{ "schemaVersion": "1.6", "slug": "geometric-graph-entropy", "title": "Universal vertices and effective dimension in geometric-graph entropy", "shortTitle": "Some points in a random geometric graph cost no bits", "url": "https://evidencepress.org/releases/geometric-graph-entropy/", "oneLine": "Storing a random geometric graph with n vertices takes d(1 − μ(Uᵣ)) n log₂ n bits up to O(n), where Uᵣ is the set of positions joined to everyone; this refutes the exact cube coefficient conjectured by Vippathalla, Coon and Badiu for d ≥ 2 and √d/2 < r < √d, and proves their torus conjecture at every radius.", "abstract": "How many bits does it take to store a random geometric graph? Scatter n labelled points in a d-dimensional region with a density bounded above and away from zero, join two of them when they are within a fixed distance r, and ask for the Shannon entropy H(Gₙ) of the resulting graph. A heuristic says d n log₂ n: each of the d coordinates of each point is pinned down to about 1/n by the others. This candidate shows exactly when that is right and what replaces it. On every compact convex domain and for every fixed radius, H(Gₙ) = d(1 − μ(Uᵣ)) n log₂ n + O(n), where Uᵣ is the set of universal positions, within distance r of the whole domain: a vertex there is joined to everyone, so its position carries no information. For translation rules of finite perimeter on the flat torus, which may have holes or several pieces, the coefficient is the rank of the lattice generated by the rule's nonzero Fourier frequencies, the number of independent directions the rule can detect. For the unit cube this refutes the exact-coefficient reading of the conjecture of Vippathalla, Coon and Badiu (ISIT 2026; as stated in arXiv:2601.10778v1) at every radius between √d/2 and √d when d ≥ 2: in the unit square at r = 1 the coefficient is 2√3 − 2π/3 ≈ 1.3697, not ≈ 1.4610, and the gap peaks at about 0.161 near r ≈ 1.13. The conjecture's order of growth holds in every case, and its exact coefficients hold for smaller radii, in dimension one and on the torus at every 0 < r < √d/2, each with an O(n) remainder. The radius dependence of every error term is explicit. All results concern a fixed radius as n grows. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.", "datePublished": "2026-10-27", "dateModified": "2026-10-27", "version": "1.1.0-candidate", "doi": "10.5281/zenodo.23686157", "doiUrl": "https://doi.org/10.5281/zenodo.23686157", "conceptDoi": "10.5281/zenodo.23686156", "pdfUrl": "https://github.com/ipitchford/geometric-graph-entropy/releases/download/v1.1.0-candidate/geometric-graph-entropy-1.1.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23686157", "repoUrl": "https://github.com/ipitchford/geometric-graph-entropy", "releaseUrl": "https://github.com/ipitchford/geometric-graph-entropy/releases/tag/v1.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/geometric-graph-entropy/index.md", "bibtexUrl": "https://evidencepress.org/releases/geometric-graph-entropy/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/geometric-graph-entropy.mp3", "imageUrl": "https://evidencepress.org/assets/og/geometric-graph-entropy.png?v=cf3b182fc5", "coverArtUrl": "https://evidencepress.org/assets/art/geometric-graph-entropy.svg?v=917f352b84", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/geometric-graph-entropy.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/geometric-graph-entropy.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted mathematical candidate. Every theorem has a written proof in the manuscript; the computations are sanity checks, not proof certificates. Internal AI-assisted five-role review, outside-model adversarial reviews before and after revision, a verification re-review and a targeted proof audit; final editor fixes were made without a further review round. External specialist review, independent reproduction and formal verification have not been established. Statements about the conjecture refer to arXiv:2601.10778v1." }, "assurance": [ { "dimension": "availability", "label": "Availability and archiving", "question": "Is the evidence package publicly retrievable from an archive under a persistent identifier?", "state": "passed", "evidenceUrl": "https://zenodo.org/records/23686157", "note": "The versioned research package and manuscript are publicly archived on GitHub and Zenodo. Provider asset parity was checked against frozen local files; availability does not establish correctness or peer review." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "From a fresh extraction of the frozen archive, the verification script, the SymPy identities and the figure script reproduced the shipped receipts and curve data, and eleven semantic negative controls rejected corrupted coefficients, laws, bounds and receipts. GitHub CI repeats the replay on Linux." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated rerun has been established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "partial", "note": "Internal reviewers recomputed the square coefficient, the gap curve, the finite-n illustration and the torus Fourier bounds with their own scripts; this is producer-side implementation diversity, not unaffiliated reimplementation." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof-assistant formalisation; the theorems are proved in prose." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "AI-assisted internal and adversarial review is not external specialist review." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "No external editorial peer review." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "partial", "note": "The main check needs only the Python standard library; the SymPy and figure checks use pinned SymPy, NumPy and Matplotlib versions listed in requirements.txt, not a locked container." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted mathematical research", "AI-assisted internal review, re-review, proof audit and outside-model adversarial review", "Claude Code: final editor fixes, publication packaging and release" ], "humanRole": "Publication authority only; no human research contribution inferred.", "reviewStatus": "Internal AI-assisted and adversarial review, not external peer review." }, "attributionBoundary": null, "problem": null, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared from the final manuscript (version 1.1, 27 October 2026, 23 pages) with the final editor fixes, its verification receipts and the frozen release archive. The versioned GitHub and Zenodo research archives were checked against the frozen release assets on 27 October 2026.", "grandfatheredAtSchemaVersion": null, "keywords": [ "random geometric graph", "Shannon entropy", "graph compression", "graphon", "universal vertex", "effective dimension", "Fourier analysis", "counterexample" ], "keyResults": [ "Convex domains: for every compact convex body D ⊂ ℝᵈ, density bounded above and below, and fixed r > 0, H(Gₙ) = d(1 − μ(Uᵣ)) n log₂ n + O(n), where Uᵣ = {x ∈ D : D ⊆ B̄(x, r)} is the set of universal positions (Theorem 3.1, proof).", "Cube correction: for d ≥ 2 and √d/2 < r < √d the earlier conjectured coefficient d[1 − β(r)] is strictly larger than the true d[1 − u_d(r)], so the exact-coefficient reading of the conjecture in arXiv:2601.10778v1 is false there; its upper bounds remain valid (Proposition 4.1, proof).", "Unit square, r = 1: coefficient 2√3 − 2π/3 ≈ 1.3697 instead of ≈ 1.4610, 6.25% lower; across the square the gap peaks at about 0.161 near r ≈ 1.13 (18.7% of the earlier coefficient), and the relative gap at about 20.9% near r ≈ 1.17 (Corollary 4.2, proof; exact SymPy identity; gap curve numerical).", "Fourier rank: for an even finite-perimeter translation rule b on the torus, the coefficient is the rank of the lattice generated by the nonzero Fourier frequencies of b, the number of directions the rule can detect (Theorem 5.1, proof).", "Torus: the coefficient is d for every 0 < r < √d/2, proving the torus conjecture of arXiv:2601.10778v1 beyond the previously proved r ≤ 1/4 (Corollary 5.6, proof).", "Confirmed elsewhere: the n log n order in every case, the cube coefficient d for r ≤ √d/2, and dimension one at every radius, each with an O(n) remainder (Theorem 3.1, Corollary 3.5).", "Radius dependence: the upper remainders are uniform in r; the proved lower remainders grow like log(1/r) as r → 0 and, on the torus, like d² log₂(1/(√d/2 − r)) per vertex near the endpoint (Propositions 3.4 and 5.7, proof)." ], "reviews": [], "evidencePackage": "Written proofs of the convex-domain coefficient theorem, the cube correction, the Fourier-rank theorem, the full-range torus corollary and the explicit radius dependence of every remainder; exact SymPy identities for the square coefficient 2√3 − 2π/3, the exact gap and the cap-variance moment; a standard-library sanity-check script (exact three-vertex laws by rational polygon clipping, closed forms against independent quadrature, two-sided torus Fourier bounds at d = 2) whose JSON receipt replays byte for byte; the figure script and its curve data; eleven semantic negative controls.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Vippathalla, Coon and Badiu (2026), On the entropy of a random geometric graph. 