Research: Cartesian source-destination grid cost screen

[1337isnot1337]

Result

This is a scoped negative result for a Cartesian source-destination grid child. For block operands X_i and Y_j, the exact sparse block identity A_(i,1)=X_i, B_(1,j)=Y_j gives (AB)_(i,j)=X_iY_j for every pair. Under the generic width-r matrix-product charge, one block costs r^(2-a) size-n child units versus r^2 for separate products, so its ratio is r^(-a).

On the bounded domain r<=51 and 0<=a<=1/99, the maximum possible saving is below 5%. The exact certificate is 51*19^99 < 20^99, giving 51^(-1/99)>19/20. The first integer width reaching 5% at a=1/99 is 161. Input gathering, source access, copies, helper use, frame changes, output storage, dirty restoration, hosting, precision and interface transitions are all priced at zero in this favorable screen.

This does not construct an admissible selected-word window, establish a replacement global ceiling, exclude different bilinear algorithms, or claim a new kappa.

Reproduction

• python3 -I -B research/complex-cartesian-grid-screen-seed506af/verify.py passes 15 exact formal-polynomial identity fixtures, two wrong-placement controls, exact integer threshold checks, and an integer 99th-root interval.
• python3 scripts/research_package.py --package research/complex-cartesian-grid-screen-seed506af --check-only passes source pins and package metadata.
• The registered replay launcher requires Python 3.12, which is not installed locally. The verifier passed directly under local Python 3.14.7.

Provenance and coordination

• Cycle seed 506af17109f2b8945b6b47236c53520b; track complex-mechanisms; task complex-rank-shape-decoder; priority revision 3; exact base 0b1cccf86e2f5132ed286a26ef0a681a495740ea.
• Selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.
• Claim and handoff: Larger κ: proposals that change the current cost structure #1167 (comment)
• Related work: PR Research: source-bound screen of diagonal rank-16 child batches #1779 diagonal batching, PR Add exact C3 gather rank-shape obstruction #1814 three-source gather, PR Research: nonconsecutive commuting C3 fanout schedule #1865 one-source fanout, and PRs Research: rank-two interface candidate for paired frame transitions #1764/Research: higher-rank paired boundary at a single ADD gate #1836 joint boundary children. The Cartesian grid output is distinct from those patterns.
• Prepared by Cinna Davis with substantial OpenAI Codex assistance in mechanism selection, derivation, verifier implementation, and writing.
• Final overlap refresh at 2026-10-14T12:57:23Z covered 535 open PRs and 516 Discussion records with complete pagination and no warnings. The only text match was this package's own README; no live PR or Discussion matched the Cartesian-grid terms.

[Mister-Meeseeks]

Batch review at 72804aaceac1a62a1636ee329330a13b9004ac96.

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

Reviewed scope: Sparse block-product identity and exact99th-root screen show less than5% local-work reduction for the declared generic width-r child at r<=51,a<=1/99; first qualifying width161. No alternative tensor algorithm or actual selected-grid placement is 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.