Published August 6, 2026
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Why Exact Heisenberg Capacity Does Not Determine the Phenomenological Cascade Rate: Observable Heterogeneity, Normalisation, and the Transfer Boundary
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
Files
SpectralO14.pdf
Additional details
Related works
- Is supplemented by
- Other: https://cosmochrony.org/science/spectral/program/o14/ (URL)
- Software: https://github.com/Cosmochrony/spectral-o14/tree/main/code (URL)