Research Article · Preprint Mathematics
Exact-rank quadratic correlation: a phase-compatible Clique reduction
- Primary subject
- Mathematics
- Subject subfields
- Combinatorics and Graph Theory · Algorithms and Complexity
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 1 · Permanent link
Overview
Abstract
We give an additive-gap reduction from Clique to correlation maximization over Boolean quadratic phases of prescribed even polar rank r ≥ 4. For the bounded function obtained by averaging graph-edge characters, an (r + 1)-clique gives exact-rank optimum 2^(−r/2) binom(r + 1, 2)/|E|, while its absence decreases the optimum by at least 2^(−r/2)/|E|. The proof combines the graphic span bound with control of nonzero affine Fourier supports and a nondegenerate quadratic Gauss witness realizing every clique-edge sign simultaneously. Consequently, an exact-rank additive optimizer with running time f(r) poly(n, ε⁻¹) would give a fixed-parameter Clique algorithm, with the corresponding randomized implication for randomized optimization. Explicit examples separate exact rank from rank at most a cap and show why the rank-two gap fails. An anchored affine-restriction lemma preserves both an output-rank cap and its rank distance from a known quadratic, providing a positive structural contrast.
Scope of the work
Additive-gap Clique reduction for Boolean quadratic phases of exact even polar rank r ≥ 4, with affine-support control, phase-compatible witnesses and anchored rank-preserving restrictions.
Reproduction notes
Editable manuscript, build instructions and exact finite verification program are included in the supplementary package.
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Cite this edition
Scott Schubr. (2026). Exact-rank quadratic correlation: a phase-compatible Clique reduction (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/exact-rank-quadratic-correlation-phase-compatible-clique-reduction/v/1
BibTeX
@techreport{luminary_exact_rank_quadratic_correlation_phase_compatible_clique_reduction_v1,
title = {Exact-rank quadratic correlation: a phase-compatible Clique reduction},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/exact-rank-quadratic-correlation-phase-compatible-clique-reduction/v/1}
}Loading discussion…