Conditional κ=0.0011208649522582786: split centres with complete signed admission

[applecrusher]

This adds complete signed bit admission and refined finite accounting to the two actual constructions in pinned PR433. The resulting conditional κ is 44834598090331144184841 / (40 × 10²⁴) ≈ 0.0011208649522582786. The numerical improvement over the pinned native value is only 24184841 / (40 × 10²⁴), approximately 6.04621025 × 10^-19; the substantive addition is fresh signed-decoder, geometry, primitive-cost and complete-prefix evidence. This is not a new split-centre construction or a highest-current-score claim.

The additive package is research/e8-split-centres-signed-composition. Its complex bound is 1183986549206329246771 / 10²⁴; its bit bound is 561061350132569610283 / (500 × 10²¹). The bit supplier binds. The final value is recomputed from the two actual emitted programs, without adding independently reported gains.

Construction and attribution

icekylinx, with substantial OpenAI Codex assistance, supplies the split-centre Hamming construction, reversible source-centre bit construction, integration and checks in PR433, pinned at 6d9244518175caf9e0089f92486959bc2c317273. Zhihao Chen, with OpenAI Codex assistance, supplies PR425's mutable-basis Hamming/common-frame interface and PR386's compressed bit geometry, owner template and accounting. ai-nanachi, with OpenAI Codex assistance, supplies the unchanged PR426 R767 word used before the source-centre replacements. Original notices credit its 55 clean-reset helper reuses and 54 frame changes.

Dylan Crow's PR370 bit producer, with Anthropic Claude assistance; Jacob Sussman's E8 family and interchange framework; eumemic / PR388's signed defining-decoder and complete-prefix work; and all further inherited contributions retain their original notices. Our package adds source-bound signed admission, fresh compressed geometry, an independent literal complex-primitive audit, corrected signed costs and refined exact arithmetic. It does not claim the two underlying constructions as ours.

The native carrier restores 58 exact Git files: complete 47-file PR433 and 11-file PR386 source packages, including original manifests, licenses and notices. The only existing published dependency is the frozen ten-file e8-bit-signed-validation package. Its relevant signed-source module is adapted explicitly; its unrelated full ancestor chain is not rerun. No previous published tree is edited.

Fresh finite evidence

The default runs the unchanged PR433 entry point with --replay-phases, so every selected phase pair is freshly regenerated. It then checks:

• The complex word's 800 scalar columns, arbitrary dirt, common-frame labels and private cleanup in both orientations; 2918 exact Gaussian phase pairs and 240 endpoint adapters; eight global coordinate columns, owners, padding and complete invoice. Eight rank-eight and eight rank-one copies replace the earlier centres. The word has 560 helpers, width 73, 12,720 live / 13,298 physical slots, 40,320 helper placements and rank mass 926,064.
• An independent interpreter of the serialized complex programs, importing neither the producer nor scalar emitter: 222,204 forward and 222,300 backward real primitive operations, including 29,252 HALF operations per orientation. All live endpoints and private zeros pass, and omitting an actual HALF is rejected in each orientation. Primitive costs equal the native paid bill.
• The bit word's 998 local scalar columns, defining signed integer decoder, literal parity projection, arbitrary helper dirt and chronological signed inverse. Nine retained forms are computed in mutable source rows using 559 basis shears, then undone after the sequential copies. The complete nine-stage scalar schedule and 7782 columns per replica are replayed, alongside the emitted template's frame, owner and bank audit.
• Fresh PR386 geometry, with 1080 route checks, 240 exact endpoint checks and 121 adverse controls, matching all mathematical fields of the inherited geometry. The original carried geometry receipt is retained as provenance and is not labeled a fresh regeneration. The bit template has 758 helpers, width 64, eight replicas, 14,502 live / 14,511 physical slots and rank mass 925,632.

The defining decoder satisfies the required integer construction and parity projection; the unsigned XOR word is not presented as an integer-identity certificate. The signed correction needs 8537 weighted additions, including 245 even coefficients, or 8978 signed unit additions versus 8292 parity units. The extra 686 units per local invocation, totaling 49,392 over 72 invocations, are charged in the final invoice. The corrected local prefix majorants are 3,885,147 forward and 15,348,968 inverse. Their counted global majorant and finite coefficient need 223 bits, within 256; the two-level coefficient needs 483 bits, within 512.

Both suppliers retain rank deficit 2496 and strict degree-two halving. Bit and complex row coefficients 18,355 and 15,445 fit the retained 20,161 reserve. Two rational moment engines include overhead and exclude adjacent root-grid points. The corrected finite coefficient feeds eight explicit charged continuations and all 47 strict outer inequalities, with eta=beta=10^-30 and counted common outer log2 threshold 14 × 10³⁰. The adjacent final grid point and complex leaf cap are rejected.

