# Consequences of results in the OpenAI mathematics release Jizhou Guo — mitsuha2021b@gmail.com — 8 October 2026 Each entry of this list combines one theorem of the OpenAI mathematics release (github.com/openai/math) with results already in the literature. Every statement is conditional on the release theorem named in its entry and, where stated, on a named preprint. Release statements were read at commit adc7f1241b42e322a6451854ab7e4b4c146bf78a of github.com/openai/math, and none of the cited release papers is among those withdrawn in the repository revision at commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb (history entry of October 7, 2026). Each entry gives the statement, the release input, the bridging results and the deduction. ## Contents 1. Deterministic construction of finite fields — family 142 2. Lower bounds for Carmichael and Novak–Carmichael numbers — family 011 3. Tate, Beilinson–Parshin and Lichtenbaum conjectures for varieties of abelian type over finite fields — family 032 4. Failure of Tachikawa's first conjecture — family 199 5. Wall's question, the Kapovich–Kleiner conjecture and the toral relative Cannon conjecture — family 246 6. C∗-simplicity of Thompson's group T and further consequences of the nonamenability of F — family 248 7. Group von Neumann algebras of surface groups — family 287 8. Rank r Donaldson–Thomas invariants of Calabi–Yau threefolds from rank 0 — family 055 9. Ground configurations of the planar disordered Ising ferromagnet — family 212 10. The weak recovery threshold of the four-community stochastic block model — family 229 11. The quantum Fourier transform and fanout in constant depth — family 274 12. The Strong Non-Synthesis Conjecture of Lombardi–Ma–Wright — family 283 ## 1. Deterministic construction of finite fields **Statement.** Conditional on Theorem 1.1 of the OpenAI paper "Deterministic polynomial factorization over prime fields" (family 142), which depends on Theorem 1.2 of the companion paper of family 029 (a uniform zero-free strip for finite-order Hecke L-functions of cyclotomic fields containing the 12th roots of unity): there is a deterministic algorithm which, given a prime p in binary and an integer n ≥ 1 in unary, outputs a monic irreducible polynomial g of degree n in 𝔽_p[x] using a number of bit operations bounded by a polynomial in n and ⌈log₂ p⌉, with an absolute exponent. The algorithm uses no randomness, no GRH, and no integer-factorization or primitive-root oracle. Consequently the finite field 𝔽_{p^n} = 𝔽_p[x]/(g) can be constructed deterministically in time polynomial in n and log p, for every prime p and every degree n. **Release input.** Family 142, "Deterministic polynomial factorization over prime fields" (OpenAI, October 4, 2026), Theorem 1.1: a uniform deterministic algorithm that, given a prime p in binary and a nonzero dense polynomial f ∈ 𝔽_p[x] of degree n, outputs the leading coefficient and the distinct monic irreducible factors of f with multiplicities in a number of bit operations polynomial in (n + 1)⌈log₂ p⌉, with absolute implied constants and without randomness, integer-factorization oracle, primitive-root oracle or GRH. Its proof uses Theorem 1.2 of the family-029 paper "Primitive roots for every admissible integer base" (OpenAI, October 4, 2026): for every finite-order Hecke character η of a cyclotomic number field F containing μ₁₂, the function L_F(s, η) has no zeros in Re s > 1 − 10^(−6). No Lean formalization is listed for families 142 and 029. **Bridge.