{ "schemaVersion": "1.6", "slug": "unbounded-bns-component-orbits", "title": "A linearly growing free-by-cyclic family with unbounded BNS component-orbit count", "shortTitle": "Unbounded BNS component orbits", "url": "https://evidencepress.org/releases/unbounded-bns-component-orbits/", "oneLine": "For every m>=2, an explicit linearly growing free-by-cyclic group has first Betti number m+2, exactly 2m+2 BNS components, and at least m+1 component orbits under its outer automorphism group.", "abstract": "For every integer m>=2, this anonymous, unrefereed theorem candidate constructs an explicit linearly growing automorphism of F_(2m+1) whose mapping torus G_m has first Betti number m+2. Cashen and Levitt's graph-of-groups criterion identifies its Bieri-Neumann-Strebel invariant as the complement of the m+1 rational hyperplanes x+i y=0 for 0<=i<=m, giving exactly 2m+2 connected components. A direct Bass-Serre calculation gives rank ker chi = 1+m|x|+sum_(i=1)^m |x+i y| for primitive integral BNS characters. The minimum kernel ranks on the outer antipodal component pair and the m interior antipodal pairs are respectively 1+2m and 1+m(m+1)+2j^2 for 0<=j=2, the stated graph map induces a linearly growing automorphism of F_(2m+1), and its mapping torus G_m has first Betti number m+2.", "Conditional on Cashen-Levitt Corollary 2.10, Sigma^1(G_m) is the complement of x+i y=0 for 0<=i<=m and has exactly 2m+2 connected components.", "Every primitive integral BNS character has free kernel of rank 1+m|x|+sum_(i=1)^m |x+i y|.", "The componentwise minimum kernel ranks are 1+2m for the outer antipodal pair and 1+m(m+1)+2j^2 for the j-th interior antipodal pair, 0<=j=2, the BNS description conditional on Cashen-Levitt Corollary 2.10, and the lower bound of m+1 component orbits; not an exact orbit count, a computation of Out(G_m), a classification of free-by-cyclic BNS invariants, a historical-priority finding or external validation." }, "bottleneckTargeted": [ "assurance", "publication", "translation" ], "semanticBridge": { "state": "explicit", "description": "The package records the archived AIM request, the exact Cashen-Levitt dependency, the weighted-chord graph map and graph-of-groups presentation, the character coordinates, the edge forms x+i y, the direct Bass-Serre kernel-rank formula, the integer chamber minima and replay predicates. This exposes every transition from the problem wording to the orbit lower bound.", "remainingRisks": [ "A producer-side mistake in the graph-of-groups conventions or in checking the hypotheses of Cashen-Levitt Corollary 2.10 could propagate through the proof and replay.", "The Python and JavaScript checks share the same producer-defined certificate and do not independently validate the universal argument.", "The component invariant separates m+1 classes but does not determine whether equal-minimum components lie in the same Out(G_m)-orbit.", "The bounded novelty search cannot establish historical priority." ] }, "humanJudgmentGates": [ "Keep the exact 2m+2 component count separate from the at-least-m+1 orbit count.", "Do not infer that each antipodal component pair is one Out(G_m)-orbit.", "Inspect the hypotheses and application of Cashen-Levitt Corollary 2.10 rather than treating the edge-form computation as self-sufficient.", "Treat replay for 2<=m<=8 as regression evidence for the formulas, not as proof of the universal theorem.", "Keep same-producer cross-language corroboration, internal editorial review, unaffiliated reconstruction, formal verification and peer review distinct.", "Make no first-discovery or historical-priority claim from a bounded search." ], "parentLinks": [ { "relation": "depends-on-claim", "workId": null, "legacyReleaseSlug": null, "externalUrl": "https://web.archive.org/web/20240828224509id_/http://aimpl.org/freebycyclic/6/", "inheritedClaim": "AIM Problem 6.2 asks for free-by-cyclic families with b1 greater than two and many BNS components up to the action of Out(G).", "inheritedAssuranceCeiling": "The archived page fixes the existence request and wording but does not establish this construction, its correctness, novelty, priority or exact orbit count." }, { "relation": "extends-result", "workId": null, "legacyReleaseSlug": null, "externalUrl": "https://doi.org/10.1515/jgth-2015-0038", "inheritedClaim": "Cashen and Levitt provide the graph-of-groups BNS criterion used here, a linearly growing circle-of-tori family with many excluded subspheres, and a general fiber-rank formula.", "inheritedAssuranceCeiling": "Their work