[Rafael-SOWNet]

Closes #2424.

(-2)^x + 4^x = 0 took the process down with a stack overflow. The cycle is Solve → SolveInverting → ExponentialSolver.SolveMultiplicative → Solve:

1. SolveLinear rewrites (-2)^x + 4^x as t^ln(-2) + t^ln(4) in t = e^x.
2. SolveMultiplicative chooses a proper power t^unit only when every exponent is real. ln(-2) = ln 2 + iπ is not real, so the substitution stays t itself.
3. Substituting t by itself only renames the variable, so the solver solved the same equation again, forever.

A negative base was only one way in. Any non-real numeric exponent overflowed the same way on 2.6.0 and on master: x^i + x = 0, x^(1+i) + x = 0, x^i + x^(2*i) = 0.

The fix declines a substitution of x by itself, since it is a renaming and never a step. With the cycle gone, Newton's method answers these equations, and two more changes keep its answers roots:

• The convergence test is relative near 0. It was absolute (last step < 1e-6). Near the zero at t = 0 that every term shares, the iteration crawled in steps under 1e-6 while still about 18% of |t| from where it settles, and returned points 0.0004 and 0.23 from the nearest root. The step is now measured against |value| below 1 and is unchanged from 1 up, so this can only drop a point, never add one.
• A point where the equation is undefined at exact arguments is not a root. x^i + x = 0 would otherwise answer 0, where 0^i has no value. A decimal argument never counts as undefined, so a root found in decimals still survives rounding and overflow. Across 2184 existing tests, the new rule fired only in its own test.

With principal powers, the roots of a^x + s·b^x = 0 are x = (ln(−s) + 2πik)/(ln a − ln b). Every root below lies on that family, within 2e-15.

equation = 0	2.6.0 and master	this branch
(-2)^x + 4^x	stack overflow	9 roots, k = −7..1
(-3)^x + 9^x	stack overflow	7 roots, k = −5..1
(-2)^x - 4^x	stack overflow	8 roots, k = −6..1, including 0
(-2)^x + 2^x	stack overflow	{ -5, -3, -1, 1, 3 }
x^i + x	stack overflow	{ -e^(π/2) i, e^(-π/2) i }
x^i + x^(2*i)	stack overflow	{ e^π }
x^(1+i) + x	stack overflow	0, e^−5π, e^−3π, e^−π, e^π (0^(1+i) is 0, so 0 is a root)

Each takes 290–470 ms. The answers are the roots inside Newton's search region, not the whole infinite family: that is #2319's question, and its rates have no common real unit here. The rows are in BREAKING-CHANGES.md.

Checked on this branch:

• Tests: UnitTests 19363 passed, plus the MCP, Terminal, InteractiveWrapper and F# wrapper suites, and the AOT smoke test. The new test runs each solve under a termination guard and asserts that every root returned has a relative residual of at most 1e-9 and lies on the family. An unsolved answer is accepted; an empty one is not.
• Harnesses: BoundCheck, RootCheck, CasBench, PropCheck, SimpSweep, RuleCheck, CanonCheck and Confluence all pass.

🤖 Generated with Claude Code

[codecov]

Codecov Report

❌ Patch coverage is 94.11765% with 1 line in your changes missing coverage. Please review.
✅ Project coverage is 86.81%. Comparing base (2eea2e9) to head (ce6aa87).
⚠️ Report is 1 commits behind head on master.

Files with missing lines	Patch %	Lines
...ontinuous/Solvers/EquationSolver/SolveStatement.cs	87.50%	0 Missing and 1 partial ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #2432      +/-   ##
==========================================
- Coverage   86.81%   86.81%   -0.01%     
==========================================
  Files         312      312              
  Lines       69551    69564      +13     
  Branches    20316    20322       +6     
==========================================
+ Hits        60383    60392       +9     
- Misses       4932     4938       +6     
+ Partials     4236     4234       -2     

Flag	Coverage Δ
cpp	37.77% <ø> (ø)
csharp	87.35% <94.11%> (-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
...inuous/Solvers/EquationSolver/ExponentialSolver.cs	86.60% <100.00%> (+0.24%)	⬆️
...ontinuous/Solvers/NumericalSolving/NewtonSolver.cs	96.06% <100.00%> (+0.09%)	⬆️
...ontinuous/Solvers/EquationSolver/SolveStatement.cs	83.85% <87.50%> (+0.13%)	⬆️

... and 7 files with indirect coverage changes

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