Prevent recursive solver factor cycles

[ChrisRackauckas-Claude]

Please ignore this PR until it has been reviewed by @ChrisRackauckas.

What changed and why

solve_univar treated a single Nemo factor as new work whenever the factor was not structurally equal to the input polynomial. Nemo returns a separate factorization unit, so an irreducible polynomial can be represented as u = -1 plus the negated polynomial; recursively factoring that result can alternate between p and -p until a stack overflow.

This PR reconstructs the single-factor result with Nemo's unit before deciding that factorization made progress. It adds a public-API regression for an irreducible degree-5 symbolic polynomial and verifies that symbolic_solve returns RootsOf rather than recursing.

Failing before

I ran the new test against unfixed master with a temporary recursion-depth diagnostic guard (not included in this PR), which turns the otherwise unbounded recursion into a prompt error:

$ julia +1.12 --startup-file=no --project=../solver-instrument-env -e 'using Test, Symbolics; using Groebner, Nemo; @variables x a b c d; @testset "Higher degree univar" begin; expr=-b+a*x+c*x-d*x^2+b*x^3-a*x^4-x^5; root=only(symbolic_solve(expr,x)); @test Symbolics.operation(root)===Symbolics.RootsOf; @test isequal(Symbolics.arguments(root),[Symbolics.unwrap(expr),Symbolics.unwrap(x)]); end'
Higher degree univar: Error During Test
  solve_univar recursion at depth 51 for b - a*x - c*x + d*(x^2) - b*(x^3) + a*(x^4) + x^5
  solve_univar(...) (repeats 51 times)
Test Summary:        | Error  Total     Time
Higher degree univar |     1      1  1m10.5s
ERROR: Some tests did not pass: 0 passed, 0 failed, 1 errored, 0 broken.

The factor trace on unfixed code alternates deterministically:

1 u=-1//1 next=b - a*x - c*x + d*(x^2) - b*(x^3) + a*(x^4) + x^5 equal=false reconstructed=true
2 u=-1//1 next=-b + a*x + c*x - d*(x^2) + b*(x^3) - a*(x^4) - (x^5) equal=false reconstructed=true
3 u=-1//1 next=b - a*x - c*x + d*(x^2) - b*(x^3) + a*(x^4) + x^5 equal=false reconstructed=true

Passing after

The identical test against this branch passes:

$ julia +1.12 --startup-file=no --project=../solver-stack-env -e 'using Test, Symbolics; using Groebner, Nemo; @variables x a b c d; @testset "Higher degree univar" begin; expr=-b+a*x+c*x-d*x^2+b*x^3-a*x^4-x^5; root=only(symbolic_solve(expr,x)); @test Symbolics.operation(root)===Symbolics.RootsOf; @test isequal(Symbolics.arguments(root),[Symbolics.unwrap(expr),Symbolics.unwrap(x)]); end'
Test Summary:        | Pass  Total     Time
Higher degree univar |    2      2  1m08.2s

The complete solver file passes, including the regression:

$ julia +1.12 --startup-file=no --project=../solver-stack-env test/solver.jl
Test Summary:        | Pass  Total  Time
Higher degree univar |    2      2  1.2s
Test Summary:         | Pass  Broken  Total   Time
Isolate/Attract solve |   32       1     33  29.0s

The full Core group reached 17,087 passes. The sole error is the independently reproduced PreallocationTools.DiffCache.any_du failure already fixed in the separate draft PR linked below; the RootFinding solver group is clean.

$ GROUP=Core julia +1.12 --startup-file=no --project=../solver-stack-env -e 'using Pkg; Pkg.test("Symbolics")'
Test Summary:                            |  Pass  Error  Broken  Total      Time
test set                                 | 17087      1     357  17445  16m54.5s
  Nested ForwardDiff Sparsity Test       |            1              1      2.4s
  RootFinding solver                     |   169              5    174   4m52.0s

Additional local checks:

$ GROUP=QA julia +1.12 --startup-file=no --project=../solver-stack-env -e 'using Pkg; Pkg.test("Symbolics")'
Testing Symbolics tests passed

$ julia +1.12 --startup-file=no --project=../formatter-env -m Runic --check src/solver/main.jl test/solver.jl
$ typos src/solver/main.jl test/solver.jl
$ git diff --check

The last three commands exited successfully without output.

Regression boundary

The recursive single-factor condition was introduced by the Symbolics commit linked below. Its parent returned the reduced polynomial as RootsOf in 13.912763 seconds. The dependency behavior that exposes the latent bug begins at the tested Nemo version boundary: Nemo 0.47.5 terminates, while 0.48.0, 0.48.1, 0.49.5, 0.52.4, 0.56.0, and 0.56.1 return alternating unit/factor representations for this polynomial. This is valid factorization behavior; consuming Nemo's unit in Symbolics is the scoped fix.

Not verified

Docs were not built because this changes no docstring, rendered documentation, or public API. GPU, downstream, and allowed-to-fail jobs were not run locally. No dependency or license changes are included.

Reviewer focus: is_unfactored reconstructs only a single factor with Nemo's unit for the progress check. Multi-factor root handling is unchanged.

Links

• Original solver task context: latexify of a Vector{Equation} with whole-array equations fails with "cannot latexify objects of type BasicSymbolic" #1961
• PreallocationTools base-failure fix: Fix symbolic cache dispatch with PreallocationTools 1.5+ #1964
• Commit that introduced recursive single-factor handling: a94771b
• First tested Nemo version exposing the alternating factor form: https://github.com/Nemocas/Nemo.jl/releases/tag/v0.48.0

🤖 Generated with Claude Code
https://chatgpt.com/codex/tasks/01a03a04-70ae-77a0-ba1b-ec7a1ed9ef47

[ChrisRackauckas-Claude]

Investigation and verification checklist:

• Reduce the recursive factor cycle to a public symbolic_solve reproducer.
• Capture failing-before evidence with a temporary diagnostic depth guard.
• Consume Nemo's factorization unit in the factorization progress check.
• Run the identical regression after the fix (2/2 pass).
• Run test/solver.jl, the full Core group, QA, Runic, typos, and git diff --check.
• Identify the Symbolics source boundary and tested Nemo behavior boundary.
• Await review by @ChrisRackauckas and CI results.

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Codecov Report

❌ Patch coverage is 0% with 3 lines in your changes missing coverage. Please review.
✅ Project coverage is 19.29%. Comparing base (acacfd2) to head (2297339).
⚠️ Report is 3 commits behind head on master.

Files with missing lines	Patch %	Lines
src/solver/main.jl	0.00%	3 Missing ⚠️
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Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1966      +/-   ##
==========================================
- Coverage   23.42%   19.29%   -4.14%     
==========================================
  Files          61       61              
  Lines        5997     5987      -10     
==========================================
- Hits         1405     1155     -250     
- Misses       4592     4832     +240     

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