{"query": "The Riemann hypothesis", "count": 20, "results": [{"id": "card_question_riemann", "title": "The Riemann hypothesis", "shelf": "millennium", "surface": "secular", "snippet": "The Riemann hypothesis. Every non-trivial zero of the Riemann zeta function has real part one half. Its chart is the tick stick stick_riemann_hypothesis: what is sealed, what is cited and what stays o", "authority_tier": "reference", "source": "Clay Mathematics Institute, the Millennium Prize Problems (2000)", "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_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_n_ebe4c25c22ae", "title": "The Riemann hypothesis — sealed all around, refused at the center", "shelf": "science", "surface": "secular", "snippet": "The deepest open question about the primes, and the cleanest demonstration of the engine's\nhonesty. The FACTS seal: zeta(2)=pi²/6 (Basel), zeta(4)=pi⁴/90, zeta(6)=pi⁶/945, zeta(-1)=-1/12,\nthe trivial ", "authority_tier": "engine_derived", "source": "Concordance assay — 2026-07-09", "readable": false, "generated": false}, {"id": "card_theory_differential_geometry", "title": "Differential geometry (curvature measured from inside)", "shelf": "theories", "surface": "secular", "snippet": "Differential geometry (curvature measured from inside) — an engine domain that can touch it: geometry. Calibration: map-only — specific curvature and geodesic computations verify; the framework is bro", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_floor_riemann", "title": "The Riemann chain - from Euler's product and Legendre's count to the critical line", "shelf": "codex", "surface": "secular", "snippet": "Two trees. The analytic: Euler's product over the primes (1737), Dirichlet's L-functions (1837). The arithmetic: Legendre's guess at the prime count (1798), Gauss's logarithmic integral (counted 1792,", "authority_tier": "engine_derived", "source": "Narrow Highway - a chain on the one map (operator seed)", "readable": false, "generated": false}, {"id": "card_theory_non_euclidean_hyperbolic_elliptic_geometry", "title": "Non-Euclidean (hyperbolic / elliptic) geometry", "shelf": "theories", "surface": "secular", "snippet": "Non-Euclidean (hyperbolic / elliptic) geometry — an engine domain that can touch it: geometry. Calibration: partial — curvature relations verify; the choice of axioms is map-only. Deny Euclid's fifth ", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_measure_theory__lebesgue_integration", "title": "Measure theory (Lebesgue integration)", "shelf": "theories", "surface": "secular", "snippet": "Measure theory (Lebesgue integration) — an engine domain that can touch it: mathematics. Calibration: map-only — a foundation, not a computation. What it means for a set to have a SIZE, done carefully", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_alm_connection_riemann_hypothesis_open_zeta_facts", "title": "Almanac: The Riemann Hypothesis -- an open question about where the zeta function vanishes, with the established facts sealed and the conjecture marked honestly", "shelf": "almanac", "surface": "secular", "snippet": "SITUATION\n  The Riemann zeta function, zeta(s) = the sum of 1/k^s, is the deepest known bridge between the smooth world of analysis and the prime numbers. Several of its truths are firmly ESTABLISHED ", "authority_tier": "reference", "source": "The Almanac — verified-only practical wisdom (sealed)", "readable": false, "generated": false}, {"id": "card_src_mill_platt_trudgian_2021", "title": "D. Platt 2021 — The Riemann hypothesis is true up to 3·10^12", "shelf": "millennium", "surface": "secular", "snippet": "D. Platt, T. Trudgian (2021). The Riemann hypothesis is true up to 3·10^12. Bull. London Math. Soc. 53 (2021) 792–797. DOI 10.1112/blms.12460. arXiv: 2004.09765. Canonical: https://doi.org/10.1112/blm", "authority_tier": "reference", "source": "D. Platt, T. Trudgian (2021), Bull. London Math. Soc. 53 (2021) 792–797", "readable": false, "generated": false}, {"id": "card_chart_weil_rh", "title": "The finite-field (Weil) Riemann hypothesis", "shelf": "codex", "surface": "secular", "snippet": "The finite-field (Weil) Riemann hypothesis. A chart on the one map: this is the tick stick