Observable-Class Mismatch and the Corrected δ→β∗ Relation on Heisenberg Graphs: Block Normalisation, Central-Phase Contribution, and Structural Correspondence

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 but a structural mismatch identified as Source~S2 in O13: the proxy-level derivation of the δ ↦ β* relation 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. This paper resolves S2. We identify the observable-class mismatch precisely: the O7 derivation 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, derive an effective exponent δeff(q) that absorbs the central-phase contribution, and obtain the corrected structural relation\[β* = {1}{δeff + 1/2} + ε(q),\]where ε(q) → 0 as q → ∞. Numerical re-fits of δeff at q ∈ \29,61,101,151,211\ determine which of two scenarios is realised: S2-A (reconciliation, β* recovered in $(0.09,0.13)$) or S2-B (structural revision of the amplification mechanism of O3$).