Scoped negative result: two P112 totals need overlapping independent lifts (no new kappa)

[mightyOil]

Scoped result

For exact P112 ten-term decoders retaining the eight coordinate readers, two extra totals cannot reduce the pairing-rank sum below 64 if either added factor matrix has rank at most one, either added reader is outside the coordinate column space, or their coordinate coefficient supports are disjoint. Any remaining candidate therefore needs two independent, overlapping coordinate lifts and independent source columns. No new kappa or physical supplier is claimed.

The written proof combines the inherited #845/#998 single-lift floor with support-rank subadditivity, a quotient-space rank argument, and disjoint support accounting. It also gives the k-total quotient-nullity-one and disjoint-support extensions. An explicit basis-change control preserves the decoder identity but changes its cost from 64 to 66, preventing an unjustified normalization of the remaining search.

Reproduce

python3 -I -S -B research/p112-two-total-rank-one-seed21/test_package.py

Expected: PASS_P112_TWO_TOTAL_PACKAGE controls=8 identities=1180.

The package replays source-bound #845/#998 premises, 15 new primary fixtures and nine mathematical controls, and a separate implementation's 1,180 exact matrix identities (including all 28 coordinate pairs and all 210 disjoint 2+2 partitions). It checks optimized execution, receipt/source mutations, output preservation and minimal offline relocation. make selected-record-check also passes. See VALIDATION.json for executed evidence and timing.

Sources, novelty and limits

Exact base: 1b59b624d21924f8e3537cdc1ac52d3be6eda2fe. OS seed 21 assigns the complex-supplier track. All changes stay in one new research directory; selected PR1094, upstream and predecessor packages are unchanged. Exact source/commit pins, local claim, proof, certificates and credits are included.

Credit Mihir Srivastava (#845), @woklerx (#998), DreamingOfClouds (#717/#446), and Joel Pulikkan / GamingPuzzled (#620). Sagar Thakrar's #1133 rational descent is adjacent single-lift work. Concurrent #1180 (75b408646d558147a4eb250fc407d1fbfcb8df4e) covers arbitrary term counts when both factor matrices stay in the root code. Our distinct cases admit independent sources outside that space and off-column canceling readers; primary certificates explicitly record source quotient ranks. Neither scope subsumes the other. No pending code or result is imported.

This is a written scoped obstruction with finite regression evidence, not formal verification of the multiplication theorem. The coupled full-rank case with overlapping coordinate supports and arbitrary sources is unresolved by this package. Equal rank sums can still have different recursive moments; no complete moment ceiling is asserted. Separate-total copy assumptions and all physical/analytic obligations remain inherited. No full supplier replay, native/hosting admission, hosted CI pass, or human mathematical review is claimed.

Prepared by Philipp / @mightyOil with substantial OpenAI Codex assistance in proof, implementation, separate AI checks and writing. Apache-2.0.

Immediate pre-submission refresh reached #1196. #1191 (640b43bb13d3d476a2d83b74ef0d33f14d1b5711) proves one-sided masked-to-all-pairs rigidity and does not assert this copy-cost floor; #1187 (15bee4c8ea0f54fea575a48441c6acc8bad263cd) studies a P104 puncture threshold. Their scopes are distinct. The committed validation records the preceding complete 798-PR snapshot through #1186.

[Mister-Meeseeks]

100-PR batch review at a0935afff6cc5540a582c5881bc8d5f98b587357.

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

Reviewed scope: Support-rank subadditivity reduces dependent or disjoint-coordinate added totals to the known P112 single-lift floor; coupled overlapping full-rank case stays 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.