[ToikY]

Change and mathematical scope

Prove the exact inverse 1-norm envelope for complex n-by-n matrices with absolute row cap rho and column cap beta*rho, for every n>=1, real beta>=1 and 0<rho<1. One interval-overlap matrix attains all partial-sum bounds simultaneously. The new recurrence handles split rows and its current-index term near beta=1. At n=3, rho=1/2, beta=5/4 the exact maximum is 38/15.

This answers Nick Petalas's fractional-capacity question in #6788. His integer theorem and transport method retain their credit. Only research/fractional-column-resolvent is added. No actual Gaussian caller, physical invoice, supplier, selected-assumption discharge or new kappa is claimed.

Task and overlap

Canonical main 7c49fe3 and priority revision 8 were refreshed at entry and before publication without drift. The full uniform-transfer-audit card is retained with explicitly changed scope: abstract conditioning. Its original actual-caller gate remains unmet.

The live hub/index, all 755 open PR bodies across eight pages, scoped helper and all-state indexed body searches were checked. #6788 remains at c1d9153; its exact proof/README are retained with Git-blob/SHA256 pins. Sparse local coverage omitted 8,881 notes; hidden work and literature-wide priority remain unknown. Canonical claim.

Validation

CPython 3.12.14, standard library only. Both registered normal and optimized commands pass with identical generated receipts:

• 3,117 exact grid inversions, 21,921 admitted cap cases, 64,731 columns and 192,291 partial-sum comparisons.
• 729 attaining matrices through dimension 128, including 216 dense inverses, 14,148 attained partial sums, 243 credited integer-formula comparisons and six new n=3 formula checks.
• 72 real sign-conjugate fixtures and 12 rejecting controls. The complex theorem rests on the written proof.

Both inverse-composition identities, interval intersections, a direct triangular solve and the recurrence are separate checks. Read-only Git-free normal/optimized exports with empty PATH/PYTHONPATH reproduce the registered receipts; all 14 export members stay unchanged. Four further source-corruption controls reject before mathematics. Committed bytes, modes, canonical source history, official export/input coverage and all 13 remote blobs pass; the public tree equals the tested tree.

Source-aware proof and code reviews found no blocker. Related roles/exposure are disclosed; AI review and finite/hosted execution are separate from human or proof-kernel acceptance. No predecessor scientific code ran. An initial wrapper rejected the README heading before science; the heading-only correction and original logs/commits are preserved.

Reproduce:

python3.12 -B scripts/research_package.py --package research/fractional-column-resolvent

Expected: PASS_FRACTIONAL_COLUMN_RESOLVENT and PASS_REGISTERED_RESEARCH_PACKAGE. Both original hosted checks passed on attempt 1 at c9f16c0 in run 42237389704: plan (711 helper tests, 24 skips) and the registered scientific replay (normal and optimized). CPython 3.12.16 hosted receipts match the complete local receipts exactly. The test checkout has the exact public tree and canonical/public parents; it is a synthetic PR test ref, not a repository merge.

Credit

Prepared for ToikY with substantial OpenAI Codex assistance across coordinating, verifier, source-aware mathematical review and metadata roles. Credit Nick Petalas (#6788), Ali Romman (#5255/#5306), Cinna Davis (#4100), James Reed (#4282), ToikY (#4209), classical Neumann/transport methods and the retained conditioning/support-envelope/Schur lineage. Original notices and Apache-2.0 license are preserved.

Open next questions

Bind both caps to an actual emitted Gaussian matrix/caller and charge all preparation/restoration costs. Heterogeneous capacities and prescribed supports are different problems. All five selected assumptions remain; no further search is reserved.

Proof, exact sources, verifier and evidence boundaries.