Strong permanent inequalities for complex two- and three-chord seven-cycle supports
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Abstract
We prove per(|A|²) ≤ per(A) for complex positive semidefinite correlation matrices supported on a seven-cycle with chords 03,15,26. More precisely, the difference dominates one fiftieth of the matching-partition loss. The proof combines elementary cycle estimates with an explicit nonnegative integer polynomial identity, supplied with a standard-library verification program. The two-chord subgraph with chords 03,15 inherits the same bound. For the crossing placement 03,14 we give a separate analytic proof with constant 1/12. All results include missing edges and singular matrices. The normalized arguments require only vertex-star magnitude bounds, and we describe equality and diagonal scaling.
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- Strong_Permanent_Multichord_Seven_Cycles.pdf92.5 KiB · pdf
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Research details
- Subjects
- Mathematics · Linear Algebra and Matrix Inequalities
- Keywords
- permanent inequality, positive semidefinite matrix, Chollet conjecture, seven-cycle support, graph matching, exact integer certificate
- Manuscript date
- 2026-10-22
- Reader reuse
- No additional permission granted
Scope
The strong bound applies to the specified three-chord seven-cycle support and its subgraphs, including the two-chord 03,15 placement, with matching-loss constant 1/50. The crossing two-chord 03,14 placement has a separate analytic 1/12 bound. Arbitrary complex phases, missing edges and singular PSD endpoints are included. The normalized proof holds on the stated Hermitian vertex-star magnitude domain.
Disclosures
AI tools assisted drafting and verification, as disclosed in the manuscript; the author is responsible for the final statements, proofs and attribution. The preceding cycle/single-chord paper, structural graph literature, cycle expansion and polynomial-positivity framework are credited.
Reproduction
The six-member package contains the complete seven-page PDF, editable manuscript/main.tex, certificate.json, verify_certificate.py, reader instructions and MANIFEST.sha256. The Python standard-library verifier reconstructs 53 matchings, 15 cycles and the exact 323-term nonnegative integer identity with zero residual. Run python -B verify_certificate.py, optionally with -O. The analytic reduction, endpoint arguments and crossing-class proof are in the manuscript; no solver or private dependencies are needed.
References and related resources
- J. Chollet, Is there a permanental analogue to Oppenheim’s inequality?, American Mathematical Monthly 89 (1982), 57–58
- S. W. Drury, A counterexample to a question of Bapat and Sunder (2016)
- P. Pant and R. Singh, Structural Classes for Chollet’s Permanent Conjecture, arXiv:2604.24192v2
- S. Schubr, Strong permanent inequalities for complex positive semidefinite cycle and single-chord supports, edition 1 (2026)
- D. Handelman, Representing polynomials by positive linear functions on compact convex polyhedra (1988)
- R. Mehatari, M. R. Kannan and A. Samanta, On the adjacency matrix of a complex unit gain graph (2022)