Native Span Growth from Capacity Decay and the Failure of the Heisenberg Capacity-to-Rate Transfer

Abstract

The spectral admissibility programme measures a projective block capacity $\sigma_c(n)$ along the breadth-first cascade of the Heisenberg Cayley graph and converts capacity exponents into growth exponents through the relation $\beta = 1/(\delta + \tfrac{1}{2})$, derived on Ramanujan–LPS relaxation graphs. This note establishes the span-growth law the Heisenberg substrate itself imposes, states its rigorous status, and proves that the expander-derived conversion law does not transfer to the substrate on which the capacities are measured. The cumulative span obeys the exact increment identity $\Delta r(n) = \sigma_c(n)\,|S_n|$, with polynomial sphere growth of homogeneous dimension $D = 4$; for real-valued weights with power-law decay, summation yields a three-branch classification in $({\delta_c}, D)$: polynomial growth $r(n) \sim n^{D-{\delta_c}}$ for ${\delta_c} < D$, logarithmic growth at ${\delta_c} = D$, saturation for ${\delta_c} > D$. Because the measured increments are integers, the pointwise power-law hypothesis is unrealizable whenever ${\delta_c} > D - 1$ — a range that includes the measured windows — so the classification reaches the measured rank only through an exact finite-window formulation with two-sided bounds; every numerical statement is window-scoped, and no joint large-$q$ limit is claimed. The transfer no-go has three independent legs, each scoped to the constructions the corpus defines: valence–exploration proportionality forces a vanishing capacity exponent and fails for both native realisations of the valence (for the span realisation outright at fixed $q$, since $r \le q$ while $|B_n| \to q^3$); the native frontier is a power of the explored volume, $|S_n| \asymp |B_n|^{3/4}$, not of the achieved span, so no constant restores the square-root form for $D > 2$ and the bounded-flux constant acquires no native carrier; and the pair observable carries no defined native growth process, while the exponent coordinates of the conversion law coincide only in the reduced filling model, which fails natively. The separation-of-variables algebra itself is coordinate-honest and returns the native growth-branch map $\beta = D - {\delta_c}$. Two block-level consequences follow: the shell-growth factor $D - 1$ is the native boundary term, additive and never inside a reciprocal; and fitting against the shifted logarithm $\log(n{+}1)$ inflates exponents by a computable $8$–$13\%$ on the production windows. Window-effective exponents make contact near $q = 211$ and separate strictly for $q \ge 307$; the contact is transitional in the accessible data and the asymptotic value of ${\delta_c}$ remains open. The polynomial span exponent is distinct from the exponential per-shell rate ${\beta^{*}}$ of the projected-Yukawa mass factorisation. Interpretive status. The classification and the no-go are proved under explicit, separately stated hypotheses; the numerical exponents are finite-window measurements; nothing here modifies a capacity measurement. As a structural reading, span growth on an emergent substrate is a competition between geometric opening (volume growth) and novelty exhaustion (capacity decay), and a conversion law between the two is a property of the substrate's geometry, not of the cascade alone: changing the geometry changes the law. This reading is an interpretation, not a result.