{"query": "d/dt (t³) = 3t²  (position → velocity) · d/dt (3t²) = 6t  (v", "count": 20, "results": [{"id": "card_theory_stochastic_processes", "title": "Stochastic processes & Ito calculus", "shelf": "theories", "surface": "secular", "snippet": "Stochastic processes & Ito calculus — an engine domain that can touch it: mathematics. 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_electromagnetic_induction", "title": "Electromagnetic induction (Faraday's law — how all electricity is made)", "shelf": "theories", "surface": "secular", "snippet": "Electromagnetic induction (Faraday's law — how all electricity is made) — an engine domain that can touch it: electrical. Calibration: seals — EMF, turns ratio and flux relations compute. A CHANGING m", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_v_99e6939d3390", "title": "d/dt (t³) = 3t²  (position → velocity) · d/dt (3t²) = 6t  (velocity → accelerati", "shelf": "mathematics", "surface": "secular", "snippet": "Verdict: HOLDS.\n• d/dt of t**3 = 3*t**2, matches 3*t**2\n• d/dt of 3*t**2 = 6*t, matches 6*t\nSealed and independently re-checkable: https://narrowhighway.com/s/aa6dbf0d3b9f9ae1a525d9b57efb2a932f9af344a", "authority_tier": "engine", "source": "Verified by the engine — mathematics", "readable": false, "generated": false}, {"id": "card_works_calculus_chain", "title": "Differentiate twice: position → velocity → acceleration", "shelf": "the-works", "surface": "secular", "snippet": "A body's position goes as t³ in this toy law. Differentiate once for velocity, again for acceleration. Each differentiation is machine-checked; the second stands on the first.  Worked & sealed by the ", "authority_tier": "verified", "source": "The Works — worked & sealed", "readable": false, "generated": false}, {"id": "card_n_8530c72a2201", "title": "Fluid probability dynamics — one continuity equation across physics, geometry, and ML", "shelf": "science", "surface": "secular", "snippet": "One equation wears many clothes. Probability is a conserved fluid: the continuity equation\ndρ/dt + ∇·J = 0 (current J = ρv) says density is never created or destroyed — it only flows. The\nsame skeleto", "authority_tier": "engine_derived", "source": "Concordance assay — 2026-07-08", "readable": false, "generated": false}, {"id": "card_maxwell_gauge_charge", "title": "U(1) gauge symmetry → conservation of charge", "shelf": "codex", "surface": "secular", "snippet": "Electromagnetism is the U(1) gauge theory, and by Noether its symmetry gives conservation of electric charge - the continuity equation d(rho)/dt + div J = 0 is built into Maxwell's equations. The symm", "authority_tier": "engine_derived", "source": "Narrow Highway — Maxwell's equations", "readable": false, "generated": false}, {"id": "card_n_041763374eca", "title": "Fokker–Planck & Liouville — probability flow in statistical mechanics", "shelf": "science", "surface": "secular", "snippet": "The stochastic instantiation of probability-as-fluid. Fokker–Planck sets the current\nJ = μρ − D∇ρ (drift + diffusion); its steady state is the Boltzmann distribution ρ ∝ e^(−U/kT),\nwhich makes the cur", "authority_tier": "engine_derived", "source": "Concordance assay — 2026-07-08", "readable": false, "generated": false}, {"id": "card_src_mill_hamilton_1982", "title": "R. S. Hamilton 1982 — Three-manifolds with positive Ricci curvature", "shelf": "millennium", "surface": "secular", "snippet": "R. S. Hamilton (1982). Three-manifolds with positive Ricci curvature. J. Differential Geom. 17 (1982) 255–306. DOI 10.4310/jdg/1214436922. Canonical: https://doi.org/10.4310/jdg/1214436922. Free copy:", "authority_tier": "reference", "source": "R. S. Hamilton (1982), J. Differential Geom. 17 (1982) 255–306", "readable": false, "generated": false}, {"id": "card_bridge_master_replicator", "title": "Master equation: dx_i/dt = x_i (f_i - <f>)", "shelf": "bridges", "surface": "secular", "snippet": "a type grows exactly when it beats the average — natural selection and game dynamics are one equation. ONE equation, 2 domains, connected by a change of variable: biology [f = Darwinian fitness; the m", "authority_tier": "reference", "source": "The Bridges — cross-domain isomorphisms", "readable": false, "generated": false}, {"id": "card_calc_em_wave", "title": "Electromagnetic waves", "shelf": "calculations", "surface": "secular", "snippet": "Electromagnetic waves — electrical. Formula: d2E/dt2 = c^2 laplacian(E). Canonical FORM: wave (d2u/dt2 = c^2 * laplacian(u)) — light is Maxwell's equations in wave form. Same form, different domain: