--- title: "A ten-by-ten trace formula for the essential Hurwitz form" date: 2026-09-07 version: "0.1.0-candidate" doi: 10.5281/zenodo.22650445 pdf: https://github.com/ipitchford/essential-hurwitz-trace-formula/releases/download/v0.1.0-candidate/paper.pdf repository: https://github.com/ipitchford/essential-hurwitz-trace-formula archive: https://zenodo.org/records/22650445 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # A ten-by-ten trace formula for the essential Hurwitz form ## Summary A generic section of the essential variety consists of ten points. Its Hurwitz form detects where that intersection degenerates. This unrefereed candidate gives a compact ten-by-ten trace formula for that large degree-30 polynomial, together with a written proof that the apparent denominator cancels globally. It also treats the related variety of symmetric four-by-four matrices of rank at most two. ## Summary for specialists For a nine-by-four frame B, substitute the affine matrix E=B(1,x,y,z) into the ten essential cubics. Split their coefficients into the ten degree-three monomials and the ten monomials of degree at most two, writing f=Ac+Cb. Set D=det A. On D nonzero, reduction by these equations produces the rank-ten quotient algebra and its multiplication matrices. With T the trace pairing in the ordered quadratic basis, the candidate formula is $$H(B)=D(B)^4\det T(B).$$ The manuscript proves that H extends to a homogeneous polynomial of frame degree 120, descending to the degree-30 Hurwitz form in Pluecker coordinates, up to a nonzero scalar. It supplies an exact-division polynomial expression and a finite interpolation fallback at D=0. The output is a compact arithmetic formula, not an expanded monomial coefficient list. ## Technical account The proof has four stages. First, the determinant D is identified with the Chow boundary: a zero detects intersection with the chosen plane at infinity. Second, the coefficient reduction gives a free algebra of rank ten uniformly over the open set D nonzero, not merely at sampled frames. Third, along a generic point of the boundary, exactly one section point tends to infinity. The quadratic evaluation basis bounds the pole of the trace determinant by four. This is why the correcting factor is precisely D to the fourth power. Fourth, the resulting global polynomial has the required degree and generic simple vanishing, identifying the irreducible Hurwitz divisor. For symmetric matrices, the manuscript specifies all ten upper-triangular coordinates, signed cofactor generators and the witness induced by the essential-to-symmetric linear identification. The rank-one singular locus has codimension three, satisfying the required generic-section condition. ## Evidence, assurance and limitations The eight-page manuscript contains the universal argument. Exact producer controls test general linear covariance, a primitive-element comparison with basis correction, a tangency, a reduced section with a point at infinity, a misleading projection collision, Pluecker input validation and the polynomial quotient identity. A separate producer-constructed Singular Jacobian computation cross-checks tangency and reducedness. A further control uses Jiang and Sturmfels's published symmetric example: the value vanishes at parameter 194 with trace rank nine, whereas parameter 195 gives a nonzero value and rank ten. This tests a published fixture, not the entire theorem. The supplied review and publisher-coordinated model reviews are disclosed as internal assurance inputs. The supplied review's linked extra audit files were unavailable and are not claimed as archived independent reconstruction. No unaffiliated whole-proof acceptance, formal verification, historical-priority clearance or demonstrated practical improvement is asserted. ## Relationship to earlier work Sturmfels's Hurwitz-form theorem provides the degree and divisor framework. Floystad, Kileel and Ottaviani provide the essential variety's Chow-form and symmetric-model setting. Classical trace discriminants and D'Andrea–Jeronimo rational trace formulas are antecedents, not inventions of this candidate. Jiang and Sturmfels already computed a one-parameter degree-30 specialization in their 2021 Example 12. The present contribution sought is a reusable compact formula and its global cancellation proof, not the first specialized Hurwitz evaluation. Fan, Kileel and Kimia's May 2025 revision identifies a necessary degree-30 five-point ill-posedness polynomial and notes the explicit polynomial was unavailable in that treatment. This does not establish practical superiority of the present exact algorithm. ## Who should care, and why | Audience | Potential use | Required caution | |---|---|---| | Computational algebraic geometers | Inspect or reuse a small trace-matrix representation of a large divisor. | The global proof remains unrefereed. | | Multiview-geometry researchers | Study the algebraic degeneracy locus underlying five-point estimation. | No physical sufficiency, numerical stability or speedup is established. | | Exact-computation developers | Independently reconstruct the finite algebra and boundary evaluator. | Finite agreement is not whole-proof validation. | ## Why the problem matters A polynomial may have an unwieldy expanded expression but a short exact evaluation recipe. Here the key issue is not merely finding a determinant that works on a convenient chart: it is proving that the chart denominator cancels and that the formula represents the intended global geometric divisor. A successful compact representation makes further scrutiny and exact experimentation accessible without expanding every coefficient. ## How to inspect or reproduce the recorded checks Download the immutable archive, verify MANIFEST.sha256, and install the pinned SymPy dependency. Run python3 verify.py and python3 -O verify.py for the Python-only suite. Add --full to either command when Singular is installed to include the separately constructed Jacobian checks. The verifier works in a temporary copy and compares exact semantic outputs while allowing timings to differ. Both modes retain their checks, and an intentional failure confirms optimized mode cannot turn