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"P_count": 0, "J_count": 3, "affine_coefficients": [ "421875000000/325234848133", "-293302860916345/325234848133" ] }, { "start": 3035, "end": 3035, "E_count": 3, "P_count": 1, "J_count": 2, "affine_coefficients": [ "32812500000/25018065241", "-23656109493565/25018065241" ] }, { "start": 3035, "end": 3035, "E_count": 3, "P_count": 2, "J_count": 1, "affine_coefficients": [ "7656250000/5773399671", "-5714450748515/5773399671" ] }, { "start": 3035, "end": 3035, "E_count": 3, "P_count": 3, "J_count": 0, "affine_coefficients": [ "5359375000/3996969003", "-4134902200895/3996969003" ] } ], "prime_valuation_certificate": { "7": [ 0, 1, 0 ], "13": [ 0, 0, -1 ], "5": [ 2, 1, 2 ] }, "counts": { "source_vertices": 3, "source_operator_edges": 6, "paths_through_six_edges": 402, "affine_classes_with_typed_endpoints": 78, "central_cross_segment_circuit_equalities": 2 }, "general_form": "For any admitted path fromu tov with counts(e,p,j), F(t)=v+E^e P^p J^j(t-u), with(v-u)/2=p+j-e. Same endpoints and counts give the same map; prime valuations show different counts cannot give the same slope.", "closed_path_form": "At every one of the three source vertices, closed paths havee=p+j and multiplier(1750/1587)^p(7500/6877)^j. Distinct pivot routes with the same counts can represent the same affine map.", "interpretation": "This is a reduced coordinate-affine grammar on the union of already admitted source edges. It preserves source/pivot provenance separately. Two successive E returns use different existing pivots, not the fixed-Noah double-E ladder or an extra Gear." }