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

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: \(K^*CF/K^*ₙᵤₗₗ = 1.0053\), 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 \(L_p(1/2,√2)\) 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 \[Φ_p(b) = {bord_p(b) - 1}{p} {p},\] and documents the essential non-equivalence with the classical Fermat quotient \(q_p(b) = (bᵖ⁻¹-1)/p p\): the two maps produce disjoint FA sets beyond \(\{3,5\}\). The known order-Fermat FA primes below \(10^7\) 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 (toroidal_sheaves_verify), provenance documentation, and SHA-256 integrity checksums.