{"query": "The Millennium floor - seven open questions, and where they", "count": 20, "results": [{"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_computational_complexity__p__np__reductions", "title": "Computational complexity (P, NP, reductions)", "shelf": "theories", "surface": "secular", "snippet": "Computational complexity (P, NP, reductions) — an engine domain that can touch it: computer_science. Calibration: map-only — P vs NP is open, and this card says so. Not whether a problem can be solved", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_floor_millennium", "title": "The Millennium floor - seven open questions, and where they connect", "shelf": "codex", "surface": "secular", "snippet": "The seven Millennium Prize Problems on the one map of reality. The FLOOR is what is proven or observed: the joints where two of the questions meet at one established thing - the GUE statistics shared ", "authority_tier": "engine_derived", "source": "Narrow Highway - the Millennium floor (operator seed)", "readable": false, "generated": false}, {"id": "card_n_dcb1a67c26e8", "title": "The evolution of language — descent traced by regular law", "shelf": "science", "surface": "secular", "snippet": "Languages descend like living lineages, and the descent is RECOVERABLE — the comparative\nmethod is one of the humanities' deterministic sciences. English 'father' descends from\nProto-Indo-European *ph", "authority_tier": "engine_derived", "source": "Concordance assay — 2026-07-09", "readable": false, "generated": false}, {"id": "card_joint_renormalization_group", "title": "The renormalization group, and the continuum limit as the question", "shelf": "codex", "surface": "secular", "snippet": "Wilson's lattice (1974) and Forster, Nelson and Stephen's randomly stirred fluid (1977) run the same renormalization group: a coupling that changes with scale, a finite system whose continuum limit is", "authority_tier": "engine_derived", "source": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "readable": false, "generated": false}, {"id": "card_joint_arnold_geodesics", "title": "Euler flow is geodesic flow on the volume-preserving diffeomorphisms", "shelf": "codex", "surface": "secular", "snippet": "Arnold (1966): the Euler equations of an ideal fluid are the geodesic equations of the group of volume-preserving diffeomorphisms with the kinetic-energy metric - the fluid as a point moving on an inf", "authority_tier": "engine_derived", "source": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "readable": false, "generated": false}, {"id": "card_src_mill_forster_nelson_stephen_1977", "title": "D. Forster 1977 — Large-distance and long-time properties of a randomly stirred fluid", "shelf": "millennium", "surface": "secular", "snippet": "D. Forster, D. R. Nelson, M. J. Stephen (1977). Large-distance and long-time properties of a randomly stirred fluid. Phys. Rev. A 16 (1977) 732–749. DOI 10.1103/PhysRevA.16.732. Canonical: https://doi", "authority_tier": "reference", "source": "D. Forster, D. R. Nelson, M. J. Stephen (1977), Phys. Rev. A 16 (1977) 732–749", "readable": false, "generated": false}, {"id": "card_n_36048aaa9a1b", "title": "Glottochronology — carbon-dating the words", "shelf": "science", "surface": "secular", "snippet": "Swadesh (1950s): a language's CORE vocabulary — the 100-200 most basic words (mother,\nwater, two, hand) — is replaced at a roughly constant rate, so its loss is EXPONENTIAL:\nretention r^t with r about", "authority_tier": "engine_derived", "source": "Concordance assay — 2026-07-09", "readable": false, "generated": false}, {"id": "card_question_p_vs_np", "title": "P versus NP", "shelf": "millennium", "surface": "secular", "snippet": "P versus NP. Is every problem whose solution can be checked in polynomial time also solvable in polynomial time? Its chart is the tick stick stick_p_versus_np: what is sealed, what is cited and what s", "authority_tier": "reference", "source": "Clay Mathematics Institute, the Millennium Prize Problems (2000)", "readable": false, "generated": false}, {"id": "card_joint_manin_algorithm", "title": "If Sha is finite, the rank is computable", "shelf": "codex", "surface": "secular", "snippet": "Manin (1971): if the Tate-Shafarevich group is finite - part of what BSD asserts - then the rank of an elliptic curve over Q is effectively computable by descent. Without it, no algorithm is known. A ", "authority_tier": "engine_derived", "source": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "readable": false, "generated": false}, {"id": "card_question_poincare", "title": "The Poincare conjecture", "shelf": "millennium", "surface": "secular", "snippet": "The Poincare conjecture. Every simply connected closed three-manifold is homeomorphic to the three-sphere. Its chart is the tick stick stick_poincare_conjecture: what is