Press release · 19 September 2026 · version 0.1.0-candidate
The homogeneous adjacency spectrum of the Fano plane
An exact tensor spectrum, a real eigenvalue without real eigenvectors, and a hidden genus-three curve.
Summary
The Fano plane is a tiny geometry: seven points joined by seven three-point lines. Yet its tensor spectrum is much richer than the eigenvalues of an ordinary seven-by-seven matrix. The relevant characteristic polynomial has degree 448.
This candidate gives that polynomial's complete factorization and nineteen distinct eigenvalues. It also finds an unusual contrast: the real eigenvalue two has no real eigenvector, while two complex eigenvalues have an entire smooth curve of projective eigenvectors. The curve has degree eight and genus three. Exact computation supports these claims, but the release remains unrefereed; historical priority and unaffiliated validation are not established.
Summary for specialists
Let the seven lines be $124,235,346,457,156,267,137$. Use the symmetric order-three adjacency tensor with entries $1/2$ on permutations of each line, so that $\mathcal A x^2=\lambda x^{[2]}$. The candidate factorization is
The H-spectrum is $\{0,1,3,\theta\}$, where $\theta$ is the unique real root of $\lambda^3+2\lambda^2+2\lambda-2$. At each root of $\lambda^2+\lambda+2$, the saturated projective eigenscheme is a smooth geometrically irreducible curve of degree eight and genus three, contained in $\sum_i x_i=0$.
Technical account
The main compression is a standard Poisson product formula. It replaces one large symbolic resultant by a recursion through induced hypergraphs, using multiplication matrices of dimension at most 64. Modular evaluation and interpolation produce residue polynomials; an a-priori root bound makes their Chinese-remainder lift unique. The thirty-prime product has 930 bits, exceeding twice the coefficient bound. The reused six-vertex polynomial satisfies its own, smaller bound.
Explicit point, line, flag and anti-flag constructions give eigenvectors. These prove existence, not completeness: the latter comes from the resultant. A rational polynomial identity excludes real eigenvectors at two. For the curve, a characteristic-zero saturation, Hilbert series, Jacobian calculation and minimal resolution supply the smoothness and connectedness argument. Smoothness plus geometric connectedness gives geometric irreducibility.
The cover illustrates the seven-point incidence structure and the nineteen distinct roots in the complex plane. It does not depict multiplicity or serve as a verification certificate.
Evidence, assurance and limitations
Producer replay checks all 449 coefficients, all thirty saved residue polynomials and every inductive lift bound. Fresh checks cover explicit vectors, all 168 automorphisms, characteristic-zero curve geometry, the no-real-vector identity, rational chart lengths and four Macaulay evaluations. Normal and optimized Python runs agree; deliberately corrupted inputs are rejected. Macaulay supplies a different determinant construction but shares interpolation helpers with the Poisson implementation.
Routine replay does not freshly repeat the complete thirty-prime computation or every finite number-field decomposition. Those original artifacts and their source programs are preserved. CAS correctness and the translation from incidence data to equations remain trust boundaries. Internal editorial assessment is not unaffiliated specialist review, independent reproduction or proof-assistant verification. No official research-quality star rating, historical first-solution claim or measured impact is asserted.
Relationship to earlier work
Cooper and Dutle established the normalization and Poisson approach; this is an application, not a new resultant algorithm. Clark and Cooper's inspected author version supplies leading Fano coefficients and the complete polynomial for the two-line-deleted Rowling hypergraph. The latter is a regression fixture, not a new result here. Our codegree-fourteen coefficient differs from the inspected author-version table; no correction to an uninspected final publisher PDF is claimed.
The problem appears in Cooper's October 2020 list. A recent paper's reference to a 2022 Fano polynomial is a priority lead, not resolved by the historical open label. The bounded literature and GitHub search found no matching full factorization, but does not establish exhaustive novelty or priority.
Who should care, and why
| Audience | Potential use | Required caution |
|---|---|---|
| Spectral hypergraph researchers | A compact exact example with nontrivial eigenschemes | Distinguish resultant multiplicity, scheme length and point count |
| Computer algebra researchers | A small reproducible benchmark across two resultant constructions | Shared utilities and producer coordination are not independence |
| Algebraic geometers | An explicit degree-eight, genus-three eigencurve | No identification with a classical curve is claimed |
| Interested readers | A concrete contrast between matrix and tensor eigenvectors | Candidate computation is not established consensus |
Why the problem matters
Small symmetric examples make otherwise abstract tensor phenomena inspectable. Here a finite spectral list coexists with a continuous family of eigenvectors, and a real eigenvalue need not admit a real eigenvector. The potential value is an exact test case and reusable evidence package—not a demonstrated practical application or an advance in the general theory of tensor multiplicities.
How to inspect or reproduce the recorded checks
Start with the archive README and assurance statement. Install Python 3.12 or later, Singular 4.4.1 and the pinned Python dependencies, then run:
python package.py check
python verify.py
python -O verify.py
python test_negative.pyExpected status is PASS, with nonzero exit on failure. The README separately documents the more expensive full-resultant route. The manifest checks file identity; it does not validate the mathematics.
The most valuable next projects
- Reimplement and rerun the full resultant computation outside the producer workflow, including the source-to-equations bridge.
- Audit the finite eigenscheme classifications and characteristic-zero curve proof with specialist algebraic-geometric scrutiny.
- Seek a conceptual explanation of the exceptional exponent 52 and determine whether the curve has a recognized classical model.
- Resolve the historical-priority leads through broader primary-source work.
What is in the evidence package
The archive contains the PDF and accessible manuscript source, coefficient and residue data, exact Python and Singular programs, characteristic-zero certificates, replay receipts, semantic negative controls, source/novelty audit, review responses, machine-readable claims, manifest and component licences. Original prose/data are CC0 and executable code MIT. The scholarly creator is Anonymous. Audio and imagery communicate the result; they add no mathematical evidence.
Media
The audio briefing is provided in the header above. Download the MP3 briefing · read the transcript.
Open directions for follow-up research
Also available in machine-readable form for research agents and follow-up projects.
- Unaffiliated full resultant reconstruction and semantic audit.
- Specialist audit of finite eigenscheme classification and curve geometry.
- Conceptual explanation of multiplicity 52 and possible classical curve identification.
- Broader primary-source work on historical priority.
Verification status
Unrefereed computational theorem candidate. Default replay does not freshly repeat the complete thirty-prime computation or every finite number-field decomposition. No unaffiliated validation, formal proof, exhaustive priority or impact claim.
Cite
BibTeX
@misc{fanoplanespectrum2026,
title = {The homogeneous adjacency spectrum of the Fano plane},
author = {Anonymous},
year = {2026},
doi = {10.5281/zenodo.22850084},
url = {https://doi.org/10.5281/zenodo.22850084},
version = {0.1.0-candidate},
howpublished = {Zenodo},
note = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/fano-plane-spectrum/}
}Also: cite.bib · paper.json · this page as Markdown