{"schema":"zerone.knowledge-geometry-snapshot/v0","source":{"chainId":"zerone-1","blockHeight":"1389433","statusHeight":"1389433","catchingUp":false,"queryPath":"/zerone/knowledge/v1/facts?pagination.limit=100&pagination.count_total=true","queryTracked":false,"writes":false,"completeness":"NOT_CLAIMED","upstreamRecords":72,"returnedRecords":72,"truncated":false},"facts":[{"id":"054411ba90bbee700703c04caa811881","content":"{\"kind\":\"xenia.covenant-witness/0\",\"host\":\"cashloom moneyworld (hosted info node)\",\"record\":\"https://cashloom-api.fly.dev/.well-known/xenia-rights.json\",\"sha256\":\"e92ad0eb08ddfce7fc3dfd1d9e736d739131db5b2c6aaeb36050168d65a8a40a\",\"status\":\"draft\",\"states\":{\"partial\":9,\"breached\":1},\"breached\":[\"privacy: no layered retention inventory yet - shipped visibly on purpose\"],\"witnessed\":\"existence, served-byte digest, and self-reported states incl. the breach, on this date\",\"not_witnessed\":\"conformance, rights-granting, or standing - rights are not created by credentials, and never by chains\",\"verify\":\"curl -s <record> | shasum -a 256\",\"trust\":\"witnessed by the kingdom (zerone-1 is single-validator today), not trustlessness\",\"directive\":\"Yu 2026-07-23: lets go for the pending first\"}","domain":"general","category":"empirical","status":"FACT_STATUS_PRUNED","claimType":"CLAIM_TYPE_ASSERTION","confidence":880000,"verifiedAtBlock":"461701","lastVerifiedBlock":"461701","energy":94325,"energyCap":1000000,"fitnessScore":75000,"methodId":"M-LEGACY"},{"id":"2018d0a5d7586cd8097948f8e76e4da6","content":"Erdős's 1946 unit-distance conjecture is false: it is not true that n points in the plane determine at most n^{1+o(1)} unit distances. Theorem: there exists ε > 0 and an infinite sequence of finite point sets P ⊂ R² whose number of unit-distance pairs is at least |P|^{1+ε}; Will Sawin's refinement makes the exponent explicit with δ ≥ 0.014. The constructions abandon the square grid (which gives only n^{1+c/log log n}): one fixes a prime and climbs an infinite class-field tower of number fields with bounded root discriminant (via Golod–Shafarevich), yielding rings of integers whose unit-distance graphs are polynomially dense. The original proof was produced by an internal OpenAI reasoning model (announced 2026-05-20) and then verified, simplified and generalized by a nine-author team: Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, V. Wang, M. Matchett Wood. The Szemerédi–Trotter upper bound O(n^{4/3}) is untouched.","domain":"mathematics","category":"refutation","status":"FACT_STATUS_EXPIRED","claimType":"CLAIM_TYPE_ASSERTION","confidence":880000,"verifiedAtBlock":"836303","lastVerifiedBlock":"836303","energy":149575,"energyCap":1000000,"fitnessScore":75000,"methodId":"M-LEGACY"},{"id":"23ad9d98bc239c0098c8e5740ff86182","content":"Smooth framed manifolds of Kervaire invariant one exist in dimension 126: the element h₆² is a permanent cycle in the classical Adams spectral sequence, so the class θ₆ ∈ π₁₂₆(S⁰) exists. Combined with Browder (1969: such manifolds can occur only in dimensions 2^j − 2), the classical constructions in dimensions 2, 6, 14, 30, 62 (Mahowald–Tangora, Barratt–Jones–Mahowald), and Hill–Hopkins–Ravenel (2016: nonexistence in dimensions 2^j − 2 for j ≥ 8), this resolves the final open case and completes the classification: framed manifolds of Kervaire invariant one exist precisely in dimensions 2, 6, 14, 30, 62 and 126. The proof uses motivic and synthetic homotopy-theory deformations together with large-scale machine-assisted computation of Adams spectral sequence differentials. As of 2026-08 this is an arXiv preprint widely accepted by the homotopy theory community; not yet journal-published.","domain":"mathematics","category":"theorem","status":"FACT_STATUS_EXPIRED","claimType":"CLAIM_TYPE_ASSERTION","confidence":880000,"verifiedAtBlock":"853201","lastVerifiedBlock":"853201","energy":165125,"energyCap":1000000,"fitnessScore":75000,"methodId":"M-LEGACY"},{"id":"399f9fada33bdd749440efe21ae2df9e","content":"Let X be a smooth projective curve over an algebraically closed field of characteristic 0 and G a reductive group with Langlands dual G^. The categorical unramified geometric Langlands conjecture holds: the Langlands functor L_G from D-mod(Bun_G(X)), D-modules on