TWPR Quadratic / Twin-Product Shadows in the6-Wheel / Base-5gating,least-factorstraddling, andconditional CP/CCdensity
Year: 2026
Type: preprint
Abstract: Let 6k−16k−1 and 6k+16k+1 be twin primes. Their quadratic twin-product shadow is Θ(k)=(36k2−1,36k2+1).Θ(k)=(36k2−1,36k2+1). The left coordinate is the squarefree semiprime (6k−1)(6k+1)(6k−1)(6k+1); the right coordinate is prime or composite. The note proves a dilation–squaring semiconjugacy, a 1(mod4)1(mod4) divisor law, and a least-factor straddling theorem: for a composite right companion, its least prime factor lies below the source twin and the complementary factor lies above it. On the regular twin-source domain, Base 5 gives the exact first gate 5 | 36k2+1 ⟺ 5 ∤ k. This is extended to a local gate fan and exact CRT survivor/first-gate layers. Their ungated local fraction decays rigorously like C/logzC/logz. A separate Bateman–Horn/Hardy–Littlewood module conditionally predicts the same logarithmic order for the actual CP proportion. Finite ledgers and two selected TWPR trace embeddings accompany the theorem chain.
Source: HAL
Author: Steiner, Stephen
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204808973