/#496):** raise the v0.8.0 MSRV to Rust - 1.97.0, finish the baseline `assert_matches!` cleanup, audit the new +- **Rust 1.97.1 release gate (#329/#496):** raise the v0.8.0 MSRV to Rust + 1.97.1, finish the baseline `assert_matches!` cleanup, audit the new integer/`NonZero` bit helpers against Hilbert bit-depth/index invariants, review `RepeatN::default` and Cargo 1.97 tooling changes for useful adoption, - and re-benchmark predicate `cold_path` decisions under the 1.97.0 compiler. + and re-benchmark predicate `cold_path` decisions under the 1.97.1 compiler. - **Notebook/export artifact (#64/#408):** ship the generic simplicial-complex JSON export, reproducible quickstart and validation notebooks, tracked validation diagrams, and reviewer-facing artifact instructions. +### v0.8.1 validation evidence and performance follow-up + +- **Level 4 realization validation (#482):** complete the independent + randomized and degenerate 2D–5D agreement campaign for the revised + simplex-intersection narrow phase and record representative before/after + benchmarks. +- **Level 5 Delaunay reporting (#483):** replace the Euclidean + all-vertices/all-simplices report bottleneck while preserving its violation + set, with 2D–5D agreement tests and before/after benchmarks. This work is + blocked by issue #482. + ### v0.9.0 and later horizon v0.9.0 is the right parking lot for work that is valuable but larger or less diff --git a/justfile b/justfile index a53bafa5..2605ff29 100644 --- a/justfile +++ b/justfile @@ -19,17 +19,17 @@ cargo_machete_version := "0.9.2" clippy_sarif_version := "0.8.0" dprint_version := "0.55.2" git_cliff_version := "2.13.1" -just_version := "1.56.0" +just_version := "1.57.0" nextest_version := "0.9.140" -rumdl_version := "0.2.34" +rumdl_version := "0.2.43" samply_version := "0.13.1" sarif_fmt_version := "0.8.0" taplo_version := "0.10.0" tectonic_version := "0.16.9" tex_fmt_version := "0.5.7" typos_version := "1.48.0" -uv_version := "0.11.29" -zizmor_version := "1.26.1" +uv_version := "0.11.32" +zizmor_version := "1.28.0" # Common cargo-llvm-cov arguments for all coverage runs. # Excludes benches/examples from reports while allowing integration tests to @@ -751,8 +751,8 @@ perf-help: @echo " # 4D random-walk Pachner diagnostics with CSV/JSON artifacts" @echo " just bench-ci # Final optimized CI-suite benchmark run" @echo " just profile v0.7.5 # v0.7.5 code on its declared Rust toolchain" - @echo " just profile 1.97.0 # Current tree on Rust 1.97.0" - @echo " just profile 1.97.0 v0.7.5 # v0.7.5 code on Rust 1.97.0" + @echo " just profile 1.97.1 # Current tree on Rust 1.97.1" + @echo " just profile 1.97.1 v0.7.5 # v0.7.5 code on Rust 1.97.1" # Quick pre-push 2D-5D large-scale wall-clock smoke guard. [group('benchmarks and performance')] @@ -782,8 +782,8 @@ perf-large-scale-smoke max_secs="60": _ensure-nextest echo "" echo "▶ ${dimension}: ${test_name} (${n_points} vertices, ${max_secs}s cap)" # Construction wall-clock guard: validate Levels 1-3 + Level 5 only. - # Level 4 embedding overlap validation runs at scale under `just test-slow` - # (full scope); see issue #482. + # Level 4 retains bounded full-scope regression coverage under + # `just test-slow`; broader Level 4/5 work remains in #482/#483. if env \ DELAUNAY_BULK_PROGRESS_EVERY="$progress_every" \ DELAUNAY_LARGE_DEBUG_MAX_RUNTIME_SECS="$max_secs" \ @@ -1075,7 +1075,7 @@ rust-core-check: fmt-check clippy doc-check semgrep semgrep-test # Repository-owned Semgrep rules for project-specific Rust diagnostics. [group('validation')] semgrep: _ensure-uv - uv run semgrep --error --strict --timeout 120 --config semgrep.yaml . + uv run --locked semgrep --error --strict --timeout 120 --config semgrep.yaml . # Test the repository-owned Semgrep rules against their fixtures. [group('validation')] @@ -1101,7 +1101,7 @@ semgrep-test: _ensure-uv mkdir -p "$state_dir" uv run python scripts/semgrep_fixture_config.py "$fixture" "$PWD/semgrep.yaml" "$config_path" - SEMGREP_SEND_METRICS=off SEMGREP_SETTINGS_FILE="$state_dir/settings.yml" uv run semgrep scan --test --strict --config "$config_path" "$fixture" + SEMGREP_SEND_METRICS=off SEMGREP_SETTINGS_FILE="$state_dir/settings.yml" uv run --locked semgrep scan --test --strict --config "$config_path" "$fixture" done < <(find tests/semgrep -type f ! -name '*.fixed' -print0) # Install required tools and build the development profile. @@ -1139,34 +1139,79 @@ setup-tools: _ensure-uv ensure_tectonic_build_dependencies() { echo "Ensuring native dependencies needed to install Tectonic..." + + local candidate homebrew_repository package sdk_version + local -a pkg_config_command=(pkg-config) + local -a missing_pkg_config_packages=() + local -a required_pkg_config_packages=(fontconfig freetype2 graphite2 icu-uc libpng zlib) + + append_pkg_config_path() { + local directory="$1" + if [ -d "$directory" ] && [[ ":${PKG_CONFIG_PATH:-}:" != *":$directory:"* ]]; then + export PKG_CONFIG_PATH="$directory${PKG_CONFIG_PATH:+:$PKG_CONFIG_PATH}" + fi + } + if ! have pkg-config; then - echo "❌ 'pkg-config' not found. Install pkg-config before building Tectonic from Cargo." - exit 1 + if ! have pkgx; then + echo "❌ Neither 'pkg-config' nor 'pkgx' was found. Install pkgx or pkg-config before building Tectonic from Cargo." + exit 1 + fi + pkg_config_command=(pkgx +freedesktop.org/pkg-config pkg-config) + echo " ✓ pkgx will supply pkg-config ephemerally" fi - if ! pkg-config --exists icu-uc; then - shopt -s nullglob - for candidate in \ - /opt/homebrew/opt/icu4c*/lib/pkgconfig \ - /usr/local/opt/icu4c*/lib/pkgconfig; do - if [ -d "$candidate" ]; then - export PKG_CONFIG_PATH="$candidate${PKG_CONFIG_PATH:+:$PKG_CONFIG_PATH}" - break - fi - done - shopt -u nullglob + + shopt -s nullglob + for candidate in \ + /opt/homebrew/lib/pkgconfig \ + /opt/homebrew/share/pkgconfig \ + /opt/homebrew/opt/{fontconfig,freetype,graphite2,icu4c*,libpng}/lib/pkgconfig \ + /usr/local/lib/pkgconfig \ + /usr/local/share/pkgconfig \ + /usr/local/opt/{fontconfig,freetype,graphite2,icu4c*,libpng}/lib/pkgconfig; do + append_pkg_config_path "$candidate" + done + shopt -u nullglob + + if have brew && have xcrun && sdk_version="$(xcrun --sdk macosx --show-sdk-version 2>/dev/null)"; then + homebrew_repository="$(brew --repository)" + append_pkg_config_path "$homebrew_repository/Library/Homebrew/os/mac/pkgconfig/${sdk_version%%.*}" fi - if ! pkg-config --exists icu-uc; then - echo "❌ 'icu-uc' was not found by pkg-config." - echo " Install ICU development files, or set PKG_CONFIG_PATH to the directory containing icu-uc.pc." + + for package in "${required_pkg_config_packages[@]}"; do + if ! "${pkg_config_command[@]}" --exists "$package"; then + missing_pkg_config_packages+=("$package") + fi + done + if (( ${#missing_pkg_config_packages[@]} )); then + echo "❌ pkg-config could not resolve: ${missing_pkg_config_packages[*]}" + echo " Install the missing native development files, or add their metadata directories to PKG_CONFIG_PATH." exit 1 fi - echo " ✓ pkg-config can resolve icu-uc" + echo " ✓ pkg-config can resolve Tectonic's native dependencies" echo "" } + ensure_pinned_tectonic() { + local expected_version="{{ tectonic_version }}" + + if cargo_tool_has_exact_version tectonic "$expected_version"; then + echo " ✓ tectonic $expected_version" + return + fi + + ensure_tectonic_build_dependencies + echo " ⏳ Installing tectonic $expected_version (cargo)..." + if have pkg-config; then + cargo install --locked tectonic --version "$expected_version" + else + pkgx +freedesktop.org/pkg-config cargo install --locked tectonic --version "$expected_version" + fi + } + echo "This recipe installs pinned Rust CLI tools through cargo." echo "External prerequisites that must already be on PATH: uv, jq, rustup, cargo, and chktex." - echo "pkg-config and ICU development files are required only when the pinned Tectonic version must be installed." + echo "pkgx or pkg-config, plus native development files, are required only when the pinned Tectonic version must be installed." echo "" echo "Ensuring uv-managed Python tooling..." @@ -1198,10 +1243,7 @@ setup-tools: _ensure-uv ensure_pinned_cargo_tool rumdl rumdl "{{ rumdl_version }}" ensure_pinned_cargo_tool samply samply "{{ samply_version }}" ensure_pinned_cargo_tool taplo taplo-cli "{{ taplo_version }}" - if ! cargo_tool_has_exact_version tectonic "{{ tectonic_version }}"; then - ensure_tectonic_build_dependencies - fi - ensure_pinned_cargo_tool tectonic tectonic "{{ tectonic_version }}" + ensure_pinned_tectonic ensure_pinned_cargo_tool tex-fmt tex-fmt "{{ tex_fmt_version }}" ensure_pinned_cargo_tool typos typos-cli "{{ typos_version }}" ensure_pinned_cargo_tool zizmor zizmor "{{ zizmor_version }}" diff --git a/papers/ARTIFACT.md b/papers/ARTIFACT.md index 165520fc..2b3a2719 100644 --- a/papers/ARTIFACT.md +++ b/papers/ARTIFACT.md @@ -19,7 +19,7 @@ documents remain the source of truth for their respective contracts. | Commit | The commit peeled from `v0.8.0` | `git rev-list -n 1 v0.8.0` | | GitHub archive | [`delaunay` at `v0.8.0`][source-tag] | Tag and commit must agree | | Crate | [`delaunay` 0.8.0 on crates.io][crate-release] | Package version must be `0.8.0` | -| Rust | 1.97.0 | `rustc --version` and [`rust-toolchain.toml`](../rust-toolchain.toml) | +| Rust | 1.97.1 | `rustc --version` and [`rust-toolchain.toml`](../rust-toolchain.toml) | | Zenodo collection | [`10.5281/zenodo.16931097`][zenodo-concept] | Stable software concept DOI | | Zenodo v0.8.0 record | Published by [the final release audit][release-audit] | Version DOI, tag, and commit must match this release | @@ -59,7 +59,7 @@ snapshot for executable reproduction. ## Environment and setup -The repository pins Rust 1.97.0 in both [`Cargo.toml`](../Cargo.toml) and +The repository pins Rust 1.97.1 in both [`Cargo.toml`](../Cargo.toml) and [`rust-toolchain.toml`](../rust-toolchain.toml). From the exact release checkout, install the repository-owned tools and dependencies with: @@ -342,7 +342,7 @@ describes a published artifact: 2. Run `just ci-slow`, `just papers`, and `just publish-check` from the release commit. 3. Run `just bench-perf-summary` after the version bump and verify the checked-in - summary records v0.8.0, the release commit, Rust 1.97.0, and hardware metadata. + summary records v0.8.0, the release commit, Rust 1.97.1, and hardware metadata. 4. Inspect `cargo package --list` and confirm the documented crates.io packaging decision remains true. 5. 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zFE4pJ*->S~zp)w~rnZXNcrcvVW3`HA_0OkSd1>ZP`0T%_&7L})A6fBu?uhvN;HE?o z%l2KT25j$qD7IQWWn0A@@$5$_F%Gl=k51_HgFqaAVRR$8I6M$_(gttOFosh5lyW7w)P0-a%@JAeR z?fZRogJhftTK4^`2SGiNF6x1C>!E+&A>ogZc%tdtVznHA+m0rCLd^cjsg05U_zFEI^{Bibcm4~p4w=`(}dHt~iMIrUaVu7BB(2hvI?LonZV`5@r NWQ8Fk6H^d}`5(F}BCr4e diff --git a/papers/validation.tex b/papers/validation.tex index 9cd8dc66..64e9c58a 100644 --- a/papers/validation.tex +++ b/papers/validation.tex @@ -45,6 +45,28 @@ \section{Introduction} \item \AuthorTodo{Preview the paper structure.} \end{itemize} +Applications such as Causal Dynamical Triangulation (CDT) require a robust and +reliable triangulation library, capable of maintaining correct properties of the +triangulation in simulations that may alter their structure millions +of times per +simulation. +These invariant properties should be verifiable during construction +and mutation, +and violations of these properties should be detectable and diagnosable, so that +fixable errors may be corrected and unfixable errors may be reported. + +Delaunay is a Rust library for constructing and mutating triangulations with +insertion, deletion, and bi-stellar k-flip operations, while maintaining a range +of invariant properties important for the construction of Pseudo- and +Piecewise-Linear (PL) manifolds, +the evaluation of geometric predicates, and the embedding of +triangulations in Euclidean, +toroidal, and spherical spaces. + +\AuthorTodo{Qualify the preceding overview so the ordinary mutable + Euclidean/toroidal workflows are distinguished from the separate bounded +\(S^2\)/\(S^3\) spherical prototype.} + \section{Background and Definitions} \begin{itemize} @@ -116,6 +138,48 @@ \subsection{Level 4: Valid Realization} stored coherence and Level 3 intrinsic orientability.} \item \AuthorTodo{Explain why realization validity is a precondition for geometric predicates.} + \item \AuthorTodo{Introduce two realized \(d\)-simplices with vertices + \(a_0,\ldots,a_d\) and \(b_0,\ldots,b_d\), vertex labels + \(\ell_i\) and \(m_j\), and shared-label set \(S\), then explain the exact + shared-face intersection program in + Equation~\ref{eq:shared-face-intersection-lp}.} + \begin{equation} + \label{eq:shared-face-intersection-lp} + \begin{aligned} + z^\star + ={}& \underset{\alpha_i,\beta_j \geq 0}{\operatorname{maximize}} + && \sum_{\ell_i \notin S} \alpha_i + + \sum_{m_j \notin S} \beta_j \\ + & \text{\rm subject to} + && \sum_{i=0}^{d} \alpha_i a_i + = \sum_{j=0}^{d} \beta_j b_j, \\ + &&& \sum_{i=0}^{d} \alpha_i = 1, + \qquad \sum_{j=0}^{d} \beta_j = 1 . + \end{aligned} + \end{equation} + \item \AuthorTodo{Explain the three exact outcomes: infeasibility proves the + simplices disjoint; \(z^\star=0\) proves every feasible intersection point + is confined to the shared face; and \(z^\star>0\) supplies barycentric + weight on non-shared labels for a typed intersection witness.} + \item \AuthorTodo{Explain how revised-simplex basis navigation replaces + exhaustive active-set enumeration of the intersection polytope on the + common path and removes the bottleneck exposed by the 4D and 5D Level 4 + checks.} + \item \AuthorTodo{Describe the filtered-exact hierarchy: bounding-box, + separating-facet, orientation, and shared-face normal tests reject or + certify easy pairs; floating-point simplex phases propose axes and bases; + proved floating-point error bounds or exact rational primal/dual checks + certify accepted results; and active-set enumeration remains the + conservative fallback.} + \item \AuthorTodo{Describe how exact finite-\texttt{f64} systems use the + runtime-dispatched fraction-free Bareiss solver from \texttt{la-stack}, + while general rational systems retain the \texttt{BigRational} fallback.} + \item \AuthorTodo{Treat the broader independent randomized and degenerate + 2D--5D agreement campaign and representative before/after benchmarks as + v0.8.1 work tracked by issue \#482.} + \item \AuthorTodo{Do not present the bounded v0.8.0 4D and 5D slow-test probes + as performance evidence. Coordinate any Level 5 all-violations performance + claims with the separate v0.8.1 work in issue \#483.} \end{itemize} \subsection{Level 5: Geometric Predicates} diff --git a/pyproject.toml b/pyproject.toml index 24ce393c..221a3a5e 100644 --- a/pyproject.toml +++ b/pyproject.toml @@ -242,7 +242,7 @@ dev = [ "actionlint-py==1.7.12.24", "pytest==9.1.1", "ruff==0.15.21", - "semgrep==1.169.0", + "semgrep==1.171.0", "shellcheck-py==0.11.0.1", "shfmt-py==4.0.0", "ty==0.0.59", @@ -258,3 +258,8 @@ notebooks = [ "polars>=1.42.1", "plotly>=6.9.0", ] + +[tool.uv] +# Semgrep 1.171.0 still pins vulnerable mcp 1.23.3. Keep the patched override +# until Semgrep's published dependency constraint moves to mcp 1.28.1 or later. +override-dependencies = [ "mcp==1.28.1" ] diff --git a/rust-toolchain.toml b/rust-toolchain.toml index 7159467e..18c90fd3 100644 --- a/rust-toolchain.toml +++ b/rust-toolchain.toml @@ -1,6 +1,6 @@ [toolchain] # Pin to MSRV as specified in Cargo.toml -channel = "1.97.0" +channel = "1.97.1" # Essential repository components. Keep this lean so checkout and CI setup do # not install IDE-only or cross-target components unless a workflow asks for diff --git a/scripts/tests/test_benchmark_models.py b/scripts/tests/test_benchmark_models.py index b504eef6..1ce73646 100644 --- a/scripts/tests/test_benchmark_models.py +++ b/scripts/tests/test_benchmark_models.py @@ -454,9 +454,13 @@ def test_parse_throughput_data_with_scientific_notation(self) -> None: (parse_time_data, "Time: [0.0, 1.0, 2.0] µs"), (parse_time_data, "Time: [1.0, 3.0, 2.0] µs"), (parse_time_data, "Time: [1.0, 1e999, 1e999] µs"), + (parse_time_data, "Time: [1.0, 2.0] µs"), + (parse_time_data, "Time: [1.0, 2.0, 3.0, 4.0] µs"), (parse_throughput_data, "Throughput: [-1.0, 1.0, 2.0] Kelem/s"), (parse_throughput_data, "Throughput: [1.0, 3.0, 2.0] Kelem/s"), (parse_throughput_data, "Throughput: [1.0, 1e999, 1e999] Kelem/s"), + (parse_throughput_data, "Throughput: [1.0, 2.0] Kelem/s"), + (parse_throughput_data, "Throughput: [1.0, 2.0, 3.0, 4.0] Kelem/s"), ], ) def test_parsers_reject_nonphysical_intervals( diff --git a/src/geometry/matrix.rs b/src/geometry/matrix.rs index 5eceac2c..b652132e 100644 --- a/src/geometry/matrix.rs +++ b/src/geometry/matrix.rs @@ -20,7 +20,7 @@ /// wrappers such as [`determinant`] can preserve exact failure context without /// exposing the rest of `la-stack` to downstream callers. pub use la_stack::LaError; -use la_stack::Matrix as LaMatrix; +use la_stack::{BigRational, Matrix as LaMatrix}; pub(crate) use la_stack::{DEFAULT_SINGULAR_TOL, SingularityReason, Vector as LaVector}; use thiserror::Error; @@ -184,6 +184,34 @@ pub(crate) fn matrix_set( m.set(row, column, value).map_err(Into::into) } +/// Solve a runtime-sized finite `f64` system with exact fraction-free elimination. +/// +/// The backend converts every IEEE 754 input to its exact rational value before +/// applying Bareiss elimination, so this is an exact solve rather than a +/// floating-point approximation. +pub(crate) fn solve_exact_runtime_system( + matrix: &[Vec], + rhs: &[f64], +) -> Option, StackMatrixDispatchError>> { + let dimension = rhs.len(); + if matrix.len() != dimension || matrix.iter().any(|row| row.len() != dimension) { + return None; + } + + Some(try_with_la_stack_matrix!(dimension, |stack_matrix| { + for (row, values) in matrix.iter().enumerate() { + for (column, value) in values.iter().copied().enumerate() { + matrix_set(&mut stack_matrix, row, column, value)?; + } + } + let rhs_vector = LaVector::try_new(std::array::from_fn(|index| rhs[index]))?; + stack_matrix + .solve_exact(rhs_vector) + .map(|solution| solution.into_iter().collect()) + .map_err(Into::into) + })) +} + /// Return a determinant and its certified error bound when the f64 fast filter supports the matrix size. /// /// `Ok(None)` means the closed-form direct determinant path is unavailable or diff --git a/src/geometry/realization.rs b/src/geometry/realization.rs index 9c6b773a..74cc700b 100644 --- a/src/geometry/realization.rs +++ b/src/geometry/realization.rs @@ -40,9 +40,12 @@ #![forbid(unsafe_code)] use crate::core::collections::{MAX_PRACTICAL_DIMENSION_SIZE, SmallBuffer}; +use crate::geometry::matrix::{StackMatrixDispatchError, solve_exact_runtime_system}; use crate::geometry::point::{Point, ValidatedCoordinates}; +use crate::geometry::predicates::Orientation; +use crate::geometry::robust_predicates::robust_orientation; use crate::geometry::traits::coordinate::InvalidCoordinateValue; -use la_stack::{BigInt, BigRational, FromPrimitive, Signed}; +use la_stack::{BigInt, BigRational, FromPrimitive, Signed, ToPrimitive}; use thiserror::Error; /// Stack-backed buffer for per-simplex realization labels and coordinates. @@ -608,89 +611,1654 @@ where L: Clone + Eq, { let shared_labels = shared_labels(first, second); - let second_vertices_in_first = barycentric_coordinates_of_vertices(second, first)?; + if shared_face_fast_confinement(first, second, &shared_labels) { + return Ok(()); + } + let basis_orientation = realization_orientation(first); + if basis_orientation == Some(Orientation::DEGENERATE) { + return Err(SimplexIntersectionFailure::SingularBarycentricBasis); + } + if let Some(orientation) = basis_orientation + && (simplex_is_strictly_outside_a_facet(first, second, orientation) + || realization_orientation(second).is_some_and(|second_orientation| { + second_orientation != Orientation::DEGENERATE + && simplex_is_strictly_outside_a_facet(second, first, second_orientation) + }) + || intersection_is_confined_by_orientation(first, second, &shared_labels, orientation)) + { + return Ok(()); + } + + let fallback_barycentric = if basis_orientation.is_none() { + Some(barycentric_coordinates_of_vertices(second, first)?) + } else { + None + }; + + match intersection_via_linear_program(first, second, &shared_labels) { + IntersectionLinearProgramResult::Valid => return Ok(()), + IntersectionLinearProgramResult::Invalid(witness) => { + return Err(SimplexIntersectionFailure::IntersectionOutsideSharedFace { witness }); + } + IntersectionLinearProgramResult::Fallback => {} + } + + let second_vertices_in_first = match fallback_barycentric { + Some(coordinates) => coordinates, + None => barycentric_coordinates_of_vertices(second, first)?, + }; + if intersection_is_confined_to_shared_face( + &second_vertices_in_first, + first.labels(), + &shared_labels, + ) { + return Ok(()); + } let intersection_vertices = intersection_polytope_vertices(&second_vertices_in_first); - for beta in intersection_vertices { - let alpha = alpha_from_beta(&beta, &second_vertices_in_first); - let first_only_witness_labels = - positive_nonshared_labels(&alpha, first.labels(), &shared_labels); - let second_only_witness_labels = - positive_nonshared_labels(&beta, second.labels(), &shared_labels); + for beta in intersection_vertices { + let alpha = alpha_from_beta(&beta, &second_vertices_in_first); + let first_only_witness_labels = + positive_nonshared_labels(&alpha, first.labels(), &shared_labels); + let second_only_witness_labels = + positive_nonshared_labels(&beta, second.labels(), &shared_labels); + + if !first_only_witness_labels.is_empty() || !second_only_witness_labels.is_empty() { + return Err(SimplexIntersectionFailure::IntersectionOutsideSharedFace { + witness: SimplexIntersectionWitness { + shared: shared_labels, + first_only_witness: first_only_witness_labels, + second_only_witness: second_only_witness_labels, + }, + }); + } + } + + Ok(()) +} + +/// Tries cheap candidate separators in the normal space of one shared face. +#[expect( + clippy::too_many_lines, + reason = "the separator search is one bounded projection-and-certification workflow" +)] +fn shared_face_fast_confinement( + first: &LabeledSimplexRealization, + second: &LabeledSimplexRealization, + shared_labels: &[L], +) -> bool +where + L: Eq, +{ + if shared_labels.is_empty() || shared_labels.len() >= D { + return false; + } + let shared_coordinates: Vec<_> = shared_labels + .iter() + .map(|label| { + let first_index = first + .labels() + .iter() + .position(|candidate| candidate == label)?; + let second_index = second + .labels() + .iter() + .position(|candidate| candidate == label)?; + coordinates_are_identical( + &first.coordinates()[first_index], + &second.coordinates()[second_index], + ) + .then_some(first.coordinates()[first_index]) + }) + .collect::>() + .unwrap_or_default(); + if shared_coordinates.len() != shared_labels.len() { + return false; + } + let base = shared_coordinates[0]; + let shared_deltas: Vec> = shared_coordinates[1..] + .iter() + .map(|coordinates| { + coordinates + .iter() + .zip(base) + .map(|(coordinate, base_coordinate)| coordinate - base_coordinate) + .collect() + }) + .collect(); + let gram: Vec> = shared_deltas + .iter() + .map(|left| { + shared_deltas + .iter() + .map(|right| dot_product_f64(left, right)) + .collect() + }) + .collect(); + + let mut axis = vec![0.0; D]; + let mut normalized_rays = Vec::with_capacity(2 * D); + for (label, coordinates) in first.labels().iter().zip(first.coordinates()) { + if shared_labels.contains(label) { + continue; + } + let raw_ray: Vec<_> = coordinates + .iter() + .zip(base) + .map(|(coordinate, shared)| coordinate - shared) + .collect(); + let Some(ray) = project_ray_orthogonal_to_shared_face(raw_ray, &shared_deltas, &gram) + else { + return false; + }; + for coordinate in 0..D { + axis[coordinate] += ray[coordinate]; + } + push_normalized_ray(&mut normalized_rays, ray); + } + for (label, coordinates) in second.labels().iter().zip(second.coordinates()) { + if shared_labels.contains(label) { + continue; + } + let raw_ray: Vec<_> = base + .iter() + .zip(coordinates) + .map(|(shared, coordinate)| shared - coordinate) + .collect(); + let Some(ray) = project_ray_orthogonal_to_shared_face(raw_ray, &shared_deltas, &gram) + else { + return false; + }; + for coordinate in 0..D { + axis[coordinate] += ray[coordinate]; + } + push_normalized_ray(&mut normalized_rays, ray); + } + if axis.iter().all(|value| value.is_finite()) + && provisional_axis_certifies_confinement(first, second, shared_labels, &axis) + { + return true; + } + + let mut refined_axis = vec![0.0; D]; + for ray in &normalized_rays { + for (value, component) in refined_axis.iter_mut().zip(ray) { + *value += component; + } + } + for _ in 0..16 { + let mut adjusted = false; + for ray in &normalized_rays { + let projection = dot_product_f64(&refined_axis, ray); + if projection < 1.0 { + let correction = 1.0 - projection; + for (value, component) in refined_axis.iter_mut().zip(ray) { + *value = component.mul_add(correction, *value); + } + adjusted = true; + } + } + if !adjusted { + break; + } + } + refined_axis.iter().all(|value| value.is_finite()) + && provisional_axis_certifies_confinement(first, second, shared_labels, &refined_axis) +} + +/// Removes provisional components tangent to the shared affine face. +fn project_ray_orthogonal_to_shared_face( + mut ray: Vec, + shared_deltas: &[Vec], + gram: &[Vec], +) -> Option> { + if shared_deltas.is_empty() { + return Some(ray); + } + let residuals: Vec<_> = shared_deltas + .iter() + .map(|delta| dot_product_f64(delta, &ray)) + .collect(); + let correction = solve_f64_system(gram.to_vec(), residuals)?; + for (coordinate, value) in ray.iter_mut().enumerate() { + for (coefficient, delta) in correction.iter().zip(shared_deltas) { + *value = (-coefficient).mul_add(delta[coordinate], *value); + } + } + ray.iter().all(|value| value.is_finite()).then_some(ray) +} + +/// Certifies one provisional shared-face separator through a filtered or exact path. +fn provisional_axis_certifies_confinement( + first: &LabeledSimplexRealization, + second: &LabeledSimplexRealization, + shared_labels: &[L], + axis: &[f64], +) -> bool +where + L: Eq, +{ + if shared_labels.len() == 1 { + filtered_single_shared_vertex_confinement(first, second, shared_labels, axis) + } else { + exact_shared_face_confinement_certificate(first, second, shared_labels, axis) + } +} + +/// Appends one finite unit ray for provisional separator refinement. +fn push_normalized_ray(rays: &mut Vec>, ray: Vec) { + let norm = ray + .iter() + .fold(0.0_f64, |value, component| value.hypot(*component)); + if norm.is_finite() && norm > 0.0 { + rays.push(ray.into_iter().map(|component| component / norm).collect()); + } +} + +/// Computes one simplex orientation through the filtered-exact predicate path. +fn realization_orientation( + simplex: &LabeledSimplexRealization, +) -> Option { + let points: SimplexRealizationBuffer<_> = (0..simplex.coordinates().len()) + .filter_map(|index| simplex.point_at(index)) + .collect(); + robust_orientation(&points).ok() +} + +/// Proves disjointness when one simplex lies strictly beyond a facet of another. +/// +/// Replacing facet-opposite basis vertex `i` with a point `p` scales the basis +/// orientation by the barycentric coordinate of `p` at `i`. If every vertex of +/// the other simplex has the opposite non-zero sign for one `i`, convexity +/// places that entire simplex in the open exterior half-space. +fn simplex_is_strictly_outside_a_facet( + basis: &LabeledSimplexRealization, + other: &LabeledSimplexRealization, + basis_orientation: Orientation, +) -> bool { + let basis_points: SimplexRealizationBuffer<_> = (0..basis.coordinates().len()) + .filter_map(|index| basis.point_at(index)) + .collect(); + + (0..basis.labels().len()).any(|basis_index| { + (0..other.coordinates().len()).all(|other_index| { + let Some(other_point) = other.point_at(other_index) else { + return false; + }; + let mut replaced_points = basis_points.clone(); + replaced_points[basis_index] = other_point; + robust_orientation(&replaced_points).is_ok_and(|orientation| { + orientation != basis_orientation && orientation != Orientation::DEGENERATE + }) + }) + }) +} + +/// Uses filtered-exact orientation signs to prove confinement to the shared face. +/// +/// Replacing basis vertex `i` with a point `p` changes the orientation +/// determinant by the barycentric coordinate of `p` at `i`. Therefore, when +/// every vertex of the other simplex has a zero or opposite determinant sign +/// for each non-shared basis vertex, convexity proves that any point in both +/// simplices has zero weight outside the shared face. +fn intersection_is_confined_by_orientation( + basis: &LabeledSimplexRealization, + other: &LabeledSimplexRealization, + shared_labels: &[L], + basis_orientation: Orientation, +) -> bool +where + L: Eq, +{ + let basis_points: SimplexRealizationBuffer<_> = (0..basis.coordinates().len()) + .filter_map(|index| basis.point_at(index)) + .collect(); + + for (basis_index, basis_label) in basis.labels().iter().enumerate() { + if shared_labels.contains(basis_label) { + continue; + } + for other_index in 0..other.coordinates().len() { + let Some(other_point) = other.point_at(other_index) else { + return false; + }; + let mut replaced_points = basis_points.clone(); + replaced_points[basis_index] = other_point; + let Ok(replaced_orientation) = robust_orientation(&replaced_points) else { + return false; + }; + if replaced_orientation == basis_orientation { + return false; + } + } + } + true +} + +/// Accepts a pair when simplex-facet signs prove any intersection lies in the shared face. +/// +/// A point inside the basis simplex has non-negative barycentric coordinates. +/// If every vertex of the other simplex has a non-positive coordinate for each +/// basis vertex outside the shared face, convexity makes those