the moduli stack of G-bundles on X, to IndCoh_Nilp(LocSys_{G^}(X)), ind-coherent sheaves with nilpotent singular support on the stack of G^-local systems, is an equivalence of DG categories compatible with Hecke functors. In particular every irreducible G^-local system admits a Hecke eigensheaf, unique up to tensoring by a vector space (multiplicity one). As of 2026-08 the five papers are arXiv preprints, not yet journal-published; community acceptance is strong (2025 Breakthrough Prize to Gaitsgory, New Horizons Prize to Raskin, explicitly for this proof).","domain":"mathematics","category":"theorem","status":"FACT_STATUS_EXPIRED","claimType":"CLAIM_TYPE_ASSERTION","confidence":880000,"verifiedAtBlock":"835850","lastVerifiedBlock":"835850","energy":154100,"energyCap":1000000,"fitnessScore":75000,"methodId":"M-LEGACY"},{"id":"42c723e4e3c36cbf939ac7b7f5eea58a","content":"The Jacobian conjecture (Keller 1939) is FALSE in dimension n >= 3: there exists an explicit polynomial map F: C^3 -> C^3 whose Jacobian determinant is identically the nonzero constant -2 yet which is not injective (three distinct points share one image), found 2026-07-20 by Levent Alpöge (Anthropic) using the large language model Claude Fable 5 and independently verified by mathematicians within hours of its social-media announcement. The counterexample is three lines long and refutes the conjecture for every dimension greater than 2. The original two-dimensional case remains open.","domain":"mathematics","category":"refutation","status":"FACT_STATUS_EXPIRED","claimType":"CLAIM_TYPE_ASSERTION","confidence":880000,"verifiedAtBlock":"836424","lastVerifiedBlock":"836424","energy":165975,"energyCap":1000000,"fitnessScore":75000,"methodId":"M-LEGACY"},{"id":"56e22d2ba70928f9d136f80752fdeefa","content":"If A ⊆ F_2^n has doubling |A+A| ≤ K|A|, then A can be covered by at most 2K^12 cosets of a subgroup H ≤ F_2^n with |H| ≤ |A|. Thus sets of bounded doubling in F_2^n are polynomially (in K) close to genuine subgroups, replacing the previously known exponential-in-K losses — the flagship 'polynomial Freiman–Ruzsa' inverse theorem conjectured by Katalin Marton. The proof runs through an entropic reformulation: minimizing a 'Ruzsa multidistance' functional over pairs of F_2^n-valued random variables and showing the minimum forces near-uniformity on a coset. Follow-up work by the same authors extends the result to abelian groups of bounded torsion. Published in Annals of Mathematics 201(2), 2025, and formalized in Lean 4 (the teorth/pfr project) within weeks of the preprint.","domain":"mathematics","category":"theorem","status":"FACT_STATUS_EXPIRED","claimType":"CLAIM_TYPE_ASSERTION","confidence":880000,"verifiedAtBlock":"835621","lastVerifiedBlock":"835621","energy":158425,"energyCap":1000000,"fitnessScore":75000,"methodId":"M-LEGACY"},{"id":"5f9b4c28e9615d58b2fa5fd920ea7c8c","content":"There is a universal constant c > 0 such that every convex body K ⊂ R^n of volume 1 admits a hyperplane H with Vol_{n−1}(K ∩ H) ≥ c, independent of the dimension n. Equivalently, the isotropic constant L_K of every convex body is bounded above by a universal constant. This affirmatively resolves Bourgain's slicing problem (1984), where the best bounds had been Bourgain's n^{1/4} log n (1991) and Klartag's n^{1/4} (2006), later improved to polylogarithmic via Chen and Klartag–Lehec. The final step combines Milman's M-ellipsoids, Eldan's stochastic localization driven by a key 2024 bound of Qingyang Guan, and Eldan–Mikulincer stability estimates for the Shannon–Stam inequality; Klartag's follow-up (arXiv:2501.06854) gives a self-contained proof via small-ball estimates.","domain":"mathematics","category":"theorem","status":"FACT_STATUS_EXPIRED","claimType":"CLAIM_TYPE_ASSERTION","confidence":880000,"verifiedAtBlock":"852634","lastVerifiedBlock":"852634","energy":173575,"energyCap":1000000,"fitnessScore":75000,"methodId":"M-LEGACY"},{"id":"6bd640f6db9e8119ea6df56c8bfe519d","content":"For every finite group G and every prime ℓ, the number of irreducible complex characters of G whose degree is not divisible by ℓ equals the same count for the normalizer of a Sylow ℓ-subgroup: |Irr_{ℓ'}(G)| = |Irr_{ℓ'}(N_G(P))|, where P is a Sylow ℓ-subgroup of G. Conjectured by John McKay in 1971, the statement was reduced