{"query": "The Legendre dual of the Hamiltonian — H = p q' − L", "count": 20, "results": [{"id": "card_lag_legendre_dual", "title": "The Legendre dual of the Hamiltonian — H = p q' − L", "shelf": "codex", "surface": "secular", "snippet": "L and H are Legendre transforms of one another: L = T - V (the generator of the path), H = T + V (the generator of time). Free particle: L = (1/2) m v^2, p = m v, H = p v - L. Sealed on this stick and", "authority_tier": "engine_derived", "source": "Narrow Highway — the Lagrangian", "readable": false, "generated": false}, {"id": "card_v_f4b827f56b3c", "title": "p ∨ ¬p is a tautology · ¬(p ∧ q) ≡ ¬p ∨ ¬q  (De Morgan)", "shelf": "formal_logic", "surface": "secular", "snippet": "Verdict: HOLDS.\n• formal_logic.tautology: formula 'p | ~p' tautology=True, matches claim\n• formal_logic.equivalence: '~(p & q)' ≡ '~p | ~q' = True, matches claim\nSealed and independently re-checkable:", "authority_tier": "engine", "source": "Verified by the engine — formal_logic", "readable": false, "generated": false}, {"id": "card_domchk_formal_logic_claimed_tautology", "title": "Formal Logic: tautology", "shelf": "formal_logic", "surface": "secular", "snippet": "A worked check in formal logic: tautology.\n\nGIVEN\n  conclusion = q\n  formula = p & q\n\nCLAIMED\n  claimed_tautology = False\n\nTHE ENGINE'S VERDICT\n  formal_logic.tautology: CONFIRMED — formula 'p & q' ta", "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_pron_q_s", "title": "q's", "shelf": "pronunciation", "surface": "secular", "snippet": "q's: pronounced (ARPABET) K Y UW1 Z. From the CMU Pronouncing Dictionary — the standard machine-readable pronunciations of North American English.", "authority_tier": "reference", "source": "CMU Pronouncing Dictionary (cmudict) — BSD-2-Clause, Carnegie Mellon", "readable": false, "generated": false}, {"id": "card_domchk_formal_logic_claimed_equivalent", "title": "Formal Logic: equivalent", "shelf": "formal_logic", "surface": "secular", "snippet": "A worked check in formal logic: equivalent.\n\nGIVEN\n  conclusion = q\n  formula_a = p | q\n  formula_b = ~(~p & ~q)\n\nCLAIMED\n  claimed_equivalent = True\n\nTHE ENGINE'S VERDICT\n  formal_logic.equivalence: ", "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_domchk_formal_logic_claimed_contradiction", "title": "Formal Logic: contradiction", "shelf": "formal_logic", "surface": "secular", "snippet": "A worked check in formal logic: contradiction.\n\nGIVEN\n  conclusion = q\n  formula = p & q\n\nCLAIMED\n  claimed_contradiction = False\n\nTHE ENGINE'S VERDICT\n  formal_logic.contradiction: CONFIRMED — formul", "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_domchk_formal_logic_claimed_satisfiable", "title": "Formal Logic: satisfiable", "shelf": "formal_logic", "surface": "secular", "snippet": "A worked check in formal logic: satisfiable.\n\nGIVEN\n  conclusion = q\n  formula = p & q\n\nCLAIMED\n  claimed_satisfiable = True\n\nTHE ENGINE'S VERDICT\n  formal_logic.satisfiability: CONFIRMED — formula 'p", "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_seal_7d19e009f60569bf6a7101837efd7dc08c029c0b13c74006bf0fdb8a00a36066", "title": "Receipt 7d19e009f605… — sealed: record", "shelf": "seals", "surface": "secular", "snippet": "{\n \"anchors\": [],\n \"axis_coords\": {\n  \"axis\": \"formal_logic\",\n  \"dimensions\": [\n   \"discreteness\",\n   \"reasoning\"\n  ]\n },\n \"gate_results\": [\n  {\n   \"details\": {\n    \"broken_at\": null,\n    \"confirmed_s", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "readable": false, "generated": false}, {"id": "card_seal_bf721976b7c16218903a1baaba8367baf8e4385c24554e75d3be20b047580a12", "title": "Receipt bf721976b7c1… — sealed: record", "shelf": "seals", "surface": "secular", "snippet": "{\n \"anchors\": [],\n \"axis_coords\": {\n  \"axis\": \"formal_logic\",\n  \"dimensions\": [\n   \"discreteness\",\n   \"reasoning\"\n  ]\n },\n \"gate_results\": [\n  {\n   \"details\": {\n    \"broken_at\": null,\n    \"confirmed_s", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "readable": false, "generated": false}, {"id": "card_seal_fcaafe2868ea3e068682458c85f3ac56068f264801a7c612654cad3f227942a6", "title": "Receipt fcaafe2868ea… — sealed: record", "shelf": "seals", "surface": "secular", "snippet": "{\n \"anchors\": [],\n \"axis_coords\": {\n  \"axis\": \"formal_logic\",\n  \"dimensions\": [\n   \"discreteness\",\n   \"reasoning\"\n  ]\n },\n \"gate_results\": [\n  {\n   \"details\": {\n    \"broken_at\": null,\n    \"confirmed_s", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "readable": false, "generated": false}, {"id": "card_seal_16be227e2c500435ad33091a129cf327a966c264ee73308d64d4e3eb2065ddb3", "title": "Receipt 16be227e2c50… — sealed: