Number--Wave Framework III: Twin Prime Counting as a HABIT Spectrum Intersection

This article applies the HABIT intersection law to the twin-prime counting function. The main construction is the Twin Prime Effective Shadow Spectrum TPESS(n)=PESS(n)∩ PESS₂(n), where \(PESS_2\) is the shifted effective-shadow spectrum corresponding to \(x+2\). The article derives an exact finite identity for the twin-prime counting function: π₂(n)=(π(n)-π(√ n)+1)(π(n-2)-π(√ n)+2)/ n/2₂(√ n)(1+vH(n))+π₂(√ n+2)-1. Here C₂(x)=∏2<p≤ xp(p-2)/(p-1)² is the finite twin-prime factor, and \(v_H(n)\) is the remaining HABIT vibration after the identical-wave and parallel-wave structure has been separated. The identity is exact once \(v_H(n)\) is defined from the finite spectrum intersection. The associated Twin-Prime HABIT Vibration Conjecture states that vH(n)=O\!(1log n). Under this conjecture, the formula gives the Hardy--Littlewood twin-prime main scale while retaining the full finite correction structure.