Sigma Geometry Readout from a Source-Locked ⁸⁸Sr Quantum-Walk Transition Section: Same-Section Registration and Discrete Noether Transport

This work establishes the first theorem-level connection between a structure derived internally by the Sigma calculus and a published experimental quantum system. The Sigma calculus predicts a canonical finite transition structure generated by four equivalent source directions. Until now, this structure had been established only within the internal mathematical framework. Here it is shown that the complete structure is realized, at the level of unweighted nearest-neighbour transition incidence, in the published site-resolved (⁸⁸Sr) quantum-walk experiment of Young et al. The result is stronger than a visual or geometric similarity. An exact finite equivalence is constructed between the Sigma source complex and the experimentally documented transition section. This equivalence preserves the full outgoing transition star, admissible paths, face relations, discrete derivatives, and the associated Noether data. It can be verified by explicit finite matrix identities and therefore provides both a reproducible certificate and a clear criterion by which the proposed registration could be falsified. After the experimental transition structure is registered, the canonical Sigma projection produces a three-dimensional carrier whose normalized Gram matrix is exactly that of a regular tetrahedral simplex: its diagonal entries are (1), all off-diagonal entries are (-1/3), and the resulting simplex residual vanishes exactly. No spatial angles, geometric coordinates, external point cloud, or fitted reconstruction are introduced to obtain this result. The significance of this package is therefore that Sigma has crossed an important boundary: a nontrivial structure derived from its own finite transition and Noether principles is no longer confined to the abstract calculus but has been identified exactly within a source-locked section of real experimental data. The scope of the result is deliberately precise. It establishes equivalence for the published unweighted nearest-neighbour effective-transition section. It does not claim microscopic identity with the experimental apparatus, exact equality of tunnelling rates, reconstruction of laboratory space, or exclusion of additional longer-range interactions.