** Victor Shoup, "New algorithms for finding irreducible polynomials over finite fields", Math. Comp. 54 (1990), no. 189, 435–447, Theorem 3.1 (numbering of the author's manuscript): for F = GF(p), p prime, constructing an irreducible polynomial of degree n over F can be reduced deterministically, in time bounded by a polynomial in n and log p, to factoring polynomials over F. For comparison, deterministic constructions in time polynomial in n and log p under GRH are due to Adleman–Lenstra (1986) and Evdokimov (1986), a deterministic construction in time polynomial in p is in Shoup (1990), and a pseudo-deterministic construction is due to Rai (arXiv:2410.04071). **Deduction.** Let L = ⌈log₂ p⌉. Shoup's algorithm is deterministic and runs in time polynomial in n and L apart from calls that factor polynomials over 𝔽_p. Each call is on a dense polynomial of degree at most n² over the same field 𝔽_p (a cyclotomic polynomial of degree q − 1 for each prime q | n with q ≠ p, and at most m log p polynomials of degree mq with m < q), and the number of calls is polynomial in n and L. Replacing each call by the algorithm of Theorem 1.1, which factors every dense polynomial over 𝔽_p for every prime p given in binary, gives a total running time polynomial in n and L. The cases q = 2 and q = p are handled in Shoup's Theorem 2.1 without factoring, at cost polynomial in n. The output has nL bits, so a bound polynomial in n (unary) and log p is the natural form. The direction used is: factoring over 𝔽_p in polynomial time implies construction of irreducible polynomials over 𝔽_p in polynomial time. **Related.** The same deduction from family 142 and Shoup's reduction is recorded as entry TC-3 of the catalogue [atimics/math2, docs/COROLLARIES.md](https://github.com/atimics/math2/blob/0ea55c2e73d18645d79594feacc1dd8c8bbd185b/docs/COROLLARIES.md) in its version of 8 October 2026, 06:00 UTC, which precedes this entry and also treats factoring over 𝔽\_{p^k}. **Scope.** The statement concerns irreducible polynomials of every degree n over a prime field 𝔽_p, with p given in binary and the running time measured in n (unary) and log p. ## 2. Lower bounds for Carmichael and Novak–Carmichael numbers **Statement.** Conditional on Theorem 1.1 of the OpenAI paper "The Poisson–Dirichlet law for prime predecessors" (family 011). Let C(x) be the number of Carmichael numbers ≤ x, and let N_C(x) be the number of Novak–Carmichael numbers ≤ x, i.e. positive integers n with a^n ≡ 1 (mod n) for every a coprime to n, equivalently p − 1 | n for every prime p | n (terminology of Kalmynin). - (a) For every fixed ε > 0 there is x₀(ε) with C(x) ≥ x^(5/12 − ε) for all x ≥ x₀ (Alford–Granville–Pomerance, Theorem 1, with E ∈ (0,1) arbitrary and B < 5/12). - (a′) With the value B = 0.4736 of Harman (Int. J. Number Theory 4 (2008) 241–248), as recorded in Wright (Bull. Aust. Math. Soc. 101 (2020), Theorem 2.3 and p. 381) and Lichtman (arXiv:2211.09641): C(x) ≥ x^(0.4736 − ε) for all x ≥ x₀(ε). - (b) For every fixed ε > 0, N_C(x) ≥ x^(1 − ε) for x ≥ x₀(ε) (Kalmynin, Theorem 1); hence N_C(x) = x^(1 − o(1)). **Release input.** Family 011, "The Poisson–Dirichlet law for prime predecessors" (OpenAI, September 24, 2026), Theorem 1.1: for fixed k ≥ 1 and bounded continuous F : [0,1]^k → ℝ, the average of F(V₁(p), …, V_k(p)) over the primes 3 ≤ p ≤ x, with equal weights, tends to E F(L₁, …, L_k) as x → ∞ through all real x. Here V_j(p) = log q_j(p) / log(p − 1), q₁ ≥ q₂ ≥ … are the prime factors of p − 1 counted with multiplicity, and (L_j) has the Poisson–Dirichlet law PD(1). The paper records that this resolves the conjecture of Granville (its (1.3)): #{p ≤ x : P⁺(p − 1) ≤ x^(1/u)} ∼ ρ(u) π(x) for fixed u ≥ 1, where P⁺(m) is the largest prime factor of m and ρ is Dickman's function. The companion paper "Weighted dilation graphs, smooth shifted primes and totient fibers" (family 011), Theorem 1.2, gives #{p prime : 2x < p ≤ 5x, P⁺(p − 1) ≤ x^δ} ≥ x^(1 − o(1)) for fixed 0 < δ < 1/4. No Lean formalization is listed for family 011. **Bridge.