stick_finite_field_the_weil_riemann_hypothesis, placed at the node it charts. Basis: the stick seals |alpha|^", "authority_tier": "engine_derived", "source": "Narrow Highway - a chart: a tick stick placed on the one map", "readable": false, "generated": false}, {"id": "card_src_mill_odlyzko_zeros_tables", "title": "A. M. Odlyzko 1988 — Tables of zeros of the Riemann zeta function", "shelf": "millennium", "surface": "secular", "snippet": "A. M. Odlyzko (1988-2001). Tables of zeros of the Riemann zeta function. University of Minnesota (zeros1: the first 100,000 zeros, accurate to 3e-9; zeros2: the first 100 to over 1000 places; zeros3/4", "authority_tier": "reference", "source": "A. M. Odlyzko (1988-2001), University of Minnesota (zeros1: the first 100,000 zeros, accurate to 3e-9; zeros2: the first 100 to over 1000 places; z", "readable": false, "generated": false}, {"id": "card_src_mill_odlyzko_schonhage_1988", "title": "A. M. Odlyzko 1988 — Fast algorithms for multiple evaluations of the Riemann zeta function", "shelf": "millennium", "surface": "secular", "snippet": "A. M. Odlyzko, A. Schönhage (1988). Fast algorithms for multiple evaluations of the Riemann zeta function. Trans. Amer. Math. Soc. 309 (1988) 797–809. DOI 10.1090/S0002-9947-1988-0961614-2. Canonical:", "authority_tier": "reference", "source": "A. M. Odlyzko, A. Schönhage (1988), Trans. Amer. Math. Soc. 309 (1988) 797–809", "readable": false, "generated": false}, {"id": "card_src_mill_keiper_1992", "title": "J. B. Keiper 1992 — Power series expansions of Riemann's xi function", "shelf": "millennium", "surface": "secular", "snippet": "J. B. Keiper (1992). Power series expansions of Riemann's xi function. Math. Comp. 58 (1992) 765–773. DOI 10.1090/S0025-5718-1992-1122072-5. Canonical: https://doi.org/10.1090/S0025-5718-1992-1122072-", "authority_tier": "reference", "source": "J. B. Keiper (1992), Math. Comp. 58 (1992) 765–773", "readable": false, "generated": false}, {"id": "card_src_mill_maslanka_2004", "title": "K. Maślanka 2004 — Effective method of computing Li's coefficients and their properties", "shelf": "millennium", "surface": "secular", "snippet": "K. Maślanka (2004). Effective method of computing Li's coefficients and their properties. arXiv (2004). arXiv: math/0402168. Canonical: https://arxiv.org/abs/math/0402168. Free copy: https://arxiv.org", "authority_tier": "reference", "source": "K. Maślanka (2004), arXiv (2004)", "readable": false, "generated": false}, {"id": "card_c_9aaf105f7dd8", "title": "Primes and zeta — Euler's bridge and the pri ↔ The Riemann hypothesis — sealed all around, ", "shelf": "connections", "surface": null, "snippet": "the Riemann zeros control exactly how the actual count wobbles around the smooth estimate  — a concord the card itself states; mined + verified.", "authority_tier": "engine_derived", "source": "Concordance miner — 2026-07-11", "readable": false, "generated": false}, {"id": "card_c_25f0eae28355", "title": "The primes and the nucleus — one statistical ↔ The Riemann hypothesis — sealed all around, ", "shelf": "connections", "surface": null, "snippet": "the SPACINGS between the Riemann zeta zeros follow the same distribution as the eigenvalue spacings of large random Hermitian matrices (the Gaussian Unitary Ensemble)  — a concord the card itself stat", "authority_tier": "engine_derived", "source": "Concordance miner — 2026-07-11", "readable": false, "generated": false}, {"id": "card_c_d721087fe2f2", "title": "The Riemann hypothesis — sealed all around,  ↔ Primes and zeta — Euler's bridge and the pri", "shelf": "connections", "surface": null, "snippet": "zeta(2)=pi²/6 (Basel), zeta(4)=pi⁴/90, zeta(6)=pi⁶/945, zeta(-1)=-1/12, the trivial zero zeta(-2)=0, and Euler's bridge zeta(2)·6 = pi².  — a concord the card itself states; mined + verified.", "authority_tier": "engine_derived", "source": "Concordance miner — 2026-07-11", "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_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}], "house": {"door": "FIND", "kind": "cards", "trail": "results", "seal": null, "next_step": {"do": "open the top card", "door": "FIND", "tool": "card_get", "params": {"id": "card_question_riemann"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}