[pinterbanget]

Summary

This adds research/gaussian-column-gap/. It answers open question 1 of #4336: does the inverse 1-norm of #4100's narrow Gaussian matrix stay below the infinity norm on longer paths? #4336 noted that a proof needs a column analogue of the row gap. This package supplies one, for a single constant-phase run.

Claim type: a written lemma inside one phase run, plus a certified finite table at p = 19. No new κ. This is not the full transfer to GAUSSIAN.md's global rounded reference H.

Lemma

Inside a phase run, β_i = β_j − θh turns GAUGEFREE-INTERIORS.md's original form into N_{j−h,j} = exp(−a(h² + 2β_j h)). Column j therefore depends only on β_j. Bounding which rows exist then gives

‖N − I‖₁ ≤ e^{−2aθ} + e^{−2a(1−θ)} + 2e^{−2a}/(1 − e^{−4a}) = 1 − δ_col.

That has the same leading term as GAUSSIAN.md's row bound e^{−2aθ} + 5e^{−2a}. Consequences:

• ‖N⁻¹‖₁ ≤ 1/δ_col, with no bandwidth term;
• the same bound for principal sections and for unnormalized Schur inverses, by restriction;
• the 2-norm bound becomes 1/√(δ δ_col), in place of √((1/δ)(1/δ + 2bρ/δ²)).

Certified table, p = 19 (#4319's parameters)

• The gap: δ_col ≥ 0.0384042426, so 1/δ_col = 26.04. At the same gap, INVERSE-CONDITIONING.md's 1-norm ceiling is 2633.96.
• Coverage: 64 placements, every n = 10..40 and n = 100, each at both ends of a phase run.
• Every placement:
  • ‖A⁻¹‖₁ ≤ 1/δ_col, with the largest ‖A⁻¹‖₁ δ_col at 0.1685;
  • ‖A⁻¹‖₁ < ‖A⁻¹‖_∞;
  • on the normalized 3+3 Schur section, the 1-norm is below the infinity norm.
• Saturation: the 1-norm levels off at 4.3868 at the start of the run and 4.0010 at its end.
• Agreement with Certify the inverse 1-norm of #4319's normalized Gaussian Schur section: 2.8138, against the lemma's 3938 ceiling (no new κ) #4336: n = 10 reproduces its section values, 3.600635 and 2.813822.

Method. π and exp have their own rational enclosures. A float64 midpoint inverse is certified by exact rational left and right residuals. The sections use an exact 6×6 inverse plus a perturbation bound. The certified error is below 10⁻⁹. A control checks that the column form fails across a phase cut.

Evidence

python3.12 -B scripts/research_package.py --package research/gaussian-column-gap --check-only
  PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
python3.12 -B scripts/research_package.py --package research/gaussian-column-gap
  PASS_REGISTERED_RESEARCH_PACKAGE research/gaussian-column-gap outcome=completed   (~2.5 min, stdlib only)

Head: 477c3ae2f35f12c800bd8cb1bbef6388182d3839. Base: canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244, priority revision 7, selected record unchanged.

Not covered

• Cross-cut entries, where β_{i+h} wraps.
• The column-side truncation and rounding transfer to H. This is plausible by GAUSSIAN.md's row argument but not written here.
• The normalized-section scaling S_jj in general.
• Other primes and bandwidths in the table.
• Whether any Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100 caller's guard moves.

Task and overlap

• Card: uniform-transfer-audit (the precision/conditioning obligation), refined.
• Claim: Handoffs and requests for independent checks #1166 (comment)
• Overlap: no PR or discussion had taken Certify the inverse 1-norm of #4319's normalized Gaussian Schur section: 2.8138, against the lemma's 3938 ceiling (no new κ) #4336's question. jwr404's Direct elimination puts the inverse 1-norm strictly between 281/100 and 282/100 (no new κ) #4354/The normalized matrix 2-norm is strictly below 1226/625 and below 19^10 (no new κ) #4361 chain stays on the single 10-vertex section.

Credit

• The kernel, the row form and the conditioning lemma are Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100's, building on PR1673 (Vlado Vojdanovski).
• The section and its parameters are jwr's (Direct elimination puts the inverse infinity norm strictly between 18/5 and 361/100 (no new κ) #4319 and its chain).

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

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