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

A Three-Branch Classification with Its Rigorous Status, and the Substrate No-Go for the Expander-Derived Cascade-Exponent Law

Cosmochrony working paper (spectral admissibility sub-programme). Cite via the Zenodo concept DOI: 10.5281/zenodo.21480521.

Result

On the breadth-first cascade of the Heisenberg Cayley graph, the cumulative fingerprint span $r(n)$ obeys the exact increment identity $\Delta r(n) = \sigma_c(n),|S_n|$, where $\sigma_c(n)$ is the measured block capacity and $|S_n|$ the exact BFS sphere, with homogeneous dimension $D = 4$ (Bass--Guivarc'h). For real-valued weight sequences with power-law decay $\sigma_c(n) \sim a,n^{-\delta_c}$, summation yields a complete three-branch classification depending only on $(\delta_c, D)$:

• $\delta_c < D$: polynomial span growth $r(n) \sim n^{D-\delta_c}$ (growth branch, $\beta_{\mathrm{Heis}} = D - \delta_c$);
• $\delta_c = D$: logarithmic growth $r(n) \sim \log n$ (marginal branch);
• $\delta_c > D$: saturation $r_\infty - r(n) \sim n^{-(\delta_c - D)}$ (strictly 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 --- and the classification reaches the measured rank only through an exact finite-window formulation with two-sided bounds ($a_\pm$, $\kappa_\pm$ sandwiching $r(n) - r(m)$ by $\sum k^{D-1-\delta_c}$). Every numerical exponent statement is window-scoped, and no joint large-$q$ limit is claimed.

The expander-derived conversion law $\beta = 1/(\delta + \tfrac{1}{2})$ does not transfer to the substrate on which the capacities are measured. The no-go has three independent legs, each scoped to the constructions the corpus defines:

1. No valence--exploration proportionality: proportionality forces a vanishing capacity exponent, and it fails for both native realisations of the LPS valence --- for the span realisation outright at fixed $q$, by the dimension count $r(n) \le q$ against $|B_n| \to q^3$; for the bounded effective valence by boundedness.
2. No square-root frontier, no native flux carrier: the native frontier is a power of the explored volume, $|S_n| \asymp |B_n|^{3/4}$ (in dyadic average, with exact two-sided constants on the enumerated range), not of the achieved span; no uniform constant restores the $\sqrt{r}$ form for $D > 2$, and the bounded-flux constant $c_{\mathrm{BI}}$ has no native carrier because the explored volume is autonomous on a fixed graph.
3. Pair and coordinate obstructions: no native growth process carrying $\sigma_{\mathrm{pair}} = \sigma_c \sigma_{q-c}$ is defined or derived in the corpus, and the exponent coordinates of the conversion law ($\alpha$ against valence, $\delta$ against shell index) coincide only in the reduced filling model $p(n) = n$, which fails natively.

The separation-of-variables algebra itself is coordinate-honest: with the native inputs it returns the growth-branch map $\beta_{\mathrm{Heis}} = D - \delta_c$. On the block-level side, the shell-growth factor $D - 1$ is the native boundary term (additive, never inside a reciprocal), intra-window slopes are invariant under constant normalisations, and fitting against the shifted logarithm $\log(n{+}1)$ inflates exponents by a computable 8--13% on the production windows; the inter-$q$ estimator layer is left untouched. The window-effective exponents of the production data make contact near $q = 211$ and separate strictly for $q \ge 307$: the contact is a transitional crossing, with the asymptotic value of $\delta_c$, and hence the asymptotic branch, left open. The polynomial span exponent is explicitly distinguished from the exponential per-shell rate carried by $\beta^*$ in the projected-Yukawa mass factorisation.

Repository layout

• tex/ — LaTeX source (SpanGrowthNote.tex) and bibliographies
• code/ — span_growth_reproduce.py, the single reproduction script
• data/ — compact copies of the block-level observables actually used (frozen windows and per-checkpoint provenance in data/provenance.json; about 47 kB total)
• figures/ — effective_exponent_crossing.pdf, produced by the script (PDF only)
• compile.sh — full compile cycle (pdflatex -> bibtex -> pdflatex x2) into out/ (generated, git-ignored)
• zenodo.json — publication metadata (concept DOI)

Reproducing the numerical results

From the repository root, with numpy and matplotlib installed:

python code/span_growth_reproduce.py

This recomputes the exact spheres of $\mathrm{Heis}_3(\mathbb{Z})$, every window-effective exponent of the paper's table (with the bootstrap standard errors of the $q \ge 307$ rows reproduced exactly under the documented seed), the 27 window-shift and knee-trimming robustness variants, the $\log(n{+}1)$ convention check, and regenerates figures/effective_exponent_crossing.pdf.

Context

Web page: https://cosmochrony.org/science/spectral/program/span-growth/. Part of the Cosmochrony research programme; see the programme registry for the full dependency structure.