{"query": "The Poincare chain - from the homology sphere and the heat f", "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_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_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_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_src_mill_perelman_2003a", "title": "G. Perelman 2003 — Ricci flow with surgery on three-manifolds", "shelf": "millennium", "surface": "secular", "snippet": "G. Perelman (2003). Ricci flow with surgery on three-manifolds. arXiv (2003). arXiv: math/0303109. Canonical: https://arxiv.org/abs/math/0303109. Free copy: https://arxiv.org/abs/math/0303109. License", "authority_tier": "reference", "source": "G. Perelman (2003), arXiv (2003)", "readable": false, "generated": false}, {"id": "card_src_mill_kleiner_lott_2008", "title": "B. Kleiner 2008 — Notes on Perelman's papers", "shelf": "millennium", "surface": "secular", "snippet": "B. Kleiner, J. Lott (2008). Notes on Perelman's papers. Geom. Topol. 12 (2008) 2587–2855. DOI 10.2140/gt.2008.12.2587. arXiv: math/0605667. Canonical: https://doi.org/10.2140/gt.2008.12.2587. Free cop", "authority_tier": "reference", "source": "B. Kleiner, J. Lott (2008), Geom. Topol. 12 (2008) 2587–2855", "readable": false, "generated": false}, {"id": "card_src_mill_morgan_tian_2007", "title": "J. Morgan 2007 — Ricci Flow and the Poincaré Conjecture", "shelf": "millennium", "surface": "secular", "snippet": "J. Morgan, G. Tian (2007). Ricci Flow and the Poincaré Conjecture. Clay Mathematics Monographs 3, AMS, 2007. arXiv: math/0607607. Canonical: https://bookstore.ams.org/CMIM/3. Free copy: https://arxiv.", "authority_tier": "reference", "source": "J. Morgan, G. Tian (2007), Clay Mathematics Monographs 3, AMS, 2007", "readable": false, "generated": false}, {"id": "card_src_mill_perelman_2002", "title": "G. Perelman 2002 — The entropy formula for the Ricci flow and its geometric applications", "shelf": "millennium", "surface": "secular", "snippet": "G. Perelman (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv (2002). arXiv: math/0211159. Canonical: https://arxiv.org/abs/math/0211159. Free copy: https://arxiv.or", "authority_tier": "reference", "source": "G. Perelman (2002), arXiv (2002)", "readable": false, "generated": false}, {"id": "card_src_mill_perelman_2003b", "title": "G. Perelman 2003 — Finite extinction time for the solutions to the Ricci flow on certain three-manifolds", "shelf": "millennium", "surface": "secular", "snippet": "G. Perelman (2003). Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv (2003). arXiv: math/0307245. Canonical: https://arxiv.org/abs/math/0307245. Free copy: ", "authority_tier": "reference", "source": "G. Perelman (2003), arXiv (2003)", "readable": false, "generated": false}, {"id": "card_src_mill_cao_zhu_2006", "title": "H.-D. Cao 2006 — A complete proof of the Poincaré and geometrization conjectures — application of the Hamilton-Perelman theory of the Ricci flow", "shelf": "millennium", "surface": "secular", "snippet": "H.-D. Cao, X.-P. Zhu (2006). A complete proof of the Poincaré and geometrization conjectures — application of the Hamilton-Perelman theory of the Ricci flow. Asian J. Math. 10 (2006) 165–492. DOI 10.4", "authority_tier": "reference", "source": "H.-D. Cao, X.-P. Zhu (2006), Asian J. Math. 10 (2006) 165–492", "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_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_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_theory_heliocentrism_kepler_s_laws", "title": "Heliocentrism & Kepler's laws", "shelf": "theories", "surface": "secular", "snippet": "Heliocentrism & Kepler's laws — an engine domain that can touch it: astronomy. Calibration: seals — a deterministic verifier can CONFIRM or BREAK checkable claims drawn from it. Three laws, from Tycho", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_hydrologic_cycle_open_channel_flow", "title": "Hydrologic cycle & open-channel flow (Manning, Darcy)", "shelf": "theories", "surface": "secular", "snippet": "Hydrologic cycle & open-channel flow (Manning, Darcy) — an engine domain that can touch it: hydrology. Calibration: seals — Manning velocity, Darcy flow and catchment volumes compute. Water circulates", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_domchk_geometry_claimed_circle_area", "title": "Geometry: circle area", "shelf": "geometry", "surface": "secular", "snippet": "A worked check in geometry: circle area.\n\nGIVEN\n  circle_radius = 5\n  pyth_a = 3\n  pyth_b = 4\n  pyth_c = 5\n  tri_a = 3\n  tri_b = 4\n  tri_c = 5\n\nCLAIMED\n  claimed_circle_area = 78.5398\n\nTHE ENGINE'S VE", "authority_tier": "reference", "source": "The verifier's own documented relation, and a run this engine performed against it (deterministic; re-runnable with tools/domain_goldens.py)", "readable": false, "generated": false}, {"id": "card_domchk_geometry_claimed_circle_circumference", "title": "Geometry: circle circumference", "shelf": "geometry", "surface": "secular", "snippet": "A worked check in geometry: circle circumference.\n\nGIVEN\n  circle_radius = 5\n  pyth_a = 3\n  pyth_b = 4\n  pyth_c = 5\n  tri_a = 3\n  tri_b = 4\n  tri_c = 5\n\nCLAIMED\n  claimed_circle_circumference = 31.415", "authority_tier": "reference", "source": "The verifier's own documented relation, and a run this engine performed against it (deterministic; re-runnable with tools/domain_goldens.py)", "readable": false, "generated": false}, {"id": "card_src_word_radius", "title": "radius", "shelf": "dictionary", "surface": "secular", "snippet": "radius: (noun) the length of a line segment between the center and circumference of a circle or sphere — syn: r · (noun) a straight line from the center to the perimeter of a circle (or from the cente", "authority_tier": "reference", "source": "WordNet 3.0, Princeton University (WordNet License)", "readable": false, "generated": false}, {"id": "card_src_etym_radius", "title": "radius", "shelf": "etymology", "surface": "secular", "snippet": "radius: etymology (Webster 1913) — n.: [L., a staff, rod, spoke of a wheel, radius, ray. See Ray a divergent line.]. From Webster's Revised Unabridged Dictionary (1913), public domain.", "authority_tier": "reference", "source": "Webster's Revised Unabridged Dictionary (1913), Project Gutenberg eBook #29765 — public domain", "readable": true, "generated": false}, {"id": "card_works_spherical_tank", "title": "Sizing a spherical tank of radius 3", "shelf": "the-works", "surface": "secular", "snippet": "For a sphere of radius 3, the volume is (4/3)πr³ and the surface area is 4πr² — both come to 36π here, decided to full precision.  Worked & sealed by the engine — V = (4/3)π·3³ = 36π ≈ 113.097 ;  A = ", "authority_tier": "verified", "source": "The Works — worked & sealed", "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"}}