Inverse-Square Critical Coverage and Constant Saturation Depth from an Exact Interval Theorem

Working note of the Cosmochrony spectral admissibility sub-programme. Citable version: Zenodo concept DOI 10.5281/zenodo.21049163. Web page: https://cosmochrony.org/science/spectral/program/critical-coverage/.

Summary

The auto-calibrated saturation depth $n_1(q)$ of the breadth-first exploration of $\mathrm{Heis}3(\mathbb{Z}/q\mathbb{Z})$ — the central asymptotic variable of the O25–O28 window calibration — converges in probability to the constant $n* = 22$ under uniform generic block sampling, and the critical coverage $x_1(q) = |B_{n_1}|/q^2$ obeys the inverse-square law $x_1(q) \asymp q^{-2}$.

The mechanism is an exact algebraic reduction. Every implemented three-step Weil fingerprint is a non-zero scalar multiple of a single Fourier character, so the cumulative rank is a sumset cardinality,

$$r_c(n) = \bigl|,T'_c + C,I_{n+1},\bigr|, \qquad T'_c = {(c_2+c_3)\eta_2 + c_3\eta_3 : \eta_i \in {-1,0,1}},$$

with $C = c_1+c_2+c_3$ and $I_m$ the reduction of $[-m,m]$; the shellwise novelty obeys the universal deterministic ceiling $\Delta r_c(n) \le 2|T'c| \le 18$, with a complete classification of the block resonances that lower it. Exact sphere counts of the infinite discrete Heisenberg group give the strict threshold crossing $s{22} = 16{,}934 < 18/\varepsilon_{\mathrm{sat}} = 18{,}000 < s_{23} = 19{,}412$.

Results

• Deterministic (every prime $q \ge 311$, every admissible block sequence): $n_1(q) \le 22$ and $5/q^2 \le x_1(q) \le 99{,}689/q^2$, so $x_1(q) \asymp q^{-2}$; the positive-edge and logarithmic scenarios are excluded without probabilistic input.
• Probabilistic (i.i.d. uniform generic blocks): $n_1(q) \to 22$ and $q^2 x_1(q) \to |B_{22}| = 99{,}689$ in probability.
• Exact depth–coverage identity: $n_1(q) = (x_1(q)/\kappa_{n_1})^{1/4}\sqrt{q}$ with $\kappa_n = |B_n|/n^4$.
• Transitional regime: the measured depths ($n_1 = 14$ at $q = 211$, $19$ at $q = 601$) are explained by block resonances; their deficits produce the observed effective exponent $\approx -0.76$ en route to the exact asymptotic $-2$.
• Window closure: the implemented auto-calibrated fitting window $[n_0, n_1]$ is deterministically empty for $q &gt; 4 \cdot 22^4 = 937{,}024$.
• Collapse degeneracy: the coverage-collapse ansatz $r_q(n) \approx q,\Phi(|B_n|/q^2)$ is degenerate near the threshold front ($\Phi \equiv 0$ there); the law of $x_1$ is combinatorial, not diffusive.
• Consequence: in the dual-window limit of the canonical Fourier filtration (Q5a), the potential coefficient $\lambda = \lim x_1/C_{\mathrm{Heis}}$ vanishes: that limit carries no oscillator potential.

Reproduction

• code/h3z_spheres.py — exact integer BFS of $H_3(\mathbb{Z})$ to depth 31: sphere sizes $s_n$, ball sizes $|B_n|$, central extent $\Gamma(n) = \lfloor n^2/4 \rfloor$, and the strict crossing $n_*(10^{-3}) = 22$, $|B_{22}| = 99{,}689$.
• Scripts in simulation/spectral/o25 of the Cosmochrony repository: n1_resumable.py, n1_parallel.py (depth $n_1$), cheis_ballgrowth.py ($C_{\mathrm{Heis}}$, $D=4$), lambda2_scaling.py. Both depth scripts reproduce the O28 values $4, 8, 11, 13, 14$ at $q = 29, 61, 101, 151, 211$ exactly.

The compiled PDF is generated into out/ by compile.sh (pdflatex → bibtex → pdflatex ×2).

Status

Working paper, deposited on Zenodo (concept DOI 10.5281/zenodo.21049163). Companion to O28: it resolves the saturation-depth asymptotics that O28 left open.