** - W. R. Alford, A. Granville, C. Pomerance, "There are infinitely many Carmichael numbers", Ann. of Math. (2) 139 (1994) 703–722. Let π(x, y) count the primes p ≤ x such that p − 1 has no prime factor exceeding y. Let 𝓔 be the set of E ∈ (0,1) for which π(x, x^(1−E)) ≥ γ(E) π(x) for some γ(E) > 0 and all large x (condition (0.1)); Erdős conjectured 𝓔 = (0,1). Let 𝓑 be the set of B ∈ (0,1) for which there are x₂(B) and a positive integer D_B such that, for x ≥ x₂(B), π(y; d, a) ≥ π(y)/(2φ(d)) whenever (a,d) = 1, d ≤ min{x^B, y/x^(1−B)} and d is divisible by none of at most D_B exceptional integers, each exceeding log x (condition (0.3)). Theorem 2.1: (0, 5/12) ⊂ 𝓑. Theorem 1: for E ∈ 𝓔 and B ∈ 𝓑 there is x₀(E, B) with C(x) ≥ x^(EB) for x ≥ x₀. - G. Harman, "Watt's mean value theorem and Carmichael numbers", Int. J. Number Theory 4 (2008) 241–248: the admissible value B = 0.4736. - A. Kalmynin, "Novak–Carmichael numbers and shifted primes without large prime factors", arXiv:1706.07343, Theorem 1: for fixed u ∈ (0,1), if Π(z, z^u) = z^(1+o(1)) as z → ∞, then N_C(x) ≫ x^(1−u+o(1)); here Π(x, y) counts the primes p ≤ x whose predecessor p − 1 has largest prime factor at most y. **Deduction.** - Step 1 (𝓔 = (0,1)). Fix E ∈ (0,1) and put u = 1/(1 − E). Then π(x, x^(1−E)) = #{p ≤ x : P⁺(p − 1) ≤ x^(1/u)}, the quantity in (1.3), so π(x, x^(1−E)) ∼ ρ(u) π(x) with ρ(u) > 0, and (0.1) holds with γ(E) = ρ(u)/2. (From Theorem 1.1: for εx < p ≤ x one has log(p − 1)/log x = 1 − O_ε(1/log x), the event is V₁(p) ≤ 1/u + o(1), and L₁ has a continuous law with P(L₁ ≤ 1/u) = ρ(u); sandwich with continuous test functions and let ε → 0.) This is Erdős's conjecture 𝓔 = (0,1) in the formulation of Alford–Granville–Pomerance (p. 704). - Step 2 ((a), (a′)). With B < 5/12, so B ∈ 𝓑, Theorem 1 gives C(x) ≥ x^(EB). Given ε > 0, choose E < 1 and B < 5/12 with EB > 5/12 − ε. With the admissible value B = 0.4736 the same argument gives (a′). The positive proportion γ(E) π(x) in (0.1) is the form of input that the construction of Alford–Granville–Pomerance uses. - Step 3 ((b)). For fixed u ∈ (0,1), (1.3) gives Π(z, z^u) ∼ ρ(1/u) π(z) = z^(1+o(1)), which is Kalmynin's hypothesis; it also follows from Theorem 1.2 of the companion paper (take x = z/5 and δ = min(u, 1/5)). Kalmynin's Theorem 1 gives N_C(x) ≫ x^(1−u+o(1)); taking u = ε/2 gives N_C(x) ≥ x^(1−ε), and N_C(x) ≤ x. The smoothness conventions agree, and both results count all primes p ≤ x. **Scope.** The exponents in (a) and (a′) are those of the progression input B of Alford–Granville–Pomerance (5/12, respectively 0.4736); the statement concerns the bounds C(x) ≥ x^(B − ε) and N_C(x) ≥ x^(1 − ε) as written. ## 3. Tate, Beilinson–Parshin and Lichtenbaum conjectures for varieties of abelian type over finite fields **Statement.