Fix false-positive in check_primality on degenerate random specialization

[ChrisRackauckas-Claude]

Ignore until reviewed by @ChrisRackauckas.

Problem

The master tests / Core (julia pre, ubuntu-latest) lane is red. The only failure is test/bodies/io_projections.jl:55:

IO-projections (+ extra projection): Test Failed at .../test/bodies/io_projections.jl:55
  Expression: !(check_primality(proj))
   Evaluated: !(check_primality(Dict(...)))   # check_primality returned true

The proj ideal for the issue-#132 PK/PD model is not prime, so the test asserts !check_primality(proj). The function returned true, failing the test. This is the prerelease (Julia 1.13.0-rc1) lane only; 1.10/1.11/1.12 pass.

Root cause (a real soundness bug, not version-specific)

check_primality(polys, extra_relations) specializes every non-leader variable of each generator to a random integer (rand(1:100)), then asks whether the characteristic polynomial of a generic multiplication operator on the resulting zero-dimensional quotient is irreducible.

Each generator here is degree 2 in its leader (y1(t)_2, y2(t)_3), with a non-constant leading coefficient in the other variables. When the random point makes a leading coefficient vanish, the generator drops to degree 1. When all generators collapse to degree 1, the quotient ring becomes 1-dimensional, its charpoly is degree 1 (trivially irreducible), and the routine reports the (non-prime) ideal as prime — a false positive.

Reproduced on 1.13.0-rc1: over seeds 0..499, check_primality(proj) returns true for 6 of them (e.g. 30, 169, 305, 393, 488, 496). The 1.13 RNG stream change merely caused the un-seeded global RNG to land on such a draw by the time the test reached line 55; the bug is RNG-dependent and present on every version.

For a bad seed the Groebner basis specializes to [y1(t)_2 - c1, y2(t)_3 - c2] (quotient dim 1, irreducible degree-1 charpoly → true); for a good seed it is [y1(t)_2^2 + ..., y2(t)_3^2 + ...] (quotient dim 4, charpoly factors → false).

Fix

Only accept random specializations that preserve each generator's degree in its leader — i.e. stay on the open set where the leading coefficients do not vanish (precisely the set the ideal is "saturated at the leading coefficient", per the docstring). Resample otherwise, with a bounded retry. extra_relations are evaluated at the same point, unchanged behavior.

Verification (run locally)

Julia 1.13.0-rc1:

• false-positive rate over seeds 0..499: 6/500 → 0/500
• io_projections body: 10/10 (was 9 pass / 1 fail)
• positive case check_primality(proj, [projection_poly]): 50/50 true (unchanged)
• negative case stress over seeds 0..299: 0/300 true (correct)

Julia 1.12 (no regression):

• io_projections body: 10/10
• check_primality_zerodim body: 5/5 (subroutine untouched)

🤖 Generated with Claude Code

[pogudingleb]

Thanks!
I added some cleanup in a separate PR (#526 ) as could not edit this one.