{ "schemaVersion": "1.4", "slug": "s6-extension-results-candidate", "title": "Extensions of a claimed complex structure on S6", "shortTitle": "Extensions around a claimed complex S6", "url": "https://evidencepress.org/releases/s6-extension-results-candidate/", "oneLine": "Exact arithmetic and symbolic checks, a standalone conductor-adjoint theorem candidate, and source-conditional Hodge and deformation calculations extend a claimed—but unverified—complex structure on S6.", "abstract": "This anonymous, unrefereed candidate collects seven extension notes and eleven deterministic verification programs around a manuscript claiming a complex structure on the smooth six-sphere. It classifies twist determinants, rank-four finite spectra, explicit triangle-monodromy families, conductor ideals and torsion specializations; formulates a conductor-adjoint nonvanishing criterion for R2 direct images independently of the global construction; and, conditional on the source construction and local calculations, derives h11=2, h12=1, tangent-cohomology dimensions (1,1,1,0) and a smooth one-dimensional Kuranishi germ. The original strongest-directions programme is only partly completed: the Hodge calculation is supplied as a conditional proof candidate, local moduli are advanced without a global equivalence classification, and adapted Hermitian metrics remain open. Producer-side replay, public CI and byte-matched GitHub and Zenodo archives pass. None of this independently proves that the source threefold exists, is smooth or is diffeomorphic to S6, and no independent reconstruction, formal verification, external specialist review, editorial peer review, priority determination or scientific-impact evidence is reported.", "datePublished": "2026-08-24", "dateModified": "2026-08-24", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.22085272", "doiUrl": "https://doi.org/10.5281/zenodo.22085272", "conceptDoi": "10.5281/zenodo.22085271", "pdfUrl": "https://github.com/ipitchford/s6-extension-results-candidate/releases/download/v0.1.0-candidate/s6-extension-results-candidate-v0.1.0-paper.pdf", "altPdfUrl": "https://zenodo.org/records/22085272/files/s6-extension-results-candidate-v0.1.0-paper.pdf?download=1", "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/22085272", "repoUrl": "https://github.com/ipitchford/s6-extension-results-candidate", "releaseUrl": "https://github.com/ipitchford/s6-extension-results-candidate/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/s6-extension-results-candidate/index.md", "bibtexUrl": "https://evidencepress.org/releases/s6-extension-results-candidate/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/s6-extension-results-candidate.mp3?v=3914ce4c5c", "imageUrl": "https://evidencepress.org/assets/og/s6-extension-results-candidate.png?v=77edd61bee", "coverArtUrl": "https://evidencepress.org/assets/art/s6-extension-results-candidate.svg?v=b2767519ed", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/s6-extension-results-candidate.mp3", "name": "Audio briefing — extensions around a claimed complex S6", "description": "Plain-English OpenAI API synthetic-voice summary of the exact, written-under-hypotheses, source-conditional and open layers; a communication aid, not additional mathematical evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/s6-extension-results-candidate.txt" }, { "type": "video", "url": "https://youtu.be/O-YxmEzrNGQ", "name": "Deconstructing the S^6 Claim An Epistemic Map of Complex Geometry", "description": "Evidence Press video overview of this release. This is a communication aid; it does not extend the release's claims or assurance status and is not additional research evidence." } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Anonymous, unrefereed multi-result extension candidate. Availability and internal replay pass for the immutable package. The arithmetic and symbolic layer is the strongest replayable evidence; written theorem candidates extend beyond what the programs check; and the Hodge, deformation, topology, exceptional-fibre and family-wide consequences remain conditional on the unverified source manuscript and stated local models. The five-role editorial gate and confirmation review are internal and model-mediated. No unaffiliated rerun, independently authored implementation, proof-assistant formalization, external complex-geometry specialist review, journal peer review, broad priority determination or impact evaluation is reported. The page-6 text overflow identified during final review was repaired and is guarded by an executable layout check; this presentation repair did not change a mathematical claim." }, "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/22085272", "note": "The tagged GitHub prerelease and published Zenodo record expose four matching assets; unauthenticated downloads from both services matched the local SHA-256 ledger." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "Public CI and the frozen local gate passed ordinary and optimized replay, four mutation controls, manifest checks, fresh extraction, TeX preflight and PDF-layout regression.", "evidenceUrl": "https://github.com/ipitchford/s6-extension-results-candidate/actions/runs/32758679287" }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated party has reported running the exact tagged package or independently checking its source-to-claim bridges." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "The notes, programs, hardening and review were coordinated within one producer workflow; no separately authored implementation has reconstructed the results." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "The release contains deterministic programs and finite symbolic checks but no proof-assistant formalization or trusted-base account of the written theorems." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "The role reports are internal model-mediated quality control, not external review by specialists in complex geometry, Hodge theory, deformation theory or arithmetic monodromy." