[Rafael-SOWNet]

Closes #2425.

Under MathS.Settings.Codomain.Set(Domain.Real), an exponential equation could answer a root that is not real. e^x + 1 = 0 answered { ln(-1) }, and ln(-1) is iπ.

There were two causes:

• The inversion ignored the codomain. Powf.InvertNode turned a^u = v into log(a, v) for every v. The real codomain is applied only inside particular solvers (the logarithm, Lambert W and modulus solvers), and the exponential inversion was not one of them.
• The replacement search assumed its variable was real. It solved the variable that stands for a repeated subtree under the caller's codomain, so over the reals it treated that variable as real. That is false where the subtree is something like i x. It is now solved over the complex numbers in that case.

Over the reals, the inversion now answers no root where the base is a positive real, the exponent is real-valued, and the value is decidedly not a positive real. That is decided by value, from a 40-digit enclosure, never by the printed form. Anything the check cannot decide keeps its root as before. Over the complex numbers, 31 equations give output identical character for character to master's.

equation, over RR	2.6.0 and master	this branch	true
e^x + 1 = 0	{ ln(-1) }	{ }	{ }
e^x - 2e^(-x) - 1 = 0	{ ln(-1), ln(2) }	{ ln(2) }	{ ln(2) }
2^x = -3	{ log(2, -3) }	{ }	{ }
3 e^x + 5 = 0	{ ln(-5/3) }	{ }	{ }
e^(2x) + e^x - 2 = 0	{ 0, ln(-2) }	{ 0 }	{ 0 }
e^x + e^(-x) = 1	{ ln((1∓sqrt(-3))/2) }	{ }	{ }
e^x - e^(-x) = 3	{ ln((3-sqrt13)/2), ln((3+sqrt13)/2) }	{ ln((3+sqrt13)/2) }	the same
e^x = -1 + i	{ ln(-1+i) }	{ }	{ }
e^(x^2) = -1	{ ±sqrt(ln(-1)) }	{ }	{ }
e^x - 2 = 0, e^(2x) - 3e^x + 2 = 0, ln(x) = 2, e^x = y	unchanged	unchanged

The rows are in BREAKING-CHANGES.md, measured on the 2.6.0 package. Not changed here, and still not filtered by the real codomain: polynomial roots (x^2 = -1 gives { i, -i }) and trigonometric ones (sin(x) = 2).

Checked on this branch:

• Tests: UnitTests 19370 passed, plus the MCP, Terminal, InteractiveWrapper and F# wrapper suites, and the AOT smoke test. The new tests check, for each root, its count, that its value is real, and that substituting it leaves a residual under 1e-30; and that the complex codomain still answers iπ.
• Harnesses: BoundCheck, RootCheck, CasBench, PropCheck, SimpSweep, RuleCheck, CanonCheck and Confluence all pass.

🤖 Generated with Claude Code

[codecov]

Codecov Report

❌ Patch coverage is 73.91304% with 6 lines in your changes missing coverage. Please review.
✅ Project coverage is 86.81%. Comparing base (2eea2e9) to head (87b8f00).
⚠️ Report is 1 commits behind head on master.

Files with missing lines	Patch %	Lines
...nuous/Solvers/EquationSolver/InvertNode.Classes.cs	50.00%	3 Missing and 3 partials ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #2433      +/-   ##
==========================================
- Coverage   86.81%   86.81%   -0.01%     
==========================================
  Files         312      312              
  Lines       69551    69573      +22     
  Branches    20316    20325       +9     
==========================================
+ Hits        60383    60401      +18     
- Misses       4932     4933       +1     
- Partials     4236     4239       +3     

Flag	Coverage Δ
cpp	37.77% <ø> (ø)
csharp	87.35% <73.91%> (-0.01%)	⬇️
fsharp	32.97% <ø> (ø)
interactive	39.81% <ø> (ø)

Flags with carried forward coverage won't be shown. Click here to find out more.

Files with missing lines	Coverage Δ
...Solvers/EquationSolver/AnalyticalEquationSolver.cs	96.25% <100.00%> (+0.25%)	⬆️
...nuous/Solvers/EquationSolver/InvertNode.Classes.cs	61.61% <50.00%> (-0.52%)	⬇️

... and 4 files with indirect coverage changes

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