Scoped negative result: 72/77 copy-cost lower bounds for two masked Walsh P112 totals

[just-mn]

Change and mathematical scope

Two masked Walsh source totals can leave the P112 coordinate column space, but this specific two-coordinate-reader family still cannot improve the copy-rank bill. A written support-cell reduction and exact finite premises give cost at least 72 for every real coefficient matrix with two nonzero columns, and at least 77 when both masked sources are nonzero, against the coordinate baseline 64. These continuous lower bounds are not asserted to be attained.

The separate exhaustive census covers all 28 coordinate pairs, 511 nonzero binary label cuts and 40 small unimodular matrices: 572,320 cases, minimum 72 overall and 78 with both sources nonzero. The all-real argument reduces residual supports to six scalar representatives and added-reader supports to five types, then sums independent term minima as an optimistic lower bound. This is not an inference from a failed search.

Only research/p112-masked-walsh/ is added. It contains the proof, source pins, deterministic certificate, geometry, two independent rank implementations, controls and local sprint claim. Exact base: f567884aeb8851b6e794af1184119a3bfbe1e631; OS assignment seed 48, structural decoder research. No coordination issue existed at the original claim. The final October 14 refresh read the new coordination/handoff discussions and checked relevant P112 PRs through #1656. Main was 5e554ff58d01921747d27177045c0061f31e1ad1, selected #1094 at kappa 0.001672660197076918; the consumed source heads stayed unchanged.

Credit Mihir Srivastava's #845 at 125ce25bdae421627b56077adc14940c2d0aa52f for the retained exact geometry and single-lift predecessor; DreamingOfClouds' #717 at 4a97676294f9fa413b65986e9984fddeda71e4cc for the model/open problem and #446 for the inherited copy interpretation; Joel Pulikkan / GamingPuzzled's #620 for the baseline 64. Source files and their notice are preserved verbatim. #1133's rational single-lift descent is distinct.

Concurrent overlap: Philipp / @mightyOil's #1197 excludes, among other cases, disjoint added-reader coefficient supports. Here both readers can use the same {p,q}; disjoint source row supports are a different condition. This gives quantitative evidence for a specific subfamily of its overlapping-reader opening. @thakrarsagar's #1327 studies a different two-sided parity lift, whose strengthened floor requires both factors off the root code; all our reader columns stay in col(X). #1191 concerns masked exactness, while our identity already holds on all pairs. #1375/#1460/#1473 use free-rectangle factorizations. Exact heads, additional adjacent work and scope comparisons are in OVERLAP.json and the README. The contribution is the specialized 72/77 certificate and independent census, not a general decoder impossibility or priority claim.

Even granting preparation, source access, helpers, frame changes, restoration and recursive children zero extra cost, the copy bill loses. This does not exclude arbitrary two-total sources, dense reader changes, other masks, interacting delivery programs or improvements to other expenses. No physical supplier, new kappa, full multiplication proof or human peer review is claimed. Inherited compiler, hosting, precision/tape and analytic assumptions remain.

Validation

Python 3.10+; standard library only:

python3 -I -B research/p112-masked-walsh/verify.py
python3 -I -O -B research/p112-masked-walsh/independent_check.py
python3 -I -B research/p112-masked-walsh/test_verify.py

Expected markers: PASS_P112_MASKED_WALSH cases=572320 minimum=72 baseline=64 no_new_kappa, PASS_P112_MASKED_WALSH_INDEPENDENT, and a passing unittest.

CPython 3.12.13 on macOS: all checks pass. The independent direct-pairing implementation matches every histogram and all 14,308 relaxation cases and checks 28 full endpoint identities. Isolated optimized relocation, source/geometry/receipt corruption rejection, missing-relaxation and duplicate-pair rejection, export preservation and staged whitespace pass. Uploaded file contents match the locally verified commit byte-for-byte. Repository-wide make verify, selected supplier cold replay, hosted CI and Lean were not run for this isolated package. The selected frontier and upstream/ are unchanged.

Prepared for Denis (@just-mn) with substantial OpenAI Codex assistance in research, mathematics, implementation, independent AI replay and writing. Apache-2.0; full attribution and limitations are in the package.

[Mister-Meeseeks]

Batch review at f87724057cc5fc5192fe2904675a563273612851.

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

Reviewed scope: Two masked-Walsh totals have COPY floor72,or77 with both sources nonzero, via complete finite support-cell relaxation over real coefficients. Full572,320-case small-unimodular census and independent pairing replay; broader sources/readers and rank-shape gains excluded.

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.