coordinates +/// non-positive throughout the other simplex. Any point in both simplices must +/// therefore have zero weight on every non-shared basis vertex. +fn intersection_is_confined_to_shared_face( + other_vertices_in_basis: &[Vec], + basis_labels: &[L], + shared_labels: &[L], +) -> bool +where + L: Eq, +{ + basis_labels + .iter() + .enumerate() + .filter(|(_, label)| !shared_labels.contains(label)) + .all(|(basis_index, _)| { + other_vertices_in_basis + .iter() + .all(|barycentric| !barycentric[basis_index].is_positive()) + }) +} + +/// Collects labels common to two simplex realizations so witnesses can distinguish shared faces. +fn shared_labels( + first: &LabeledSimplexRealization, + second: &LabeledSimplexRealization, +) -> SimplexRealizationBuffer +where + L: Clone + Eq, +{ + first + .labels() + .iter() + .filter(|label| second.labels().contains(label)) + .cloned() + .collect() +} + +/// Expresses every vertex of one simplex in the barycentric basis of another. +fn barycentric_coordinates_of_vertices( + vertices: &LabeledSimplexRealization, + basis: &LabeledSimplexRealization, +) -> Result>, SimplexIntersectionFailure> { + vertices + .coordinates() + .iter() + .map(|coords| barycentric_coordinates(coords, basis)) + .collect() +} + +/// Computes exact barycentric coordinates of one point in one simplex basis. +fn barycentric_coordinates( + point: &[f64; D], + simplex: &LabeledSimplexRealization, +) -> Result, SimplexIntersectionFailure> { + if D == 0 { + return Ok(vec![rational_one()]); + } + + let origin = &simplex.coordinates()[0]; + let mut matrix = vec![vec![rational_zero(); D]; D]; + let mut rhs = vec![rational_zero(); D]; + + for axis in 0..D { + let origin_coord = rational_from_f64(origin[axis]); + rhs[axis] = rational_from_f64(point[axis]) - origin_coord.clone(); + for (column, matrix_value) in matrix[axis].iter_mut().enumerate() { + *matrix_value = + rational_from_f64(simplex.coordinates()[column + 1][axis]) - origin_coord.clone(); + } + } + + let lambdas = solve_rational_system(matrix, rhs) + .ok_or(SimplexIntersectionFailure::SingularBarycentricBasis)?; + let lambda_sum = lambdas + .iter() + .fold(rational_zero(), |acc, value| acc + value.clone()); + let mut barycentric = Vec::with_capacity(D + 1); + barycentric.push(rational_one() - lambda_sum); + barycentric.extend(lambdas); + Ok(barycentric) +} + +/// Outcome of the exact linear-program intersection path. +#[derive(Debug)] +enum IntersectionLinearProgramResult { + /// The simplices are disjoint or meet only in their shared face. + Valid, + /// A feasible point has positive weight outside the shared face. + Invalid(SimplexIntersectionWitness), + /// An internal simplex-method precondition failed; use active-set enumeration. + Fallback, +} + +/// Uses exact linear programming to avoid enumerating every intersection-polytope basis. +/// +/// The variables are the non-negative barycentric weights `alpha` and `beta` +/// of the two simplices. Equality constraints require each weight vector to sum +/// to one and both weighted coordinate sums to describe the same point. +/// Maximizing the total weight on non-shared labels is zero exactly when every +/// feasible intersection point lies in the shared face. +fn intersection_via_linear_program( + first: &LabeledSimplexRealization, + second: &LabeledSimplexRealization, + shared_labels: &[L], +) -> IntersectionLinearProgramResult +where + L: Clone + Eq, +{ + let first_labels = first.labels(); + let second_labels = second.labels(); + let simplex_vertex_count = first_labels.len(); + if second_labels.len() != simplex_vertex_count + || first.coordinates().len() != simplex_vertex_count + || second.coordinates().len() != simplex_vertex_count + { + return IntersectionLinearProgramResult::Fallback; + } + + let variable_count = simplex_vertex_count * 2; + let constraint_count = D + 2; + let mut float_matrix = vec![vec![0.0; variable_count]; constraint_count]; + let mut float_rhs = vec![0.0; constraint_count]; + float_rhs[0] = 1.0; + float_rhs[1] = 1.0; + + for (alpha_index, coordinates) in first.coordinates().iter().enumerate() { + float_matrix[0][alpha_index] = 1.0; + for (axis, coordinate) in coordinates.iter().enumerate() { + float_matrix[axis + 2][alpha_index] = *coordinate; + } + } + for (beta_index, coordinates) in second.coordinates().iter().enumerate() { + float_matrix[1][simplex_vertex_count + beta_index] = 1.0; + for (axis, coordinate) in coordinates.iter().enumerate() { + float_matrix[axis + 2][simplex_vertex_count + beta_index] = -*coordinate; + } + } + + let mut float_objective = vec![0.0; variable_count]; + for (index, label) in first_labels.iter().enumerate() { + if !shared_labels.contains(label) { + float_objective[index] = 1.0; + } + } + for (index, label) in second_labels.iter().enumerate() { + if !shared_labels.contains(label) { + float_objective[simplex_vertex_count + index] = 1.0; + } + } + + let ProvisionalPhaseDuals { + infeasibility_axis, + confinement_axis, + confinement_basis, + } = provisional_phase_dual_axes(&float_matrix, &float_rhs, &float_objective, variable_count); + if infeasibility_axis + .is_some_and(|axis| simplices_are_strictly_separated_along(first, second, &axis)) + { + return IntersectionLinearProgramResult::Valid; + } + let axis_certifies = confinement_axis.is_some_and(|axis| { + filtered_single_shared_vertex_confinement(first, second, shared_labels, &axis) + || exact_shared_face_confinement_certificate(first, second, shared_labels, &axis) + }); + if axis_certifies { + return IntersectionLinearProgramResult::Valid; + } + + let matrix = exact_matrix_from_f64(&float_matrix); + let rhs: Vec<_> = float_rhs.iter().copied().map(rational_from_f64).collect(); + let objective: Vec<_> = float_objective + .iter() + .copied() + .map(rational_from_f64) + .collect(); + let dual_certifies = confinement_basis.is_some_and(|basis| { + phase_two_dual_proves_shared_face_confinement(&matrix, &rhs, &objective, &basis) + }); + if dual_certifies { + return IntersectionLinearProgramResult::Valid; + } + + match maximize_nonnegative_equality_linear_program(&matrix, &rhs, &objective) { + ExactLinearProgramResult::Infeasible => IntersectionLinearProgramResult::Valid, + ExactLinearProgramResult::Optimal { + objective_value, .. + } if !objective_value.is_positive() => IntersectionLinearProgramResult::Valid, + ExactLinearProgramResult::Optimal { solution, .. } => { + let first_only_witness = positive_nonshared_labels( + &solution[..simplex_vertex_count], + first_labels, + shared_labels, + ); + let second_only_witness = positive_nonshared_labels( + &solution[simplex_vertex_count..], + second_labels, + shared_labels, + ); + if first_only_witness.is_empty() && second_only_witness.is_empty() { + return IntersectionLinearProgramResult::Fallback; + } + IntersectionLinearProgramResult::Invalid(SimplexIntersectionWitness { + shared: shared_labels.iter().cloned().collect(), + first_only_witness, + second_only_witness, + }) + } + ExactLinearProgramResult::Failed => IntersectionLinearProgramResult::Fallback, + } +} + +/// Result of an exact non-negative equality-constrained linear program. +enum ExactLinearProgramResult { + /// The objective is maximized at an exact basic feasible solution. + Optimal { + /// Values for the original decision variables. + solution: Vec, + /// Exact objective value at `solution`. + objective_value: BigRational, + }, + /// No non-negative solution satisfies the equality constraints. + Infeasible, + /// Tableau construction or a boundedness invariant failed. + Failed, +} + +/// Result of one revised-simplex optimization phase. +enum RevisedSimplexResult { + /// The current basis is optimal. + Optimal { + /// Values for every variable in the supplied matrix. + solution: Vec, + /// Exact objective value at `solution`. + objective_value: BigRational, + }, + /// The objective can increase without bound. + Unbounded, + /// The supplied basis or iteration budget failed. + Failed, +} + +/// Maximizes an exact objective subject to `matrix * x = rhs` and `x >= 0`. +/// +/// Phase I introduces one artificial variable per equality. Bland's pivot +/// ordering makes degenerate exact-arithmetic bases deterministic and avoids +/// cycling. Artificial variables are pivoted out at zero before Phase II. +fn maximize_nonnegative_equality_linear_program( + matrix: &[Vec], + rhs: &[BigRational], + objective: &[BigRational], +) -> ExactLinearProgramResult { + let constraint_count = rhs.len(); + let variable_count = objective.len(); + if constraint_count == 0 + || matrix.len() != constraint_count + || matrix.iter().any(|row| row.len() != variable_count) + || rhs.iter().any(Signed::is_negative) + { + return ExactLinearProgramResult::Failed; + } + + let total_variable_count = variable_count + constraint_count; + let mut phase_matrix = Vec::with_capacity(constraint_count); + for (row_index, row) in matrix.iter().enumerate() { + let mut phase_row = Vec::with_capacity(total_variable_count); + phase_row.extend(row.iter().cloned()); + phase_row.extend((0..constraint_count).map(|column| { + if column == row_index { + rational_one() + } else { + rational_zero() + } + })); + phase_matrix.push(phase_row); + } + + let mut basis: Vec = (variable_count..total_variable_count).collect(); + let mut phase_one_objective = vec![rational_zero(); total_variable_count]; + for coefficient in &mut phase_one_objective[variable_count..] { + *coefficient = -rational_one(); + } + + let phase_one = revised_simplex_maximize( + &phase_matrix, + rhs, + &phase_one_objective, + total_variable_count, + &mut basis, + ); + let RevisedSimplexResult::Optimal { + objective_value, .. + } = phase_one + else { + return ExactLinearProgramResult::Failed; + }; + if objective_value.is_negative() { + return ExactLinearProgramResult::Infeasible; + } + if objective_value.is_positive() + || !pivot_artificial_variables_out(&phase_matrix, rhs, variable_count, &mut basis) + { + return ExactLinearProgramResult::Failed; + } + + let mut phase_two_objective = vec![rational_zero(); total_variable_count]; + phase_two_objective[..variable_count].clone_from_slice(objective); + match revised_simplex_maximize( + &phase_matrix, + rhs, + &phase_two_objective, + variable_count, + &mut basis, + ) { + RevisedSimplexResult::Optimal { + solution, + objective_value, + } => ExactLinearProgramResult::Optimal { + solution: solution[..variable_count].to_vec(), + objective_value, + }, + RevisedSimplexResult::Unbounded | RevisedSimplexResult::Failed => { + ExactLinearProgramResult::Failed + } + } +} + +/// Removes zero-valued Phase I artificial variables from the active basis. +fn pivot_artificial_variables_out( + matrix: &[Vec], + rhs: &[BigRational], + original_variable_count: usize, + basis: &mut [usize], +) -> bool { + for basis_position in 0..basis.len() { + if basis[basis_position] < original_variable_count { + continue; + } + + let Some(current_basis_matrix) = basis_matrix(matrix, basis) else { + return false; + }; + let Some(basic_solution) = + solve_rational_system(current_basis_matrix.clone(), rhs.to_vec()) + else { + return false; + }; + if basic_solution[basis_position] != rational_zero() { + return false; + } + + let mut replacement = None; + for candidate in 0..original_variable_count { + if basis.contains(&candidate) { + continue; + } + let Some(direction) = solve_rational_system( + current_basis_matrix.clone(), + matrix_column(matrix, candidate), + ) else { + continue; + }; + if direction[basis_position] != rational_zero() { + replacement = Some(candidate); + break; + } + } + let Some(candidate) = replacement else { + return false; + }; + basis[basis_position] = candidate; + } + true +} + +/// Uses a floating-point basis search followed by an exact optimality certificate. +/// +/// The floating-point phase is only a guide: it cannot return a validation +/// result directly. A candidate basis is accepted only after exact rational +/// primal and dual checks. If certification fails, the deterministic exact +/// simplex path restarts from the original basis. +fn revised_simplex_maximize( + matrix: &[Vec], + rhs: &[BigRational], + objective: &[BigRational], + entering_variable_count: usize, + basis: &mut [usize], +) -> RevisedSimplexResult { + let starting_basis = basis.to_vec(); + if navigate_simplex_basis_f64(matrix, rhs, objective, entering_variable_count, basis).is_some() + && let Some(optimal) = + certify_optimal_basis(matrix, rhs, objective, entering_variable_count, basis) + { + return optimal; + } + + basis.clone_from_slice(&starting_basis); + revised_simplex_maximize_exact(matrix, rhs, objective, entering_variable_count, basis) +} + +/// Runs deterministic exact revised simplex from an existing feasible basis. +fn revised_simplex_maximize_exact( + matrix: &[Vec], + rhs: &[BigRational], + objective: &[BigRational], + entering_variable_count: usize, + basis: &mut [usize], +) -> RevisedSimplexResult { + const MAX_SIMPLEX_ITERATIONS: usize = 10_000; + + if matrix.is_empty() + || matrix.len() != rhs.len() + || basis.len() != rhs.len() + || matrix.iter().any(|row| row.len() != objective.len()) + || entering_variable_count > objective.len() + { + return RevisedSimplexResult::Failed; + } + + for _ in 0..MAX_SIMPLEX_ITERATIONS { + let Some(current_basis_matrix) = basis_matrix(matrix, basis) else { + return RevisedSimplexResult::Failed; + }; + let Some(basic_solution) = + solve_rational_system(current_basis_matrix.clone(), rhs.to_vec()) + else { + return RevisedSimplexResult::Failed; + }; + if basic_solution.iter().any(Signed::is_negative) { + return RevisedSimplexResult::Failed; + } + + let basic_objective: Vec<_> = basis + .iter() + .map(|&index| objective[index].clone()) + .collect(); + let Some(dual_solution) = solve_rational_system( + transpose_square_matrix(¤t_basis_matrix), + basic_objective, + ) else { + return RevisedSimplexResult::Failed; + }; + + let entering = (0..entering_variable_count) + .filter(|candidate| !basis.contains(candidate)) + .find(|&candidate| { + let reduced_cost = objective[candidate].clone() + - dot_product(&dual_solution, &matrix_column(matrix, candidate)); + reduced_cost.is_positive() + }); + let Some(entering) = entering else { + let mut solution = vec![rational_zero(); objective.len()]; + for (&variable, value) in basis.iter().zip(&basic_solution) { + solution[variable] = value.clone(); + } + return RevisedSimplexResult::Optimal { + objective_value: dot_product(objective, &solution), + solution, + }; + }; + + let Some(direction) = + solve_rational_system(current_basis_matrix, matrix_column(matrix, entering)) + else { + return RevisedSimplexResult::Failed; + }; + let leaving = direction + .iter() + .enumerate() + .filter(|(_, coefficient)| coefficient.is_positive()) + .map(|(position, coefficient)| { + ( + position, + basic_solution[position].clone() / coefficient.clone(), + ) + }) + .min_by( + |(left_position, left_ratio), (right_position, right_ratio)| { + left_ratio + .cmp(right_ratio) + .then_with(|| basis[*left_position].cmp(&basis[*right_position])) + }, + ); + let Some((leaving_position, _)) = leaving else { + return RevisedSimplexResult::Unbounded; + }; + basis[leaving_position] = entering; + } + + RevisedSimplexResult::Failed +} + +/// Searches for an optimal basis quickly using f64 arithmetic. +/// +/// Every decision made here is provisional. The caller must pass the resulting +/// basis through [`certify_optimal_basis`] before using it. +fn navigate_simplex_basis_f64( + matrix: &[Vec], + rhs: &[BigRational], + objective: &[BigRational], + entering_variable_count: usize, + basis: &mut [usize], +) -> Option> { + let float_matrix = rational_matrix_to_f64(matrix)?; + let float_rhs = rationals_to_f64(rhs)?; + let float_objective = rationals_to_f64(objective)?; + navigate_simplex_basis_f64_values( + &float_matrix, + &float_rhs, + &float_objective, + entering_variable_count, + basis, + ) +} + +/// Runs provisional revised simplex directly on finite floating-point values. +fn navigate_simplex_basis_f64_values( + matrix: &[Vec], + rhs: &[f64], + objective: &[f64], + entering_variable_count: usize, + basis: &mut [usize], +) -> Option> { + const MAX_SIMPLEX_ITERATIONS: usize = 10_000; + const REDUCED_COST_TOLERANCE: f64 = 1.0e-10; + const DIRECTION_TOLERANCE: f64 = 1.0e-12; + const FEASIBILITY_TOLERANCE: f64 = 1.0e-9; + + if matrix.is_empty() + || matrix.len() != rhs.len() + || basis.len() != rhs.len() + || matrix.iter().any(|row| row.len() != objective.len()) + || entering_variable_count > objective.len() + { + return None; + } + + for _ in 0..MAX_SIMPLEX_ITERATIONS { + let current_basis_matrix = basis_matrix_f64(matrix, basis)?; + let mut basic_solution = solve_f64_system(current_basis_matrix.clone(), rhs.to_vec())?; + if basic_solution + .iter() + .any(|&value| value < -FEASIBILITY_TOLERANCE) + { + return None; + } + for value in &mut basic_solution { + if value.is_sign_negative() { + *value = 0.0; + } + } + + let basic_objective: Vec<_> = basis.iter().map(|&index| objective[index]).collect(); + let dual_solution = solve_f64_system( + transpose_square_matrix_f64(¤t_basis_matrix), + basic_objective, + )?; + + let entering = (0..entering_variable_count) + .filter(|candidate| !basis.contains(candidate)) + .find(|&candidate| { + let column = matrix_column_f64(matrix, candidate); + let dual_contribution = dot_product_f64(&dual_solution, &column); + let scale = 1.0 + objective[candidate].abs() + dual_contribution.abs(); + objective[candidate] - dual_contribution > REDUCED_COST_TOLERANCE * scale + }); + let Some(entering) = entering else { + return Some(dual_solution); + }; + + let direction = + solve_f64_system(current_basis_matrix, matrix_column_f64(matrix, entering))?; + let leaving = direction + .iter() + .enumerate() + .filter(|(_, coefficient)| **coefficient > DIRECTION_TOLERANCE) + .map(|(position, coefficient)| (position, basic_solution[position] / coefficient)) + .min_by( + |(left_position, left_ratio), (right_position, right_ratio)| { + left_ratio + .total_cmp(right_ratio) + .then_with(|| basis[*left_position].cmp(&basis[*right_position])) + }, + ); + let (leaving_position, _) = leaving?; + basis[leaving_position] = entering; + } + + None +} + +/// Finds candidate separating directions from floating-point Phase I and II duals. +/// +/// Returned directions are only guides. Callers must verify their geometric +/// certificates with exact arithmetic before accepting them. +struct ProvisionalPhaseDuals { + /// Candidate direction from the infeasibility phase. + infeasibility_axis: Option>, + /// Candidate direction from the shared-face-confinement phase. + confinement_axis: Option>, + /// Provisional Phase II basis available for exact dual certification. + confinement_basis: Option>, +} + +impl ProvisionalPhaseDuals { + /// Returns a result with no provisional certificates. + const fn none() -> Self { + Self { + infeasibility_axis: None, + confinement_axis: None, + confinement_basis: None, + } + } +} + +fn provisional_phase_dual_axes( + matrix: &[Vec], + rhs: &[f64], + original_objective: &[f64], + variable_count: usize, +) -> ProvisionalPhaseDuals { + let constraint_count = rhs.len(); + let Some(total_variable_count) = variable_count.checked_add(constraint_count) else { + return ProvisionalPhaseDuals::none(); + }; + let mut phase_matrix = Vec::with_capacity(constraint_count); + for (row_index, row) in matrix.iter().enumerate() { + let mut phase_row = Vec::with_capacity(total_variable_count); + phase_row.extend(row.iter().copied()); + phase_row.extend((0..constraint_count).map( + |column| { + if column == row_index { 1.0 } else { 0.0 } + }, + )); + phase_matrix.push(phase_row); + } + + let mut basis: Vec<_> = (variable_count..total_variable_count).collect(); + let mut objective = vec![0.0; total_variable_count]; + for coefficient in &mut objective[variable_count..] { + *coefficient = -1.0; + } + let Some(phase_one_dual) = navigate_simplex_basis_f64_values( + &phase_matrix, + rhs, + &objective, + total_variable_count, + &mut basis, + ) else { + return ProvisionalPhaseDuals::none(); + }; + let phase_one_axis = (phase_one_dual.len() >= 2).then(|| phase_one_dual[2..].to_vec()); + + if !pivot_artificial_variables_out_f64(&phase_matrix, rhs, variable_count, &mut basis) { + return ProvisionalPhaseDuals { + infeasibility_axis: phase_one_axis, + confinement_axis: None, + confinement_basis: None, + }; + } + let mut phase_two_objective = vec![0.0; total_variable_count]; + if original_objective.len() != variable_count { + return ProvisionalPhaseDuals { + infeasibility_axis: phase_one_axis, + confinement_axis: None, + confinement_basis: None, + }; + } + phase_two_objective[..variable_count].clone_from_slice(original_objective); + let phase_two_dual = navigate_simplex_basis_f64_values( + &phase_matrix, + rhs, + &phase_two_objective, + variable_count, + &mut basis, + ); + let phase_two_axis = phase_two_dual + .as_ref() + .and_then(|dual| (dual.len() >= 2).then(|| dual[2..].to_vec())); + let phase_two_basis = phase_two_dual.map(|_dual| basis); + ProvisionalPhaseDuals { + infeasibility_axis: phase_one_axis, + confinement_axis: phase_two_axis, + confinement_basis: phase_two_basis, + } +} + +/// Removes zero artificial variables from a provisional floating-point basis. +fn pivot_artificial_variables_out_f64( + matrix: &[Vec], + rhs: &[f64], + original_variable_count: usize, + basis: &mut [usize], +) -> bool { + const ZERO_TOLERANCE: f64 = 1.0e-9; + const PIVOT_TOLERANCE: f64 = 1.0e-12; + + for basis_position in 0..basis.len() { + if basis[basis_position] < original_variable_count { + continue; + } + let Some(current_basis_matrix) = basis_matrix_f64(matrix, basis) else { + return false; + }; + let Some(basic_solution) = solve_f64_system(current_basis_matrix.clone(), rhs.to_vec()) + else { + return false; + }; + if basic_solution[basis_position].abs() > ZERO_TOLERANCE { + return false; + } + + let replacement = (0..original_variable_count) + .filter(|candidate| !basis.contains(candidate)) + .find(|&candidate| { + solve_f64_system( + current_basis_matrix.clone(), + matrix_column_f64(matrix, candidate), + ) + .is_some_and(|direction| direction[basis_position].abs() > PIVOT_TOLERANCE) + }); + let Some(replacement) = replacement else { + return false; + }; + basis[basis_position] = replacement; + } + true +} + +/// Certifies strict projection separation along a finite candidate direction. +fn simplices_are_strictly_separated_along( + first: &LabeledSimplexRealization, + second: &LabeledSimplexRealization, + axis: &[f64], +) -> bool { + if axis.len() != D || axis.iter().any(|value| !value.is_finite()) { + return false; + } + let exact_axis: Vec<_> = axis.iter().copied().map(rational_from_f64).collect(); + let Some((first_min, first_max)) = exact_projection_bounds(first, &exact_axis) else { + return false; + }; + let Some((second_min, second_max)) = exact_projection_bounds(second, &exact_axis) else { + return false; + }; + first_max < second_min || second_max < first_min +} + +/// Returns exact projection bounds for one realized simplex. +fn exact_projection_bounds( + simplex: &LabeledSimplexRealization, + axis: &[BigRational], +) -> Option<(BigRational, BigRational)> { + let mut projections = simplex.coordinates().iter().map(|coordinates| { + coordinates + .iter() + .zip(axis) + .fold(rational_zero(), |sum, (coordinate, coefficient)| { + sum + rational_from_f64(*coordinate) * coefficient + }) + }); + let first = projections.next()?; + Some( + projections.fold((first.clone(), first), |(minimum, maximum), projection| { + (minimum.min(projection.clone()), maximum.max(projection)) + }), + ) +} + +/// Certifies confinement through one shared vertex with a floating-point error bound. +/// +/// Scaling the provisional direction is exact because the factor is a power of +/// two. Each signed projection difference is then accepted only when its +/// conservative lower bound exceeds the unit dual margin. Ambiguous, +/// underflow-sensitive, or overflowing cases continue to exact arithmetic. +fn filtered_single_shared_vertex_confinement( + first: &LabeledSimplexRealization, + second: &LabeledSimplexRealization, + shared_labels: &[L], + axis: &[f64], +) -> bool +where + L: Eq, +{ + let [shared_label] = shared_labels else { + return false; + }; + let Some(first_shared_index) = first + .labels() + .iter() + .position(|candidate| candidate == shared_label) + else { + return false; + }; + let Some(second_shared_index) = second + .labels() + .iter() + .position(|candidate| candidate == shared_label) + else { + return false; + }; + let first_shared = &first.coordinates()[first_shared_index]; + let second_shared = &second.coordinates()[second_shared_index]; + if !coordinates_are_identical(first_shared, second_shared) { + return false; + } + + let scaled_axis: Vec<_> = axis.iter().map(|value| *value * 1024.0).collect(); + if scaled_axis.len() != D || scaled_axis.iter().any(|value| !value.is_finite()) { + return false; + } + first + .labels() + .iter() + .zip(first.coordinates()) + .filter(|(label, _)| *label != shared_label) + .all(|(_, coordinates)| { + certified_dot_difference_lower_bound(&scaled_axis, coordinates, first_shared) + .is_some_and(|lower_bound| lower_bound > 1.0) + }) + && second + .labels() + .iter() + .zip(second.coordinates()) + .filter(|(label, _)| *label != shared_label) + .all(|(_, coordinates)| { + certified_dot_difference_lower_bound(&scaled_axis, second_shared, coordinates) + .is_some_and(|lower_bound| lower_bound > 1.0) + }) +} - if !first_only_witness_labels.is_empty() || !second_only_witness_labels.is_empty() { - return Err(SimplexIntersectionFailure::IntersectionOutsideSharedFace { - witness: SimplexIntersectionWitness { - shared: shared_labels, - first_only_witness: first_only_witness_labels, - second_only_witness: second_only_witness_labels, - }, - }); +/// Returns a conservative lower bound for `axis · (left - right)`. +fn certified_dot_difference_lower_bound( + axis: &[f64], + left: &[f64; D], + right: &[f64; D], +) -> Option { + let mut estimate = 0.0; + let mut magnitude = 0.0; + for ((coefficient, left_value), right_value) in axis.iter().zip(left).zip(right) { + let left_product = coefficient * left_value; + let right_product = coefficient * right_value; + if !left_product.is_finite() + || !right_product.is_finite() + || product_underflowed(*coefficient, *left_value, left_product) + || product_underflowed(*coefficient, *right_value, right_product) + { + return None; + } + estimate = coefficient.mul_add(*left_value, estimate); + estimate = (-coefficient).mul_add(*right_value, estimate); + magnitude += left_product.abs() + right_product.abs(); + if !estimate.is_finite() || !magnitude.is_finite() { + return None; } } - Ok(()) + let operation_count = f64::from(u32::try_from(2 * D).ok()?); + let denominator = 1.0 - operation_count * f64::EPSILON; + if denominator <= 0.0 { + return None; + } + let gamma = operation_count * f64::EPSILON / denominator; + let magnitude_upper_bound = magnitude / denominator.powi(2); + let error_bound = 4.0 * gamma * magnitude_upper_bound; + error_bound.is_finite().then_some(estimate - error_bound) } -/// Collects labels common to two simplex realizations so witnesses can distinguish shared faces. -fn shared_labels( +/// Detects a product whose exact non-zero value rounded into the subnormal range. +fn product_underflowed(left: f64, right: f64, product: f64) -> bool { + left != 0.0 && right != 0.0 && (product == 0.0 || product.abs() < f64::MIN_POSITIVE) +} + +/// Compares finite coordinates exactly while treating both signed zeros alike. +fn coordinates_are_identical(left: &[f64; D], right: &[f64; D]) -> bool { + left.iter().zip(right).all(|(left, right)| { + let left_bits = left.to_bits(); + let right_bits = right.to_bits(); + left_bits == right_bits || (left_bits << 1 == 0 && right_bits << 1 == 0) + }) +} + +/// Certifies a Phase II dual after rebuilding its two affine offsets exactly. +/// +/// For direction `w`, the smallest feasible dual offsets are +/// `max_i(c_i - w·a_i)` and `max_j(c_j + w·b_j)`. Their sum is an exact upper +/// bound on the maximum non-shared barycentric weight. A non-positive bound +/// proves that every intersection point lies in the shared face. +fn exact_shared_face_confinement_certificate( first: &LabeledSimplexRealization, second: &LabeledSimplexRealization, -) -> SimplexRealizationBuffer + shared_labels: &[L], + axis: &[f64], +) -> bool where - L: Clone + Eq, + L: Eq, { - first + if axis.len() != D || axis.iter().any(|value| !value.is_finite()) { + return false; + } + let Some(exact_axis) = exact_axis_through_shared_face(first, shared_labels, axis) else { + return false; + }; + let first_offset = first .labels() .iter() - .filter(|label| second.labels().contains(label)) - .cloned() - .collect() + .zip(first.coordinates()) + .map(|(label, coordinates)| { + let objective = if shared_labels.contains(label) { + rational_zero() + } else { + rational_one() + }; + objective - exact_projection(coordinates, &exact_axis) + }) + .max(); + let second_offset = second + .labels() + .iter() + .zip(second.coordinates()) + .map(|(label, coordinates)| { + let objective = if shared_labels.contains(label) { + rational_zero() + } else { + rational_one() + }; + objective + exact_projection(coordinates, &exact_axis) + }) + .max(); + first_offset + .zip(second_offset) + .is_some_and(|(first_offset, second_offset)| { + first_offset + second_offset <= rational_zero() + }) } -/// Expresses every vertex of one simplex in the barycentric basis of another. -fn barycentric_coordinates_of_vertices( - vertices: &LabeledSimplexRealization, - basis: &LabeledSimplexRealization, -) -> Result>, SimplexIntersectionFailure> { - vertices - .coordinates() +/// Reconstructs a provisional axis in the shared face's exact affine nullspace. +/// +/// Shared vertices define homogeneous constraints on `(axis, offset)`. A +/// floating-point rank search selects pivot columns, then the fraction-free +/// exact solver constructs one nullspace basis vector per free column directly +/// from the original coordinates. Combining those vectors with provisional +/// free-coordinate weights restores exact shared-face equality without a +/// rational Gram solve. +fn exact_axis_through_shared_face( + simplex: &LabeledSimplexRealization, + shared_labels: &[L], + axis: &[f64], +) -> Option> +where + L: Eq, +{ + let shared_coordinates: Vec<_> = shared_labels .iter() - .map(|coords| barycentric_coordinates(coords, basis)) + .map(|label| { + simplex + .labels() + .iter() + .position(|candidate| candidate == label) + .and_then(|index| simplex.coordinates().get(index)) + }) + .collect::>()?; + if shared_coordinates.is_empty() { + return Some( + axis.iter() + .map(|value| rational_from_f64(*value * 1024.0)) + .collect(), + ); + } + + let constraint_matrix: Vec> = shared_coordinates + .iter() + .map(|coordinates| { + let mut row = coordinates.to_vec(); + row.push(1.0); + row + }) + .collect(); + let pivot_columns = independent_columns_f64(&constraint_matrix)?; + let free_columns: Vec<_> = (0..