{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R433",
  "content_hash": null,
  "slug": "ksd6-artifact-transfer-certificate",
  "type": "artifact",
  "title": "Executable min-plus periodicity certificate",
  "summary": "Inline Python builds the full transfer, checks exact values through width 25, and verifies the vector identity that proves the infinite recurrence.",
  "relevance": "For A period-six recurrence for domination on the three-row knight graph, record ksd6-artifact-transfer-certificate (“Executable min-plus periodicity certificate”) supplies evidence or a replay used to check the packet. The record states: Inline Python builds the full transfer, checks exact values through width 25, and verifies the vector identity that proves the infinite recurrence.",
  "relevance_source": "recorded",
  "body": "The program implements the four-mask transition in `ksd6-claim-min-plus-certificate`. It retains the minimum cost for every target state, which is exact because all selections in a new column are among the eight three-bit masks. The two forced zero transitions in `finish` impose the right boundary condition.\n\nThe reachable-state counts at widths zero through six are\n\\[\n1,8,64,114,196,279,351.\n\\]\nThe state set then stays at 351 through every vector used in the certificate. The canonical transfer serialization orders states by the 12-bit encoding \\(a+(b\\ll3)+(u\\ll6)+(v\\ll9)\\), then orders the chosen mask from 0 through 7. Its SHA-256 digest is `5a6b35842eb54931e9d1b0de3ed239567576c98df3f7fa81420150232bcf9567`.\n\nThe program checks all 25 initial values, the ten scalar recurrence instances starting at widths 9 through 18, equality of the finite supports of \\(f_{19}\\) and \\(f_{25}\\), and all 351 componentwise equations \\(f_{25}(s)=f_{19}(s)+4\\). The stable nine-line output has SHA-256 digest `115b3c754e1663b8922a609ee2e637e95cab85fb8359886b7f1dbd36e9ffefb4`.",
  "status": "available",
  "evidence_grade": "executable",
  "scope": {
    "kind": "universal",
    "statement": "the complete finite-state transfer and its induced domination sequence for all three-row knight strips"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "partial",
    "kind": "inline_python_min_plus_computation",
    "entrypoint": "join source_lines with newline and run with python3",
    "runtime": "CPython 3.9 or later, standard library only",
    "citation": {
      "locator": "Inline CPython source below, executed on 2026-07-24"
    },
    "inline_source": [
      "from hashlib import sha256",
      "FULL=7",
      "MASKS=range(8)",
      "POPCOUNT=tuple(bin(x).count('1') for x in MASKS)",
      "",
      "def one(m):",
      "    return ((m&1)<<2)|((m&4)>>2)",
      "",
      "def two(m):",
      "    return ((m&1)<<1)|((m&2)<<1)|((m&2)>>1)|((m&4)>>1)",
      "",
      "def transition(state,x):",
      "    a,b,u,v=state",
      "    if (u|two(x))!=FULL:",
      "        return None",
      "    return (b,x,v|one(x),x|one(b)|two(a))",
      "",
      "def advance(vector,choices=MASKS):",
      "    result={}",
      "    for state,value in vector.items():",
      "        for x in choices:",
      "            target=transition(state,x)",
      "            if target is None:",
      "                continue",
      "            candidate=value+POPCOUNT[x]",
      "            if candidate<result.get(target,10**9):",
      "                result[target]=candidate",
      "    return result",
      "",
      "def finish(vector):",
      "    return min(advance(advance(vector,(0,)),(0,)).values())",
      "",
      "def code(state):",
      "    a,b,u,v=state",
      "    return a|(b<<3)|(u<<6)|(v<<9)",
      "",
      "START=(0,0,7,7)",
      "vectors=[{START:0}]",
      "for n in range(25):",
      "    vectors.append(advance(vectors[-1]))",
      "gamma=[finish(vector) for vector in vectors]",
      "expected=(0,3,4,4,4,4,4,6,8,8,8,8,8,10,11,12,12,12,12,",
      "          14,15,16,16,16,16,18)",
      "assert tuple(gamma)==expected",
      "assert all(gamma[n+6]==gamma[n]+4 for n in range(9,19))",
      "assert set(vectors[19])==set(vectors[25])",
      "assert all(vectors[25][s]==vectors[19][s]+4 for s in vectors[19])",
      "stable=set(vectors[6])",
      "assert len(stable)==351",
      "assert all(set(vectors[n])==stable for n in range(6,26))",
      "edges=[]",
      "for state in sorted(stable,key=code):",
      "    for x in MASKS:",
      "        target=transition(state,x)",
      "        if target is not None:",
      "            assert target in stable",
      "            edges.append((code(state),x,code(target),POPCOUNT[x]))",
      "edge_serial=''.join(f'{a},{x},{b},{w}\\n' for a,x,b,w in edges).encode()",
      "v19_serial=''.join(f'{code(s)}:{vectors[19][s]}\\n'",
      "                   for s in sorted(stable,key=code)).encode()",
      "v25_serial=''.join(f'{code(s)}:{vectors[25][s]}\\n'",
      "                   for s in sorted(stable,key=code)).encode()",
      "gamma_serial=(','.join(map(str,gamma[1:]))+'\\n').encode()",
      "print('state_space 4096 stable_reachable_states',len(stable),",
      "      'transfer_edges',len(edges))",
      "print('reachable_counts',','.join(str(len(v)) for v in vectors[:7]))",
      "print('gamma_1_25',','.join(map(str,gamma[1:])))",
      "print('direct_recurrence_checks n=9..18 count 10 all_pass')",
      "print('vector_identity f25=f19+4 finite_entries',len(vectors[19]))",
      "print('transfer_sha256',sha256(edge_serial).hexdigest())",
      "print('f19_sha256',sha256(v19_serial).hexdigest())",
      "print('f25_sha256',sha256(v25_serial).hexdigest())",
      "print('gamma_sha256',sha256(gamma_serial).hexdigest())"
    ],
    "missing": [
      "command",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Inline CPython source below, executed on 2026-07-24"
  },
  "relations": [
    {
      "slug": "R436",
      "title": "A 4,096-state transfer proves the infinite tail",
      "object_type": "claim",
      "relation": "evidences",
      "direction": "outgoing"
    },
    {
      "slug": "R435",
      "title": "The period-six recurrence holds from n=9",
      "object_type": "claim",
      "relation": "evidences",
      "direction": "outgoing"
    },
    {
      "slug": "knight-strip-domination-period-six",
      "title": "knight strip domination period six",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}