Inverse-Square Critical Coverage and Constant Saturation Depth

Overview

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 — is determined completely by an exact algebraic reduction.

Fingerprints are Fourier characters. Every implemented three-step Weil fingerprint is a non-zero scalar multiple of a single Fourier character, so the cumulative Gram–Schmidt rank is exactly 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 spheres decide the crossing. Exact sphere counts of the infinite discrete Heisenberg group give the strict threshold crossing \(s_{22}=16{,}934 < 18/\varepsilon_{\mathrm{sat}} = 18{,}000 < s_{23}=19{,}412\), so the limiting depth is \(n_*(10^{-3})=22\) and the limiting coverage constant is \(|B_{22}|=99{,}689\).

Scope statement. The deterministic bounds hold for every prime \(q\ge 311\) and every admissible block sequence; the convergence \(n_1(q)\to 22\) is probabilistic, under uniform generic block sampling. The result is spectral only: no mass-hierarchy mechanism and no spatial-limit existence claim follows from it.

Main results

Exact interval theorem: the cumulative rank is the sumset cardinality \(r_c(n)=|T'_c+C\,I_{n+1}|\) for every prime \(q\), every block with \(C\ne 0\), and every depth.
Universal ceiling and resonances: \(2\le\Delta r_c(n)\le 2|T'_c|\le 18\) before saturation; block resonances \(t'-t=C\,m\) activate at depth \(\lceil (m-1)/2\rceil\) and explain the slow blocks seen in the O25 pipeline.
Deterministic bounds: for every prime \(q\ge 311\), \(n_1(q)\le 22\) and \(5/q^2\le x_1(q)\le 99{,}689/q^2\) — the positive-edge and logarithmic scenarios are excluded without probabilistic input.
Convergence in probability: under uniform generic block sampling, \(n_1(q)\to 22\) and \(q^2 x_1(q)\to |B_{22}|=99{,}689\).
Transitional regime: the measured depths (\(14\) at \(q=211\), \(19\) at \(q=601\)) are explained by resonance deficits; their shadow is 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 \(q > 4\cdot 22^4 = 937{,}024\).

Why the coverage-collapse route fails

A natural structural route would posit the scaling collapse \(r_q(n)\approx q\,\Phi(|B_n|/q^2)\) and solve the front equation \(\Phi'(x_1)=\varepsilon_{\mathrm{sat}}\,q\). The interval theorem shows this collapse is degenerate near the threshold front: at fixed coverage the rank is \(O(\sqrt q)=o(q)\), so \(\Phi\equiv 0\) there. The law of \(x_1\) is combinatorial — interval growth against exact sphere counts — not diffusive: the spectral gap \(\lambda_2\) never enters the rank dynamics.

Relation to the Cosmochrony programme

The note is a companion to O28 within the spectral admissibility sub-programme: it resolves the saturation-depth asymptotics that O28 established as sub-linear, and explains the measured transitional values. It depends on O9 (Heisenberg growth, \(D=4\)), O12 (the Weil-block construction), and O25–O28 (the saturation-depth measurements).

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.

Reference

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