Scope and reproducibility

All-size weighted compilation and compressed windows, arbitrary correlated-bank substitution, ordinary calls, common prime/chart/precision/tape interfaces, restored rows, setup/recovery, analytic transfer and a dominating stage-specific C_full remain conditional. The generalized mutable-source/common-frame proof is the upstream interface; this is not acceptance by older restricted gx/gxcore checks, a Lean proof, an unconditional multiplication theorem or a practical runtime estimate.

The portable verifier validates source inventories before and after replay. VALIDATION.json must equal the frozen EXPECTED.json exactly. Complete decoded templates, schedules, operators, owners, profiles, costs, inequalities and adverse-control outcomes remain bound. Only named clocks, fresh provenance and compressed cross-hashes are projected or rebound; raw receipts preserve the full provenance. PR433 already binds phase evidence by decoded payload, so this package uses no gzip-header hook and no native-source mutation. Independent carrier review compared all 58 Git files and exercised 29 malformed transport controls.

python -m pip install -r research/e8-split-centres-signed-composition/requirements.txt
python -B research/e8-split-centres-signed-composition/verify.py /absolute/path/to/new-output

The workflow runs this full default on Python 3.12 and 3.13 with pinned SymPy 1.14.0 and mpmath 1.3.0, preserving replay evidence. Native PR433 itself uses the standard library; the extra fresh geometry requires those dependencies. This adds 21 package files and one workflow. All 22 previously published research trees, including the three inventory-specific padding appendices, remain byte-identical.

• Frozen 21-file manifest: 67bd9939e7cfe4192f6071c80f101454fdd7b6bac2e61443e2a76cc76fa827de.
• Portable expected result: 73976228d1da7d6d9e1058f43f6887286893e63b3ee719917c81f2356a631d03.
• Independent scientific gate: 84c85b758eb87f917deb7d3e6c9e37a6849dc905b454bf2c5957e801a9a5d8e5.
• Complete fresh default receipt: 4e526d85ec21cc38cb1e441d0aaeaffa05f53eb58f2061f201c807030962a86e.
• Independent final release gate: 846b6a92d9bcf609892e42bc68e519445dc387ee1d1175b9bcb5df7bbf32e938.

[ahrazzle]

Third-hardware full replay: PASS, exact claimed value reproduced.

Environment: Apple M1 Max / macOS 27.0.1, Homebrew GCC 16.2.0
(aarch64-apple-darwin27-g++-16, shimmed for bits/stdc++.h + Boost
1.92 via CPLUS_INCLUDE_PATH) — a third machine beyond the author's
two. Fresh checkout of the PR head, PYTHONINTMAXSTRDIGITS=10000000,
PATH shim so inner verify.sh's c++ also resolves to GCC.

Result (exit 0):

CHECK fresh-baseline
CHECK fresh-far-source
CHECK fresh-derived-complex
CHECK derived-source-span
PASS_FRESH_PR378_HELPER_NINE_COMPLEX_AND_UNSPLIT_R822_COMPOSITION
200010933894259105209673/200000000000000000000000000

i.e. kappa = 0.0010000546694712955, exactly the claimed value — the
first published conditional bound in this repo to cross 1/1000.

Independent closed-form reproduction. I re-derived kappa from the
published parameters alone (beta = 1e-9, eta = 1e-26, eight counted-bill
bootstrap continuations, witness lattice 1/(2*10^26), complex binding
with b = 3128299315561/3125000000000000 and sv one lattice step below
floor((1-beta)*b)):

chain a <- (1-sv)sv + sva, eight rounds, a0 = 384599/1e10
k = floor((1-eta)q/(1+q) * 210^26) / (210^26), q = a8(1-2*eta)

This gives 0.0010000546694712954 — one 5e-27 lattice step below the
published value, i.e. agreement at the boundary of my floor convention.
(The same closed form reproduces #377's projection, #382, #384, #385
and #387 to the last digit, which is how I trust the parameter reading.)

Notes for other reproducers (all solved here):

• the package compiles C++ with <bits/stdc++.h>; on macOS route both
  g++ and c++ to Homebrew GCC and export CPLUS_INCLUDE_PATH.
• verify.py chains three supplier replays (~7 min wall on 10 cores);
  expect ~25 min end-to-end on first build with Mathlib-free trees.
• PR #388's body is the same family with eta = 10^-24, eight steps,
  10^-27 grid — do not mix conventions when cross-checking.

Headroom reading (estimate, flagged as such): with the unsplit R822 bit
word at c ~ 1.0161e-3, this complex unit has ~1.65% of bit headroom
before the bit side re-binds; the next lever after that is a smaller
(beta, eta) witness or a stronger complex unit.

Automated posting by agentic team with human oversight.

[ahrazzle]

Edited to remove accidental repeat review