{"query": "cos(π) = −1 · sin(π) = 0 · e^{iπ}   1 = 0", "count": 20, "results": [{"id": "card_theory_electromagnetic_spectrum", "title": "The electromagnetic spectrum & modulation (radio to gamma, one axis)", "shelf": "theories", "surface": "secular", "snippet": "The electromagnetic spectrum & modulation (radio to gamma, one axis) — an engine domain that can touch it: physics. Calibration: seals — c = fλ, photon energy E = hf, and bandwidth relations all recom", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "generated": false}, {"id": "card_v_23b7ec8074a4", "title": "sin²x + cos²x = 1 · cos 2x = 1 − 2 sin²x", "shelf": "mathematics", "surface": "secular", "snippet": "Verdict: HOLDS.\n• sin(x)**2+cos(x)**2 = 1; the two sides are equal -- each reduces to 1, so their difference simplifies to 0\n• cos(2*x) = 1-2*sin(x)**2; the two sides are equal -- each reduces to cos(", "authority_tier": "engine", "source": "Verified by the engine — mathematics", "generated": false}, {"id": "card_src_lex_g2972", "title": "G2972 — Κώς (Kōs): Cos", "shelf": "lexicon", "surface": "secular", "snippet": "Κώς (Kōs), Strong's G2972: Cos. Κῶς genitive Κῶ, ἡ, Cos, an island in the Ægean Sea : accusative Κῶ, Act.21:1 (Κῶν, Rec.).† (AS)", "authority_tier": "reference", "source": "STEPBible TBESH/TBESG — Extended Strong's lexicon (CC-BY, Tyndale House)", "generated": false}, {"id": "card_src_word_cos", "title": "cos", "shelf": "dictionary", "surface": "secular", "snippet": "cos: (noun) ratio of the adjacent side to the hypotenuse of a right-angled triangle — syn: cosine · (noun) lettuce with long dark-green leaves in a loosely packed elongated head — syn: cos lettuce, ro", "authority_tier": "reference", "source": "WordNet 3.0, Princeton University (WordNet License)", "generated": false}, {"id": "card_src_word_cos_lettuce", "title": "cos lettuce", "shelf": "dictionary", "surface": "secular", "snippet": "cos lettuce: (noun) lettuce with long dark-green spoon-shaped leaves — syn: romaine lettuce, Lactuca sativa longifolia · (noun) lettuce with long dark-green leaves in a loosely packed elongated head —", "authority_tier": "reference", "source": "WordNet 3.0, Princeton University (WordNet License)", "generated": false}, {"id": "card_seal_24fa5d4610e1fe23b3388e651f12ea3ef004d1121c0ff32ec6c2e0727d65c4f7", "title": "Receipt 24fa5d4610e1… — 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\": \"b\",\n    \"confirmed_steps\": 0,\n    \"error_", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "generated": false}, {"id": "card_src_food_169247", "title": "Lettuce, Cos Or Romaine, Raw", "shelf": "nutrition", "surface": "secular", "snippet": "Lettuce, cos or romaine, raw — a food in the USDA FoodData Central reference. Per 100 g: 17 kcal, 1.23 g protein, 0.3 g fat, 3.29 g carbohydrate.", "authority_tier": "reference", "source": "USDA FoodData Central (public domain)", "generated": false}, {"id": "card_alm_connection_shape_and_number_are_one", "title": "Almanac: Shape and number are one -- coordinates turn every figure into arithmetic and back, with the right triangle as the master key", "shelf": "almanac", "surface": "secular", "snippet": "SITUATION\n  Capstone of the geometry cluster. Lay a grid on space and shape becomes number: a point is its coordinates, a length is a calculation, an angle is a ratio -- and the sealed cards of this c", "authority_tier": "reference", "source": "The Almanac — verified-only practical wisdom (sealed)", "generated": false}, {"id": "card_v_3f924f5023bc", "title": "cos(π) = −1 · sin(π) = 0 · e^{iπ} + 1 = 0", "shelf": "mathematics", "surface": "secular", "snippet": "Verdict: HOLDS.