Overview

A graph-spectral conformance agent needs the complete Ihara zeta denominator, which encodes nonbacktracking primitive closed walks of every length; a minimum binary cycle-space basis or spanning-tree count cannot determine this generating function.

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Indexed 2026-11-02 17:26:22

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Input Schema

{ "body": { "artifact": { "ihara": { "claim": [ "2" ], "edges": [ [ 0, 1 ], [ 1, 2 ], [ 2, 3 ], [ 3, 4 ] ], "vertices": 5 } } }, "bodyType": "json", "method": "POST", "type": "http" }

Output Schema

{ "example": { "billable": true, "findings": [ { "confidence": 95, "evidence": [ { "page_indices": [], "page_url": null, "pattern": "finite-graph-ihara-zeta-complete-inverse-polynomial_scoped_evidence", "type": "artifact" } ], "limits": [ "Finite simple undirected graph,1–6 vertices,≤9 edges; disconnected graphs and isolated/tree/pendant vertices supported. Edges unordered, loops and duplicate edges unsupported. All integer coefficients ascending powers, trailing zeros removed.", "Zeta counts oriented primitive reduced tailless closed walks up to cyclic shift; reversing orientation is a separate prime class. Trees have reciprocal1. Only leaves are stripped: a bridge joining cyclic cores can contribute to reduced closed walks and is retained. Not Tutte/ribbon or a binary cycle-space basis." ], "module": "finite-graph-ihara-zeta-complete-inverse-polynomial", "state": "observed", "value": { "claim_matches": false, "degree": 0, "inverse_zeta_coefficients": [ "1" ] }, "warnings": [] } ], "known_count": 1, "scope": { "contact_values_returned": false, "network": false, "supplied_artifact_only": true }, "warnings": [] }, "type": "json" }