Observable-Class Mismatch in the Proposed δ ↦ β^* Relation on Heisenberg Graphs: Block Normalisation, Central-Phase Diagnostics, and the Transfer Boundary

Papers O12–O13 established that the exact Weil-block capacity exponent δ̂exact(q) decreases monotonically from $4.80$ to $3.59$ as q ranges over \29,61,101,151,211\, refuting the finite-size hypothesis S1 and ruling out convergence toward the phenomenological target δ ∈ [7.4,10.6]. The tension between δ̂exact and the window β* ∈ (0.09,0.13) is therefore not a numerical artefact. Source S2 in O13 identified an observable-class mismatch: the proxy-level δ ↦ β* prescription does not account for the block-normalisation by |Sₙ|, the internal block dimension q, or the central-phase contribution ψc(γ + bx) present in every exact Weil fingerprint. We identify that mismatch precisely: the O7 argument operates on a scalar global capacity, while δ̂exact is extracted from a block-by-block mean over representations of internal dimension q. We decompose the exact observable as Σₙ⁽ᶜ⁾ = Σₙᵍᵉᵒᵐ + Σₙγ, and isolate the central-phase contribution. A factor q-η is constant in an intra-q regression and therefore cannot change its fitted slope; the earlier subtraction ηlog q/log n* belongs only to an endpoint or inter-q estimator. Conditional on importing the LPS capacity-to-rate prescription, that estimator defines the phenomenological diagnostic \[ βdiag = {1}{δend + 1/2} + ε(q), \] where ε(q) remains a modelled variance term. This is not a native Heisenberg derivation of β*. The numerical diagnostic at q ∈ \29,61,101,151,211\ selects the paper's S2-B branch; the native transfer obstruction is independent of that numerical outcome.