[Rafael-SOWNet]

Closes #2428.

Four rewrite rules were wrong as written, each the same on 2.6.0. Each fix is its own commit, and each keeps the hand-written switch copy of the rule in Patterns.*.cs in step with the data rule, as MatchedRulesAgreeWithTheSwitchTest requires.

1. a^n / b^(c n) → (a / b^c)^n, and its mirror. Wrong through Simplify.
   • x^n/y^(2*n) became (x / y^2)^n: at x = 1, y = −1, n = 1/2 the input is −1 and the output 1.
   • x^(2n)/y^n → (x^2/y)^n fails the same way at x = −1, y = 1.
   • Fix: the guard their sibling a^n/b^n → (a/b)^n carries (a^n / b^n is gathered into (a/b)^n unconditionally, which is wrong for bases of opposite sign #802): n whole, or both bases positive reals. Gathering numeric bases still works: 2^n/3^(2n) → (2/9)^n.
2. (a > b and b > c) implies a > c → True. Wrong through Simplify: at a = i, b = 1, c = 0 the input is NaN.
   • Attaching a in RR, as p or not p depends on which side the not is written, and reduces to True where the proposition is undefined #876 does for x > x, would narrow instead: at a = i, b = 0, c = 1 the input is True, because NaN and False is False.
   • Fix: transitivity is decided only where every operand is known to be ordered (the real reading, or a known-real operand); otherwise the input is left as written.
   • Two BooleanSimplify.TestInequality rows that pinned x > y > z → x > z for symbols on the complex plane move to a theory over the reals, where they still hold.
3. arcsin(n/d) → arccsc(d/n) at n = 0. Reachable through Transformation.Rewriting(RewriteRules.Trigonometric): arcsin(0/x) → arccsc(x/0) turns 0 into NaN, and arccos(0/x) → arcsec(x/0) likewise. Simplify folds 0/x first and is not affected.
   • Fix: a nonzero-numerator guard on all four rules.
4. sgn(a) · a / |a| cancels. Reachable through Transformation.Rewriting(RewriteRules.Common): sgn(x)*(y*x)/abs(x) → y for non-real x, where at x = i the input is −y. sgn(a)·a/|a| is a^2/|a|^2, which is 1 only for real a. Simplify is not affected, because the canonical order flattens the product first.
   • Fix: cancel only where the operand is ordered.

soundcheck on master: 12 widen, 1 narrows, 80 none found. On this branch: 12 widen, 0 narrows, 81 none found. The arcsine rule moved from narrows to none found; the 12 widenings are other rules, deferred to #1019 §14. Each changed answer has a BREAKING-CHANGES.md row measured on the 2.6.0 package, each under its own heading.

Checked on this branch:

• Tests: UnitTests 19381 passed, plus the MCP, Terminal, InteractiveWrapper and F# wrapper suites, and the AOT smoke test. Each fix has its own test.
• Harnesses: BoundCheck, RootCheck, CasBench, PropCheck, SimpSweep, RuleCheck, CanonCheck and Confluence all pass, with reports identical to master's.
• limitcheck at DEPTH=3, COUNT=600, on 42 seeds against master: no row changed.

🤖 Generated with Claude Code

[codecov]

Codecov Report

❌ Patch coverage is 89.28571% with 6 lines in your changes missing coverage. Please review.
✅ Project coverage is 86.86%. Comparing base (2eea2e9) to head (6178967).
⚠️ Report is 2 commits behind head on master.

Files with missing lines	Patch %	Lines
...Math/Core/Transformations/Matching/MatchedRules.cs	91.17%	0 Missing and 3 partials ⚠️
...lification/Patterns/Patterns.EqualityInequality.cs	33.33%	0 Missing and 2 partials ⚠️
...nctions/Simplification/Patterns/Patterns.Common.cs	50.00%	0 Missing and 1 partial ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #2434      +/-   ##
==========================================
+ Coverage   86.81%   86.86%   +0.04%     
==========================================
  Files         312      312              
  Lines       69551    69766     +215     
  Branches    20316    20387      +71     
==========================================
+ Hits        60383    60604     +221     
+ Misses       4932     4916      -16     
- Partials     4236     4246      +10     

Flag	Coverage Δ
cpp	37.77% <ø> (ø)
csharp	87.40% <89.28%> (+0.04%)	⬆️
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
...unctions/Simplification/Patterns/Patterns.Power.cs	92.09% <100.00%> (+0.20%)	⬆️
...s/Simplification/Patterns/Patterns.Trigonometry.cs	96.75% <100.00%> (ø)
...nctions/Simplification/Patterns/Patterns.Common.cs	91.27% <50.00%> (-0.34%)	⬇️
...lification/Patterns/Patterns.EqualityInequality.cs	87.81% <33.33%> (-0.96%)	⬇️
...Math/Core/Transformations/Matching/MatchedRules.cs	99.47% <91.17%> (-0.08%)	⬇️

... and 13 files with indirect coverage changes

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