--- title: "A six-dimensional counterexample to cross-polytope maximality in zonoid approximation" date: 2026-09-08 version: "1.0.0-candidate" doi: 10.5281/zenodo.22660175 pdf: https://github.com/ipitchford/six-dimensional-zonoid-counterexample/releases/download/v1.0.0-candidate/six-dimensional-zonoid-counterexample-1.0.0-candidate.pdf repository: https://github.com/ipitchford/six-dimensional-zonoid-counterexample archive: https://zenodo.org/records/22660175 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # A six-dimensional counterexample to cross-polytope maximality in zonoid approximation ## Summary The cross-polytope is a natural candidate for the symmetric shape hardest to enclose by sums of line segments and their limits. This release gives a counterexample: adding one antipodal pair to a six-dimensional cross-polytope increases the exact enclosure factor. The gain is small but strictly positive, and rational certificates prove it without relying on numerical tolerances. ## Summary for specialists The cross-polytope does not maximize the zonoid enclosure factor among all origin-symmetric convex bodies. This candidate gives a concrete counterexample in six dimensions: \[ \begin{aligned} K&=\operatorname{conv}\{\pm v_1,\ldots,\pm v_7\},\\ v_j&=4e_j\quad(1\le j\le6),\\ v_7&=(1,1,1,1,2,2),\\ \lambda(K)&=\frac{122}{65}=\frac{15}{8}+\frac1{520}. \end{aligned} \] Here \(\lambda(K)\) is the smallest factor \(t\) for which some zonoid lies between \(K\) and \(tK\). A zonoid is a Hausdorff limit of Minkowski sums of segments. The benchmark \(\lambda(C_6)=15/8\) is the value recorded in Schneider’s AIM question. The strict gap is small but exact; no floating-point tolerance enters the final certificate checks. ## Technical account: why the finite witness covers every zonoid The lower witness supplies 64 rational signed-sum identities. Negation covers the remaining sign choices. These identities imply a support inequality for every generator direction, not just a sampled list. Summing over generators proves it for zonotopes; continuity of support functions extends it to every centered zonoid. Symmetrization preserves the required enclosure, so allowing translated zonoids does not evade the bound. The matching upper witness contains 53 rational generators and a \(7\times53\) coefficient matrix. Exact vertex representations prove the inner inclusion. Exact support checks at every facet prove the outer inclusion. A second enumeration describes the polar as a clipped cube: 54 retained cube vertices plus 16 new edge intersections give 70 polar vertices, or 35 antipodal facet pairs. ## Evidence and replay The linked paper contains the full argument and the nonzero lower weights. The evidence archive includes both rational certificates, standard-library Python checkers, historical discovery records and the review-response record. ```sh python3 replay_review.py python3 -O replay_review.py python3 verify_polar.py ``` The replay rejects eight corrupted lower witnesses and eight corrupted upper witnesses, including surplus and missing coordinates. The polar checker verifies all 70 upper-support equalities. These rejection controls test the checkers; they are not substitutes for the written proof or external validation. ## What changed after review The supplied review recommended minor revisions without identifying a fatal proof defect. This release adds explicit dimension checks, four corresponding rejection tests, a corrected accessible manuscript, related-work distinctions, the numerator calculation and the polar-body explanation. The supplied review’s separately linked audit ZIP was unavailable here, so its reported arithmetic checks are not counted as inspected independent reproduction. ## Scope and limitations This is an unrefereed computer-assisted candidate. Producer replay and five internal model-assisted editorial reports do not constitute unaffiliated specialist review, independent reproduction or formal verification. A bounded search for the final object found no matching earlier example, but does not clear historical priority. The result does not identify the largest factor in dimension six, the smallest counterexample dimension, a unique enclosure or the minimum possible generator count. It provides no measured application benefit or research-speed comparison. ## Relationship to earlier work Schneider’s support-function arguments are antecedents of the method. Henk, Linke and Wills study enclosures of a fixed cross-polytope; Siegel studies finite approximation of bodies already known to be zonoids. Neither formulation should be conflated with maximizing the enclosure factor over arbitrary symmetric bodies. ## Who should care, and why | Audience | Potential use | Required caution | |---|---|---| | Convex geometers | An explicit test object for zonoid enclosure extremality | Not a classification of worst bodies | | Optimization researchers | A rational dual witness with an all-direction interpretation | Discovery LP alone would not prove the lower bound | | Proof-verification researchers | Small exact certificates and two facet checks | No proof-assistant formalization is supplied | ## Why the problem matters An extremal question asks more than whether one symmetric example is difficult. A single exact counterexample can separate a plausible benchmark from the actual extremal value. Here the matching enclosure also determines the new body's factor, not merely a lower estimate. ## The most valuable next checks The next useful assurance steps are unaffiliated scrutiny of the universal support-function argument, independent reconstruction of the rational certificates and a broader contribution-specific priority assessment. Historical research-goal metrics are separately labelled in the archive. The publication attempt below measures only its prospectively registered assurance-and-publication scope. ## What is in the evidence package The PDF and accessible Markdown give the full proof; the two root JSON certificates contain the rational witnesses. The replay scripts check identities, containments and malformed-input rejection. The separate polar checker explains the facet count. The archive also includes internal editorial reports, the supplied-review response, a bounded final-object search log, component licences and historical discovery records clearly separated from the authoritative