Capacity Decay Determines Span-Growth Universality on Heisenberg Graphs

The spectral admissibility programme measures a projective block capacity $\sigma_c(n)$ along the breadth-first cascade of the Heisenberg Cayley graph, decaying as a power law $\sigma_c(n) \sim a\,n^{-{\delta_c}}$, and converts capacity exponents into cascade growth exponents through the structural relation $\beta = 1/(\delta + \tfrac{1}{2})$, derived on Ramanujan–Lubotzky–Phillips–Sarnak (LPS) relaxation graphs. The present note derives the span-growth law that the Heisenberg substrate itself imposes. The cumulative fingerprint span $r(n)$ obeys the exact increment identity $\Delta r(n) = \sigma_c(n)\,|S_n|$, and the exact breadth-first spheres of ${\mathrm{Heis}_3(\mathbb{Z})}$ grow polynomially with homogeneous dimension $D = 4$. Asymptotic summation then yields a complete three-branch classification, depending only on $({\delta_c}, D)$: polynomial span growth $r(n) \sim n^{D-{\delta_c}}$ for ${\delta_c} < D$; logarithmic growth for ${\delta_c} = D$; saturation $r_\infty - r(n) \sim n^{-({\delta_c} - D)}$ for ${\delta_c} > D$. Two structural consequences follow. First, the span is never proportional to the number of configurations reached: $r(n)/|B_n| \to 0$ for every ${\delta_c} > 0$, so the valence–exploration proportionality that closes the LPS growth equation does not transpose to this substrate. Second, no uniform constant $c$ can sustain the LPS-form bound $\Delta r \le c\,\sigma_c \sqrt{r}$: since $r(n) \le |B_n|$, any such constant must exceed $|S_n|/\sqrt{|B_n|} \asymp n^{D/2-1}$, which diverges for every polynomial growth dimension $D > 2$. The square-root frontier factor — and with it the additive $\tfrac{1}{2}$ of the cascade-exponent law — is therefore specific to expander-type substrates; on Heisenberg graphs the capacity-to-growth map on the growth branch is $\beta = D - {\delta_c}$. Pipeline data illustrate the classification: the window-effective exponents make contact near $q = 211$ (${\delta_c}^{\mathrm{eff}} = 3.748$ vs $D_{\mathrm{eff}} = 3.753$) and separate strictly for $q \ge 307$, with ${\delta_c}^{\mathrm{eff}} - D_{\mathrm{eff}}$ between $-0.21$ and $-0.49$ at $6.7$–$14.5$ standard errors of the inter-pair bootstrap, a sign stable across all $27$ window-shift and knee-trimming variants and under both logarithmic conventions. The contact is therefore transitional in the accessible data — the span sits in the growth branch at the largest measured primes — and no current indication forces ${\delta_c} = D$; the statement concerns window-effective exponents only, with no asymptotic claim. The polynomial span exponent studied here is distinct from the exponential per-shell rate carried by the symbol ${\beta^{*}}$ in the projected-Yukawa mass factorisation; the two objects are separated explicitly. Interpretive status. The classification theorem and the two no-go corollaries are proved under explicit power-law hypotheses; the numerical exponents are finite-window measurements; the $q = 211$ near-contact is resolved, within the accessible data, as a transitional crossing of window-effective exponents, while the asymptotic value of ${\delta_c}$ remains open. As a structural reading, span growth on an emergent relational substrate is governed by a competition between geometric opening (volume growth) and capacity decay (novelty exhaustion); at the measured windows the two pass through near-balance transitionally and separate at the largest measured primes. This reading is an interpretation, not a result.