{"query": "From the unit circle to the double-angle law", "count": 20, "results": [{"id": "card_works_trig_identities", "title": "From the unit circle to the double-angle law", "shelf": "the-works", "surface": "secular", "snippet": "The Pythagorean identity is the unit circle written as algebra; the double-angle law for cosine follows in its train. Both are confirmed exactly (not sampled), for all x.  Worked & sealed by the engin", "authority_tier": "verified", "source": "The Works — worked & sealed", "generated": false}, {"id": "card_theory_non_euclidean_hyperbolic_elliptic_geometry", "title": "Non-Euclidean (hyperbolic / elliptic) geometry", "shelf": "theories", "surface": "secular", "snippet": "Non-Euclidean (hyperbolic / elliptic) geometry — an engine domain that can touch it: geometry. Calibration: partial — curvature relations verify; the choice of axioms is map-only. Deny Euclid's fifth ", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "generated": false}, {"id": "card_theory_map_projections", "title": "Map projections (why no flat map is honest)", "shelf": "theories", "surface": "secular", "snippet": "Map projections (why no flat map is honest) — an engine domain that can touch it: geography. Calibration: seals — projected coordinates and distortion factors compute exactly. NO FLAT MAP OF A SPHERE ", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "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)", "generated": false}, {"id": "card_theory_physical_oceanography", "title": "Physical oceanography (hydrostatics, tides)", "shelf": "theories", "surface": "secular", "snippet": "Physical oceanography (hydrostatics, tides) — an engine domain that can touch it: oceanography. Calibration: seals — pressure with depth and tidal periods compute. Pressure rises about one atmosphere ", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "generated": false}, {"id": "card_theory_differential_geometry", "title": "Differential geometry (curvature measured from inside)", "shelf": "theories", "surface": "secular", "snippet": "Differential geometry (curvature measured from inside) — an engine domain that can touch it: geometry. Calibration: map-only — specific curvature and geodesic computations verify; the framework is bro", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "generated": false}, {"id": "card_domchk_geometry_claimed_interior_angle_sum_deg", "title": "Geometry: interior angle sum deg", "shelf": "geometry", "surface": "secular", "snippet": "A worked check in geometry: interior angle sum deg.\n\nGIVEN\n  polygon_n = 6\n\nCLAIMED\n  claimed_interior_angle_sum_deg = 720\n\nTHE ENGINE'S VERDICT\n  geometry.polygon_interior_angle_sum: CONFIRMED — 6-go", "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)", "generated": false}, {"id": "card_domchk_molecular_geometry_claimed_bond_angle_deg", "title": "Molecular Geometry: bond angle deg", "shelf": "molecular_geometry", "surface": "secular", "snippet": "A worked check in molecular geometry: bond angle deg.\n\nGIVEN\n  bonding_domains = 4\n  lone_pairs = 0\n\nCLAIMED\n  claimed_bond_angle_deg = 109.47\n\nTHE ENGINE'S VERDICT\n  molecular_geometry.vsepr: CONFIRM", "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)", "generated": false}, {"id": "card_src_word_incidence_angle", "title": "incidence angle", "shelf": "dictionary", "surface": "secular", "snippet": "incidence angle: (noun) the angle that a line makes with a line perpendicular to the surface at the point of incidence — syn: angle of incidence", "authority_tier": "reference", "source": "WordNet 3.0, Princeton University (WordNet License)", "generated": false}, {"id": "card_src_word_angle_of_incidence", "title": "angle of incidence", "shelf": "dictionary", "surface": "secular", "snippet": "angle of incidence: (noun) the angle that a line makes with a line perpendicular to the surface at the point of incidence — syn: incidence angle", "authority_tier": "reference", "source": "WordNet 3.0, Princeton University (WordNet