PureOne/VESPER-QUORUM-WDD

VESPER–QUORUM

Exact Half-Translation Masks for STFT Phase Retrieval and PSD Recovery

Version 6.0.0 · 3 October 2026 · AI-assisted mathematical research

Manuscript attribution: Eve — prepared for Maciej Nowicki.
Release preparation and publication: Maciej Nowicki, with AI assistance.

VESPER–QUORUM presents an exact arithmetic classification of a structured family of finite cyclic STFT phase-retrieval systems with ambiguity-function zeros. For every even dimension (d=2a\ge4), three gcd tests and three nonemptiness checks decide uniform uniqueness for every admissible erasure mask confined to shifts (0) and (a). Within this family, the criterion also characterizes strict rank-one positive-semidefinite completeness. The manuscript connects these results to sharp paired geometric stability and globally justified affine/PSD reconstruction.

The release includes a self-contained manuscript, LaTeX source, executable algorithms, exact certificates, numerical experiments, original reproduction evidence, and a fresh release-preparation reproduction run. This dataset repository distributes research artifacts; it does not contain a trained model or a conventional train/validation/test dataset.

Scope: the stated half-translation family is classified in the supplied manuscript. The unrestricted structural problem for arbitrary ambiguity-zero masks remains open. Mathematical review by independent specialists and novelty priority are unconfirmed.

Read and reproduce

File or directory	Purpose
research/manuscript.pdf	Complete manuscript and proofs
research/manuscript.tex	Editable LaTeX source
THEOREM_LEDGER.md	Claims, evidence, and boundaries
REPRODUCING.md	Installation, checks, and noisy demo
research/README.md	Original API and certificate guide
research/STATUS.json	Original machine-readable research status
release_checks/RELEASE_REPORT.json	Fresh release checks and completion status
release_checks/reproduction_report.json	Fresh executed mathematical verification
research/SOURCE_AUDIT.md	Supplied source and contribution audit
archives/VESPER_QUORUM_WDD_Research_v6.0.0.zip	Unmodified original archive
CITATION.cff	Citation metadata without an invented DOI
LICENSE.md	License scope

For Python 3.10 or later:

python -m venv .venv
# Activate .venv for your operating system, then:
python -m pip install -r research/requirements.txt
python research/reproduce.py
python research/demo.py

The reproduction scripts write reports into research/. Run them in a working copy if you want to preserve the delivered checksums. A passing computation supports the particular assertions it checks; it does not establish novelty or replace the manuscript's analytic proofs.

The exact mask criterion

For a known window (w), use the ambiguity convention

[ A_w(m,r)=\sum_{j\in\mathbb Z_d}w_j\overline{w_{j-m}}e^{-2\pi i rj/d}. ]

Assume the exact zero mask satisfies (Z\subseteq{0,a}\times\mathbb Z_{2a}), excludes the origin, and is symmetric under ((m,r)\mapsto(-m,-r)). An abstract erasure mask obeying these conditions is also covered by the arithmetic theorem. Realizability by one window is a separate requirement.

Split the observed strip indices into four sectors:

[ \begin{aligned} R_0&={r\text{ even}:(0,r)\notin Z},\ R_1&={r\text{ even}:(a,r)\notin Z},\ R_2&={r\text{ odd}:(a,r)\notin Z},\ R_3&={r\text{ odd}:(0,r)\notin Z}. \end{aligned} ]

For each nonempty (R_\nu), (\nu=1,2,3), choose a reference (r_\nu\in R_\nu) and define

[ g_\nu=\gcd!\left(a,r_\nu,\left{r/2:r\in R_0\right},\left{(r-r_\nu)/2:r\in R_\nu\right}\right). ]

Every argument is integral. The supplied theorem states

[ \text{uniform phase retrieval} \iff\text{strict rank-one PSD completeness} \iff R_1,R_2,R_3\ne\varnothing\ \text{and}\ g_1=g_2=g_3=1. ]

This is a uniform statement over all complex signals, including signals with coordinate zeros. The implementation rejects masks outside the family and does not declare exact zeros from a floating-point tolerance.

import sys
sys.path.insert(0, "research")
from vesper_quorum import classify_strip, mixed_mask10

decision = classify_strip(mixed_mask10())
print(decision.injective)  # True
print(decision.gcds)       # (1, 1, 1)

Two realizable windows, opposite outcomes

Let (\zeta=e^{2\pi i/3}). The good window is

[ w_+=\tfrac15(1,2\zeta,1,1,-2,2,1,2\zeta^2,2,1). ]

Its exact six-point zero set is

[ Z_+={(0,2),(0,4),(0,6),(0,8),(5,0),(5,5)}. ]

Exact cyclotomic identities and rational outward intervals certify all 100 ambiguity coefficients. The minimum nonzero modulus and squared geometric gaps are

[ \alpha=\frac{3\sqrt5-5}{25},\qquad \Gamma=\frac3{10},\qquad \rho=\frac1{10}. ]

(\Gamma) is the optimal unweighted gap for rank-one competitors; (\rho) is the optimal unweighted strict gap for a rank-one PSD target against arbitrary PSD competitors. Window-weighted measurement bounds derived from (\alpha) are certified lower bounds, not claimed optimal weighted constants.

