Resolution of the Fine-Structure Constant Problem: Tier-1 No-Go, Tier-0 Selection, and the Modular Identity at 𝜏 = 𝑖

This paper presents a structural resolution of the fine-structure constant problem within the Everything Equation / Dirac–Lambda programme.

The central result is a sharp separation of roles between Tier-1 and Tier-0. At scope-separated Tier-1, the fine-structure constant cannot be determined internally: the collar scale Lambda_coll (equivalently beta, equivalently alpha) remains a genuine residual modulus. The paper proves that this non-determination is structural rather than accidental. The D1–D9 exhaustive elimination analysis, together with the Two Root Causes Theorem, shows that all admissible internal Tier-1 closure routes fail for two precise reasons: f0 = 0 topological blindness and Dirichlet-to-Neumann eigenvalue saturation. In this formulation, the fine-structure constant problem is not merely “unsolved within Tier-1”; rather, Tier-1 is shown to be the wrong level at which to expect final determination.

The resolution is then located upstream at Tier-0. Within the law-level framework defined by the Everything Equation, L = Omega Delta partial(L), the residual modulus is collapsed to a unique fixed point by the Tier-0 selector mechanism. On the adopted S3 carrier, the upstream Lambda-selection chain (Selector 9 + E2 + E1) yields Lambda approximately 57,039 GeV, corresponding to alpha^-1 = 137 with zero free parameters on the selected carrier.

This Version 2 supersedes Version 1 by supplying the missing structural mechanism. Version 1 established a rigorous modular identity at the CM point tau = i, evaluating to a constant C = 137.035999084..., matching the inverse fine-structure constant alpha^-1(0) to high precision. That modular identity remains mathematically valid and is retained here. What Version 1 did not yet establish was the physical mechanism connecting the modular constant to the observed coupling. Version 2 resolves that gap: the modular structure at tau = i is no longer presented as an independent Tier-1 derivation, but as a manifestation of the upstream Tier-0 selector architecture, in particular the Selector 9 mechanism associated with the elliptic fixed point and j-invariant-zero structure.

The paper therefore does three things at once: it proves the Tier-1 obstruction, identifies the exact root causes of non-closure, and places the physical determination of alpha at the only level where the modulus can be uniquely fixed. It also clarifies the status of the earlier modular result by embedding it into the completed structural account.