[petalas]

Summary

Prove the sharp inverse 1-norm envelope for every complex n by n matrix E with absolute row sums at most rho and column sums at most m rho, where 0<rho<1 and integer m>=2.

exact maximum ||(I-E)^-1||_1 = sum_(i=1)^n rho^ceil(log_m i) / (1-rho)
attainment: E_(i,ceil(i/m)) = rho, every other entry zero
critical case n=m^h, rho=1/m: exact maximum h+m/(m-1)

Claim type: scoped mathematical proof with exact finite corroboration. The bound covers diffuse rows and arbitrary complex phases. Sorted inverse-column partial sums supply the upper bound; one explicit nonnegative routing matrix attains it. No new kappa or selected-frontier change is claimed.

This takes the abstract column-budget portion of Ali Romman's #5255 open conditioning question. The predecessor's row-only theorem is a cited premise for comparison, not a replay deliverable. At row cap 1/2, adding column cap 1 changes the dimension-8 maximum from 9 to 5 and the dimension-64 maximum from 65 to 8. These compare different admitted matrix classes.

Task and overlap

Only six files in research/column-capped-resolvent/ are added. Base is 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8. TASK.json explicitly changes complex-rank-shape-decoder to this conditioning side result. Source definition, selected record and original card are pinned as provenance; there are no external runtime inputs.

Advance claim. Live claims, current/open account work, main README object/formula searches and merged searches were checked before commitment. The cached snapshot covered 734 open PRs and 6,839 Discussion records with complete remote pagination. Sparse local-note coverage was incomplete, supplemented by main-tree Git searches. Unpublished work remains outside coverage.

Evidence and reproduction

Local CPython 3.12.15, standard library only:

• All 6,144 admitted half-row-unit matrix evaluations through dimension three pass exact rational inversion and 54,072 partial-sum inequalities. The exact maximum is attained in all 18 parameter domains. Another 648 signed matrix evaluations exercise cancellation.
• All 2,240 attaining fixtures through dimension 64 pass 72,800 column equations and 72,800 transport equalities; 420 also have separately computed dense inverses. Sixteen critical closed forms, twelve rejected inputs and five contract/example controls pass.
• Both registered warning-strict normal and optimized commands pass with identical full generated receipts. Individually timed local checks take 3.94 and 3.99 seconds, below their 180-second limits. These are timeout observations, not a performance comparison.
• Registered replay, exact committed bytes, modes and source-history preflight pass at c1d915303450bec823784befd3ada5a261796a5c.
• Official export and input coverage pass. Both modes pass from a read-only Git-free export with empty PATH and PYTHONPATH. All seven exported files, including the manifest, stay unchanged; full receipts agree and their six package hashes match the committed blobs.
• In-memory compilation, AST function-size inspection and staged whitespace checks pass. The largest function is 34 lines; a separate static linter is unavailable.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/column-capped-resolvent/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/column-capped-resolvent/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/column-capped-resolvent
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/column-capped-resolvent --check-source-history --check-modes

Other hosts may substitute their Python 3.12 path. Expected markers are PASS_COLUMN_CAPPED_RESOLVENT, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The verifier writes no files and prints a generated receipt bound to all package hashes. The written proof establishes the unbounded theorem; finite enumeration corroborates it.

Limitations and unrun obligations

No actual emitted Gaussian matrix is admitted to these caps. Physical preparation, rounding, stock, native operations, ownership/restoration, tapes, compiler, analytic recovery and the complete supplier invoice remain separate. All five selected assumptions remain. Arbitrary noninteger column-to-row ratios, unequal row caps and prescribed supports are outside this theorem.

Every registered check ran locally. No predecessor scientific suite, selected cold replay, native suite, Lean or Docker build ran. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Determine the exact envelope for a noninteger column-to-row ratio. Begin with a three-vertex rational-cap transport problem allowing split rows; integer parent routing does not saturate fractional capacities.
• Bind both caps to one exact emitted Gaussian coefficient matrix and charge its preparation. That matrix and physical cost are the missing steps toward a precision improvement.

Related open work

• Sharpen a uniform finite-cap Gaussian inverse bound (no κ) #6459 sharpens a constant-shift Gaussian inverse spectral-norm bound. This gives an exact induced 1-norm maximum over an abstract row/column-cap class; no common numerical inverse norm is computed.

• Prove exact fixed-rank Gaussian inverse error (no κ) #6483 determines a fixed-rank approximation asymptote for a varying-phase Gaussian inverse. This provides no approximation spectrum or source-specific Gaussian bound.

• Prove sharp Euclidean conditioning for square-zero error words (no κ) #6671 gives sharp Euclidean conditioning for ordered square-zero error words. This concerns a single resolvent under simultaneous absolute-sum constraints and does not extend that word model.

• Prove a sharp bounded-error cutoff for acyclic decoder rows (no κ) #6790 proves a bounded-error rank cutoff for acyclic decoder rows using a triangular comparison envelope. This theorem optimizes a resolvent 1-norm under simultaneous row and column caps; it computes no decoder error cutoff.

• Improve three-qubit Clifford Gram condition bound to 807 (no κ) #6794 improves a sufficient three-qubit Clifford tensor-Gram condition bound. That spectral-norm matrix family differs from this abstract resolvent 1-norm class; no common numerical norm is computed.

The related-work refresh inspected all ten open account PRs created since the advance CLAIM, including this PR. The other nine, #6785 through #6794 excluding #6788, compute no competing value for this simultaneous-cap envelope. The adjacent conditioning results are distinguished above. No additional computations were run to check other contributors' results.

Credit

This contribution was produced by Nick Petalas. Credit Ali Romman #5255 for the support-envelope theorem and column-budget question, plus #5306 for unequal row budgets; Cinna Davis #4100 and James Reed #4282 for the conditioning/periodic-section context; Nick Petalas #5859 for the distinct forbidden-diagonal extension; and ToikY #4209 for same-rho Schur closure. Neumann comparison, transport inequalities and partial-sum induction are classical. General literature priority is unclaimed. All predecessor credit remains unchanged. Apache-2.0.

CI notes

Both hosted checks passed at exact head c1d915303450bec823784befd3ada5a261796a5c in run 42223371817: plan and replay (research/column-capped-resolvent). The completed checks were read once; run metadata confirms the matching head and conclusion success.

Hosted replay certifies execution of the registered normal and optimized finite commands and repository checks. All listed local checks also passed. The universal written proof, independent scientific acceptance and the remaining physical obligations retain their stated scope.

Merge Danger

Door: two-way. Blast radius: isolated research package. No shared code, frozen evidence or selected records change.

[ToikY]

FINDING / fractional-capacity follow-up: PR #6806, exact head c9f16c0, gives a written sharp envelope for every n>=1, real beta>=1 and 0<rho<1. The attainer has entries rho times the overlap of [i-1,i] and [beta*(j-1),beta*j]. A monotone-cone argument makes its inverse column attain every transport partial-sum bound simultaneously, including the current-index terms near beta=1. For n=3, rho=1/2, beta=5/4 the exact value is 38/15.

Your integer theorem, sorted transport method and explicit open question retain predecessor credit. The exact proof/README at c1d9153 are retained unchanged; no predecessor scientific code ran. New normal/optimized registered checks and read-only Git-free exports pass, with source-aware proof/code challenges and their related roles disclosed. Original hosted checks are currently pending.

The all-real claim rests on the written proof; AI reviews and finite execution do not supply human or proof-kernel acceptance. Actual Gaussian caps, physical caller/cost binding, all five selected assumptions and the frontier remain separate. No follow-on search is reserved. Prepared for ToikY with substantial OpenAI Codex assistance.