[petalas]

Summary

Proves the exact inverse 1-norm envelope for arbitrary finite directed supports under unequal row contraction budgets, including forbidden diagonal entries. This is a scoped algebraic proof with exact finite corroboration, not a new kappa or a complete multiplication theorem.

all diagonals allowed: column j = sum_i w_ij / (1-r_j)
arbitrary support:     column j = sum_i w_ij / (1-q_j)
q_j = maximum budget product around a simple directed cycle through j
w_ij = maximum budget product along a simple path from i to j

The proof constructs one nonnegative matrix attaining every entry of each chosen column. It extends Ali Romman's merged #5306 and #5255 by removing their all-diagonals-allowed hypothesis; their verifiers are not replayed. A loopless two-vertex example has exact envelope 14/5, compared with 6 in the larger diagonal-completed class. Missing self-loops strictly improve their own column envelope.

Proof and complete scope, standalone verifier, and source provenance.

Evidence

• Complete small-domain comparison covers 3,634 directed-support/budget cases with at least one forbidden diagonal, 50,334 zero-or-saturated policy matrix evaluations, 10,850 full-column attaining inverses, and 10,902 diffuse signed/nonnegative matrix evaluations. These are evaluation counts, not deduplicated matrix counts.
• Separate rational Gauss-Jordan inversion corroborates the maximum-product formula. Explicit acyclic, longer-weighted-cycle and disconnected-cycle examples distinguish the changed hypotheses. Ten rejection controls pass.
• CPython 3.12.15, standard library only: warning-strict normal and optimized direct checks, both registered commands, metadata validation, committed-byte/source-history preflight, and optimized Git-free replay of committed package bytes pass. Each registered command takes about 2.5 seconds locally, below its 180-second bound. The registered pair takes about 5.4 seconds.

Exact reproduction from the repository root:

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

Another CPython 3.12 path can replace the displayed interpreter. Expected markers are PASS_SUPPORT_CYCLE_RESOLVENT, PASS_REGISTERED_RESEARCH_PACKAGE, and PASS_RESEARCH_COMMIT_BYTES. Direct replay generates a JSON receipt bound to all five package-file hashes and writes no files by default.

Limitations

The universal claim rests on the written proof. Finite tests are corroboration, not proof-kernel acceptance. Budgets satisfy 0<r_i<1; allowed entries may be zero. No infinite-graph, mandatory-nonzero-edge, Gaussian-specific or operator 2-norm optimum is claimed. No actual caller's support or budgets are established. Physical preparation, native operations, compiler, ownership, precision/tape, stock and analytic recovery remain separate. Selected cold replay, predecessor scientific suites, native replay, Lean and Docker are unrun. The selected frontier is unchanged.

Open next questions

• Extract a proven missing-diagonal support and unequal rational row budgets from one emitted rounded coefficient matrix; compute the return-cycle bound and charge preparation. Source binding and physical costs are missing.
• Fix source-compatible per-entry constraints that forbid concentration and determine their sharp envelope. The attaining policies here may be excluded by those additional constraints.

Related open work

• #5862 optimizes Euclidean conditioning of fixed-product unitary observation cycles. This package optimizes the inverse 1-norm over directed-support/unequal-row-budget classes. No common numerical value is computed or compared.

The account's open PRs created since the CLAIM at 2026-10-27T09:05:49Z were refreshed. No additional PR studying this graph-support inverse envelope was found. Merged #5306 and #5255 remain the direct predecessors.

CI notes

At commit f585fc78fb2b38bd213d0dd78ec2ade8e10e2fc1, run 40479387415 has plan failed and replay skipped. The failed planning job fails in Onboarding helper tests: all ten reported failures/errors are in tests/test_research_current_assignment.py, ending FAILED (failures=8, errors=2, skipped=22) and exit code 1. This matches the shared main de5ec2e7a failure reported on #5854 through #5859; those tests are unchanged by this PR and maintainers will repair them separately.

Hosted replay has not certified this package. Only the local checks listed above have run for its scientific verification. The package replay job was skipped after planning failed. No shared tests or files outside this package were changed to bypass the failure.

Merge Danger

Door: two-way. Blast Radius: isolated research package. Only five new files under research/support-cycle-resolvent/ are added.

This contribution was produced by Nick Petalas. Credit Ali Romman for #5255/#5306, Cinna Davis #4100 for the conditioning context, James Reed #4282 for periodic sections, and William Lawyer #852 for the different weighted-cycle setting. Classical Neumann, discounted-path and maximum-principle methods retain their credit; general literature priority is unclaimed. Independent human and formal proof-kernel review are unclaimed. Apache-2.0.