[petalas]

Summary

Prove an exact criterion for the specified nonnegative coupled bill and a sharp finite-depth divergence certificate. Claim type: scoped mathematical proof with exact finite corroboration.

A >= 0, b >= 0, c >= 0
H = vertices on a path from supp(b) to supp(c), r = |H|

sum c^T A^k b is finite
  iff H is empty or A[H,H] has positive strict contraction weights.

Otherwise an exact rational witness gives, for every N >= 0,
  sum_(k<N) c^T A^k b >= floor(N/r) h, h > 0.

A unit r-cycle attains equality for every N. The window r is sharp.

Ali Romman's #5785 supplies the whole-matrix M-matrix alternative and explicitly leaves forcing/reachability applicability open. This result changes the contract to a specified forcing and observed output, trims exactly the irrelevant blocks, and makes an applicable obstruction quantitative. The classical alternative and nonnegative matrix theory retain their credit. No predecessor fixture, implementation or numerical result is re-audited.

The seven new files under research/observable-forced-recurrence/ contain the proof, a bounded exact producer with a separate arithmetic checker, an exhaustive verifier and pinned package metadata. The proof includes the classical lemma, so the pending predecessor is a citation rather than a runtime Git dependency.

Evidence

• Before: global noncontraction alone cannot decide a specified observed bill. For example, A=diag(1/2,2), b=(1,1), c=(1,0) has an unstable forced component and observed total exactly 2. After: the new path reduction supplies the finite certificate for that bill and observable divergent certificates when the relevant block fails contraction.
• All 36,864 enumerated cases pass. The two-type rational grid has 1,920 zero, 385 finite nonzero and 1,791 divergent cases. The three-type binary grid has 9,600 zero, 2,995 finite nonzero and 20,173 divergent cases. Every forcing and output vector is binary; zero vectors are included.
• Independent permutation determinants check every principal-minor decision, and Cramer's rule checks every finite total. Every case also checks seven literal full/restricted terms and the corresponding prefix bounds. Four further fixtures check hidden unstable components and exact geometric tails. Eight critical cycles give 152 exact depth checks. Twelve forged certificates and four invalid inputs are rejected normally and under optimization.
• CPython 3.12.15, standard library only: warning-as-error normal and optimized direct replay, both registered commands, package metadata, committed bytes and declared source ancestry pass. Each registered scientific command takes about five seconds locally, below its 240-second timeout. A read-only Git-free export of the committed package passes optimized replay. Generated JSON receipts bind the package file hashes.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/observable-forced-recurrence/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/observable-forced-recurrence/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/observable-forced-recurrence
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/observable-forced-recurrence --check-source-history

Another CPython 3.12 interpreter path may replace the displayed one. Expected markers are PASS_OBSERVABLE_FORCED_RECURRENCE, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations

The written theorem holds in every finite dimension; the reference implementation accepts at most eight types and 16-bit rational input parts, with an 8,192-bit certificate-vector limit. Finite enumeration corroborates the proof. Independent human and formal proof-kernel review are unclaimed.

No complete source recurrence, child/adapter/restoration ledger, supplier, native implementation, compiler, precision/tape, stock or outer recovery is admitted. A divergent upper-bound matrix does not disprove an actual caller. No new kappa or selected-frontier change is claimed. Predecessor scientific suites, selected cold replay, native replay, Lean and Docker were not run. This is an explicit analytical scope change from the decoder card, recorded in TASK.json.

Open next questions

• Bind one complete source recurrence, its forcing and every genuinely billed output. Begin with its type graph and justify each nonnegative coefficient and cost; missing source bounds remain the blocker.
• Determine exact polynomial or exponential growth of divergent observed bills. Begin with a chain of critical strongly connected components. The result here stops at the sharp universal linear certificate.

Related open work

• #5785 constructs the classical whole-matrix alternative. This contribution extends its forcing/reachability boundary to an exact observed bill and a quantitative divergence window.
• #5859 optimizes full inverse 1-norms over directed-support and row-budget classes. This contribution fixes A,b,c and decides the observed Neumann bill; no common numerical optimum is computed or compared.
• #5803 develops phase-dependent nonnegative potentials and broader analytical budgets. This contribution treats a stationary declared input/output bill with exact convergence and divergence alternatives.

The account's open PRs created since the advance CLAIM at 2026-10-27T11:13:24Z were refreshed. None of the eight other PRs studies the specified observed coupled recurrence bill. No additional common numerical quantity was found for comparison.

CI notes

At commit 9f0cc0214c17c906817c70a041109195a3fbaed4, run 40496008742 finished with plan failed; replay skipped. The sole failed step is plan / Onboarding helper tests. All ten failure/error entries are in unchanged tests/test_research_current_assignment.py, ending FAILED (failures=8, errors=2, skipped=22) and exit code 1. The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'; the eight failures concern frozen-card and current/historical first-step expectations.

The same failed step, test file and final summary were directly confirmed in PR #5891's planning job, also based on main de5ec2e7a. This PR changes only its seven package files. No shared tests or other files were changed to bypass the failure; maintainers will handle the shared failure.

Hosted replay has not certified this package. Only the local scientific and publication checks listed above have run. The package replay job was skipped after planning failed.

Merge Danger

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

This contribution was produced by Nick Petalas. Credit Ali Romman #5785 and the classical M-matrix, nonnegative-matrix, path-expansion and Cramer's-rule ingredients. Apache-2.0.