The Photon-Edge Gate: Uniqueness and Minimality of the Residual Five-Rail Complex for the Fine-Structure Constant

This paper executes the rigor gate named by the Quantum Traction Theory (QTT) corpus itself (Main Book v10.01, scorecard p. 35): not another expression for the fine-structure constant, but a proof of the uniqueness and minimality of the residual five-rail photon-edge complex behind it.

The theorem object is α⁻¹ = 4π[8 + ρ/2 + λ_γ], with ρ = 2πcos(π/8). After quotienting the bulk photon access house (8) and the neutral projected half-loop (ρ/2), the residual edge λ_γ is the signed Schur trace of a finite five-rail complex — spectral, work/noise–heat, ultraviolet-capacity, Artian area-accounting, and compact boundary rails — with signed metric diag(+,−,−,−,+) and denominators (64ρ, 1024ρ, 24²·64ρ², 24²·64ρ⁴, 24·8·32·64) derived from counted incidence atoms. Numerically λ_γ = 0.00252517226324577 and α⁻¹ = 137.035999165998, landing at −0.524σ against CODATA 2022 (137.035999177(21)), with no observed electromagnetic quantity anywhere in the constructor.

The proof runs R0–R7: quotient isolation; access lemmas for 8 and ρ/2; one lemma per denominator; the forced sign metric; minimality in both directions; and a published adjacent-exclusion overlay in which every nearby illegal alternative is priced (pulls from 69σ to 1.6×10⁶σ). Version 2.0 adds the isolation theorem — the declared grammar is quantized, with the nearest legal neighbor 13.8σ away, so there is no continuous knob to tune — plus a six-channel observation-closure ledger (pp. 324–331), a quantified inheritance audit through the Higgs ruler, the Artian–UEL master matrix (both of whose gauge inputs are now theorem-grade together with the strong companion, DOI 10.5281/zenodo.20625550), and the baryon-asymmetry row; verified page anchors with a full concordance appendix; live deep links into quantumtraction.org; three figures; recount recipes operationalizing falsifier F3; a six-step reproduction path; and a machine-verification appendix whose script regenerates every printed number from π and 1/8 alone.

Falsifiers F1–F7 are printed.