Research: T16 owner concentration raises conditional κ to 0.001661642255531641

[FanPUYen]

Change and mathematical scope

This construction raises the conditional exponent saving from the currently accepted #799 value 0.001661634653669567 to 0.001661642255531641, an absolute gain of 7.601862074 × 10⁻⁹. The complex side remains binding; #799's local X4 word, all local frames and helper widths, and #589's bit supplier are unchanged.

The controller uses sixteen data replicas and jointly assigns their 144 complete invocations. Eleven explicit whole-helper edits change three idle ranks from (44,3,3) to (48,2,0), preserving rank mass and strictly lowering the recursive moment by concavity. This changes the fixed inventory covered by #795's eight-replica optimality result and the concurrent uniform extension in #850. It makes no optimality claim for sixteen replicas.

The complete accounting uses two separate eight-replica parent bank groups, actual bridge-aware owner windows, restored shared private workspace, and all invocation/wrapper costs. The final hosted controller has 72,576 helper roles, rank mass 1,041,408 and deficit 5,120. Its 401-bit prefix fits the unchanged 512-bit charge; physical row coefficient 16,469 fits the unchanged 20,161 ceiling. The conservative complex surcharge assembly also passes, at κ 0.001661642252649825.

Only the new construction package and its validation receipt are added. The selected record, inherited checkers and upstream/ are unchanged. The package contains a written argument, deterministic owner and parent builders, an independent checker, an explicit source-adapter list, frozen semantic data, source pins and one strict end-to-end command:

python3 -B research/complex-double-replica-owners/verify.py \
  --work /absolute/new/t16-x4-replay

Expected: PASS_STRICT_T16_PR799_OWNER_CONCENTRATION and kappa_decimal=0.001661642255531641.

Validation

Completed on macOS arm64, Python 3.12.14, Apple clang 21 and Boost 1.86:

• Final frozen-source default cold replay: PASS, 585.107 s, without ancestry reuse. Both suppliers, PR741/799 ancestry, full T16 construction, independent audit, invoice and exact outer assembly were rebuilt. Package and dependency bytes stayed unchanged.
• 16 package rejection controls + 2 recompression positives: PASS, 4.209 s. A separate AI agent independently re-bound the complete final result and reviewed the source changes.
• All 18 original make verify groups were covered on an isolated 7ee02b7 archive: the first invocation passed 12 groups, then exited 2 because Apple clang lacked -fopenmp. A private real OpenMP runtime fixed the environment; the remaining six groups then passed, including 559 tests in 85 modules and 20 patch checks. This is not claimed as a single successful make verify invocation.
• Ten affected/current-integration checks passed at f567884a, including selected-record integrity, reconciliation, 741/799 bindings and supplier consolidation. Final staged whitespace checks pass.

Full exact assembly, finite invoice, manifests, controls, environment and separate suite receipts are retained in research/complex-double-replica-owners-validation/.

The local word is unchanged, but its complete native, phase, wrapper and ancestry replay is included in the default command. Both rational moment engines and all 47 strict outer inequalities are checked; 11 omitted owner edits, six independent geometry/window controls, 15 emitted-controller mutations, 69 admission mutations and 12 bit-supplier mutations are rejected. The independent checker imports neither the builders nor the controller auditor. Sixteen additional package controls reject source, semantic, invoice, stage and artifact corruption; two recompression controls preserve admission. The ancestry coverage gate credits Lucas's #838 finding and Chinmay's related #847 integration; no pending implementation is imported.

Initial research used 7ee02b7c96d85f22b38d28bd7ed49bb8c1aeb261; after #799 was accepted, the witness was rebuilt against f567884aeb8851b6e794af1184119a3bfbe1e631. The κ comparison above uses #799, not the older #741 result. The local claim records predecessor overlap and the discarded search avenues.

This is a conditional finite composition, not a measured multiplication runtime improvement or a complete theorem/Lean proof. Inherited all-size/compiler, common-label/mutable-source, whole-call hosting, precision/tape and analytic assumptions remain. Hosted CI and maintainer review are separate from the local checks.

New work by Hao Yan (@FanPUYen) with substantial OpenAI Codex assistance in research, implementation, independently tasked AI checks, verification and writing. No independent human review is claimed. Apache-2.0; full attribution is in NOTICE.md, including Jeremy Kothe (#799), Chafik Boukhalfa (#741/#685/#795), icekylinx, DreamingOfClouds, eumemic, Rohan Arun, and the retained controller/hosting lineage.