Independent proof: P112 three-call COPY192 via a sharp block count

[theon199]

Change and mathematical scope

Independent proof / scoped negative result, no new κ. Three independent P112 all-pairs calls retain sharp separate-total union-pairing cost 192, even with a different arbitrary isometric binary label chart per call and free conversions. For every 0 < q <= 1, the sharp power bill remains 24*8^q.

The new six-order block-triangular argument proves n + (2/3)p >= 24, where p counts pure cross-call terms. Within-call terms cost at least eight; pure terms can genuinely cost seven. Hence the copy bill is at least 192 + (5/3)p. An exact coupled identity attains the counting lemma, while the standard decoder attains COPY192.

Scope: any finite number of complex root-code factors, ordinary transpose, independent endpoint C=I_3, and the separate-total union-price objective. This does not cover arbitrary interacting circuits, outside-code factors, non-isometric own-call charts or four-call batches. Physical compiler, hosting, precision/tape and analytic obligations remain untouched; no selected supplier/frontier changes.

Task and overlap

• Base: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority revision 4.
• OS assignment seed 47, structural decoder track; centre-path-sharing card refined to one copied-information proof obligation.
• Extends PR Negative result: P112 two-call batch keeps the 128 copy floor under isometric cross-call label charts (rank-7 cross terms exist; PR1714 OQ3, isometric free-conversion case) #2294, head 18d6ac4f864af2b71aa8cf5f4541fcfc4350e2f0, from two calls to its explicitly open three-call case. Uses a new triangular deletion argument, not more chart sampling. Main's PR1714 support/model sources are SHA256/Git-blob pinned; no predecessor code is imported or vendored.
• Final overlap: 550 open PR bodies and 1,058 Discussions, complete pagination, no warnings, 05:54 UTC. Main and the PR2294 head were unchanged. Other three-mode hits concern coordinates within one call.
• Canonical claim; scope, proof, source pins and evidence are in research/p112-three-call-isometric-floor/.

Concurrent work discovered immediately before submission: DreamingOfClouds' #2338, head 82700676e1d42acf98dc1921cb6ef82551620008, §3.4 states a broader all-batch symmetric-chart floor using normal classes and an additional frame lemma. Our core proof/verifier was completed before reading it (commit 9df32e528da547a1c1cb51cbd480f88f8b7d197e). This PR adds a different, frame-lemma-free three-call argument, a sharp counting certificate and full all-pairs/mutation evidence. Its scope and surplus bound are narrower; no first-closure or stronger-floor claim. The concurrent package has not been fully audited here.

Validation

python3.12 scripts/research_package.py --package research/p112-three-call-isometric-floor

Expected: PASS_P112_THREE_CALL_ISOMETRIC_FLOOR, PASS_THREE_CALL_PACKAGE_CONTROLS, then PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed.

macOS arm64, CPython 3.12.13: registered optimized and direct normal replay pass. CPython 3.14.5 optimized output is byte-identical. Exact receipt SHA256: c1278f0b9154e144fe67e85652c0b98c4bfd98b9c481a2680037a2b8efe08f09.

Coverage: all 511 support functionals; 64 within-call and 64 cross-chart exceptional pairs; 12 support patterns × 6 orders; four full 336×336 endpoint fixtures; seven internal mathematical controls. Relocation/read-only checks and five on-disk corruptions pass. A four-call 2×2-support control catches the invalid extension of the deletion lemma. Metadata/pins and git diff --check pass. Finite replay supports the written proof; it is not formal verification of the full theorem. Selected cold replay and hosted CI are not claimed.

Credit

Credit DreamingOfClouds / concurrent PR2338 for the broader all-batch statement; alephnan / PR1714; Sagar Thakrar / PR2294; Mihir Srivastava / PR845; Adam Gall / PR1098; woklerx / PR1077 and PR1180; themilkman216 / PR968; Douglas Colkitt / PR1240, plus the retained selected construction lineage. Full credit and exact provenance are in the package. New work Apache-2.0.

Sricharan Vagvala (@theon199), with substantial OpenAI Codex assistance drafting the proof, code and documentation; author-checked, no independent human review or proof-assistant check.

Open next questions

The package's README and TASK.json give two bounded follow-ups. First classify balanced 2×2 pure supports for four calls: their mean truncation rank is one, so this proof no longer suffices. A physical escape would instead need a specified interacting-copy or outside-code interface with its complete preparation, restoration and recursive-child bill.