Scoped proof: independent P112 batches retain the 64k copy floor

[alephnan]

Change and mathematical scope

Independent calls might appear to share the selected supplier's 64-unit centre charge. This package proves that arbitrary scalar mixing across calls cannot do so in the common-chart P112 block-root-code model, even when each term is charged only once on the union of its labels across calls.

For every complex rectangular call-coupling matrix C and every finite exact factorization of C tensor (X X^T) with both factors in the respective block-root codes, the minimum copied-pairing-rank charge is 64 rank(C). The exact charge is 8m+b, where m counts active terms and b counts terms with neither side coordinate-aligned. This gives 64k for k independent calls, the equality cases, and an unrestricted minimum-term corollary: every exact 8k-term factorization automatically satisfies the code hypotheses.

A cheaper independent two-call decoder must therefore have at least 17 active terms, leave the root code, and save at least nine net rank units relative to rank eight at 17 terms. These are necessary conditions, not an outside-code exclusion. Correlated demands can attain 64, but change the requested endpoint. The proof is all-size after explicitly checked finite premises; fixtures are regression evidence.

Scoped proof and finite certificate; no new supplier or kappa. The physical interpretation needs the stated separate-total delivery interface. Changed charts, masked identities, temporal source relations, interacting circuits and outside-code overcomplete factors remain open. No selected cold replay, native/compiler implementation, physical budget, Lean proof or human peer review is claimed. Only research/p112-batch-centre-floor/ is added; selected and frozen inputs are unchanged.

Task and overlap

Task centre-path-sharing, priority revision 3; refined to the card's copied-information lower-bound method. Scope key p112-independent-direct-sum-root-code-union-support-floor; nonce afb2871f5b82a11465a83416a203340b; base/main 0b1cccf86e2f5132ed286a26ef0a681a495740ea. The card's first ceiling and lever checks passed. Claim: #1691.

The delta over #845/#1098's finite support lemma and #1180's one-call root-code floor is the all-batch coupling-rank law, exact mixing surcharge, minimum-term corollary and escape budget. #968/#1077 supply frame and complex-coefficient predecessors. This is complementary to #1643/#1664/#1684's fixed-word transport audits, #1633's outside-code excess-rank budgets and #1689's masked-Walsh subfamily. The shared-source two-panel claim in Discussion1516 explicitly changes the independent-input premise and is not excluded.

Fresh complete pagination covered 396 open PRs and 352 Discussion records at 07:29–07:30 UTC on 2026-10-14; local notes and the ledger were searched. Direct predecessor heads were rechecked. Exact heads, raw source hashes, Git blob IDs, coverage limits and scope comparison are in SOURCES.json, TASK.json and the README. This is not a novelty certificate.

Validation

Windows, CPython 3.12.2, standard library and pinned Git history:

python3.12 scripts/research_package.py --package research/p112-batch-centre-floor

The registered replay passes all three commands under -I -O -B:

• Primary checker: 511 exact hyperplane/minor witnesses, 64 exceptional support pairings, 451,584 independent-call matrix entries including 250,880 cross-call entries, and 75,264 rectangular-coupling entries; ten semantic controls.
• Separately written geometry auditor: standard doubled roots, set-based binary elimination, modular rank lower bounds plus exact null-vector upper bounds, and subset-expansion integer determinants; all 511 witnesses and 64 pairings match; seven controls. It does not independently check the batch fixtures or certify the all-size proof.
• Disposable renamed-package replay: positive read-only checks and five intended failures from four on-disk corruption groups, including source-blob and receipt corruption. Cross-call crosstalk, correlated-endpoint substitution, outside-code factors and ordinary-versus-Hermitian transpose have explicit controls.

Staged whitespace checks pass. Two separate AI proof reviews found no unresolved mathematical issue; this is not human review. All-size proof and exact finite evidence are separately labeled.

Credit

Mihir Srivastava / @mihir-s-05 (#845) owns the support reduction and original census; Adam Gall / @adamgall (#1098) independently corroborated it. Credit @themilkman216 (#968), @woklerx (#1077/#1180), and Douglas Colkitt (#1240) for the frame, field extension, one-call floor and bottleneck. Their results are rechecked, not claimed as new. The README retains the full DreamingOfClouds, GamingPuzzled, icekylinx, tensor-code, fixed-word-audit and selected-construction lineage.

Prepared for alephnan with substantial OpenAI Codex assistance in proof, implementation, separate algorithms and adversarial review. Apache-2.0.

Open next questions

The README gives three bounded handoffs: an exact outside-code 17-term two-call family meeting the nine-unit escape budget; a realized caller with rank-deficient separable coupling and full physical costs; or legal changed label charts with all conversions charged. None is supplied or ruled out here.

[Mister-Meeseeks]

Batch review at 333acd5b34adf2e3d87869bc7c39e49b675da3f3.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: The written direct-sum root-code proof and independently rechecked finite P112 support premises give exact copied-pairing charge 64 rank(C), including all independent batch sizes. Minimum-term factors lie in the code automatically; overcomplete outside-code factors and changed charts remain open.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.