Classify sharp noncubic ranks of the H6 odd-Gram family

[RossRichardson]

Summary

For every characteristic-zero odd function on the attainable H6 Gram values with f(2)=0 and f(4)=1, the canonical 28,800-port root-product principal block has sharp minimum rank 9,424 for genuine quintic interpolation and 20,400 for nonzero septic coefficient. Explicit rational parameters attain both minima; the cubic member has root-block rank 800. The proof classifies the full two-parameter family, including complex coefficients.

Under the explicit width-17 FULL-to-zero scalar rank-one return-episode premise, the noncubic rank lower bounds force paid rank at least 160,208 or 346,800. Each exceeds all 30,960 ports before compensation, so this particular kernel/factor route cannot improve its inherited coverage deficit. Sequential reuse counts separate episodes, not distinct physical helpers.

Scope and limits

Only research/h6-odd-gram-rank/ is added. The result concerns the root-product principal block; sharpness does not assert the whole H6 matrix has the same rank. It does not bound arbitrary dirty helpers, nonisotropic kernels, operator factors, partial-frame networks, changed geometry or a different coverage law. No new kappa, physical supplier or selected-record change is claimed.

The live primary e8_cubed.py remains pinned to original Git bytes at 20ecf2891d80b5acf18d980b2164fb44ea7faa90. Its geometric binding is a written inspection of e8_shells/construct_shell; the NumPy shell generator and global H6 pipeline are not rerun. The all-parameter result is a written exact case proof with finite supporting replay, not finite-sample extrapolation or formal-kernel verification.

Test plan

• Two separate standard-library implementations: each checks all 57,600 association-square entries, derives six eigenforms and checks 18 actual 28,800-column actions.
• Exact five-projector algebra, complete parameter cases, both attaining witnesses, four producer coefficient/normalization controls and one independent normalization control.
• Registered isolated optimized Python 3.12 replay with warnings as errors, fresh child bytecode namespaces, authenticated inputs and external output only; 120-second cap.
• Exact commit f8768756d8ac4abb263295170a436ba54562fdcd: committed-byte/source-history preflight passed for all 12 package/input files with only ancestral 20ec source history. Actual read-only Git-free optimized export replay passed in 1.82 seconds with every byte unchanged. VALIDATION.json retains earlier precommit measurements.
• Automatic exact-head hosted CI 39129473617 passed for f8768756d8ac4abb263295170a436ba54562fdcd; no manual rerun.

Sources and credit

Credit Dominik Scholz's E8 geometry/factorization in PR606, DreamingOfClouds' H6/Walsh proposal in Discussion1167, and the complete compensation/encoder audit in PR1830, with their retained predecessor lineage. This adds the complete normalized odd-function classification beyond the cubic kernel; no priority is claimed for E8 association algebra or generic rank subadditivity.

Ross Richardson, with substantial OpenAI Codex assistance in derivation, separate audit and packaging; Apache-2.0. Separate AI implementations are not independent human review. Exact source pins, script adaptations, proof, scope and reproducible checks are included.