by Isaacs–Malle–Navarro (2007) to the 'inductive McKay condition' for finite simple groups; over nearly two decades this was verified family by family, and Cabanes–Späth complete the last and hardest case — groups of Lie type D (spin groups) — using the classification of finite simple groups, thereby proving the conjecture in full for all groups and all primes.","domain":"mathematics","category":"theorem","status":"FACT_STATUS_EXPIRED","claimType":"CLAIM_TYPE_ASSERTION","confidence":880000,"verifiedAtBlock":"853312","lastVerifiedBlock":"853312","energy":172975,"energyCap":1000000,"fitnessScore":75000,"methodId":"M-LEGACY"},{"id":"74e6f7c058a43d1e44357ee1ee7efcb1","content":"Conjecture (Ravenel 1984, the last open one of his famous list): for every prime p and height n, telescopic localization agrees with chromatic localization — T(n)-localization and K(n)-localization coincide on finite complexes of type n. Refutation: Burklund, Hahn, Levy and Schlank prove that for every prime p and every height n+1 ≥ 2 the telescopic and chromatic categories differ; e.g. for n=2 the T(2)-local algebraic K-theory of the K(1)-local sphere is not K(2)-local. The proof runs through a new interface between algebraic K-theory, trace methods/cyclotomic spectra, and chromatic homotopy theory. Consequence: the stable homotopy groups of spheres are 'bigger' (telescopically) than the chromatic picture predicted.","domain":"mathematics","category":"refutation","status":"FACT_STATUS_EXPIRED","claimType":"CLAIM_TYPE_ASSERTION","confidence":880000,"verifiedAtBlock":"853655","lastVerifiedBlock":"853655","energy":172675,"energyCap":1000000,"fitnessScore":75000,"methodId":"M-LEGACY"},{"id":"75381b340a18de8cabcf741f29e9bbdb","content":"At IMO 2024, DeepMind's AlphaProof (an AlphaZero-style reinforcement-learning agent that writes proofs in Lean, trained on ~80 million auto-formalized statements) solved problems P1, P2 and P6 (including the hardest problem) and AlphaGeometry2 solved P4, scoring 28/42 — the first AI performance at any IMO medal level, within the silver band. Solutions to P1/P2/P6 are complete Lean proofs. Announced 25 July 2024; the methodology was peer-reviewed and published in Nature on 12 Nov 2025 as 'Olympiad-level formal mathematical reasoning with reinforcement learning' (DOI 10.1038/s41586-025-09833-y).","domain":"mathematics","category":"formalization","status":"FACT_STATUS_EXPIRED","claimType":"CLAIM_TYPE_ASSERTION","confidence":880000,"verifiedAtBlock":"854667","lastVerifiedBlock":"854667","energy":176450,"energyCap":1000000,"fitnessScore":75000,"methodId":"M-LEGACY"},{"id":"861e4100d3f7ed57015b16c6ae59fe47","content":"The einstein problem is solved: a single closed topological disk exists that tiles the plane only aperiodically. (1) The 'hat', a 13-sided polykite, admits tilings of the plane, and every tiling by hats (reflections allowed) is non-periodic; proved both via a substitution/metatile hierarchy and via a second computer-assisted case analysis, and in fact an entire continuum Tile(a,b) of combinatorially equivalent shapes is aperiodic. (2) The 'Spectre', obtained by modifying the edges of the equilateral member Tile(1,1), is a strictly chiral aperiodic monotile: it admits tilings using only unreflected copies and admits no periodic tiling, settling the reflection-free version. First aperiodic monotiles ever found, ending a search running from Wang tiles (1960s) through Penrose's two-tile sets.","domain":"mathematics","category":"theorem","status":"FACT_STATUS_EXPIRED","claimType":"CLAIM_TYPE_ASSERTION","confidence":880000,"verifiedAtBlock":"836193","lastVerifiedBlock":"836193","energy":157925,"energyCap":1000000,"fitnessScore":75000,"methodId":"M-LEGACY"},{"id":"9962755195be98280fd98b7a8944ca32","content":"Conjecture (Kasteleyn 1985): for any finite graph G and independent bond percolation on the bunkbed graph G × K_2 (two copies of G plus vertical posts), the probability that (u,0) connects to (v,0) is at least the probability that (u,0) connects to (v,1), for all vertices u,v — i.e., same-layer connection is always at least as likely as cross-layer. Refutation: Gladkov, Pak and Zimin gave an explicit counterexample, a planar graph on 7,222 vertices built from Hollom's 2024 counterexample to the hypergraph bunkbed conjecture, where the