[petalas]

Summary

Prove a sharp scoped obstruction for Ross Richardson's PR #3649 nonisotropic-kernel question. On the canonical 28,800 H6 E8 root-product ports, every characteristic-zero fitting kernel F(s,t) that depends separately on the two root inner products and is odd under simultaneous sign reversal has rank at least 800. Equality uniquely forces the existing cubic F(s,t)=((s+t)^3-4(s+t))/48.

PR3649: odd f(s+t), two free parameters
This result: odd F(s,t), six free parameters, no exchange symmetry
Exact minimum: 800, uniquely the original cubic

Claim type: scoped mathematical proof with an exact finite support certificate. The new work is the minimum and equality theorem for the larger family. The existing cubic witness and five-projector root algebra are cited premises, not re-verification deliverables. Only research/h6-bivariate-rank/ is added. Advance CLAIM.

This contribution was produced by Nick Petalas. Credit Ross Richardson #3649/#1830, Dominik Scholz #606, DreamingOfClouds' H6/Walsh proposal and the retained predecessor lineage. Independent human and formal proof-kernel review are unclaimed.

Evidence

Before, the retained odd-Gram theorem left nonisotropic kernels outside its scope. After, the twelve tensor sectors and six fitting equations give a complete exact optimization for the separate-inner-product family. Any sector of rank at least 3920 loses immediately. The remaining family has two independent parameters; each contributes at least 400, with a unique attaining value.

• Complete rational feasibility checks enumerate all 4,096 sector supports. Exactly 1,764 permitted supports admit a solution; these supports may include zero coefficients. The unique minimum permitted support costs 800 and determines all coefficients uniquely.
• The displayed nonzero minor, affine solution identities, all 25 cubic entries, 300 basis-parity identities and 13 explicit controls pass. RESULT.json is generated and binds all five other package files by SHA256.
• Direct normal and optimized isolated verification pass. Both VERIFY.json commands were timed locally at 0.594 and 0.535 seconds, respectively; the full registered package invocation passed in 1.365 seconds. Each registered command has a 120-second timeout. These are check durations, not performance comparisons.
• Metadata/pins, exact six-file committed-byte/source-history preflight, committed export integrity and an optimized Git-free exported replay pass. Ruff 0.14.0 E/F/B/UP/I passes. AST inspection confirms every function is below 60 lines. Local interpreter is CPython 3.12.15, macOS arm64.

From the repository root, replace the local interpreter path with any CPython 3.12 installation if needed:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/h6-bivariate-rank/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/h6-bivariate-rank/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/h6-bivariate-rank
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/h6-bivariate-rank --check-source-history

Expected markers: PASS_H6_BIVARIATE_RANK minimum=800 unique=cubic supports=4096, PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed, and PASS_RESEARCH_COMMIT_BYTES. All sources are pinned historical provenance at main ec2780e. No external runtime input or predecessor executable is imported. The selected-source metadata baseline passed as orientation only.

Limitations and unrun obligations

This theorem assumes characteristic zero, simultaneous oddness, dependence only on the two inner products, fixed root-product ports and the retained fitting zeros. It does not bound arbitrary coordinate-dependent or operator-valued kernels, nonodd kernels, changed geometry or positive characteristic. Rank 800 is a principal-block scalar rank, not the full 30,960-port H6 rank or a physical helper count.

Physical/native realization, partial-frame visits, copies, compensation, restoration, hosting, stock, precision/tape, compiler and complete outer transfer remain unpriced. No new supplier, kappa, selected-frontier change, full selected cold replay, Lean or Docker result is claimed. All registered local checks ran; independent human and proof-kernel review remain unclaimed.

Open next questions

Can a coordinate-dependent kernel distinguish root pairs with the same (s,t) and reduce the rank? Start with one explicit orbit refinement and all fitting constraints. The present projector diagonalization no longer applies. Neither such a fitting kernel nor its complete physical invoice is supplied here.

Related open work

No open scientific PR studying this H6 bivariate kernel or root-product minimum-rank quantity was found in the refreshed open-PR/claim searches. The open H6 search returned #1903, a voluntary documentation proposal; it computes no kernel rank. Merged #3649 supplies the cited isotropic result and open question. Sparse local notes and unpublished work limit overlap coverage.

CI notes

CI pending for head 635db41. Hosted checks have not yet certified this package. Local checks above passed. No CI failure has been observed or attributed.

Merge danger

Two-way door. Blast radius: research. Six files in one new opt-in package; no shared code, predecessor files or selected records change.