Published August 6, 2026 | Version 1.1

Why Exact Heisenberg Capacity Does Not Determine the Phenomenological Cascade Rate: Observable Heterogeneity, Normalisation, and the Transfer Boundary

Authors/Creators

  • 1. Independent Researcher

Description

O12 gives an exact formula for $\dim W_{\leq n}^{(c)}$, hence for $\Sigma_n^{(c)}=\Delta r_n^{(c)}/|S_n|$, and the exact asymptotic capacity exponent $\delta=3$. The five reported finite-window slopes $4.42,4.80,4.51,4.27,3.59$ are non-monotone crossover statistics, not candidate asymptotic exponents for a capacity-to-rate map. We determine which parts of the proposed O7 transfer survive for the exact observable. The mean over heterogeneous Weil blocks supplies only a scalar power-law premise. A factor $q^{-\eta}$ is constant in any fixed-$q$ regression and changes its intercept, not its slope. Most decisively, the central coordinate $\gamma$ multiplies each exact Fourier mode by a scalar phase and is algebraically invisible to rank, so the proposed central-phase correction vanishes identically, $\delta_{\gamma,\mathrm{corr}}=0$. A factor $q^{-\eta}$ cannot define an endpoint correction from a fitted slope alone: a true endpoint exponent contains $+\eta\log q/\log n^*$ and also depends on the fitted amplitude. No endpoint or inter-$q$ estimator is selected by the exact pipeline. Consequently the LPS reciprocal prescription produces no numerical $\beta^{*}$ diagnostic from these data. Interpretive outlook. Exact spectral capacity and a phenomenological cascade rate remain distinct observables until a growth carrier and a capacity-to-rate identification are independently constructed.

Notes

Version 1.1 establishes exact central-phase invariance of the rank observable, removes the phase-bias mechanism, classifies the five finite-window slopes as non-monotone crossover statistics toward the exact asymptotic exponent 3, and proves that no endpoint or cascade-rate diagnostic follows from fixed-q amplitude normalisation without a fitted amplitude, a selected estimator, and a native transfer carrier.

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