[Rafael-SOWNet]

Fixes #2137, and the first row of #2133, which is the same shape.

The defect: Simplify gathers fractions a pair or a sort level at a time, and left some sums apart that are a constant over one denominator. Transformation.RationalCanonicalization decides every row of #2137, and Simplify never asked it.

The change: the canonical form is offered as one more candidate, last, as the Boolean minimiser's form is. It is offered only where:

• the expression is a rational function with a quotient in it,
• its complexity is at most 64,
• the integrator is not running.

The complexity metric decides, and a tie keeps the form the rules reached. The candidate is written with whole coefficients, 2/(4x^2 + 1) and not (1/2)/(x^2 + 1/4), through a flag on TryCanonicalize. The transformation keeps its monic form.

Simplify	2.6.0	this PR
x/(x+y) + y/(x+y)	unchanged	1 provided not x + y = 0
1/(1 - (b/a)^2) - a^2/(a^2 - b^2)	unchanged	0 provided not a ^ 3 - a * b ^ 2 = 0
x/c + (y*c - x*d)/(c^2*(z + d/c)) - (x*z + y)/(c*z + d)	unchanged	0 provided not c * d + c ^ 2 * z = 0
R^2 - R^2*(a+b)*(1/a+1/b)/4 - R^2*(a-b)*(1/a-1/b)/4 (#2133)	R ^ 2 + -1/2 * (R ^ 2 + R ^ 2) provided …	0 provided not a * b = 0

Each condition is the input's domain, written canonically: square-free, and without what the answer's own denominator already excludes. Three tests pinned a longer condition over the same points, (b - 1)(1 - b) for b - 1, and a b c x for a b c beside a quotient over x. Their expectations are updated with a comment saying why.

Why each restriction is there, measured against master 8a85f8ccf:

• Last: offered first, the candidate won ties. That changed forms across the suite, and an acoth integral lost its answer. Offered last, both are back.
• A walk that allocates nothing: asked through Nodes, the check cost SimplifyEasy 3.4% more allocation and failed the performance gate. Now it is byte-identical, 80,350 B on both.
• Not inside the integrator: there, 1/(b cosh x + c sinh x + sqrt(b^2 - c^2))^4 took 4.8 s to a 737-character answer, against master's 1.1 s to 427. The canonicalization itself cost 2.9 ms in all. What cost the time was the route the gathered quotient sent the integrator down. With the integrator excluded, that integral is identical to master.

Checked:

• The intbench chain: identical verdicts in all nine families, 0 moves, allocation within 0.05%.
• Performance gate: passed on all 19 gated benchmarks.
• Unit suite: 17,147 passed, 0 failed.
• SimpSweep 10,463/10,463; CanonCheck matches its baseline; BoundCheck 0 disagreements; CasBench 120/121, unchanged; RootCheck 596/596.
• SumOfFractionsOverOneDenominatorTest: 7 rows. Each checks the value, and that the answer is defined where the input is.
• BREAKING-CHANGES entry added, measured on the 2.6.0 package.

Cost where it applies: a sum of fractions that is a rational function pays for the canonicalization even when its form loses. 1/a + 1/b + 1/c + 1/d allocates 952 KB against 743 KB, and x/(x+y) + y/(x+y) 1.80 MB against 1.72 MB. None of the gated benchmarks is such an expression, which is why the gate does not see it.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 86.43%. Comparing base (8a85f8c) to head (ce80455).
⚠️ Report is 1 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #2174      +/-   ##
==========================================
- Coverage   86.43%   86.43%   -0.01%     
==========================================
  Files         295      295              
  Lines       64881    64905      +24     
  Branches    19084    19092       +8     
==========================================
+ Hits        56081    56100      +19     
- Misses       4770     4777       +7     
+ Partials     4030     4028       -2     

Flag	Coverage Δ
cpp	37.77% <ø> (ø)
csharp	87.00% <100.00%> (-0.01%)	⬇️
fsharp	32.97% <ø> (ø)
interactive	40.72% <ø> (ø)

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

Files with missing lines	Coverage Δ
.../Functions/Algebra/Polynomials/RationalFunction.cs	73.33% <100.00%> (+3.21%)	⬆️
...s/Continuous/Integration/Integration.Definition.cs	90.59% <100.00%> (+0.01%)	⬆️
...ouriMath/Functions/Simplification/Simplificator.cs	96.27% <100.00%> (+0.18%)	⬆️

... and 8 files with indirect coverage changes

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