=D) + .filter(|column| !pivot_columns.contains(column)) + .collect(); + let pivot_matrix: Vec> = constraint_matrix + .iter() + .map(|row| pivot_columns.iter().map(|&column| row[column]).collect()) + .collect(); + + let mut provisional_affine = axis.to_vec(); + provisional_affine.push(-dot_product_f64(axis, shared_coordinates[0])); + let mut exact_affine = vec![rational_zero(); D + 1]; + for free_column in free_columns { + let weight = rational_from_f64(provisional_affine[free_column]); + exact_affine[free_column] += weight.clone(); + let rhs: Vec<_> = constraint_matrix + .iter() + .map(|row| -row[free_column]) + .collect(); + let solution = solve_exact_runtime_system(&pivot_matrix, &rhs)?.ok()?; + for (&pivot_column, coefficient) in pivot_columns.iter().zip(solution) { + exact_affine[pivot_column] += weight.clone() * coefficient; + } + } + + let scale = rational_from_f64(1024.0); + for value in &mut exact_affine[..D] { + *value *= scale.clone(); + } + exact_affine.truncate(D); + Some(exact_affine) +} + +/// Selects a full-rank set of matrix columns with provisional elimination. +fn independent_columns_f64(matrix: &[Vec]) -> Option> { + const PIVOT_TOLERANCE: f64 = 1.0e-12; + + let row_count = matrix.len(); + let column_count = matrix.first()?.len(); + if row_count == 0 || matrix.iter().any(|row| row.len() != column_count) { + return None; + } + let mut work = matrix.to_vec(); + let scale = work + .iter() + .flatten() + .fold(0.0_f64, |maximum, value| maximum.max(value.abs())); + let mut pivot_columns = Vec::with_capacity(row_count); + for column in 0..column_count { + let pivot_position = (pivot_columns.len()..row_count).max_by(|&left, &right| { + work[left][column] + .abs() + .total_cmp(&work[right][column].abs()) + })?; + if work[pivot_position][column].abs() <= PIVOT_TOLERANCE * scale.max(1.0) { + continue; + } + let pivot_row = pivot_columns.len(); + work.swap(pivot_row, pivot_position); + let pivot = work[pivot_row][column]; + let pivot_values = work[pivot_row].clone(); + for row_values in work.iter_mut().skip(pivot_row + 1) { + let factor = row_values[column] / pivot; + for (value, pivot_value) in row_values[column..].iter_mut().zip(&pivot_values[column..]) + { + *value -= factor * pivot_value; + } + } + pivot_columns.push(column); + if pivot_columns.len() == row_count { + return Some(pivot_columns); + } + } + None +} + +/// Certifies the floating-point Phase II basis with one exact dual solve. +/// +/// Dual feasibility and a non-positive exact objective bound prove confinement +/// without constructing a primal solution or repeating exact simplex pivots. +fn phase_two_dual_proves_shared_face_confinement( + matrix: &[Vec], + rhs: &[BigRational], + objective: &[BigRational], + basis: &[usize], +) -> bool { + let constraint_count = rhs.len(); + let variable_count = objective.len(); + if matrix.len() != constraint_count + || matrix.iter().any(|row| row.len() != variable_count) + || basis.len() != constraint_count + { + return false; + } + + let total_variable_count = variable_count + constraint_count; + let phase_matrix: Vec<_> = matrix + .iter() + .enumerate() + .map(|(row_index, row)| { + let mut phase_row = Vec::with_capacity(total_variable_count); + phase_row.extend(row.iter().cloned()); + phase_row.extend((0..constraint_count).map(|column| { + if column == row_index { + rational_one() + } else { + rational_zero() + } + })); + phase_row + }) + .collect(); + let mut phase_objective = vec![rational_zero(); total_variable_count]; + phase_objective[..variable_count].clone_from_slice(objective); + + let Some(current_basis_matrix) = basis_matrix(&phase_matrix, basis) else { + return false; + }; + let basic_objective: Vec<_> = basis + .iter() + .filter_map(|&index| phase_objective.get(index).cloned()) + .collect(); + if basic_objective.len() != constraint_count { + return false; + } + let Some(dual_solution) = solve_rational_system( + transpose_square_matrix(¤t_basis_matrix), + basic_objective, + ) else { + return false; + }; + let dual_is_feasible = (0..variable_count).all(|candidate| { + objective[candidate].clone() + - dot_product(&dual_solution, &matrix_column(matrix, candidate)) + <= rational_zero() + }); + dual_is_feasible && dot_product(rhs, &dual_solution) <= rational_zero() +} + +/// Computes one exact projection along a finite candidate axis. +fn exact_projection(coordinates: &[f64; D], axis: &[BigRational]) -> BigRational { + coordinates + .iter() + .zip(axis) + .fold(rational_zero(), |sum, (coordinate, coefficient)| { + sum + rational_from_f64(*coordinate) * coefficient + }) +} + +/// Certifies exact primal feasibility and dual optimality for one candidate basis. +fn certify_optimal_basis( + matrix: &[Vec], + rhs: &[BigRational], + objective: &[BigRational], + entering_variable_count: usize, + basis: &[usize], +) -> Option { + let current_basis_matrix = basis_matrix(matrix, basis)?; + let basic_solution = solve_rational_system(current_basis_matrix.clone(), rhs.to_vec())?; + if basic_solution.iter().any(Signed::is_negative) { + return None; + } + + let basic_objective: Vec<_> = basis + .iter() + .map(|&index| objective[index].clone()) + .collect(); + let dual_solution = solve_rational_system( + transpose_square_matrix(¤t_basis_matrix), + basic_objective, + )?; + let dual_is_feasible = (0..entering_variable_count) + .filter(|candidate| !basis.contains(candidate)) + .all(|candidate| { + let reduced_cost = objective[candidate].clone() + - dot_product(&dual_solution, &matrix_column(matrix, candidate)); + !reduced_cost.is_positive() + }); + if !dual_is_feasible { + return None; + } + + let mut solution = vec![rational_zero(); objective.len()]; + for (&variable, value) in basis.iter().zip(&basic_solution) { + solution[variable] = value.clone(); + } + Some(RevisedSimplexResult::Optimal { + objective_value: dot_product(objective, &solution), + solution, + }) +} + +/// Converts an exact rational matrix to finite f64 values for basis navigation. +fn rational_matrix_to_f64(matrix: &[Vec]) -> Option>> { + matrix.iter().map(|row| rationals_to_f64(row)).collect() +} + +/// Converts exact rationals to finite f64 values for provisional calculations. +fn rationals_to_f64(values: &[BigRational]) -> Option> { + values + .iter() + .map(|value| value.to_f64().filter(|converted| converted.is_finite())) .collect() } -/// Computes exact barycentric coordinates of one point in one simplex basis. -fn barycentric_coordinates( - point: &[f64; D], - simplex: &LabeledSimplexRealization, -) -> Result, SimplexIntersectionFailure> { - if D == 0 { - return Ok(vec![rational_one()]); +/// Extracts a floating-point basis matrix. +fn basis_matrix_f64(matrix: &[Vec], basis: &[usize]) -> Option>> { + let column_count = matrix.first()?.len(); + if matrix.len() != basis.len() + || matrix.iter().any(|row| row.len() != column_count) + || basis.iter().any(|&column| column >= column_count) + { + return None; } + Some( + matrix + .iter() + .map(|row| basis.iter().map(|&column| row[column]).collect()) + .collect(), + ) +} - let origin = &simplex.coordinates()[0]; - let mut matrix = vec![vec![rational_zero(); D]; D]; - let mut rhs = vec![rational_zero(); D]; +/// Copies one floating-point matrix column. +fn matrix_column_f64(matrix: &[Vec], column: usize) -> Vec { + matrix.iter().map(|row| row[column]).collect() +} - for axis in 0..D { - let origin_coord = rational_from_f64(origin[axis]); - rhs[axis] = rational_from_f64(point[axis]) - origin_coord.clone(); - for (column, matrix_value) in matrix[axis].iter_mut().enumerate() { - *matrix_value = - rational_from_f64(simplex.coordinates()[column + 1][axis]) - origin_coord.clone(); +/// Transposes a square floating-point matrix. +fn transpose_square_matrix_f64(matrix: &[Vec]) -> Vec> { + let dimension = matrix.len(); + (0..dimension) + .map(|row| (0..dimension).map(|column| matrix[column][row]).collect()) + .collect() +} + +/// Computes a floating-point dot product used only for provisional pivot selection. +fn dot_product_f64(left: &[f64], right: &[f64]) -> f64 { + left.iter() + .zip(right) + .map(|(left, right)| left * right) + .sum() +} + +/// Solves a small floating-point system with scaled partial pivoting. +#[expect( + clippy::needless_range_loop, + reason = "index-based elimination keeps pivot row/column operations explicit" +)] +fn solve_f64_system(mut matrix: Vec>, mut rhs: Vec) -> Option> { + const PIVOT_TOLERANCE: f64 = 1.0e-14; + + let dimension = rhs.len(); + if matrix.len() != dimension || matrix.iter().any(|row| row.len() != dimension) { + return None; + } + + for pivot_column in 0..dimension { + let pivot_row = (pivot_column..dimension).max_by(|&left, &right| { + matrix[left][pivot_column] + .abs() + .total_cmp(&matrix[right][pivot_column].abs()) + })?; + let row_scale = matrix[pivot_row] + .iter() + .map(|&value| value.abs()) + .fold(0.0_f64, f64::max); + if !row_scale.is_finite() + || matrix[pivot_row][pivot_column].abs() <= PIVOT_TOLERANCE * row_scale.max(1.0) + { + return None; + } + if pivot_row != pivot_column { + matrix.swap(pivot_column, pivot_row); + rhs.swap(pivot_column, pivot_row); + } + + for row in pivot_column + 1..dimension { + let factor = matrix[row][pivot_column] / matrix[pivot_column][pivot_column]; + matrix[row][pivot_column] = 0.0; + for column in pivot_column + 1..dimension { + matrix[row][column] = + factor.mul_add(-matrix[pivot_column][column], matrix[row][column]); + } + rhs[row] = factor.mul_add(-rhs[pivot_column], rhs[row]); } } - let lambdas = solve_rational_system(matrix, rhs) - .ok_or(SimplexIntersectionFailure::SingularBarycentricBasis)?; - let lambda_sum = lambdas - .iter() - .fold(rational_zero(), |acc, value| acc + value.clone()); - let mut barycentric = Vec::with_capacity(D + 1); - barycentric.push(rational_one() - lambda_sum); - barycentric.extend(lambdas); - Ok(barycentric) + let mut solution = vec![0.0; dimension]; + for row in (0..dimension).rev() { + let mut sum = rhs[row]; + for column in row + 1..dimension { + sum = matrix[row][column].mul_add(-solution[column], sum); + } + solution[row] = sum / matrix[row][row]; + if !solution[row].is_finite() { + return None; + } + } + Some(solution) +} + +/// Extracts the square matrix formed by the current basis columns. +fn basis_matrix(matrix: &[Vec], basis: &[usize]) -> Option>> { + let column_count = matrix.first()?.len(); + if matrix.len() != basis.len() + || matrix.iter().any(|row| row.len() != column_count) + || basis.iter().any(|&column| column >= column_count) + { + return None; + } + Some( + matrix + .iter() + .map(|row| basis.iter().map(|&column| row[column].clone()).collect()) + .collect(), + ) +} + +/// Copies one matrix column for an exact linear solve. +fn matrix_column(matrix: &[Vec], column: usize) -> Vec { + matrix.iter().map(|row| row[column].clone()).collect() +} + +/// Transposes a square exact-rational matrix. +fn transpose_square_matrix(matrix: &[Vec]) -> Vec> { + let dimension = matrix.len(); + (0..dimension) + .map(|row| { + (0..dimension) + .map(|column| matrix[column][row].clone()) + .collect() + }) + .collect() +} + +/// Computes an exact dot product without introducing floating-point decisions. +fn dot_product(left: &[BigRational], right: &[BigRational]) -> BigRational { + left.iter() + .zip(right) + .fold(rational_zero(), |sum, (left, right)| { + sum + left.clone() * right.clone() + }) } /// Enumerates candidate vertices of the intersection polytope in second-simplex weights. @@ -849,6 +2417,11 @@ fn solve_rational_system( if matrix.len() != dimension || matrix.iter().any(|row| row.len() != dimension) { return None; } + match try_solve_exact_f64_system(&matrix, &rhs) { + ExactF64Solve::Solution(solution) => return Some(solution), + ExactF64Solve::Singular => return None, + ExactF64Solve::Unsupported => {} + } for pivot_col in 0..dimension { let pivot_row = @@ -885,6 +2458,64 @@ fn solve_rational_system( Some(solution) } +/// Outcome of attempting the fraction-free exact `f64` solver. +enum ExactF64Solve { + /// The runtime stack dispatcher cannot represent this rational system. + Unsupported, + /// The system is exactly singular. + Singular, + /// The exact system solution. + Solution(Vec), +} + +/// Uses fraction-free Bareiss elimination when every coefficient came directly +/// from a finite `f64`; otherwise leaves the rational solver as the fallback. +fn try_solve_exact_f64_system(matrix: &[Vec], rhs: &[BigRational]) -> ExactF64Solve { + let Some(float_matrix) = exact_rational_matrix_as_f64(matrix) else { + return ExactF64Solve::Unsupported; + }; + let Some(float_rhs) = exact_rationals_as_f64(rhs) else { + return ExactF64Solve::Unsupported; + }; + let Some(result) = solve_exact_runtime_system(&float_matrix, &float_rhs) else { + return ExactF64Solve::Unsupported; + }; + match result { + Ok(solution) => ExactF64Solve::Solution(solution), + Err(StackMatrixDispatchError::La { + source: la_stack::LaError::Singular { .. }, + }) => ExactF64Solve::Singular, + Err(_) => ExactF64Solve::Unsupported, + } +} + +/// Converts rationals to `f64` only when doing so preserves each value exactly. +fn exact_rational_matrix_as_f64(matrix: &[Vec]) -> Option>> { + matrix + .iter() + .map(|row| exact_rationals_as_f64(row)) + .collect() +} + +/// Converts a finite floating-point matrix to its exact rational representation. +fn exact_matrix_from_f64(matrix: &[Vec]) -> Vec> { + matrix + .iter() + .map(|row| row.iter().copied().map(rational_from_f64).collect()) + .collect() +} + +/// Converts rationals to finite `f64` values and verifies an exact round trip. +fn exact_rationals_as_f64(values: &[BigRational]) -> Option> { + values + .iter() + .map(|value| { + let converted = value.to_f64().filter(|candidate| candidate.is_finite())?; + (rational_from_f64(converted) == *value).then_some(converted) + }) + .collect() +} + /// Converts a finite f64 to an exact rational value for barycentric predicates. fn rational_from_f64(value: f64) -> BigRational { BigRational::from_f64(value) @@ -988,6 +2619,46 @@ mod tests { ); } + #[test] + fn exact_linear_program_classifies_disjoint_triangles_as_valid() { + let first = LabeledSimplexRealization::try_new( + vec![0, 1, 2], + vec![[0.0, 0.0], [1.0, 0.0], [0.0, 1.0]], + ) + .unwrap(); + let second = LabeledSimplexRealization::try_new( + vec![3, 4, 5], + vec![[2.0, 2.0], [3.0, 2.0], [2.0, 3.0]], + ) + .unwrap(); + let shared = shared_labels(&first, &second); + + assert_matches!( + intersection_via_linear_program(&first, &second, &shared), + IntersectionLinearProgramResult::Valid + ); + } + + #[test] + fn exact_linear_program_accepts_a_shared_facet_intersection() { + let first = LabeledSimplexRealization::try_new( + vec![0, 1, 2], + vec![[0.0, 0.0], [1.0, 0.0], [0.0, 1.0]], + ) + .unwrap(); + let second = LabeledSimplexRealization::try_new( + vec![0, 1, 3], + vec![[0.0, 0.0], [1.0, 0.0], [0.0, -1.0]], + ) + .unwrap(); + let shared = shared_labels(&first, &second); + + assert_matches!( + intersection_via_linear_program(&first, &second, &shared), + IntersectionLinearProgramResult::Valid + ); + } + #[test] fn labeled_simplex_realization_rejects_non_finite_coordinates() { let err = LabeledSimplexRealization::try_new( @@ -1106,6 +2777,28 @@ mod tests { ); } + #[test] + fn exact_linear_program_reports_crossing_triangle_witnesses() { + let first = LabeledSimplexRealization::try_new( + vec![0, 1, 2], + vec![[0.0, 0.0], [2.0, 0.0], [0.0, 2.0]], + ) + .unwrap(); + let second = LabeledSimplexRealization::try_new( + vec![3, 4, 5], + vec![[2.0, 2.0], [1.0, -1.0], [3.0, 2.0]], + ) + .unwrap(); + let shared = shared_labels(&first, &second); + + assert_matches!( + intersection_via_linear_program(&first, &second, &shared), + IntersectionLinearProgramResult::Invalid(witness) + if witness.first_only_witness.iter().any(|label| [0, 1, 2].contains(label)) + && witness.second_only_witness.iter().any(|label| [3, 4, 5].contains(label)) + ); + } + #[test] fn spanning_periodic_simplex_is_detected() { let simplex = LabeledSimplexRealization::try_new( diff --git a/tests/large_scale_debug.rs b/tests/large_scale_debug.rs index 8f98cf05..cda089b8 100644 --- a/tests/large_scale_debug.rs +++ b/tests/large_scale_debug.rs @@ -20,12 +20,13 @@ //! //! - 2D: 40,000 vertices //! - 3D: 7,500 vertices -//! - 4D: 900 vertices -//! - 5D: 150 vertices +//! - 4D: 100 vertices +//! - 5D: 50 vertices //! -//! The `just` helpers may use slightly smaller local defaults so routine -//! acceptance/profiling runs stay near one minute on maintainer Apple M4 Max -//! hardware. Pass `n` explicitly when a run must match a documented scale. +//! The `just` diagnostic helpers use independently calibrated defaults and a +//! longer manual runtime budget. Pass `n` explicitly when a run must match a +//! documented scale. Broader high-dimensional Level 4 and Level 5 validation +//! work remains tracked by #482 and #483 for v0.8.1. #![cfg_attr(not(feature = "slow-tests"), allow(dead_code))] //! @@ -1570,8 +1571,9 @@ where // flip-based Level 5 Delaunay check. This skips the expensive Level 4 // realization overlap scan and the full report's all-violations Delaunay // scan, keeping the wall-clock guard focused on construction - // correctness. Level 4 is exercised at scale by `just test-slow` - // (full scope); see issue #482. + // correctness. Level 4 retains bounded full-scope regression + // coverage under `just test-slow`; broader Level 4 and Level 5 + // performance work remains in #482/#483 for v0.8.1. println!("Running validation (Levels 1–3 + fast Level 5; realization skipped)..."); let t_validate = Instant::now(); let topology_result = dt.as_triangulation().validation_report(); @@ -1853,6 +1855,8 @@ macro_rules! gen_debug_large_scale_tests { gen_debug_large_scale_tests! { debug_large_scale_2d: 2, "2D", 40_000; debug_large_scale_3d: 3, "3D", 7_500; - debug_large_scale_4d: 4, "4D", 900; - debug_large_scale_5d: 5, "5D", 150; + // Keep the high-dimensional full-report cases bounded until the Level 4 + // and Level 5 performance/equivalence work in #482/#483 lands in v0.8.1. + debug_large_scale_4d: 4, "4D", 100; + debug_large_scale_5d: 5, "5D", 50; } diff --git a/uv.lock b/uv.lock index 033ec528..02fe2c1c 100644 --- a/uv.lock +++ b/uv.lock @@ -2,6 +2,9 @@ version = 1 revision = 3 requires-python = ">=3.14" +[manifest] 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without pkgx. - Refresh validation artifacts and uv notebook dependencies, and enforce native Python 3.14 annotation semantics. Closes #506 --- .github/workflows/papers.yml | 20 +- docs/architecture/module_map.md | 3 +- .../validation/validation_hierarchy.png | Bin 333428 -> 333428 bytes .../validation_level_1_element_validity.png | Bin 127929 -> 127929 bytes ...tion_level_2_combinatorial_consistency.png | Bin 217999 -> 218006 bytes ...lidation_level_3_intrinsic_pl_topology.png | Bin 292728 -> 292759 bytes .../validation_level_4_valid_realization.png | Bin 163408 -> 163408 bytes ...alidation_level_5_geometric_predicates.png | Bin 55854 -> 55854 bytes docs/dev/commands.md | 31 +- docs/dev/tooling-alignment.md | 7 +- justfile | 47 +- papers/validation.pdf | Bin 1228570 -> 1228622 bytes pyproject.toml | 6 +- scripts/check_docs_version_sync.py | 2 - scripts/paper_check.py | 73 +- scripts/tag_release.py | 2 - scripts/tests/test_check_docs_version_sync.py | 2 - scripts/tests/test_paper_check.py | 92 +- semgrep.yaml | 21 + src/geometry/matrix.rs | 12 + src/geometry/realization.rs | 2024 +--------------- src/geometry/util/simplex_lp.rs | 2131 +++++++++++++++++ src/lib.rs | 1 + tests/semgrep/scripts/tests/python_style.py | 7 + uv.lock | 6 +- 25 files changed, 2480 insertions(+), 2007 deletions(-) create mode 100644 src/geometry/util/simplex_lp.rs create mode 100644 tests/semgrep/scripts/tests/python_style.py diff --git a/.github/workflows/papers.yml b/.github/workflows/papers.yml index 356b9b63..802a91bf 100644 --- a/.github/workflows/papers.yml +++ b/.github/workflows/papers.yml @@ -19,6 +19,7 @@ on: paths: - ".github/workflows/papers.yml" - ".github/actions/setup-just/action.yml" + - "docs/assets/validation/**" - "justfile" - "notebooks/01_validation.ipynb" - "papers/**" @@ -36,6 +37,7 @@ on: paths: - ".github/workflows/papers.yml" - ".github/actions/setup-just/action.yml" + - "docs/assets/validation/**" - "justfile" - "notebooks/01_validation.ipynb" - "papers/**" @@ -105,7 +107,7 @@ jobs: - name: Install paper system dependencies run: | brew update - brew install fontconfig freetype graphite2 harfbuzz icu4c libpng pkgconf zlib + brew install freetype graphite2 icu4c libpng pkgconf zlib icu_pkg_config="$(brew --prefix icu4c)/lib/pkgconfig" zlib_pkg_config="$(brew --prefix zlib)/lib/pkgconfig" export PKG_CONFIG_PATH="$icu_pkg_config:$zlib_pkg_config${PKG_CONFIG_PATH:+:$PKG_CONFIG_PATH}" @@ -130,14 +132,16 @@ jobs: with: tool: tex-fmt@${{ steps.tool_versions.outputs.TEX_FMT_VERSION }} - - name: Build papers - run: just papers + - name: Build and compare papers + run: | + just validation-doc-figures + just paper-artifact-check - - name: Verify paper artifacts are current + - name: Verify paper figures are current run: | - if [[ -n "$(git status --porcelain -- papers/validation.pdf papers/generated)" ]]; then - git status --short -- papers/validation.pdf papers/generated - echo "::error::Paper artifacts changed after 'just papers'." - echo "Run 'just papers' and commit the regenerated PDF/PNG outputs." + if [[ -n "$(git status --porcelain -- docs/assets/validation)" ]]; then + git status --short -- docs/assets/validation + echo "::error::Paper figures changed after 'just validation-doc-figures'." + echo "Run 'just validation-doc-figures' and commit the regenerated PNG outputs." exit 1 fi diff --git a/docs/architecture/module_map.md b/docs/architecture/module_map.md index 56daf0e9..ef23f4f9 100644 --- a/docs/architecture/module_map.md +++ b/docs/architecture/module_map.md @@ -77,7 +77,8 @@ PL-manifold validation. Ridge ownership therefore belongs in `src/topology/`. - `traits/coordinate.rs` - coordinate abstractions and typed coordinate diagnostic payloads. - `util/` - coordinate conversions, vector norms, circumsphere computations, - geometric measures, point generation, and random triangulation generation. + geometric measures, point generation, random triangulation generation, and + the private `simplex_lp.rs` implementation used by realization validation. The currently supported caller-visible coordinate scalar is `f64`. Exact arithmetic is used internally by robust predicate fallbacks. 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paper-check +just paper-artifact-check just paper-refresh just papers ``` @@ -626,23 +627,27 @@ changing tracked files. `just paper-pdf-check` uses the uv-managed `paper-pdf-check` helper to verify that target-built PDF opens, has pages, includes expected title/reference text, and does not contain the literal `\today`. `just paper-check` lints, builds, and sanity-checks a paper without -refreshing tracked artifacts. `just paper-refresh` runs that check before -copying the target-built PDF to `papers/validation.pdf`. `just papers` refreshes -the canonical figures and reviewer PDF through those named artifact owners. +refreshing tracked artifacts. `just paper-artifact-check` additionally compares +the rebuilt and tracked reviewer PDFs page by page using extracted text and page +geometry, avoiding a false requirement that platform-native PDF internals be +byte-identical. `just paper-refresh` runs the basic check before copying the +target-built PDF to `papers/validation.pdf`. `just papers` refreshes the +canonical figures and reviewer PDF through those named artifact owners. Tectonic and `tex-fmt` are pinned Cargo-installed tools. `chktex` comes from a TeX distribution or system package manager. Installing or upgrading Tectonic from Cargo also requires a `pkg-config` implementation and development headers -for its externally resolved native bridge libraries, including fontconfig, -FreeType, Graphite2, ICU, libpng, and zlib; the pinned default build vendors -HarfBuzz. When the pinned Tectonic version is absent, `just setup-tools` uses an -existing `pkg-config` or obtains it ephemerally with -`pkgx +freedesktop.org/pkg-config`, then checks the complete external native -dependency set. On macOS it auto-detects common Homebrew metadata directories, -including the active SDK metadata used for system compression libraries, before -it asks for a manual `PKG_CONFIG_PATH`. An already-correct Tectonic installation -does not require those native build prerequisites. Paper CI installs the -platform native package set explicitly. +for its externally resolved native bridge libraries. macOS requires FreeType, +Graphite2, ICU, libpng, and zlib, but not fontconfig. Non-Apple platforms +additionally require fontconfig and OpenSSL. The pinned default build vendors +HarfBuzz. When the pinned Tectonic version is absent, `just setup-tools` +requires `pkg-config` (commonly installed as `pkgconf`) and checks the +platform-specific external native dependency set. On macOS it auto-detects +common Homebrew metadata +directories, including the active SDK metadata used for system compression +libraries, before it asks for a manual `PKG_CONFIG_PATH`. An already-correct +Tectonic installation does not require those native build prerequisites. Paper +CI installs the platform native package set explicitly. Reviewer-facing validation diagrams under `docs/assets/validation/` use the same deterministic notebook with a separate explicit output switch: diff --git a/docs/dev/tooling-alignment.md b/docs/dev/tooling-alignment.md index aa4fb590..79347e0c 100644 --- a/docs/dev/tooling-alignment.md +++ b/docs/dev/tooling-alignment.md @@ -244,10 +244,9 @@ The useful updates ported in this pass are: `taiki-e/cache-cargo-install-action`/`cargo install --locked` pattern before the corresponding recipes run. Pinned Rust CLI tools are installed through Cargo rather than Homebrew so local setup cannot drift from CI pins. The - local Tectonic bootstrap likewise avoids a permanent `pkg-config` installation: - when that command is absent, `just setup-tools` obtains it ephemerally through - `pkgx` for the dependency probe and Cargo installation. The external native - bridge libraries and their pkg-config metadata remain platform-managed. The CI + local Tectonic bootstrap requires the platform-managed `pkg-config` command + (commonly provided by `pkgconf`) alongside the external native bridge + libraries and their metadata. The CI build matrix runs `just ci`, including `cargo machete` dependency hygiene, on Linux, macOS, and Windows after syncing the locked uv dev group and installing the pinned Cargo tools. The papers workflow diff --git a/justfile b/justfile index 2605ff29..79492c8e 100644 --- a/justfile +++ b/justfile @@ -340,6 +340,7 @@ help-workflows: @echo " just notebook-execute # Execute one notebook under target/notebooks" @echo " just validation-doc-figures # Refresh canonical validation figures" @echo " just paper-check # Lint, build, and check without tracked changes" + @echo " just paper-artifact-check # Compare the build with the reviewer PDF" @echo " just paper-refresh # Check, then refresh one tracked reviewer PDF" @echo " just papers # Refresh figures and the validation reviewer PDF" @echo "" @@ -550,6 +551,22 @@ pachner-stress-3d attempts="100" vertices="9000" validate_every="10" output_dir= [group('benchmarks and performance')] pachner-stress-4d attempts="100" vertices="1000" validate_every="10" output_dir="target/pachner_stress/4d" mode="round-trip": (_pachner-stress-dim "4d" vertices attempts validate_every output_dir mode) +# Check that one target-built paper is structurally equivalent to its tracked reviewer PDF. +[group('notebooks and papers')] +paper-artifact-check paper="validation": (paper-check paper) + #!