Judge a root against the terms it cancels, and judge it either way it is written

[Rafael-SOWNet]

Two faults in the check that drops roots which do not satisfy the equation they came from. Found while testing whether #214 could be closed.

A quartic answered with the empty set

"1/210 - (17*x)/210 + (101*x^2)/210 - (247*x^3)/210 + x^4 = 0".Solve("x")
// master: {  }

It has four real roots. They come back from the numeric search as decimals, and a root found that way is only as accurate as the search that found it — the terms cancel at it to within that accuracy rather than exactly. These leave a residual of 5.2e-6 against terms of about 1.2e-2. The check read that residual on its own size against a fixed 1e-6 and threw all four away.

The residual is now read against the largest of the terms it is the sum of.

The threshold is measured, not picked. A spurious root is not a near miss — it comes from a rewrite that widened the domain, so its residual is a whole quantity in its own right: 2*pi*i for a logarithm sum, or the difference between −3 and 9 for an exponential. Across the equations I measured:

	relative residual
genuine roots	0, or 1e-15, worst case 4.5e-4
spurious roots	1.55 – 1.98

The threshold is a hundredth — twenty times above the worst genuine root, a hundred and fifty times below the mildest spurious one.

The same equation answered two ways

The check only ran when the caller wrote the equation out. Handed the bare expression it went through a different entry point that checked only for finiteness:

("2^x + 2^(2*x) - 6 = 0").Solve("x")       // { 1 }
"2^x + 2^(2*x) - 6".SolveEquation("x")     // { 1, ln((-3) ^ (1 / ln(2))) }

The second answer is a root of nothing. The exponential solver substitutes t = e^x and inverts with t = v^(1/p); composing that with x = ln(t) is only the same number while the imaginary part stays inside the principal branch. For a negative v it does not — ln((-3)^(1/ln 2)) is 1.58 - 1.75i, exactly one turn away from the actual root ln(-3)/ln(2) = 1.58 + 4.53i, and satisfies nothing. Both entry points now check.

The genuine complex root is still not found; this only stops a non-root being reported in its place. Recovering it needs the exponential solver to invert the composition rather than the two steps separately, which is a separate piece of work.

Verification

18 tests, 5 of which fail without the change. 3898 unit tests and 127 F# tests on both target frameworks, none failing — including MomoTest, which is the quartic above and which fails without the first half of this. 117-problem self-verifying corpus at 88/117 with nothing wrong, in error or timing out; 1320 property checks over 151 expressions, none failing.

One thing I did not pin: x^5 + 3*x + 1 comes back with 28 roots for a quintic, near-duplicates from the numeric search. That is untouched here and worth its own look.

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Codecov Report

❌ Patch coverage is 88.23529% with 2 lines in your changes missing coverage. Please review.
✅ Project coverage is 80.57%. Comparing base (90c00a8) to head (f1460df).
⚠️ Report is 57 commits behind head on master.

Files with missing lines	Patch %	Lines
...ontinuous/Solvers/EquationSolver/SolveStatement.cs	86.66%	0 Missing and 2 partials ⚠️
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Additional details and impacted files

@@            Coverage Diff             @@
##           master     #685      +/-   ##
==========================================
- Coverage   80.99%   80.57%   -0.43%     
==========================================
  Files         155      156       +1     
  Lines       13687    12930     -757     
  Branches     1957     2125     +168     
==========================================
- Hits        11086    10418     -668     
+ Misses       1990     1906      -84     
+ Partials      611      606       -5     

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