{ "schemaVersion": "1.6", "slug": "multiplane-lensing-sparse-bound", "title": "A tighter ceiling on gravitationally lensed images", "shortTitle": "The multiplane lensing bound", "url": "https://evidencepress.org/releases/multiplane-lensing-sparse-bound/", "oneLine": "Keeping neighbouring lens planes in the equations lowers the two-binary image ceiling from 41 to 33; the known 25-image construction leaves a sharp question open.", "abstract": "The manuscript proves an upper bound N ≤ U_K for regular, unobstructed images in K-plane positive point-mass lenses with positive nearest-neighbour distance coefficients, no external shear and no continuous matter. The recurrence U_i=(1+g_i²)U_{i−1}+2g_iV_{i−1}, V_i=g_iU_{i−1}+V_{i−1}, starting at (1,0), evaluates a sparse mixed-volume envelope. It gives 33 for two binary planes and 213 for three, compared with Perry’s 41 and 365. The general prose proof is accompanied by exact finite checks and a certificate for at least 25 images in a known example. The sharp maximum and exhaustive priority remain unresolved.", "datePublished": "2026-10-01", "dateModified": "2026-10-01", "version": "0.2.0-candidate", "doi": "10.5281/zenodo.23075650", "doiUrl": "https://doi.org/10.5281/zenodo.23075650", "conceptDoi": "10.5281/zenodo.23075649", "pdfUrl": "https://github.com/ipitchford/multiplane-lensing-sparse-bound/releases/download/v0.2.0-candidate/paper.pdf", "altPdfUrl": "https://zenodo.org/records/23075650/files/paper.pdf?download=1", "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23075650", "repoUrl": "https://github.com/ipitchford/multiplane-lensing-sparse-bound", "releaseUrl": "https://github.com/ipitchford/multiplane-lensing-sparse-bound/releases/tag/v0.2.0-candidate", "markdownUrl": "https://evidencepress.org/releases/multiplane-lensing-sparse-bound/index.md", "bibtexUrl": "https://evidencepress.org/releases/multiplane-lensing-sparse-bound/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/multiplane-lensing-sparse-bound.mp3?v=217e272609", "imageUrl": "https://evidencepress.org/assets/og/multiplane-lensing-sparse-bound.png?v=057153848f", "coverArtUrl": "https://evidencepress.org/assets/art/multiplane-lensing-sparse-bound.svg?v=1e7c62753b", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/multiplane-lensing-sparse-bound.mp3", "name": "The multiplane lensing bound", "description": "A synthetic OpenAI fable briefing in the house British style. This explanation is not additional scientific evidence.", "durationLabel": "1 min 55 sec", "transcriptUrl": "https://evidencepress.org/assets/audio/multiplane-lensing-sparse-bound.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Complete prose proof with exact finite checks; unrefereed candidate. The sharp physical-image maximum, exhaustive priority, formal verification and unaffiliated reproduction are not 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/23075650", "note": "At 2026-10-01T06:01:59Z, public archive availability was confirmed: full downloads of all four versioned GitHub and published Zenodo assets matched the frozen local files. Canonical site delivery is recorded separately from this archive-availability assessment." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "Producer exact finite combinatorial checks, normal/optimised Python, semantic negative controls and 25-root rational interval replay passed. These checks support specified computations, not a formal proof of the general theorem.", "evidenceUrl": "https://github.com/ipitchford/multiplane-lensing-sparse-bound/blob/v0.2.0-candidate/REPLAY_RECEIPT.md" }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "Supplied ChatGPT review-session reruns do not establish unaffiliated reproduction." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "Separately written review code supplies implementation diversity; independent institutional or human reproduction is not established." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "The universal proof is mathematical prose, not an end-to-end proof-assistant artifact." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "The supplied ChatGPT report is not authenticated external human 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 journal editorial peer review is established." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "partial", "note": "Standard-library Python checkers, exact rational inputs and replay commands are supplied. Their producer and review-session execution does not establish an unaffiliated environment reproduction.", "evidenceUrl": "https://github.com/ipitchford/multiplane-lensing-sparse-bound/blob/v0.2.0-candidate/baseline/README.md" } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "Codex research, proof exposition, exact replay and release preparation", "Supplied ChatGPT review and accompanying checking code; institutional independence unestablished" ], "humanRole": "The user authorised publication. No human scientific contribution or formal review role is inferred.", "disclosure": "Anonymous is the scholarly creator and Evidence Press the publisher. Additional AI-assisted implementations remain separate from unaffiliated reproduction and human peer review." }, "attributionBoundary": null, "problem": { "name": "The maximum number of images in multiplane point-mass lensing", "url": "https://arxiv.org/abs/2302.11735v1" }, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared from the v0.2.0-candidate proof package and supplied ChatGPT review, with exact baseline parameter readback. Reviewer independence and human specialist status are not inferred.", "grandfatheredAtSchemaVersion": null, "keywords": [ "gravitational lensing", "multiplane point-mass lenses", "image counting", "mixed volume", "Bernstein theorem", "sparse polynomial systems", "transfer recurrence" ], "keyResults": [ "A recurrence bounds regular unobstructed images for any K and positive mass counts g_i.", "Two binary planes: known attained 25, upper bound 33, previous general upper bound 41.", "Three binary planes: upper bound 213 versus Perry’s 365; the sharp product conjecture remains open." ], "reviews": [], "evidencePackage": "Written sparse mixed-volume proof; producer checks of 17,577 subset pairs, 340 weighted cases and 80 seeded cases; supplied ChatGPT review code checking 250,953 pairs through K=10; exact rational certificates for 25 regular roots in a known construction.