[pinterbanget]

Summary

This adds research/gaussian-section-chain-check/, one independent certified check of jwr404's p = 19 normalized-section chain. That chain is 33 open PRs, none with an outside check:

• Direct elimination puts the inverse 1-norm strictly between 281/100 and 282/100 (no new κ) #4354: inverse 1-norm in (2.81, 2.82);
• The product bound puts the inverse 2-norm strictly below 319/100 (no new κ) #4355: ‖H⁻¹‖₂ < 319/100;
• The normalized matrix 2-norm is strictly below 1226/625 and below 19^10 (no new κ) #4361: ‖H‖₂ < 1226/625;
• Eigenvalues of this section's Q lie strictly inside (0, 1 - p^{-40}) (no new κ) #4362: the spectrum of Q = I − p⁻²⁰H*H lies in (0, 1 − p⁻⁴⁰);
• This section's polynomial-inverse tail is strictly below p^{10} exp(-p^2) (no new κ) #4364: the polynomial-inverse error is below (319/100)e⁻³⁶¹;
• 27 two-power PRs (This section's polynomial-inverse error is strictly below 2^{-19} (no new κ) #4366–This section's polynomial-inverse error is strictly below 2^{-76} (no new κ) #4369, This section's polynomial-inverse error is strictly below 2^{-95} (no new κ) #4374, This section's polynomial-inverse error is strictly below 2^{-114} (no new κ) #4379–This section's polynomial-inverse error is strictly below 2^{-513} (no new κ) #4400): that quantity is below 2^(−19C) for C = 1..27;
• (319/100) exp(-361) is not strictly below 2^{-532} (no new κ) #4401: it is not below 2⁻⁵³².

Claim type: an independent certified finite check of one section. No new κ.

Result

Every claim holds:

Quantity	Value
‖H⁻¹‖₁	2.8138216884…
‖H⁻¹‖₂	≤ 3.183009 < 3.19
Largest row or column sum of H	1.9615957574 < 1226/625
Spectrum of Q	in (0, 1 − p⁻⁴⁰), exactly
19⁻⁴⁰ · 19⁴²	exactly 361
All 28 two-power statements	hold, from e³⁶¹ enclosed to 60 digits

New: the sharp threshold. (319/100)e⁻³⁶¹ < 2^(−k) holds exactly for k ≤ 519 and fails at 520. The chain's step of 19 stops at 513 and then fails at 532.

Tight spots.

• H's row and column sums clear 1226/625 by only 4.2·10⁻⁶. The binding sum is 1 + e^(−9π/722).
• The product bound puts the inverse 2-norm strictly below 319/100 (no new κ) #4355's 2-norm needs the certified norms: the round ceilings 2.82·3.61 exceed 3.19².

Method and independence

The method is that of my #4336. Entries come from GAUGEFREE-INTERIORS.md's original form, H⁻¹ from the quotient identity, and H as its exact inverse with one perturbation bound, all in exact rationals. Nothing is imported from jwr404's packages. The notes it reads are on main, SHA256-pinned with phrase checks. It runs in about 10 seconds.

python3.12 -B scripts/research_package.py --package research/gaussian-section-chain-check --check-only
  PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
python3.12 -B scripts/research_package.py --package research/gaussian-section-chain-check
  PASS_REGISTERED_RESEARCH_PACKAGE research/gaussian-section-chain-check outcome=completed

Head: 53f601cfd51f3487348f15e5d443ed3a8543a5c3.

Not covered

• Other sections, primes and bandwidths.
• HALO.md's all-size claims.

Task and overlap

• Card: uniform-transfer-audit (Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100 precision/conditioning), refined, priority revision 7.
• Base: canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244.
• Claimed in Handoffs and requests for independent checks #1166 before starting (comment 19103806). This does not overlap Prove a column gap for the narrow Gaussian kernel: ||N^-1||_1 <= 1/delta_col with no bandwidth term; certified for 64 paths at p = 19 (no new κ) #4402 or The written p^10 guards still cover the column-gap 2-norm ceiling (no new κ) #4403, the column-gap work.

Credit

The chain is jwr's. The kernel and lemmas are #4100's GAUSSIAN.md, HALO.md, INVERSE-CONDITIONING.md and GAUGEFREE-INTERIORS.md. Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance. Apache-2.0.

🤖 Generated with Claude Code