{"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_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}, {"id": "card_theory_periodic_law", "title": "Periodic law (Mendeleev)", "shelf": "theories", "surface": "secular", "snippet": "Periodic law (Mendeleev) — an engine domain that can touch it: periodic_table. Calibration: seals — element properties and group trends look up and compute. Arrange elements by atomic number and their", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_bohr_quantum_model_of_the_atom", "title": "Bohr / quantum model of the atom", "shelf": "theories", "surface": "secular", "snippet": "Bohr / quantum model of the atom — an engine domain that can touch it: atomic. Calibration: seals — energy levels and shell capacities compute. Electrons occupy discrete orbits with QUANTISED energies", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_saros_eclipse_cycle", "title": "The Saros cycle (eclipse prediction)", "shelf": "theories", "surface": "secular", "snippet": "The Saros cycle (eclipse prediction) — an engine domain that can touch it: ephemeris. Calibration: seals — a deterministic verifier can CONFIRM or BREAK checkable claims drawn from it, and seal them. ", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (lone-domain seeding)", "readable": false, "generated": false}, {"id": "card_theory_epidemiology_models", "title": "Compartmental epidemiology (SIR models and R₀)", "shelf": "theories", "surface": "secular", "snippet": "Compartmental epidemiology (SIR models and R₀) — an engine domain that can touch it: medicine. Calibration: seals — R0, herd-immunity thresholds and epidemic curves compute. Divide a population into S", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_haber_bosch", "title": "Haber–Bosch nitrogen fixation (bread from air)", "shelf": "theories", "surface": "secular", "snippet": "Haber–Bosch nitrogen fixation (bread from air) — an engine domain that can touch it: chemistry. Calibration: seals — stoichiometry, yield and equilibrium shifts compute. N₂ + 3H₂ ⇌ 2NH₃, run at roughl", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_floor_the_lagrangian", "title": "The Lagrangian — the Hamiltonian's dual, the path chosen", "shelf": "codex", "surface": "secular", "snippet": "L = T - V, and the action S = integral L dt. Its law is the principle of least action: of all paths, nature takes the one where the action is stationary. Four pillars rest on the floors already seeded", "authority_tier": "engine_derived", "source": "Narrow Highway — the Lagrangian", "readable": false, "generated": false}, {"id": "card_theory_fluid_mechanics", "title": "Fluid mechanics (Bernoulli, Reynolds, Navier–Stokes)", "shelf": "theories", "surface": "secular", "snippet": "Fluid mechanics (Bernoulli, Reynolds, Navier–Stokes) — an engine domain that can touch it: hydrology. Calibration: seals — continuity, Bernoulli and Reynolds number compute directly. CONTINUITY first:", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_normative_ethics", "title": "Normative ethics (consequentialism / deontology / virtue)", "shelf": "theories", "surface": "secular", "snippet": "Normative ethics (consequentialism / deontology / virtue) — an engine domain that can touch it: philosophy. Calibration: map-only — interpretive by nature, and the card says so plainly. Three families", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "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"}}