--- title: "The homogeneous adjacency spectrum of the Fano plane" date: 2026-09-19 version: "0.1.0-candidate" doi: 10.5281/zenodo.22850084 pdf: https://github.com/ipitchford/fano-plane-spectrum/releases/download/v0.1.0-candidate/fano_spectrum.pdf repository: https://github.com/ipitchford/fano-plane-spectrum archive: https://zenodo.org/records/22850084 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # The homogeneous adjacency spectrum of the Fano plane ## Summary The Fano plane is a tiny geometry: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 $$ \begin{aligned} \phi_F(\lambda)={}&\lambda^{35}(\lambda-1)^{35}(\lambda-2)^{28}(\lambda-3)\\ &\cdot(\lambda^2-\lambda+1)^7(\lambda^2+2\lambda+6)^7(\lambda^2+\lambda+1)^{21}\\ &\cdot(\lambda^2+\lambda+2)^{52}(\lambda^3+2\lambda^2+2\lambda-2)^{21}\\ &\cdot(\lambda^4-\lambda^3-\lambda^2+\lambda+1)^{28}. \end{aligned} $$ 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: ```sh python package.py check python verify.py python -O verify.py python test_negative.py ``` Expected 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 1. Reimplement and rerun the full resultant computation outside the producer workflow, including the source-to-equations bridge. 2. Audit the finite eigenscheme classifications and characteristic-zero curve proof with specialist algebraic-geometric scrutiny. 3. Seek a conceptual explanation of the exceptional exponent 52 and determine whether the curve has a recognized classical model. 4. 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. ## Open directions for follow-up research - 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. ## Research process, metrics and reusable methods This is prospective process metadata under the Evidence Press operating model and research-metrics policy. It records the intended handoff, measured scope and claim boundary; it is not evidence that the method accelerated this work. - Work ID: ep-work:fano-plane-spectrum - Attempt and metric receipts: ep-attempt:fano-plane-spectrum-assurance-publication: published / positive; scope assurance-through-publication; target Complete review revisions and GitHub, Zenodo and guarded Evidence Press publication with media and public readback.; active forecast 150 minutes (90-240); Fermi components Scientific integration, verifier hardening and editorial gate: 1 x 30/50/80 minutes low/central/high (Existing exact package.); Immutable archives and communication assets: 1 x 30/50/80 minutes low/central/high (Standard release route.); Composite CI and deployment/readback cycles: 1 x 30/50/80 minutes low/central/high (Standard guarded route.); positive-signal/closure probabilities 0.95/0.85 within 240 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 6/unknown/unknown/0/5/2; cycles positive/negative/inconclusive 0/0/0; falsification gates 5; architectures tested/rejected 1/0; result target-closed; target reached true; forecast error -144 minutes; ratio 0.04; inside interval false; positive-signal/target-closure Brier scores 0.0025/0.0225; missing telemetry activeHumanMinutes: Human effort not instrumented.; computeMinutes: Substantive computation not separately instrumented; ordinary producer replay and build/CI are excluded.; deduplicatedModelTokens: Root response-level deduplication available locally, but complete all-agent accounting unavailable; no partial total presented as complete.; uncachedInputTokens: Root-only cache accounting available, complete all-agent total unavailable.; appended measurement corrections measurement.agentRuns -> metrics.outcome.agentRuns: 7 (reason: Root coordination plus five internal role executions and one focused confirmation.); measurement.reworkMinutes -> metrics.outcome.reworkMinutes: 2 (reason: Previously recorded two-minute seal/log race repair; other repair time not separately instrumented.). Work ledger: https://evidencepress.org/api/work-ledger.json. Metrics policy: https://evidencepress.org/api/research-metrics-policy.json - Intended aims: science - Artifact roles: research-output, evidence-assessment, communication - Decision object: certificate — Exact Fano homogeneous resultant and eigencurve certificates. Scope: One specified order-three normalized tensor in characteristic zero; no general multiplicity or priority theorem. - Reusable methods: Certificate-first, proof-carrying research (certificate-first); Structural compression (structural-compression); Adversarial scientific controls (adversarial-controls); Assurance as a vector (assurance-vector); Agent-readable research objects (agent-readable-research-object). Registry: https://evidencepress.org/api/method-registry.json - Targeted clocks: assurance, publication - Semantic bridge: explicit — Seven-line incidence data determines quadratic eigen-equations; Poisson recursion, CRT and characteristic-zero algebra support the declared claims. Remaining risks: CAS and source-to-equations correctness.; Saved versus fresh computational provenance.; Finite decomposition and historical-priority coverage.. - Human judgement gates: Audit mathematical arguments and equation encoding.; Assess prior art and priority separately.; Retain rights and assurance boundaries. - Next assurance action: Unaffiliated full computation and specialist proof audit. - Claim ceiling: 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. - Aim-scoped impact evidence: - science: NO_IMPACT_EVIDENCE — Inspectable exact spectral and eigenscheme example in Producer-coordinated mathematical publication; design none; comparator None.; estimand No acceleration or impact effect estimated.; no real-world effect evidence asserted ## 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. ## References 1. Cooper and Dutle (2012), Spectra of uniform hypergraphs. Normalization and spectral framework. 2. Cooper and Dutle (2015), Computing hypermatrix spectra with the Poisson product formula. Established algorithmic precedent. 3. Clark and Cooper, Applications of the Harary–Sachs theorem for hypergraphs. Inspected author version: leading Fano coefficients and complete Rowling polynomial. 4. Doğan, Tsigaridas and Zafeirakopoulos (2026), Resultant multiplicity via projective degrees and applications to tensor eigenvalues. Recent preprint, contextual bound.