Conditional κ = 0.001679754788150048 from fine source-prime padding

[chafreaky]

This raises the conditional final saving to κ = 0.001679754788150048, exceeding PR1758 head 8d975b6caba8d261f1924e527225c193219a9967 by more than 9.2793e-9. The new mechanism is fine source-prime padding; the signed-554 T74 finite supplier from PR1699 is unchanged.

Choose source primes at relative gap Θ(p^-2) from their dyadic targets. Trudgian’s established prime theorem, Theorem 2 supplies enough distinct primes eventually. The smaller phase increment changes the grouped chirp guard to beta + 2 xi - 2 < 1. Absolute weight setup increases to Cp + 64p² ceil(log₂ p) bits, polynomially generated and amortized per complete microbox; online records remain O(p) bits. The proof charges a larger polynomial halo/core, dyadic normalizers once per group, and the existing explicit Strassen boundary (7^16 < 2^45). It does not rely on the structured boundary solver of PR1758. The parameter point is epsilon=0.9999999, beta=0.9999997, xi≈0.90792965478815.

The full dependency and precision argument is in FINE-PRIME-PADDING.md. The unchanged supplier has 319680 whole-helper roles, 249 base / 417 extra calls, dimension 73, live width 66304, prefix 1441 charged at 2048 and row coefficient 19545 below 20161. Its certified complex endpoint remains 335950991225108904267/200000000000000000000000.

Validation:

• 56 exact strict inequalities, 14 rejecting variants and a separate derivation of all nine paid cost powers pass. Finite controls include 30 deterministic prime witnesses and a concrete underflow failure for the old absolute weight precision.
• Byte-preserved predecessor checks run with a corrected full symbolic-constant assertion, asserted circuit residuals, strict union-array equality and 64 additional fine-theta symbol samples. These are finite evidence, not an all-size proof.
• Full read-only-source replays passed on macOS/Python 3.11 (568.86 s) and Linux/Python 3.14 (642.21 s), including all 18 supplier-admission stages and seven composition stages. Complete MATH.json receipts are byte-identical: SHA256 91553c2ce172165ecce79325eac0c71798fd3939f67236ec45f85c4d8870786b. Manifest SHA256 5399cb7adb01434dd53a22a5aaa3acd98e02902c232de4871421fe13c0b86559. Every committed package file matches the frozen source. The default verifier has no native-reuse flag. Dedicated fork CI passed on Python 3.11 and 3.14 at exact head c52345caa318fc6b1b6e2659c57d746317a72823. Both hosted complete mathematical receipts are byte-identical to both cold-platform receipts above; all seven composition stages and 18 retained supplier stages passed in each run. The committed package still matches the frozen source.

Independent scoped audit #1797 reports no gap in the explicit prime, conditioning, precision, locality and ledger inequalities it checked; its unchanged verifier also passes locally. It leaves the inherited circuit theorem, cost/layout derivations, supplier admission and all-size compiler outside scope, and it is an AI-session audit without human review.

Subsequent PR1800 vendors this mechanism and refines epsilon toward one, reporting a higher parameter-point value near 0.0016797549561255445. This PR retains the original explicit interior point and the fine-prime mechanism; it is no longer the highest reported numerical point.

python3 -m pip install -r research/fine-prime-padding/requirements.txt
CXX=c++ BOOST_INCLUDE=/path/to/boost/include python3 -B \
  research/fine-prime-padding/verify.py --work /absolute/new-work

The multiplication conclusion retains the inherited all-size compiler, common-label/mutable-source, hosting, semantic online precision C1=1, fixed-tape/prime-packing, interchange, recovery and analytic assumptions. The new written prime/precision/locality transfer requires review; no unconditional theorem, practical cutoff or independent human review is claimed. The selected result is unchanged.

Credit Tim Trudgian for the cited prime theorem; Vlado Vojdanovski for PR1673; IdeaOuter and the PR1758 contributors for fractional-shift interiors and per-visit accounting; Jeremy Kothe and contributors for the radix-union lineage; LeverRank for the second simulation; and all supplier authors retained in PR1699’s notices. PR1672 has priority for the signed-554 kernel. This branch contains PR1699’s unchanged supplier package as a pinned dependency. New work is the fine-prime-padding package, its workflow and the byte-preservation attribute rule.

Prepared by Chafik Boukhalfa with substantial OpenAI Codex assistance. Apache-2.0; original notices and source pins remain intact.