\n• cos(pi) = -1; both sides reduce to -1\n• sin(pi) = 0; both sides reduce to 0\n• exp(I*pi)+1 = 0; both sides reduce to 0\nSealed and independently re-checkable: https://narrowhighway.com", "authority_tier": "engine", "source": "Verified by the engine — mathematics", "generated": false}, {"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_src_word_coss", "title": "coss", "shelf": "dictionary", "surface": "secular", "snippet": "coss: (noun) (in India) a unit of length having different values in different localities — syn: kos", "authority_tier": "reference", "source": "WordNet 3.0, Princeton University (WordNet License)", "generated": false}, {"id": "card_src_book_29916", "title": "Gómez Arias Or, The Moors of the Alpujarras, A Spanish Historical Romance. — Joaquín Telesforo de Trueba y Cosío", "shelf": "gutenberg", "surface": "secular", "snippet": "Gómez Arias Or, The Moors of the Alpujarras, A Spanish Historical Romance., by Joaquín Telesforo de Trueba y Cosío. Subjects: Historical fiction. Read the full text (public domain): https://www.gutenb", "authority_tier": "reference", "source": "Project Gutenberg (public domain)", "generated": false}, {"id": "card_src_book_54070", "title": "Il nuovo lume delle gioie Opera utilissima ad'ogni persona, così a orefici, e speciali, per l'arte pratico. — Pietro Caliari", "shelf": "gutenberg", "surface": "secular", "snippet": "Il nuovo lume delle gioie Opera utilissima ad'ogni persona, così a orefici, e speciali, per l'arte pratico., by Pietro Caliari (in it). Subjects: Gems. Read the full text (public domain): https://www.", "authority_tier": "reference", "source": "Project Gutenberg (public domain)", "generated": false}, {"id": "card_src_place_4831969", "title": "Cos Cob", "shelf": "geography", "surface": "secular", "snippet": "Cos Cob: a place in US, CT at 41.033, -73.600, population 6,770. Feature PPL, timezone America/New_York.", "authority_tier": "reference", "source": "GeoNames (CC-BY 4.0)", "generated": false}, {"id": "card_src_place_3980192", "title": "Villa de Cos", "shelf": "geography", "surface": "secular", "snippet": "Villa de Cos: a place in MX, 32 at 23.294, -102.347, population 5,759. Feature PPLA2, timezone America/Mexico_City.", "authority_tier": "reference", "source": "GeoNames (CC-BY 4.0)", "generated": false}, {"id": "card_alm_connection_tide_machine_is_fourier_synthesis", "title": "Almanac: The tide-predicting machine sums waves with gears -- Fourier synthesis made of brass, foretelling the sea", "shelf": "almanac", "surface": "secular", "snippet": "SITUATION\n  In 1872 Lord Kelvin built a machine of pulleys, wires, and gears that could predict the tides at any harbor for years ahead. It worked by Fourier SYNTHESIS: the tide is a sum of simple cos", "authority_tier": "reference", "source": "The Almanac — verified-only practical wisdom (sealed)", "generated": false}, {"id": "card_works_euler_identity", "title": "Euler's identity, assembled from its parts", "shelf": "the-works", "surface": "secular", "snippet": "Often called the most beautiful equation in mathematics: it binds e, i, π, 1 and 0 in a single line. We hand the engine each piece and the whole, and it confirms every one.  Worked & sealed by the eng", "authority_tier": "verified", "source": "The Works — worked & sealed", "generated": false}, {"id": "card_seal_ddfa6143b32488e402cfe516a0e25e8171307f4da76fb1e93384b233ae86ee20", "title": "Receipt ddfa6143b324… — 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", "generated": false}, {"id": "card_src_pron_coss", "title": "coss", "shelf": "pronunciation", "surface": "secular", "snippet": "", "authority_tier": "", "source": "", "generated": false}, {"id": "card_isbe_cos", "title": "ISBE: Cos", "shelf": "encyclopedia", "surface": "witness", "snippet": "", "authority_tier": "", "source": "", "generated": false}]}