Conditional κ = 0.001661961039655941: one E4 call fills four T37 gaps

[davidNbreslauer]

Change and mathematical scope

PR942's T37 construction leaves six rank-one idle dimensions. This changes one of its 333 U8 calls by reversing a four-helper entrance-promotion group and places the four enlarged helpers into four of those gaps. The resulting conditional exponent is κ = 0.001661961039655941, an increase of 0.000000000491941994 over the selected PR942 value, 0.001661960547713947.

The change keeps the scalar word and endpoints fixed. Four helper widths increase from four to five; four whole-helper exchanges and the associated offset updates consume four idle dimensions. The exact global profile becomes:

{1:848478, 2:112562, 3:148522, 4:53942, 5:23976, 8:2664, 48:11086}

There are 33,152 live families, 33,666 physical families, rank mass 2,408,256 and deficit 11,840. The complex root is certified between the adjacent rational endpoints 0.001664727752321934979327 and 0.001664727752321934979328. The outer reduction charges the full 1024-bit complex allowance and gives the exact nominated κ, 1661961039655941/10^18.

The package contains the literal local edits, all 20 changed owner records, a deterministic reconstruction, a written dependency argument, pinned sources, strict assembly and separate arithmetic certificates. The research note explains the construction and proof obligations. Changing a local supplier's width inventory changes a hypothesis of PR942's homogeneous U8 ceiling.

The tested base is 407d52ff15b2217b6e22bce44728b4c3907b6515. Before submission, upstream abdc6a70 still selects PR942 and has no changes to this package's pinned source files. This adds only research/u8-e4-gap-fill/; the selected record and upstream/ remain unchanged. Larger concurrent claims, including #987, #1004, #1005, #1018 and their later T16 descendants, are acknowledged. This contribution is the separate entrance-frame and gap-filling mechanism on the fully replayed T37 baseline; it makes no largest-pending-κ claim.

Validation

The complete local finite replay passed on macOS arm64 with Python 3.12. A fresh cold PR942 ancestry replay passed first; the candidate verifier then revalidated that complete evidence closure before reuse. VALIDATION.json records the environment, receipts, hashes and limits.

• Native endpoint, scalar, frame, wrapper and phase checks passed, including 25 negative controls and 2,949 phase edges.
• The separate event/owner/chart recount checked all 167,832 helper roles, 333 calls and 9,628 actual chart types. It rejected eight owner mutations and omission of each of the 20 owner edits.
• Emitted controller audit, complete invoice, all nine finite-admission checks and all 47 strict outer inequalities passed. Required prefix: 779 bits against a 1024-bit allowance. Physical row coefficient: 17,495 against the retained limit of 20,161.
• All 23 controller mutations, 114 strict admission mutations, 12 bit-supplier controls, seven payload boundaries and 11 payload mutations passed. The two-provider control counts are derived from the pinned generator and actual dispatch; verifier predicates are unchanged.
• Mercator, base-two and separate positive-binomial arithmetic agreed. Final binding rechecked 122 generated artifacts. The selected-source inventory and whitespace checks passed.

Cold reproduction, from the repository root with a new output directory:

python3 -B research/u8-e4-gap-fill/verify.py --work /absolute/path/to/new-e4-replay

The research note also documents reuse of a completed, source-bound verify_frontier942.py output. Assertions must remain enabled. Repository-wide make verify, hosted CI, independent human review and a full proof-assistant formalization are not claimed.

The inherited all-size/compiler, common-label and mutable-source realization, physical hosting, precision/tape, analytic-transfer and bit H1–H5 assumptions remain open. Finite certificate success does not establish the full multiplication theorem.

Credit: BlockInvert's PR664-style p6n-k4 entrance promotions; Jeremy Kothe's PR799 and PR869/870 U8 lineage; Jonathan Capalbo's PR942 owners and 1024-bit integration; Chafik Boukhalfa's PR685/890 and Hao Yan's PR858 owner lineage; and the native, bit, arithmetic and outer suppliers retained in the linked notices. Prepared for David Breslauer with substantial OpenAI Codex assistance. The new material is Apache-2.0; retained source licenses and notices remain in force.