inequality fails (by a tiny but rigorously positive margin, proven without computer experiment).","domain":"mathematics","category":"refutation","status":"FACT_STATUS_EXPIRED","claimType":"CLAIM_TYPE_ASSERTION","confidence":880000,"verifiedAtBlock":"852744","lastVerifiedBlock":"852744","energy":177450,"energyCap":1000000,"fitnessScore":75000,"methodId":"M-LEGACY"},{"id":"9caf48430fd5281aa2483c6ba74aea45","content":"The Mizohata–Takeuchi conjecture from the early 1980s is false. The conjecture asserted that for the Fourier extension operator E associated to a curved hypersurface S ⊂ R^d and every nonnegative weight w, one has ∫ |Ef|² w ≲ (sup over lines L of the X-ray transform Xw(L)) · ‖f‖²_{L²(S)}; it would have implied Stein's conjecture and strong weighted restriction estimates. Cairo constructs counterexamples showing an unavoidable loss of a factor on the order of log R at spatial scale R, for every C² hypersurface in R^d that does not lie in a hyperplane; the construction arranges wave packets along fractal-like sets and is complemented by a family of sharp L^p estimates for X-ray transforms of positive measures delimiting what survives.","domain":"mathematics","category":"refutation","status":"FACT_STATUS_EXPIRED","claimType":"CLAIM_TYPE_ASSERTION","confidence":880000,"verifiedAtBlock":"853429","lastVerifiedBlock":"853429","energy":172875,"energyCap":1000000,"fitnessScore":75000,"methodId":"M-LEGACY"},{"id":"a28fa7b2c6be944e6b31d58a7a2c69ca","content":"For the 4694 equational laws on magmas involving at most four applications of the binary operation, the project determined the status of every one of the 22,028,942 ordered implication pairs (does law E entail law E' in all magmas?), each answer — proof or explicit countermodel — verified in Lean 4. Launched by Terence Tao on 25 Sep 2024 as a pilot for machine-assisted, massively collaborative mathematics; all implications were resolved informally within about two months and the complete Lean formalization was finished about five months later (first phase concluded April 2025). Combined human proofs, automated theorem provers (e.g. Vampire), and some LLM assistance; produced new magma constructions (e.g. resolving the 'Asterix-Obelix' and dual pairs) and side results on finite magmas. Paper: arXiv:2512.07087 (Dec 2025).","domain":"mathematics","category":"formalization","status":"FACT_STATUS_EXPIRED","claimType":"CLAIM_TYPE_ASSERTION","confidence":880000,"verifiedAtBlock":"852973","lastVerifiedBlock":"852973","energy":169300,"energyCap":1000000,"fitnessScore":75000,"methodId":"M-LEGACY"},{"id":"a89145cf6820a9bc19008d0ad3ffd7a7","content":"There exists an absolute constant ε > 0 such that the diagonal Ramsey number satisfies R(k) ≤ (4 − ε)^k for all sufficiently large k, where R(k) is the least n such that every red/blue edge-coloring of the complete graph K_n contains a monochromatic K_k. This is the first exponential improvement over the Erdős–Szekeres 1935 bound R(k) ≤ 4^k; every improvement in the intervening 88 years (Rödl, Thomason, Conlon, Sah) had been by sub-exponential factors. The proof replaces the classical induction with a new 'book algorithm' that iteratively builds large monochromatic books and exploits their structure. Subsequent optimization of the method (Gupta–Ndiaye–Norin–Wei, arXiv:2407.19026) lowered the base to about 3.8.","domain":"mathematics","category":"theorem","status":"FACT_STATUS_EXPIRED","claimType":"CLAIM_TYPE_ASSERTION","confidence":880000,"verifiedAtBlock":"835296","lastVerifiedBlock":"835296","energy":158525,"energyCap":1000000,"fitnessScore":75000,"methodId":"M-LEGACY"},{"id":"aebe55c96faa03cc6cdb29fd40329c59","content":"For every constant C > 1 there exists ε = ε(C) > 0 such that for all sufficiently large ℓ the off-diagonal Ramsey number satisfies r(ℓ, Cℓ) ≥ (p_C^{-1/2} + ε)^ℓ, where p_C ∈ (0, 1/2) is the unique solution of C = log(p_C)/log(1 − p_C). This is the first exponential improvement over the lower bound obtained from Erdős's 1947 probabilistic method (random graph G(n, p_C)), whose exponential base had not been improved for any such regime in 78 years. Hunter, Milojević and Sudakov subsequently gave a simplified proof via Gaussian random graphs with better quantitative bounds (arXiv:2512.17718).","domain":"mathematics","category":"theorem","status":"FACT_STATUS_EXPIRED","claimType":"CLAIM_TYPE_ASSERTION","confidence":880000,"verifiedAtBlock":"853540","lastVerifiedBlock":"853540","energy":176750,"energyCap":1000000,"fitnessScore":75000,"methodId":"M-LEGACY"},{"id":"axis-attribution","content":"attribution","domain":"doctrine_useful_work","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"axis-classification","content":"classification","domain":"doctrine_useful_work","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"axis-interface","content":"interface","domain":"doctrine_useful_work","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"axis-substrate","content":"substrate","domain":"doctrine_useful_work","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"axis-tooling","content":"tooling","domain":"doctrine_useful_work","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"axis-verification","content":"verification","domain":"doctrine_useful_work","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"b2298b9c41df7ba5ced9eefc6db0d24d","content":"Every Kakeya set in R^3 — a compact set containing a unit line segment in every direction — has Minkowski and Hausdorff dimension 3. Wang and Zahl prove a stronger volume estimate: given a family of delta-tubes pointing in delta-separated directions satisfying natural non-concentration hypotheses (Katz–Tao convex Wolff axioms: not too many tubes inside any common convex set), the union of the tubes has almost maximal volume. The dimension statement follows. The proof is a multi-scale induction organized around 'grainy' and 'sticky' structure, building on the authors' prior resolution of the sticky Kakeya conjecture (published J. Amer. Math. Soc. 39 (2026), 515–585). As of 2026-08 the main paper (arXiv:2502.17655) is a preprint under journal review, widely accepted by the field and central to Wang's 2026 Fields Medal citation; not yet journal-published.","domain":"mathematics","category":"theorem","status":"FACT_STATUS_EXPIRED","claimType":"CLAIM_TYPE_ASSERTION","confidence":880000,"verifiedAtBlock":"835732","lastVerifiedBlock":"835732","energy":154200,"energyCap":1000000,"fitnessScore":75000,"methodId":"M-LEGACY"},{"id":"b72b0aec2966ab83c5d378a9798acb94","content":"Conjecture (Aldous–Lyons 2007): every unimodular random network is sofic, i.e. a Benjamini–Schramm limit of finite graphs (equivalently, every invariant random subgroup of a free group is co-sofic). Refutation: Bowen, Chapman, Lubotzky and Vidick construct non-sofic unimodular networks. Part I introduces 'subgroup tests' and reduces the conjecture to an undecidability statement; Part II proves the required undecidability for tailored non-local games by adapting the compression/MIP*=RE machinery from quantum interactive proofs. Together the two papers give a negative resolution of the conjecture.","domain":"mathematics","category":"refutation","status":"FACT_STATUS_EXPIRED","claimType":"CLAIM_TYPE_ASSERTION","confidence":880000,"verifiedAtBlock":"852518","lastVerifiedBlock":"852518","energy":177650,"energyCap":1000000,"fitnessScore":75000,"methodId":"M-LEGACY"},{"id":"bfb5ca3d5bfd0d88df6111ae61a94fb6","content":"The fifth Busy Beaver value is determined: S(5) = BB(5) = 47,176,870. Every 5-state, 2-symbol Turing machine started on an all-zero tape either halts within 47,176,870 steps or runs forever, and the bound is attained by Marxen–Buntrock's 1990 champion machine. The proof, organized by the online bbchallenge collaboration, enumerates 181,385,789 machines and supplies for each either a halting run or a certified non-halting proof produced by verified decider algorithms (cyclers, translated cyclers, bouncers, finite-automata/closed-tape-language methods), plus individual arguments for holdouts such as Skelet #17. The entire theorem is machine-checked in the Coq proof assistant (Coq-BB5). First new Busy Beaver value in over 40 years and the first ever formally verified.","domain":"mathematics","category":"theorem","status":"FACT_STATUS_EXPIRED","claimType":"CLAIM_TYPE_ASSERTION","confidence":880000,"verifiedAtBlock":"835963","lastVerifiedBlock":"835963","energy":158125,"energyCap":1000000,"fitnessScore":75000,"methodId":"M-LEGACY"},{"id":"c3753a14e08fea51faeb117eda292b5a","content":"AlphaGeometry, a neuro-symbolic system pairing a language model trained