Number--Wave Framework III: Twin Prime Counting Function by HABIT

Number--Wave Framework III: Twin Prime Counting Function by HABIT applies the Number--Wave Framework and the HABIT intersection law to twin-prime counting. The article constructs the Twin Prime Effective Shadow Spectrum as the intersection of the Prime Effective Shadow Spectrum and its shift by two. This gives a spectrum-based representation of twin-prime endpoints and connects the effective-shadow count to the classical twin-prime counting function. Using HABIT, the article derives a twin-prime counting formula involving the Prime Effective Shadow Spectrum, the shifted Prime Effective Shadow Spectrum, the common-wave correction, the finite twin-prime factor, and the parallel-wave correction factor. The remaining finite variation is expressed by the HABIT vibration (v_H(n)), for which a structural bound is given. The article also states the Strong HABIT Vibration Conjecture, which gives a conditional asymptotic reading matching the Hardy--Littlewood twin-prime scale. In addition to the effective-shadow route, the article presents a Prime Spectrum form of the twin-prime counting function using an integral independent companion.