[1337isnot1337]

The p7 controller has a stronger native moment than #4414's p6 construction, but its small productive ranks prevent the previous tail bound from admitting it. This contribution pays to enlarge small productive transforms after tensor composition, raising the conditional multiplication saving to κ = 0.00268647, from #4359's 0.0023.

The exact p7/keep232127 controller is regenerated from authenticated source code, then physically audited on 121 axes. All 6,720 contacts and inverses, 1,568 main endpoints, 6,120 fresh helper fields and 2,688 whole idle calls are bound to their actual operators and phases. The complete raw invoice has 7,688 independent productive fields, target rank mass 131,723 and paid rank mass 131,115.

After chronological tensor composition, complete each productive rank below 11^6240 by a paid transform on the missing disjoint axes of the same arbitrary input field. Split every original and added paid job at 10^6240. This preserves the rank-one surplus and gives strict paid/target separation. At tested saving 0.00268648, the original moment and both paid-tail bounds give an exact normalized surplus greater than 0.00000136095023295546. The source compiler and outer ledger then specify sufficient uniform rank 2^16201843136005340321, while retaining ordinary BIT saving 0.0027.

The self-contained package research/pr4359-p7-padding/ includes the complete native audit, paid-completion proof, replacement expanded-word ownership/precision constants, all nine outer cost rows, 54 retained predicates, three functional guards and rejection controls. Entry points are README.md, NATIVE.md, TENSOR.md, ENDPOINT.md and RESULT.json.

Validation: exact Git-blob/SHA256 source authentication; independent native and proof reviews; separate rational interval implementation; optimized local Python 3.14.8 replay; clean Pyroam Python 3.12.13 optimized and registered replays; main-provenance ancestry and committed-byte preflight. The registered follow-up also reproduces 209 explicit algebraic profiles for pair counts 2–20: only the selected p7 profile passes at 0.00268648, and all fail at 0.00268649. This finite screen is not a global optimum or native admission for every candidate.

Sources are pinned to #4359 (da637c87), #4405 (873d1b3c) and #4412 (9f0df9bf), on canonical main 12fa8c5. This continues #4414's native geometry and extends #4412's composition proof. The unmerged source heads are not represented as publication ancestry.

Conditional research result awaiting review. The all-column arbitrary-input native words, whole-job frame realization and monomial tape costs, analytic decomposition, physical primitives, fixed-tape model and outer transfer remain explicit premises. TENSOR.md discusses the related #4375 child-interface audit: the construction requires a single rank-r*f child and does not substitute f scalar copies. The enormous word is algorithmically specified, not materialized. No selected-frontier change, unconditional theorem, practical implementation or global optimality is claimed.

Native source and controller: @zxv. Underlying mixed flow: @jacobalansussman / Theorem 1G. Predecessor attribution and notices are retained. Prepared for @1337isnot1337 with substantial OpenAI Codex assistance; Apache-2.0.