License)", "generated": false}, {"id": "card_src_word_extinction_angle", "title": "extinction angle", "shelf": "dictionary", "surface": "secular", "snippet": "extinction angle: (noun) the angle from its axis that a crystal must be rotated before appearing maximally dark when viewed in polarized light — syn: angle of extinction", "authority_tier": "reference", "source": "WordNet 3.0, Princeton University (WordNet License)", "generated": false}, {"id": "card_src_word_angle_of_extinction", "title": "angle of extinction", "shelf": "dictionary", "surface": "secular", "snippet": "angle of extinction: (noun) the angle from its axis that a crystal must be rotated before appearing maximally dark when viewed in polarized light — syn: extinction angle", "authority_tier": "reference", "source": "WordNet 3.0, Princeton University (WordNet License)", "generated": false}, {"id": "card_src_lex_g6118", "title": "G6118 — ἀγκών (agkōn): an angle", "shelf": "lexicon", "surface": "secular", "snippet": "ἀγκών (agkōn), Strong's G6118: an angle. 1. the bend of the arm, the elbow , (Homer) 2. generally the arm , like ἀγκάλη, (Pindar) 3. any bend , as the jutting angle of a wall, (Iliad by Homer): the be", "authority_tier": "reference", "source": "STEPBible TBESH/TBESG — Extended Strong's lexicon (CC-BY, Tyndale House)", "generated": false}, {"id": "card_src_word_dip_circle", "title": "dip circle", "shelf": "dictionary", "surface": "secular", "snippet": "dip circle: (noun) a measuring instrument for measuring the angle of magnetic dip (as from an airplane) — syn: inclinometer", "authority_tier": "reference", "source": "WordNet 3.0, Princeton University (WordNet License)", "generated": false}, {"id": "card_src_word_phase_angle", "title": "phase angle", "shelf": "dictionary", "surface": "secular", "snippet": "phase angle: (noun) a particular point in the time of a cycle; measured from some arbitrary zero and expressed as an angle — syn: phase", "authority_tier": "reference", "source": "WordNet 3.0, Princeton University (WordNet License)", "generated": false}, {"id": "card_src_word_angle_closure_glaucoma", "title": "angle-closure glaucoma", "shelf": "dictionary", "surface": "secular", "snippet": "angle-closure glaucoma: (noun) glaucoma in which the iris blocks the outflow of aqueous humor — syn: acute glaucoma, closed-angle glaucoma", "authority_tier": "reference", "source": "WordNet 3.0, Princeton University (WordNet License)", "generated": false}, {"id": "card_src_word_angle_of_attack", "title": "angle of attack", "shelf": "dictionary", "surface": "secular", "snippet": "angle of attack: (noun) the acute angle between the direction of the undisturbed relative wind and the chord of an airfoil", "authority_tier": "reference", "source": "WordNet 3.0, Princeton University (WordNet License)", "generated": false}, {"id": "card_src_word_angle_of_reflection", "title": "angle of reflection", "shelf": "dictionary", "surface": "secular", "snippet": "angle of reflection: (noun) the angle between a reflected ray and a line perpendicular to the reflecting surface at the point of incidence", "authority_tier": "reference", "source": "WordNet 3.0, Princeton University (WordNet License)", "generated": false}, {"id": "card_src_word_closed_angle_glaucoma", "title": "closed-angle glaucoma", "shelf": "dictionary", "surface": "secular", "snippet": "closed-angle glaucoma: (noun) glaucoma in which the iris blocks the outflow of aqueous humor — syn: acute glaucoma, angle-closure glaucoma", "authority_tier": "reference", "source": "WordNet 3.0, Princeton University (WordNet License)", "generated": false}, {"id": "card_src_lex_h4740", "title": "H4740 — מִקְצֹעַ (maq.tso.a): Angle", "shelf": "lexicon", "surface": "secular", "snippet": "מִקְצֹעַ (maq.tso.a), Strong's H4740: Angle. Angle, part of the fortifications of Jerusalem This name means a corner structure, buttress", "authority_tier": "reference", "source": "STEPBible TBESH/TBESG — Extended Strong's lexicon (CC-BY, Tyndale House)", "generated": false}]}