A Sphere-Helix Construction for the Twin Prime Conjecture: The First Chord and the Prime Supply of the Gaussian Spiral — Three Projections in Mutual Certification (V1)

This paper gives a construction for the infinitude of twin primes on the geometric equivalent of the ζ function — the sphere helix (V14, DOI: 10.5281/zenodo.20637946). The method is mutual certification of the three projections: the three orthogonal projections of the sphere helix describe one object and certify one another, each with its own emphasis. In the diameter cross-section, a twin pair (p, p+2) is the first chord of the even-diameter sphere D=2(p+1) — the conjugate pair nearest the reflection's fixed point, whose chord product realizes the extremal case m²−1 of the conservation law pq = m²−ε². The law holds in the same form for every offset ε, and the model distinguishes no ε: the first chord has no special terrain. In the yz projection the same object appears as an adjacent prime pair on an arm of the Gaussian spiral: the arms are infinitely many (one per magnetic line), every arm carries every integer position — hence every prime — by construction; the double sum identity ζ·β (V14, Path A) is the analytic identity of the projection. The central symmetry of the sphere welds the two projections into one object. Following the causal template of Goldbach V2 (DOI: 10.5281/zenodo.20622870) — symmetry is the cause of the prime pair — the reflection symmetry of the functional equation acts in the same form at the first layer ε=1 as at any layer; with the supply unbounded and the pairing intrinsic, both-prime first chords recur along the equator sequence without end. A twin pair is the Goldbach decomposition of D=2p+2 nearest the equator: the two conjectures are two quantifiers on one conservation law. No sieve, no Hardy–Littlewood asymptotics, no object external to the model. The argument stands on the same rigid object as RH (V14) and Goldbach (V2), with the first chord as the first neighboring layer of the 1/2 self-conjugate locus. Bilingual: Chinese and English PDFs with figures.