Motivation
PR #84 strengthens the exact finite-cell negative result substantially: axis L=2,3,4,5 matching polynomials are irreducible over Q, have degrees 4,9,16,25=L^2, and pairwise gcd 1. That rules out a hidden bounded-degree factor carrying the physical root.
The next exact algebraic question is whether these polynomials also have large Galois groups. If the degree-9/16/25 examples have S_n (or comparably large) Galois groups, the finite-cell physical roots are not merely high degree; they also lack a simple radicals/fixed-solvable-group explanation.
This would still say nothing about transcendence or algebraicity of the infinite-volume p_c, but it would sharply constrain a common finite-polynomial closed-form mechanism.
Program
For each exact axis polynomial L=2..5:
- Compute the integer discriminant or at least certify whether it is a square.
- Factor modulo many good primes
q not dividing the discriminant/leading coefficient.
- Record factor-degree partitions; by Dedekind/Frobenius these give cycle types present in the Galois group.
- Combine transitivity (already from irreducibility) with certified cycle types and standard finite permutation-group criteria to prove
S_n, A_n, or the strongest justified lower bound on the group.
- Keep the proof certificate independent of a CAS black box where practical: modular factorization records + explicit group-theory criterion.
Useful sufficient patterns
Do not hard-code one theorem without checking hypotheses, but look for combinations such as:
- an odd permutation (nonsquare discriminant or suitable factorization cycle type), excluding
A_n;
- a long prime cycle /
(n-1)-cycle plus transitivity/primitivity;
- a transposition or cycle structure that, together with a long cycle, forces
S_n.
For degree 25 in particular, modular factorizations across modest primes should give many Frobenius cycle types cheaply.
Outputs
scripts/certify_axis_matching_galois.py;- certificate JSON listing primes, exact modular factor-degree partitions and the group-theory deductions;
- regression tests for every arithmetic certificate;
- concise statement of what is proved for each L.
Interpretation boundary
A full S_25 result would imply the finite L=5 physical root is not solvable by radicals, but must not be presented as evidence that the infinite threshold is transcendental or not expressible by special functions.
Gate
This is a low-compute exact side route. It should not consume the server campaign reserved for #43/#57.
Motivation
PR #84 strengthens the exact finite-cell negative result substantially: axis
L=2,3,4,5matching polynomials are irreducible overQ, have degrees4,9,16,25=L^2, and pairwise gcd 1. That rules out a hidden bounded-degree factor carrying the physical root.The next exact algebraic question is whether these polynomials also have large Galois groups. If the degree-9/16/25 examples have
S_n(or comparably large) Galois groups, the finite-cell physical roots are not merely high degree; they also lack a simple radicals/fixed-solvable-group explanation.This would still say nothing about transcendence or algebraicity of the infinite-volume
p_c, but it would sharply constrain a common finite-polynomial closed-form mechanism.Program
For each exact axis polynomial
L=2..5:qnot dividing the discriminant/leading coefficient.S_n,A_n, or the strongest justified lower bound on the group.Useful sufficient patterns
Do not hard-code one theorem without checking hypotheses, but look for combinations such as:
A_n;(n-1)-cycle plus transitivity/primitivity;S_n.For degree 25 in particular, modular factorizations across modest primes should give many Frobenius cycle types cheaply.
Outputs
scripts/certify_axis_matching_galois.py;Interpretation boundary
A full
S_25result would imply the finiteL=5physical root is not solvable by radicals, but must not be presented as evidence that the infinite threshold is transcendental or not expressible by special functions.Gate
This is a low-compute exact side route. It should not consume the server campaign reserved for #43/#57.