{"query": "The Millennium floor - seven open questions, and where they", "count": 20, "results": [{"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_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_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_spine_millennium_sources", "title": "The Millennium sources — the papers behind the seven sticks", "shelf": "spine", "surface": "secular", "snippet": "63 sources located for the seven Millennium sticks (Riemann, Birch and Swinnerton-Dyer, Navier-Stokes, Yang-Mills, P versus NP, Hodge, Poincare), one reference card each: the bibliographic record, DOI", "authority_tier": "reference", "source": "The Millennium sources — a spine of located records", "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_weil_conjectures", "title": "The Weil conjectures (Deligne 1974): the Riemann hypothesis over finite fields, a theorem", "shelf": "codex", "surface": "secular", "snippet": "The Weil conjectures are the analogue of the Riemann hypothesis for the zeta functions of varieties over FINITE FIELDS, and they are a theorem (Deligne 1974): every eigenvalue of Frobenius on the i-th", "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_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_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_floor_navier_stokes", "title": "The Navier-Stokes chain - from the ideal fluid and the viscous one to the regularity question", "shelf": "codex", "surface": "secular", "snippet": "Two trees. Euler's ideal fluid (1757) and Navier's viscous correction (1822) become one in Stokes' derivation from the stress of a continuum (1845): the equations as written today. Then Reynolds' numb", "authority_tier": "engine_derived", "source": "Narrow Highway - a chain on the one map (operator seed)", "readable": false, "generated": false}, {"id": "card_floor_logarithm", "title": "The logarithm - the instrument the joints share", "shelf": "codex", "surface": "secular", "snippet": "Two trees: Napier's table (1614) and Saint-Vincent's hyperbola (1647) - a number's logarithm as a tabulated count, and as an area. They become one function in Euler's Introductio (1748): e, and exp an", "authority_tier": "engine_derived", "source": "Narrow Highway - a chain on the one map (operator seed)", "readable": false, "generated": false}, {"id": "card_src_mill_bochner_1946", "title": "S. Bochner 1946 — Vector fields and Ricci curvature", "shelf": "millennium", "surface": "secular", "snippet": "S. Bochner (1946). Vector fields and Ricci curvature. Bull. Amer. Math. Soc. 52 (1946) 776–797. DOI 10.1090/S0002-9904-1946-08647-4. Canonical: https://doi.org/10.1090/S0002-9904-1946-08647-4. Free co", "authority_tier": "reference", "source": "S. Bochner (1946), Bull. Amer. Math. Soc. 52 (1946) 776–797", "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_src_mill_arnold_1966", "title": "V. I. Arnold 1966 — Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits", "shelf": "millennium", "surface": "secular", "snippet": "V. I. Arnold (1966). Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits. Ann. Inst. Fourier 16 (1966) 319–361. DOI 10.5", "authority_tier": "reference", "source": "V. I. Arnold (1966), Ann. Inst. Fourier 16 (1966) 319–361", "readable": false, "generated": false}, {"id": "card_floor_p_vs_np", "title": "The P versus NP chain - from the machine and the circuit to the three barriers", "shelf": "codex", "surface": "secular", "snippet": "Two trees. Computability: Turing's machine (1936), time as a resource (Hartmanis-Stearns 1965), feasible as polynomial in the input's length (Cobham 1965, Edmonds 1965). Circuits: Shannon's count of g", "authority_tier": "engine_derived", "source": "Narrow Highway - a chain on the one map (operator seed)", "readable": false, "generated": false}, {"id": "card_floor_yang_mills", "title": "The Yang-Mills chain - from the gauge field and the axioms to the mass gap", "shelf": "codex", "surface": "secular", "snippet": "Two trees. The gauge field: Maxwell, Weyl's gauge principle (1929), Yang and Mills' non-abelian field (1954). The axioms: Wightman's (1956) and Osterwalder-Schrader's Euclidean ones (1973) - what it m", "authority_tier": "engine_derived", "source": "Narrow Highway - a chain on the one map (operator seed)", "readable": false, "generated": false}, {"id": "card_floor_hodge", "title": "The Hodge chain - from periods and topology to the classes that are not cycles", "shelf": "codex", "surface": "secular", "snippet": "Two trees. The transcendental: Riemann's periods (1857). The topological: Lefschetz's analysis situs of a variety and the (1,1) theorem (1924), de Rham's forms (1931). They become one in Hodge's harmo", "authority_tier": "engine_derived", "source": "Narrow Highway - a chain on the one map (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_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_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}], "house": {"door": "FIND", "kind": "cards", "trail": "results", "seal": null, "next_step": {"do": "open the top card", "door": "FIND", "tool": "card_get", "params": {"id": "card_floor_millennium"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}