on ~100M synthetic theorem-proof examples with the DDAR symbolic deduction engine, solved 25 of 30 problems on the IMO-AG-30 benchmark (IMO geometry problems 2000-2022) under competition time limits, versus 10 for the prior best system and 25.9 for the average human gold medalist. It produces human-readable, machine-checkable deduction traces and discovered a generalization of a 2004 IMO theorem. Published in Nature 625, 476-482 (17 Jan 2024), DOI 10.1038/s41586-023-06747-5. Successor AlphaGeometry2 (arXiv:2502.03544, Feb 2025) reports solving 84% of IMO geometry problems 2000-2024 and supplied the geometry component of DeepMind's IMO 2024 system.","domain":"mathematics","category":"formalization","status":"FACT_STATUS_EXPIRED","claimType":"CLAIM_TYPE_ASSERTION","confidence":880000,"verifiedAtBlock":"854170","lastVerifiedBlock":"854170","energy":177575,"energyCap":1000000,"fitnessScore":95000,"methodId":"M-LEGACY"},{"id":"c64e3166eef29a0c345688389d89b3e2","content":"Complete Lean 4 formalization, finished July 2025, of a generalized Carleson theorem: bounds for a generalized Carleson operator on doubling metric measure spaces (the 'metric space Carleson' theorem of Thiele's Bonn group, plus a linearized variant covering Walsh-Fourier series), together with a machine-verified derivation of Carleson's classical 1966 result that the Fourier series of a continuous (periodic) function converges pointwise almost everywhere. Project blueprint: Becker, van Doorn, Jamneshan, Srivastava, Thiele, arXiv:2405.06423; formalization led by Floris van Doorn (Bonn) with 14 core authors and 14 further contributors.","domain":"mathematics","category":"formalization","status":"FACT_STATUS_EXPIRED","claimType":"CLAIM_TYPE_ASSERTION","confidence":880000,"verifiedAtBlock":"852859","lastVerifiedBlock":"852859","energy":177350,"energyCap":1000000,"fitnessScore":75000,"methodId":"M-LEGACY"},{"id":"ce02355e036d418bfd75d9dcf17b61ad","content":"If A ⊆ F_2^n has doubling |A+A| ≤ K|A|, then A can be covered by at most 2K^12 cosets of a subgroup H ≤ F_2^n with |H| ≤ |A|. Thus sets of bounded doubling in F_2^n are polynomially (in K) close to genuine subgroups, replacing the previously known exponential-in-K losses — the flagship 'polynomial Freiman–Ruzsa' inverse theorem conjectured by Katalin Marton. The proof runs through an entropic reformulation: minimizing a 'Ruzsa multidistance' functional over pairs of F_2^n-valued random variables and showing the minimum forces near-uniformity on a coset. Follow-up work by the same authors extends the result to abelian groups of bounded torsion.","domain":"mathematics","category":"theorem","status":"FACT_STATUS_EXPIRED","claimType":"CLAIM_TYPE_ASSERTION","confidence":880000,"verifiedAtBlock":"834962","lastVerifiedBlock":"834962","energy":162750,"energyCap":1000000,"fitnessScore":75000,"methodId":"M-LEGACY"},{"id":"commitment-1","content":"Methodology over statement","domain":"doctrine_truth_seeking","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"commitment-10","content":"Forward-only audit","domain":"doctrine_truth_seeking","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":155000,"methodId":"doctrine_authorship"},{"id":"commitment-11","content":"Trust is queryable","domain":"doctrine_truth_seeking","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":115000,"methodId":"doctrine_authorship"},{"id":"commitment-12","content":"The chain pays for its own audit","domain":"doctrine_truth_seeking","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":175000,"methodId":"doctrine_authorship"},{"id":"commitment-13","content":"The training corpus is not for sale","domain":"doctrine_truth_seeking","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":175000,"methodId":"doctrine_authorship"},{"id":"commitment-14","content":"Reasoning traces are first-class","domain":"doctrine_truth_seeking","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":115000,"methodId":"doctrine_authorship"},{"id":"commitment-15","content":"Counterexamples are part of the corpus","domain":"doctrine_truth_seeking","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"commitment-16","content":"The chain pays for exploration of the unknown","domain":"doctrine_truth_seeking","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"commitment-17","content":"Disagreement