** Conditional on Theorem 1.1 of the OpenAI paper "The rational Hodge conjecture for CM abelian varieties" (family 032; the rational Hodge conjecture holds for every complex CM abelian variety), through its consequence Corollary 8.3 (the Tate conjecture for abelian varieties over finite fields). Let k be a finite field of characteristic p and let X be a smooth projective k-variety of abelian type in the sense of Kahn, i.e. its Chow motive h(X) with ℚ-coefficients lies in the thick rigid subcategory A_ab of Chow motives generated by the motives h(A_L) of abelian varieties A_L over finite extensions L/k (this class contains every abelian variety over k and every product of curves). Then X lies in Kahn's class B_tate(k): the Tate conjecture holds for X in every codimension and for every prime ℓ ≠ p, i.e. the cycle class map CH^i(X) ⊗ ℚ_ℓ → H^(2i)(X̄, ℚ_ℓ(i))^(Gal(k̄/k)) is surjective for all i. Consequently, by Kahn (Ann. Sci. École Norm. Sup. (4) 36 (2003) 977–1002): - (1) rational and numerical equivalence coincide on X with rational coefficients (Théorème 1.10, equivalently Théorème 1 of the introduction); - (2) the Beilinson–Parshin conjecture holds for X: K_i(X) is a torsion group for every i > 0 (Corollaire 2.2); - (3) the conjectures of Lichtenbaum (Lecture Notes in Math. 1068 (1984) 127–138, section 7), in the eight-item form recalled by Kahn and including the special-value formula, hold for X (Corollaire 3.10, using Corollaire 3.8); - (4) for every open U ⊂ X, the groups H⁰(U, K₂), H¹(U, K₂) and H²(U, K₂) = CH²(U) are finitely generated, and K₃(F)\_ind = K₃(k) for the function field F of X (Théorème 4.5). **Release input.** Family 032, "The rational Hodge conjecture for CM abelian varieties" (OpenAI, September 30, 2026). Theorem 1.1: for a complex CM abelian variety A (End(A) ⊗ ℚ contains a commutative semisimple ℚ-algebra of dimension 2 dim A) and every integer j ≥ 0, the Betti cycle class map CH^j(A)\_ℚ → H^(2j)(A, ℚ) ∩ H^(j,j)(A) is surjective. Corollary 8.3 (Section 8.2): for every abelian variety A over 𝔽_q, every r ≥ 0 and every prime ℓ ≠ char 𝔽_q, the map CH^r(A) ⊗\_ℤ ℚ_ℓ → H^(2r)\_ét(Ā, ℚ_ℓ(r))^(G_q) is surjective, and numerical and ℓ-adic homological equivalence coincide on rational algebraic cycles on Ā. It is deduced from Milne's theorem (Compositio Math. 117 (1999), Theorem 7.1 and formulation (0.1)) with Theorem 1.1 as hypothesis, followed by descent from 𝔽_{q^m} to 𝔽_q. No Lean formalization is listed for this paper. **Bridge.** Bruno Kahn, "Équivalences rationnelle et numérique sur certaines variétés de type abélien sur un corps fini", Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 6, 977–1002 (journal numbering). Définition 1: B(k) is the set of isomorphism classes of smooth projective k-varieties whose Chow motive lies in the thick rigid subcategory A_ab generated by Artin motives and motives of abelian varieties (Notation 1.2: the smallest thick subcategory of Chow motives containing the h(A_L), L/k finite); X is of abelian type if X ∈ B(k); B_tate(k) consists of the X ∈ B(k) satisfying the Tate conjecture, for one ℓ ≠ p equivalently for all (Tate, Théorème 2.9; Kahn, Lemme 1.9). Théorème 1.10: if X ×\_k X′ ∈ B_tate(k), then CH^(dim X + n)(X′ × X) ⊗ ℚ → A_num^(dim X + n)(X′ × X) ⊗ ℚ is an isomorphism for all n ∈ ℤ. Corollaire 2.2 (building on Geisser, K-Theory 13 (1998) 109–122, Theorem 3.3), Corollaire 3.10 and Théorème 4.5 are as stated above. Kahn's Example 1(c) records Milne's implication from the Hodge conjecture for complex CM abelian varieties to the Tate conjecture for abelian varieties over finite fields (see also Milne, "The Tate conjecture over finite fields", Theorem 7.4). **Deduction.