supplies the structural criterion and closest located architecture, but not the present weighted-chord family, displayed chamber-minimum sequence or claimed m+1 orbit lower bound." } ], "assuranceTarget": { "dimensions": [ "independentRerun", "independentReimplementation", "formalVerification", "specialistReview", "dataEnvironmentReproducibility", "semanticValidation", "noveltyAssessment", "priorityAssessment" ], "nextAction": "Obtain an authenticated unaffiliated reconstruction of the Cashen-Levitt application and Bass-Serre minimum argument, then compute or further constrain Out(G_m) without promoting the present lower bound to an exact orbit count.", "claimCeiling": "An anonymous, AI-assisted, unrefereed theorem candidate constructing a linearly growing free-by-cyclic family with exactly 2m+2 BNS components and at least m+1 Out(G_m)-orbits, supported by a complete written proof, one cited structural theorem, public immutable assets, producer replay and internal review; not an exact orbit classification, historical-priority finding, unaffiliated validation, formal verification, authenticated external specialist review or editorial peer review." }, "impactClaims": [ { "id": "unbounded-bns-component-orbits-science-no-impact", "aim": "science", "outcome": "Faster or more reliable construction and assurance of geometric-group-theory examples addressing open problem-list requests", "setting": "AI-assisted explicit construction, symbolic proof, source correspondence, adversarial replay, internal review and guarded candidate publication", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "No matched conventional research or publication workflow was registered.", "estimand": "No effect on discovery time, active human effort, compute, correction rate, proof quality, independent-assurance time, uptake, citation or field outcomes was 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:unbounded-bns-component-orbits-publication", "status": "published", "resultClass": "positive", "metrics": { "schemaVersion": "1.0", "measurementScope": "publication-only", "scopeBoundary": "From prospective registration before public repository and DOI creation through terminal canonical Evidence Press readback or a preserved stop. Earlier theorem discovery, proof construction, source recovery, supplied-review repair, deterministic replay and internal editorial closure are excluded and not reconstructed.", "forecast": { "frozenAt": "2026-08-31T08:27:44Z", "procedureClass": "reviewed-candidate-identity-through-full-publication", "targetOutcome": "Publish synchronized immutable GitHub and Zenodo assets and a reader-first Evidence Press release with provenance-bound media, two composite seals, guarded zero-cost deployment and exact canonical readback without broadening the at-least-m+1 orbit lower bound.", "expectedActiveMinutes": 120, "plausibleLowMinutes": 75, "plausibleHighMinutes": 180, "expectedUnattendedWaitMinutes": 45, "referenceClass": { "label": "Recent reviewed exact-mathematics identity-through-publication closures", "sampleSize": 4, "basis": "Recent Evidence Press candidates used the same immutable GitHub and Zenodo identity, reader-first media, composite sealing, hosted CI, guarded deployment and canonical-readback architecture; this twelve-page universal family also needs careful theorem-to-page translation." }, "fermiComponents": [ { "component": "Public repository, release CI and Zenodo identity", "count": 1, "lowMinutesPerUnit": 15, "centralMinutesPerUnit": 25, "highMinutesPerUnit": 35, "basis": "One dependency-light reviewed package with established GitHub and Zenodo workflows, DOI-bearing metadata and exact cross-service asset comparison." }, { "component": "Reader-first page and operating records", "count": 1, "lowMinutesPerUnit": 20, "centralMinutesPerUnit": 30, "highMinutesPerUnit": 45, "basis": "One twelve-page geometric-group-theory candidate whose chamber geometry, lower-bound invariant and Cashen--Levitt dependency need precise translation." }, { "component": "Deterministic art, provenance-bound audio, Open Graph image and thumbnail", "count": 1, "lowMinutesPerUnit": 15, "centralMinutesPerUnit": 25, "highMinutesPerUnit": 35, "basis": "One release using established generators but requiring a new weighted-chord and BNS-wall visual that must not imply an exact orbit count." }, { "component": "Composite A/B and C/D seals, CI, deployment and exact readback", "count": 1, "lowMinutesPerUnit": 25, "centralMinutesPerUnit": 40, "highMinutesPerUnit": 65, "basis": "One new