{"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)", "readable": false, "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", "readable": false, "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", "readable": false, "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)", "readable": false, "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)", "readable": false, "generated": false}, {"id": "card_calc_molecular_vibration", "title": "Molecular bond vibration", "shelf": "calculations", "surface": "secular", "snippet": "Molecular bond vibration — chemistry. Formula: omega = sqrt(k/mu). Canonical FORM: periodic (x = A*cos(omega*t + phi)) — a chemical bond is a spring — the harmonic oscillator with the reduced mass mu ", "authority_tier": "reference", "source": "The Calculation Map — every calculation, mapped by form", "readable": false, "generated": false}, {"id": "card_floor_standard_model", "title": "The floor of the Standard Model - where two trees meet", "shelf": "codex", "surface": "secular", "snippet": "The Standard Model of particle physics stands on two roots: Maxwell's electromagnetism (the electromagnetic tree) and Fermi's 1933 theory of the weak interaction (the weak tree), both on quantum mecha", "authority_tier": "engine_derived", "source": "Narrow Highway - the Standard-Model chain (operator seed)", "readable": false, "generated": false}, {"id": "card_calc_v_ephemeris_moon_phase", "title": "Moon phase", "shelf": "calculations", "surface": "secular", "snippet": "Moon phase — ephemeris. Formula: phase = age mod 29.53 d. Canonical FORM: periodic (x = A*cos(omega*t + phi)) — engine verifier ephemeris.moon_phase — the deterministic check the concordance runs. Sam", "authority_tier": "reference", "source": "The Calculation Map — every calculation, mapped by form", "readable": false, "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", "readable": false, "generated": false}, {"id": "card_calc_v_ephemeris_sunrise_sunset", "title": "Sunrise sunset", "shelf": "calculations", "surface": "secular", "snippet": "Sunrise sunset — ephemeris. Formula: cos(H) = -tan(lat) tan(dec). Canonical FORM: trigonometric (a sin(theta) = b sin(phi)) — engine verifier ephemeris.sunrise_sunset — the deterministic check the con", "authority_tier": "reference", "source": "The Calculation Map — every calculation, mapped by form", "readable": false, "generated": false}, {"id": "card_calc_v_ephemeris_equinox_solstice", "title": "Equinox solstice", "shelf": "calculations", "surface": "secular", "snippet": "Equinox solstice — ephemeris. Formula: solar longitude = 0,90,180,270. Canonical FORM: periodic (x = A*cos(omega*t + phi)) — engine verifier ephemeris.equinox_solstice — the deterministic check the co", "authority_tier": "reference", "source": "The Calculation Map — every calculation, mapped by form", "readable": false, "generated": false}, {"id": "card_bridge_form_periodic", "title": "Bridge: the periodic form across 6 domains", "shelf": "bridges", "surface": "secular", "snippet": "The canonical form periodic (x = A*cos(omega*t + phi)) is the SAME computation in 6 different domains — chemistry, condensed_matter, electrical, ephemeris, photonics, physics. One calculation, many fi", "authority_tier": "reference", "source": "The Bridges — cross-domain isomorphisms", "readable": false, "generated": false}, {"id": "card_calc_lattice_vibration", "title": "Lattice phonons", "shelf": "calculations", "surface": "secular", "snippet": "Lattice phonons — condensed_matter. Formula: omega(k) = 2 sqrt(K/m) |sin(ka/2)|. Canonical FORM: periodic (x = A*cos(omega*t + phi)) — a crystal's vibrations are a chain of coupled harmonic oscillator", "authority_tier": "reference", "source": "The Calculation Map — every calculation, mapped by form", "readable": false, "generated": false}, {"id": "card_calc_shm", "title": "Simple harmonic motion", "shelf": "calculations", "surface": "secular", "snippet": "Simple harmonic motion — physics. Formula: x = A cos(omega t + phi). Canonical FORM: periodic (x = A*cos(omega*t + phi)) — a restoring force proportional to displacement. Same form, different domain: ", "authority_tier": "reference", "source": "The Calculation Map — every calculation, mapped by form", "readable": false, "generated": false}, {"id": "card_calc_quantum_oscillator", "title": "Quantum harmonic oscillator", "shelf": "calculations", "surface": "secular", "snippet": "Quantum harmonic oscillator — physics. Formula: E_n = hbar omega (n + 1/2). Canonical FORM: periodic (x = A*cos(omega*t + phi)) — the oscillator quantized — evenly-spaced energy levels above a zero-po", "authority_tier": "reference", "source": "The Calculation Map — every calculation, mapped by form", "readable": false, "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", "readable": false, "generated": false}, {"id": "card_seal_01484e4a87bbc2e5872d248c03a50dd553e4dca8326a21ad4e8391e1cd5f7fd5", "title": "Receipt 01484e4a87bb… — 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_src_etym_coss", "title": "coss", "shelf": "etymology", "surface": "secular", "snippet": "coss: etymology (Webster 1913) — n.: [Cf. Pers. k a road measure of about two miles; or Skr. kr.]; n.: [It. cosa.]; [It. regola di cosa rule of thing, the unknown quantity being called the cosa, or th", "authority_tier": "reference", "source": "Webster's Revised Unabridged Dictionary (1913), Project Gutenberg eBook #29765 — public domain", "readable": true, "generated": false}, {"id": "card_src_etym_kiss", "title": "kiss", "shelf": "etymology", "surface": "secular", "snippet": "kiss: etymology (Webster 1913) — v. t.: [OE. kissen, cussen, AS. cyssan, fr. coss a kiss; of uncertain origin; akin to D. kus, G. kuss, Icel. koss.]; n.: [OE. kiss, derived under the influence of the ", "authority_tier": "reference", "source": "Webster's Revised Unabridged Dictionary (1913), Project Gutenberg eBook #29765 — public domain", "readable": true, "generated": false}, {"id": "card_seal_443ffdc38fbeb8b8f16ee717f3a3edd803c263598507453649e1c3e730d52297", "title": "Receipt 443ffdc38fbe… — 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}], "house": {"door": "FIND", "kind": "cards", "trail": "results", "seal": null, "next_step": {"do": "open the top card", "door": "FIND", "tool": "card_get", "params": {"id": "card_theory_electromagnetic_spectrum"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}