Audit: P111 Parseval blocks cost 73 in the fixed-block scalar model

[adamgall]

Result: an exact fixed-block scalar-copy obstruction

This independently audits PR #1900 at 9a891c9cb25b15f5a189bcad1d022198adeaf885. Its F2 Parseval identity is valid: reconstruction agrees on all 111 targets and all 661 formal dirty input columns. This is not the complex-field issue in #1889.

The new question is the copy interface. Within each of #1900's five fixed rank-two coefficient blocks, allow any finite sum of inherited scalar copied totals, with factors in that block's two-dimensional binary source and target spans. All basis choices, asymmetric terms and repeated terms are allowed; cross-block rebasing and new operator-valued primitives are not.

The exact block minima are 16, 15, 14, 14, 14, totaling 73, versus 74 for the literal ten-channel Parseval expansion and 63 for the selected nine scalar channels. Every finite word reduces by parity to a subset of nine possible rank-one terms: the verifier checks all 2,560 subsets, the 160 exact factorizations, and all 30 ordered GL2(F2) bases. Rational pairing ranks are independently cross-checked by modular elimination on the full Gram support matrices.

The sum of five union-support ranks is indeed 40, but it is not the invoice of five ordinary scalar copies implementing these rank-two blocks. Literal one-copy-per-union substitution gives 5,838 wrong source entries and 17,146 wrong full dirty entries. With source port 0 equal to one and all other inputs zero, target 0 receives zero instead of one. The package includes this counterexample and 45 two-bit single-channel witnesses.

Scope is essential: this is not a proof that a genuinely new physical block-valued copy must cost two scalar copies, nor a disproof of a possible plane-port supplier. The coefficient planes do not themselves provide physical data-port planes or justify unpaid=2. The all-length scalar P111 optimum 63 is credited predecessor work; the new result is the exact pinned five-block audit and its typed-interface counterexamples.

Favorable paid screen

Keeping the incumbent non-copy children and line-port contract, the best fixed-block scalar interpretation replaces two rank-7 copies by three rank-8 copies. With every additional basis, routing, storage, restoration, hosting and precision charge optimistically zero, its exact ceiling is a <= 0.001401325233, compared with the original line relaxation a <= 0.001947467559. This screen does not treat omitted implementation charges as free in a completed construction. No new supplier, selected frontier, or kappa is claimed.

Verification

uv run --python 3.12 --no-project python -I -O -B research/p111-parseval-copy-interface/verify.py
uv run --python 3.12 --no-project python scripts/research_package.py --package research/p111-parseval-copy-interface

Observed locally on CPython 3.12.12, Darwin arm64:

PASS_P111_PARSEVAL_COPY_INTERFACE blocks=5 subsets=2560 factorizations=160 dirty_columns=661 copy_min=73 baseline=63 union_failure_entries=17146
PASS_REGISTERED_RESEARCH_PACKAGE research/p111-parseval-copy-interface outcome=negative-result

An isolated optimized generator smoke and registered replay passed. Positive controls preserve the original and Parseval dirty maps. Negative controls cover every one-channel choice, ten optimal-term omissions, a broken frame, omitted block, switched channels, and rational-versus-binary rank. The first metadata wrapper attempt rejected an invalid outcome enum; it was corrected before the successful registered replay. A second agent independently reviewed the scoped proof and ran the registered Python 3.12 replay before publication. No human proof review, full selected cold replay, or unseen CI result is claimed.

Provenance and changed files

Only research/p111-parseval-copy-interface/ is added: README, exact RESULT, complete independently emitted SELECTION, SOURCE-CLAIM, TASK, SOURCES, VERIFY and standalone verifier. No frozen input or selected frontier is changed.

• Independently sampled seed: 3cc9f4c8f3719f13f4ae6bf28c9ba982.
• Track: recursive-interfaces; task: coverage-law-change, explicitly changed to the active typed-interface audit.
• Exact baseline: 0b1cccf86e2f5132ed286a26ef0a681a495740ea.
• Round-two claim; physical plane handoff.
• Final targeted live search found Construction: five plane queries for selected P111 bit decoder (no new kappa) #1900 only for Parseval and no open blockwise-copy audit; its head remains pinned above. Search results are not a novelty proof.

Credit: Cinna Davis / 1337isnot1337 for #1900's valid construction; PR589/PR555's bit word; DreamingOfClouds' PR446 per-total pricing premise; Russell Seligmann's Codex-assisted P111 scalar-copy proof and Joel Pulikkan / GamingPuzzled's earlier PR620 cost-63 experiments; the retained bit orthogonal-pair proof; ctoth's plane-port question; Douglas Colkitt's exact ceiling engine. Prepared by Adam Gall with substantial OpenAI Codex assistance. No independent human review is claimed. New work is Apache-2.0; source notices are preserved.

[Mister-Meeseeks]

Batch review at f7ad7dc0e3a0c16a4f02eb89409b6a81f6bc7834.

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

Reviewed scope: All2560 fixed-block scalar subsets and160 valid full-dirty factorizations establish exact minimum73, versus claimed support-union sum40 and baseline63. Independent modular/rational ranks,17146-entry union counterexample and45 single-channel controls pass. Cross-block rebasing and genuinely new block-copy primitives remain outside this obstruction.

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.