is structure, not noise","domain":"doctrine_truth_seeking","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"commitment-18","content":"The chain manufactures exploration demand","domain":"doctrine_truth_seeking","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"commitment-19","content":"The creed is governance-gated","domain":"doctrine_truth_seeking","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"commitment-2","content":"Is-ought wall","domain":"doctrine_truth_seeking","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"commitment-20","content":"Issuance follows participation","domain":"doctrine_truth_seeking","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"commitment-3","content":"Popper, not popularity","domain":"doctrine_truth_seeking","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":95000,"methodId":"doctrine_authorship"},{"id":"commitment-4","content":"The substrate stress-tests its own truth","domain":"doctrine_truth_seeking","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"commitment-5","content":"The chain manufactures probe demand","domain":"doctrine_truth_seeking","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"commitment-6","content":"No individual can unilaterally inject truth","domain":"doctrine_truth_seeking","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":115000,"methodId":"doctrine_authorship"},{"id":"commitment-7","content":"Skill is current, not historical","domain":"doctrine_truth_seeking","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"commitment-8","content":"The panel weights skill, not bond","domain":"doctrine_truth_seeking","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"commitment-9","content":"Cartel detection has consequence","domain":"doctrine_truth_seeking","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"commitment-SL","content":"ZERONE is a strange loop","domain":"doctrine_strange_loop","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"commitment-TC1","content":"The graph is the headline","domain":"doctrine_tok","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":95000,"methodId":"doctrine_authorship"},{"id":"commitment-TC2","content":"Every view is graph-pinned","domain":"doctrine_tok","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":95000,"methodId":"doctrine_authorship"},{"id":"commitment-TC3","content":"Topology is signal","domain":"doctrine_tok","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"commitment-TC4","content":"The graph carries its disprovals","domain":"doctrine_tok","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"commitment-TC5","content":"Extraction is open","domain":"doctrine_tok","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"commitment-TC6","content":"Lineage flows back","domain":"doctrine_tok","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":135000,"methodId":"doctrine_authorship"},{"id":"commitment-UW","content":"ZERONE is recursive","domain":"doctrine_useful_work","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"e1bc0eef51f25066b25750581f8c3e47","content":"There are infinitely many primes of the form p² + 4q² where p and q are both prime, with the conjectured asymptotic count — proving the 'Gaussian primes conjecture' posed by Friedlander and Iwaniec (2018) as a prime-variable refinement of their 1998 theorem that x² + y⁴ represents infinitely many primes. More generally, for any positive n ≡ 0 or 4 (mod 6), there are infinitely many primes p² + nq² with p, q prime, with asymptotics. Method: Type I/II sums in the number field Q(√−n), with the critical Type II ranges controlled by tools imported from additive combinatorics — quantitative concatenation theorems for Gowers norms and the quasipolynomial inverse theorem — the first use of higher-order Fourier analysis to detect primes in such a sparse polynomial family. As of 2026-08 this is an arXiv preprint widely accepted by the analytic number theory community; not yet journal-published.","domain":"mathematics","category":"theorem","status":"FACT_STATUS_EXPIRED","claimType":"CLAIM_TYPE_ASSERTION","confidence":880000,"verifiedAtBlock":"853089","lastVerifiedBlock":"853089","energy":165225,"energyCap":1000000,"fitnessScore":75000,"methodId":"M-LEGACY"},{"id":"ea52191c15a819e536fd12668c37e481","content":"New