[pinterbanget]

Summary

This adds research/free-identity-allp/, addressing open question 2 of Dugongue's #4534: does the noncommuting-stage controller identity hold for all p, or only for #4470's fixed p16 word?

Claim type: a reduction lemma plus exact finite evidence. No new κ.

Results

• Reduction lemma. If abelianization is injective on the monomials occurring in the controller word W (for instance, if all of them are strictly descending words), then the commutative identity L W_ab R = diag implies the free identity L W R = diag for the same scalar L and R.
• Evidence. Conditional κ = 0.00296 via query-thinned productive controllers (+10.18%) #4440's constructor on main emits controllers and their commutative boundaries:
  • for every p = 2…14 and every prefix length a (91 cases; 45 rerun in the verifier), those commutative boundaries diagonalize W in the free algebra ℚ⟨d_s⟩[q, q^{−1}], with single monomials matching the commutative diagonal;
  • W always has exactly 2p monomials, all strictly descending;
  • Conditional kappa = 0.00302015 (+0.6304% over #4447): joint helper boundaries #4470's actual p16 word has the same property.
• Not proved. An all-p proof of the descending invariant, which would give Frame-aware native construction lemmas and exact p16 certificates (no new κ) #4534's all-p theorem via the lemma.

Evidence

python3.12 -B scripts/research_package.py --package research/free-identity-allp --check-only
  PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
python3.12 -B scripts/research_package.py --package research/free-identity-allp
  PASS_REGISTERED_RESEARCH_PACKAGE research/free-identity-allp outcome=completed   (~1 min, stdlib only)

Head: 59a4595eeb90810c6c4b56749824e793d170eb85. Base: canonical main f9f85c1e.

Credit: Dugongue (#4534) and Cinna Davis (#4440, #4470). Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance.

🤖 Generated with Claude Code