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "No journal or comparable scholarly venue has conducted peer review to a decision." }, { "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/s6-extension-results-candidate/blob/v0.1.0-candidate/ENVIRONMENT.txt", "note": "Versions and dependencies are recorded, the principal Python checks use a simple entry point, and public CI passes, but no independently recreated container or functional environment is reported." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI systems under human direction", "OpenAI Codex", "Python standard library", "Singular computer algebra system", "Pandoc and LaTeX toolchain" ], "humanRole": "Research direction, source selection, mediation, claim-boundary decisions and publication authorization by the human creator; repository maintenance and Evidence Press publication are not scholarly authorship, and scholarly attribution remains Anonymous.", "disclosure": "AI systems contributed research exploration, proof drafting, computation, checking, editorial review, exposition and release engineering in one producer-coordinated workflow. Exact replay, internal model agreement, public CI and matching hashes are not independent reconstruction, external specialist review or editorial peer review." }, "attributionBoundary": null, "problem": { "name": "Existence and geometry of a complex structure on the smooth six-sphere", "url": "https://alpo.ge/s6.pdf" }, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "reader-first-reference", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Authored prospectively for the reader-first schema 2.0 against the immutable GitHub and Zenodo identities, the exact source-PDF hash, the frozen internal review target and the confirmation receipt.", "grandfatheredAtSchemaVersion": null, "keywords": [ "complex six-sphere", "complex threefolds", "Hodge numbers", "Kuranishi space", "higher direct images", "normal-crossing fibres", "triangle monodromy", "Neron-Severi loci", "computer-assisted mathematics", "unrefereed candidate" ], "keyResults": [ "Conditional on the source construction and local calculations, the package derives h11=2 and h12=1, tangent-cohomology dimensions (1,1,1,0), and a smooth one-dimensional Kuranishi germ locally parametrized by c0.", "A standalone conductor-adjoint criterion proposes that a nonzero normalized adjoint section forces nonvanishing of top tangent cohomology and the R2 direct-image stalk for a reduced normal-crossing fibre.", "The attainable signed twist determinants are exactly the integers coprime to 12, each fixed determinant has a rank-two affine lattice of twist triples, and the homotopy-sphere locus is conditionally p=plus or minus 1.", "Rank-four finite local monodromy can have exact order only in {1,2,3,4,5,6,8,10,12}; 314 determinant-compatible spectral pairs split into 58 two-dimensional irreducible, 72 rigid irreducible and 184 irreducible-excluded cases after two scalar exclusions.", "The exceptional Neron-Severi locus splits into moving graph branches and a torsion branch; the first smooth torsion specialization found is t=32/5 of exact order four, conditional on the source period interpretation.", "The original extension programme is only partly complete: the Hodge goal has a conditional candidate answer, global equivalence of c0 values and sphere-locus twists remains open, and the Hermitian-metric programme was not undertaken." ], "reviews": [], "evidencePackage": "The immutable GitHub and Zenodo releases contain a deterministic source ZIP, the nine-page synthesis PDF, publication confirmation and SHA-256 ledger. The package preserves seven mathematical notes, eleven verification programs, claim and dependency ledgers, source and citation audits, five internal journal-like role reports, a confirmation review, four semantic mutation controls, ordinary and optimized entry points, fresh-extraction verification and a PDF-layout regression checker. The final seven-part gate passed, GitHub Actions passed, the live source PDF re-download matched its recorded SHA-256, and all four GitHub and Zenodo release assets matched local hashes after unauthenticated download. These establish public availability, producer-side replay, source-version identity and scoped encoded checks. They do not verify the sheaf-theoretic proofs, the source-to-formula bridge in full, the existence or smoothness of the proposed compact threefold, its S6 diffeomorphism type, or any external-assurance dimension.", "openProblems": [ "Give the conductor-adjoint theorem an external specialist referee pass against the precise R2 direct-image step in the corrected Campana-Demailly-Peternell argument.", "Independently audit the source construction, especially the infinite toric quotient, gluing, smoothness, topology and recognition of the underlying smooth S6.", "Reconstruct every elementary modification, Kodaira-Spencer vanishing order, residue map and branch-normal calculation in the Hodge and deformation proof without using the source's expected Hodge values as premises.", "Determine the global equivalence relation on c0 and on the affine lattices of p=plus or minus 1 twist triples.", "Carry out the originally proposed balanced, strongly Gauduchon, pluriclosed/SKT and canonical Hermitian-metric programme.", "Compute Bott-Chern and Aeppli cohomology along the proposed c0 deformation curve.", "Complete the compactified nonsplit (12,12,infinity) period problem and determine which arithmetic monodromy orbits admit compact holomorphic or polarizable families.", "Conduct a broad object-level prior-art and priority audit before any first-discovery language is used." ], "relatedWorks": [ { "citation": "Anonymous source manuscript. The (3,4,infinity) modular family of 2-tori, completed at its three special points, is a complex structure on S6.", "url": "https://alpo.ge/s6.pdf" }, { "citation": "Campana, F., Demailly, J.