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 $\hat{\delta}_{\mathrm{exact}}(q)$ 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 $\delta \in [7.4,10.6]$. The tension between $\hat{\delta}_{\mathrm{exact}}$ and the window $\beta^{*} \in (0.09,0.13)$ is therefore not a numerical artefact. Source S2 in O13 identified an observable-class mismatch: the proxy-level $\delta \mapsto \beta^{*}$ prescription does not account for the block-normalisation by $|S_n|$, the internal block dimension $q$, or the central-phase contribution $\psi_c(\gamma + bx)$ present in every exact Weil fingerprint. We identify that mismatch precisely: the O7 argument operates on a scalar global capacity, while $\hat{\delta}_{\mathrm{exact}}$ is extracted from a block-by-block mean over representations of internal dimension $q$. We decompose the exact observable as $\Sigma_n^{(c)} = \Sigma_n^{\mathrm{geom}} + \Sigma_n^{\gamma}$, and isolate the central-phase contribution. A factor $q^{-\eta}$ is constant in an intra-$q$ regression and therefore cannot change its fitted slope; the earlier subtraction $\eta\log q/\log n^{*}$ belongs only to an endpoint or inter-$q$ estimator. Conditional on importing the LPS capacity-to-rate prescription, that estimator defines the phenomenological diagnostic \[ \beta_{\mathrm{diag}} = \frac{1}{\delta_{\mathrm{end}} + \tfrac{1}{2}} + \epsilon(q), \] where $\epsilon(q)$ remains a modelled variance term. This is not a native Heisenberg derivation of $\beta^{*}$. The numerical diagnostic at $q \in \{29,61,101,151,211\}$ selects the paper's S2-B branch; the native transfer obstruction is independent of that numerical outcome.