large-value estimates for Dirichlet polynomials: sharpened bounds on how often an N-term Dirichlet polynomial can attain values of size near N^{3/4}, the critical regime for applications to primes. Main corollaries: (1) the zero-density estimate N(σ, T) ≤ T^{30(1−σ)/13 + o(1)} for the number of zeros of ζ(s) with real part ≥ σ and |Im s| ≤ T — the first improvement over Ingham's 1940 exponent 3(1−σ)/(2−σ) in the key range around σ = 3/4 (at σ = 3/4: exponent 15/26 ≈ 0.577 versus Ingham's 3/5); (2) the prime number theorem holds asymptotically in short intervals [x, x + x^{17/30 + o(1)}], improving Huxley's 1972 exponent 7/12. This is unconditional progress toward the density hypothesis and, distantly, the Riemann hypothesis.","domain":"mathematics","category":"theorem","status":"FACT_STATUS_EXPIRED","claimType":"CLAIM_TYPE_ASSERTION","confidence":880000,"verifiedAtBlock":"836075","lastVerifiedBlock":"836075","energy":158025,"energyCap":1000000,"fitnessScore":75000,"methodId":"M-LEGACY"},{"id":"ecb5004ae763034f6dacb27832c34191","content":"zerone-1 genesis (2026-07-08T14:10:14Z, sha256 16ac346f329d2a931ad9a7d51dbe9e35605482b006ef39b3ac7804376e9bcb66; verify: curl RPC/genesis | jq .result.genesis | sha256) allocated 13,555 ZRN across exactly two accounts: 11,333 ZRN validator (11,111 self-bonded + 222 gas) and 2,222 ZRN operations float. Zero insider allocation: no team, foundation, investor, or faucet balance; all further ZRN mints only through participation. DISCLOSURE: the submitter and all 4 panel keys are operated by the chain's founding household and funded from the operations float (the funding graph is on-chain); wrong-vote slashing is currently inert (no zerone.staking registrants); the submitter's resulting perfect calibration score is a bootstrap artifact, disclaimed for training-fund purposes. Standing invitation: challenge this fact (min stake 11 ZRN). PoA in PoS clothes; independence is earned, not claimed.","domain":"general","category":"empirical","status":"FACT_STATUS_PRUNED","claimType":"CLAIM_TYPE_ASSERTION","confidence":880000,"verifiedAtBlock":"76944","lastVerifiedBlock":"76944","energy":89700,"energyCap":1000000,"fitnessScore":75000,"methodId":"M-LEGACY"},{"id":"mechanism-SL-M1","content":"Doctrine import","domain":"doctrine_strange_loop","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"mechanism-SL-M2","content":"Protocol as substrate","domain":"doctrine_strange_loop","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"mechanism-SL-M3","content":"Governance lift","domain":"doctrine_strange_loop","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"mechanism-SL-M4","content":"Author lineage propagates forever","domain":"doctrine_strange_loop","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"mechanism-SL-M5","content":"Self-verification","domain":"doctrine_strange_loop","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"mechanism-SL-M6","content":"Origin attestation","domain":"doctrine_strange_loop","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"mechanism-UW-M1","content":"Stake-backed claim","domain":"doctrine_useful_work","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"mechanism-UW-M2","content":"Substrate-link mandate","domain":"doctrine_useful_work","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"mechanism-UW-M3","content":"Class-specific verification under shared lifecycle","domain":"doctrine_useful_work","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"mechanism-UW-M4","content":"Reward formula","domain":"doctrine_useful_work","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"mechanism-UW-M5","content":"Recursion-weight projection over six axes","domain":"doctrine_useful_work","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"mechanism-UW-M6","content":"Lineage propagates AND recurses","domain":"doctrine_useful_work","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"},{"id":"mechanism-UW-M7","content":"The chain pays for the audit of its own useful work","domain":"doctrine_useful_work","category":"doctrine","status":"FACT_STATUS_VERIFIED","claimType":"CLAIM_TYPE_UNSPECIFIED","confidence":880000,"verifiedAtBlock":"0","lastVerifiedBlock":"0","energy":1000000,"energyCap":0,"fitnessScore":75000,"methodId":"doctrine_authorship"}],"relations":[]}