{"query": "The integers, examined: a prime, a divisor, a factorial", "count": 14, "results": [{"id": "card_works_number_theory", "title": "The integers, examined: a prime, a divisor, a factorial", "shelf": "the-works", "surface": "secular", "snippet": "Exact facts about whole numbers, decided by algorithm, not by eye: 97 is prime; 48 and 36 share a greatest common divisor of 12; and 6! counts the orderings of six things.  Worked & sealed by the engi", "authority_tier": "verified", "source": "The Works — worked & sealed", "readable": false, "generated": false}, {"id": "card_domchk_number_theory_claimed_factorial", "title": "Number Theory: factorial", "shelf": "number_theory", "surface": "secular", "snippet": "A worked check in number theory: factorial.\n\nGIVEN\n  factorial_n = 5\n  gcd_a = 12\n\nCLAIMED\n  claimed_factorial = 120\n\nTHE ENGINE'S VERDICT\n  number_theory.factorial: CONFIRMED — 5! = 120 (matches clai", "authority_tier": "reference", "source": "The verifier's own documented relation, and a run this engine performed against it (deterministic; re-runnable with tools/domain_goldens.py)", "readable": false, "generated": false}, {"id": "card_src_word_factorial", "title": "factorial", "shelf": "dictionary", "surface": "secular", "snippet": "factorial: (noun) the product of all the integers up to and including a given integer · (adjective) of or relating to factorials", "authority_tier": "reference", "source": "WordNet 3.0, Princeton University (WordNet License)", "readable": false, "generated": false}, {"id": "card_seq_double_factorials", "title": "Double factorials", "shelf": "sequences", "surface": "secular", "snippet": "Double factorials: 1, 1, 2, 3, 8, 15, 48, 105, 384, 945, 3840, 10395, 46080, 135135, 645120, 2027025, 10321920, 34459425, … — n!! — the product of every other integer down to 1 or 2.", "authority_tier": "reference", "source": "Computed — classical number theory (public domain)", "readable": false, "generated": false}, {"id": "card_src_book_212", "title": "A List of Factorial Math Constants — Unknown", "shelf": "gutenberg", "surface": "secular", "snippet": "A List of Factorial Math Constants, by Unknown. Subjects: Mathematics; Factorials. Read the full text (public domain): https://www.gutenberg.org/ebooks/212", "authority_tier": "reference", "source": "Project Gutenberg (public domain)", "readable": true, "generated": false}, {"id": "card_seq_factorials", "title": "Factorials", "shelf": "sequences", "surface": "secular", "snippet": "Factorials: 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, … — n! = 1·2·…·n.", "authority_tier": "reference", "source": "Computed — classical number theory (public domain)", "readable": false, "generated": false}, {"id": "card_v_84e08f40a749", "title": "97 is prime · gcd(48, 36) = 12 · 6! = 720", "shelf": "number_theory", "surface": "secular", "snippet": "Verdict: HOLDS.\n• number_theory.primality: 97 is prime (matches claim)\n• number_theory.gcd: gcd(48, 36) = 12 (matches claim)\n• number_theory.factorial: 6! = 720 (matches claim)\nSealed and independentl", "authority_tier": "engine", "source": "Verified by the engine — number_theory", "readable": false, "generated": false}, {"id": "card_v_a082ae65129d", "title": "4 factorial is 30", "shelf": "mathematics", "surface": "secular", "snippet": "Verdict: BROKEN.\n• factorial(4) - (30.0) simplifies to -6.00000000000000\nSealed and independently re-checkable: https://narrowhighway.org/s/6d72e7036ec06563230fcadbea71c11d15a3993c757c7127fe493072ac7e", "authority_tier": "engine", "source": "Verified by the engine — mathematics", "readable": false, "generated": false}, {"id": "card_v_fed08f175040", "title": "5 factorial is 120", "shelf": "mathematics", "surface": "secular", "snippet": "Verdict: HOLDS.\n• factorial(5) = 120.0; the two sides are equal -- each reduces to 120, so their difference simplifies to 0\nSealed and independently re-checkable: https://narrowhighway.org/s/93927bf56", "authority_tier": "engine", "source": "Verified by the engine — mathematics", "readable": false, "generated": false}, {"id": "card_seal_6d72e7036ec06563230fcadbea71c11d15a3993c757c7127fe493072ac7e2cd2", "title": "Receipt 6d72e7036ec0… — sealed: record", "shelf": "seals", "surface": "secular", "snippet": "{\n \"anchors\": [],\n \"axis_coords\": {\n  \"axis\": \"mathematics\",\n  \"dimensions\": [\n   \"reasoning\"\n  ]\n },\n \"gate_results\": [\n  {\n   \"details\": {\n    \"broken_at\": \"a1\",\n    \"confirmed_steps\": 0,\n    \"error", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "readable": false, "generated": false}, {"id": "card_seal_93927bf567f0074b27cc35887c4a262a104765a7fb4e6e84ddcd7cb4f66b6a34", "title": "Receipt 93927bf567f0… — sealed: record", "shelf": "seals", "surface": "secular", "snippet": "{\n \"anchors\": [],\n \"axis_coords\": {\n  \"axis\": \"mathematics\",\n  \"dimensions\": [\n   \"reasoning\"\n  ]\n },\n \"gate_results\": [\n  {\n   \"details\": {\n    \"broken_at\": null,\n    \"confirmed_steps\": 1,\n    \"error", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "readable": false, "generated": false}, {"id": "card_n_bc8d104f3593", "title": "Language as a tree — the method biology and linguistics share", "shelf": "science", "surface": "secular", "snippet": "Darwin himself drew the parallel: languages branch like species. And the SAME mathematics\ncounts both — for n languages OR n genes, the number of possible rooted binary trees is the\ndouble factorial (", "authority_tier": "engine_derived", "source": "Concordance assay — 2026-07-09", "readable": false, "generated": false}, {"id": "card_c_03bb9eec2540", "title": "Language as a tree — the method biology and  ↔ The evolution of language — descent traced b", "shelf": "connections", "surface": null, "snippet": "the number of possible rooted binary trees is the double factorial (2n-3)!!, sealed: 105 for five, 34,459,425 for ten  — a concord the card itself states; mined + verified.", "authority_tier": "engine_derived", "source": "Concordance miner — 2026-07-11", "readable": false, "generated": false}, {"id": "card_n_dcb1a67c26e8", "title": "The evolution of language — descent traced by regular law", "shelf": "science", "surface": "secular", "snippet": "Languages descend like living lineages, and the descent is RECOVERABLE — the comparative\nmethod is one of the humanities' deterministic sciences. English 'father' descends from\nProto-Indo-European *ph", "authority_tier": "engine_derived", "source": "Concordance assay — 2026-07-09", "readable": false, "generated": false}]}