canonical URL requiring two ledger-separated integration cycles, hosted multi-version checks, accessibility, guarded deployment and convergence verification." } ], "probabilityPositiveSignal": 0.96, "probabilityTargetClosure": 0.82, "probabilityHorizonMinutes": 180, "assumptions": [ "GitHub, Zenodo, hosted CI, OpenAI fable TTS and zero-cost Cloudflare credentials remain available without weakening a gate or exposing a secret.", "The final DOI-bearing metadata and reader page preserve the lower-bound, imported-theorem, novelty, language and external-assurance ceilings.", "Origin main does not acquire an irreconcilable overlapping release or operating-ledger change.", "The established package and site generators accept the new candidate without a release-contract migration." ], "reforecastTriggers": [ "GitHub, Zenodo, TTS, CI, Cloudflare or canonical readback remains unavailable after bounded recovery.", "A public replay or archive-integrity gate disagrees with the confirmed local package.", "A site, citation, licence, accessibility or preservation gate finds a claim-boundary conflict.", "Origin main advances with overlapping release, method-registry or work-ledger edits requiring reconciliation." ], "stopRule": "Stop on a fatal claim, source, licence, authentication, preservation, CI, accessibility or exact-readback failure. At 120 active minutes prioritize load-bearing release gates; at 180 active minutes preserve the exact blocker rather than weaken a gate." }, "outcome": { "measuredAt": "2026-08-31T09:15:48Z", "calendarElapsedMinutes": 48, "activeAgentMinutes": 41, "activeHumanMinutes": null, "computeMinutes": 1, "unattendedWaitMinutes": 7, "blockedMinutes": 0, "reworkMinutes": 1, "agentRuns": 1, "maxParallelAgents": 1, "modelTurns": 1, "deduplicatedModelTokens": null, "uncachedInputTokens": null, "researchCycles": { "positive": 0, "negative": 0, "inconclusive": 0 }, "falsificationGatesRun": 0, "candidateArchitecturesTested": 0, "candidateArchitecturesRejected": 0, "substantiveReviewRounds": 0, "p0Findings": 0, "p1Findings": 0, "prepublicationClaimCorrections": 0, "resultState": "target-closed", "resultSummary": "The review-repaired candidate reached synchronized immutable GitHub and Zenodo identity, a reader-first canonical Evidence Press release, byte-bound fable audio and transcript, deterministic weighted-chord and BNS-fan art, an inspected Open Graph card, an exact mirrored thumbnail, append-only operating records, protocol revision 191, green Node 18, 20 and 22 plus accessibility CI, zero-cost guarded deployment, bounded custom-domain protocol convergence, exact 44-release and 31-institutional-artifact readback, full publication preservation and accepted IndexNow submission. One manifest-selection defect and one bounded social-card presentation issue were repaired before publication without changing the research claim. The immutable machine resource fields retain the first live release-candidate snapshot; an appended correction records later observed supplemental totals of four compute minutes, four agent runs and four rework minutes. The result remains an anonymous unrefereed theorem candidate proving an exact 2m+2 component count conditional on Cashen--Levitt Corollary 2.10 and only an at-least-m+1 outer-orbit lower bound, with no exact orbit classification, historical-priority, unaffiliated-validation, formal-verification, external-specialist-review, peer-review or impact promotion.", "positiveSignalObserved": true, "targetReached": true, "forecastErrorMinutes": -79, "forecastRatio": 0.342, "withinForecastInterval": false, "varianceReason": "The established identity, page, media, composite sealing, hosted-CI and guarded-deployment tooling closed the publication route below the 75-minute lower active-time bound; GitHub and Zenodo credentials were already healthy, the candidate package was review-complete at intake, and only bounded manifest-selection and social-card presentation repairs were required.", "missingFields": [ { "field": "activeHumanMinutes", "reason": "No instrument captured human direction or review time at the prospective publication-only boundary." }, { "field": "deduplicatedModelTokens", "reason": "No active fork-aware goal counter exposed exact task-local model-token usage at or after the 08:27:44Z publication-ledger boundary, so tokens are not reconstructed from conversation context." }, { "field": "uncachedInputTokens", "reason": "The runtime does not expose an uncached-input token counter." } ] } }, "missingnessReason": null } ] } }