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Perry (2021), An upper bound for the number of gravitationally lensed images in a multiplane point-mass ensemble: the general E²+O² comparator.", "url": "https://doi.org/10.1007/s13324-021-00478-4", "doi": "10.1007/s13324-021-00478-4" }, { "citation": "Keeton, Lundberg and Perry (2023), Multiplane gravitational lenses with an abundance of images: the attained product construction and physical recurrence.", "url": "https://arxiv.org/pdf/2302.11735v1", "doi": null }, { "citation": "Mao, Witt and An (2014), Three-dimensional microlensing: Bernstein-method precedent and the known two-one-mass six-image case.", "url": "https://doi.org/10.1093/mnras/stt1988", "doi": "10.1093/mnras/stt1988" }, { "citation": "Petters (1997), Multiplane gravitational lensing. III. Upper bound on number of images: the earlier one-mass-per-plane upper bound.", "url": "https://doi.org/10.1063/1.531818", "doi": "10.1063/1.531818" }, { "citation": "Bernshtein (1975), The number of roots of a system of equations: the classical sparse root-counting theorem used in the proof.", "url": "https://doi.org/10.1007/BF01075595", "doi": "10.1007/BF01075595" }, { "citation": "Verschelde (2007), Homotopy methods for solving polynomial systems: isolated-root and multihomogeneous bound formulations.", "url": "https://homepages.math.uic.edu/~jan/tutorial.pdf", "doi": null } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:multiplane-lensing-sparse-bound", "attemptIds": [ "ep-attempt:multiplane-lensing-sparse-bound-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "evidence-assessment" ], "lineageId": null, "accelerationPrimitives": [ "certificate-first", "structural-compression", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "bound", "description": "The explicit recurrence N ≤ U_K for regular unobstructed images in positive multiplane point-mass lenses.", "scope": "Positive nearest-neighbour distance coefficients, distinct masses within each plane, no external shear or continuous matter; not a sharpness or attainment theorem." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "The proof injects regular unobstructed physical images into simple polynomial roots, encloses their sparse supports and evaluates the mixed-volume envelope through feasible subset pairs on a path.", "remainingRisks": [ "The prose reduction and use of classical algebraic root bounds are not formally verified.", "Finite recurrence checks and the 25-root example do not prove the universal theorem or global exhaustion.", "Priority is not settled by bounded literature searches; Perry’s relevant dissertation remains unread." ] }, "humanJudgmentGates": [ "Publication was authorised by the user; no human scientific or referee role is inferred.", "A future qualified review can assess the prose proof, physical-to-algebraic interpretation and unresolved priority boundary." ], "parentLinks": [ { "relation": "reuses-evidence", "workId": null, "legacyReleaseSlug": null, "externalUrl": "https://arxiv.org/pdf/2302.11735v1", "inheritedClaim": "Published existence of product_i(5g_i−5) nondegenerate images for g_i≥2, including the 25-image two-binary construction.", "inheritedAssuranceCeiling": "Cited mathematical construction. The local interval replay proves at least 25 roots at the specified parameters; it does not establish global exhaustion or the sharp maximum." } ], "assuranceTarget": { "dimensions": [ "independentRerun", "independentReimplementation", "formalVerification", "specialistReview", "semanticValidation", "noveltyAssessment", "priorityAssessment" ], "nextAction": "Obtain a qualified unaffiliated check of the full sparse proof and physical-root bridge, and resolve the remaining dissertation and citation-network priority checks.", "claimCeiling": "Unrefereed proof candidate with scoped exact checks; no sharp-maximum, formal-verification, human-peer-review or exhaustive-priority claim." }, "impactClaims": [ { "id": "multiplane-lensing-sparse-bound-no-impact", "aim": "science", "outcome": "A reusable general upper bound and auditable sparse counting argument", "setting": "AI-assisted mathematical research and candidate publication", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "No matched-assurance research or workflow comparator is registered.", "estimand": "No discovery-speed, publication-speed, scientific-adoption or real-world impact effect is estimated.", "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:multiplane-lensing-sparse-bound-publication", "status": "result-ready", "resultClass": "positive", "metrics": { "schemaVersion": "1.0", "measurementScope": "publication-only", "scopeBoundary": "Remaining archive, site authoring, standard media and public delivery only. 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Forecast is not a completion cap." }, "outcome": null }, "missingnessReason": null } ] } }