PrimSpace — Supplementary Note S-3: Twin Prime Neighbor Factorization Structure

This supplementary note analyzes the factorization structure of integers adjacent to twin prime pairs (p, p+2) at two scales: N=10⁹ (3, 424, 506 twin pairs) and N=10¹0 (27, 412, 679 twin pairs), computed from the primeₐtlasN1e10full. prmc dataset (455, 052, 511 primes, PRMC v4 format). Main results: Theorem 1 (conditional on Hardy–Littlewood Conjecture B): For any prime q > 3, the probability that q divides the shared middle neighbor p+1 = q-1 of a twin prime pair equals exactly 1/ (q−2), strictly exceeding the naive Dirichlet expectation 1/q. Verified to 2¹024), lpf (p−1) ~ p/60 grows with p, making Pollard's p−1 attack structurally infeasible. This is relevant to specialized constructions (TPF-FIPS protocol) exploiting twin prime structure for deterministic prime generation and forward secrecy. Empirical validation: All results verified on 27, 412, 679 twin prime pairs up to 10¹0 using the twinₐnalyzer tool (statically compiled C, streaming PRMC v4 decoder, O (1) memory, 77 seconds for 455M primes). --- ### Keywords twin primes, prime factorization, neighbor factorization, Hardy-Littlewoodconjecture, middle neighbor theorem, 2-adic valuation, squarefree density, smooth numbers, Pollard p-1, characteristic large factor law, Mertensconstant, prime gap structure, PrimSpace, PRMC format, empirical number theory, cryptographic prime generation, TPF-FIPS