Conditional κ = .003641239650184 (+4.78276042e-7 over #7122): recharted helper split

research/p104-recharted-rank8-split/APPLICATION.md

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+# Conditional application of the rechart and added helper split
+
+This applies Rohan Arun's #7113 component proof and the retained changed-native adapters to the exact #7122 rechart plus this additional rank-eight split. The full frozen source closure remains intact. PROOF.md supplies the source-specific normalized-coordinate rechart argument and the arbitrary-input helper replacement. Both stages' target/paid changes are recounted; no source's numeric receipt substitutes for current verification. The five assumptions and selected completed BIT predecessor remain. The template is the [#7000 application review](https://github.com/CrocSwap/integer-mult-bounds/blob/e54523517d0f03e58352e647c403a5af3e525a57/docs/research/pr-intake-evidence/pr7000-application/PROOF-REVIEW.md), including its BIT ambient 134 repair. The current substitutions follow.
+
+## Native and caller operator substitution
+
+Rohan Arun's rechart changes seven helper inputs and 22 public scalar contact charts, preserves scalar coefficients/order and rebuilds every causal public transport. This removes four paid MOVEs per native while changing both paid and target rank masses by seven. The additional split neutrally advances 454's 757→924 MOVE and splits its 3→246 rank-eight target into 3→924 (rank 7) and 924→246 (rank 1). It adds 22 ADDs + 4 SCALEs and one fresh input. All owned transitions/lifetimes are retained. The normalized-coordinate rechart proof and the exact prefix/suffix relations with boundary/cut bases 7/6 are in PROOF.md. Its complete forward and adjoint replays include every owned row. All 3,048 current paid/helper chart pairs and 104 main boundary pairs are then rebuilt with the pinned compiler and distinct signed-Pauli/absolute-phase interpreter. For these Clifford maps, the signed Pauli action fixes the operator up to global phase, which the exact Gaussian check fixes. Thus every column of each chart map is determined. The fresh replay includes a deliberately wrong absolute phase.
+
+The unchanged 17-stage caller has five noncommuting quarter turns and twelve padding turns in inverse pairs. Its grouped idle bases lie over Z[1/30]. Its exact rational and integer Laurent engines agree on all 72 output fields for every formal central/body variable. In the physical substitution, seventeen disjoint eight-axis stage blocks and nine central axes use ambient 145. For each invocation, exchange the eight native complement axes with the fresh stage block while fixing the routed central axis and spectator axes. The phase-polynomial checks verify both directions. Literal A and B routes then apply commuting body operators on disjoint stage blocks.
+
+A signed rank-nine idle uses one positive rank-nine recursive request plus the charged ordinary conjugation/phase implementing the inverse on the protected central axis. Current finite signed-child checks cover widths 1,2,3; the symbolic phase geometry supplies the disjoint-axis substitution. The ordered occurrence digest is recomputed from the literal caller and current native geometry. It binds 13,600 native contact boundaries, 110,655 paid positions and 22,291 productive fields. Main-owner aliasing is rejected. Allocate 18,547 distinct (stage,helper) public inputs and 867 reserved native owned banks, plus the retained serial query/LAMBDA bank. The surplus is 1,424. The ownership discipline and owned-erasure hypothesis remain explicit.
+
+## Common-grid and fixed-root induction
+
+Scale each of the 3,744 raw caller main input fields by 675 and divide each completed caller main output by 675. The independent integer Laurent replay checks the necessary valuations 3³ and 5², rejects a lower exponent of either prime, and uses safe binary valuation 381. Every normalized main coefficient and pre-ADD product is integral on that odd grid. Substituting actual binary Clifford address operators introduces only powers of two, so the odd-divisibility invariant transfers to whole arrays and all spectators.
+
+Within each native invocation, scale all 1,491 public fields by 1125 and unscale the completed returns. The separate exact prefix/pre-product derivation and integer replay establish the native grid at fractional height 24. Fresh helpers receive their own independent inputs. The exact native endpoint ensures these helper inputs do not perturb the normalized main caller invariant. Both odd scales are included simultaneously in the unfinished-context mass allowance.
+
+The fresh forward/adjoint envelope has height 405 and context mass exponent 1172, giving c=2344. This is recomputed from the current word and caller; #7122's allowance was 2344. It remains below 4096. The same conservative ambient 153 compiler threshold funds it: Kstar has 14,093 bits and c·Kstar has 14,104 bits, below H=16495300000000000000000. The regenerated compiler gap is 1/100000000000000000. Every rounded caller bound, outer and fixed-tape gate is checked with these current values. Apply the retained full-root common-format induction with fixed padding, signed width p+h+2D+3, fractional width p+D, and 26 temporary bits. Completed child returns preserve this format. Exact finite-state odd division retains the reviewed carry/remainder invariant.
+
+## Physical upper-cost substitution
+
+The source-unit ordinary bound 777,763,024 includes the caller and native conversion services. It is distinct from the physical-machine bound. The fresh raw-occurrence counts give 1,146,361 native scalar macro inputs, 117,856 caller macro inputs and 58,182 conversions. The retained conservative factor 32 gives 42,316,768 scalar services; with the reviewed 2²⁰ service constant this is 44,372,347,322,368 times the root scalar-width envelope. Numerators of all actual native/caller instructions and both conversion scales fit 24 bits.
+
+The current occurrence count gives 298,740 monomial map slots. At actual ambient 145, the retained elementary bound 4(145+1)² is 85,264 records per map, hence at most 25,471,767,360 elementary records per raw occurrence. Use actual owner widths and these finite coefficients in the same fixed-tape stock and movement envelopes from the selected review. The positive compiler/outer slack permits enlarging the eventual cutoffs. This is a conditional asymptotic substitution, not an enumerated machine trace.
+
+The recomputed complete recurrence has the bracket and claim in README.md; both exact price engines, the compiler and 54 strict plus 4 functional outer inequalities pass. These checks establish the stated conditional component application under the five assumptions. Independent mathematical review, formal-kernel validation and maintainer admission are separate obligations.
+
+Credit Rohan Arun for the #7113 proof and implementation, Douglas Colkitt and the #7000 application reviewers, the retained compiler/precision/source authors, and the complete lineage in README.md. Adapted for this exact rechart plus helper split, retaining all peer and predecessor credit, by dillona with substantial OpenAI Codex assistance. Apache-2.0.

research/p104-recharted-rank8-split/DISCOVERY.json

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+    [
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+  "stop_reason": "First declared group found smooth positive nomination; verification takes priority.",
+  "local_only": "Discovery scripts/negative scopes remain local. Public WITNESS and constructor are independent of search. Full incumbent and candidate evidence is in the registered package."
+}

research/p104-recharted-rank8-split/ENVIRONMENT.json

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+{
+  "initial_python": "3.14.7",
+  "initial_platform": "macOS-27.0.1-arm64-arm-64bit-Mach-O",
+  "registered_replay": {
+    "python": "3.12.16",
+    "platform": "macOS-27.0.1-arm64-arm-64bit",
+    "result": "PASS both complete commands; every native and composition receipt field matched"
+  },
+  "dependencies": "Python standard library",
+  "interpreter_origin": "Official Astral python-build-standalone release20261026; isolated /tmp installation",
+  "interpreter_archive_sha256": "fbf4afef53c2a6b72bb741bc06d5671a9dfaf87bd84e5aa73ac9b3d0d589f14e",
+  "hosted_ci": "Not yet published; no hosted scientific pass claimed.",
+  "completed_utc": "2026-11-10T21:15:02.771080+00:00"
+}

research/p104-recharted-rank8-split/INPUTS.json

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