Weil-Block Projective Capacity on Heisenberg Graphs: Resolving the Representation Obstruction and Extracting the Capacity Exponent

We introduce a bidimensional Fourier-character proxy for the Weil-block decomposition of the three-step projective fingerprint on Heisenberg Cayley graphs over Z/qZ, designed to overcome the representation-level obstruction identified in O10. While dense TensorSketch fingerprints fail to produce a stable pre-saturation window at large q, the bidimensional proxy — projecting onto channels indexed by generic sextuples (τ,σ) = (t₁,t₂,t₃,s₁,s₂,s₃) ∈ (Z/qZ)⁶, acting on the abelianised $(a,b)$-coordinates of endpoint triples — yields structured observables with a proxy ambient dimension q² that matches the pre-saturation window. We define incremental block-wise projective capacities Σₙ(τ,σ) for each generic channel (τ,σ) and show that their mean value Σ̄ₙ exhibits a measurable pre-saturation decay regime of the form Σ̄ₙ ~ C\,n^-δcap. Numerical experiments at q ∈ \101, 151, 211, 307\ establish conditions (B1)--(B3) under which the decay exponent δcap is extractable, and yield δcap ≈ 3.4 at $q = 307$ with R² = 0.987, stabilising from $q = 211$ onward. These results provide the first operational framework in which a decay exponent can be extracted from discrete Heisenberg geometries, thereby testing the common-slope hypothesis of the Cosmochrony spectral framework.