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 (, ) = (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 ₙ^ (, ) for each generic channel (, ) and show that their mean value ₙ exhibits a measurable pre-saturation decay regime of the form ₙ C\, n^-₂₀₏. Numerical experiments at q \101, 151, 211, 307\ establish conditions (B1) -- (B3) under which the decay exponent ₂₀₏ is extractable, and yield ₂₀₏ 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.