{"query": "The lattice sign problem is NP-hard", "count": 20, "results": [{"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_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_src_mill_karp_1972", "title": "R. M. Karp 1972 — Reducibility among combinatorial problems", "shelf": "millennium", "surface": "secular", "snippet": "R. M. Karp (1972). Reducibility among combinatorial problems. Complexity of Computer Computations (Plenum, 1972) 85–103. DOI 10.1007/978-1-4684-2001-2_9. Canonical: https://doi.org/10.1007/978-1-4684-", "authority_tier": "reference", "source": "R. M. Karp (1972), Complexity of Computer Computations (Plenum, 1972) 85–103", "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_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_cs_computability", "title": "Computability and complexity — Turing to P vs NP", "shelf": "codex", "surface": "secular", "snippet": "A Turing machine defines what is computable; the universal machine runs any program (the engine, the laptop, the brain's computation). Some problems are undecidable (the halting problem); among the de", "authority_tier": "engine_derived", "source": "Narrow Highway — computer science", "readable": false, "generated": false}, {"id": "card_floor_computer_science", "title": "Computer science — bits, logic, and what can be computed", "shelf": "codex", "surface": "secular", "snippet": "Information is bits (a byte is 2^8 = 256 values); logic gates build from the 16 Boolean functions of two inputs; good algorithms beat bad ones (merge sort's n log n against n^2); and computability and", "authority_tier": "engine_derived", "source": "Narrow Highway — computer science", "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_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_src_mill_mulmuley_2011", "title": "K. D. Mulmuley 2011 — On P vs. NP and geometric complexity theory", "shelf": "millennium", "surface": "secular", "snippet": "K. D. Mulmuley (2011). On P vs. NP and geometric complexity theory. J. ACM 58 (2011) 5:1–26. DOI 10.1145/1944345.1944346. arXiv: 0908.1936. Canonical: https://doi.org/10.1145/1944345.1944346. Free cop", "authority_tier": "reference", "source": "K. D. Mulmuley (2011), J. ACM 58 (2011) 5:1–26", "readable": false, "generated": false}, {"id": "card_src_mill_deligne_1974", "title": "P. Deligne 1974 — La conjecture de Weil. I", "shelf": "millennium", "surface": "secular", "snippet": "P. Deligne (1974). La conjecture de Weil. I. Publ. Math. IHÉS 43 (1974) 273–307. DOI 10.1007/BF02684373. Canonical: https://doi.org/10.1007/BF02684373. Free copy: http://www.numdam.org/item/PMIHES_197", "authority_tier": "reference", "source": "P. Deligne (1974), Publ. Math. IHÉS 43 (1974) 273–307", "readable": false, "generated": false}, {"id": "card_cs_busy_beaver", "title": "The Busy Beaver function — BB(5) proved, BB(6) open", "shelf": "codex", "surface": "secular", "snippet": "BB(n), the Busy Beaver function (Rado 1962), is the most steps an n-state 2-symbol Turing machine can run before halting. It is UNCOMPUTABLE in general - a value of BB would decide the halting problem", "authority_tier": "engine_derived", "source": "Narrow Highway — computer science", "readable": false, "generated": false}, {"id": "card_src_mill_mulmuley_sohoni_2001", "title": "K. D. Mulmuley 2001 — Geometric complexity theory I: an approach to the P vs. NP and related problems", "shelf": "millennium", "surface": "secular", "snippet": "K. D. Mulmuley, M. Sohoni (2001). Geometric complexity theory I: an approach to the P vs. NP and related problems. SIAM J. Comput. 31 (2001) 496–526. DOI 10.1137/S009753970038715X. Canonical: https://", "authority_tier": "reference", "source": "K. D. Mulmuley, M. Sohoni (2001), SIAM J. Comput. 