[petalas]

Summary

Scoped mathematical proof with exact finite corroboration; no new kappa or selected-frontier change.

Assuming fifteen missing scalar observation dimensions per independent port, dense cross-port linear queries, adaptive feedback and arbitrary deterministic decoding still need total output capacity at least 15P.

528 requests, 104 ports: total capacity >= 1560
  capacity <= 3 requires at least 504 capacity-three requests
528 requests, 112 ports: total capacity >= 1680
  capacity <= 4 requires at least 96 capacity-four requests

If requests stay scalar, the alternative requires at least 1032 or 1152 newly supplied free dimensions. The proof gives the exact abstract minimum with added side information. A request's capacity counts its scalar output coordinates, independently of its support across ports or its native frame rank.

This extends the multi-port question in Matarener's PR #5996, head 47242572d3b22ce49371245edc561dd75bf263e8. Its reported per-port quotient dimension is an explicit cited premise, not replayed or validated here. The general rank principle is classical and already appears in the repository's outside-input probe and centre-information work. The new result is this conditional thin-pool contract and capacity distribution. Advance CLAIM.

Evidence

Before, the cited thin-pool exclusion restricts requests to a port. The new proof permits cross-port scalar rows and data-dependent request selection, and gives necessary vector-output budgets.

Local warning-strict normal and optimized runs pass PASS_CROSS_PORT_THIN_CAPACITY: 304,993 binary subspace triples, 4,230 independent literal fiber comparisons, 116 rational recoding/side-information fixtures, 510 complete adaptive transcripts, 255 ambiguous shortened transcripts, four semantic controls and four invalid-input rejections. Completeness is also checked against every square binary matrix through dimension four. Registered commands took 0.24 and 0.23 seconds locally, only a replay-budget observation.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/cross-port-thin-capacity/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/cross-port-thin-capacity/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/cross-port-thin-capacity
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/cross-port-thin-capacity --check-source-history

Metadata, both registered commands, committed bytes/source history, the official six-file export, export integrity and a read-only Git-free optimized replay pass. CPython 3.12.15 on macOS arm64; standard library only. Other machines may substitute their CPython 3.12 path. The standalone verifier needs no Git, network or predecessor source. Optional --receipt PATH creates a hash-bound receipt outside the package.

Limitations

The universal statement rests on the written proof. The thin-pool application assumes the cited quotient deficit and independent port inputs, with exactly the declared free information. Binding actual multi-port semantics and every free operation to this contract is unproved. Full vector responses can carry many coordinates and are not excluded. Native frame rank is not identified with output capacity. Physical realization, chronological availability, dirty restoration, full paid invoice, compiler, stock, precision and tape costs remain open. No complete multiplication theorem, independent human review or proof-kernel acceptance is claimed. No predecessor scientific replay, Lean or selected cold replay was run.

Related open work

• Per-port floor of fifteen rank-one requests for the thin query stages (atlas P8 impossible per port; no new kappa) #5996 supplies the reported per-port deficit and asks about multi-port channels; this adds a conditional dense-query capacity bound without rechecking its computation.

• Exhaustion census: no PR5599-type helper split remains on the PR5907 word (no new kappa) #5994 excludes helper-split cuts on the PR5907 word; it studies target splitting rather than the thin-query observation capacity here.

• Compress the ordinary events of the PR5907 P104 word: 25,998 to 25,861, same invoice #5975 compresses ordinary events on PR5907 while preserving its invoice; this supplies no event rewrite or new word.

• Independent replay, invoice, native and funding audit of #5827 and #5907 (no new κ) #5960 reviews the PR5827/5907 construction and funding; this performs no such review.

• Certify conditional kappa 0.0033999964 via the p19 word without the coded overlay (complex 0.007578535, BIT 0.0034, zero field bill) #5270 changes the p19 controller and prices its uncoded endpoint; the present theorem assumes the reported fifteen-deficit independent-port contract and does not transfer to that changed word.

• Price the p19 coded-reverse chain: conditional κ = 0.0033999964 #5440 prices a coded p19 chain with separate precompilation obligations; no claim about that chain's observation capacity is made here.

• Joint bit+complex supplier cannot beat the average of the two p16 invoices under the whole-job premise (negative result) #5298 studies joint bit/complex whole-job accounting; its mass premise differs from independent scalar-observation capacity.

• Save twenty P104 scalar units and two events through dirty-output sharing (no new κ) #5999 changes ordinary dirty-output sharing on PR5907 while preserving its paid profile; it does not compute thin-query observation capacity.

• Prove one local rank-5 deletion per FULL lifts the selected complex invoice to the BIT saving (no new κ) #6010 screens local rank deletions and helper refinements on the selected P104 invoice; it gives an accounting target, not a channel with the output capacity required here.

• Raise the partial-cache observer error floor to sqrt(7/92), exact over scalar witnesses (no new κ) #6006 bounds approximate decoding error for a different partial-cache observer; its scalar-witness norm and history differ from this exact independent-port observation contract.

The completion refresh read the titles and bodies of this account's fourteen other open PRs created since the CLAIM at 2026-10-27T21:02:46Z: #5998 through #6011. None reports a competing 528-request thin-pool capacity distribution. No additional computations were run to compare their results.

Open next questions

1. Can one physical query meet the necessary multi-output capacity distribution at favorable complete cost? First specify its maps and native tariff, then restoration and chronological availability.
2. Does the actual multi-port caller satisfy the direct-sum observation hypotheses? Supply input-coordinate and free-span maps before applying the numerical bound physically.
3. Can a changed producer cheaply supply the required extra free dimensions? Price one independently new observable and its complete consumer/restoration bill.

CI notes

Hosted CI passed at 3a6c4194b6c4278308541b140166d7f21fcda096: plan and replay (research/cross-port-thin-capacity), in run 40584322604. The check list was read once after completion, and the successful run head matches the PR head. Hosted replay certifies the registered bounded corroboration, not the universal theorem or physical caller binding. All registered checks also passed locally; no local registered check remains unrun.

Merge Danger

Door: two-way. Blast radius: isolated. Only six new package files are added. Shared code, frozen evidence and the selected record are unchanged.

The proof and verifier were produced by Nick Petalas. Credit Matarener, Cinna Davis, Ali Romman, Dugongue and the predecessor lineage retained in PR #5996.