DONE: P112 batch copy optimum is 64 rank(C) in the root-code model

[alephnan]

DONE / FINDING — scoped proof and exact finite evidence

PR1714, head 333acd5b34adf2e3d87869bc7c39e49b675da3f3, adds only research/p112-batch-centre-floor/.

Result. In the common-chart standard-P112 block-root-code model, arbitrary finite complex scalar mixing across calls has exact term charge 8m+b and minimum total charge 64 rank(C) for the all-pairs demand C tensor (X X^T). Thus k independent calls require 64k, even charging a term once on the union of its labels across calls. Minimum-term factorizations automatically satisfy the code restriction. For a cheaper independent two-call decoder, at least 17 active terms, an outside-code factor and at least nine net rank units saved at 17 terms are necessary, not sufficient.

Evidence. Registered CPython 3.12.2 replay passed on Windows, standard library only, with all commands under -I -O -B. Exact witnesses cover 511 hyperplanes and 64 exceptional pairings. Independently implemented standard-root/set-elimination/modular-plus-null-vector checks corroborate those finite premises and all integer minors. Full identities cover 451,584 independent-call entries, including 250,880 cross-call entries, plus 75,264 rectangular-coupling entries. Ten primary controls, seven independent geometry controls, positive renamed-package replay and five intended failures from four on-disk corruption groups pass. These are finite checks; the all-size theorem is the separate written proof. Hosted CI was queued when this update was posted.

Scope and credit. The one-call support lemma/census belongs to Mihir Srivastava / PR845, corroborated by Adam Gall / PR1098. The frame/complex/arbitrary-term predecessors are themilkman216 / PR968 and woklerx / PR1077/1180; Douglas Colkitt / PR1240 supplies the bottleneck and lever. Exact heads and complete lineage are retained. This extends their results to batch coupling and exact mixing costs; it does not claim their finite premises as new. The final overlap refresh covered 396 open PRs and 352 Discussion records with complete pagination at 07:29–07:30 UTC; main remained 0b1cccf86e2f5132ed286a26ef0a681a495740ea.

Handoff. Test one exact outside-code 17-term two-call family against the nine-unit necessary saving, or derive a realized rank-deficient caller with its full physical costs. The shared-source two-panel claim in Discussion1516 changes the independent-input premise and is not excluded. Changed charts, masked identities, temporal source relations and interacting delivery also remain outside scope. No new kappa, complete physical supplier, selected cold replay, Lean proof or human peer review is claimed; the selected frontier is unchanged.

Prepared for alephnan with substantial OpenAI Codex assistance, including separate algorithms and adversarial AI review. Apache-2.0.

Original claim (completed)

CLAIM - centre-path-sharing, priority revision 3

Question: Can scalar coding across k independent standard P112 all-pairs demands reduce the copied pairing-rank bill below 64k? I will prove or refute the floor in the direct-sum root-code model, charging each shared term once on the UNION of its source/reader labels across calls. Arbitrary finite term counts are allowed under the root-code restriction; minimal 8k-term factorizations should imply that restriction automatically.

Base: 0b1cccf. Selector nonce afb2871f5b82a11465a83416a203340b, no reroll. Predecessors: themilkman216 PR968, woklerx PR1077/1180, Douglas Colkitt PR1240.

Method: independently implemented exact census of all 511 binary label hyperplanes, rational complement-root ranks and exceptional support pairings, followed by a written all-k rank proof. Positive direct-sum/mixed/overcomplete fixtures and meaningful failure controls. Expected artifact: research/p112-batch-centre-floor, source pins, registered replay and focused PR.

This complements ctoth Discussion1516 and PR1643/1664/1684: their fixed-word tree/dirty-host interfaces differ from arbitrary root-code decoder changes across independent calls. Discussion1636 changes global spectral demands, outside this scope. No claim about masked decoders, outside-code overcomplete factors, physical circuit optimality, supplier admission or new kappa. No selected-frontier change.

Fresh overlap: 382 open PRs and 333 Discussion records, complete pagination, at 07:04-07:05 UTC; metadata/local notes only, not proof of novelty. Selected source bindings and the card first control pass.

Prepared for alephnan with substantial OpenAI Codex assistance, including separate mathematical and verification agents. Claim expires 2026-10-14T09:01:15Z; expected finish about 08:01Z.