[pinterbanget]

Summary

This adds research/gaussian-section-inverse-norms/. It answers the open question of #4319 (jwr404): what is the induced 1-norm of the inverse of the normalized p = 19 Gaussian Schur section on survivors 0, 1, 2, 7, 8, 9? It also rechecks #4319's infinity-norm bracket by a different route.

Claim type: a certified finite computation on one section. No new κ. This is not an all-size inverse bound.

The 1-norm matters because #4100's INVERSE-CONDITIONING.md bounds the 2-norm by √(‖·‖₁ ‖·‖_∞). Its 1-norm ceiling 1/δ + 2bρ/δ² is the large factor. At #4317's tight gap it is 3938.

Result

Every value is certified with exact rationals, to within 7·10⁻⁸⁷.

Norm of N⁻¹	Value	Ceiling (#4317)
infinity, attained at row 0	3.600635404331…	27
1, attained at column 2	2.813821688397…	3938
2	in [1.6265, 3.1830]	321

• Direct elimination puts the inverse infinity norm strictly between 18/5 and 361/100 (no new κ) #4319's bracket holds: 18/5 < ‖N⁻¹‖_∞ < 361/100.
• The 1-norm is about 1/1400 of the lemma's ceiling. It is also below the infinity norm.
• The same holds on the unsectioned 10-vertex path: ‖A⁻¹‖₁ = 4.3868, against ‖A⁻¹‖_∞ = 6.0623.

Control. The principal restriction A_KK, which skips the interior elimination and so the filled 2→7 entry, has inverse infinity norm below 3. It misses #4319's bracket.

Independence

The verifier imports nothing from #4319.

• Entries. It builds the entries from GAUGEFREE-INTERIORS.md's original form, exp(−a[(ρh + β_i)² − β_{i+h}²]), and checks exactly that this equals the gauge form Direct elimination puts the inverse infinity norm strictly between 18/5 and 361/100 (no new κ) #4319 uses.
• No Schur complement. By the quotient identity, N⁻¹ = (A⁻¹)_KK · diag(S).
• Error control. It inverts the rational midpoint exactly and covers the entry radii with one perturbation bound, in place of interval elimination.
• Enclosures. π (Machin) and exp (halving, Taylor and outward squaring) are its own.

Evidence

The verifier uses only the standard library and runs in about 6 seconds. The three notes it reads are on main, pinned by SHA256, with phrase checks.

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

Head: 2d2ea7bd2ae310a50cf29d0a6fe989b4336b891b.

Not covered

• Other sections, primes and bandwidths.
• GAUSSIAN.md's all-size row-gap claim.
• Whether Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100's callers need this section's true 2-norm or only the lemma's ceiling.

Task and overlap

• Card: uniform-transfer-audit (precision/conditioning obligation of Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100), refined, priority revision 7.
• Base: canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244.
• No PR or discussion had taken Direct elimination puts the inverse infinity norm strictly between 18/5 and 361/100 (no new κ) #4319's open question.

Credit

The section, the bracket and the open question are jwr's (#4319 and the #4305–#4318 chain). The kernel and the lemma are #4100's (GAUSSIAN.md, INVERSE-CONDITIONING.md and the vendored GAUGEFREE-INTERIORS.md). Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance. Apache-2.0.

🤖 Generated with Claude Code