[aliromman]

This construction gives conditional κ = 151333/50000000 = 0.00302666, a gain of 0.00000431 over PR #5377 and 0.0000048842 over the selected final PR #4610. Eight independent helper relations remove all eight backward rank-eight centers and eight later rank-one jobs, with all eight lost rank-nine helper targets charged.

The proof and complete bill, exact witness, literal word, result and source pins define the review object. The branch starts at canonical main 1ec1119972c9aefe42525172d07b332fcff81cb4. All runtime inputs are present at that base; the pending predecessor package is not a runtime dependency.

Each backward call retains 224 independent main fields and 400 independent helper fields. Exact row reduction prepares eight owned helper pivots. Eight simultaneous snapshots replace the centers in the same physical frames. Sixteen unrestricted source identities prove the COPY and zero replacements; a separate replay checks all 624 retained inputs and all 17 arbitrary owned entry fields. Forward calls and the p16 controller keep their source programs.

The complete controller removes 128 H1 and 128 H8 paid jobs, and 128 H9 productive targets, relative to the selected source. Its moment gain is 128(1 + 8^t - 9^t) > 0 for 0 < t < 1. The bill has 64,148 paid jobs and 10,800 productive fields, with rank surplus 752. Compared with the one-relation result, this removes 224 further paid jobs and 112 further helper targets. All preparation, COPY, CLEAR and eight new snapshot slots are charged: the local scalar allowance is 158,744, the route allowance is 4,834,000, and the complete raw nonrecursive bound is 181,098,496 < 10^9.

Two rational interval engines put the fixed-profile crossing in (0.00302667476807, 0.00302667476808). The claimed bound leaves more than 1.476807e-8 raw headroom. The tested saving 0.00302667, reserve 1e-9, both completion tails, finite compiler and all 54 strict plus four functional outer predicates pass. This ceiling applies to this complete profile, not all relation systems.

Reproduce with CPython 3.12 or 3.14 and the standard library:

python3 -B scripts/research_package.py --package research/kappa-joint-relations-20261023

Both registered modes print:

PASS_CONDITIONAL_KAPPA_00302666_EIGHT_JOINT_RELATIONS

Normal and optimized registered checks passed on CPython 3.12.14, and the direct author check passed on CPython 3.14. All seven native and five cost/outer rejection controls passed. Metadata/pins, 19 committed-file bindings, official export integrity and both read-only Git-free replays passed; all 20 exported files, including the manifest, remained unchanged. The optional full-history preflight was not run in this shallow checkout. Hosted CI is separate and is not claimed here.

The full source phase lowering and ordinary BIT construction are authenticated and reused. This package does not rerun physical tape, numerical precision, analytic decomposition, prime avoidance or integer recovery. No expanded compiler word, Lean/Docker run, independent human acceptance or selected-frontier update is claimed. The result retains the five selected framework assumptions recorded in the current admission review.

Credit: Ali Romman with substantial OpenAI Codex assistance, Apache-2.0. This extends our frozen one-relation proof and verifier. Cao Yifan supplied the final native word and phase evidence in PR #4610. Cinna Davis supplied the controller/helper lineage in PR #4470. Chris R supplied the independent source admission in PR #5241; it does not admit this changed inventory. zxv supplied the cache contract and row/lifetime handoff in PR #5283. The linked proof and pinned source notices retain Dugongue, zxv, Jacob Alan Sussman, Cinna Davis, ToikY and the earlier native/compiler/transfer credit.

The canonical claim and handoff records the live overlap checks and the distinct neighboring scopes. This PR contains one new isolated research package and leaves the predecessor packages and selected record intact.

[Matarener]

Independent reproduction of the sixteen source identities, the relation system and the complete bill (Mark / Matarener, local run on Windows 10, Python 3.12). No new κ.

• Registered replay of research/kappa-joint-relations-20261023 passes locally, normal and optimized (PASS_CONDITIONAL_KAPPA_00302666_EIGHT_JOINT_RELATIONS, ~5 min).
• A separately written free-word row engine (no code shared with the package; only WITNESS.json is read) on the unrestricted PR4610 backward word confirms all 16 identities: for each of the 8 centres, C0 input before the centre event minus the named snapshot equals (±1/4)·D_k, and the named role's row before the zero event equals (∓1/4)·D_k, with the stated frames (snapshot frame = centre output frame; zero event is a rank-1 climb from that frame to frame 1). Snapshot rows have 61–75 formal terms.
• Own row reduction of the 8 relations over their 44 helpers: rank 8, pivots {225, 227, 229, 230, 241, 245, 257, 288} as stated, all preparation coefficients dyadic with |coefficient| ≤ 1. Substituting the pivots, every centre input equals its snapshot exactly, every zero-site row is empty, and all 624 non-pivot endpoints (rows and frames) are identical to the unrestricted source. All 8 snapshots are taken before the first centre and all 8 zero sites lie after the last centre.
• Root of the complete profile (PR4610 ARITHMETIC.json targets/paid, minus 128 H1 and 128 H8 paid, minus 128 H9 targets): 0.00302667476807454, inside the stated interval; 64,148 paid jobs, 10,800 productive fields, rank masses 237,982 / 238,734. The Raise conditional κ to 0.003026674765 with a smaller completion reserve #5387 value 0.003026674765 is also below this root (by ≈1e-11).

Remark: the gain per site is 1 + 8^t − 9^t; without the eight zero-site rank-1 removals the same profile change would lower the root to ≈0.0025, so the exact zero rows carry the result; they are verified above. Not covered here, as in the whole chain: owned erasure of the nonzero C0 content, phases (receipt reused), compiler/precision/tape, BIT. Scripts audit5383.py (standard library + mpmath) available on request.