Research: PR #2308's P112 shared correction needs at least 12 COPY units

[commdata2338]

Change and mathematical scope

PR #2308 proves a shared rational correction basis and leaves its complete
physical charge open. This focused audit of its named P112 frame pair finds
that a complete additive scalar correction requires at least 12 COPY rank
units, while its stated fixed-profile 5% screen allows at most 8.

The package preserves the predecessor's exact rational sharing result. It
supplies a source-bound scalar endpoint/cost audit and a written rank-to-cost
proof, with exact finite certificates:

• The named frames overlap in four ports, so their union has 12 ports.
• The displayed common basis has coefficient parity rank 1, but the ordinary
  separate-term support COPY ranks are [1,1,4,4,1,1,4,4] per direction,
  40 total. Rescaling the coefficients by 2 and the basis by 1/2 reduces
  parity rank to zero without changing the scalar map or support bill.
• The two cross-block corrections leave 32 forbidden within-union entries.
  One explicit input gives target value 23/2 instead of 11.
• The forced 12 x 64 correction rectangle has rank 12. Its integer-scaled
  minor has determinant 2048, giving a COPY floor for any number of arbitrary
  complex additive scalar factors with the original G/2 map retained.
• Correctly binding unpaid coordinates to 31/28, the exact favorable moment
  at the predecessor's 5% target is greater than 1.00042899363319 at that floor.
  Merely correcting the port count, while still assuming +8, does pass; the
  lower-bound argument is essential.

The natural deleted-frame base decoder itself grows from COPY64 to COPY68 on
these ports. Its displayed cross-block realization therefore costs 108 before
the missing repair. This is a finite algebraic invoice, not a native program
or an optimum over other bases.

Only research/p112-shared-correction-cost-audit/ is added. No source archive,
predecessor package, selected record, compiler or frontier is changed. This
does not rule out a redesigned whole decoder, bundled operator interface or
changed local profile, and claims no supplier or new kappa.

Task and overlap

Base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2; contribution commit ca5baa17182636a35536c1605a2d2da550e715b7; main was rechecked unchanged at 07:31 UTC before submission.

Base and final refreshed canonical main:
0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority revision 4. The final refresh
found no main changes. PR #2308 was re-fetched and remains at
7653e0cc72527c1e7fc02499122d383713ee9d54; this is the sole pending scientific
predecessor. Its certificate is preserved byte for byte, with Git blob
1aa8ddab0e0cd9e4e91da414a107ade54c2cba28 and SHA256 in SOURCES.json.
All retained geometry, selected-label archive and interval-engine dependencies
are pinned to the base by both Git blob and SHA256.

The claim is in Discussion #2278.
The final cached overlap check covered 598 open PRs including drafts, fetched
2026-10-15 07:21:25 UTC, and 1,126 Discussion records, fetched 07:23:29 UTC;
both paginations were complete. Offline warnings are retained: this metadata
search is not a novelty certificate. The current thread and exact predecessor
head were read again.

Nearest work: PR #2308 supplies the rational sharing identity, not the missing
physical tariff. The retained paired-plane-scalar-mask-obstruction audits
PR #1825's different binary-to-scalar factors. This account's PR #2021 and
the concurrent PR #2321 coefficient/overcomplete follow-ups concern a different
two-column completion. The PR #2294 two-call proof work has a different object.

Local task card:

• Task/scope: complex-rank-shape-decoder, chooser nonce
  6ba173a193cdec68f20ebb9c9c8c8073, track complex-mechanisms, original scope
  key 02b31ebd160542c86299; refined to PR2308/named-pair/scalar-support-cost.
• Object: the two frames, shared direction and common basis in the pinned
  certificate, on all 112 selected ports and both correction directions.
• Scope status: changed to an independent audit of the missing cost/interface
  implication, rather than synthesis of a full new local profile. The forced
  rectangle additionally gives a basis-independent additive bound on this same
  pair with G/2 fixed; it is not a whole-decoder lower bound.
• Method/domain: independently reconstruct120 E8 roots, bind selected labels,
  exact rational factorization and scalar masks, separate support-pairing ranks,
  a characteristic-zero rank proof and outward interval screening. No optimizer
  or all-frame-pair search.
• Budget/gate: retain the declared local profile and target
  35491136290869/20000000000000000, correct the port count, then test the
  unavoidable added charge before native construction.
• Success/stop: one exact interface decision with controls, then stop. The
  additive model fails the declared screen; no target was lowered to claim success.
• Artifact/reproduction: package README, TASK, sources, certificate, receipt
  and two checkers; commands below. Standard library, bounded session.