The bad window

[ w_-=\tfrac1{\sqrt{110}}(1,2,3,4,5,i,4i,3i,2i,5i) ]

also has full coordinate support and six ambiguity zeros, but fails uniform phase retrieval. The signals ((e_0+e_5)/\sqrt2) and ((e_0-e_5)/\sqrt2) have identical intensities. Equal zero counts therefore do not determine recoverability.

The good v6 window is not strictly rank-two complete. Its uniform PSD rank threshold is one. The rank-three guarantee of an earlier NACRE–CHOIR window belongs to a different construction and is not transferred to this example.

Recovery and noise

Let the measured intensity data be (\widetilde y=\mathcal M_w(X)+e), with (|e|_F\le\varepsilon). After WDD inversion on observed coefficients, form

[ \widetilde X_0=P_{Z^c}X+E_0,\qquad |E_0|_F\le\frac{\varepsilon}{\alpha\sqrt d}. ]

From any PSD initial matrix, iterate

[ Y_{n+1}=\Pi_{\mathrm{PSD}}\left(\widetilde X_0+P_ZY_n\right). ]

For an injective mask in the classified family and a rank-one PSD target, the analytic theorem gives

[ |Y_n-X|_F^2\le(1-\rho_Z)^n|Y_0-X|_F^2+ \frac{1-(1-\rho_Z)^n}{d\alpha^2\rho_Z}\varepsilon^2. ]

For the good ten-dimensional window, the squared-error contraction factor is (9/10), and the asymptotic bound is

[ \limsup_n|Y_n-X|_F\le\frac{25}{3\sqrt5-5}\varepsilon<14.636\varepsilon. ]

The theorem uses exact arithmetic. The delivered implementation uses complex128; roundoff is outside this theorem and a small numerical step is not an error certificate. Conditioning depends on the smallest observed window coefficient. All (d^2) cyclic intensities are retained.

A certified boundary outside the family

The seven-dimensional example is an 18-point abstract Weyl erasure mask. Two finite-field ideal certificates exclude every nonzero complex rank-at-most-two matrix in its span. A separate exact Hermitian witness has inertia ((6,1)), so a rank-one PSD target has a rank-six PSD competitor with the same observed coefficients.

This shows why pure-state uniqueness alone cannot justify arbitrary-mask PSD recovery. The exact realization of this particular mask by a single STFT window is unestablished. It is not presented as a realizable-window counterexample.

Verification evidence

Audit	Delivered assertion checked
verify_masks.py	44,739,232 strip masks in even dimensions 4–24; gcd and congruence criteria agree
verify_exact_certificates.py	Two ten-dimensional windows; 188 nonzero coefficients certified in total; exact inertia certificate
verify_complex7.py	Degree-four ideal certificates modulo 29 and 43; 29,400 independent minor evaluations
verify_numerics.py	900 mask/projector checks, 900 sharp paired witnesses, 1,230 dual checks, 2,700 paired inequalities, and 80 recovery cases

The original run and fresh release run are kept separately. Timing and floating-point errors may differ by platform. Exact zero identities, modular ranks, interval exclusions, and numerical experiments are identified separately in the reports.

Research status and reuse

The precisely stated family result is presented as proved in the manuscript, with executable supporting evidence. The release package can be complete while the broader research problem remains unresolved; no measured percentage of that unrestricted problem is assigned. Independent review, an exhaustive novelty search, reduced measurement complexity, laboratory validation, and unknown-window calibration recovery are not established.

The manuscript credits established lifting, algebraic injectivity, strict-completeness, and PSD-projection methods. See research/SOURCE_AUDIT.md and the manuscript bibliography for the contribution boundary. This release-preparation check does not certify mathematical priority.

Code is distributed under MIT; research documents and certificate data under CC BY 4.0, as specified in LICENSE.md. Cite version 6.0.0 and its manuscript attribution. Add the actual repository URL and any DOI after publication; neither is invented here.