record", "shelf": "seals", "surface": "secular", "snippet": "{\n \"anchors\": [],\n \"axis_coords\": {\n  \"axis\": \"formal_logic\",\n  \"dimensions\": [\n   \"discreteness\",\n   \"reasoning\"\n  ]\n },\n \"gate_results\": [\n  {\n   \"details\": {\n    \"broken_at\": null,\n    \"confirmed_s", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "readable": false, "generated": false}, {"id": "card_seal_38cd5cf7f77e86267c823148eaab5be06c4f600e5d84ed19f3c38fe0ff308c47", "title": "Receipt 38cd5cf7f77e… — sealed: record", "shelf": "seals", "surface": "secular", "snippet": "{\n \"anchors\": [],\n \"axis_coords\": {\n  \"axis\": \"formal_logic\",\n  \"dimensions\": [\n   \"discreteness\",\n   \"reasoning\"\n  ]\n },\n \"gate_results\": [\n  {\n   \"details\": {\n    \"broken_at\": null,\n    \"confirmed_s", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "readable": false, "generated": false}, {"id": "card_seal_694d46006594fab30c52dfcbade66eec8b3e177a47d016b3da92411decf799c6", "title": "Receipt 694d46006594… — sealed: record", "shelf": "seals", "surface": "secular", "snippet": "{\n \"anchors\": [],\n \"axis_coords\": {\n  \"axis\": \"formal_logic\",\n  \"dimensions\": [\n   \"discreteness\",\n   \"reasoning\"\n  ]\n },\n \"gate_results\": [\n  {\n   \"details\": {\n    \"broken_at\": null,\n    \"confirmed_s", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "readable": false, "generated": false}, {"id": "card_seal_6cb27b4dc3a0f1d3cd1dd9697066c1ba2ef48a5f77e33d06f138e7872a76b241", "title": "Receipt 6cb27b4dc3a0… — sealed: record", "shelf": "seals", "surface": "secular", "snippet": "{\n \"anchors\": [],\n \"axis_coords\": {\n  \"axis\": \"formal_logic\",\n  \"dimensions\": [\n   \"discreteness\",\n   \"reasoning\"\n  ]\n },\n \"gate_results\": [\n  {\n   \"details\": {\n    \"broken_at\": null,\n    \"confirmed_s", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "readable": false, "generated": false}, {"id": "card_seal_eea2da4c2ab3eea32e4c86a66f6fd1ea3d6f5d9780570cf5c6bb6fd460952d18", "title": "Receipt eea2da4c2ab3… — sealed: record", "shelf": "seals", "surface": "secular", "snippet": "{\n \"anchors\": [],\n \"axis_coords\": {\n  \"axis\": \"formal_logic\",\n  \"dimensions\": [\n   \"discreteness\",\n   \"reasoning\"\n  ]\n },\n \"gate_results\": [\n  {\n   \"details\": {\n    \"broken_at\": null,\n    \"confirmed_s", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "readable": false, "generated": false}, {"id": "card_theory_sabermetrics_sports_analytics", "title": "Sabermetrics / sports analytics (Pythagorean expectation, Elo)", "shelf": "theories", "surface": "secular", "snippet": "Sabermetrics / sports analytics (Pythagorean expectation, Elo) — an engine domain that can touch it: sports_analytics. Calibration: seals — expected win ratios and Elo updates compute. Bill James's Py", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_vsepr_theory", "title": "VSEPR theory (molecular geometry)", "shelf": "theories", "surface": "secular", "snippet": "VSEPR theory (molecular geometry) — an engine domain that can touch it: molecular_geometry. Calibration: seals — shapes and bond angles predict directly. Valence Shell Electron Pair Repulsion: electro", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_digital_analog", "title": "Digital versus analog (why discreteness survives copying)", "shelf": "theories", "surface": "secular", "snippet": "Digital versus analog (why discreteness survives copying) — an engine domain that can touch it: computer_science. Calibration: map-only — sampling rates and bit depth are ordinary arithmetic, and THIS", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_numerical_analysis", "title": "Numerical analysis (why the computer's answer is not the answer)", "shelf": "theories", "surface": "secular", "snippet": "Numerical analysis (why the computer's answer is not the answer) — an engine domain that can touch it: computer_science. Calibration: map-only — error bounds and condition numbers are computable in pr", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_molecular_symmetry_point_groups", "title": "Molecular symmetry & point groups", "shelf": "theories", "surface": "secular", "snippet": "Molecular symmetry & point groups — an engine domain that can touch it: molecular_geometry. Calibration: seals — a deterministic verifier can CONFIRM or BREAK checkable claims drawn from it, and seal ", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (lone-domain seeding)", "readable": false, "generated": false}], "house": {"door": "FIND", "kind": "cards", "trail": "results", "seal": null, "next_step": {"do": "open the top card", "door": "FIND", "tool": "card_get", "params": {"id": "card_lag_legendre_dual"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}