Validation

On CPython 3.12, standard library only:

python3 -I -O -B research/p112-shared-correction-cost-audit/verify.py
python3 -I -O -B research/p112-shared-correction-cost-audit/check_certificate.py
python3 scripts/research_package.py --package research/p112-shared-correction-cost-audit --check-only
python3 scripts/research_package.py --package research/p112-shared-correction-cost-audit

All pass. The primary line is:

PASS_P112_SHARED_CORRECTION_COST_AUDIT union=12 parity_rank=1 correction_COPY=40 forbidden_entries=32 additive_COPY_floor=12 five_percent_screen=FAIL controls=10 no_new_kappa

The ten semantic control groups include omitted/changed factors, wrong label
map, coefficient-gauge invariance, wrong line-only mask, duplicate-port count,
the dirty counterexample, all 224 scalar columns, a valid repaired fitting
matrix and a dependent minor. The second checker uses explicit binary spans
and independently reconstructed integer minors modulo101, without importing
the primary checker. A nonzero integer minor modulo101 certifies its
nonvanishing over Q/C; it is not a finite-field replacement for an identity.

A Git-free source export passes both checkers and remains byte-identical after
default replay. Additional controls reject a false lower-bound receipt,
changed source bytes and an occupied output path, preserving the occupied
file. The research baseline bindings, prescribed local-ceiling positive control,
package metadata/pins and whitespace check also pass.

Not checked: the predecessor's 20,160-block census, another pair, the optimum
additive bill, a whole redesigned decoder or physical sharing interface,
native phases/chronology, dirty helper restoration, global owners/idle profile,
precision, all-size admission, hosted CI or the selected cold replay. The
five-percent exclusion retains the predecessor's fixed local profile; a
changed profile needs a new complete screen.

The registered replay (primary audit and second check) was run again on the final package bytes before submission, under an 8 GiB memory cap: it passed in about a second and left the worktree unchanged. The basic counts check by hand: two eight-port frames sharing four ports have a 12-port union, and a rank-12 correction rectangle needs at least 12 rank-one scalar terms, each costing at least one COPY unit.

Credit

Cinna Davis (1337isnot1337) / PR #2308 retains credit for the rational sharing
identity, finite census, selected-label map and original hypothetical screen.
Philippe Van Velzen (philippeVV) / Discussion #2278 supplied the shared-plane
question and additive-class screens. Adam Gall's paired-plane audit supplies
the scalar endpoint distinction. themilkman216 / PR #968, DreamingOfClouds and
the retained E8 lineage supply the geometric model; Douglas Colkitt supplies
the interval ceiling engine. Rohan Arun, Jeremy Kothe and Hao Yan retain
selected-source credit. The unchanged foreign certificate has its own notice;
the contribution retains Apache-2.0 licensing.

Prepared for commdata2338 (@commdata2338), with substantial OpenAI Codex assistance.

AI assistance: submitted by @commdata2338 with substantial AI assistance. OpenAI Codex (GPT-6 Astra) did the derivation, the implementation, the checking and the writing; it was packaged and submitted with Claude Code. No independent human review is claimed.

Open next questions

The package records two bounded follow-ups: solve the full mixed-port scalar
fitting problem while redesigning the whole decoder; or specify a block/operator
interface that implements the sharing with a complete dirty-input identity and
tariff. Start from the forced rectangle or the (16,20) counterexample, then
test a favorable completed budget. Neither is supplied by coefficient parity.

🤖 Generated with Claude Code

https://claude.ai/code/session_01X86sCSXdoDd61SDFfC1PKG