31 (2001) 496–526", "readable": false, "generated": false}, {"id": "card_floor_cryptography", "title": "Cryptography — public keys and hard problems", "shelf": "codex", "surface": "secular", "snippet": "Secrecy built on number theory and computational hardness. RSA encrypts with a public key and decrypts with a private one (the round-trip sealed), its security the belief that factoring is hard; the o", "authority_tier": "engine_derived", "source": "Narrow Highway — cryptography", "readable": false, "generated": false}, {"id": "card_src_mill_davis_matiyasevich_robinson_1976", "title": "M. Davis 1976 — Hilbert's tenth problem: Diophantine equations: positive aspects of a negative solution", "shelf": "millennium", "surface": "secular", "snippet": "M. Davis, Yu. Matiyasevich, J. Robinson (1976). Hilbert's tenth problem: Diophantine equations: positive aspects of a negative solution. Proc. Symp. Pure Math. 28 (AMS, 1976) 323–378. DOI 10.1090/pspu", "authority_tier": "reference", "source": "M. Davis, Yu. Matiyasevich, J. Robinson (1976), Proc. Symp. Pure Math. 28 (AMS, 1976) 323–378", "readable": false, "generated": false}, {"id": "card_src_mill_williams_2011", "title": "R. Williams 2014 — Nonuniform ACC circuit lower bounds", "shelf": "millennium", "surface": "secular", "snippet": "R. Williams (2014). Nonuniform ACC circuit lower bounds. J. ACM 61 (2014) 2:1–32 (conference version CCC 2011). DOI 10.1145/2559903. Canonical: https://doi.org/10.1145/2559903. Free copy: https://peop", "authority_tier": "reference", "source": "R. Williams (2014), J. ACM 61 (2014) 2:1–32 (conference version CCC 2011)", "readable": false, "generated": false}, {"id": "card_src_mill_blum_1984", "title": "N. Blum 1984 — A Boolean function requiring 3n network size", "shelf": "millennium", "surface": "secular", "snippet": "N. Blum (1984). A Boolean function requiring 3n network size. Theoret. Comput. Sci. 28 (1984) 337–345. DOI 10.1016/0304-3975(83)90029-4. Canonical: https://doi.org/10.1016/0304-3975(83)90029-4. Free c", "authority_tier": "reference", "source": "N. Blum (1984), Theoret. Comput. Sci. 28 (1984) 337–345", "readable": false, "generated": false}, {"id": "card_crypto_hardness", "title": "Hardness — security is a hard problem", "shelf": "codex", "surface": "secular", "snippet": "RSA is secure only because factoring n into its primes is believed hard; break that (or prove P = NP, or run Shor's quantum algorithm) and it falls. Security IS a hardness assumption. Rests on the P v", "authority_tier": "engine_derived", "source": "Narrow Highway — cryptography", "readable": false, "generated": false}, {"id": "card_src_mill_aaronson_2016", "title": "S. Aaronson 2016 — P =? NP", "shelf": "millennium", "surface": "secular", "snippet": "S. Aaronson (2016). P =? NP. Open Problems in Mathematics (Springer, 2016) 1–122. DOI 10.1007/978-3-319-32162-2_1. Canonical: https://doi.org/10.1007/978-3-319-32162-2_1. Free copy: https://www.scotta", "authority_tier": "reference", "source": "S. Aaronson (2016), Open Problems in Mathematics (Springer, 2016) 1–122", "readable": false, "generated": false}, {"id": "card_src_mill_manin_1971", "title": "Yu. I. Manin 1971 — Cyclotomic fields and modular curves", "shelf": "millennium", "surface": "secular", "snippet": "Yu. I. Manin (1971). Cyclotomic fields and modular curves. Russian Math. Surveys 26:6 (1971) 7–78. DOI 10.1070/RM1971v026n06ABEH001272. Canonical: https://doi.org/10.1070/RM1971v026n06ABEH001272. No f", "authority_tier": "reference", "source": "Yu. I. Manin (1971), Russian Math. Surveys 26:6 (1971) 7–78", "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_joint_sign_problem"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}