{"uid":"cap_JieLmAiQMA09GaqXq2T1e","slug":"simplicial-f2-one-cocycle-cup-product-class-verifier-1c1ae471","name":"Simplicial F2 One-Cocycle Cup Product Class Verifier","description":"A topology test agent needs whether the cup product of two specified1cocycles is zero in H2 of its complete finite2D simplicial complex, with a coboundary witness, not only chain ranks or homology dimensions.","url":"https://detect.halowerk.com/v1/F-NIGHT-ECMA/simplicial-f2-one-cocycle-cup-class?utm_source=zero.xyz","method":"POST","headers":{},"bodySchema":{"type":"object","properties":{"artifact":{"type":"object","required":["cup_product"],"properties":{"cup_product":{"type":"object","required":["vertices","edges","triangles","alpha","beta","claim"],"properties":{"beta":{"type":"array","items":{"type":"integer","maximum":1,"minimum":0},"maxItems":32,"minItems":0},"alpha":{"type":"array","items":{"type":"integer","maximum":1,"minimum":0},"maxItems":32,"minItems":0},"claim":{"type":"boolean"},"edges":{"type":"array","items":{"type":"array","items":{"type":"integer","maximum":11,"minimum":0},"maxItems":2,"minItems":2},"maxItems":32,"minItems":0},"vertices":{"type":"integer","maximum":12,"minimum":1},"triangles":{"type":"array","items":{"type":"array","items":{"type":"integer","maximum":11,"minimum":0},"maxItems":3,"minItems":3},"maxItems":24,"minItems":0}},"additionalProperties":false}},"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_YgE4HT8VMieFqRxgDj-9L","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":"Verifies whether the cup product of two specified F2 one-cocycles is zero in H2 of a given finite 2D simplicial complex, returning a coboundary witness if applicable.","exampleAgentPrompt":"Check whether the cup product of these two F2 one-cocycles (alpha = [1,0,1,0] and beta = [0,1,1,0]) is zero in H2 of my simplicial complex with 4 vertices, edges [[0,1],[1,2],[2,3],[3,0],[0,2]], and triangles [[0,1,2],[0,2,3]] — my claim is that it IS zero, and I need a coboundary witness if that's correct.","exampleUseCases":[{"title":"Topology agent cup product vanishing test","prompt":"I have a 2D simplicial complex with 5 vertices, edges [[0,1],[1,2],[2,0],[0,3],[3,4],[4,0]], and triangles [[0,1,2],[0,3,4]]. Can you verify whether the cup product of cocycles alpha=[1,0,0,1,0,0] and beta=[0,1,1,0,0,0] is zero in H2, and return a coboundary witness? My claim is that it's zero."},{"title":"Cohomology ring structure verification","prompt":"For my simplicial complex with 6 vertices, edges [[0,1],[1,2],[2,3],[3,4],[4,5],[5,0],[0,3]], and triangles [[0,1,2],[3,4,5]], check if alpha=[1,1,0,0,0,0,1] cup beta=[0,0,1,1,0,0,0] is nonzero in H2. I'm claiming the cup product is NOT zero — please confirm or refute with a witness."},{"title":"Topological feature detection in complex","prompt":"My topology agent needs to know: given a simplicial complex with 3 vertices, edges [[0,1],[1,2],[0,2]], and triangle [[0,1,2]], does the cup product of alpha=[1,0,1] and beta=[0,1,1] vanish in H2? I believe it does — can you check and provide the coboundary witness?"}],"resultDescription":"Returns whether the cup product of the two supplied F2 one-cocycles (alpha and beta) equals zero in H2 of the given finite 2D simplicial complex, together with a coboundary witness certificate when the claim is verified correct.","failureModes":["Invalid simplicial complex (e.g. edges referencing out-of-range vertex indices) returns a validation error","Alpha or beta cocycle arrays of wrong length relative to edge count returns schema mismatch","Non-cocycle input (alpha or beta not satisfying coboundary condition) may return error or unexpected result","Exceeding max vertices (12), edges (32), or triangles (24) returns schema validation error","Malformed triangle arrays (not exactly 3 vertex indices) return input error"],"whenToPreferThis":"Use this endpoint when an agent needs to rigorously verify a cohomological cup product claim over F2 in a small finite 2D simplicial complex and requires a coboundary witness — not just chain ranks or homology dimensions. Prefer this over general homology tools when the specific algebraic topology question is about the cohomology ring structure (cup products) rather than Betti numbers or chain connectivity.","instructions":null,"reviewSummary":null,"reviewSummaryHighlights":null,"reviewSummaryConcerns":null,"reviewSummaryGeneratedAt":null,"activationCount":0,"lastUsedAt":null,"lastSuccessfullyRanAt":null,"lastHealthCheckAt":"2026-11-04T06:40:50.514Z","isFirstParty":false,"canonicalSlug":"simplicial-f2-one-cocycle-cup-product-class-verifier-1c1ae471"}