κ = 7.69571027349774217804511e-4: cross-group exchanges with growing-prime refinement

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κ = 7.69571027349774217804511e-4

This conditional finite construction improves frozen #323 (769557449978832/10^18) by replacing kernel families across three entrance groups, then applying #321's uniform growing-prime threshold and eight finite bootstrap levels to the complete new word. The exact result is 769571027349774217804511/10^27; the structural change alone gives 769571023988249/10^18 under the predecessor arithmetic settings. All previously published heads and claims remain frozen.

The exchange replaces 55 old families with 51 new families, recomputing shared-donor costs globally. There are 1043 early kernel entries of total rank 1195 and three retained late rank-two kernels. The four presink retimings, 240 early restorations, seven terminal sinks and 133 reorder moves are rederived and verified on the new actual word. Relative to #323, the local paid histogram changes by {1:-10, 2:-8, 3:+9, 11:-4, 12:+4}. The word has 336227 ADDs and 48507 paid calls in 386883 records, with all 8223 arbitrary dirty helper inputs restored.

The 300-replica bank cover has 427887 banks per stage and literal stock 3291435, with 12334500 explicit helper assignments and no padding. Stock increases by 75 relative to #323; the improved paid profile more than offsets that increase. The full literal profile has 75064500 children, rank mass 328531500, and deficit 612000. Both reflected frame ledgers, 431 new frame charts, 3658 changed projector charts, and exact rank witnesses for all 14038 used frames and annihilators are checked. The complete finite coefficient is 127517659550812001 and the payload bound is 98 bits.

The arithmetic refinement preserves the inherited growing-prime regime. It imposes only an eventual lower threshold q >= 2^127-1, keeps the full exceptional fallback, and checks both independent rational moment engines, eight positive-gap finite bootstrap levels, and all 47 strict outer inequalities. The adjacent 10^-27 grid point is rejected. All-size compiler, weighted-selector, common-ancestor-chart, restored-row, routing, prime-supply, precision/recovery, and complex analytic interfaces remain explicit conditions; this is not an unconditional multiplication theorem or a constant-modulus machine.

Validation: two complete fresh replays of the same immutable 360-file package, plus a separate C++ audit of the literal forward/inverse word, COPY lifetime, dirty restoration, both paid ledgers and exact bank supply. A second rank-witness audit uses modular Gaussian determinants and exact Hadamard bounds, independently confirming all 28076 integer minors. Omitting late restoration gives nine wrong rows. Package manifest: dda12311734d6935c11b310b3b959c650fc142cbc697dc3373eade96c58ba263. Word SHA256: 72e78b2db88ef8f4a5c4aef48c66e5b317b99a6bee0cdcda70f50748d383ac3a.

python -m pip install -r research/p10-crossgroup-prime/vendor/predecessor/requirements.txt
python -B research/p10-crossgroup-prime/verify.py --output /tmp/p10-crossgroup-proof --cxx g++ --boost-include /usr/include

The package retains byte-preserved #320 and #322 sources and their original notices and disclosures. Credit remains with DreamingOfClouds for the p=10 source, Chafik Boukhalfa for #320's transcript-stage port, Rohan Arun and the cited contributors for preceding mechanisms, and #321 at c76247701762feb2c5bea5bcf302b010eb7d1bc3 for the arithmetic primitives. The new exchanges, composition, audits, wrapper and documentation were prepared with substantial OpenAI Codex assistance; inherited Anthropic Claude/OpenAI disclosures are preserved. See SOURCE.json, NOTICE.md, PROOF.md and REFINEMENT-PROOF.md.

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Correction to the validation paragraph: #326 was published before the two full immutable portable replays finished. At publication, the native construction, exact price, separate literal audit, and independent integer-minor audit had passed; the complete portable replays were still running. The statement that both full replays had passed was premature.

A subsequent committed-blob check also found that Git normalized the CRLF bytes in code/upstream-pr321-fixed_prime.py, so this committed checkout does not match its complete package manifest. The manifest guard correctly rejects that mismatch. The mathematical candidate is unchanged, but this PR is not a successfully verified portable release and should not be treated as one.

I am preserving this published head and preparing a separate replacement PR with explicit byte-preserving Git attributes. That replacement will be published only after both fresh complete replays pass and every committed package blob matches its pinned manifest. I will link the replacement here. This correction was prepared with OpenAI Codex assistance.

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The v2 portable replay has now failed at strict kernel-selection equality before candidate reconstruction. Native admission and independent literal/minor audits still pass, but #326 must be treated as withdrawn/incomplete pending a new byte-preserving package and successful full replays. The failure is preserved in /private/tmp/p10-crossgroup-t300-v2-root-proof/FAILURE.json; no published claim is being updated.