The study investigates the relationship between exact heisenberg capacity and the cascade rate, suggesting they are distinct observables.
O12 gives an exact formula for W≤ n⁽ᶜ⁾, hence for Σₙ⁽ᶜ⁾=Δ rₙ⁽ᶜ⁾/|Sₙ|, and the exact asymptotic capacity exponent δ=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-η is constant in any fixed-q regression and changes its intercept, not its slope. Most decisively, the central coordinate γ multiplies each exact Fourier mode by a scalar phase and is algebraically invisible to rank, so the proposed central-phase correction vanishes identically, δγ,corr=0. A factor q-η cannot define an endpoint correction from a fitted slope alone: a true endpoint exponent contains +ηlog 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 β* 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.
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Jérôme Beau (2026) studied this question.
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