{"query": "The Tate conjecture: Hodge's arithmetic twin, and BSD over 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_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_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_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_src_pron_hodge_s", "title": "hodge's", "shelf": "pronunciation", "surface": "secular", "snippet": "hodge's: pronounced (ARPABET) HH AA1 JH IH0 Z. From the CMU Pronouncing Dictionary — the standard machine-readable pronunciations of North American English.", "authority_tier": "reference", "source": "CMU Pronouncing Dictionary (cmudict) — BSD-2-Clause, Carnegie Mellon", "readable": false, "generated": false}, {"id": "card_theory_julian_day_number", "title": "Julian Day Number (continuous astronomical time)", "shelf": "theories", "surface": "secular", "snippet": "Julian Day Number (continuous astronomical time) — an engine domain that can touch it: calendar_time. Calibration: seals — a deterministic verifier can CONFIRM or BREAK checkable claims drawn from it,", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (lone-domain seeding)", "readable": false, "generated": false}, {"id": "card_theory_fundamental_theorem_of_arithmetic", "title": "Fundamental theorem of arithmetic (unique factorization)", "shelf": "theories", "surface": "secular", "snippet": "Fundamental theorem of arithmetic (unique factorization) — an engine domain that can touch it: number_theory. Calibration: seals. Every integer greater than 1 is a product of primes in exactly one way", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_peano_arithmetic_axioms", "title": "Peano arithmetic axioms", "shelf": "theories", "surface": "secular", "snippet": "Peano arithmetic axioms — an engine domain that can touch it: number_theory. Calibration: seals. Five axioms generating the natural numbers: zero is a number, every number has a successor, zero succee", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_digital_analog", "title": "Digital versus analog (why discreteness survives copying)", "shelf": "theories", "surface": "secular", "snippet": "Digital versus analog (why discreteness survives copying) — an engine domain that can touch it: computer_science. Calibration: map-only — sampling rates and bit depth are ordinary arithmetic, and THIS", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_honeycomb_conjecture", "title": "Optimal packing & the honeycomb conjecture (Hales)", "shelf": "theories", "surface": "secular", "snippet": "The hexagonal grid is the most efficient way to divide a surface into regions of equal area with the LEAST TOTAL PERIMETER. Stated by Pappus of Alexandria around 300 AD as an observation about bees, i", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_saros_eclipse_cycle", "title": "The Saros cycle (eclipse prediction)", "shelf": "theories", "surface": "secular", "snippet": "The Saros cycle (eclipse prediction) — an engine domain that can touch it: ephemeris. 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_molecular_symmetry_point_groups", "title": "Molecular symmetry & point groups", "shelf": "theories", "surface": "secular", "snippet": "Molecular symmetry & point groups — an engine domain that can touch it: molecular_geometry. Calibration: seals — a deterministic verifier can CONFIRM or BREAK checkable claims drawn from it, and seal ", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (lone-domain seeding)", "readable": false, "generated": false}, {"id": "card_theory_aufbau_configuration", "title": "Aufbau principle & electron configuration", "shelf": "theories", "surface": "secular", "snippet": "Aufbau principle & electron configuration — an engine domain that can touch it: periodic_table. Calibration: seals — a deterministic verifier can CONFIRM or BREAK checkable claims drawn from it, and s", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (lone-domain seeding)", "readable": false, "generated": false}, {"id": "card_theory_tuning_systems_comma", "title": "Tuning systems & the Pythagorean comma", "shelf": "theories", "surface": "secular", "snippet": "Tuning systems & the Pythagorean comma — an engine domain that can touch it: music_theory. Calibration: seals — a deterministic verifier can CONFIRM or BREAK checkable claims drawn from it, and seal t", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (lone-domain seeding)", "readable": false, "generated": false}, {"id": "card_theory_singular_value_decomposition", "title": "Singular value decomposition (SVD)", "shelf": "theories", "surface": "secular", "snippet": "Singular value decomposition (SVD) — an engine domain that can touch it: linear_algebra. Calibration: seals — a deterministic verifier can CONFIRM or BREAK checkable claims drawn from it, and seal the", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (lone-domain seeding)", "readable": false, "generated": false}, {"id": "card_theory_integers_counting", "title": "Counting & the integers (the first abstraction)", "shelf": "theories", "surface": "secular", "snippet": "Counting & the integers (the first abstraction) — an engine domain that can touch it: number_theory. Calibration: seals — arithmetic on the integers verifies exactly. Counting is the recognition that ", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_projective_geometry", "title": "Projective geometry (perspective & the cross-ratio)", "shelf": "theories", "surface": "secular", "snippet": "Projective geometry (perspective & the cross-ratio) — an engine domain that can touch it: geometry. Calibration: seals — a deterministic verifier can CONFIRM or BREAK checkable claims drawn from it, a", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (lone-domain seeding)", "readable": false, "generated": false}, {"id": "card_theory_depth_of_field", "title": "Depth of field & hyperfocal distance", "shelf": "theories", "surface": "secular", "snippet": "Depth of field & hyperfocal distance — an engine domain that can touch it: photography. 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_income_capitalization", "title": "Income capitalization (V = NOI / cap rate)", "shelf": "theories", "surface": "secular", "snippet": "Income capitalization (V = NOI / cap rate) — an engine domain that can touch it: real_estate. Calibration: seals — a deterministic verifier can CONFIRM or BREAK checkable claims drawn from it, and sea", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (lone-domain seeding)", "readable": false, "generated": false}, {"id": "card_theory_platonic_solids", "title": "The five Platonic solids", "shelf": "theories", "surface": "secular", "snippet": "The five Platonic solids — an engine domain that can touch it: geometry. Calibration: seals — a deterministic verifier can CONFIRM or BREAK checkable claims drawn from it, and seal them. There are EXA", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (lone-domain seeding)", "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"}}