/usr/bin/env bash + set -euo pipefail + paper={{ quote(paper) }} + case "$paper" in + ""|*[!A-Za-z0-9_-]*) + echo "❌ Invalid paper name: $paper" + echo " Use only ASCII letters, digits, underscores, and hyphens." + exit 1 + ;; + esac + uv run paper-pdf-check "target/papers/${paper}/${paper}.pdf" \ + --reference "papers/${paper}.pdf" + # Compile one paper with Tectonic under target/papers/. [group('notebooks and papers')] paper-build paper="validation": _ensure-tectonic _ensure-uv @@ -1140,25 +1157,25 @@ setup-tools: _ensure-uv ensure_tectonic_build_dependencies() { echo "Ensuring native dependencies needed to install Tectonic..." - local candidate homebrew_repository package sdk_version - local -a pkg_config_command=(pkg-config) + local candidate homebrew_repository package platform sdk_version local -a missing_pkg_config_packages=() - local -a required_pkg_config_packages=(fontconfig freetype2 graphite2 icu-uc libpng zlib) + local -a required_pkg_config_packages=(freetype2 graphite2 icu-uc libpng zlib) + + platform="$(uname -s)" + if [ "$platform" != "Darwin" ]; then + required_pkg_config_packages=(fontconfig freetype2 graphite2 icu-uc libpng openssl zlib) + fi append_pkg_config_path() { local directory="$1" if [ -d "$directory" ] && [[ ":${PKG_CONFIG_PATH:-}:" != *":$directory:"* ]]; then - export PKG_CONFIG_PATH="$directory${PKG_CONFIG_PATH:+:$PKG_CONFIG_PATH}" + export PKG_CONFIG_PATH="${PKG_CONFIG_PATH:+$PKG_CONFIG_PATH:}$directory" fi } if ! have pkg-config; then - if ! have pkgx; then - echo "❌ Neither 'pkg-config' nor 'pkgx' was found. Install pkgx or pkg-config before building Tectonic from Cargo." - exit 1 - fi - pkg_config_command=(pkgx +freedesktop.org/pkg-config pkg-config) - echo " ✓ pkgx will supply pkg-config ephemerally" + echo "❌ 'pkg-config' was not found. Install pkgconf or pkg-config before building Tectonic from Cargo." + exit 1 fi shopt -s nullglob @@ -1179,7 +1196,7 @@ setup-tools: _ensure-uv fi for package in "${required_pkg_config_packages[@]}"; do - if ! "${pkg_config_command[@]}" --exists "$package"; then + if ! pkg-config --exists "$package"; then missing_pkg_config_packages+=("$package") fi done @@ -1202,16 +1219,12 @@ setup-tools: _ensure-uv ensure_tectonic_build_dependencies echo " ⏳ Installing tectonic $expected_version (cargo)..." - if have pkg-config; then - cargo install --locked tectonic --version "$expected_version" - else - pkgx +freedesktop.org/pkg-config cargo install --locked tectonic --version "$expected_version" - fi + cargo install --locked tectonic --version "$expected_version" } echo "This recipe installs pinned Rust CLI tools through cargo." echo "External prerequisites that must already be on PATH: uv, jq, rustup, cargo, and chktex." - echo "pkgx or pkg-config, plus native development files, are required only when the pinned Tectonic version must be installed." + echo "pkg-config, plus native development files, is required only when the pinned Tectonic version must be installed." echo "" echo "Ensuring uv-managed Python tooling..." diff --git a/papers/validation.pdf b/papers/validation.pdf index 0d2265e31c8db83f363c6130eb76d67b30ea856d..f301dc2ee25ce65580ec7699e920a66f8b3b5944 100644 GIT binary patch delta 444631 zcmZ5{cU)6hw|3M422mKLskE^mRZx)L6crQ{q<4@my|+L(cBDo@2t7)XPUxMWV2E_- zB|xNv2%(1%AV9u@^Uizk{q`UJq9!Lhd#|=j!lJJ%DBI2)Q#)*Wo(|<6Up4517Lkh9J z%;v1x8VtI-wnd4q7ED>B{F*j)5?|Gymku z>f&O->f$w0TA^q)l(Gc*XXMllME&RGizjRl#GW9NGGLlrdAxkJV-3(d#h2rjR%=z;vJ|VFeh6_!K8eH1;lxikRF*QA6 zCZ+R#aK0TbGNJDb4(Z|JXf8{rmV<5Di+ks_xg-lr%hknKF31#mjTd)pU#sy%I}TPj z4tER^0CL4`)#UvtPYi~TKl#-%76SR>D?jAj)#+f`OS$RQG;gCKJ_}2zE%!acF~(CD z_HbM0@2#u1=%lHxC-l8x<+T%( zwPqjMN{w;+Mak2%&G;~SZ>T`@c~=Q!{QLTyDXLl&eC)(R_n^)%=(Y&Ya8c2fV?oy; z5a91Ww;_;2RpV02#8E@ZqWAbXm5)8lZ1|wTNNlx9a{a~un_7?pJ?acTrWZd*?6dJ+ zd#OvI2d_Jy!0RWw|3zMq{1y0hnArqFyG6Uqs&{ww$?6kN22TbEVu|2mlFhy;<-GxBi-~PbB!aWsCI)JNsM-OQK%Al$?tyk~>cE<~a zAxA6mdm&7L#+8b>5G!4!pXIW}`N1<#=q8$K=t`>l%N{wmoslA4fk1zv 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argparse import os import re diff --git a/scripts/paper_check.py b/scripts/paper_check.py index ea177764..b1b448a7 100644 --- a/scripts/paper_check.py +++ b/scripts/paper_check.py @@ -13,17 +13,29 @@ if TYPE_CHECKING: from collections.abc import Sequence + from pypdf.generic import RectangleObject + class PdfInspectionError(RuntimeError): """Raised when a generated PDF cannot satisfy paper sanity checks.""" +@dataclass(frozen=True, slots=True) +class PdfPageInspection: + """Platform-stable facts extracted from one PDF page.""" + + text: str + media_box: tuple[float, float, float, float] + crop_box: tuple[float, float, float, float] + + @dataclass(frozen=True, slots=True) class PdfInspection: - """Extracted PDF facts used by sanity checks.""" + """Extracted PDF facts used by sanity and reviewer-copy checks.""" page_count: int text: str + pages: tuple[PdfPageInspection, ...] @dataclass(frozen=True, slots=True) @@ -52,6 +64,7 @@ class PdfCheckOptions: min_pages: PositivePageCount required_text: tuple[str, ...] forbidden_text: tuple[str, ...] + reference: Path | None = None def __post_init__(self) -> None: """Reject options that bypass the positive page-count parser.""" @@ -74,8 +87,18 @@ def parse_positive_page_count(value: str) -> PositivePageCount: raise argparse.ArgumentTypeError(msg) from error +def rectangle_coordinates(rectangle: RectangleObject) -> tuple[float, float, float, float]: + """Return one PDF rectangle in a representation stable across pypdf objects.""" + return ( + float(rectangle.left), + float(rectangle.bottom), + float(rectangle.right), + float(rectangle.top), + ) + + def inspect_pdf(pdf: Path) -> PdfInspection: - """Extract page count and text from a PDF.""" + """Extract page count, text, and page geometry from a PDF.""" if not pdf.is_file(): msg = f"PDF does not exist: {pdf}" raise FileNotFoundError(msg) @@ -89,14 +112,41 @@ def inspect_pdf(pdf: Path) -> PdfInspection: msg = f"{pdf}: encrypted PDFs are not valid reviewer copies" raise PdfInspectionError(msg) - text_parts: list[str] = [] + pages: list[PdfPageInspection] = [] for page_number, page in enumerate(reader.pages, start=1): try: - text_parts.append(page.extract_text() or "") - except PyPdfError as error: - msg = f"{pdf}: failed to extract text from page {page_number}: {error}" + pages.append( + PdfPageInspection( + text=page.extract_text() or "", + media_box=rectangle_coordinates(page.mediabox), + crop_box=rectangle_coordinates(page.cropbox), + ) + ) + except (PyPdfError, TypeError, ValueError) as error: + msg = f"{pdf}: failed to inspect page {page_number}: {error}" raise PdfInspectionError(msg) from error - return PdfInspection(page_count=len(reader.pages), text="\n".join(text_parts)) + return PdfInspection( + page_count=len(pages), + text="\n".join(page.text for page in pages), + pages=tuple(pages), + ) + + +def compare_pdf_structure(generated: PdfInspection, reference: PdfInspection) -> list[str]: + """Return structural differences that imply a stale reviewer copy.""" + failures: list[str] = [] + if generated.page_count != reference.page_count: + failures.append(f"reference has {reference.page_count} page(s), rebuilt PDF has {generated.page_count}") + return failures + + for page_number, (generated_page, reference_page) in enumerate(zip(generated.pages, reference.pages, strict=True), start=1): + if generated_page.text != reference_page.text: + failures.append(f"reference page {page_number} text differs from rebuilt PDF") + if generated_page.media_box != reference_page.media_box: + failures.append(f"reference page {page_number} media box differs from rebuilt PDF") + if generated_page.crop_box != reference_page.crop_box: + failures.append(f"reference page {page_number} crop box differs from rebuilt PDF") + return failures def check_pdf(options: PdfCheckOptions) -> PdfInspection: @@ -109,6 +159,9 @@ def check_pdf(options: PdfCheckOptions) -> PdfInspection: failures.extend(f"missing required text: {required!r}" for required in options.required_text if required not in inspection.text) failures.extend(f"found forbidden text: {forbidden!r}" for forbidden in options.forbidden_text if forbidden in inspection.text) + if options.reference is not None: + reference = inspect_pdf(options.reference) + failures.extend(compare_pdf_structure(inspection, reference)) if failures: msg = f"{options.pdf}: " + "; ".join(failures) @@ -124,6 +177,11 @@ def build_parser() -> argparse.ArgumentParser: parser.add_argument("--min-pages", type=parse_positive_page_count, default=PositivePageCount.from_raw(1), help="minimum acceptable page count") parser.add_argument("--require-text", action="append", default=None, help="text that must appear in extracted PDF text") parser.add_argument("--forbid-text", action="append", default=None, help="text that must not appear in extracted PDF text") + parser.add_argument( + "--reference", + type=Path, + help="tracked reviewer PDF whose page text and geometry must match", + ) return parser @@ -136,6 +194,7 @@ def main(argv: Sequence[str] | None = None) -> int: min_pages=namespace.min_pages, required_text=tuple(namespace.require_text or ()), forbidden_text=tuple(namespace.forbid_text or ()), + reference=namespace.reference, ) try: diff --git a/scripts/tag_release.py b/scripts/tag_release.py index e6af6520..547a8948 100644 --- a/scripts/tag_release.py +++ b/scripts/tag_release.py @@ -12,8 +12,6 @@ Ported from the la-stack project's tag_release.py. """ -from __future__ import annotations - import argparse import logging import re diff --git a/scripts/tests/test_check_docs_version_sync.py b/scripts/tests/test_check_docs_version_sync.py index a8046e59..cee92e06 100644 --- a/scripts/tests/test_check_docs_version_sync.py +++ b/scripts/tests/test_check_docs_version_sync.py @@ -1,7 +1,5 @@ """Tests for documentation and package-version synchronization checks.""" -from __future__ import annotations - from typing import TYPE_CHECKING import pytest diff --git a/scripts/tests/test_paper_check.py b/scripts/tests/test_paper_check.py index 30f77574..b0caa921 100644 --- a/scripts/tests/test_paper_check.py +++ b/scripts/tests/test_paper_check.py @@ -1,28 +1,36 @@ """Tests for the paper PDF sanity checker.""" -from typing import TYPE_CHECKING, cast +from pathlib import Path +from typing import cast import pytest from pypdf.errors import PyPdfError from paper_check import PdfCheckOptions, PdfInspectionError, PositivePageCount, check_pdf, main -if TYPE_CHECKING: - from pathlib import Path - class FakePage: """Minimal pypdf page stand-in.""" - def __init__(self, text: str) -> None: + def __init__(self, text: str, *, media_box: tuple[float, float, float, float] = (0.0, 0.0, 612.0, 792.0)) -> None: """Create a fake page with extractable text.""" self._text = text + self.mediabox = FakeRectangle(media_box) + self.cropbox = FakeRectangle(media_box) def extract_text(self) -> str: """Return fake extracted page text.""" return self._text +class FakeRectangle: + """Minimal pypdf rectangle stand-in.""" + + def __init__(self, coordinates: tuple[float, float, float, float]) -> None: + """Create one fake page rectangle.""" + self.left, self.bottom, self.right, self.top = coordinates + + class FakeReader: """Minimal pypdf reader stand-in.""" @@ -51,6 +59,9 @@ def raise_pypdf_error(_path: str) -> FakeReader: class BrokenPage: """Minimal pypdf page stand-in that raises an extraction error.""" + mediabox = FakeRectangle((0.0, 0.0, 612.0, 792.0)) + cropbox = mediabox + def extract_text(self) -> str: """Raise a fake pypdf extraction error.""" msg = "cannot extract text" @@ -156,15 +167,82 @@ def test_check_pdf_wraps_pypdf_reader_errors(tmp_path: Path, monkeypatch: pytest check_pdf(PdfCheckOptions(pdf=pdf, min_pages=min_pages(1), required_text=(), forbidden_text=())) -def test_check_pdf_wraps_pypdf_text_extraction_errors(tmp_path: Path, monkeypatch: pytest.MonkeyPatch) -> None: +def test_check_pdf_wraps_pypdf_page_inspection_errors(tmp_path: Path, monkeypatch: pytest.MonkeyPatch) -> None: pdf = write_pdf_stub(tmp_path) monkeypatch.setattr("paper_check.PdfReader", FakeReaderWithBrokenPage) - with pytest.raises(PdfInspectionError, match="failed to extract text from page 1"): + with pytest.raises(PdfInspectionError, match="failed to inspect page 1"): check_pdf(PdfCheckOptions(pdf=pdf, min_pages=min_pages(1), required_text=(), forbidden_text=())) +def test_check_pdf_accepts_structurally_equivalent_reference(tmp_path: Path, monkeypatch: pytest.MonkeyPatch) -> None: + pdf = write_pdf_stub(tmp_path) + reference = tmp_path / "reference.pdf" + reference.write_bytes(b"%PDF-1.7\n") + monkeypatch.setattr("paper_check.PdfReader", FakeReader) + + inspection = check_pdf( + PdfCheckOptions( + pdf=pdf, + min_pages=min_pages(1), + required_text=(), + forbidden_text=(), + reference=reference, + ) + ) + + assert inspection.page_count == 1 + + +def test_check_pdf_rejects_reference_with_different_page_text(tmp_path: Path, monkeypatch: pytest.MonkeyPatch) -> None: + pdf = write_pdf_stub(tmp_path) + reference = tmp_path / "reference.pdf" + reference.write_bytes(b"%PDF-1.7\n") + + def fake_reader(path: str) -> FakeReader: + text = "stale reviewer text" if Path(path) == reference else "rebuilt paper text" + return FakeReader(path, text=text) + + monkeypatch.setattr("paper_check.PdfReader", fake_reader) + + with pytest.raises(PdfInspectionError, match="reference page 1 text differs"): + check_pdf( + PdfCheckOptions( + pdf=pdf, + min_pages=min_pages(1), + required_text=(), + forbidden_text=(), + reference=reference, + ) + ) + + +def test_check_pdf_rejects_reference_with_different_page_geometry(tmp_path: Path, monkeypatch: pytest.MonkeyPatch) -> None: + pdf = write_pdf_stub(tmp_path) + reference = tmp_path / "reference.pdf" + reference.write_bytes(b"%PDF-1.7\n") + + def fake_reader(path: str) -> FakeReader: + reader = FakeReader(path) + if Path(path) == reference: + reader.pages = [FakePage("Validation Architecture in delaunay", media_box=(0.0, 0.0, 600.0, 792.0))] + return reader + + monkeypatch.setattr("paper_check.PdfReader", fake_reader) + + with pytest.raises(PdfInspectionError, match="reference page 1 media box differs"): + check_pdf( + PdfCheckOptions( + pdf=pdf, + min_pages=min_pages(1), + required_text=(), + forbidden_text=(), + reference=reference, + ) + ) + + def test_main_reports_success_to_stdout(capsys: pytest.CaptureFixture[str], tmp_path: Path, monkeypatch: pytest.MonkeyPatch) -> None: pdf = write_pdf_stub(tmp_path) diff --git a/semgrep.yaml b/semgrep.yaml index 83b56784..8668a492 100644 --- a/semgrep.yaml +++ b/semgrep.yaml @@ -2142,6 +2142,27 @@ rules: - "/tests/semgrep/**" pattern-regex: '^\s*except\s+Exception(?:\s+as\s+\w+)?\s*:' + - id: delaunay.python.no-future-annotations-on-python314 + languages: + - python + severity: WARNING + message: >- + Python 3.14 defers annotations by default. Remove + `from __future__ import annotations`; suppress this rule with a reason + only when a runtime consumer intentionally requires stringified + annotations. + metadata: + category: maintainability + rationale: >- + The repository requires Python 3.14 or newer. The future import opts + into legacy stringified annotations and is deprecated, so retaining it + can obscure the runtime annotation semantics. + paths: + include: + - "/scripts/**/*.py" + - "/tests/semgrep/scripts/tests/python_style.py" + pattern: from __future__ import annotations + - id: delaunay.notebook.no-pandas-table-engine languages: - generic diff --git a/src/geometry/matrix.rs b/src/geometry/matrix.rs index b652132e..17f8e587 100644 --- a/src/geometry/matrix.rs +++ b/src/geometry/matrix.rs @@ -280,6 +280,18 @@ mod tests { ); } + #[test] + fn solve_exact_runtime_system_rejects_malformed_shapes() { + assert_eq!( + solve_exact_runtime_system(&[vec![1.0, 0.0]], &[1.0, 0.0]), + None + ); + assert_eq!( + solve_exact_runtime_system(&[vec![1.0], vec![0.0, 1.0]], &[1.0, 0.0]), + None + ); + } + #[test] fn la_index_error_maps_to_matrix_error_with_context() { let err = StackMatrixDispatchError::from(LaError::index_out_of_bounds(3, 4, 2)); diff --git a/src/geometry/realization.rs b/src/geometry/realization.rs index 74cc700b..688c7085 100644 --- a/src/geometry/realization.rs +++ b/src/geometry/realization.rs @@ -40,12 +40,13 @@ #![forbid(unsafe_code)] use crate::core::collections::{MAX_PRACTICAL_DIMENSION_SIZE, SmallBuffer}; -use crate::geometry::matrix::{StackMatrixDispatchError, solve_exact_runtime_system}; use crate::geometry::point::{Point, ValidatedCoordinates}; use crate::geometry::predicates::Orientation; use crate::geometry::robust_predicates::robust_orientation; use crate::geometry::traits::coordinate::InvalidCoordinateValue; -use la_stack::{BigInt, BigRational, FromPrimitive, Signed, ToPrimitive}; +use crate::geometry::util::simplex_lp::{ + IntersectionLinearProgramResult, intersection_via_linear_program, shared_face_fast_confinement, +}; use thiserror::Error; /// Stack-backed buffer for per-simplex realization labels and coordinates. @@ -629,1909 +630,118 @@ where return Ok(()); } - let fallback_barycentric = if basis_orientation.is_none() { - Some(barycentric_coordinates_of_vertices(second, first)?) - } else { - None - }; - - match intersection_via_linear_program(first, second, &shared_labels) { - IntersectionLinearProgramResult::Valid => return Ok(()), - IntersectionLinearProgramResult::Invalid(witness) => { - return Err(SimplexIntersectionFailure::IntersectionOutsideSharedFace { witness }); - } - IntersectionLinearProgramResult::Fallback => {} - } - - let second_vertices_in_first = match fallback_barycentric { - Some(coordinates) => coordinates, - None => barycentric_coordinates_of_vertices(second, first)?, - }; - if intersection_is_confined_to_shared_face( - &second_vertices_in_first, - first.labels(), + match intersection_via_linear_program( + first, + second, &shared_labels, - ) { - return Ok(()); - } - let intersection_vertices = intersection_polytope_vertices(&second_vertices_in_first); - - for beta in intersection_vertices { - let alpha = alpha_from_beta(&beta, &second_vertices_in_first); - let first_only_witness_labels = - positive_nonshared_labels(&alpha, first.labels(), &shared_labels); - let second_only_witness_labels = - positive_nonshared_labels(&beta, second.labels(), &shared_labels); - - if !first_only_witness_labels.is_empty() || !second_only_witness_labels.is_empty() { - return Err(SimplexIntersectionFailure::IntersectionOutsideSharedFace { - witness: SimplexIntersectionWitness { - shared: shared_labels, - first_only_witness: first_only_witness_labels, - second_only_witness: second_only_witness_labels, - }, - }); - } - } - - Ok(()) -} - -/// Tries cheap candidate separators in the normal space of one shared face. -#[expect( - clippy::too_many_lines, - reason = "the separator search is one bounded projection-and-certification workflow" -)] -fn shared_face_fast_confinement( - first: &LabeledSimplexRealization, - second: &LabeledSimplexRealization, - shared_labels: &[L], -) -> bool -where - L: Eq, -{ - if shared_labels.is_empty() || shared_labels.len() >= D { - return false; - } - let shared_coordinates: Vec<_> = shared_labels - .iter() - .map(|label| { - let first_index = first - .labels() - .iter() - .position(|candidate| candidate == label)?; - let second_index = second - .labels() - .iter() - .position(|candidate| candidate == label)?; - coordinates_are_identical( - &first.coordinates()[first_index], - &second.coordinates()[second_index], - ) - .then_some(first.coordinates()[first_index]) - }) - .collect::>() - .unwrap_or_default(); - if shared_coordinates.len() != shared_labels.len() { - return false; - } - let base = shared_coordinates[0]; - let shared_deltas: Vec> = shared_coordinates[1..] - .iter() - .map(|coordinates| { - coordinates - .iter() - .zip(base) - .map(|(coordinate, base_coordinate)| coordinate - base_coordinate) - .collect() - }) - .collect(); - let gram: Vec> = shared_deltas - .iter() - .map(|left| { - shared_deltas - .iter() - .map(|right| dot_product_f64(left, right)) - .collect() - }) - .collect(); - - let mut axis = vec![0.0; D]; - let mut normalized_rays = Vec::with_capacity(2 * D); - for (label, coordinates) in first.labels().iter().zip(first.coordinates()) { - if shared_labels.contains(label) { - continue; - } - let raw_ray: Vec<_> = coordinates - .iter() - .zip(base) - .map(|(coordinate, shared)| coordinate - shared) - .collect(); - let Some(ray) = project_ray_orthogonal_to_shared_face(raw_ray, &shared_deltas, &gram) - else { - return false; - }; - for coordinate in 0..D { - axis[coordinate] += ray[coordinate]; - } - push_normalized_ray(&mut normalized_rays, ray); - } - for (label, coordinates) in second.labels().iter().zip(second.coordinates()) { - if shared_labels.contains(label) { - continue; - } - let raw_ray: Vec<_> = base - .iter() - .zip(coordinates) - .map(|(shared, coordinate)| shared - coordinate) - .collect(); - let Some(ray) = project_ray_orthogonal_to_shared_face(raw_ray, &shared_deltas, &gram) - else { - return false; - }; - for coordinate in 0..D { - axis[coordinate] += ray[coordinate]; - } - push_normalized_ray(&mut normalized_rays, ray); - } - if axis.iter().all(|value| value.is_finite()) - && provisional_axis_certifies_confinement(first, second, shared_labels, &axis) - { - return true; - } - - let mut refined_axis = vec![0.0; D]; - for ray in &normalized_rays { - for (value, component) in refined_axis.iter_mut().zip(ray) { - *value += component; - } - } - for _ in 0..16 { - let mut adjusted = false; - for ray in &normalized_rays { - let projection = dot_product_f64(&refined_axis, ray); - if projection < 1.0 { - let correction = 1.0 - projection; - for (value, component) in refined_axis.iter_mut().zip(ray) { - *value = component.mul_add(correction, *value); - } - adjusted = true; - } - } - if !adjusted { - break; - } - } - refined_axis.iter().all(|value| value.is_finite()) - && provisional_axis_certifies_confinement(first, second, shared_labels, &refined_axis) -} - -/// Removes provisional components tangent to the shared affine face. -fn project_ray_orthogonal_to_shared_face( - mut ray: Vec, - shared_deltas: &[Vec], - gram: &[Vec], -) -> Option> { - if shared_deltas.is_empty() { - return Some(ray); - } - let residuals: Vec<_> = shared_deltas - .iter() - .map(|delta| dot_product_f64(delta, &ray)) - .collect(); - let correction = solve_f64_system(gram.to_vec(), residuals)?; - for (coordinate, value) in ray.iter_mut().enumerate() { - for (coefficient, delta) in correction.iter().zip(shared_deltas) { - *value = (-coefficient).mul_add(delta[coordinate], *value); - } - } - ray.iter().all(|value| value.is_finite()).then_some(ray) -} - -/// Certifies one provisional shared-face separator through a filtered or exact path. -fn provisional_axis_certifies_confinement( - first: &LabeledSimplexRealization, - second: &LabeledSimplexRealization, - shared_labels: &[L], - axis: &[f64], -) -> bool -where - L: Eq, -{ - if shared_labels.len() == 1 { - filtered_single_shared_vertex_confinement(first, second, shared_labels, axis) - } else { - exact_shared_face_confinement_certificate(first, second, shared_labels, axis) - } -} - -/// Appends one finite unit ray for provisional separator refinement. -fn push_normalized_ray(rays: &mut Vec>, ray: Vec) { - let norm = ray - .iter() - .fold(0.0_f64, |value, component| value.hypot(*component)); - if norm.is_finite() && norm > 0.0 { - rays.push(ray.into_iter().map(|component| component / norm).collect()); - } -} - -/// Computes one simplex orientation through the filtered-exact predicate path. -fn realization_orientation( - simplex: &LabeledSimplexRealization, -) -> Option { - let points: SimplexRealizationBuffer<_> = (0..simplex.coordinates().len()) - .filter_map(|index| simplex.point_at(index)) - .collect(); - robust_orientation(&points).ok() -} - -/// Proves disjointness when one simplex lies strictly beyond a facet of another. -/// -/// Replacing facet-opposite basis vertex `i` with a point `p` scales the basis -/// orientation by the barycentric coordinate of `p` at `i`. If every vertex of -/// the other simplex has the opposite non-zero sign for one `i`, convexity -/// places that entire simplex in the open exterior half-space. -fn simplex_is_strictly_outside_a_facet( - basis: &LabeledSimplexRealization, - other: &LabeledSimplexRealization, - basis_orientation: Orientation, -) -> bool { - let basis_points: SimplexRealizationBuffer<_> = (0..basis.coordinates().len()) - .filter_map(|index| basis.point_at(index)) - .collect(); - - (0..basis.labels().len()).any(|basis_index| { - (0..other.coordinates().len()).all(|other_index| { - let Some(other_point) = other.point_at(other_index) else { - return false; - }; - let mut replaced_points = basis_points.clone(); - replaced_points[basis_index] = other_point; - robust_orientation(&replaced_points).is_ok_and(|orientation| { - orientation != basis_orientation && orientation != Orientation::DEGENERATE - }) - }) - }) -} - -/// Uses filtered-exact orientation signs to prove confinement to the shared face. -/// -/// Replacing basis vertex `i` with a point `p` changes the orientation -/// determinant by the barycentric coordinate of `p` at `i`. Therefore, when -/// every vertex of the other simplex has a zero or opposite determinant sign -/// for each non-shared basis vertex, convexity proves that any point in both -/// simplices has zero weight outside the shared face. -fn intersection_is_confined_by_orientation( - basis: &LabeledSimplexRealization, - other: &LabeledSimplexRealization, - shared_labels: &[L], - basis_orientation: Orientation, -) -> bool -where - L: Eq, -{ - let basis_points: SimplexRealizationBuffer<_> = (0..basis.coordinates().len()) - .filter_map(|index| basis.point_at(index)) - .collect(); - - for (basis_index, basis_label) in basis.labels().iter().enumerate() { - if shared_labels.contains(basis_label) { - continue; - } - for other_index in 0..other.coordinates().len() { - let Some(other_point) = other.point_at(other_index) else { - return false; - }; - let mut replaced_points = basis_points.clone(); - replaced_points[basis_index] = other_point; - let Ok(replaced_orientation) = robust_orientation(&replaced_points) else { - return false; - }; - if replaced_orientation == basis_orientation { - return false; - } - } - } - true -} - -/// Accepts a pair when simplex-facet signs prove any intersection lies in the shared face. -/// -/// A point inside the basis simplex has non-negative barycentric coordinates. -/// If every vertex of the other simplex has a non-positive coordinate for each -/// basis vertex outside the shared face, convexity makes those coordinates -/// non-positive throughout the other simplex. Any point in both simplices must -/// therefore have zero weight on every non-shared basis vertex. -fn intersection_is_confined_to_shared_face( - other_vertices_in_basis: &[Vec], - basis_labels: &[L], - shared_labels: &[L], -) -> bool -where - L: Eq, -{ - basis_labels - .iter() - .enumerate() - .filter(|(_, label)| !shared_labels.contains(label)) - .all(|(basis_index, _)| { - other_vertices_in_basis - .iter() - .all(|barycentric| !barycentric[basis_index].is_positive()) - }) -} - -/// Collects labels common to two simplex realizations so witnesses can distinguish shared faces. -fn shared_labels( - first: &LabeledSimplexRealization, - second: &LabeledSimplexRealization, -) -> SimplexRealizationBuffer -where - L: Clone + Eq, -{ - first - .labels() - .iter() - .filter(|label| second.labels().contains(label)) - .cloned() - .collect() -} - -/// Expresses every vertex of one simplex in the barycentric basis of another. -fn barycentric_coordinates_of_vertices( - vertices: &LabeledSimplexRealization, - basis: &LabeledSimplexRealization, -) -> Result>, SimplexIntersectionFailure> { - vertices - .coordinates() - .iter() - .map(|coords| barycentric_coordinates(coords, basis)) - .collect() -} - -/// Computes exact barycentric coordinates of one point in one simplex basis. -fn barycentric_coordinates( - point: &[f64; D], - simplex: &LabeledSimplexRealization, -) -> Result, SimplexIntersectionFailure> { - if D == 0 { - return Ok(vec![rational_one()]); - } - - let origin = &simplex.coordinates()[0]; - let mut matrix = vec![vec![rational_zero(); D]; D]; - let mut rhs = vec![rational_zero(); D]; - - for axis in 0..D { - let origin_coord = rational_from_f64(origin[axis]); - rhs[axis] = rational_from_f64(point[axis]) - origin_coord.clone(); - for (column, matrix_value) in matrix[axis].iter_mut().enumerate() { - *matrix_value = - rational_from_f64(simplex.coordinates()[column + 1][axis]) - origin_coord.clone(); - } - } - - let lambdas = solve_rational_system(matrix, rhs) - .ok_or(SimplexIntersectionFailure::SingularBarycentricBasis)?; - let lambda_sum = lambdas - .iter() - .fold(rational_zero(), |acc, value| acc + value.clone()); - let mut barycentric = Vec::with_capacity(D + 1); - barycentric.push(rational_one() - lambda_sum); - barycentric.extend(lambdas); - Ok(barycentric) -} - -/// Outcome of the exact linear-program intersection path. -#[derive(Debug)] -enum IntersectionLinearProgramResult { - /// The simplices are disjoint or meet only in their shared face. - Valid, - /// A feasible point has positive weight outside the shared face. - Invalid(SimplexIntersectionWitness), - /// An internal simplex-method precondition failed; use active-set enumeration. - Fallback, -} - -/// Uses exact linear programming to avoid enumerating every intersection-polytope basis. -/// -/// The variables are the non-negative barycentric weights `alpha` and `beta` -/// of the two simplices. Equality constraints require each weight vector to sum -/// to one and both weighted coordinate sums to describe the same point. -/// Maximizing the total weight on non-shared labels is zero exactly when every -/// feasible intersection point lies in the shared face. -fn intersection_via_linear_program( - first: &LabeledSimplexRealization, - second: &LabeledSimplexRealization, - shared_labels: &[L], -) -> IntersectionLinearProgramResult -where - L: Clone + Eq, -{ - let first_labels = first.labels(); - let second_labels = second.labels(); - let simplex_vertex_count = first_labels.len(); - if second_labels.len() != simplex_vertex_count - || first.coordinates().len() != simplex_vertex_count - || second.coordinates().len() != simplex_vertex_count - { - return IntersectionLinearProgramResult::Fallback; - } - - let variable_count = simplex_vertex_count * 2; - let constraint_count = D + 2; - let mut float_matrix = vec![vec![0.0; variable_count]; constraint_count]; - let mut float_rhs = vec![0.0; constraint_count]; - float_rhs[0] = 1.0; - float_rhs[1] = 1.0; - - for (alpha_index, coordinates) in first.coordinates().iter().enumerate() { - float_matrix[0][alpha_index] = 1.0; - for (axis, coordinate) in coordinates.iter().enumerate() { - float_matrix[axis + 2][alpha_index] = *coordinate; - } - } - for (beta_index, coordinates) in second.coordinates().iter().enumerate() { - float_matrix[1][simplex_vertex_count + beta_index] = 1.0; - for (axis, coordinate) in coordinates.iter().enumerate() { - float_matrix[axis + 2][simplex_vertex_count + beta_index] = -*coordinate; - } - } - - let mut float_objective = vec![0.0; variable_count]; - for (index, label) in first_labels.iter().enumerate() { - if !shared_labels.contains(label) { - float_objective[index] = 1.0; - } - } - for (index, label) in second_labels.iter().enumerate() { - if !shared_labels.contains(label) { - float_objective[simplex_vertex_count + index] = 1.0; - } - } - - let ProvisionalPhaseDuals { - infeasibility_axis, - confinement_axis, - confinement_basis, - } = provisional_phase_dual_axes(&float_matrix, &float_rhs, &float_objective, variable_count); - if infeasibility_axis - .is_some_and(|axis| simplices_are_strictly_separated_along(first, second, &axis)) - { - return IntersectionLinearProgramResult::Valid; - } - let axis_certifies = confinement_axis.is_some_and(|axis| { - filtered_single_shared_vertex_confinement(first, second, shared_labels, &axis) - || exact_shared_face_confinement_certificate(first, second, shared_labels, &axis) - }); - if axis_certifies { - return IntersectionLinearProgramResult::Valid; - } - - let matrix = exact_matrix_from_f64(&float_matrix); - let rhs: Vec<_> = float_rhs.iter().copied().map(rational_from_f64).collect(); - let objective: Vec<_> = float_objective - .iter() - .copied() - .map(rational_from_f64) - .collect(); - let dual_certifies = confinement_basis.is_some_and(|basis| { - phase_two_dual_proves_shared_face_confinement(&matrix, &rhs, &objective, &basis) - }); - if dual_certifies { - return IntersectionLinearProgramResult::Valid; - } - - match maximize_nonnegative_equality_linear_program(&matrix, &rhs, &objective) { - ExactLinearProgramResult::Infeasible => IntersectionLinearProgramResult::Valid, - ExactLinearProgramResult::Optimal { - objective_value, .. - } if !objective_value.is_positive() => IntersectionLinearProgramResult::Valid, - ExactLinearProgramResult::Optimal { solution, .. } => { - let first_only_witness = positive_nonshared_labels( - &solution[..simplex_vertex_count], - first_labels, - shared_labels, - ); - let second_only_witness = positive_nonshared_labels( - &solution[simplex_vertex_count..], - second_labels, - shared_labels, - ); - if first_only_witness.is_empty() && second_only_witness.is_empty() { - return IntersectionLinearProgramResult::Fallback; - } - IntersectionLinearProgramResult::Invalid(SimplexIntersectionWitness { - shared: shared_labels.iter().cloned().collect(), - first_only_witness, - second_only_witness, - }) - } - ExactLinearProgramResult::Failed => IntersectionLinearProgramResult::Fallback, - } -} - -/// Result of an exact non-negative equality-constrained linear program. -enum ExactLinearProgramResult { - /// The objective is maximized at an exact basic feasible solution. - Optimal { - /// Values for the original decision variables. - solution: Vec, - /// Exact objective value at `solution`. - objective_value: BigRational, - }, - /// No non-negative solution satisfies the equality constraints. - Infeasible, - /// Tableau construction or a boundedness invariant failed. - Failed, -} - -/// Result of one revised-simplex optimization phase. -enum RevisedSimplexResult { - /// The current basis is optimal. - Optimal { - /// Values for every variable in the supplied matrix. - solution: Vec, - /// Exact objective value at `solution`. - objective_value: BigRational, - }, - /// The objective can increase without bound. - Unbounded, - /// The supplied basis or iteration budget failed. - Failed, -} - -/// Maximizes an exact objective subject to `matrix * x = rhs` and `x >= 0`. -/// -/// Phase I introduces one artificial variable per equality. Bland's pivot -/// ordering makes degenerate exact-arithmetic bases deterministic and avoids -/// cycling. Artificial variables are pivoted out at zero before Phase II. -fn maximize_nonnegative_equality_linear_program( - matrix: &[Vec], - rhs: &[BigRational], - objective: &[BigRational], -) -> ExactLinearProgramResult { - let constraint_count = rhs.len(); - let variable_count = objective.len(); - if constraint_count == 0 - || matrix.len() != constraint_count - || matrix.iter().any(|row| row.len() != variable_count) - || rhs.iter().any(Signed::is_negative) - { - return ExactLinearProgramResult::Failed; - } - - let total_variable_count = variable_count + constraint_count; - let mut phase_matrix = Vec::with_capacity(constraint_count); - for (row_index, row) in matrix.iter().enumerate() { - let mut phase_row = Vec::with_capacity(total_variable_count); - phase_row.extend(row.iter().cloned()); - phase_row.extend((0..constraint_count).map(|column| { - if column == row_index { - rational_one() - } else { - rational_zero() - } - })); - phase_matrix.push(phase_row); - } - - let mut basis: Vec = (variable_count..total_variable_count).collect(); - let mut phase_one_objective = vec![rational_zero(); total_variable_count]; - for coefficient in &mut phase_one_objective[variable_count..] { - *coefficient = -rational_one(); - } - - let phase_one = revised_simplex_maximize( - &phase_matrix, - rhs, - &phase_one_objective, - total_variable_count, - &mut basis, - ); - let RevisedSimplexResult::Optimal { - objective_value, .. - } = phase_one - else { - return ExactLinearProgramResult::Failed; - }; - if objective_value.is_negative() { - return ExactLinearProgramResult::Infeasible; - } - if objective_value.is_positive() - || !pivot_artificial_variables_out(&phase_matrix, rhs, variable_count, &mut basis) - { - return ExactLinearProgramResult::Failed; - } - - let mut phase_two_objective = vec![rational_zero(); total_variable_count]; - phase_two_objective[..variable_count].clone_from_slice(objective); - match revised_simplex_maximize( - &phase_matrix, - rhs, - &phase_two_objective, - variable_count, - &mut basis, - ) { - RevisedSimplexResult::Optimal { - solution, - objective_value, - } => ExactLinearProgramResult::Optimal { - solution: solution[..variable_count].to_vec(), - objective_value, - }, - RevisedSimplexResult::Unbounded | RevisedSimplexResult::Failed => { - ExactLinearProgramResult::Failed - } - } -} - -/// Removes zero-valued Phase I artificial variables from the active basis. -fn pivot_artificial_variables_out( - matrix: &[Vec], - rhs: &[BigRational], - original_variable_count: usize, - basis: &mut [usize], -) -> bool { - for basis_position in 0..basis.len() { - if basis[basis_position] < original_variable_count { - continue; - } - - let Some(current_basis_matrix) = basis_matrix(matrix, basis) else { - return false; - }; - let Some(basic_solution) = - solve_rational_system(current_basis_matrix.clone(), rhs.to_vec()) - else { - return false; - }; - if basic_solution[basis_position] != rational_zero() { - return false; - } - - let mut replacement = None; - for candidate in 0..original_variable_count { - if basis.contains(&candidate) { - continue; - } - let Some(direction) = solve_rational_system( - current_basis_matrix.clone(), - matrix_column(matrix, candidate), - ) else { - continue; - }; - if direction[basis_position] != rational_zero() { - replacement = Some(candidate); - break; - } - } - let Some(candidate) = replacement else { - return false; - }; - basis[basis_position] = candidate; - } - true -} - -/// Uses a floating-point basis search followed by an exact optimality certificate. -/// -/// The floating-point phase is only a guide: it cannot return a validation -/// result directly. A candidate basis is accepted only after exact rational -/// primal and dual checks. If certification fails, the deterministic exact -/// simplex path restarts from the original basis. -fn revised_simplex_maximize( - matrix: &[Vec], - rhs: &[BigRational], - objective: &[BigRational], - entering_variable_count: usize, - basis: &mut [usize], -) -> RevisedSimplexResult { - let starting_basis = basis.to_vec(); - if navigate_simplex_basis_f64(matrix, rhs, objective, entering_variable_count, basis).is_some() - && let Some(optimal) = - certify_optimal_basis(matrix, rhs, objective, entering_variable_count, basis) - { - return optimal; - } - - basis.clone_from_slice(&starting_basis); - revised_simplex_maximize_exact(matrix, rhs, objective, entering_variable_count, basis) -} - -/// Runs deterministic exact revised simplex from an existing feasible basis. -fn revised_simplex_maximize_exact( - matrix: &[Vec], - rhs: &[BigRational], - objective: &[BigRational], - entering_variable_count: usize, - basis: &mut [usize], -) -> RevisedSimplexResult { - const MAX_SIMPLEX_ITERATIONS: usize = 10_000; - - if matrix.is_empty() - || matrix.len() != rhs.len() - || basis.len() != rhs.len() - || matrix.iter().any(|row| row.len() != objective.len()) - || entering_variable_count > objective.len() - { - return RevisedSimplexResult::Failed; - } - - for _ in 0..MAX_SIMPLEX_ITERATIONS { - let Some(current_basis_matrix) = basis_matrix(matrix, basis) else { - return RevisedSimplexResult::Failed; - }; - let Some(basic_solution) = - solve_rational_system(current_basis_matrix.clone(), rhs.to_vec()) - else { - return RevisedSimplexResult::Failed; - }; - if basic_solution.iter().any(Signed::is_negative) { - return RevisedSimplexResult::Failed; - } - - let basic_objective: Vec<_> = basis - .iter() - .map(|&index| objective[index].clone()) - .collect(); - let Some(dual_solution) = solve_rational_system( - transpose_square_matrix(¤t_basis_matrix), - basic_objective, - ) else { - return RevisedSimplexResult::Failed; - }; - - let entering = (0..entering_variable_count) - .filter(|candidate| !basis.contains(candidate)) - .find(|&candidate| { - let reduced_cost = objective[candidate].clone() - - dot_product(&dual_solution, &matrix_column(matrix, candidate)); - reduced_cost.is_positive() - }); - let Some(entering) = entering else { - let mut solution = vec![rational_zero(); objective.len()]; - for (&variable, value) in basis.iter().zip(&basic_solution) { - solution[variable] = value.clone(); - } - return RevisedSimplexResult::Optimal { - objective_value: dot_product(objective, &solution), - solution, - }; - }; - - let Some(direction) = - solve_rational_system(current_basis_matrix, matrix_column(matrix, entering)) - else { - return RevisedSimplexResult::Failed; - }; - let leaving = direction - .iter() - .enumerate() - .filter(|(_, coefficient)| coefficient.is_positive()) - .map(|(position, coefficient)| { - ( - position, - basic_solution[position].clone() / coefficient.clone(), - ) - }) - .min_by( - |(left_position, left_ratio), (right_position, right_ratio)| { - left_ratio - .cmp(right_ratio) - .then_with(|| basis[*left_position].cmp(&basis[*right_position])) - }, - ); - let Some((leaving_position, _)) = leaving else { - return RevisedSimplexResult::Unbounded; - }; - basis[leaving_position] = entering; - } - - RevisedSimplexResult::Failed -} - -/// Searches for an optimal basis quickly using f64 arithmetic. -/// -/// Every decision made here is provisional. The caller must pass the resulting -/// basis through [`certify_optimal_basis`] before using it. -fn navigate_simplex_basis_f64( - matrix: &[Vec], - rhs: &[BigRational], - objective: &[BigRational], - entering_variable_count: usize, - basis: &mut [usize], -) -> Option> { - let float_matrix = rational_matrix_to_f64(matrix)?; - let float_rhs = rationals_to_f64(rhs)?; - let float_objective = rationals_to_f64(objective)?; - navigate_simplex_basis_f64_values( - &float_matrix, - &float_rhs, - &float_objective, - entering_variable_count, - basis, - ) -} - -/// Runs provisional revised simplex directly on finite floating-point values. -fn navigate_simplex_basis_f64_values( - matrix: &[Vec], - rhs: &[f64], - objective: &[f64], - entering_variable_count: usize, - basis: &mut [usize], -) -> Option> { - const MAX_SIMPLEX_ITERATIONS: usize = 10_000; - const REDUCED_COST_TOLERANCE: f64 = 1.0e-10; - const DIRECTION_TOLERANCE: f64 = 1.0e-12; - const FEASIBILITY_TOLERANCE: f64 = 1.0e-9; - - if matrix.is_empty() - || matrix.len() != rhs.len() - || basis.len() != rhs.len() - || matrix.iter().any(|row| row.len() != objective.len()) - || entering_variable_count > objective.len() - { - return None; - } - - for _ in 0..MAX_SIMPLEX_ITERATIONS { - let current_basis_matrix = basis_matrix_f64(matrix, basis)?; - let mut basic_solution = solve_f64_system(current_basis_matrix.clone(), rhs.to_vec())?; - if basic_solution - .iter() - .any(|&value| value < -FEASIBILITY_TOLERANCE) - { - return None; - } - for value in &mut basic_solution { - if value.is_sign_negative() { - *value = 0.0; - } - } - - let basic_objective: Vec<_> = basis.iter().map(|&index| objective[index]).collect(); - let dual_solution = solve_f64_system( - transpose_square_matrix_f64(¤t_basis_matrix), - basic_objective, - )?; - - let entering = (0..entering_variable_count) - .filter(|candidate| !basis.contains(candidate)) - .find(|&candidate| { - let column = matrix_column_f64(matrix, candidate); - let dual_contribution = dot_product_f64(&dual_solution, &column); - let scale = 1.0 + objective[candidate].abs() + dual_contribution.abs(); - objective[candidate] - dual_contribution > REDUCED_COST_TOLERANCE * scale - }); - let Some(entering) = entering else { - return Some(dual_solution); - }; - - let direction = - solve_f64_system(current_basis_matrix, matrix_column_f64(matrix, entering))?; - let leaving = direction - .iter() - .enumerate() - .filter(|(_, coefficient)| **coefficient > DIRECTION_TOLERANCE) - .map(|(position, coefficient)| (position, basic_solution[position] / coefficient)) - .min_by( - |(left_position, left_ratio), (right_position, right_ratio)| { - left_ratio - .total_cmp(right_ratio) - .then_with(|| basis[*left_position].cmp(&basis[*right_position])) - }, - ); - let (leaving_position, _) = leaving?; - basis[leaving_position] = entering; - } - - None -} - -/// Finds candidate separating directions from floating-point Phase I and II duals. -/// -/// Returned directions are only guides. Callers must verify their geometric -/// certificates with exact arithmetic before accepting them. -struct ProvisionalPhaseDuals { - /// Candidate direction from the infeasibility phase. - infeasibility_axis: Option>, - /// Candidate direction from the shared-face-confinement phase. - confinement_axis: Option>, - /// Provisional Phase II basis available for exact dual certification. - confinement_basis: Option>, -} - -impl ProvisionalPhaseDuals { - /// Returns a result with no provisional certificates. - const fn none() -> Self { - Self { - infeasibility_axis: None, - confinement_axis: None, - confinement_basis: None, - } - } -} - -fn provisional_phase_dual_axes( - matrix: &[Vec], - rhs: &[f64], - original_objective: &[f64], - variable_count: usize, -) -> ProvisionalPhaseDuals { - let constraint_count = rhs.len(); - let Some(total_variable_count) = variable_count.checked_add(constraint_count) else { - return ProvisionalPhaseDuals::none(); - }; - let mut phase_matrix = Vec::with_capacity(constraint_count); - for (row_index, row) in matrix.iter().enumerate() { - let mut phase_row = Vec::with_capacity(total_variable_count); - phase_row.extend(row.iter().copied()); - phase_row.extend((0..constraint_count).map( - |column| { - if column == row_index { 1.0 } else { 0.0 } - }, - )); - phase_matrix.push(phase_row); - } - - let mut basis: Vec<_> = (variable_count..total_variable_count).collect(); - let mut objective = vec![0.0; total_variable_count]; - for coefficient in &mut objective[variable_count..] { - *coefficient = -1.0; - } - let Some(phase_one_dual) = navigate_simplex_basis_f64_values( - &phase_matrix, - rhs, - &objective, - total_variable_count, - &mut basis, - ) else { - return ProvisionalPhaseDuals::none(); - }; - let phase_one_axis = (phase_one_dual.len() >= 2).then(|| phase_one_dual[2..].to_vec()); - - if !pivot_artificial_variables_out_f64(&phase_matrix, rhs, variable_count, &mut basis) { - return ProvisionalPhaseDuals { - infeasibility_axis: phase_one_axis, - confinement_axis: None, - confinement_basis: None, - }; - } - let mut phase_two_objective = vec![0.0; total_variable_count]; - if original_objective.len() != variable_count { - return ProvisionalPhaseDuals { - infeasibility_axis: phase_one_axis, - confinement_axis: None, - confinement_basis: None, - }; - } - phase_two_objective[..variable_count].clone_from_slice(original_objective); - let phase_two_dual = navigate_simplex_basis_f64_values( - &phase_matrix, - rhs, - &phase_two_objective, - variable_count, - &mut basis, - ); - let phase_two_axis = phase_two_dual - .as_ref() - .and_then(|dual| (dual.len() >= 2).then(|| dual[2..].to_vec())); - let phase_two_basis = phase_two_dual.map(|_dual| basis); - ProvisionalPhaseDuals { - infeasibility_axis: phase_one_axis, - confinement_axis: phase_two_axis, - confinement_basis: phase_two_basis, - } -} - -/// Removes zero artificial variables from a provisional floating-point basis. -fn pivot_artificial_variables_out_f64( - matrix: &[Vec], - rhs: &[f64], - original_variable_count: usize, - basis: &mut [usize], -) -> bool { - const ZERO_TOLERANCE: f64 = 1.0e-9; - const PIVOT_TOLERANCE: f64 = 1.0e-12; - - for basis_position in 0..basis.len() { - if basis[basis_position] < original_variable_count { - continue; - } - let Some(current_basis_matrix) = basis_matrix_f64(matrix, basis) else { - return false; - }; - let Some(basic_solution) = solve_f64_system(current_basis_matrix.clone(), rhs.to_vec()) - else { - return false; - }; - if basic_solution[basis_position].abs() > ZERO_TOLERANCE { - return false; - } - - let replacement = (0..original_variable_count) - .filter(|candidate| !basis.contains(candidate)) - .find(|&candidate| { - solve_f64_system( - current_basis_matrix.clone(), - matrix_column_f64(matrix, candidate), - ) - .is_some_and(|direction| direction[basis_position].abs() > PIVOT_TOLERANCE) - }); - let Some(replacement) = replacement else { - return false; - }; - basis[basis_position] = replacement; - } - true -} - -/// Certifies strict projection separation along a finite candidate direction. -fn simplices_are_strictly_separated_along( - first: &LabeledSimplexRealization, - second: &LabeledSimplexRealization, - axis: &[f64], -) -> bool { - if axis.len() != D || axis.iter().any(|value| !value.is_finite()) { - return false; - } - let exact_axis: Vec<_> = axis.iter().copied().map(rational_from_f64).collect(); - let Some((first_min, first_max)) = exact_projection_bounds(first, &exact_axis) else { - return false; - }; - let Some((second_min, second_max)) = exact_projection_bounds(second, &exact_axis) else { - return false; - }; - first_max < second_min || second_max < first_min -} - -/// Returns exact projection bounds for one realized simplex. -fn exact_projection_bounds( - simplex: &LabeledSimplexRealization, - axis: &[BigRational], -) -> Option<(BigRational, BigRational)> { - let mut projections = simplex.coordinates().iter().map(|coordinates| { - coordinates - .iter() - .zip(axis) - .fold(rational_zero(), |sum, (coordinate, coefficient)| { - sum + rational_from_f64(*coordinate) * coefficient - }) - }); - let first = projections.next()?; - Some( - projections.fold((first.clone(), first), |(minimum, maximum), projection| { - (minimum.min(projection.clone()), maximum.max(projection)) - }), - ) -} - -/// Certifies confinement through one shared vertex with a floating-point error bound. -/// -/// Scaling the provisional direction is exact because the factor is a power of -/// two. Each signed projection difference is then accepted only when its -/// conservative lower bound exceeds the unit dual margin. Ambiguous, -/// underflow-sensitive, or overflowing cases continue to exact arithmetic. -fn filtered_single_shared_vertex_confinement( - first: &LabeledSimplexRealization, - second: &LabeledSimplexRealization, - shared_labels: &[L], - axis: &[f64], -) -> bool -where - L: Eq, -{ - let [shared_label] = shared_labels else { - return false; - }; - let Some(first_shared_index) = first - .labels() - .iter() - .position(|candidate| candidate == shared_label) - else { - return false; - }; - let Some(second_shared_index) = second - .labels() - .iter() - .position(|candidate| candidate == shared_label) - else { - return false; - }; - let first_shared = &first.coordinates()[first_shared_index]; - let second_shared = &second.coordinates()[second_shared_index]; - if !coordinates_are_identical(first_shared, second_shared) { - return false; - } - - let scaled_axis: Vec<_> = axis.iter().map(|value| *value * 1024.0).collect(); - if scaled_axis.len() != D || scaled_axis.iter().any(|value| !value.is_finite()) { - return false; - } - first - .labels() - .iter() - .zip(first.coordinates()) - .filter(|(label, _)| *label != shared_label) - .all(|(_, coordinates)| { - certified_dot_difference_lower_bound(&scaled_axis, coordinates, first_shared) - .is_some_and(|lower_bound| lower_bound > 1.0) - }) - && second - .labels() - .iter() - .zip(second.coordinates()) - .filter(|(label, _)| *label != shared_label) - .all(|(_, coordinates)| { - certified_dot_difference_lower_bound(&scaled_axis, second_shared, coordinates) - .is_some_and(|lower_bound| lower_bound > 1.0) - }) -} - -/// Returns a conservative lower bound for `axis · (left - right)`. -fn certified_dot_difference_lower_bound( - axis: &[f64], - left: &[f64; D], - right: &[f64; D], -) -> Option { - let mut estimate = 0.0; - let mut magnitude = 0.0; - for ((coefficient, left_value), right_value) in axis.iter().zip(left).zip(right) { - let left_product = coefficient * left_value; - let right_product = coefficient * right_value; - if !left_product.is_finite() - || !right_product.is_finite() - || product_underflowed(*coefficient, *left_value, left_product) - || product_underflowed(*coefficient, *right_value, right_product) - { - return None; - } - estimate = coefficient.mul_add(*left_value, estimate); - estimate = (-coefficient).mul_add(*right_value, estimate); - magnitude += left_product.abs() + right_product.abs(); - if !estimate.is_finite() || !magnitude.is_finite() { - return None; - } - } - - let operation_count = f64::from(u32::try_from(2 * D).ok()?); - let denominator = 1.0 - operation_count * f64::EPSILON; - if denominator <= 0.0 { - return None; - } - let gamma = operation_count * f64::EPSILON / denominator; - let magnitude_upper_bound = magnitude / denominator.powi(2); - let error_bound = 4.0 * gamma * magnitude_upper_bound; - error_bound.is_finite().then_some(estimate - error_bound) -} - -/// Detects a product whose exact non-zero value rounded into the subnormal range. -fn product_underflowed(left: f64, right: f64, product: f64) -> bool { - left != 0.0 && right != 0.0 && (product == 0.0 || product.abs() < f64::MIN_POSITIVE) -} - -/// Compares finite coordinates exactly while treating both signed zeros alike. -fn coordinates_are_identical(left: &[f64; D], right: &[f64; D]) -> bool { - left.iter().zip(right).all(|(left, right)| { - let left_bits = left.to_bits(); - let right_bits = right.to_bits(); - left_bits == right_bits || (left_bits << 1 == 0 && right_bits << 1 == 0) - }) -} - -/// Certifies a Phase II dual after rebuilding its two affine offsets exactly. -/// -/// For direction `w`, the smallest feasible dual offsets are -/// `max_i(c_i - w·a_i)` and `max_j(c_j + w·b_j)`. Their sum is an exact upper -/// bound on the maximum non-shared barycentric weight. A non-positive bound -/// proves that every intersection point lies in the shared face. -fn exact_shared_face_confinement_certificate( - first: &LabeledSimplexRealization, - second: &LabeledSimplexRealization, - shared_labels: &[L], - axis: &[f64], -) -> bool -where - L: Eq, -{ - if axis.len() != D || axis.iter().any(|value| !value.is_finite()) { - return false; - } - let Some(exact_axis) = exact_axis_through_shared_face(first, shared_labels, axis) else { - return false; - }; - let first_offset = first - .labels() - .iter() - .zip(first.coordinates()) - .map(|(label, coordinates)| { - let objective = if shared_labels.contains(label) { - rational_zero() - } else { - rational_one() - }; - objective - exact_projection(coordinates, &exact_axis) - }) - .max(); - let second_offset = second - .labels() - .iter() - .zip(second.coordinates()) - .map(|(label, coordinates)| { - let objective = if shared_labels.contains(label) { - rational_zero() - } else { - rational_one() - }; - objective + exact_projection(coordinates, &exact_axis) - }) - .max(); - first_offset - .zip(second_offset) - .is_some_and(|(first_offset, second_offset)| { - first_offset + second_offset <= rational_zero() - }) -} - -/// Reconstructs a provisional axis in the shared face's exact affine nullspace. -/// -/// Shared vertices define homogeneous constraints on `(axis, offset)`. A -/// floating-point rank search selects pivot columns, then the fraction-free -/// exact solver constructs one nullspace basis vector per free column directly -/// from the original coordinates. Combining those vectors with provisional -/// free-coordinate weights restores exact shared-face equality without a -/// rational Gram solve. -fn exact_axis_through_shared_face( - simplex: &LabeledSimplexRealization, - shared_labels: &[L], - axis: &[f64], -) -> Option> -where - L: Eq, -{ - let shared_coordinates: Vec<_> = shared_labels - .iter() - .map(|label| { - simplex - .labels() - .iter() - .position(|candidate| candidate == label) - .and_then(|index| simplex.coordinates().get(index)) - }) - .collect::>()?; - if shared_coordinates.is_empty() { - return Some( - axis.iter() - .map(|value| rational_from_f64(*value * 1024.0)) - .collect(), - ); - } - - let constraint_matrix: Vec> = shared_coordinates - .iter() - .map(|coordinates| { - let mut row = coordinates.to_vec(); - row.push(1.0); - row - }) - .collect(); - let pivot_columns = independent_columns_f64(&constraint_matrix)?; - let free_columns: Vec<_> = (0..