What question did this study set out to answer?

This research aims to analyze integer factorization using an arithmetic section system over CRT products.

May 12, 2026Open Access

Arithmetic Section Systems and Local-to-Global Transfer in CF-AKS-Elliptic Factorization: An Apophatic Delimitation of Structure, Power, Certification, and Anomaly

Key Points

This research aims to analyze integer factorization using an arithmetic section system over CRT products.
Studied transfer depth under various transfer modalities: direct, factorial, and elliptic.
Conducted 17,692 tests for direct transfer and 17,193 for factorial classification to assess performance and accuracy.
Analyzed performance using the order-Fermat quotient and assessed prime independence in several tests.
CF convergents yield a ratio of 1.0053 with no statistical advantage over random residues (bootstrap 95% CI).
GCD detection achieved 6,272/6,272 successes across tested scenarios.
Notable distinctions between the order-Fermat quotient and classical Fermat quotient were established with disjoint FA sets.

Abstract

We study integer factorization through an arithmetic section system over the CRT product decomposition of \ (Z/nZ\). The framework introduces transfer depth \ (K^*\), the first iteration at which local incompatibility becomes visible through a nontrivial gcd. Under direct multiplicative transfer, \ (K^*\) equals the smaller local order (Proposition 3. 3) ; under factorial transfer, the exact criterion is uniqueness of the component order absorbed by \ (k!\) (Proposition 3. 4) ; under elliptic transfer, detection is governed by local point-order smoothness. The main contribution is apophatic. CF convergents provide structural organization but show no detectable statistical advantage over uniform random residues (Experiment E5: \ (K^*₂₅/K^*₍ₔ₋₋ = 1. 0053\), bootstrap 95% CI \ (-0. 0617, +0. 1700\) ). The computational gain is localized in the choice of group, as in ECM, not in the CF probe. The architecture matches ECM complexity \ (Lₚ (1/2, 2) \) but does not improve it. Version 0. 5 integrates a corrected full computational run (598. 8s, 8 workers). The algebraic transfer core is regression-stable: direct transfer matches Proposition 3. 3 in 17, 692/17, 692 tests; factorial classification matches Proposition 3. 4 in all 17, 193 decidable tests; sign-flip desynchronization is observed in 4, 823/4, 823 tests; gcd detection succeeds in 6, 272/6, 272 tests; and the predicted desynchronization count is observed for \ (k=2, , 10\) over 180/180 composites. The corrected run also sharpens three open interfaces: (i) Syracuse modular statistics support approximate CRT spectral transfer (mean ratio 0. 9552, range 0. 8093, 1. 0180) — computational support, not an exact theorem; (ii) a one-path active-hit proxy exhibits empirical exponent ≈ 0. 4920, supporting conditioned sublinear pruning but not a tree-level theorem; (iii) the expected FA-prime apophatic independence fails — several arithmetic-section invariants separate known full-absorption primes from controls, even after excluding 3, 5 and using size-matched controls. This version introduces a formal definition of full-absorption (FA) primes via the order-Fermat quotient map \ₚ (b) = b^{ordₚ (b) - 1}p p, \ and documents the essential non-equivalence with the classical Fermat quotient \ (qₚ (b) = (b^p-1-1) /p p\): the two maps produce disjoint FA sets beyond \ (\3, 5\\). The known order-Fermat FA primes below \ (10⁷\) are \ (\3, 5, 11, 19, 71, 733, 855719\\). The deposit package contains: final paper (PDF, LaTeX, Markdown), full machine-readable verification results (JSON), technical reports, a compatible Python software snapshot (toroidalₛheavesᵥerify), provenance documentation, and SHA-256 integrity checksums.

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