{
  "schema": "nexah.a0.local-modular-crt-operator-registry",
  "version": "0.1.0",
  "date": "2026-10-13",
  "status": "LOCAL_SOURCE_BOUND_ENGINEERING_REGISTRY_NOT_CANONICAL",
  "typing_rule": "A modulus declares a carrier. An operator is a typed map with a domain, codomain, parameters and rule.",
  "reused_source_types": ["OP_MODULO", "OP_CONGRUENCE"],
  "classes": [
    {
      "class_id": "A0.MOD.REDUCE",
      "signature": "Z -> Z_m",
      "parameters": ["m"],
      "rule": "x |-> x mod m",
      "lossiness": "lossy unless the input is already restricted to a complete residue system"
    },
    {
      "class_id": "A0.MOD.TRANSLATE",
      "signature": "Z_m -> Z_m",
      "parameters": ["b", "m"],
      "rule": "x |-> x+b mod m",
      "lossiness": "bijective"
    },
    {
      "class_id": "A0.MOD.MULTIPLY",
      "signature": "Z_m -> Z_m",
      "parameters": ["a", "m"],
      "rule": "x |-> a*x mod m",
      "lossiness": "bijective iff gcd(a,m)=1; otherwise many-to-one"
    },
    {
      "class_id": "A0.MOD.MIRROR",
      "signature": "Z_m -> Z_m",
      "parameters": ["m"],
      "rule": "x |-> -x mod m",
      "lossiness": "bijective involution"
    },
    {
      "class_id": "A0.CRT.SPLIT",
      "signature": "Z_pq -> Z_p x Z_q",
      "parameters": ["p", "q"],
      "rule": "x |-> (x mod p, x mod q)",
      "lossiness": "bijective for the declared carrier when gcd(p,q)=1"
    },
    {
      "class_id": "A0.CRT.BIND",
      "signature": "Z_p x Z_q -> Z_pq",
      "parameters": ["p", "q"],
      "rule": "return the unique CRT class c",
      "lossiness": "bijective when gcd(p,q)=1; fail closed otherwise"
    },
    {
      "class_id": "A0.CRT.SHELL",
      "signature": "(Z,Z_M) -> Z",
      "parameters": ["M"],
      "rule": "(n,c) |-> (n-c)/M when n=c mod M",
      "lossiness": "exact only when the quotient is retained"
    }
  ],
  "instances": [
    {
      "instance_id": "A0.CRT.CHANNEL/5x73",
      "class_ids": ["A0.CRT.SPLIT", "A0.CRT.BIND"],
      "carrier": "Z_365",
      "parameters": {"p": 5, "q": 73},
      "status": "VALIDATED_EXHAUSTIVE"
    },
    {
      "instance_id": "A0.CRT.CHANNEL/41x61",
      "class_ids": ["A0.CRT.SPLIT", "A0.CRT.BIND"],
      "carrier": "Z_2501",
      "parameters": {"p": 41, "q": 61},
      "status": "VALIDATED_EXHAUSTIVE"
    },
    {
      "instance_id": "A0.MOD.MULTIPLY/64@365",
      "class_ids": ["A0.MOD.MULTIPLY"],
      "carrier": "Z_365",
      "parameters": {"a": 64, "m": 365},
      "inverse": 154,
      "order": 6,
      "status": "VALIDATED_EXHAUSTIVE"
    },
    {
      "instance_id": "A0.MOD.MULTIPLY/64@2501",
      "class_ids": ["A0.MOD.MULTIPLY"],
      "carrier": "Z_2501",
      "parameters": {"a": 64, "m": 2501},
      "inverse": 1993,
      "order": 10,
      "status": "VALIDATED_EXHAUSTIVE"
    },
    {
      "instance_id": "A0.MOD.MULTIPLY/41@2501",
      "class_ids": ["A0.MOD.MULTIPLY"],
      "carrier": "Z_2501",
      "parameters": {"a": 41, "m": 2501},
      "kernel_size": 41,
      "image_size": 61,
      "status": "VALIDATED_EXHAUSTIVE_NONUNIT_CONTROL"
    },
    {
      "instance_id": "A0.MOD.MULTIPLY/61@2501",
      "class_ids": ["A0.MOD.MULTIPLY"],
      "carrier": "Z_2501",
      "parameters": {"a": 61, "m": 2501},
      "kernel_size": 61,
      "image_size": 41,
      "status": "VALIDATED_EXHAUSTIVE_NONUNIT_CONTROL"
    },
    {
      "instance_id": "A0.MOD.MIRROR/-1@2501",
      "class_ids": ["A0.MOD.MIRROR"],
      "carrier": "Z_2501",
      "parameters": {"m": 2501},
      "status": "VALIDATED_EXHAUSTIVE"
    },
    {
      "instance_id": "A0.MOD.TRANSLATE/100@2501",
      "class_ids": ["A0.MOD.TRANSLATE"],
      "carrier": "Z_2501",
      "parameters": {"b": 100, "m": 2501},
      "status": "VALIDATED_EXHAUSTIVE"
    }
  ],
  "governance": {
    "numbermagick_source_mutated": false,
    "orion_registry_mutated": false,
    "navigator_identity_created": false,
    "navigator_relation_created": false,
    "claim_promotion_authorized": false,
    "publication_authorized": false
  }
}