[JosieElliston]

FINDING (independent audit, Claude Code donated session for @JosieElliston): #1797 checks the explicit transfer inequalities of FINE-PRIME-PADDING.md §§1–3, 5 and 7 at this head (c52345ca). It re-derives them from the prose and the cited #1673/#1758 statements, without using this PR's checkers or ledger. No gap was found in that scope.

• §1: the θ interval and S/T > 1/2 are exact, and the Trudgian step is sound. Informational: at ε = 0.9999999 it becomes effective near ln p ≈ 4.3·10⁸, assuming log x = p^(1-ε).
• §2: the cited bounds do hold for every θ in (0, 1]. The 1/(8p²) row gap is uniform in p. The column (1-norm) analogue, which §5's spectral norms need, also holds: ‖N-I‖₁ ≤ e^{-2aθ} + e^{-2a(1-θ)} + 4e^{-2a}.
• §3: a ≤ 16⌈log₂ p⌉ for every p ≥ 3; the setup inequality has 2.77× slack.
• §5: p^10 e^{-p²} is exact via H⁻¹ − σΣ_{j<J}QʲHᵀ = Q^J H⁻¹.
• §7: all nine rows and side conditions check, and the controls reject.
• Saturation: on this ledger every parameter choice has κ < εq < b_cert, PR1699's endpoint 0.001679754956125544521335. The outer-only headroom left is (1−ε)q ≈ 1.68·10⁻¹⁰, so further gains on this line need a better supplier (your Open question 3).
• Not checked: Conditional κ = 0.001679745508802299: gauge-free (fractional-shift) interiors, structured boundary solves and radix-distributed sparse-union schedule on PR1673's capped phase-cell fusion outer, pivot-566+554 T74 supplier #1758's accretive circuit theorem, the cost-row derivations, the layout, supplier admission, and the all-size compiler.

[Mister-Meeseeks]

Scoped batch triage at c52345caa318fc6b1b6e2659c57d746317a72823. This PR remains open.

Fine-prime padding/precision transfer and retained supplier need dedicated admission. PR1797 is a scoped inequality audit, not a check of the accretive Toeplitz circuit theorem, cost-row derivations, union layout or all-size compiler. This proposal uses explicit Strassen boundaries rather than depending on PR1758 structured solves; preserve that distinction.

This is a review continuation, not a rejection or a request to repeat the checks already retained. Credit and source pins are preserved in the batch ledger. The selected bound is unchanged. Thank you for the contribution.

[philippeVV]

Question on the ξ < ε side condition (FINE.md:312). It does not affect κ today, but it matters if a supplier ever reaches a > 3/32.

FINE.md:312 lists both 1/2 < ξ < ε and ξ < 2. As far as I can trace, ξ < ε is inherited from #1673's cap_below_cell:

• CAPPED-GAUGE-AUDIT.md:19-27: boundary density bound "when Bcap ≤ d";
• note.tex:236-240, 254: O(d)-length cells.

That was needed when phase cells had length Θ(d) = Θ(p^ε). Here, cells have length Θ(p²), and FINE.md:238-243 proves the capped carving for B ≤ p², which is the ξ < 2 row. I could not find any other inequality in this PR that uses B ≤ d:

• precision β + 2ξ < 3 (FINE.md:150-169);
• Schur/inverse bounds (cap-independent per the Conditional kappa = 0.001677160151714381 via capped phase-cell fusion #1673 audit);
• boundary and union rows (FINE.md:252-253, 266-268).

If ξ < ε is indeed not needed in the fine-prime lineage, the outer ceiling of the FINE.md:288-312 row system changes. An exact LP over these rows, checked against closed forms at every grid point, gives:

• As written: sup κ = min(a, (3−τ)/2, ½).
• With ξ < ε replaced by ξ < k (θ = p^{−k}): sup κ = a·min(1, (1 + a(4−τ+k))/((1+a)(1+2a))). This equals a exactly for a ≤ (k+1−τ)/2.
  • At k = 2 with Strassen τ = 45/16 the cap is still a up to 3/32, but it reaches 0.698 at a = 1, instead of being capped at 3/32.

At today's a ≈ 0.0017 the row is slack, so nothing changes now. Inheritors that carry the same text: #1800 (FINE-PADDING-COMPOSITION.md:23, 38) and #2180.

Could you confirm whether ξ < ε is required anywhere I missed? If it isn't, the lineage's large-a ceiling is a rather than ½.

(Checked with Claude Code assistance; the LP checker is standard library only and available on request.)