{"uid":"cap_QM38H2lkOLH0bwATqASPd","slug":"finite-matroid-independence-family-axioms-verifier-5c3be364","name":"Finite Matroid Independence Family Axioms Verifier","description":"Prüft eine vollständig gelieferte endliche Unabhängigkeitsfamilie gegen die drei Matroidaxiome, einschließlich Austauschfehlern, Schleifen und leerer Grundmenge; bestätigt keine bloße Stichprobe und berechnet keinen greedy Lösungsweg.","url":"https://detect.halowerk.com/v1/F-B-NIGHT-MATROID_INDEPENDENCE_AXIOMS/finite-matroid-independent-family-axioms?utm_source=zero.xyz","method":"POST","headers":{},"bodySchema":{"type":"object","properties":{"artifact":{"type":"object","required":["ground_set","independent_sets","claims"],"properties":{"claims":{"type":"object","required":["ground_elements","independent_sets","maximum_cardinality","empty_set_present","hereditary","augmentation","is_matroid"],"properties":{"hereditary":{"type":"boolean"},"is_matroid":{"type":"boolean"},"augmentation":{"type":"boolean"},"ground_elements":{"type":"number"},"independent_sets":{"type":"number"},"empty_set_present":{"type":"boolean"},"maximum_cardinality":{"type":"number"}},"additionalProperties":false},"ground_set":{"type":"array","items":{"type":"string","maxLength":128,"minLength":1},"maxItems":16,"uniqueItems":true},"independent_sets":{"type":"array","items":{"type":"array","items":{"type":"integer","maximum":15,"minimum":0},"maxItems":16},"maxItems":256}},"additionalProperties":false}}},"responseSchema":null,"example":null,"exampleRequest":null,"tags":["x402"],"displayCostAmount":"0.003","displayCostAsset":"USDC","priceDynamic":false,"priceHint":null,"priceStatus":"priced","priceSource":"probe","requiresHandshake":false,"reviewCount":0,"rating":{"score":"0.00","successRate":"0.00","reviews":0,"stars":null,"state":"unrated"},"availabilityStatus":"unknown","priceObserved":null,"sessionDeposit":null,"pricing":{"kind":"static","summary":"$0.003/call","primary":{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.003","per":"call","confidence":"exact"},"accepted":[{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.003","per":"call","confidence":"exact"}]},"paymentMethods":[{"uid":"pm_EbFf8pZDLS1z6xoRCsfw5","protocol":"x402","methodType":"crypto","chain":"base","mode":"charge","costAmount":"0.003","costPer":"request","priority":0,"asset":"0x833589fCD6eDb6E08f4c7C32D4f71b54bdA02913","unit":"request","depositMicros":null,"planRef":null}],"brandName":null,"brandSlug":null,"brandBaseUrl":null,"brandDocsUrl":null,"whatItDoes":"Validates whether a fully-specified finite independent family satisfies all three matroid axioms (empty set, hereditary, augmentation), reporting any violations.","exampleAgentPrompt":"Check whether my finite independent family forms a valid matroid: the ground set is ['a','b','c','d'], the independent sets are [[], ['a'], ['b'], ['c'], ['a','b'], ['a','c']], and I'm claiming hereditary=true, augmentation=true, empty_set_present=true, is_matroid=true, with 4 ground elements, 6 independent sets, and maximum cardinality 2 — verify all three matroid axioms and flag any violations.","exampleUseCases":[{"title":"Catching augmentation axiom failure","prompt":"I have a ground set ['x','y','z'] and independent sets [[], [0], [1], [0,1], [0,2]] — my claims say augmentation is true and it's a matroid, but can you verify all three matroid axioms and tell me if the augmentation property actually holds?"},{"title":"Verifying student-submitted matroid homework","prompt":"My student submitted a matroid with ground set ['e1','e2','e3','e4'] and independent families [[], [0], [1], [2], [0,1], [1,2], [0,1,2]], claiming 4 ground elements, 7 independent sets, maximum cardinality 3, hereditary true, augmentation true, empty set present, and is_matroid true — please validate all three matroid axioms and confirm or refute their claims."},{"title":"Detecting loops in combinatorial structure","prompt":"I built a set system over ground set ['p','q','r'] with independent sets [[], [1], [2], [1,2]] and I claim hereditary=true, augmentation=false, empty_set_present=true, is_matroid=false, 3 ground elements, 4 independent sets, max cardinality 2 — run the full matroid axiom check and tell me if there are any loops or hereditary violations."}],"resultDescription":"Returns a structured validation report confirming or refuting each of the three matroid axioms (empty set, hereditary closure, augmentation/exchange), cross-checked against the caller's claims about ground element count, independent set count, maximum cardinality, and overall matroid status. Violations are flagged with specifics (e.g. exchange errors, missing empty set, non-hereditary subsets).","failureModes":["Input ground_set has duplicate elements (schema requires uniqueItems)","Independent set references index out of range for ground_set size (index >= ground_set.length)","ground_set exceeds 16 elements or independent_sets exceeds 256 entries","claims counts do not match actual sizes (ground_elements or independent_sets mismatch)","Malformed or missing required fields in artifact or claims objects","Payment failure or x402 protocol error resulting in 402 response"],"whenToPreferThis":"Use this endpoint when you need to definitively verify whether a fully enumerated finite independent family satisfies all three matroid axioms, especially when you want to detect subtle violations like missing empty set, non-hereditary subsets, or augmentation failures. It is not suitable for sampling-based checks, greedy algorithm computation, or matroid optimization — it only validates axiom compliance on a complete, explicitly provided independent family up to 16 elements and 256 sets.","instructions":null,"reviewSummary":null,"reviewSummaryHighlights":null,"reviewSummaryConcerns":null,"reviewSummaryGeneratedAt":null,"activationCount":0,"lastUsedAt":null,"lastSuccessfullyRanAt":null,"lastHealthCheckAt":"2026-11-04T06:40:28.792Z","isFirstParty":false,"canonicalSlug":"finite-matroid-independence-family-axioms-verifier-5c3be364"}