them off. No historical machine path, remote solver or generic expanded polynomial computation is required. PDF rebuilding separately needs Pandoc and pdfLaTeX. ## The most valuable next projects The highest-value next step is unaffiliated scrutiny of the generic Chow-boundary valuation and divisor identification, together with an independently reconstructed exact implementation. A specialist prior-art search should test for equivalent earlier constructions. Numerically stable evaluation, practical estimator benchmarks and applications to real camera configurations are separate projects, not results already established here. ## What is in the evidence package The package includes the PDF and accessible manuscript, formula and global evaluator, symmetric fixture, exact receipts, nonmutating verifier, source comparison, claims index, review response, internal editorial records, environment guidance and complete manifest. Original prose and data use CC0; original code uses MIT. GitHub and the version DOI identify the same release assets. Publication and communication media improve access, not mathematical assurance. ## Open directions for follow-up research - Unaffiliated scrutiny of the Chow-boundary valuation and global divisor identification. - Independent implementation and specialist priority comparison. - Numerically stable evaluation and empirical camera-estimation benchmarks are separate targets. ## 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:essential-hurwitz-trace-formula - Attempt and metric receipts: ep-attempt:essential-hurwitz-trace-formula-assurance-publication: published / positive; scope assurance-through-publication; target Review repairs, deterministic baseline, five-role internal acceptance, immutable GitHub and Zenodo assets, and first canonical Evidence Press publication.; active forecast 150 minutes (120-210); Fermi components Review repairs and sources: 1 x 25/30/40 minutes low/central/high (Minor attribution and metadata revisions.); Package baseline and internal review: 1 x 35/45/65 minutes low/central/high (Compact proof with fast exact replay.); Public identity and media: 1 x 30/35/50 minutes low/central/high (Existing publishing tools.); Site seals and canonical readback: 1 x 30/40/55 minutes low/central/high (Mandatory new-slug two-deployment cycle.); positive-signal/closure probabilities 0.95/0.85 within 240 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 8/unknown/unknown/3/0/0; cycles positive/negative/inconclusive 0/0/0; falsification gates 1; architectures tested/rejected 0/0; result target-closed; target reached true; forecast error -142 minutes; ratio 0.05333333333333334; inside interval false; positive-signal/target-closure Brier scores 0.0025/0.0225; missing telemetry activeHumanMinutes: No contemporaneous human active-time instrument.; computeMinutes: No complete scoped substantive-computation wall clock; ordinary CI/build time is not research compute.; appended measurement corrections measurement.agentRuns -> metrics.outcome.agentRuns: Terminal receipt records six agent runs; original intake snapshot retained. (reason: Prospective snapshot predates five completed internal reviewers.); measurement.reworkMinutes -> metrics.outcome.reworkMinutes: Actual repairs are disclosed in outcome varianceReason; zero does not mean no repair. (reason: No complete repair-specific active-time instrument; terminal zero is an explicit lower bound.); metrics.outcome integer-minute projection: Floor rather than round up incomplete lower bounds; derived calibration fields recalculated. (reason: Prepublication schema check requires integer minutes; precise first-cut receipt retained separately.). 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: reusable-method — Compact trace formula and global polynomial cancellation argument. Scope: Essential Hurwitz form and symmetric rank-two four-by-four corollary. - Reusable methods: Structural compression (structural-compression); Certificate-first, proof-carrying research (certificate-first); 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 — Coefficient reduction constructs the quotient algebra; Chow geometry bounds the denominator; the escaping-point valuation and degree identify the global Hurwitz form. Remaining risks: Uniform written proof awaits unaffiliated scrutiny.; Finite controls cannot establish universal correctness.; Earlier equivalent formulas are not excluded.. - Human judgement gates: Audit the generic boundary valuation and multiplicity-one identification.; Assess novelty against classical trace/Chow formulas and specialized computations.; Preserve attribution, rights and assurance boundaries. - Next assurance action: Unaffiliated mathematical audit and direct prior-art comparison. - Claim ceiling: Unrefereed compact formula and uniform written proof candidate. Producer finite replay and internal AI review do not establish unaffiliated whole-proof acceptance, formal verification, historical priority or practical improvement. - Aim-scoped impact evidence: - science: NO_IMPACT_EVIDENCE — Inspectable compact global formula candidate in Producer-coordinated publication; design none; comparator No matched comparison; estimand No acceleration or downstream effect estimated; no real-world effect evidence asserted - Parent handoffs: depends-on-claim https://arxiv.org/abs/1410.6703; inherited claim: Hurwitz degree and irreducible divisor framework.; inherited ceiling: Antecedent does not externally validate this explicit construction. ## Verification status Unrefereed compact formula and uniform written proof candidate. Producer finite replay and internal AI review do not establish unaffiliated whole-proof acceptance, formal verification, historical priority or practical improvement. ## References 1. Sturmfels, The Hurwitz Form of a Projective Variety (2017), Theorem 1.1. 2. Floystad, Kileel and Ottaviani, The Chow Form of the Essential Variety in Computer Vision (2018), section 2. 3. Jiang and Sturmfels, Bad projections of the PSD cone (2021), Example 12. 4. D Andrea and Jeronimo, Rational Formulas for Traces in Zero-Dimensional Algebras (2008). 5. Fan, Kileel and Kimia (2025), arXiv:2310.02719v2, Theorem 22 and Remark 23.