root result. ## Open directions for follow-up research - What is the largest zonoid enclosure factor in dimension six? - What is the smallest dimension admitting a counterexample? - Are optimal enclosures unique, or can fewer than 53 generators suffice? - Assess this particular example and proof against earlier literature independently. ## 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:six-dimensional-zonoid-counterexample - Attempt and metric receipts: ep-attempt:six-dimensional-zonoid-counterexample-remaining-assurance-publication: published / positive; scope assurance-through-publication; target Bounded partial-results candidate with fail-closed replay, five-role internal review, public immutable archives and guarded canonical readback.; active forecast 150 minutes (100-230); Fermi components Consolidated package and source checks: 1 x 30/45/65 minutes low/central/high (Large existing dossier, bounded revisions.); Internal five-role review: 1 x 20/30/50 minutes low/central/high (One differentiated round.); Public archives and media: 1 x 30/45/65 minutes low/central/high (Existing publisher tools.); Gates and public readback: 1 x 20/30/50 minutes low/central/high (Two mandatory seal/deploy loops.); positive-signal/closure probabilities 0.95/0.85 within 300 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 28/unknown/unknown/0/0/12; cycles positive/negative/inconclusive 0/0/0; falsification gates 1; architectures tested/rejected 0/0; result target-closed; target reached true; forecast error -122 minutes; ratio 0.19; inside interval false; positive-signal/target-closure Brier scores 0.0025/0.0225; missing telemetry activeHumanMinutes: Human effort not instrumented.; computeMinutes: No complete substantive compute meter across model calls; ordinary publication builds do not stand for research compute.; deduplicatedModelTokens: No authoritative task-local fork-aware counter; inherited token events not summed.; uncachedInputTokens: No authoritative task-local uncached input counter.; appended measurement corrections question: Can the reviewed exact six-dimensional zonoid counterexample pass remaining assurance and full publication gates? (reason: Copied prior-task descriptor survived intake templating; corrected before first publication. Original Git history and ATTEMPT_FORECAST.json retained.); selectionBasis: User-authorized minor-revision exact-counterexample release. (reason: Copied prior-task descriptor survived intake templating; corrected before first publication. Original Git history and ATTEMPT_FORECAST.json retained.); decisionObjectTarget: Inspectable exact lower certificate and matching 53-generator zonotope enclosure. (reason: Copied prior-task descriptor survived intake templating; corrected before first publication. Original Git history and ATTEMPT_FORECAST.json retained.); metrics.forecast.targetOutcome (interpretation only): Read the frozen target as the reviewed exact-certificate candidate with fail-closed replay, internal review, immutable archives and canonical readback. No numerical reforecast. (reason: Original frozen forecast accidentally says partial-results. Preserve the original forecast bytes and all numeric probabilities, time estimates and registration time. The user-authorized target throughout was publication of the exact counterexample, not new research.); measurement.agentRuns -> metrics.outcome.agentRuns: Opening 1 remains; terminal total 6. (reason: Opening snapshot retained, terminal count includes coordinator plus five separate role reviewers.); measurement.reworkMinutes -> metrics.outcome.reworkMinutes: Terminal approximate operational repair subtotal is twelve minutes, not scientific rework. (reason: Initial zero was an opening snapshot, not a terminal measurement; original retained.). 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 - Decision object: counterexample — A fourteen-vertex symmetric convex body with exact factor 122/65. Scope: Specified six-dimensional body, compared with cross-polytope factor 15/8. - 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 — Signed-sum identities imply a direction-universal inequality, summed over zonotope generators and extended to zonoids by support-function continuity; exact upper containment closes equality. Remaining risks: Written semantic bridge awaits unaffiliated specialist scrutiny.; No formal proof-assistant verification.; Historical priority remains bounded uncertainty.. - Human judgement gates: Audit the all-zonoid extension and facet completeness.; Assess contribution-specific priority and significance.; Preserve candidate status and component rights. - Next assurance action: Obtain unaffiliated scrutiny of the proof and independent exact reconstruction. - Claim ceiling: Unrefereed computer-assisted candidate. Exact producer replay and five internal model-assisted editorial reports are not unaffiliated specialist review, independent reproduction or formal verification. Historical priority, least counterexample dimension and global dimension-six extremality are not established. - Aim-scoped impact evidence: - science: NO_IMPACT_EVIDENCE — Inspectable exact mathematical counterexample in Producer-coordinated publication; design none; comparator No matched comparator.; estimand No speed or impact effect estimated.; no real-world effect evidence asserted - Parent handoffs: depends-on-claim https://aimath.org/WWN/fourierconvex/fourierconvex.pdf; inherited claim: Definition of lambda and the stated cross-polytope benchmark.; inherited ceiling: Imported source benchmark, not external validation of this counterexample. ## Verification status Unrefereed computer-assisted candidate. Exact producer replay and five internal model-assisted editorial reports are not unaffiliated specialist review, independent reproduction or formal verification. Historical priority, least counterexample dimension and global dimension-six extremality are not established. ## References 1. AIM Fourier analytic methods in convex geometry, Question 2 (Schneider), page 1. 2. Schneider (2001), On the Busemann area in Minkowski spaces: antecedent support-function methods. 3. Henk, Linke and Wills (2010), Minimal zonotopes containing the crosspolytope: fixed-body enclosure questions. 4. Siegel (2025), Optimal approximation of zonoids and uniform approximation by shallow neural networks: already-zonoid inputs.