t", "authority_tier": "reference", "source": "The Calculation Map — every calculation, mapped by form", "readable": false, "generated": false}, {"id": "card_bridge_master_harmonic_oscillator", "title": "Master equation: d2x/dt2 = -omega^2 x", "shelf": "bridges", "surface": "secular", "snippet": "a restoring force proportional to displacement — every oscillation is this, with omega^2 = restoring/inertia. ONE equation, 4 domains, connected by a change of variable: physics [omega^2 = k/m (spring", "authority_tier": "reference", "source": "The Bridges — cross-domain isomorphisms", "readable": false, "generated": false}, {"id": "card_bridge_master_wave_equation", "title": "Master equation: d2u/dt2 = c^2 * laplacian(u)", "shelf": "bridges", "surface": "secular", "snippet": "a disturbance propagating at a fixed speed — the parent of every wave. ONE equation, 6 domains, connected by a change of variable: electrical [u = E field, c = speed of light (from Maxwell)]; acoustic", "authority_tier": "reference", "source": "The Bridges — cross-domain isomorphisms", "readable": false, "generated": false}, {"id": "card_calc_fermi_dirac", "title": "Fermi-Dirac occupation", "shelf": "calculations", "surface": "secular", "snippet": "Fermi-Dirac occupation — thermodynamics. Formula: n = 1/(1 + e^((E-mu)/kT)). Canonical FORM: logistic (dP/dt = rP(1 - P/K)) — the fraction of quantum states filled — the sigmoid, from Pauli exclusion.", "authority_tier": "reference", "source": "The Calculation Map — every calculation, mapped by form", "readable": false, "generated": false}, {"id": "card_c_c1d4f07c75cd", "title": "Fokker–Planck & Liouville — probability flow ↔ Fluid probability dynamics — one continuity ", "shelf": "connections", "surface": null, "snippet": "The stochastic instantiation of probability-as-fluid. ... In Hamiltonian phase space, Liouville's theorem gives dρ/dt = 0 — the probability fluid is incompressible.  — a concord the card itself states", "authority_tier": "engine_derived", "source": "Concordance miner — 2026-07-11", "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_bridge_master_diffusion_equation", "title": "Master equation: du/dt = D * laplacian(u)", "shelf": "bridges", "surface": "secular", "snippet": "spreading by random walk — first in time, second in space. ONE equation, 8 domains, connected by a change of variable: thermodynamics [u = temperature, D = thermal diffusivity]; chemistry [u = concent", "authority_tier": "reference", "source": "The Bridges — cross-domain isomorphisms", "readable": false, "generated": false}, {"id": "card_calc_replicator_payoff", "title": "Replicator dynamics (game)", "shelf": "calculations", "surface": "secular", "snippet": "Replicator dynamics (game) — game_theory. Formula: dx/dt = x (payoff - avg). Canonical FORM: logistic (dP/dt = rP(1 - P/K)) — a strategy spreads when its payoff beats the average — the same equation, ", "authority_tier": "reference", "source": "The Calculation Map — every calculation, mapped by form", "readable": false, "generated": false}, {"id": "card_calc_carrying_capacity", "title": "Logistic population (carrying capacity)", "shelf": "calculations", "surface": "secular", "snippet": "Logistic population (carrying capacity) — ecology. Formula: dP/dt = rP(1 - P/K). Canonical FORM: logistic (dP/dt = rP(1 - P/K)) — growth that slows as it nears the limit. Same form, different domain: ", "authority_tier": "reference", "source": "The Calculation Map — every calculation, mapped by form", "readable": false, "generated": false}, {"id": "card_calc_titration_curve", "title": "Titration curve (Henderson-Hasselbalch)", "shelf": "calculations", "surface": "secular", "snippet": "Titration curve (Henderson-Hasselbalch) — chemistry. Formula: f = 1/(1 + 10^(pKa - pH)). Canonical FORM: logistic (dP/dt = rP(1 - P/K)) — the fraction deprotonated is a sigmoid in pH — a two-state occ", "authority_tier": "reference", "source": "The Calculation Map — every calculation, mapped by form", "readable": false, "generated": false}, {"id": "card_calc_v_ecology_logistic_growth", "title": "Logistic growth", "shelf": "calculations", "surface": "secular", "snippet": "Logistic growth — ecology. Formula: N = K/(1+((K-N0)/N0) e^-rt). Canonical FORM: logistic (dP/dt = rP(1 - P/K)) — engine verifier ecology.logistic_growth — the deterministic check the concordance runs", "authority_tier": "reference", "source": "The Calculation Map — every calculation, mapped by form", "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_stochastic_processes"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}