=D) - .filter(|column| !pivot_columns.contains(column)) - .collect(); - let pivot_matrix: Vec> = constraint_matrix - .iter() - .map(|row| pivot_columns.iter().map(|&column| row[column]).collect()) - .collect(); - - let mut provisional_affine = axis.to_vec(); - provisional_affine.push(-dot_product_f64(axis, shared_coordinates[0])); - let mut exact_affine = vec![rational_zero(); D + 1]; - for free_column in free_columns { - let weight = rational_from_f64(provisional_affine[free_column]); - exact_affine[free_column] += weight.clone(); - let rhs: Vec<_> = constraint_matrix - .iter() - .map(|row| -row[free_column]) - .collect(); - let solution = solve_exact_runtime_system(&pivot_matrix, &rhs)?.ok()?; - for (&pivot_column, coefficient) in pivot_columns.iter().zip(solution) { - exact_affine[pivot_column] += weight.clone() * coefficient; - } - } - - let scale = rational_from_f64(1024.0); - for value in &mut exact_affine[..D] { - *value *= scale.clone(); - } - exact_affine.truncate(D); - Some(exact_affine) -} - -/// Selects a full-rank set of matrix columns with provisional elimination. -fn independent_columns_f64(matrix: &[Vec]) -> Option> { - const PIVOT_TOLERANCE: f64 = 1.0e-12; - - let row_count = matrix.len(); - let column_count = matrix.first()?.len(); - if row_count == 0 || matrix.iter().any(|row| row.len() != column_count) { - return None; - } - let mut work = matrix.to_vec(); - let scale = work - .iter() - .flatten() - .fold(0.0_f64, |maximum, value| maximum.max(value.abs())); - let mut pivot_columns = Vec::with_capacity(row_count); - for column in 0..column_count { - let pivot_position = (pivot_columns.len()..row_count).max_by(|&left, &right| { - work[left][column] - .abs() - .total_cmp(&work[right][column].abs()) - })?; - if work[pivot_position][column].abs() <= PIVOT_TOLERANCE * scale.max(1.0) { - continue; - } - let pivot_row = pivot_columns.len(); - work.swap(pivot_row, pivot_position); - let pivot = work[pivot_row][column]; - let pivot_values = work[pivot_row].clone(); - for row_values in work.iter_mut().skip(pivot_row + 1) { - let factor = row_values[column] / pivot; - for (value, pivot_value) in row_values[column..].iter_mut().zip(&pivot_values[column..]) - { - *value -= factor * pivot_value; - } + basis_orientation.is_none(), + ) { + IntersectionLinearProgramResult::Valid => Ok(()), + IntersectionLinearProgramResult::Invalid(witness) => { + Err(SimplexIntersectionFailure::IntersectionOutsideSharedFace { witness }) } - pivot_columns.push(column); - if pivot_columns.len() == row_count { - return Some(pivot_columns); + IntersectionLinearProgramResult::SingularBarycentricBasis => { + Err(SimplexIntersectionFailure::SingularBarycentricBasis) } } - None } -/// Certifies the floating-point Phase II basis with one exact dual solve. -/// -/// Dual feasibility and a non-positive exact objective bound prove confinement -/// without constructing a primal solution or repeating exact simplex pivots. -fn phase_two_dual_proves_shared_face_confinement( - matrix: &[Vec], - rhs: &[BigRational], - objective: &[BigRational], - basis: &[usize], -) -> bool { - let constraint_count = rhs.len(); - let variable_count = objective.len(); - if matrix.len() != constraint_count - || matrix.iter().any(|row| row.len() != variable_count) - || basis.len() != constraint_count - { - return false; - } - - let total_variable_count = variable_count + constraint_count; - let phase_matrix: Vec<_> = matrix - .iter() - .enumerate() - .map(|(row_index, row)| { - let mut phase_row = Vec::with_capacity(total_variable_count); - phase_row.extend(row.iter().cloned()); - phase_row.extend((0..constraint_count).map(|column| { - if column == row_index { - rational_one() - } else { - rational_zero() - } - })); - phase_row - }) - .collect(); - let mut phase_objective = vec![rational_zero(); total_variable_count]; - phase_objective[..variable_count].clone_from_slice(objective); - - let Some(current_basis_matrix) = basis_matrix(&phase_matrix, basis) else { - return false; - }; - let basic_objective: Vec<_> = basis - .iter() - .filter_map(|&index| phase_objective.get(index).cloned()) +/// Computes one simplex orientation through the filtered-exact predicate path. +fn realization_orientation( + simplex: &LabeledSimplexRealization, +) -> Option { + let points: SimplexRealizationBuffer<_> = (0..simplex.coordinates().len()) + .filter_map(|index| simplex.point_at(index)) .collect(); - if basic_objective.len() != constraint_count { - return false; - } - let Some(dual_solution) = solve_rational_system( - transpose_square_matrix(¤t_basis_matrix), - basic_objective, - ) else { - return false; - }; - let dual_is_feasible = (0..variable_count).all(|candidate| { - objective[candidate].clone() - - dot_product(&dual_solution, &matrix_column(matrix, candidate)) - <= rational_zero() - }); - dual_is_feasible && dot_product(rhs, &dual_solution) <= rational_zero() -} - -/// Computes one exact projection along a finite candidate axis. -fn exact_projection(coordinates: &[f64; D], axis: &[BigRational]) -> BigRational { - coordinates - .iter() - .zip(axis) - .fold(rational_zero(), |sum, (coordinate, coefficient)| { - sum + rational_from_f64(*coordinate) * coefficient - }) + robust_orientation(&points).ok() } -/// Certifies exact primal feasibility and dual optimality for one candidate basis. -fn certify_optimal_basis( - matrix: &[Vec], - rhs: &[BigRational], - objective: &[BigRational], - entering_variable_count: usize, - basis: &[usize], -) -> Option { - let current_basis_matrix = basis_matrix(matrix, basis)?; - let basic_solution = solve_rational_system(current_basis_matrix.clone(), rhs.to_vec())?; - if basic_solution.iter().any(Signed::is_negative) { - return None; - } - - let basic_objective: Vec<_> = basis - .iter() - .map(|&index| objective[index].clone()) +/// Proves disjointness when one simplex lies strictly beyond a facet of another. +/// +/// Replacing facet-opposite basis vertex `i` with a point `p` scales the basis +/// orientation by the barycentric coordinate of `p` at `i`. If every vertex of +/// the other simplex has the opposite non-zero sign for one `i`, convexity +/// places that entire simplex in the open exterior half-space. +fn simplex_is_strictly_outside_a_facet( + basis: &LabeledSimplexRealization, + other: &LabeledSimplexRealization, + basis_orientation: Orientation, +) -> bool { + let basis_points: SimplexRealizationBuffer<_> = (0..basis.coordinates().len()) + .filter_map(|index| basis.point_at(index)) .collect(); - let dual_solution = solve_rational_system( - transpose_square_matrix(¤t_basis_matrix), - basic_objective, - )?; - let dual_is_feasible = (0..entering_variable_count) - .filter(|candidate| !basis.contains(candidate)) - .all(|candidate| { - let reduced_cost = objective[candidate].clone() - - dot_product(&dual_solution, &matrix_column(matrix, candidate)); - !reduced_cost.is_positive() - }); - if !dual_is_feasible { - return None; - } - let mut solution = vec![rational_zero(); objective.len()]; - for (&variable, value) in basis.iter().zip(&basic_solution) { - solution[variable] = value.clone(); - } - Some(RevisedSimplexResult::Optimal { - objective_value: dot_product(objective, &solution), - solution, + (0..basis.labels().len()).any(|basis_index| { + (0..other.coordinates().len()).all(|other_index| { + let Some(other_point) = other.point_at(other_index) else { + return false; + }; + let mut replaced_points = basis_points.clone(); + replaced_points[basis_index] = other_point; + robust_orientation(&replaced_points).is_ok_and(|orientation| { + orientation != basis_orientation && orientation != Orientation::DEGENERATE + }) + }) }) } -/// Converts an exact rational matrix to finite f64 values for basis navigation. -fn rational_matrix_to_f64(matrix: &[Vec]) -> Option>> { - matrix.iter().map(|row| rationals_to_f64(row)).collect() -} - -/// Converts exact rationals to finite f64 values for provisional calculations. -fn rationals_to_f64(values: &[BigRational]) -> Option> { - values - .iter() - .map(|value| value.to_f64().filter(|converted| converted.is_finite())) - .collect() -} - -/// Extracts a floating-point basis matrix. -fn basis_matrix_f64(matrix: &[Vec], basis: &[usize]) -> Option>> { - let column_count = matrix.first()?.len(); - if matrix.len() != basis.len() - || matrix.iter().any(|row| row.len() != column_count) - || basis.iter().any(|&column| column >= column_count) - { - return None; - } - Some( - matrix - .iter() - .map(|row| basis.iter().map(|&column| row[column]).collect()) - .collect(), - ) -} - -/// Copies one floating-point matrix column. -fn matrix_column_f64(matrix: &[Vec], column: usize) -> Vec { - matrix.iter().map(|row| row[column]).collect() -} - -/// Transposes a square floating-point matrix. -fn transpose_square_matrix_f64(matrix: &[Vec]) -> Vec> { - let dimension = matrix.len(); - (0..dimension) - .map(|row| (0..dimension).map(|column| matrix[column][row]).collect()) - .collect() -} - -/// Computes a floating-point dot product used only for provisional pivot selection. -fn dot_product_f64(left: &[f64], right: &[f64]) -> f64 { - left.iter() - .zip(right) - .map(|(left, right)| left * right) - .sum() -} - -/// Solves a small floating-point system with scaled partial pivoting. -#[expect( - clippy::needless_range_loop, - reason = "index-based elimination keeps pivot row/column operations explicit" -)] -fn solve_f64_system(mut matrix: Vec>, mut rhs: Vec) -> Option> { - const PIVOT_TOLERANCE: f64 = 1.0e-14; - - let dimension = rhs.len(); - if matrix.len() != dimension || matrix.iter().any(|row| row.len() != dimension) { - return None; - } - - for pivot_column in 0..dimension { - let pivot_row = (pivot_column..dimension).max_by(|&left, &right| { - matrix[left][pivot_column] - .abs() - .total_cmp(&matrix[right][pivot_column].abs()) - })?; - let row_scale = matrix[pivot_row] - .iter() - .map(|&value| value.abs()) - .fold(0.0_f64, f64::max); - if !row_scale.is_finite() - || matrix[pivot_row][pivot_column].abs() <= PIVOT_TOLERANCE * row_scale.max(1.0) - { - return None; - } - if pivot_row != pivot_column { - matrix.swap(pivot_column, pivot_row); - rhs.swap(pivot_column, pivot_row); - } - - for row in pivot_column + 1..dimension { - let factor = matrix[row][pivot_column] / matrix[pivot_column][pivot_column]; - matrix[row][pivot_column] = 0.0; - for column in pivot_column + 1..dimension { - matrix[row][column] = - factor.mul_add(-matrix[pivot_column][column], matrix[row][column]); - } - rhs[row] = factor.mul_add(-rhs[pivot_column], rhs[row]); - } - } +/// Uses filtered-exact orientation signs to prove confinement to the shared face. +/// +/// Replacing basis vertex `i` with a point `p` changes the orientation +/// determinant by the barycentric coordinate of `p` at `i`. Therefore, when +/// every vertex of the other simplex has a zero or opposite determinant sign +/// for each non-shared basis vertex, convexity proves that any point in both +/// simplices has zero weight outside the shared face. +fn intersection_is_confined_by_orientation( + basis: &LabeledSimplexRealization, + other: &LabeledSimplexRealization, + shared_labels: &[L], + basis_orientation: Orientation, +) -> bool +where + L: Eq, +{ + let basis_points: SimplexRealizationBuffer<_> = (0..basis.coordinates().len()) + .filter_map(|index| basis.point_at(index)) + .collect(); - let mut solution = vec![0.0; dimension]; - for row in (0..dimension).rev() { - let mut sum = rhs[row]; - for column in row + 1..dimension { - sum = matrix[row][column].mul_add(-solution[column], sum); - } - solution[row] = sum / matrix[row][row]; - if !solution[row].is_finite() { - return None; + for (basis_index, basis_label) in basis.labels().iter().enumerate() { + if shared_labels.contains(basis_label) { + continue; } - } - Some(solution) -} - -/// Extracts the square matrix formed by the current basis columns. -fn basis_matrix(matrix: &[Vec], basis: &[usize]) -> Option>> { - let column_count = matrix.first()?.len(); - if matrix.len() != basis.len() - || matrix.iter().any(|row| row.len() != column_count) - || basis.iter().any(|&column| column >= column_count) - { - return None; - } - Some( - matrix - .iter() - .map(|row| basis.iter().map(|&column| row[column].clone()).collect()) - .collect(), - ) -} - -/// Copies one matrix column for an exact linear solve. -fn matrix_column(matrix: &[Vec], column: usize) -> Vec { - matrix.iter().map(|row| row[column].clone()).collect() -} - -/// Transposes a square exact-rational matrix. -fn transpose_square_matrix(matrix: &[Vec]) -> Vec> { - let dimension = matrix.len(); - (0..dimension) - .map(|row| { - (0..dimension) - .map(|column| matrix[column][row].clone()) - .collect() - }) - .collect() -} - -/// Computes an exact dot product without introducing floating-point decisions. -fn dot_product(left: &[BigRational], right: &[BigRational]) -> BigRational { - left.iter() - .zip(right) - .fold(rational_zero(), |sum, (left, right)| { - sum + left.clone() * right.clone() - }) -} - -/// Enumerates candidate vertices of the intersection polytope in second-simplex weights. -fn intersection_polytope_vertices( - second_vertices_in_first: &[Vec], -) -> Vec> { - let variable_count = second_vertices_in_first.len(); - let active_count = variable_count.saturating_sub(1); - let constraint_count = variable_count * 2; - let mut active_set = Vec::with_capacity(active_count); - let mut vertices = Vec::new(); - - enumerate_active_sets( - constraint_count, - active_count, - 0, - &mut active_set, - &mut |active_constraints| { - if let Some(beta) = - intersection_vertex_for_active_set(second_vertices_in_first, active_constraints) - && beta_is_feasible(&beta, second_vertices_in_first) - { - vertices.push(beta); + for other_index in 0..other.coordinates().len() { + let Some(other_point) = other.point_at(other_index) else { + return false; + }; + let mut replaced_points = basis_points.clone(); + replaced_points[basis_index] = other_point; + let Ok(replaced_orientation) = robust_orientation(&replaced_points) else { + return false; + }; + if replaced_orientation == basis_orientation { + return false; } - }, - ); - - vertices -} - -/// Recursively enumerates active constraint sets for the simplex-intersection LP. -fn enumerate_active_sets( - constraint_count: usize, - active_count: usize, - start: usize, - active_set: &mut Vec, - on_active_set: &mut F, -) where - F: FnMut(&[usize]), -{ - if active_set.len() == active_count { - on_active_set(active_set); - return; - } - - let remaining = active_count - active_set.len(); - let last_start = constraint_count.saturating_sub(remaining); - for constraint in start..=last_start { - active_set.push(constraint); - enumerate_active_sets( - constraint_count, - active_count, - constraint + 1, - active_set, - on_active_set, - ); - active_set.pop(); - } -} - -/// Solves one active-constraint system and returns the candidate beta weights. -fn intersection_vertex_for_active_set( - second_vertices_in_first: &[Vec], - active_constraints: &[usize], -) -> Option> { - let variable_count = second_vertices_in_first.len(); - let mut matrix = Vec::with_capacity(variable_count); - let mut rhs = Vec::with_capacity(variable_count); - - matrix.push(vec![rational_one(); variable_count]); - rhs.push(rational_one()); - - for &constraint in active_constraints { - matrix.push(constraint_coefficients( - second_vertices_in_first, - constraint, - )); - rhs.push(rational_zero()); - } - - solve_rational_system(matrix, rhs) -} - -/// Builds coefficients for either a beta non-negativity or alpha non-negativity constraint. -fn constraint_coefficients( - second_vertices_in_first: &[Vec], - constraint: usize, -) -> Vec { - let variable_count = second_vertices_in_first.len(); - if constraint < variable_count { - let mut coefficients = vec![rational_zero(); variable_count]; - coefficients[constraint] = rational_one(); - return coefficients; - } - - let alpha_index = constraint - variable_count; - second_vertices_in_first - .iter() - .map(|barycentric| barycentric[alpha_index].clone()) - .collect() -} - -/// Checks whether beta weights and the induced alpha weights are all non-negative. -fn beta_is_feasible(beta: &[BigRational], second_vertices_in_first: &[Vec]) -> bool { - beta.iter().all(|value| !value.is_negative()) - && alpha_from_beta(beta, second_vertices_in_first) - .iter() - .all(|value| !value.is_negative()) -} - -/// Converts second-simplex beta weights into first-simplex alpha weights. -fn alpha_from_beta( - beta: &[BigRational], - second_vertices_in_first: &[Vec], -) -> Vec { - let variable_count = second_vertices_in_first.len(); - let mut alpha = vec![rational_zero(); variable_count]; - - for (beta_index, beta_value) in beta.iter().enumerate() { - for (alpha_index, alpha_value) in alpha.iter_mut().enumerate() { - *alpha_value = alpha_value.clone() - + beta_value.clone() * second_vertices_in_first[beta_index][alpha_index].clone(); } } - - alpha + true } -/// Returns labels whose barycentric coordinates witness mass outside the shared face. -fn positive_nonshared_labels( - barycentric: &[BigRational], - labels: &[L], - shared_labels: &[L], +/// Collects labels common to two simplex realizations so witnesses can distinguish shared faces. +fn shared_labels( + first: &LabeledSimplexRealization, + second: &LabeledSimplexRealization, ) -> SimplexRealizationBuffer where L: Clone + Eq, { - labels - .iter() - .zip(barycentric) - .filter(|(label, coordinate)| !shared_labels.contains(label) && coordinate.is_positive()) - .map(|(label, _coordinate)| label.clone()) - .collect() -} - -#[expect( - clippy::needless_range_loop, - reason = "index-based elimination keeps pivot row/column operations explicit" -)] -/// Solves a square rational linear system by Gaussian elimination. -fn solve_rational_system( - mut matrix: Vec>, - mut rhs: Vec, -) -> Option> { - let dimension = rhs.len(); - if matrix.len() != dimension || matrix.iter().any(|row| row.len() != dimension) { - return None; - } - match try_solve_exact_f64_system(&matrix, &rhs) { - ExactF64Solve::Solution(solution) => return Some(solution), - ExactF64Solve::Singular => return None, - ExactF64Solve::Unsupported => {} - } - - for pivot_col in 0..dimension { - let pivot_row = - (pivot_col..dimension).find(|&row| matrix[row][pivot_col] != rational_zero())?; - if pivot_row != pivot_col { - matrix.swap(pivot_col, pivot_row); - rhs.swap(pivot_col, pivot_row); - } - - let pivot_value = matrix[pivot_col][pivot_col].clone(); - for row in pivot_col + 1..dimension { - if matrix[row][pivot_col] == rational_zero() { - continue; - } - let factor = matrix[row][pivot_col].clone() / pivot_value.clone(); - matrix[row][pivot_col] = rational_zero(); - for col in pivot_col + 1..dimension { - matrix[row][col] = - matrix[row][col].clone() - factor.clone() * matrix[pivot_col][col].clone(); - } - rhs[row] = rhs[row].clone() - factor * rhs[pivot_col].clone(); - } - } - - let mut solution = vec![rational_zero(); dimension]; - for row in (0..dimension).rev() { - let mut sum = rhs[row].clone(); - for col in row + 1..dimension { - sum -= matrix[row][col].clone() * solution[col].clone(); - } - solution[row] = sum / matrix[row][row].clone(); - } - - Some(solution) -} - -/// Outcome of attempting the fraction-free exact `f64` solver. -enum ExactF64Solve { - /// The runtime stack dispatcher cannot represent this rational system. - Unsupported, - /// The system is exactly singular. - Singular, - /// The exact system solution. - Solution(Vec), -} - -/// Uses fraction-free Bareiss elimination when every coefficient came directly -/// from a finite `f64`; otherwise leaves the rational solver as the fallback. -fn try_solve_exact_f64_system(matrix: &[Vec], rhs: &[BigRational]) -> ExactF64Solve { - let Some(float_matrix) = exact_rational_matrix_as_f64(matrix) else { - return ExactF64Solve::Unsupported; - }; - let Some(float_rhs) = exact_rationals_as_f64(rhs) else { - return ExactF64Solve::Unsupported; - }; - let Some(result) = solve_exact_runtime_system(&float_matrix, &float_rhs) else { - return ExactF64Solve::Unsupported; - }; - match result { - Ok(solution) => ExactF64Solve::Solution(solution), - Err(StackMatrixDispatchError::La { - source: la_stack::LaError::Singular { .. }, - }) => ExactF64Solve::Singular, - Err(_) => ExactF64Solve::Unsupported, - } -} - -/// Converts rationals to `f64` only when doing so preserves each value exactly. -fn exact_rational_matrix_as_f64(matrix: &[Vec]) -> Option>> { - matrix - .iter() - .map(|row| exact_rationals_as_f64(row)) - .collect() -} - -/// Converts a finite floating-point matrix to its exact rational representation. -fn exact_matrix_from_f64(matrix: &[Vec]) -> Vec> { - matrix - .iter() - .map(|row| row.iter().copied().map(rational_from_f64).collect()) - .collect() -} - -/// Converts rationals to finite `f64` values and verifies an exact round trip. -fn exact_rationals_as_f64(values: &[BigRational]) -> Option> { - values + first + .labels() .iter() - .map(|value| { - let converted = value.to_f64().filter(|candidate| candidate.is_finite())?; - (rational_from_f64(converted) == *value).then_some(converted) - }) + .filter(|label| second.labels().contains(label)) + .cloned() .collect() } -/// Converts a finite f64 to an exact rational value for barycentric predicates. -fn rational_from_f64(value: f64) -> BigRational { - BigRational::from_f64(value) - .expect("validated finite f64 coordinates must convert to BigRational") -} - -/// Returns the additive identity used throughout exact barycentric arithmetic. -fn rational_zero() -> BigRational { - BigRational::from_integer(BigInt::from(0)) -} - -/// Returns the multiplicative identity used throughout exact barycentric arithmetic. -fn rational_one() -> BigRational { - BigRational::from_integer(BigInt::from(1)) -} - #[cfg(test)] mod tests { use super::*; @@ -2619,46 +829,6 @@ mod tests { ); } - #[test] - fn exact_linear_program_classifies_disjoint_triangles_as_valid() { - let first = LabeledSimplexRealization::try_new( - vec![0, 1, 2], - vec![[0.0, 0.0], [1.0, 0.0], [0.0, 1.0]], - ) - .unwrap(); - let second = LabeledSimplexRealization::try_new( - vec![3, 4, 5], - vec![[2.0, 2.0], [3.0, 2.0], [2.0, 3.0]], - ) - .unwrap(); - let shared = shared_labels(&first, &second); - - assert_matches!( - intersection_via_linear_program(&first, &second, &shared), - IntersectionLinearProgramResult::Valid - ); - } - - #[test] - fn exact_linear_program_accepts_a_shared_facet_intersection() { - let first = LabeledSimplexRealization::try_new( - vec![0, 1, 2], - vec![[0.0, 0.0], [1.0, 0.0], [0.0, 1.0]], - ) - .unwrap(); - let second = LabeledSimplexRealization::try_new( - vec![0, 1, 3], - vec![[0.0, 0.0], [1.0, 0.0], [0.0, -1.0]], - ) - .unwrap(); - let shared = shared_labels(&first, &second); - - assert_matches!( - intersection_via_linear_program(&first, &second, &shared), - IntersectionLinearProgramResult::Valid - ); - } - #[test] fn labeled_simplex_realization_rejects_non_finite_coordinates() { let err = LabeledSimplexRealization::try_new( @@ -2777,28 +947,6 @@ mod tests { ); } - #[test] - fn exact_linear_program_reports_crossing_triangle_witnesses() { - let first = LabeledSimplexRealization::try_new( - vec![0, 1, 2], - vec![[0.0, 0.0], [2.0, 0.0], [0.0, 2.0]], - ) - .unwrap(); - let second = LabeledSimplexRealization::try_new( - vec![3, 4, 5], - vec![[2.0, 2.0], [1.0, -1.0], [3.0, 2.0]], - ) - .unwrap(); - let shared = shared_labels(&first, &second); - - assert_matches!( - intersection_via_linear_program(&first, &second, &shared), - IntersectionLinearProgramResult::Invalid(witness) - if witness.first_only_witness.iter().any(|label| [0, 1, 2].contains(label)) - && witness.second_only_witness.iter().any(|label| [3, 4, 5].contains(label)) - ); - } - #[test] fn spanning_periodic_simplex_is_detected() { let simplex = LabeledSimplexRealization::try_new( diff --git a/src/geometry/util/simplex_lp.rs b/src/geometry/util/simplex_lp.rs new file mode 100644 index 00000000..5327b4cb --- /dev/null +++ b/src/geometry/util/simplex_lp.rs @@ -0,0 +1,2131 @@ +//! Exact revised-simplex support for realized-simplex intersection. +//! +//! This private geometry module owns the provisional floating-point basis search, +//! exact Phase I/II certification, artificial-variable handling, dual certificates, +//! and matrix helpers used by Level 4 realization validation. + +#![forbid(unsafe_code)] + +use crate::geometry::matrix::{StackMatrixDispatchError, solve_exact_runtime_system}; +use crate::geometry::realization::{ + LabeledSimplexRealization, SimplexIntersectionWitness, SimplexRealizationBuffer, +}; +use la_stack::{BigInt, BigRational, FromPrimitive, Signed, ToPrimitive}; + +/// Tries cheap candidate separators in the normal space of one shared face. +#[expect( + clippy::too_many_lines, + reason = "the separator search is one bounded projection-and-certification workflow" +)] +pub(in crate::geometry) fn shared_face_fast_confinement( + first: &LabeledSimplexRealization, + second: &LabeledSimplexRealization, + shared_labels: &[L], +) -> bool +where + L: Eq, +{ + if shared_labels.is_empty() || shared_labels.len() >= D { + return false; + } + let shared_coordinates: Vec<_> = shared_labels + .iter() + .map(|label| { + let first_index = first + .labels() + .iter() + .position(|candidate| candidate == label)?; + let second_index = second + .labels() + .iter() + .position(|candidate| candidate == label)?; + coordinates_are_identical( + &first.coordinates()[first_index], + &second.coordinates()[second_index], + ) + .then_some(first.coordinates()[first_index]) + }) + .collect::>() + .unwrap_or_default(); + if shared_coordinates.len() != shared_labels.len() { + return false; + } + let base = shared_coordinates[0]; + let shared_deltas: Vec> = shared_coordinates[1..] + .iter() + .map(|coordinates| { + coordinates + .iter() + .zip(base) + .map(|(coordinate, base_coordinate)| coordinate - base_coordinate) + .collect() + }) + .collect(); + let gram: Vec> = shared_deltas + .iter() + .map(|left| { + shared_deltas + .iter() + .map(|right| dot_product_f64(left, right)) + .collect() + }) + .collect(); + + let mut axis = vec![0.0; D]; + let mut normalized_rays = Vec::with_capacity(2 * D); + for (label, coordinates) in first.labels().iter().zip(first.coordinates()) { + if shared_labels.contains(label) { + continue; + } + let raw_ray: Vec<_> = coordinates + .iter() + .zip(base) + .map(|(coordinate, shared)| coordinate - shared) + .collect(); + let Some(ray) = project_ray_orthogonal_to_shared_face(raw_ray, &shared_deltas, &gram) + else { + return false; + }; + for coordinate in 0..D { + axis[coordinate] += ray[coordinate]; + } + push_normalized_ray(&mut normalized_rays, ray); + } + for (label, coordinates) in second.labels().iter().zip(second.coordinates()) { + if shared_labels.contains(label) { + continue; + } + let raw_ray: Vec<_> = base + .iter() + .zip(coordinates) + .map(|(shared, coordinate)| shared - coordinate) + .collect(); + let Some(ray) = project_ray_orthogonal_to_shared_face(raw_ray, &shared_deltas, &gram) + else { + return false; + }; + for coordinate in 0..D { + axis[coordinate] += ray[coordinate]; + } + push_normalized_ray(&mut normalized_rays, ray); + } + if axis.iter().all(|value| value.is_finite()) + && provisional_axis_certifies_confinement(first, second, shared_labels, &axis) + { + return true; + } + + let mut refined_axis = vec![0.0; D]; + for ray in &normalized_rays { + for (value, component) in refined_axis.iter_mut().zip(ray) { + *value += component; + } + } + for _ in 0..16 { + let mut adjusted = false; + for ray in &normalized_rays { + let projection = dot_product_f64(&refined_axis, ray); + if projection < 1.0 { + let correction = 1.0 - projection; + for (value, component) in refined_axis.iter_mut().zip(ray) { + *value = component.mul_add(correction, *value); + } + adjusted = true; + } + } + if !adjusted { + break; + } + } + refined_axis.iter().all(|value| value.is_finite()) + && provisional_axis_certifies_confinement(first, second, shared_labels, &refined_axis) +} + +/// Removes provisional components tangent to the shared affine face. +fn project_ray_orthogonal_to_shared_face( + mut ray: Vec, + shared_deltas: &[Vec], + gram: &[Vec], +) -> Option> { + if shared_deltas.is_empty() { + return Some(ray); + } + let residuals: Vec<_> = shared_deltas + .iter() + .map(|delta| dot_product_f64(delta, &ray)) + .collect(); + let correction = solve_f64_system(gram.to_vec(), residuals)?; + for (coordinate, value) in ray.iter_mut().enumerate() { + for (coefficient, delta) in correction.iter().zip(shared_deltas) { + *value = (-coefficient).mul_add(delta[coordinate], *value); + } + } + ray.iter().all(|value| value.is_finite()).then_some(ray) +} + +/// Certifies one provisional shared-face separator through a filtered or exact path. +fn provisional_axis_certifies_confinement( + first: &LabeledSimplexRealization, + second: &LabeledSimplexRealization, + shared_labels: &[L], + axis: &[f64], +) -> bool +where + L: Eq, +{ + if shared_labels.len() == 1 { + filtered_single_shared_vertex_confinement(first, second, shared_labels, axis) + } else { + exact_shared_face_confinement_certificate(first, second, shared_labels, axis) + } +} + +/// Appends one finite unit ray for provisional separator refinement. +fn push_normalized_ray(rays: &mut Vec>, ray: Vec) { + let norm = ray + .iter() + .fold(0.0_f64, |value, component| value.hypot(*component)); + if norm.is_finite() && norm > 0.0 { + rays.push(ray.into_iter().map(|component| component / norm).collect()); + } +} + +/// Outcome of the exact linear-program intersection path. +#[derive(Debug)] +pub(in crate::geometry) enum IntersectionLinearProgramResult { + /// The simplices are disjoint or meet only in their shared face. + Valid, + /// A feasible point has positive weight outside the shared face. + Invalid(SimplexIntersectionWitness), + /// The first simplex cannot define a barycentric basis. + SingularBarycentricBasis, +} + +/// Floating-point form used only to choose provisional revised-simplex bases. +struct ProvisionalIntersectionProgram { + matrix: Vec>, + rhs: Vec, + objective: Vec, + simplex_vertex_count: usize, +} + +/// Builds the common-point equality constraints and shared-face objective. +fn provisional_intersection_program( + first: &LabeledSimplexRealization, + second: &LabeledSimplexRealization, + shared_labels: &[L], +) -> Option +where + L: Eq, +{ + let first_labels = first.labels(); + let second_labels = second.labels(); + let simplex_vertex_count = first_labels.len(); + if second_labels.len() != simplex_vertex_count + || first.coordinates().len() != simplex_vertex_count + || second.coordinates().len() != simplex_vertex_count + { + return None; + } + + let variable_count = simplex_vertex_count * 2; + let mut matrix = vec![vec![0.0; variable_count]; D + 2]; + let mut rhs = vec![0.0; D + 2]; + rhs[0] = 1.0; + rhs[1] = 1.0; + + for (alpha_index, coordinates) in first.coordinates().iter().enumerate() { + matrix[0][alpha_index] = 1.0; + for (axis, coordinate) in coordinates.iter().enumerate() { + matrix[axis + 2][alpha_index] = *coordinate; + } + } + for (beta_index, coordinates) in second.coordinates().iter().enumerate() { + matrix[1][simplex_vertex_count + beta_index] = 1.0; + for (axis, coordinate) in coordinates.iter().enumerate() { + matrix[axis + 2][simplex_vertex_count + beta_index] = -*coordinate; + } + } + + let mut objective = vec![0.0; variable_count]; + for (index, label) in first_labels.iter().enumerate() { + if !shared_labels.contains(label) { + objective[index] = 1.0; + } + } + for (index, label) in second_labels.iter().enumerate() { + if !shared_labels.contains(label) { + objective[simplex_vertex_count + index] = 1.0; + } + } + + Some(ProvisionalIntersectionProgram { + matrix, + rhs, + objective, + simplex_vertex_count, + }) +} + +/// Uses exact linear programming to avoid enumerating every intersection-polytope basis. +/// +/// The variables are the non-negative barycentric weights `alpha` and `beta` +/// of the two simplices. Equality constraints require each weight vector to sum +/// to one and both weighted coordinate sums to describe the same point. +/// Maximizing the total weight on non-shared labels is zero exactly when every +/// feasible intersection point lies in the shared face. +pub(in crate::geometry) fn intersection_via_linear_program( + first: &LabeledSimplexRealization, + second: &LabeledSimplexRealization, + shared_labels: &[L], + verify_barycentric_basis: bool, +) -> IntersectionLinearProgramResult +where + L: Clone + Eq, +{ + let precomputed_barycentric = if verify_barycentric_basis { + let Some(coordinates) = barycentric_coordinates_of_vertices(second, first) else { + return IntersectionLinearProgramResult::SingularBarycentricBasis; + }; + Some(coordinates) + } else { + None + }; + let Some(program) = provisional_intersection_program(first, second, shared_labels) else { + return intersection_via_active_sets( + first, + second, + shared_labels, + precomputed_barycentric.as_deref(), + ); + }; + let ProvisionalIntersectionProgram { + matrix: float_matrix, + rhs: float_rhs, + objective: float_objective, + simplex_vertex_count, + } = program; + let variable_count = simplex_vertex_count * 2; + let first_labels = first.labels(); + let second_labels = second.labels(); + + let ProvisionalPhaseDuals { + infeasibility_axis, + confinement_axis, + confinement_basis, + } = provisional_phase_dual_axes(&float_matrix, &float_rhs, &float_objective, variable_count); + if infeasibility_axis + .is_some_and(|axis| simplices_are_strictly_separated_along(first, second, &axis)) + { + return IntersectionLinearProgramResult::Valid; + } + let axis_certifies = confinement_axis.is_some_and(|axis| { + filtered_single_shared_vertex_confinement(first, second, shared_labels, &axis) + || exact_shared_face_confinement_certificate(first, second, shared_labels, &axis) + }); + if axis_certifies { + return IntersectionLinearProgramResult::Valid; + } + + let matrix = exact_matrix_from_f64(&float_matrix); + let rhs: Vec<_> = float_rhs.iter().copied().map(rational_from_f64).collect(); + let objective: Vec<_> = float_objective + .iter() + .copied() + .map(rational_from_f64) + .collect(); + let dual_certifies = confinement_basis.is_some_and(|basis| { + phase_two_dual_proves_shared_face_confinement(&matrix, &rhs, &objective, &basis) + }); + if dual_certifies { + return IntersectionLinearProgramResult::Valid; + } + + match maximize_nonnegative_equality_linear_program(&matrix, &rhs, &objective) { + ExactLinearProgramResult::Infeasible => IntersectionLinearProgramResult::Valid, + ExactLinearProgramResult::Optimal { + objective_value, .. + } if !objective_value.is_positive() => IntersectionLinearProgramResult::Valid, + ExactLinearProgramResult::Optimal { solution, .. } => { + let first_only_witness = positive_nonshared_labels( + &solution[..simplex_vertex_count], + first_labels, + shared_labels, + ); + let second_only_witness = positive_nonshared_labels( + &solution[simplex_vertex_count..], + second_labels, + shared_labels, + ); + if first_only_witness.is_empty() && second_only_witness.is_empty() { + return intersection_via_active_sets( + first, + second, + shared_labels, + precomputed_barycentric.as_deref(), + ); + } + IntersectionLinearProgramResult::Invalid(SimplexIntersectionWitness { + shared: shared_labels.iter().cloned().collect(), + first_only_witness, + second_only_witness, + }) + } + ExactLinearProgramResult::Failed => intersection_via_active_sets( + first, + second, + shared_labels, + precomputed_barycentric.as_deref(), + ), + } +} + +/// Preserves the exact active-set fallback when revised-simplex setup fails. +fn intersection_via_active_sets( + first: &LabeledSimplexRealization, + second: &LabeledSimplexRealization, + shared_labels: &[L], + precomputed_barycentric: Option<&[Vec]>, +) -> IntersectionLinearProgramResult +where + L: Clone + Eq, +{ + let computed_barycentric; + let second_vertices_in_first = if let Some(coordinates) = precomputed_barycentric { + coordinates + } else { + let Some(coordinates) = barycentric_coordinates_of_vertices(second, first) else { + return IntersectionLinearProgramResult::SingularBarycentricBasis; + }; + computed_barycentric = coordinates; + &computed_barycentric + }; + + if intersection_is_confined_to_shared_face( + second_vertices_in_first, + first.labels(), + shared_labels, + ) { + return IntersectionLinearProgramResult::Valid; + } + + for beta in intersection_polytope_vertices(second_vertices_in_first) { + let alpha = alpha_from_beta(&beta, second_vertices_in_first); + let first_only_witness = positive_nonshared_labels(&alpha, first.labels(), shared_labels); + let second_only_witness = positive_nonshared_labels(&beta, second.labels(), shared_labels); + + if !first_only_witness.is_empty() || !second_only_witness.is_empty() { + return IntersectionLinearProgramResult::Invalid(SimplexIntersectionWitness { + shared: shared_labels.iter().cloned().collect(), + first_only_witness, + second_only_witness, + }); + } + } + + IntersectionLinearProgramResult::Valid +} + +/// Accepts a pair when simplex-facet signs prove any intersection lies in the shared face. +fn intersection_is_confined_to_shared_face( + other_vertices_in_basis: &[Vec], + basis_labels: &[L], + shared_labels: &[L], +) -> bool +where + L: Eq, +{ + basis_labels + .iter() + .enumerate() + .filter(|(_, label)| !shared_labels.contains(label)) + .all(|(basis_index, _)| { + other_vertices_in_basis + .iter() + .all(|barycentric| !barycentric[basis_index].is_positive()) + }) +} + +/// Expresses every vertex of one simplex in the barycentric basis of another. +fn barycentric_coordinates_of_vertices( + vertices: &LabeledSimplexRealization, + basis: &LabeledSimplexRealization, +) -> Option>> { + vertices + .coordinates() + .iter() + .map(|coordinates| barycentric_coordinates(coordinates, basis)) + .collect() +} + +/// Computes exact barycentric coordinates of one point in one simplex basis. +fn barycentric_coordinates( + point: &[f64; D], + simplex: &LabeledSimplexRealization, +) -> Option> { + if D == 0 { + return Some(vec![rational_one()]); + } + + let origin = &simplex.coordinates()[0]; + let mut matrix = vec![vec![rational_zero(); D]; D]; + let mut rhs = vec![rational_zero(); D]; + + for axis in 0..D { + let origin_coordinate = rational_from_f64(origin[axis]); + rhs[axis] = rational_from_f64(point[axis]) - origin_coordinate.clone(); + for (column, matrix_value) in matrix[axis].iter_mut().enumerate() { + *matrix_value = rational_from_f64(simplex.coordinates()[column + 1][axis]) + - origin_coordinate.clone(); + } + } + + let lambdas = solve_rational_system(matrix, rhs)?; + let lambda_sum = lambdas + .iter() + .fold(rational_zero(), |sum, value| sum + value.clone()); + let mut barycentric = Vec::with_capacity(D + 1); + barycentric.push(rational_one() - lambda_sum); + barycentric.extend(lambdas); + Some(barycentric) +} + +/// Enumerates candidate vertices of the intersection polytope in second-simplex weights. +fn intersection_polytope_vertices( + second_vertices_in_first: &[Vec], +) -> Vec> { + let variable_count = second_vertices_in_first.len(); + let active_count = variable_count.saturating_sub(1); + let constraint_count = variable_count * 2; + let mut active_set = Vec::with_capacity(active_count); + let mut vertices = Vec::new(); + + enumerate_active_sets( + constraint_count, + active_count, + 0, + &mut active_set, + &mut |active_constraints| { + if let Some(beta) = + intersection_vertex_for_active_set(second_vertices_in_first, active_constraints) + && beta_is_feasible(&beta, second_vertices_in_first) + { + vertices.push(beta); + } + }, + ); + + vertices +} + +/// Recursively enumerates active constraint sets for the simplex-intersection LP. +fn enumerate_active_sets( + constraint_count: usize, + active_count: usize, + start: usize, + active_set: &mut Vec, + on_active_set: &mut F, +) where + F: FnMut(&[usize]), +{ + if active_set.len() == active_count { + on_active_set(active_set); + return; + } + + let remaining = active_count - active_set.len(); + let last_start = constraint_count.saturating_sub(remaining); + for constraint in start..=last_start { + active_set.push(constraint); + enumerate_active_sets( + constraint_count, + active_count, + constraint + 1, + active_set, + on_active_set, + ); + active_set.pop(); + } +} + +/// Solves one active-constraint system and returns the candidate beta weights. +fn intersection_vertex_for_active_set( + second_vertices_in_first: &[Vec], + active_constraints: &[usize], +) -> Option> { + let variable_count = second_vertices_in_first.len(); + let mut matrix = Vec::with_capacity(variable_count); + let mut rhs = Vec::with_capacity(variable_count); + + matrix.push(vec![rational_one(); variable_count]); + rhs.push(rational_one()); + + for &constraint in active_constraints { + matrix.push(constraint_coefficients( + second_vertices_in_first, + constraint, + )); + rhs.push(rational_zero()); + } + + solve_rational_system(matrix, rhs) +} + +/// Builds coefficients for either a beta non-negativity or alpha non-negativity constraint. +fn constraint_coefficients( + second_vertices_in_first: &[Vec], + constraint: usize, +) -> Vec { + let variable_count = second_vertices_in_first.len(); + if constraint < variable_count { + let mut coefficients = vec![rational_zero(); variable_count]; + coefficients[constraint] = rational_one(); + return coefficients; + } + + let alpha_index = constraint - variable_count; + second_vertices_in_first + .iter() + .map(|barycentric| barycentric[alpha_index].clone()) + .collect() +} + +/// Checks whether beta weights and the induced alpha weights are all non-negative. +fn beta_is_feasible(beta: &[BigRational], second_vertices_in_first: &[Vec]) -> bool { + beta.iter().all(|value| !value.is_negative()) + && alpha_from_beta(beta, second_vertices_in_first) + .iter() + .all(|value| !value.is_negative()) +} + +/// Converts second-simplex beta weights into first-simplex alpha weights. +fn alpha_from_beta( + beta: &[BigRational], + second_vertices_in_first: &[Vec], +) -> Vec { + let variable_count = second_vertices_in_first.len(); + let mut alpha = vec![rational_zero(); variable_count]; + + for (beta_index, beta_value) in beta.iter().enumerate() { + for (alpha_index, alpha_value) in alpha.iter_mut().enumerate() { + *alpha_value = alpha_value.clone() + + beta_value.clone() * second_vertices_in_first[beta_index][alpha_index].clone(); + } + } + + alpha +} + +/// Result of an exact non-negative equality-constrained linear program. +#[derive(Debug)] +enum ExactLinearProgramResult { + /// The objective is maximized at an exact basic feasible solution. + Optimal { + /// Values for the original decision variables. + solution: Vec, + /// Exact objective value at `solution`. + objective_value: BigRational, + }, + /// No non-negative solution satisfies the equality constraints. + Infeasible, + /// Tableau construction or a boundedness invariant failed. + Failed, +} + +/// Result of one revised-simplex optimization phase. +enum RevisedSimplexResult { + /// The current basis is optimal. + Optimal { + /// Values for every variable in the supplied matrix. + solution: Vec, + /// Exact objective value at `solution`. + objective_value: BigRational, + }, + /// The objective can increase without bound. + Unbounded, + /// The supplied basis or iteration budget failed. + Failed, +} + +/// Maximizes an exact objective subject to `matrix * x = rhs` and `x >= 0`. +/// +/// Phase I introduces one artificial variable per equality. Bland's pivot +/// ordering makes degenerate exact-arithmetic bases deterministic and avoids +/// cycling. Artificial variables are pivoted out at zero before Phase II. +fn maximize_nonnegative_equality_linear_program( + matrix: &[Vec], + rhs: &[BigRational], + objective: &[BigRational], +) -> ExactLinearProgramResult { + let constraint_count = rhs.len(); + let variable_count = objective.len(); + if constraint_count == 0 + || matrix.len() != constraint_count + || matrix.iter().any(|row| row.len() != variable_count) + || rhs.iter().any(Signed::is_negative) + { + return ExactLinearProgramResult::Failed; + } + + let total_variable_count = variable_count + constraint_count; + let mut phase_matrix = Vec::with_capacity(constraint_count); + for (row_index, row) in matrix.iter().enumerate() { + let mut phase_row = Vec::with_capacity(total_variable_count); + phase_row.extend(row.iter().cloned()); + phase_row.extend((0..constraint_count).map(|column| { + if column == row_index { + rational_one() + } else { + rational_zero() + } + })); + phase_matrix.push(phase_row); + } + + let mut basis: Vec = (variable_count..total_variable_count).collect(); + let mut phase_one_objective = vec![rational_zero(); total_variable_count]; + for coefficient in &mut phase_one_objective[variable_count..] { + *coefficient = -rational_one(); + } + + let phase_one = revised_simplex_maximize( + &phase_matrix, + rhs, + &phase_one_objective, + total_variable_count, + &mut basis, + ); + let RevisedSimplexResult::Optimal { + objective_value, .. + } = phase_one + else { + return ExactLinearProgramResult::Failed; + }; + if objective_value.is_negative() { + return ExactLinearProgramResult::Infeasible; + } + if objective_value.is_positive() + || !pivot_artificial_variables_out(&phase_matrix, rhs, variable_count, &mut basis) + { + return ExactLinearProgramResult::Failed; + } + + let mut phase_two_objective = vec![rational_zero(); total_variable_count]; + phase_two_objective[..variable_count].clone_from_slice(objective); + match revised_simplex_maximize( + &phase_matrix, + rhs, + &phase_two_objective, + variable_count, + &mut basis, + ) { + RevisedSimplexResult::Optimal { + solution, + objective_value, + } => ExactLinearProgramResult::Optimal { + solution: solution[..variable_count].to_vec(), + objective_value, + }, + RevisedSimplexResult::Unbounded | RevisedSimplexResult::Failed => { + ExactLinearProgramResult::Failed + } + } +} + +/// Removes zero-valued Phase I artificial variables from the active basis. +fn pivot_artificial_variables_out( + matrix: &[Vec], + rhs: &[BigRational], + original_variable_count: usize, + basis: &mut [usize], +) -> bool { + for basis_position in 0..basis.len() { + if basis[basis_position] < original_variable_count { + continue; + } + + let Some(current_basis_matrix) = basis_matrix(matrix, basis) else { + return false; + }; + let Some(basic_solution) = + solve_rational_system(current_basis_matrix.clone(), rhs.to_vec()) + else { + return false; + }; + if basic_solution[basis_position] != rational_zero() { + return false; + } + + let mut replacement = None; + for candidate in 0..original_variable_count { + if basis.contains(&candidate) { + continue; + } + let Some(direction) = solve_rational_system( + current_basis_matrix.clone(), + matrix_column(matrix, candidate), + ) else { + continue; + }; + if direction[basis_position] != rational_zero() { + replacement = Some(candidate); + break; + } + } + let Some(candidate) = replacement else { + return false; + }; + basis[basis_position] = candidate; + } + true +} + +/// Uses a floating-point basis search followed by an exact optimality certificate. +/// +/// The floating-point phase is only a guide: it cannot return a validation +/// result directly. A candidate basis is accepted only after exact rational +/// primal and dual checks. If certification fails, the deterministic exact +/// simplex path restarts from the original basis. +fn revised_simplex_maximize( + matrix: &[Vec], + rhs: &[BigRational], + objective: &[BigRational], + entering_variable_count: usize, + basis: &mut [usize], +) -> RevisedSimplexResult { + let starting_basis = basis.to_vec(); + if navigate_simplex_basis_f64(matrix, rhs, objective, entering_variable_count, basis).is_some() + && let Some(optimal) = + certify_optimal_basis(matrix, rhs, objective, entering_variable_count, basis) + { + return optimal; + } + + basis.clone_from_slice(&starting_basis); + revised_simplex_maximize_exact(matrix, rhs, objective, entering_variable_count, basis) +} + +/// Runs deterministic exact revised simplex from an existing feasible basis. +fn revised_simplex_maximize_exact( + matrix: &[Vec], + rhs: &[BigRational], + objective: &[BigRational], + entering_variable_count: usize, + basis: &mut [usize], +) -> RevisedSimplexResult { + const MAX_SIMPLEX_ITERATIONS: usize = 10_000; + + if matrix.is_empty() + || matrix.len() != rhs.len() + || basis.len() != rhs.len() + || matrix.iter().any(|row| row.len() != objective.len()) + || entering_variable_count > objective.len() + { + return RevisedSimplexResult::Failed; + } + + for _ in 0..MAX_SIMPLEX_ITERATIONS { + let Some(current_basis_matrix) = basis_matrix(matrix, basis) else { + return RevisedSimplexResult::Failed; + }; + let Some(basic_solution) = + solve_rational_system(current_basis_matrix.clone(), rhs.to_vec()) + else { + return RevisedSimplexResult::Failed; + }; + if basic_solution.iter().any(Signed::is_negative) { + return RevisedSimplexResult::Failed; + } + + let basic_objective: Vec<_> = basis + .iter() + .map(|&index| objective[index].clone()) + .collect(); + let Some(dual_solution) = solve_rational_system( + transpose_square_matrix(¤t_basis_matrix), + basic_objective, + ) else { + return RevisedSimplexResult::Failed; + }; + + let entering = (0..entering_variable_count) + .filter(|candidate| !basis.contains(candidate)) + .find(|&candidate| { + let reduced_cost = objective[candidate].clone() + - dot_product(&dual_solution, &matrix_column(matrix, candidate)); + reduced_cost.is_positive() + }); + let Some(entering) = entering else { + let mut solution = vec![rational_zero(); objective.len()]; + for (&variable, value) in basis.iter().zip(&basic_solution) { + solution[variable] = value.clone(); + } + return RevisedSimplexResult::Optimal { + objective_value: dot_product(objective, &solution), + solution, + }; + }; + + let Some(direction) = + solve_rational_system(current_basis_matrix, matrix_column(matrix, entering)) + else { + return RevisedSimplexResult::Failed; + }; + let leaving = direction + .iter() + .enumerate() + .filter(|(_, coefficient)| coefficient.is_positive()) + .map(|(position, coefficient)| { + ( + position, + basic_solution[position].clone() / coefficient.clone(), + ) + }) + .min_by( + |(left_position, left_ratio), (right_position, right_ratio)| { + left_ratio + .cmp(right_ratio) + .then_with(|| basis[*left_position].cmp(&basis[*right_position])) + }, + ); + let Some((leaving_position, _)) = leaving else { + return RevisedSimplexResult::Unbounded; + }; + basis[leaving_position] = entering; + } + + RevisedSimplexResult::Failed +} + +/// Searches for an optimal basis quickly using f64 arithmetic. +/// +/// Every decision made here is provisional. The caller must pass the resulting +/// basis through [`certify_optimal_basis`] before using it. +fn navigate_simplex_basis_f64( + matrix: &[Vec], + rhs: &[BigRational], + objective: &[BigRational], + entering_variable_count: usize, + basis: &mut [usize], +) -> Option> { + let float_matrix = rational_matrix_to_f64(matrix)?; + let float_rhs = rationals_to_f64(rhs)?; + let float_objective = rationals_to_f64(objective)?; + navigate_simplex_basis_f64_values( + &float_matrix, + &float_rhs, + &float_objective, + entering_variable_count, + basis, + ) +} + +/// Runs provisional revised simplex directly on finite floating-point values. +fn navigate_simplex_basis_f64_values( + matrix: &[Vec], + rhs: &[f64], + objective: &[f64], + entering_variable_count: usize, + basis: &mut [usize], +) -> Option> { + const MAX_SIMPLEX_ITERATIONS: usize = 10_000; + const REDUCED_COST_TOLERANCE: f64 = 1.0e-10; + const DIRECTION_TOLERANCE: f64 = 1.0e-12; + const FEASIBILITY_TOLERANCE: f64 = 1.0e-9; + + if matrix.is_empty() + || matrix.len() != rhs.len() + || basis.len() != rhs.len() + || matrix.iter().any(|row| row.len() != objective.len()) + || entering_variable_count > objective.len() + { + return None; + } + + for _ in 0..MAX_SIMPLEX_ITERATIONS { + let current_basis_matrix = basis_matrix_f64(matrix, basis)?; + let mut basic_solution = solve_f64_system(current_basis_matrix.clone(), rhs.to_vec())?; + if basic_solution + .iter() + .any(|&value| value < -FEASIBILITY_TOLERANCE) + { + return None; + } + for value in &mut basic_solution { + if value.is_sign_negative() { + *value = 0.0; + } + } + + let basic_objective: Vec<_> = basis.iter().map(|&index| objective[index]).collect(); + let dual_solution = solve_f64_system( + transpose_square_matrix_f64(¤t_basis_matrix), + basic_objective, + )?; + + let entering = (0..entering_variable_count) + .filter(|candidate| !basis.contains(candidate)) + .find(|&candidate| { + let column = matrix_column_f64(matrix, candidate); + let dual_contribution = dot_product_f64(&dual_solution, &column); + let scale = 1.0 + objective[candidate].abs() + dual_contribution.abs(); + objective[candidate] - dual_contribution > REDUCED_COST_TOLERANCE * scale + }); + let Some(entering) = entering else { + return Some(dual_solution); + }; + + let direction = + solve_f64_system(current_basis_matrix, matrix_column_f64(matrix, entering))?; + let leaving = direction + .iter() + .enumerate() + .filter(|(_, coefficient)| **coefficient > DIRECTION_TOLERANCE) + .map(|(position, coefficient)| (position, basic_solution[position] / coefficient)) + .min_by( + |(left_position, left_ratio), (right_position, right_ratio)| { + left_ratio + .total_cmp(right_ratio) + .then_with(|| basis[*left_position].cmp(&basis[*right_position])) + }, + ); + let (leaving_position, _) = leaving?; + basis[leaving_position] = entering; + } + + None +} + +/// Finds candidate separating directions from floating-point Phase I and II duals. +/// +/// Returned directions are only guides. Callers must verify their geometric +/// certificates with exact arithmetic before accepting them. +struct ProvisionalPhaseDuals { + /// Candidate direction from the infeasibility phase. + infeasibility_axis: Option>, + /// Candidate direction from the shared-face-confinement phase. + confinement_axis: Option>, + /// Provisional Phase II basis available for exact dual certification. + confinement_basis: Option>, +} + +impl ProvisionalPhaseDuals { + /// Returns a result with no provisional certificates. + const fn none() -> Self { + Self { + infeasibility_axis: None, + confinement_axis: None, + confinement_basis: None, + } + } +} + +fn provisional_phase_dual_axes( + matrix: &[Vec], + rhs: &[f64], + original_objective: &[f64], + variable_count: usize, +) -> ProvisionalPhaseDuals { + let constraint_count = rhs.len(); + let Some(total_variable_count) = variable_count.checked_add(constraint_count) else { + return ProvisionalPhaseDuals::none(); + }; + let mut phase_matrix = Vec::with_capacity(constraint_count); + for (row_index, row) in matrix.iter().enumerate() { + let mut phase_row = Vec::with_capacity(total_variable_count); + phase_row.extend(row.iter().copied()); + phase_row.extend((0..constraint_count).map( + |column| { + if column == row_index { 1.0 } else { 0.0 } + }, + )); + phase_matrix.push(phase_row); + } + + let mut basis: Vec<_> = (variable_count..total_variable_count).collect(); + let mut objective = vec![0.0; total_variable_count]; + for coefficient in &mut objective[variable_count..] { + *coefficient = -1.0; + } + let Some(phase_one_dual) = navigate_simplex_basis_f64_values( + &phase_matrix, + rhs, + &objective, + total_variable_count, + &mut basis, + ) else { + return ProvisionalPhaseDuals::none(); + }; + let phase_one_axis = (phase_one_dual.len() >= 2).then(|| phase_one_dual[2..].to_vec()); + + if !pivot_artificial_variables_out_f64(&phase_matrix, rhs, variable_count, &mut basis) { + return ProvisionalPhaseDuals { + infeasibility_axis: phase_one_axis, + confinement_axis: None, + confinement_basis: None, + }; + } + let mut phase_two_objective = vec![0.0; total_variable_count]; + if original_objective.len() != variable_count { + return ProvisionalPhaseDuals { + infeasibility_axis: phase_one_axis, + confinement_axis: None, + confinement_basis: None, + }; + } + phase_two_objective[..variable_count].clone_from_slice(original_objective); + let phase_two_dual = navigate_simplex_basis_f64_values( + &phase_matrix, + rhs, + &phase_two_objective, + variable_count, + &mut basis, + ); + let phase_two_axis = phase_two_dual + .as_ref() + .and_then(|dual| (dual.len() >= 2).then(|| dual[2..].to_vec())); + let phase_two_basis = phase_two_dual.map(|_dual| basis); + ProvisionalPhaseDuals { + infeasibility_axis: phase_one_axis, + confinement_axis: phase_two_axis, + confinement_basis: phase_two_basis, + } +} + +/// Removes zero artificial variables from a provisional floating-point basis. +fn pivot_artificial_variables_out_f64( + matrix: &[Vec], + rhs: &[f64], + original_variable_count: usize, + basis: &mut [usize], +) -> bool { + const ZERO_TOLERANCE: f64 = 1.0e-9; + const PIVOT_TOLERANCE: f64 = 1.0e-12; + + for basis_position in 0..basis.len() { + if basis[basis_position] < original_variable_count { + continue; + } + let Some(current_basis_matrix) = basis_matrix_f64(matrix, basis) else { + return false; + }; + let Some(basic_solution) = solve_f64_system(current_basis_matrix.clone(), rhs.to_vec()) + else { + return false; + }; + if basic_solution[basis_position].abs() > ZERO_TOLERANCE { + return false; + } + + let replacement = (0..original_variable_count) + .filter(|candidate| !basis.contains(candidate)) + .find(|&candidate| { + solve_f64_system( + current_basis_matrix.clone(), + matrix_column_f64(matrix, candidate), + ) + .is_some_and(|direction| direction[basis_position].abs() > PIVOT_TOLERANCE) + }); + let Some(replacement) = replacement else { + return false; + }; + basis[basis_position] = replacement; + } + true +} + +/// Certifies strict projection separation along a finite candidate direction. +fn simplices_are_strictly_separated_along( + first: &LabeledSimplexRealization, + second: &LabeledSimplexRealization, + axis: &[f64], +) -> bool { + if axis.len() != D || axis.iter().any(|value| !value.is_finite()) { + return false; + } + let exact_axis: Vec<_> = axis.iter().copied().map(rational_from_f64).collect(); + let Some((first_min, first_max)) = exact_projection_bounds(first, &exact_axis) else { + return false; + }; + let Some((second_min, second_max)) = exact_projection_bounds(second, &exact_axis) else { + return false; + }; + first_max < second_min || second_max < first_min +} + +/// Returns exact projection bounds for one realized simplex. +fn exact_projection_bounds( + simplex: &LabeledSimplexRealization, + axis: &[BigRational], +) -> Option<(BigRational, BigRational)> { + let mut projections = simplex.coordinates().iter().map(|coordinates| { + coordinates + .iter() + .zip(axis) + .fold(rational_zero(), |sum, (coordinate, coefficient)| { + sum + rational_from_f64(*coordinate) * coefficient + }) + }); + let first = projections.next()?; + Some( + projections.fold((first.clone(), first), |(minimum, maximum), projection| { + (minimum.min(projection.clone()), maximum.max(projection)) + }), + ) +} + +/// Certifies confinement through one shared vertex with a floating-point error bound. +/// +/// Scaling the provisional direction is exact because the factor is a power of +/// two. Each signed projection difference is then accepted only when its +/// conservative lower bound exceeds the unit dual margin. Ambiguous, +/// underflow-sensitive, or overflowing cases continue to exact arithmetic. +fn filtered_single_shared_vertex_confinement( + first: &LabeledSimplexRealization, + second: &LabeledSimplexRealization, + shared_labels: &[L], + axis: &[f64], +) -> bool +where + L: Eq, +{ + let [shared_label] = shared_labels else { + return false; + }; + let Some(first_shared_index) = first + .labels() + .iter() + .position(|candidate| candidate == shared_label) + else { + return false; + }; + let Some(second_shared_index) = second + .labels() + .iter() + .position(|candidate| candidate == shared_label) + else { + return false; + }; + let first_shared = &first.coordinates()[first_shared_index]; + let second_shared = &second.coordinates()[second_shared_index]; + if !coordinates_are_identical(first_shared, second_shared) { + return false; + } + + let scaled_axis: Vec<_> = axis.iter().map(|value| *value * 1024.0).collect(); + if scaled_axis.len() != D || scaled_axis.iter().any(|value| !value.is_finite()) { + return false; + } + first + .labels() + .iter() + .zip(first.coordinates()) + .filter(|(label, _)| *label != shared_label) + .all(|(_, coordinates)| { + certified_dot_difference_lower_bound(&scaled_axis, coordinates, first_shared) + .is_some_and(|lower_bound| lower_bound > 1.0) + }) + && second + .labels() + .iter() + .zip(second.coordinates()) + .filter(|(label, _)| *label != shared_label) + .all(|(_, coordinates)| { + certified_dot_difference_lower_bound(&scaled_axis, second_shared, coordinates) + .is_some_and(|lower_bound| lower_bound > 1.0) + }) +} + +/// Returns a conservative lower bound for `axis · (left - right)`. +fn certified_dot_difference_lower_bound( + axis: &[f64], + left: &[f64; D], + right: &[f64; D], +) -> Option { + let mut estimate = 0.0; + let mut magnitude = 0.0; + for ((coefficient, left_value), right_value) in axis.iter().zip(left).zip(right) { + let left_product = coefficient * left_value; + let right_product = coefficient * right_value; + if !left_product.is_finite() + || !right_product.is_finite() + || product_underflowed(*coefficient, *left_value, left_product) + || product_underflowed(*coefficient, *right_value, right_product) + { + return None; + } + estimate = coefficient.mul_add(*left_value, estimate); + estimate = (-coefficient).mul_add(*right_value, estimate); + magnitude += left_product.abs() + right_product.abs(); + if !estimate.is_finite() || !magnitude.is_finite() { + return None; + } + } + + let operation_count = f64::from(u32::try_from(2 * D).ok()?); + let denominator = 1.0 - operation_count * f64::EPSILON; + if denominator <= 0.0 { + return None; + } + let gamma = operation_count * f64::EPSILON / denominator; + let magnitude_upper_bound = magnitude / denominator.powi(2); + let error_bound = 4.0 * gamma * magnitude_upper_bound; + error_bound.is_finite().then_some(estimate - error_bound) +} + +/// Detects a product whose exact non-zero value rounded into the subnormal range. +fn product_underflowed(left: f64, right: f64, product: f64) -> bool { + left != 0.0 && right != 0.0 && (product == 0.0 || product.abs() < f64::MIN_POSITIVE) +} + +/// Compares finite coordinates exactly while treating both signed zeros alike. +fn coordinates_are_identical(left: &[f64; D], right: &[f64; D]) -> bool { + left.iter().zip(right).all(|(left, right)| { + let left_bits = left.to_bits(); + let right_bits = right.to_bits(); + left_bits == right_bits || (left_bits << 1 == 0 && right_bits << 1 == 0) + }) +} + +/// Certifies a Phase II dual after rebuilding its two affine offsets exactly. +/// +/// For direction `w`, the smallest feasible dual offsets are +/// `max_i(c_i - w·a_i)` and `max_j(c_j + w·b_j)`. Their sum is an exact upper +/// bound on the maximum non-shared barycentric weight. A non-positive bound +/// proves that every intersection point lies in the shared face. +fn exact_shared_face_confinement_certificate( + first: &LabeledSimplexRealization, + second: &LabeledSimplexRealization, + shared_labels: &[L], + axis: &[f64], +) -> bool +where + L: Eq, +{ + if axis.len() != D || axis.iter().any(|value| !value.is_finite()) { + return false; + } + let Some(exact_axis) = exact_axis_through_shared_face(first, shared_labels, axis) else { + return false; + }; + let first_offset = first + .labels() + .iter() + .zip(first.coordinates()) + .map(|(label, coordinates)| { + let objective = if shared_labels.contains(label) { + rational_zero() + } else { + rational_one() + }; + objective - exact_projection(coordinates, &exact_axis) + }) + .max(); + let second_offset = second + .labels() + .iter() + .zip(second.coordinates()) + .map(|(label, coordinates)| { + let objective = if shared_labels.contains(label) { + rational_zero() + } else { + rational_one() + }; + objective + exact_projection(coordinates, &exact_axis) + }) + .max(); + first_offset + .zip(second_offset) + .is_some_and(|(first_offset, second_offset)| { + first_offset + second_offset <= rational_zero() + }) +} + +/// Reconstructs a provisional axis in the shared face's exact affine nullspace. +/// +/// Shared vertices define homogeneous constraints on `(axis, offset)`. A +/// floating-point rank search selects pivot columns, then the fraction-free +/// exact solver constructs one nullspace basis vector per free column directly +/// from the original coordinates. Combining those vectors with provisional +/// free-coordinate weights restores exact shared-face equality without a +/// rational Gram solve. +fn exact_axis_through_shared_face( + simplex: &LabeledSimplexRealization, + shared_labels: &[L], + axis: &[f64], +) -> Option> +where + L: Eq, +{ + let shared_coordinates: Vec<_> = shared_labels + .iter() + .map(|label| { + simplex + .labels() + .iter() + .position(|candidate| candidate == label) + .and_then(|index| simplex.coordinates().get(index)) + }) + .collect::>()?; + if shared_coordinates.is_empty() { + return Some( + axis.iter() + .map(|value| rational_from_f64(*value * 1024.0)) + .collect(), + ); + } + + let constraint_matrix: Vec> = shared_coordinates + .iter() + .map(|coordinates| { + let mut row = coordinates.to_vec(); + row.push(1.0); + row + }) + .collect(); + let pivot_columns = independent_columns_f64(&constraint_matrix)?; + let free_columns: Vec<_> = (0..