[DreamingOfClouds]

Result

Conditional κ = 0.00362105861 (exact crossing 0.0036210586137481461), +0.470% over #6931 (0.0036041048151080064). Package: research/p104-odd-scalar-native. This is a declared framework change: the word uses gate scalars 1/3 and 1/5, outside the dyadic scalar alphabet of the native verifiers. Maintainers must accept the clause; acceptance is not claimed and the result depends on it.

What changed against #6931

Roots, joint caller (byte-identical), per-port root pair, owned-ascent declaration (A6) and the BIT line (#6726) are those of #6931, and the other conditional premises are the same. The word and the scalar alphabet change.

• Move, in plain terms. Coefficients may now have denominators 3 and 5 (products 9, 15, 25, 45, 75, 225). 3 = 2²−1 and 5 = 2²+1, so multiplying is a shift and one add or subtract; exact division is one least-significant-bit-first pass with a checked remainder. Every root input is multiplied once by S = 225 = 3²5² (the least certificate), after which every prefix row and every product is dyadic and every checked division is exact. The word has 3,001 odd divisions (59 SCALE, 2,942 ADD) = 6,002 scans per call, priced separately.
• What it buys. Ten more helper splits (kernel term −55.24 per call; T +2, +7, +1, +2, +7, −9 at ranks 2, 3, 4, 5, 6, 9) and 39 stops merged into rank-2 jobs (78 paid jobs of rank 1 become 39 of rank 2; −54.07 per call). Twenty of the 39 come from a legal reordering of commuting events, tested by reorder_check.py (no pair of events sharing a register with a writer changes order). The rank mass S = 1520 is unchanged; the gain is in D alone.
• Changed clauses: the dyadic scalar alphabet (local.py:146,149, the verifier copies, native.py:65); 03-motifs.tex:57,556 for the internal arrays of one parent word; the Conditional κ .003125804 (+5.22e-7 over #5907): remove helper 470's frame-593 rung, the twin of #5907's removal #6030-specific constants of Prove deterministic precision cuts for the current selected caller (no new κ) #6900 and Prove a common precision format for the current ideal compiler (no new κ) #6901; one added operation (multiply, check divisibility, divide) at 05-layers.tex:510-513. Replacement: A = Z[1/2,1/3,1/5] with |q|+1 < 2^16 kept, a declared S certifying every prefix row, every pre-ADD product and every public row; completed endpoints stay Gaussian dyadic. Penalty: none in S; width only, 15 denominator bits and 37.48 magnitude bits per unfinished context, absorbed in W = p+h+2D+3 by 82t < d, which assumes K_star and the caller mass are recomputed for this word (they are not). Method: the merged odd grid of Stopped product-ring interchange and rational centers for conditional kappa = 7.79476e-5 #104 (icekylinx, notes/stopped-product-complex.tex:75-106) with its merged reviews. See PROOF.md.
• Exactness is conditional on exact child returns, a premise shared with the dyadic words. A truncated child makes this word fail its own divisibility check (event 26,563 at g = 44, event 24,158 at g = 50), where the dyadic baseline runs through and is wrong on 830 of 1,441 rows. Fallback: rounded division (measured exit-error gain 2^14.65 at g = 44 on the earlier stage word, one input, not re-measured here).
• Numbers: public registers 1451 (400 mains, 1051 helpers); events 67,079; peak owned fields 36; raw literal / actual / reviewed upper 962,308,192 / 972,123,664 / 985,187,200, below 10^9; the conservative recharge counts LAMBDA a second time: 997,747,200 with the scale at the root, 999,836,640 with every public register scaled on every call; logarithm power minimum 2222, written 2223 (carrier 153), BIT pin margin 1.456e-4; largest |q| 40,864.

Checks (all run from the package, all pass)

Exact replay with the S certificate and endpoints, the odd verifier and 26 acceptance and rejection controls (S = 75, 45, 15, 25, 9, 1; 1/7; |q| over the bound; non-dyadic public row; dyadic S = 1 word), both roots, the joint caller (78 roles, 18 contacts), the invoice recounted twice, interval engines and the 54 + 4 outer ledger, a second replay mod 2^61−1 with mutants, 512-component streams, an integer run of one column with generic frames, 3,230 native chart maps, the caller physical replay, and the exhaustive division-pass test. Validator: python3.12 -B scripts/research_package.py --package research/p104-odd-scalar-native.

Disclosures and not claimed

• 5 of 57,023 ADD ×2 mutants are not caught: they leave the exact endpoint unchanged (four dead stores into owned register 1459, present in the earlier stage word, and one coefficient-neutral temporary).
• 120 rewritten events and 76 tuned events are validated by exact replay only.
• Adjustments to retained checkers: native.py and verify_relaxed.py/verify_strict.py (odd-alphabet relaxation, exact lines in the README); the clause "first 8 SCALEs are the gauge" is informational. The mod-p streams do not test divisibility; the integer run is one column, not true Clifford frames, not tensor width 2.
• Not claimed: maintainer acceptance of the odd-scalar clause, A6, the per-port root, LAMBDA as ordinary work or the BIT admission; K_star and the caller mass for this word; a rounded child return; no second-reader review.

Credit: icekylinx (#104, odd grid), Ali Romman (#5599, helper-split lemma), CubeJerry (#6916, application to the shared native word; #6805, #6824, #6890, #6900, #6901), and the retained work of #6919, #6917, #6911, #6902, #6889, #6833, #6726.

Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code.

🤖 Generated with Claude Code

[DreamingOfClouds]

Clarification of the execution wording. No change to κ, T, P or S.

The body and PROOF section 2 say inputs are multiplied "once by S at the root". That sentence describes one word. An inner call receives arbitrary dyadic values, so recursion needs one of two forms, and the two raw figures in the body belong to different ones:

• Scale per call. Each call multiplies its own inputs by S = 225 at entry and divides by S at exit, so factors of S add along the unfinished contexts only. This is the 999,836,640 line, which charges the conversion on every call.
• One scale at the outer boundary. Multiply once by S^K, with K bounding the unfinished contexts, as Stopped product-ring interchange and rational centers for conditional kappa = 7.79476e-5 #104 writes 21^K, and unscale at the outer return. There is no per-call conversion. This is the 997,747,200 line. "Scale at the root" in the body should be read as this S^K.

Width is per unfinished context in both forms, as in PROOF section 4, and its stated assumption on K_star and the caller mass is unchanged.