-P., & Peternell, T. (2020). The algebraic dimension of compact complex threefolds with vanishing second Betti number: corrigendum.", "url": "https://doi.org/10.1112/S0010437X19007802" }, { "citation": "Lehn, C., Rollenske, S., & Schinko, C. (2019). The complex geometry of a hypothetical complex structure on S6.", "url": "https://arxiv.org/abs/1912.09719" }, { "citation": "Angella, D. (2018). Hodge numbers of a hypothetical complex structure on S6.", "url": "https://doi.org/10.1016/j.difgeo.2017.10.012" }, { "citation": "Kostov, V. P. (2001). On the Deligne-Simpson problem.", "url": "https://arxiv.org/abs/math/0011013" }, { "citation": "Miranda, R., & Persson, U. (1986). On extremal rational elliptic surfaces.", "url": "https://doi.org/10.1007/BF01160474" } ], "operatingModel": { "version": "1.0", "workId": "ep-work:s6-extension-results-candidate", "attemptIds": [ "ep-attempt:s6-extension-results-candidate" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "evidence-assessment", "method-demonstration", "communication" ], "lineageId": null, "accelerationPrimitives": [ "certificate-first", "structural-compression", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "executable-open-problem", "description": "A claim-classified extension object whose first adjudication target is the conductor-adjoint nonvanishing criterion, alongside exact arithmetic and symbolic certificates and explicitly conditional Hodge, deformation, topology and exceptional-fibre conclusions.", "scope": "The seven extension notes and their encoded checks around the exact source PDF; not a verification that the source compact complex threefold exists, is smooth or is diffeomorphic to S6." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "The synthesis separates exact arithmetic and symbolic statements, written results under hypotheses independent of the global construction, source-conditional consequences and open problems. CLAIMS.json identifies the source theorem or local-model dependency for each conditional claim, and the replay route states which encoded predicates it does and does not check.", "remainingRisks": [ "The global source construction, smoothness, compactification and S6 topology have not been independently reconstructed.", "The Hodge and deformation arguments depend on dense direct-image splittings, local eigencharacters, Kodaira-Spencer maps, residue ranks and branch-normal calculations not checked by the finite programs.", "The conductor criterion and its relation to the corrected Campana-Demailly-Peternell argument require external specialist adjudication.", "The bounded citation audit cannot establish novelty or absolute priority, and one producer workflow may propagate a shared semantic error across notes and checkers." ] }, "humanJudgmentGates": [ "Assess the source construction, smoothness, compactification and S6 recognition independently before treating any conditional extension as a result about an existing complex manifold.", "Check the logarithmic direct-image, residue and deformation arguments as mathematics rather than inferring them from the displayed Hodge numbers or successful finite replay.", "Referee the standalone conductor-adjoint map and its comparison with the corrected CDP argument before using conflict or refutation language.", "Keep local Kuranishi variation separate from global inequivalence of c0 values or twist triples, and keep exact period calculations separate from compactification existence.", "Require a broad object-level literature and priority audit before first, new, definitive or field-changing language.", "Confirm Anonymous scholarly attribution, component licences, immutable identifiers and release authority before publication." ], "parentLinks": [ { "relation": "extends-result", "workId": null, "legacyReleaseSlug": null, "externalUrl": "https://alpo.ge/s6.pdf", "inheritedClaim": "The external source manuscript proposes a holomorphic torus fibration whose completed compact complex threefold is claimed to be diffeomorphic to S6 for a specified twist; the present release develops arithmetic, geometric and deformation extensions around that construction.", "inheritedAssuranceCeiling": "The source is an unrefereed external manuscript. Matching its exact PDF hash establishes source identity, not the existence, smoothness, topology or correctness of the claimed complex threefold, and every source-conditional conclusion inherits that ceiling." } ], "assuranceTarget": { "dimensions": [ "independentRerun", "independentReimplementation", "formalVerification", "specialistReview", "dataEnvironmentReproducibility", "semanticValidation", "noveltyAssessment", "priorityAssessment" ], "nextAction": "Commission an external specialist adjudication of the standalone conductor-adjoint criterion and the exact CDP/S6 incompatibility, then independently reconstruct the source construction and the logarithmic Hodge calculation before evaluating the global moduli and Hermitian-metric programmes.", "claimCeiling": "An anonymous unrefereed extension candidate with replayed finite arithmetic and symbolic claims, written theorem candidates under explicit hypotheses, and source-conditional Hodge, deformation, topology and family consequences; not verification that the source complex threefold exists, is smooth or is diffeomorphic to S6, and not independent reproduction, formal verification, external specialist or editorial peer review, absolute priority or demonstrated workflow impact." }, "impactClaims": [ { "id": "s6-extension-science-no-impact", "aim": "science", "outcome": "Faster or more reliable extension and adjudication of difficult conditional complex-geometry claims", "setting": "AI-assisted arithmetic, Hodge, deformation, monodromy and conductor research around a claimed complex structure on S6", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "No matched conventional mathematical-research, theorem-review or publication workflow was registered.", "estimand": "No effect on discovery time, active human effort, compute, correction rate, proof quality, independent-assurance time, uptake or citation was estimated.", "evidenceRefs": [], "registeredDesignRef": null, "independentAssessment": null } ] } }