2026 IEEE International Symposium on Information Theory (source of the conjecture; extended version arXiv:2601.10778v1).", "url": "https://doi.org/10.1109/ISIT62367.2026.11654001", "doi": "10.1109/ISIT62367.2026.11654001" }, { "citation": "Hatami and Norine (2013), The entropy of random-free graphons and properties. Combinatorics, Probability and Computing 22(4), 517–526 (order of growth for random-free graphons).", "url": "https://doi.org/10.1017/S0963548313000175", "doi": "10.1017/S0963548313000175" }, { "citation": "McDiarmid and Müller (2014), The number of disk graphs. European Journal of Combinatorics 35, 413–431 (the Warren sign-pattern count used for the upper bounds).", "url": "https://doi.org/10.1016/j.ejc.2013.06.037", "doi": "10.1016/j.ejc.2013.06.037" }, { "citation": "Baker and Dettmann (2025), Entropy of random geometric graphs in high and low dimensions. Journal of Complex Networks 13(5), cnaf033.", "url": "https://doi.org/10.1093/comnet/cnaf033", "doi": "10.1093/comnet/cnaf033" }, { "citation": "Coon, Dettmann and Georgiou (2018), Entropy of spatial network ensembles. Physical Review E 97(4), 042319.", "url": "https://doi.org/10.1103/PhysRevE.97.042319", "doi": "10.1103/PhysRevE.97.042319" }, { "citation": "Paton, Hartle, Stepanyants, van der Hoorn and Krioukov (2022), Entropy of labeled versus unlabeled networks. Physical Review E 106(5), 054308.", "url": "https://doi.org/10.1103/PhysRevE.106.054308", "doi": "10.1103/PhysRevE.106.054308" }, { "citation": "Choi and Szpankowski (2012), Compression of graphical structures: Fundamental limits, algorithms, and experiments. IEEE Transactions on Information Theory 58(2), 620–638.", "url": "https://doi.org/10.1109/TIT.2011.2173710", "doi": "10.1109/TIT.2011.2173710" }, { "citation": "Myyryläinen, Pérez and Weigt (2024), Weighted fractional Poincaré inequalities via isoperimetric inequalities. Calculus of Variations and PDE 63, 205 (relative isoperimetry on cubes, Lemma 2.2).", "url": "https://doi.org/10.1007/s00526-024-02813-6", "doi": "10.1007/s00526-024-02813-6" }, { "citation": "Penrose (2003), Random Geometric Graphs. Oxford University Press.", "url": "https://doi.org/10.1093/acprof:oso/9780198506263.001.0001", "doi": "10.1093/acprof:oso/9780198506263.001.0001" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:geometric-graph-entropy", "attemptIds": [ "ep-attempt:geometric-graph-entropy-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "communication" ], "lineageId": null, "accelerationPrimitives": [ "counterexample-proxy-first", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "counterexample", "description": "A written proof refutes the exact-coefficient reading of the 2026 Vippathalla–Coon–Badiu cube conjecture for d ≥ 2 and √d/2 < r < √d (unit square at r = 1: 2√3 − 2π/3 instead of about 1.461), determines the entropy coefficient on every convex domain and for finite-perimeter torus rules, and proves their torus conjecture at every radius.", "scope": "Labelled hard geometric graphs with fixed dimension and fixed radius, independent positions with a density bounded above and away from zero, n → ∞; conjecture statements as in arXiv:2601.10778v1. The coefficient of n, shrinking radii, structural entropy for d ≥ 2 and efficient encoders are not addressed." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "The conjecture is quoted verbatim from the abstract and conclusion of arXiv:2601.10778v1; β(r) is used with the source's intersection with the cube; uniform density on the cube or torus is exactly the source's setting.", "remainingRisks": [ "The ISIT conference PDF was not inspected.", "No theorem-by-theorem comparison with general random-free graphon entropy results.", "The bounded literature check is not exhaustive priority certification." ] }, "humanJudgmentGates": [ "Publication is authorised; scientific endorsement is not inferred.", "External specialist review and independent reproduction remain open." ], "parentLinks": [], "assuranceTarget": { "dimensions": [ "specialistReview", "independentRerun", "formalVerification", "priorityAssessment" ], "nextAction": "Obtain specialist review of the quantitative separation lemma and the Fourier-rank theorem, and compare the coefficients against general random-free graphon entropy results theorem by theorem.", "claimCeiling": "Proved coefficient theorems for fixed radius with O(n) remainders, a proved refutation of the exact cube coefficient in the intermediate range and a proved torus confirmation. 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