{ "schemaVersion": "1.6", "slug": "small-exponent-percolation", "title": "First correction to the critical mean degree in extreme long-range percolation", "shortTitle": "The cost of stepping straight back", "url": "https://evidencepress.org/releases/small-exponent-percolation/", "oneLine": "In one-dimensional long-range percolation, the threshold sits above the branching prediction by a two-step return weight; the same criterion handles summable additions and mixtures of tail exponents.", "abstract": "For independent undirected bonds on the integers with exact probabilities C|x−y|^(−1−σ), the critical expected degree satisfies pc=1+ζ(2+2σ)/(2ζ(1+σ)²)+O(σ³) as σ decreases to zero. In particular, (pc−1)/σ² tends to π²/12. A positive first-order lace remainder, a nonbacktracking lower bound and uniform Fourier estimates also yield a two-scale kernel criterion, arbitrary bounded-mass nonnegative summable additions, and compact mixtures of exponent ratios. Constants are not evaluated: this is not a numerical threshold estimate at a specified positive exponent.", "datePublished": "2026-10-11", "dateModified": "2026-10-11", "version": "1.1.0-candidate", "doi": "10.5281/zenodo.23294525", "doiUrl": "https://doi.org/10.5281/zenodo.23294525", "conceptDoi": "10.5281/zenodo.23294524", "pdfUrl": "https://github.com/ipitchford/small-exponent-percolation/releases/download/v1.1.0-candidate/small-exponent-percolation-v1.1.0-candidate.pdf", "altPdfUrl": "https://zenodo.org/records/23294525/files/small-exponent-percolation-v1.1.0-candidate.pdf?download=1", "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23294525", "repoUrl": "https://github.com/ipitchford/small-exponent-percolation", "releaseUrl": "https://github.com/ipitchford/small-exponent-percolation/releases/tag/v1.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/small-exponent-percolation/index.md", "bibtexUrl": "https://evidencepress.org/releases/small-exponent-percolation/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/small-exponent-percolation.mp3", "imageUrl": "https://evidencepress.org/assets/og/small-exponent-percolation.png?v=d180622038", "coverArtUrl": "https://evidencepress.org/assets/art/small-exponent-percolation.svg?v=4f3169855f", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/small-exponent-percolation.mp3", "name": "Audio briefing — The cost of stepping straight back", "description": "Synthetic plain-English explanation; not additional scientific evidence." } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed written-proof candidate with internal checking; no formal verification, external specialist review or unaffiliated reproduction established." }, "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/23294525", "note": "Public GitHub and Zenodo assets verified by full downloads against the frozen archive; availability is not proof 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": "Fresh-extraction producer-side replay: 26 exact algebraic checks, ten rejected semantic controls, normal and optimized Python; not an infinite-lattice proof verification.", "evidenceUrl": "https://github.com/ipitchford/small-exponent-percolation/blob/v1.1.0-candidate/VERIFICATION.md" }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "Not established by coordinated internal AI checks or publication." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "Not established by coordinated internal AI checks or publication." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "Not established by coordinated internal AI checks or publication." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "Not established by coordinated internal AI checks or publication." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "Not established by coordinated internal AI checks or publication." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "partial", "evidenceUrl": "https://github.com/ipitchford/small-exponent-percolation/blob/v1.1.0-candidate/ENVIRONMENT.txt", "note": "Current stdlib diagnostics and sources shipped; four historical external cache files and original packaging tests are absent." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted Codex research workflow", "OpenAI Codex assistant: review actions, extension audits and publication" ], "humanRole": "Publication authority only; no human research contribution asserted.", "reviewStatus": "Unrefereed written-proof candidate with internal checking; no formal verification, external specialist review or unaffiliated reproduction established." }, "attributionBoundary": null, "problem": { "name": "The small-exponent critical threshold for undirected one-dimensional long-range percolation", "url": "https://arxiv.org/abs/2610.11872v1" }, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Supplied percolation research and review, revised with internally audited kernel extensions on 11 October 2026. No external review or discovery duration inferred.", "grandfatheredAtSchemaVersion": null, "keywords": [ "long-range percolation", "critical mean degree", "lace expansion", "nonbacktracking paths", "small-exponent asymptotics" ], "keyResults": [ "The critical expected degree is 1 plus the exact two-step return weight, with a uniform cubic remainder as the exponent tends to zero.", "A reusable two-scale criterion transfers the conclusion to even one-dimensional kernels satisfying the stated Fourier and admissibility assumptions.", "Summable positive additions alter the leading coefficient explicitly; compact exponent mixtures replace it by π²/12 times the squared mean exponent ratio." ], "reviews": [], "evidencePackage": "Written analytic proof, source applicability audit, response to the supplied review, and scoped finite exact algebraic diagnostics.