Finite Anisotropic Weighted Transition-Carrier Normal Form on a Source-Locked ⁸⁸Sr Quantum-Walk Section: Sigma Reviewer Package

This work extends the theorem-level connection established in the Sigma Transition/Noether Same-Section Reviewer Package (doi:10.5281/zenodo.21639274) from the unweighted transition-incidence structure to the additional anisotropic rate metric measured on the same source-locked ⁸⁸Sr quantum-walk section. For a positive opposite-channel rate assignment (Jₓ, Jᵧ, Jₓ, Jᵧ), compression to the canonical rank-three Sigma augmentation carrier produces the exact spectrum Jₓ, Jᵧ, and (Jₓ + Jᵧ)/2. The prescribed source quarter-turn is an isometry if and only if Jₓ = Jᵧ, whereas the corresponding half-turn remains an isometry for every positive pair. Consequently, unequal measured axis rates do not invalidate the previously established unweighted C₄ same-section registration. Instead, they reduce the symmetry of the weighted carrier metric canonically from C₄ to C₂. An explicit rank-three metric realization, its edge-length classes, and a scale-independent anisotropy certificate are derived without fitting spatial directions, angles, auxiliary points, or laboratory coordinates. For the published Young et al. central tunnelling rates, Jₓ/ℏ = 2π × 178.1 Hz, Jᵧ/ℏ = 2π × 148.3 Hz, the resulting residual is Ωᵣₐₜₑ = 22201/1331712 ≈ 0.01667102196270665. Using only the reported marginal standard errors, the exact weighted-C₄ condition Jₓ = Jᵧ is incompatible with the central estimates by at least 149/11 standard-error units for every admissible correlation. The already registered finite transition-complex equivalence transports the supplied rate-induced Dirichlet form to the Sigma-labelled transition star. This closes the weighted application-form route once the experimental rate assignment has been registered. It also provides an exact falsification interface: the unweighted source symmetry, the weighted metric symmetry, and their respective residuals are mathematically distinct and can be tested independently. The accompanying reproducibility layer reconstructs the published central rates from the public analysis material and evaluates a disjoint even/odd repetition split of the official experimental record. The split supports conditional same-experiment reproducibility of the anisotropic rate shape. It is not an independent experimental confirmation, because it uses the same experiment and retains the published nearest-neighbour opposite-axis model. The principal significance is that the Sigma-to-experiment correspondence now survives a nontrivial weighted deformation: the experimentally registered transition section retains its exact source-generated combinatorial structure while its measured rate metric acquires a rigorously determined anisotropic normal form. The tunnelling rates Jₓ and Jᵧ enter this application as externally supplied effective-Hamiltonian characterization data; predicting their numerical values independently of the chosen experimental preparation is neither required nor intended here. The result does not identify the carrier with laboratory space, establish microscopic identity with the apparatus, exclude longer-range interactions, or constitute an independent experimental validation of the Sigma framework.