The spectral admissibility programme measures a projective block capacity c (n) along the breadth-first cascade of the Heisenberg Cayley graph and converts capacity exponents into growth exponents through the relation = 1/ (+ 12), 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 r (n) = c (n) \, |Sₙ|, with polynomial sphere growth of homogeneous dimension D = 4; for real-valued weights with power-law decay, summation yields a three-branch classification in (c, D): polynomial growth r (n) n^D-{c} for c D. Because the measured increments are integers, the pointwise power-law hypothesis is unrealizable whenever 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 q while |Bₙ| q³) ; the native frontier is a power of the explored volume, |Sₙ| |Bₙ|^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 = D - 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 (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 307; the contact is transitional in the accessible data and the asymptotic value of c remains open. The polynomial span exponent is distinct from the exponential per-shell rate ^{*} 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.
Jérôme Beau (Wed,) studied this question.