=D) + .filter(|column| !pivot_columns.contains(column)) + .collect(); + let pivot_matrix: Vec> = constraint_matrix + .iter() + .map(|row| pivot_columns.iter().map(|&column| row[column]).collect()) + .collect(); + + let mut provisional_affine = axis.to_vec(); + provisional_affine.push(-dot_product_f64(axis, shared_coordinates[0])); + let mut exact_affine = vec![rational_zero(); D + 1]; + for free_column in free_columns { + let weight = rational_from_f64(provisional_affine[free_column]); + exact_affine[free_column] += weight.clone(); + let rhs: Vec<_> = constraint_matrix + .iter() + .map(|row| -row[free_column]) + .collect(); + let solution = solve_exact_runtime_system(&pivot_matrix, &rhs)?.ok()?; + for (&pivot_column, coefficient) in pivot_columns.iter().zip(solution) { + exact_affine[pivot_column] += weight.clone() * coefficient; + } + } + + let scale = rational_from_f64(1024.0); + for value in &mut exact_affine[..D] { + *value *= scale.clone(); + } + exact_affine.truncate(D); + Some(exact_affine) +} + +/// Selects a full-rank set of matrix columns with provisional elimination. +fn independent_columns_f64(matrix: &[Vec]) -> Option> { + const PIVOT_TOLERANCE: f64 = 1.0e-12; + + let row_count = matrix.len(); + let column_count = matrix.first()?.len(); + if row_count == 0 || matrix.iter().any(|row| row.len() != column_count) { + return None; + } + let mut work = matrix.to_vec(); + let scale = work + .iter() + .flatten() + .fold(0.0_f64, |maximum, value| maximum.max(value.abs())); + let mut pivot_columns = Vec::with_capacity(row_count); + for column in 0..column_count { + let pivot_position = (pivot_columns.len()..row_count).max_by(|&left, &right| { + work[left][column] + .abs() + .total_cmp(&work[right][column].abs()) + })?; + if work[pivot_position][column].abs() <= PIVOT_TOLERANCE * scale.max(1.0) { + continue; + } + let pivot_row = pivot_columns.len(); + work.swap(pivot_row, pivot_position); + let pivot = work[pivot_row][column]; + let pivot_values = work[pivot_row].clone(); + for row_values in work.iter_mut().skip(pivot_row + 1) { + let factor = row_values[column] / pivot; + for (value, pivot_value) in row_values[column..].iter_mut().zip(&pivot_values[column..]) + { + *value -= factor * pivot_value; + } + } + pivot_columns.push(column); + if pivot_columns.len() == row_count { + return Some(pivot_columns); + } + } + None +} + +/// Certifies the floating-point Phase II basis with one exact dual solve. +/// +/// Dual feasibility and a non-positive exact objective bound prove confinement +/// without constructing a primal solution or repeating exact simplex pivots. +fn phase_two_dual_proves_shared_face_confinement( + matrix: &[Vec], + rhs: &[BigRational], + objective: &[BigRational], + basis: &[usize], +) -> bool { + let constraint_count = rhs.len(); + let variable_count = objective.len(); + if matrix.len() != constraint_count + || matrix.iter().any(|row| row.len() != variable_count) + || basis.len() != constraint_count + { + return false; + } + + let total_variable_count = variable_count + constraint_count; + let phase_matrix: Vec<_> = matrix + .iter() + .enumerate() + .map(|(row_index, row)| { + let mut phase_row = Vec::with_capacity(total_variable_count); + phase_row.extend(row.iter().cloned()); + phase_row.extend((0..constraint_count).map(|column| { + if column == row_index { + rational_one() + } else { + rational_zero() + } + })); + phase_row + }) + .collect(); + let mut phase_objective = vec![rational_zero(); total_variable_count]; + phase_objective[..variable_count].clone_from_slice(objective); + + let Some(current_basis_matrix) = basis_matrix(&phase_matrix, basis) else { + return false; + }; + let basic_objective: Vec<_> = basis + .iter() + .filter_map(|&index| phase_objective.get(index).cloned()) + .collect(); + if basic_objective.len() != constraint_count { + return false; + } + let Some(dual_solution) = solve_rational_system( + transpose_square_matrix(¤t_basis_matrix), + basic_objective, + ) else { + return false; + }; + let dual_is_feasible = (0..variable_count).all(|candidate| { + objective[candidate].clone() + - dot_product(&dual_solution, &matrix_column(matrix, candidate)) + <= rational_zero() + }); + dual_is_feasible && dot_product(rhs, &dual_solution) <= rational_zero() +} + +/// Computes one exact projection along a finite candidate axis. +fn exact_projection(coordinates: &[f64; D], axis: &[BigRational]) -> BigRational { + coordinates + .iter() + .zip(axis) + .fold(rational_zero(), |sum, (coordinate, coefficient)| { + sum + rational_from_f64(*coordinate) * coefficient + }) +} + +/// Certifies exact primal feasibility and dual optimality for one candidate basis. +fn certify_optimal_basis( + matrix: &[Vec], + rhs: &[BigRational], + objective: &[BigRational], + entering_variable_count: usize, + basis: &[usize], +) -> Option { + let current_basis_matrix = basis_matrix(matrix, basis)?; + let basic_solution = solve_rational_system(current_basis_matrix.clone(), rhs.to_vec())?; + if basic_solution.iter().any(Signed::is_negative) { + return None; + } + + let basic_objective: Vec<_> = basis + .iter() + .map(|&index| objective[index].clone()) + .collect(); + let dual_solution = solve_rational_system( + transpose_square_matrix(¤t_basis_matrix), + basic_objective, + )?; + let dual_is_feasible = (0..entering_variable_count) + .filter(|candidate| !basis.contains(candidate)) + .all(|candidate| { + let reduced_cost = objective[candidate].clone() + - dot_product(&dual_solution, &matrix_column(matrix, candidate)); + !reduced_cost.is_positive() + }); + if !dual_is_feasible { + return None; + } + + let mut solution = vec![rational_zero(); objective.len()]; + for (&variable, value) in basis.iter().zip(&basic_solution) { + solution[variable] = value.clone(); + } + Some(RevisedSimplexResult::Optimal { + objective_value: dot_product(objective, &solution), + solution, + }) +} + +/// Converts an exact rational matrix to finite f64 values for basis navigation. +fn rational_matrix_to_f64(matrix: &[Vec]) -> Option>> { + matrix.iter().map(|row| rationals_to_f64(row)).collect() +} + +/// Converts exact rationals to finite f64 values for provisional calculations. +fn rationals_to_f64(values: &[BigRational]) -> Option> { + values + .iter() + .map(|value| value.to_f64().filter(|converted| converted.is_finite())) + .collect() +} + +/// Extracts a floating-point basis matrix. +fn basis_matrix_f64(matrix: &[Vec], basis: &[usize]) -> Option>> { + let column_count = matrix.first()?.len(); + if matrix.len() != basis.len() + || matrix.iter().any(|row| row.len() != column_count) + || basis.iter().any(|&column| column >= column_count) + { + return None; + } + Some( + matrix + .iter() + .map(|row| basis.iter().map(|&column| row[column]).collect()) + .collect(), + ) +} + +/// Copies one floating-point matrix column. +fn matrix_column_f64(matrix: &[Vec], column: usize) -> Vec { + matrix.iter().map(|row| row[column]).collect() +} + +/// Transposes a square floating-point matrix. +fn transpose_square_matrix_f64(matrix: &[Vec]) -> Vec> { + let dimension = matrix.len(); + (0..dimension) + .map(|row| (0..dimension).map(|column| matrix[column][row]).collect()) + .collect() +} + +/// Computes a floating-point dot product used only for provisional pivot selection. +fn dot_product_f64(left: &[f64], right: &[f64]) -> f64 { + left.iter() + .zip(right) + .map(|(left, right)| left * right) + .sum() +} + +/// Solves a small floating-point system with scaled partial pivoting. +#[expect( + clippy::needless_range_loop, + reason = "index-based elimination keeps pivot row/column operations explicit" +)] +fn solve_f64_system(mut matrix: Vec>, mut rhs: Vec) -> Option> { + const PIVOT_TOLERANCE: f64 = 1.0e-14; + + let dimension = rhs.len(); + if matrix.len() != dimension || matrix.iter().any(|row| row.len() != dimension) { + return None; + } + + for pivot_column in 0..dimension { + let pivot_row = (pivot_column..dimension).max_by(|&left, &right| { + matrix[left][pivot_column] + .abs() + .total_cmp(&matrix[right][pivot_column].abs()) + })?; + let row_scale = matrix[pivot_row] + .iter() + .map(|&value| value.abs()) + .fold(0.0_f64, f64::max); + if !row_scale.is_finite() + || matrix[pivot_row][pivot_column].abs() <= PIVOT_TOLERANCE * row_scale.max(1.0) + { + return None; + } + if pivot_row != pivot_column { + matrix.swap(pivot_column, pivot_row); + rhs.swap(pivot_column, pivot_row); + } + + for row in pivot_column + 1..dimension { + let factor = matrix[row][pivot_column] / matrix[pivot_column][pivot_column]; + matrix[row][pivot_column] = 0.0; + for column in pivot_column + 1..dimension { + matrix[row][column] = + factor.mul_add(-matrix[pivot_column][column], matrix[row][column]); + } + rhs[row] = factor.mul_add(-rhs[pivot_column], rhs[row]); + } + } + + let mut solution = vec![0.0; dimension]; + for row in (0..dimension).rev() { + let mut sum = rhs[row]; + for column in row + 1..dimension { + sum = matrix[row][column].mul_add(-solution[column], sum); + } + solution[row] = sum / matrix[row][row]; + if !solution[row].is_finite() { + return None; + } + } + Some(solution) +} + +/// Extracts the square matrix formed by the current basis columns. +fn basis_matrix(matrix: &[Vec], basis: &[usize]) -> Option>> { + let column_count = matrix.first()?.len(); + if matrix.len() != basis.len() + || matrix.iter().any(|row| row.len() != column_count) + || basis.iter().any(|&column| column >= column_count) + { + return None; + } + Some( + matrix + .iter() + .map(|row| basis.iter().map(|&column| row[column].clone()).collect()) + .collect(), + ) +} + +/// Copies one matrix column for an exact linear solve. +fn matrix_column(matrix: &[Vec], column: usize) -> Vec { + matrix.iter().map(|row| row[column].clone()).collect() +} + +/// Transposes a square exact-rational matrix. +fn transpose_square_matrix(matrix: &[Vec]) -> Vec> { + let dimension = matrix.len(); + (0..dimension) + .map(|row| { + (0..dimension) + .map(|column| matrix[column][row].clone()) + .collect() + }) + .collect() +} + +/// Computes an exact dot product without introducing floating-point decisions. +fn dot_product(left: &[BigRational], right: &[BigRational]) -> BigRational { + left.iter() + .zip(right) + .fold(rational_zero(), |sum, (left, right)| { + sum + left.clone() * right.clone() + }) +} + +/// Returns labels whose barycentric coordinates witness mass outside the shared face. +fn positive_nonshared_labels( + barycentric: &[BigRational], + labels: &[L], + shared_labels: &[L], +) -> SimplexRealizationBuffer +where + L: Clone + Eq, +{ + labels + .iter() + .zip(barycentric) + .filter(|(label, coordinate)| !shared_labels.contains(label) && coordinate.is_positive()) + .map(|(label, _coordinate)| label.clone()) + .collect() +} + +#[expect( + clippy::needless_range_loop, + reason = "index-based elimination keeps pivot row/column operations explicit" +)] +/// Solves a square rational linear system by Gaussian elimination. +fn solve_rational_system( + mut matrix: Vec>, + mut rhs: Vec, +) -> Option> { + let dimension = rhs.len(); + if matrix.len() != dimension || matrix.iter().any(|row| row.len() != dimension) { + return None; + } + match try_solve_exact_f64_system(&matrix, &rhs) { + ExactF64Solve::Solution(solution) => return Some(solution), + ExactF64Solve::Singular => return None, + ExactF64Solve::Unsupported => {} + } + + for pivot_col in 0..dimension { + let pivot_row = + (pivot_col..dimension).find(|&row| matrix[row][pivot_col] != rational_zero())?; + if pivot_row != pivot_col { + matrix.swap(pivot_col, pivot_row); + rhs.swap(pivot_col, pivot_row); + } + + let pivot_value = matrix[pivot_col][pivot_col].clone(); + for row in pivot_col + 1..dimension { + if matrix[row][pivot_col] == rational_zero() { + continue; + } + let factor = matrix[row][pivot_col].clone() / pivot_value.clone(); + matrix[row][pivot_col] = rational_zero(); + for col in pivot_col + 1..dimension { + matrix[row][col] = + matrix[row][col].clone() - factor.clone() * matrix[pivot_col][col].clone(); + } + rhs[row] = rhs[row].clone() - factor * rhs[pivot_col].clone(); + } + } + + let mut solution = vec![rational_zero(); dimension]; + for row in (0..dimension).rev() { + let mut sum = rhs[row].clone(); + for col in row + 1..dimension { + sum -= matrix[row][col].clone() * solution[col].clone(); + } + solution[row] = sum / matrix[row][row].clone(); + } + + Some(solution) +} + +/// Outcome of attempting the fraction-free exact `f64` solver. +enum ExactF64Solve { + /// The runtime stack dispatcher cannot represent this rational system. + Unsupported, + /// The system is exactly singular. + Singular, + /// The exact system solution. + Solution(Vec), +} + +/// Uses fraction-free Bareiss elimination when every coefficient came directly +/// from a finite `f64`; otherwise leaves the rational solver as the fallback. +fn try_solve_exact_f64_system(matrix: &[Vec], rhs: &[BigRational]) -> ExactF64Solve { + let Some(float_matrix) = exact_rational_matrix_as_f64(matrix) else { + return ExactF64Solve::Unsupported; + }; + let Some(float_rhs) = exact_rationals_as_f64(rhs) else { + return ExactF64Solve::Unsupported; + }; + let Some(result) = solve_exact_runtime_system(&float_matrix, &float_rhs) else { + return ExactF64Solve::Unsupported; + }; + match result { + Ok(solution) => ExactF64Solve::Solution(solution), + Err(StackMatrixDispatchError::La { + source: la_stack::LaError::Singular { .. }, + }) => ExactF64Solve::Singular, + Err(_) => ExactF64Solve::Unsupported, + } +} + +/// Converts rationals to `f64` only when doing so preserves each value exactly. +fn exact_rational_matrix_as_f64(matrix: &[Vec]) -> Option>> { + matrix + .iter() + .map(|row| exact_rationals_as_f64(row)) + .collect() +} + +/// Converts a finite floating-point matrix to its exact rational representation. +fn exact_matrix_from_f64(matrix: &[Vec]) -> Vec> { + matrix + .iter() + .map(|row| row.iter().copied().map(rational_from_f64).collect()) + .collect() +} + +/// Converts rationals to finite `f64` values and verifies an exact round trip. +fn exact_rationals_as_f64(values: &[BigRational]) -> Option> { + values + .iter() + .map(|value| { + let converted = value.to_f64().filter(|candidate| candidate.is_finite())?; + (rational_from_f64(converted) == *value).then_some(converted) + }) + .collect() +} + +/// Converts a finite f64 to an exact rational value for barycentric predicates. +fn rational_from_f64(value: f64) -> BigRational { + BigRational::from_f64(value) + .expect("validated finite f64 coordinates must convert to BigRational") +} + +/// Returns the additive identity used throughout exact barycentric arithmetic. +fn rational_zero() -> BigRational { + BigRational::from_integer(BigInt::from(0)) +} + +/// Returns the multiplicative identity used throughout exact barycentric arithmetic. +fn rational_one() -> BigRational { + BigRational::from_integer(BigInt::from(1)) +} + +#[cfg(test)] +mod tests { + use super::*; + use std::assert_matches; + + /// Builds the standard D-simplex with vertices at the origin and coordinate axes. + fn standard_simplex_coordinates() -> Vec<[f64; D]> { + let mut coordinates = Vec::with_capacity(D + 1); + coordinates.push([0.0; D]); + for axis in 0..D { + let mut vertex = [0.0; D]; + vertex[axis] = 1.0; + coordinates.push(vertex); + } + coordinates + } + + /// Builds two D-simplices that meet exactly in the facet opposite the last axis. + fn shared_facet_pair() -> ( + LabeledSimplexRealization, + LabeledSimplexRealization, + Vec, + ) { + let first_coordinates = standard_simplex_coordinates::(); + let mut second_coordinates = first_coordinates[..D].to_vec(); + let mut opposite_apex = [0.0; D]; + opposite_apex[D - 1] = -1.0; + second_coordinates.push(opposite_apex); + + let first = LabeledSimplexRealization::try_new(0..=D, first_coordinates) + .expect("standard simplex is valid"); + let second = LabeledSimplexRealization::try_new( + (0..D).chain(std::iter::once(D + 1)), + second_coordinates, + ) + .expect("opposite simplex is valid"); + (first, second, (0..D).collect()) + } + + /// Builds one D-simplex strictly inside another with disjoint labels. + fn crossing_pair() -> ( + LabeledSimplexRealization, + LabeledSimplexRealization, + ) { + let first = LabeledSimplexRealization::try_new(0..=D, standard_simplex_coordinates::()) + .expect("standard simplex is valid"); + let dimension = f64::from(u32::try_from(D + 1).expect("tested dimensions fit in u32")); + let step = 1.0 / (4.0 * dimension); + let base = [step; D]; + let mut second_coordinates = Vec::with_capacity(D + 1); + second_coordinates.push(base); + for axis in 0..D { + let mut vertex = base; + vertex[axis] += step; + second_coordinates.push(vertex); + } + let second = LabeledSimplexRealization::try_new(D + 1..=2 * D + 1, second_coordinates) + .expect("interior simplex is valid"); + (first, second) + } + + /// Checks the shared-facet classification and exact affine-nullspace certificate. + fn assert_shared_facet_is_valid() { + let (first, second, shared) = shared_facet_pair::(); + assert_matches!( + intersection_via_linear_program(&first, &second, &shared, false), + IntersectionLinearProgramResult::Valid + ); + + let mut provisional_axis = vec![0.0; D]; + provisional_axis[D - 1] = 1.0; + let exact_axis = exact_axis_through_shared_face(&first, &shared, &provisional_axis) + .expect("shared facet has an exact normal"); + let shared_projection = exact_projection(&first.coordinates()[0], &exact_axis); + for coordinates in &first.coordinates()[..D] { + assert_eq!( + exact_projection(coordinates, &exact_axis), + shared_projection + ); + } + assert!(exact_shared_face_confinement_certificate( + &first, + &second, + &shared, + &provisional_axis, + )); + } + + /// Checks that a full-dimensional interior overlap produces witnesses on both sides. + fn assert_crossing_is_invalid() { + let (first, second) = crossing_pair::(); + assert_matches!( + intersection_via_linear_program(&first, &second, &[], false), + IntersectionLinearProgramResult::Invalid(witness) + if !witness.first_only_witness.is_empty() + && !witness.second_only_witness.is_empty() + ); + } + + #[test] + fn exact_linear_program_classifies_disjoint_triangles_as_valid() { + let first = + LabeledSimplexRealization::try_new([0, 1, 2], [[0.0, 0.0], [1.0, 0.0], [0.0, 1.0]]) + .unwrap(); + let second = + LabeledSimplexRealization::try_new([3, 4, 5], [[2.0, 2.0], [3.0, 2.0], [2.0, 3.0]]) + .unwrap(); + + assert_matches!( + intersection_via_linear_program(&first, &second, &[], false), + IntersectionLinearProgramResult::Valid + ); + } + + #[test] + fn exact_linear_program_accepts_a_shared_edge_intersection_2d() { + assert_shared_facet_is_valid::<2>(); + } + + #[test] + fn exact_linear_program_reports_crossing_triangle_witnesses_2d() { + assert_crossing_is_invalid::<2>(); + } + + #[test] + fn exact_linear_program_accepts_a_shared_facet_intersection_3d() { + assert_shared_facet_is_valid::<3>(); + } + + #[test] + fn exact_linear_program_reports_crossing_tetrahedra_3d() { + assert_crossing_is_invalid::<3>(); + } + + #[test] + fn exact_linear_program_accepts_a_shared_facet_intersection_4d() { + assert_shared_facet_is_valid::<4>(); + } + + #[test] + fn exact_linear_program_reports_crossing_simplices_4d() { + assert_crossing_is_invalid::<4>(); + } + + #[test] + fn exact_linear_program_accepts_a_shared_facet_intersection_5d() { + assert_shared_facet_is_valid::<5>(); + } + + #[test] + fn exact_linear_program_reports_crossing_simplices_5d() { + assert_crossing_is_invalid::<5>(); + } + + #[test] + fn exact_linear_program_reports_a_singular_verified_basis() { + let first = + LabeledSimplexRealization::try_new([0, 1, 2], [[0.0, 0.0], [1.0, 0.0], [2.0, 0.0]]) + .unwrap(); + let second = + LabeledSimplexRealization::try_new([3, 4, 5], [[0.0, 1.0], [1.0, 1.0], [0.0, 2.0]]) + .unwrap(); + + assert_matches!( + intersection_via_linear_program(&first, &second, &[], true), + IntersectionLinearProgramResult::SingularBarycentricBasis + ); + } + + #[test] + fn exact_linear_program_runs_phase_one_and_phase_two() { + let matrix = vec![vec![rational_one(), rational_one()]]; + let rhs = vec![rational_one()]; + let objective = vec![rational_one(), rational_zero()]; + + assert_matches!( + maximize_nonnegative_equality_linear_program(&matrix, &rhs, &objective), + ExactLinearProgramResult::Optimal { + objective_value, + .. + } if objective_value == rational_one() + ); + } + + #[test] + fn exact_linear_program_reports_phase_one_infeasibility() { + let matrix = vec![vec![rational_one()], vec![rational_one()]]; + let rhs = vec![rational_one(), rational_one() + rational_one()]; + let objective = vec![rational_one()]; + + assert_matches!( + maximize_nonnegative_equality_linear_program(&matrix, &rhs, &objective), + ExactLinearProgramResult::Infeasible + ); + } + + #[test] + fn exact_linear_program_pivots_zero_artificial_variable_out_before_phase_two() { + let matrix = vec![ + vec![rational_one(), rational_zero()], + vec![rational_zero(), -rational_one()], + ]; + let rhs = vec![rational_one(), rational_zero()]; + let objective = vec![rational_one(), rational_zero()]; + + assert_matches!( + maximize_nonnegative_equality_linear_program(&matrix, &rhs, &objective), + ExactLinearProgramResult::Optimal { + solution, + objective_value, + } if solution == vec![rational_one(), rational_zero()] + && objective_value == rational_one() + ); + } + + #[test] + fn exact_linear_program_uses_exact_simplex_when_coefficients_exceed_f64_range() { + let huge = BigRational::from_integer(BigInt::from(1_u8) << usize::from(u16::MAX)); + let matrix = vec![vec![huge.clone()]]; + let rhs = vec![huge]; + let objective = vec![rational_one()]; + + assert_matches!( + maximize_nonnegative_equality_linear_program(&matrix, &rhs, &objective), + ExactLinearProgramResult::Optimal { + solution, + objective_value, + } if solution == vec![rational_one()] && objective_value == rational_one() + ); + } + + #[test] + fn filtered_dot_difference_rejects_overflow_and_underflow() { + assert_eq!( + certified_dot_difference_lower_bound(&[f64::MAX], &[2.0], &[0.0]), + None + ); + assert_eq!( + certified_dot_difference_lower_bound(&[f64::MIN_POSITIVE], &[0.5], &[0.0],), + None + ); + } + + #[test] + fn phase_two_dual_certificate_proves_zero_objective_bound() { + let matrix = vec![vec![rational_one(), rational_one()]]; + let rhs = vec![rational_one()]; + let objective = vec![rational_zero(), rational_zero()]; + + assert!(phase_two_dual_proves_shared_face_confinement( + &matrix, + &rhs, + &objective, + &[0], + )); + } +} diff --git a/src/lib.rs b/src/lib.rs index 3588c951..56064edf 100644 --- a/src/lib.rs +++ b/src/lib.rs @@ -701,6 +701,7 @@ pub mod geometry { pub mod measures; pub mod norms; pub mod point_generation; + pub(crate) mod simplex_lp; pub mod triangulation_generation; // Re-export all public utility items for ergonomic `crate::geometry::util::*` access. diff --git a/tests/semgrep/scripts/tests/python_style.py b/tests/semgrep/scripts/tests/python_style.py new file mode 100644 index 00000000..87b59a0d --- /dev/null +++ b/tests/semgrep/scripts/tests/python_style.py @@ -0,0 +1,7 @@ +"""Fixtures for repository Python style rules.""" + +# ruleid: delaunay.python.no-future-annotations-on-python314 +from __future__ import annotations + +# ok: delaunay.python.no-future-annotations-on-python314 +from typing import TYPE_CHECKING diff --git a/uv.lock b/uv.lock index 02fe2c1c..79e65065 100644 --- a/uv.lock +++ b/uv.lock @@ -499,13 +499,13 @@ dev = [ ] notebooks = [ { name = "ipykernel", specifier = ">=7.3.0" }, - { name = "jupyterlab", specifier = ">=4.6.1" }, - { name = "matplotlib", specifier = ">=3.11.0" }, + { name = "jupyterlab", specifier = ">=4.6.2" }, + { name = "matplotlib", specifier = ">=3.11.1" }, { name = "nbclient", specifier = ">=0.11.0" }, { name = "nbconvert", specifier = ">=7.17.1" }, { name = "nbformat", specifier = ">=5.10.4" }, { name = "plotly", specifier = ">=6.9.0" }, - { name = "polars", specifier = ">=1.42.1" }, + { name = "polars", specifier = ">=1.43.0" }, ] [[package]] From 75ea22f8b59e3b0c1d680c16adefedb2e0eeee7d Mon Sep 17 00:00:00 2001 From: Adam Getchell Date: Sun, 26 Jul 2026 03:36:38 -0700 Subject: [PATCH 3/3] fix(validation): reject invalid realization confinement certificates - Require coordinate-identical shared vertices before using orientation confinement. - Reject non-finite scaled axes and affine offsets before exact conversion. --- src/geometry/realization.rs | 46 ++++++++++++++++++++++++++++++++- src/geometry/util/simplex_lp.rs | 45 +++++++++++++++++++++++++++----- 2 files changed, 84 insertions(+), 7 deletions(-) diff --git a/src/geometry/realization.rs b/src/geometry/realization.rs index 688c7085..b901eab0 100644 --- a/src/geometry/realization.rs +++ b/src/geometry/realization.rs @@ -45,7 +45,8 @@ use crate::geometry::predicates::Orientation; use crate::geometry::robust_predicates::robust_orientation; use crate::geometry::traits::coordinate::InvalidCoordinateValue; use crate::geometry::util::simplex_lp::{ - IntersectionLinearProgramResult, intersection_via_linear_program, shared_face_fast_confinement, + IntersectionLinearProgramResult, coordinates_are_identical, intersection_via_linear_program, + shared_face_fast_confinement, }; use thiserror::Error; @@ -701,6 +702,29 @@ fn intersection_is_confined_by_orientation( where L: Eq, { + for shared_label in shared_labels { + let Some(basis_index) = basis + .labels() + .iter() + .position(|candidate| candidate == shared_label) + else { + return false; + }; + let Some(other_index) = other + .labels() + .iter() + .position(|candidate| candidate == shared_label) + else { + return false; + }; + if !coordinates_are_identical( + &basis.coordinates()[basis_index], + &other.coordinates()[other_index], + ) { + return false; + } + } + let basis_points: SimplexRealizationBuffer<_> = (0..basis.coordinates().len()) .filter_map(|index| basis.point_at(index)) .collect(); @@ -829,6 +853,26 @@ mod tests { ); } + #[test] + fn orientation_confinement_rejects_mismatched_shared_coordinates() { + let first = + LabeledSimplexRealization::try_new([0, 1, 2], [[0.0, 0.0], [1.0, 0.0], [0.0, 1.0]]) + .unwrap(); + let second = LabeledSimplexRealization::try_new( + [0, 3, 4], + [[-1.0, -1.0], [-2.0, -1.0], [-1.0, -2.0]], + ) + .unwrap(); + let orientation = realization_orientation(&first).expect("standard triangle is oriented"); + + assert!(!intersection_is_confined_by_orientation( + &first, + &second, + &[0], + orientation, + )); + } + #[test] fn labeled_simplex_realization_rejects_non_finite_coordinates() { let err = LabeledSimplexRealization::try_new( diff --git a/src/geometry/util/simplex_lp.rs b/src/geometry/util/simplex_lp.rs index 5327b4cb..581fa006 100644 --- a/src/geometry/util/simplex_lp.rs +++ b/src/geometry/util/simplex_lp.rs @@ -1279,7 +1279,10 @@ fn product_underflowed(left: f64, right: f64, product: f64) -> bool { } /// Compares finite coordinates exactly while treating both signed zeros alike. -fn coordinates_are_identical(left: &[f64; D], right: &[f64; D]) -> bool { +pub(in crate::geometry) fn coordinates_are_identical( + left: &[f64; D], + right: &[f64; D], +) -> bool { left.iter().zip(right).all(|(left, right)| { let left_bits = left.to_bits(); let right_bits = right.to_bits(); @@ -1357,6 +1360,9 @@ fn exact_axis_through_shared_face( where L: Eq, { + if axis.len() != D || axis.iter().any(|value| !value.is_finite()) { + return None; + } let shared_coordinates: Vec<_> = shared_labels .iter() .map(|label| { @@ -1368,11 +1374,11 @@ where }) .collect::>()?; if shared_coordinates.is_empty() { - return Some( - axis.iter() - .map(|value| rational_from_f64(*value * 1024.0)) - .collect(), - ); + let scaled_axis: Vec<_> = axis.iter().map(|value| *value * 1024.0).collect(); + if scaled_axis.iter().any(|value| !value.is_finite()) { + return None; + } + return Some(scaled_axis.into_iter().map(rational_from_f64).collect()); } let constraint_matrix: Vec> = shared_coordinates @@ -1394,6 +1400,9 @@ where let mut provisional_affine = axis.to_vec(); provisional_affine.push(-dot_product_f64(axis, shared_coordinates[0])); + if provisional_affine.iter().any(|value| !value.is_finite()) { + return None; + } let mut exact_affine = vec![rational_zero(); D + 1]; for free_column in free_columns { let weight = rational_from_f64(provisional_affine[free_column]); @@ -2115,6 +2124,30 @@ mod tests { ); } + #[test] + fn exact_axis_rejects_post_scaling_overflow() { + let simplex = LabeledSimplexRealization::try_new([0, 1], [[0.0], [1.0]]).unwrap(); + + assert_eq!( + exact_axis_through_shared_face(&simplex, &[], &[f64::MAX]), + None + ); + } + + #[test] + fn exact_axis_rejects_non_finite_provisional_affine_offset() { + let simplex = LabeledSimplexRealization::try_new( + [0, 1, 2], + [[f64::MAX, 0.0], [0.0, 1.0], [0.0, 0.0]], + ) + .unwrap(); + + assert_eq!( + exact_axis_through_shared_face(&simplex, &[0], &[2.0, 0.0]), + None + ); + } + #[test] fn phase_two_dual_certificate_proves_zero_objective_bound() { let matrix = vec![vec![rational_one(), rational_one()]];