This paper establishes that the admissible sector in the Cosmochrony spectral programme is spectrally atomic: each conjugate pair spans exactly three pure Fourier modes. This result closes key analytical gaps in Q5a by removing the need for Nash-type inequalities and completing the control of admissibility forms.

Quick Summary

• Spectral atomicity: each admissible pair is supported on exactly three pure Fourier modes
• No-mode-mixing: admissible fingerprints behave as single-frequency eigenmodes
• Closure of hypotheses:
  • closes [H-E1] (uniform Poincaré on admissible sector)
  • closes [C] (spectral tightening → Mosco convergence)
• Elimination of Nash inequalities: replaced by exact Fourier structure
• Structural refinement of Q5a: strengthens convergence results without additional assumptions

Main Results

1. Atomic Fourier structure

Each admissible fingerprint vector is concentrated on a single frequency:

• empirical concentration: $R_{99}$_% = 1
• no dispersion across modes

This implies:

• exact diagonalisation of the admissibility form
• absence of hidden mixing effects

2. Three-mode structure per conjugate pair

For each pair ${c, q-c}$:

• admissible subspace: $\mathrm{span}{ e_0, e_{\xi_c}, e_{\xi_c}^\ast }$
• dimension: $\dim H_{\text{eff}} = 3$

This matches the structural result:

• $\Sigma_c(n_3) = 3$ (O23)
• rank $r_{\mathrm{eff}} = 3$ (O28)

3. Closure of analytical hypotheses

The atomic structure implies:

• [H-E1] (Poincaré inequality)
  → holds automatically on each frequency block

• [C] (spectral tightening)
  → follows from exact mode separation

Thus:

• Mosco convergence becomes structurally controlled
• no functional-analytic workaround (e.g. Nash) is needed

4. Structural interpretation

Admissibility does not produce a diffuse spectrum but a minimal discrete support:

• one neutral mode
• one conjugate pair of oscillatory modes

This defines the admissible sector as a minimal coherent triplet, not a continuum.

Context in the Programme

Q5a establishes convergence of admissibility forms to a continuum operator:

• Hilbert limit $\mathbb{C}_q \to L^2(\mathbb{R})$
• Mosco convergence of $\mathcal{E}_q \to \mathcal{E}$

Q5a-O2 strengthens this by proving:

• the admissible sector is already fully resolved spectrally
• no hidden degrees of freedom remain at the discrete level

This aligns with:

• O23: quaternionic minimality → 3 directions
• O28: effective dimension $r_{\mathrm{eff}} = 3$
• Q7: identification $H_{\text{eff}} \simeq \mathrm{Sym}^2(V_\rho)$

Conceptual Implications

• Admissibility = spectral selection, not diffusion
• Emergence is low-rank, not high-dimensional
• Continuum limit acts on already minimal structures

This supports the core principle:

Observable structure lives in $\mathrm{Im},\Pi$, not in the full configuration space.

Numerical Evidence

• primes tested: $q = 29, 61, 101, 151$
• all conjugate pairs:
  • exact single-frequency dominance
  • no rank inflation
  • strict monotonicity of admissible projections