[pinterbanget]

Summary

This adds research/forward-fine-theta-conditioning/. It answers the open question of Matarener's #4356: does FINE-PRIME.md §2's reuse of FORWARD-SMALL-A-AUDIT.md §1–2 stay polynomial at θ = 1/(4p²)? That bound is ‖K⁻¹‖ = O(α⁵ρ²θ⁻²e^{aρ/3}).

Claim type: a written re-derivation plus rigorous floating-point bounds on actual blocks. No new κ.

Results

• Yes, it stays polynomial. θ enters only through 1/(1−e^{−bρθ}) ≤ 2α²ρ/(πθ). So ‖K⁻¹‖ ≤ (4/π²)α⁵ρ²e^{3π/(4α²)}e^{aρ/3}θ⁻² for every θ in (0, 1], which is 16p⁴·poly at the fine-prime θ.
• Measured blocks. The actual blocks K_{k,i} = α⁻¹exp(−b₀(i−λk−c)²) were measured at p = 5, 7, 11, 13, 19 (α = 3), with three starting phases and every dyadic length up to a full cell. Rigorous ‖K⁻¹‖_∞ and ‖K⁻¹‖₁ bounds lie 3·10⁻⁵ to 7·10⁻⁷ times the ceiling. At p = 19 the bound is 6634, against a ceiling of 9.1·10⁹.
• Scaling. ‖K⁻¹‖₂·√θ stays between 107.5 and 109.3 for θ from 1/100 to 1/3200. The true dependence is about θ^{−1/2}, not θ⁻²; that is O(p) rather than O(p⁴) at the fine-prime θ. This is an observation, not a proof.

Evidence

python3.12 -B scripts/research_package.py --package research/forward-fine-theta-conditioning --check-only
  PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
python3.12 -B scripts/research_package.py --package research/forward-fine-theta-conditioning
  PASS_REGISTERED_RESEARCH_PACKAGE research/forward-fine-theta-conditioning outcome=completed   (~20 s, numpy 1.26.4)

The residual bounds use the standard γ_{n+2} rounding bound and include entry errors. The 2-norm is a plain float value. Head: 09cf06535cbc26c3f2bc7b126e9141a9f04b2919. Base: canonical main 12fa8c5c, priority revision 7.

Not covered

• BILATERAL-LDU.md's base inequality.
• α ≥ 4, which first occurs at p ≥ 74.
• The narrow kernel.
• A proof of the θ^{−1/2} law, which is the README's open question 1.

Task and overlap

• Card: uniform-transfer-audit, refined.
• Claim: Handoffs and requests for independent checks #1166 (comment)
• Overlap: no other work had taken Audit #4100's fine-prime interval step: Trudgian bound holds at epsilon = 1 - 1e-30 (no new κ) #4356's question.

Credit

The bound is PR1673's FORWARD-SMALL-A-AUDIT.md. Its fine-prime reuse is #4100's FINE-PRIME.md. The question is @Matarener's (#4356).

Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance; no independent human review. Apache-2.0. This PR does not merge anything or change the frontier.

🤖 Generated with Claude Code