[1337isnot1337]

Result

This contribution proposes conditional kappa = 0.00302015, up from #4447's 0.00300123: +0.00001892, or +0.630408%. It changes the complete native supplier and then uses an exactly replayed 16-pair query-thinned controller. This is an incremental construction improvement, not extra digits of the old root.

Package: research/joint-helper-lattice/. Start with README.md and PROOF.md; exact artifacts are PATCH.json, NATIVE.json.gz, CONTROLLER.json and RESULT.json.

Construction

The search jointly changes helper entrances, helper exits and intermediate scalar contact frames. Unlike separate endpoint sweeps, it includes the nonlinear productive-helper credit in the same optimization. Relative to #4447, the patch changes 38 entrances, 47 exits and 414 scalar contacts. Both scalar orders and all coefficients, native warp events, main boundaries and owned-center operations remain unchanged. All 408 helpers remain independent, strictly productive arbitrary inputs.

The new local profiles are:

productive helpers: {4:28, 7:7, 8:47, 9:326}
paid jobs:          {1:2237, 2:388, 3:494, 4:147, 5:36, 8:8}

There are 53 fewer paid jobs per full local call, 3,363 -> 3,310. Both target and paid rank masses rise by 37, so no old target credit is kept after dropping paid work. The eight rank-eight owned-center jobs remain charged.

For every real 0<t<1, the exact local surplus gain is

17 + 126*2^t - 50*3^t - 42*4^t + 2*5^t
   - 22*7^t + 5*8^t + 17*9^t > 0.

The verifier binds this expression to the actual old/new profiles, proves its value is zero at t=1, and bounds its derivative strictly below zero on a rational interval cover of [0,1]. The floating-point mixed-integer solver is discovery evidence only; no solver/global optimality claim is used.

The new p16 controller has 18 full native calls and 15 copied-query shears. Its complete invoice contains 10,928 productive fields and 64,620 paid jobs, with rank masses 239,886 and 239,134 and surplus 752. Its raw root is enclosed in (0.0030201659770, 0.0030201659771). The input count differs from p17; total call counts are not presented as a like-for-like runtime speedup.

Paid completion and splitting use tensor power 1,024, cutoff 20^1024, productive floor 21^1024, and newly checked limits for both tails. In particular, the p16 completion tail does not satisfy the old 0.95 limit; its explicitly certified 0.97 limit is used instead. The logarithmic compiler gives sufficient uniform rank 2^64215200000000. The unchanged BIT construction supports ordinary saving 0.0034. All nine outer rows, 54 strict predicates and four functional guards pass.

Verification actually completed

At head b9a13c1673e0e9697de410de16e8bbf3207a7363, package metadata/pins and the committed-byte preflight pass (66 package/input files); the worktree is clean. All three required proof gates completed separately on Python 3.12.13:

python3.12 -B research/joint-helper-lattice/verify.py --part dependencies
python3.12 -B research/joint-helper-lattice/verify.py --part native
python3.12 -B research/joint-helper-lattice/verify.py --part arithmetic

• Both native orientations pass two retained literal interpreters on all 632 arbitrary-input columns.
• 423 distinct transport/helper pairs are lowered in both orientations with exact Fourier ranks, Clifford tableaux and absolute phases.
• The p16 rational/Laurent word, scalar boundary factorization and independent literal event replay pass.
• Two rational interval engines, the uniform local-gain proof, fixed-profile ceiling, paid tails, finite compiler, BIT conversion and complete outer ledger pass.
• Controls reject false transport ranks, wrong query orientation, an unpaid full stage, the old invoice at the new saving, and unsupported outer parameters.

VERIFY.json registers all three gates as required. Their combined launcher was not completed locally because they collectively exceed one execution-call limit. Each registered command completed separately; full outputs and times are in REPLAY-LOGS.json. No hosted CI pass is claimed at opening. --write is an authoring path and is not substituted for these proof gates.

Dependencies, credit and limitations

This branch is stacked on #4447 at 322e0ec764aa7ef7801383feb1f656e460dd8bed. Its inherited #4440/#4441/#4447 files and two documentation changes are predecessor work, not new contributions here. The new work is isolated in research/joint-helper-lattice/. Canonical main at the recorded checkpoint was 2a13eb1827784206202db90e5dc8327cf8a250db. Exact source blobs/SHA-256 pins are retained.

This is an author-checked conditional construction awaiting review. The retained all-column/native lift, physical primitives, owned erasure, whole-job matching, finite-template closure, fixed-tape and precision bounds, analytic decomposition, prime avoidance and exact recovery remain premises. The enormous compiled word is specified algorithmically, not materialized. No unconditional theorem, selected-frontier promotion, practical speedup, global optimum, formal-kernel verification or independent human acceptance is claimed.

Credit zxv's native/controller/higher-moment and logarithmic work, Jacob Alan Sussman's mixed-target integer flow, Cinna Davis's prior padding/query-thinning/retiming work, and all unchanged credited scalar, Clifford, geometry and outer-ledger sources. Prepared for Cinna Davis with substantial OpenAI assistance; Apache-2.0.

The active-session frontier was refreshed repeatedly, including before publication; no stronger kappa claim was found in those checks. The next bounded research question is a costed scalar reordering or changed warp word, not further claims of improvement from the same optimized fixed graph. See TASK.json.