Fourier Structure of the Pre-Saturation Admissible Pipeline

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 analyses the Fourier content of the pre-saturation admissible pipeline and proves: the pre-saturation fingerprints and the selected Gram–Schmidt vectors are pure Fourier modes, with displacements of order \(o(q)\). Frequencies are \(o(q)\) only for block parameters held fixed independently of \(q\); the implemented pipeline samples block parameters uniformly modulo \(q\), so macroscopic per-\(q\) frequencies occur (e.g. \(\xi/q\) near \(1/2\)) and no \(q\)-independent limit frequency exists in either regime.

It also proves two delimiting results: the three-coordinate reduction used in the numerical pipeline is a truncation, from which the rank of the admissible sector cannot be measured; and the energy form is \(O(q^{-2})\) for every unit vector — a consequence of the form's \(q^{-2}\) normalisation prefactor — which rules out the \(q^{-1}\) coercivity scaling. The spectral-gap question for the unnormalised form is untouched and open.

The paper does not close the hypotheses [H-E1] or [C] of Q5a. The canonical filtration of the admissible sector is Q5a's growing Fourier window, of dimension \(\min(2n+1,q)\), whose Mosco limit is the zero form.

Scope statement. This page summarizes Q5a-O2: pure-Fourier-mode structure of pre-saturation fingerprints and selected Gram–Schmidt vectors, the two-regime frequency statement (\(o(q)\) displacements; macroscopic per-\(q\) frequencies under uniform block-parameter sampling), the truncation status of the three-coordinate reduction, and the \(O(q^{-2})\) bound for every unit vector ruling out \(q^{-1}\) coercivity.

Main contributions

Pure-mode structure: the pre-saturation fingerprints and the selected Gram–Schmidt vectors are pure Fourier modes.
Two-regime frequency statement: displacements are \(o(q)\); frequencies are \(o(q)\) only for block parameters held fixed independently of \(q\). The implemented pipeline samples block parameters uniformly modulo \(q\), so macroscopic per-\(q\) frequencies occur (e.g. \(\xi/q\) near \(1/2\)) and no \(q\)-independent limit frequency exists in either regime.
Truncation status: the three-coordinate reduction is a truncation of the pipeline; the rank of the admissible sector cannot be measured from it.
Energy-form scaling: the form is \(O(q^{-2})\) for every unit vector — a consequence of the form's \(q^{-2}\) normalisation prefactor — ruling out the \(q^{-1}\) coercivity scaling; the spectral-gap question for the unnormalised form is untouched and open.
Canonical filtration: the admissible sector is filtered by Q5a's growing Fourier window, of dimension \(\min(2n+1,q)\), whose Mosco limit is the zero form.

What Q5a-O2 does not establish

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

Q5a-O2 does not replace that route. Spectral atomicity of the admissible sector, the mean-zero lemma, a \(q\)-independent limit frequency, \(q^{-1}\) coercivity, and Mosco compactness are not results of this paper.

In particular, the hypotheses [H-E1] and [C] of Q5a remain open: the paper delimits what the pipeline measures, it does not close the continuum-limit programme.

Interpretation

The three-coordinate projection used in the numerical pipeline is a truncation, not a measurement of the admissible sector's rank. What it faithfully records is the pure-Fourier-mode structure of the pre-saturation fingerprints.

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: the three-coordinate reduction, a truncation of the pipeline from which the rank of \(\Pi_q\) cannot be inferred

Relation to the Cosmochrony programme

Q5a-O2 delimits the pre-saturation side of Q5a's pipeline. It connects with the following papers:

O23: conditional adjoint-dimension theorem on a supplied spinor carrier; \(\Sigma_c(n_3)=3\) is a supplied selection rule
O28: \(H_{\mathrm{eff}}=\mathbb{C}^3\) and rank-3 covariance
Q5a: continuum-limit framework and Mosco programme, whose analytic hypotheses remain open
Q5b: geometric reading of the Heisenberg large-\(q\) limit

Updated status

[H-w]: proved in W1.
[H-E1]: open; the \(q^{-1}\) coercivity scaling is ruled out by the \(O(q^{-2})\) bound, which holds for every unit vector; the spectral-gap question for the unnormalised form is untouched.
[C]: open; the canonical filtration is Q5a's growing Fourier window, with zero-form Mosco limit.
[H2]: remains open: convergence of rescaled generators.

Conclusion

The pre-saturation pipeline has an exact Fourier structure, but the three-coordinate reduction is a truncation: the rank of the admissible sector and the continuum limit of Q5a remain open questions, the canonical filtration being the growing Fourier window with zero-form Mosco limit.

Reference

Jérôme Beau. Fourier Structure of the Pre-Saturation Admissible Pipeline: What the Three-Coordinate Projection Does and Does Not Measure.