{
  "artifact_schema": "fprd.result-review-certificate.v1",
  "review_date": "2026-09-01",
  "review_units": [
    {
      "title": "binary n-minus-one witness-pattern refutation",
      "claims": ["FPRD-LT-T03", "FPRD-LT-E02", "FPRD-LT-X02"]
    },
    {
      "title": "target-isolating poDFA quotient criterion",
      "claims": ["FPRD-LT-T01", "FPRD-LT-T02", "FPRD-LT-X01"]
    }
  ],
  "mathematical_decisions": [
    {
      "claims": ["FPRD-LT-T03"],
      "decision": "confirmed with boundary clarification",
      "detail": "The construction, minimality proof, J-triviality chain, and exact length-n lower bound are correct for n at least five. At n=4 the same target has the shorter representative bab, so the stated lower endpoint is necessary."
    },
    {
      "claims": ["FPRD-LT-T01", "FPRD-LT-T02"],
      "decision": "corrected a target-domain ambiguity",
      "detail": "The target is an arbitrary transformation and may lie outside M, so rho(f) is not defined on every input. The exact condition is now stated for every g in M as rho(g)=t_f iff g=f; the faithful special case likewise requires a computable code outside the image for negative targets."
    },
    {
      "claims": ["FPRD-LT-X01"],
      "decision": "confirmed",
      "detail": "All nine products preserve the two-state action, the principal left ideals are distinct, and the identity fibre is the singleton {1} although a and 0 collapse."
    }
  ],
  "independent_audit": {
    "checker": "l_trivial_quotient_family_review.py",
    "checker_sha256": "042228fb1ef40d638a15488c20ab03dd0f714180333e44978a1efc7044eceb7a",
    "output": "independent-output.json",
    "output_sha256": "a2d1a87ccfb16426ea40b338229a93a239f35a33898361049be2e5c00c58d54e",
    "result": "pass",
    "coverage": {
      "target_example_products": 9,
      "family_parameters": 15,
      "target_distances_checked": "n=4 through n=18",
      "left_and_right_green_triviality_checked": "n=4 through n=12"
    },
    "hostile_controls": [
      "n=4, where the displayed length-four word is not shortest and bab reaches the target in length three",
      "a negative target code, whose fibre inside M must be empty rather than invoking rho outside its domain"
    ]
  },
  "source_artifacts_inspected": [
    "fprd-ltrivial-fixed-binary-attack-2026-09-01.md",
    "ltrivial_linear_family_audit.py",
    "ltrivial_linear_family_audit_output.txt",
    "ltrivial_five_state_attack.cpp",
    "ltrivial_five_state_canonical_output.txt",
    "ltrivial_counterexample_verify.py",
    "ltrivial_counterexample_verify_output.txt",
    "fprd-l-trivial-opportunity-hunt-2026-09-01.md",
    "l_trivial_target_quotient_audit.py",
    "l_trivial_target_quotient_audit_output.txt"
  ],
  "primary_sources": [
    {
      "citation": "Andrew Ryzhikov and Petra Wolf, Monoids of Upper Triangular Matrices over the Boolean Semiring, MFCS 2024, LIPIcs 306, article 81, pages 81:1-81:18",
      "doi": "10.4230/LIPIcs.MFCS.2024.81",
      "use": "Proposition 9 supplies the quadratic representative-length bound for poDFAs."
    },
    {
      "citation": "Tomáš Masopust and Markus Krötzsch, Partially Ordered Automata and Piecewise Testability, Logical Methods in Computer Science 17(2) (2021), article 14, pages 14:1-14:36",
      "doi": "10.23638/LMCS-17(2:14)2021",
      "arxiv": "1907.13115",
      "use": "Confluent partially ordered minimal DFAs characterize piecewise-testable languages."
    },
    {
      "citation": "Imre Simon, Piecewise Testable Events, Automata Theory and Formal Languages, LNCS 33 (1975), pages 214-222",
      "doi": "10.1007/3-540-07407-4_23",
      "use": "Piecewise testability is equivalent to J-triviality of the syntactic monoid."
    }
  ],
  "maturity_changes": [],
  "unresolved_items": [
    "No polynomial-size target-isolating construction is known for all compact L-trivial inputs.",
    "The exact linear family does not decide whether all positive instances have polynomial-length witnesses.",
    "The five-state census remains a finite computational classification rather than a general upper bound."
  ]
}