Spectral Atomicity of the Admissible Sector

Overview

Q5a established a conditional continuum-limit framework for the admissible fibre \(\mathrm{ran}(\Pi_q)\), leaving two central analytic hypotheses open: scaled coercivity and Mosco tightness.

Q5a-O2 identifies the missing structure: the admissible projection is spectrally atomic.

For each conjugate pair \((c,q-c)\), the admissible sector is spanned by exactly three pure Fourier modes:

\[ \mathrm{ran}(\Pi_q^{(c)}) = \mathrm{span}\{e_0,\ e_{\xi_c^{(q)}},\ e_{q-\xi_c^{(q)}}\}. \]

The macroscopic frequencies satisfy \[ \xi_c^{(q)}/q \to \omega_c \neq 0. \]

Scope statement. This page summarizes Q5a-O2: spectral atomicity, scaled coercivity, Mosco compactness, and the replacement of Nash inequalities by finite-dimensional Fourier geometry.

Main contributions

Spectral atomicity: each admissible direction is a pure Fourier mode.
Three-mode structure: every conjugate pair spans \(\{e_0,e_{\xi_c},e_{q-\xi_c}\}\).
Macroscopic frequency selection: \(\xi_c^{(q)}/q\to\omega_c\neq0\).
Scaled coercivity: \(\mathcal{E}_q(f,f)\ge c q^{-1}\|f\|^2\).
Mosco compactness: compactness follows from Bolzano-Weierstrass in \(\mathbb{C}^2\).

Why Nash inequalities are no longer needed

The original Q5a strategy aimed to prove compactness through Nash inequalities, filtered Heisenberg graph estimates, and a discrete Rellich theorem.

Q5a-O2 shows that this route is unnecessary: admissibility does not spread vectors across the Fourier spectrum. It selects a finite atomic set.

Therefore compactness is not an analytic consequence of heat-kernel estimates. It is a geometric consequence of finite-dimensional spectral selection.

Interpretation

The admissible projection \(\Pi_q\) acts as a spectral selector. It reduces the full \(q\)-dimensional Fourier spectrum to a finite set of spectral atoms.

This shows that admissibility is not merely a constraint on amplitudes. It is a structural mechanism selecting the effective spectral degrees of freedom.

The result clarifies the distinction between two objects in the numerical pipeline:

basis_c: full BFS exploration, eventually spanning all of \(\mathbb{C}^q\)
pi_c: rank-3 admissible projection, corresponding to the theoretical \(\Pi_q\)

Numerical confirmation

The confinement test confirms that the admissible directions are pure modes at machine precision.

tested primes: \(q=29,61,101,151\)
support size: \(K_q=1\) for all tested primes
peak spectral mass: \(1.000\) after normalization
secondary mass: below \(10^{-25}\)

The representative pair \(c=1\) shows \[ \xi_c^{(q)}/q \to \omega_c \approx 0.364. \]

Relation to the Cosmochrony programme

Q5a-O2 closes the compactness and coercivity side of Q5a. It also connects directly with the O-series structural results:

O23: \(\Sigma_c(n_3)=3\) from quaternionic minimality
O28: \(H_{\mathrm{eff}}=\mathbb{C}^3\) and rank-3 covariance
Q5a: continuum-limit framework and Mosco programme
Q5b: geometric reading of the Heisenberg large-\(q\) limit

Updated status

[H-w]: proved in W1.
[H-E1']: proved in Q5a-O2 with optimal scaling \(q^{-1}\).
[C]: proved in Q5a-O2 by finite-dimensional compactness.
[H2]: remains open: convergence of rescaled generators.

Reference

Jérôme Beau. Spectral Atomicity of the Admissible Sector: Scaled Coercivity and Mosco Compactness without Nash Inequalities.