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Amit (2026), Explicit upper bounds on the threshold of one-dimensional long-range percolation: motivating threshold question and a different monotone-path approach.", "url": "https://arxiv.org/abs/2610.11872v1", "doi": "10.48550/arXiv.2610.11872" }, { "citation": "van der Hofstad and Sakai (2005), Critical points for spread-out self-avoiding walk, percolation and the contact process above the upper critical dimensions: established two-step correction in a different asymptotic regime.", "url": "https://arxiv.org/abs/math/0402050", "doi": null }, { "citation": "Heydenreich, van der Hofstad and Sakai (2008), Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk: imported infrared theorem.", "url": "https://arxiv.org/abs/0712.0312v2", "doi": "10.48550/arXiv.0712.0312" }, { "citation": "Borgs, Chayes, van der Hofstad, Slade and Spencer (2005), Random subgraphs of finite graphs II: inherited lace-event decomposition.", "url": "https://arxiv.org/abs/math/0401070", "doi": null } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:small-exponent-percolation", "attemptIds": [ "ep-attempt:small-exponent-percolation-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "evidence-assessment", "communication" ], "lineageId": null, "parentLinks": [], "accelerationPrimitives": [ "structural-compression", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "bound", "description": "Small-exponent first critical-mean-degree correction and a transferable two-scale kernel criterion.", "scope": "Independent undirected linear Bernoulli percolation on the integers, with the stated symmetry, Fourier and admissibility hypotheses." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "The normalized parameter is expected degree; weighted percolation event inequalities connect the Fourier diagram bounds to the actual threshold, not merely a random-walk surrogate.", "remainingRisks": [ "Analytic event and limiting arguments are not formally verified.", "The asymptotic constants do not supply a numerical exponent cutoff.", "The bounded source search does not establish exhaustive novelty or priority." ] }, "humanJudgmentGates": [ "External specialist assessment remains open; no human research contribution asserted." ], "assuranceTarget": { "dimensions": [ "specialistReview", "formalVerification" ], "nextAction": "Audit the uniform small-exponent estimates and the weighted deleted-bond event inequality against the cited sources.", "claimCeiling": "Unrefereed written-proof candidate with internal checking; no formal verification, external specialist review or unaffiliated reproduction established." }, "impactClaims": [ { "id": "small-exponent-percolation-no-impact", "aim": "science", "outcome": "Scoped asymptotic threshold and reusable sufficient criterion", "setting": "Producer-coordinated theoretical research", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "None.", "estimand": "No measured acceleration or downstream benefit.", "evidenceRefs": [], "registeredDesignRef": null, "independentAssessment": null } ] }, "researchMetrics": { "schemaVersion": "1.0", "policyUrl": "https://evidencepress.org/api/research-metrics-policy.json", "workLedgerUrl": "https://evidencepress.org/api/work-ledger.json", "claimCeiling": "These records describe scoped workflow inputs and outcomes. They support calibration and reference-class construction, but do not by themselves establish research acceleration, theorem correctness, novelty, independent assurance or impact.", "attempts": [ { "attemptId": "ep-attempt:small-exponent-percolation-publication", "status": "result-ready", "resultClass": "positive", "metrics": { "schemaVersion": "1.0", "measurementScope": "assurance-through-publication", "scopeBoundary": "Work after this registration; original research and initial extension work are left-censored. No historical clocks reconstructed.", "forecast": { "frozenAt": "2026-10-11T08:02:35.328Z", "procedureClass": "percolation-extension-publication", "targetOutcome": "Immutable research publication and canonical site readback", "expectedActiveMinutes": 110, "plausibleLowMinutes": 90, "plausibleHighMinutes": 165, "expectedUnattendedWaitMinutes": 20, "referenceClass": { "label": "Targeted extension and full candidate publication", "sampleSize": 0, "basis": "Procedural estimate, not measured benchmark" }, "fermiComponents": [ { "component": "Research closure, archive, media and guarded publication", "count": 1, "lowMinutesPerUnit": 90, "centralMinutesPerUnit": 110, "highMinutesPerUnit": 165, "basis": "Mandatory stages with new extension audits" } ], "probabilityPositiveSignal": 0.9, "probabilityTargetClosure": 0.8, "probabilityHorizonMinutes": 480, "assumptions": [ "The audited extensions remain valid in final text; providers remain accessible." ], "reforecastTriggers": [ "A substantive proof issue or provider failure." ], "stopRule": "No hard cap; complete mandatory gates or report a precise blocker." }, "outcome": null }, "missingnessReason": null } ] } }