sealed, what is cited and what", "authority_tier": "reference", "source": "Clay Mathematics Institute, the Millennium Prize Problems (2000)", "readable": false, "generated": false}, {"id": "card_joint_diophantine_rh", "title": "The Riemann hypothesis is one Diophantine equation with no solutions", "shelf": "codex", "surface": "secular", "snippet": "Davis, Matiyasevich and Robinson (1976): the Riemann hypothesis is equivalent to a specific polynomial Diophantine equation having no solutions in the integers. The hypothesis is a Pi-1 statement; its", "authority_tier": "engine_derived", "source": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "readable": false, "generated": false}, {"id": "card_src_mill_tate_1965", "title": "J. Tate 1965 — Algebraic cycles and poles of zeta functions", "shelf": "millennium", "surface": "secular", "snippet": "J. Tate (1965). Algebraic cycles and poles of zeta functions. Arithmetical Algebraic Geometry (Purdue 1963), Harper & Row, 1965, 93–110. Canonical: https://www.claymath.org/millennium/hodge-conjecture", "authority_tier": "reference", "source": "J. Tate (1965), Arithmetical Algebraic Geometry (Purdue 1963), Harper & Row, 1965, 93–110", "readable": false, "generated": false}, {"id": "card_joint_sign_problem", "title": "The lattice sign problem is NP-hard", "shelf": "codex", "surface": "secular", "snippet": "Troyer and Wiese (2005): the fermion sign problem of quantum Monte Carlo - the obstacle to simulating lattice gauge theory at finite density - is NP-hard. A proof that P = NP would remove it; its hard", "authority_tier": "engine_derived", "source": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "readable": false, "generated": false}, {"id": "card_joint_bochner_vanishing", "title": "Positive Ricci curvature kills harmonic one-forms", "shelf": "codex", "surface": "secular", "snippet": "Bochner (1946): a closed manifold with positive Ricci curvature carries no non-zero harmonic one-form, so its first Betti number is zero - a Hodge-theoretic conclusion from the very hypothesis Hamilto", "authority_tier": "engine_derived", "source": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "readable": false, "generated": false}, {"id": "card_joint_tate_conjecture", "title": "The Tate conjecture: Hodge's arithmetic twin, and BSD over function fields", "shelf": "codex", "surface": "secular", "snippet": "Tate's conjecture (1965) is the Hodge conjecture with Galois representations in place of Hodge structures. For an elliptic surface over a finite field, the Tate conjecture is equivalent to the Birch a", "authority_tier": "engine_derived", "source": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "readable": false, "generated": false}, {"id": "card_joint_l_functions", "title": "One machinery: Euler product, functional equation, critical line", "shelf": "codex", "surface": "secular", "snippet": "Zeta is the simplest L-function; L(E,s) of an elliptic curve is another, given its analytic continuation by modularity (Wiles 1995). Both have an Euler product, a functional equation about a centre, a", "authority_tier": "engine_derived", "source": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "readable": false, "generated": false}, {"id": "card_joint_gue_statistics", "title": "Zeta zeros and the lattice Dirac operator share GUE statistics", "shelf": "codex", "surface": "secular", "snippet": "The nearest-neighbour spacings of the zeros of zeta follow the Gaussian Unitary Ensemble of random matrix theory (Montgomery's pair correlation, 1973; Odlyzko's computation, 1987), and so does the low", "authority_tier": "engine_derived", "source": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "readable": false, "generated": false}, {"id": "card_floor_poincare", "title": "The Poincare chain - from the homology sphere and the heat flow to the closed question", "shelf": "codex", "surface": "secular", "snippet": "Two trees. Topology: Poincaré's homology sphere and the question (1904), Thurston's geometrization (1982). Geometric analysis: the heat-flow method of Eells and Sampson (1964), Hamilton's Ricci flow (", "authority_tier": "engine_derived", "source": "Narrow Highway - a chain on the one map (operator seed)", "readable": false, "generated": false}, {"id": "card_src_mill_myers_1941", "title": "S. B. Myers 1941 — Riemannian manifolds with positive mean curvature", "shelf": "millennium", "surface": "secular", "snippet": "S. B. Myers (1941). Riemannian manifolds with positive mean curvature. Duke Math. J. 8 (1941) 401–404. DOI 10.1215/S0012-7094-41-00832-3. Canonical: https://doi.org/10.1215/S0012-7094-41-00832-3. No f", "authority_tier": "reference", "source": "S. B. Myers (1941), Duke Math. J. 8 (1941) 401–404", "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_fluid_mechanics"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}