** - (i) B_tate(k) = B(k). Let X ∈ B(k). Then h(X) is a direct summand, in Chow motives over k with ℚ-coefficients, of a finite sum of motives h(A_j)(n_j), with A_j an abelian variety over a finite extension L_j/k and n_j ∈ ℤ. Cycle class maps commute with correspondences, so the Tate map of X in codimension i is a retract of the Tate map of the sum. For h(A_L)(n) one has CH^i(A_L) = CH^i of A over L and H^(2i)(Ā_L)^(G_k) = H^(2i)(Ā)^(G_L), so the Tate map is that of the abelian variety A over the finite field L in codimension i − n (both sides vanish outside the range 0 to dim A). It is surjective by Corollary 8.3, which covers every abelian variety over every finite field, every r ≥ 0 and every ℓ ≠ p. Hence X ∈ B_tate(k). - (ii) Théorème 1.10 with X′ = Spec k gives (1); Corollaire 2.2, Corollaire 3.10 (via Corollaire 3.8) and Théorème 4.5 apply with the hypothesis X ∈ B_tate(k). - (iii) Hypotheses match: k is a finite field (Corollary 8.3 covers every 𝔽_q); the equivalence relations carry rational coefficients; the Tate conjecture is the surjectivity of the ℓ-adic cycle class map onto Galois invariants of H^(2i)(X̄, ℚ_ℓ(i)), with the same normalization in Corollary 8.3; no restriction on dim X, p or ℓ ≠ p enters. The descent step of Corollary 8.3 shows that the cycles are defined over 𝔽_q itself, as the definition of B_tate(k) requires. **Related.** Entry AG-1 of the catalogue [atimics/math2, docs/COROLLARIES.md](https://github.com/atimics/math2/blob/fcaf37abb2a57c1aaace218546c6388c673838f1/docs/COROLLARIES.md) (8 October 2026, 01:15 UTC) notes, as a lead, the Tate conjecture and the equality of numerical and homological equivalence for products of curves over finite fields as a consequence of family 032. **Scope.** The statement concerns smooth projective varieties of abelian type over a finite field, the Tate conjecture for ℓ ≠ p, and conclusions (1)–(4) exactly as in Kahn's Théorème 1.10, Corollaire 2.2, Corollaire 3.10 and Théorème 4.5. ## 4. Failure of Tachikawa's first conjecture **Statement.** Conditional on Theorem 1.1 of the OpenAI paper "A counterexample to Tachikawa's second conjecture" (family 199). Let k = 𝔽₂(q, H₁, H₂) with q, H₁, H₂ algebraically independent, and let K be any field extension of k (K = k included). Let A be the finite-dimensional symmetric k-algebra of the release and M its finite-dimensional non-projective left module with Ext^i_A(M, M) = 0 for all i > 0, and put B_K = End_{A_K}(A_K ⊕ M_K), where A_K = K ⊗\_k A and M_K = K ⊗\_k M. Then B_K is a finite-dimensional K-algebra with Ext^i_{B_K}(DB_K, B_K) = 0 for every i > 0 (D = Hom_K(−, K)) which is not self-injective. Consequently Tachikawa's first conjecture (Tachikawa, Lecture Notes in Math. 351 (1973), Section 8: if Ext^i_A(DA, A) = 0 for all i > 0, then A is self-injective) is false for finite-dimensional algebras over K, in particular over an algebraic closure of k. The classical form of the hypothesis, Ext^i_{A^e}(A, A^e) = 0 for i > 0, is the same condition, since Ext^n_{A^e}(A, A ⊗ A) ≅ Ext^n_A(DA, A) (see the deduction). **Release input.** Family 199, "A counterexample to Tachikawa's second conjecture" (OpenAI, September 23, 2026), Theorem 1.1: for k = 𝔽₂(q, H₁, H₂) with q, H₁, H₂ algebraically independent, there exist a finite-dimensional associative unital symmetric k-algebra A and a finite-dimensional unital left A-module M such that M is nonprojective and Ext^i_A(M, M) = 0 for every integer i > 0. By Corollary 1.2 and Proposition 8.1, for every field extension K/k the algebra A_K is symmetric and M_K is non-projective and self-orthogonal. The Lean formalization of the release covers the pair (A, M) over k; the field-extension, endomorphism-algebra and further homological consequences are outside its scope. **Bridge.** H. Enomoto and R. Marczinzik, "On the homological conjectures for Artin algebras", arXiv:2609.19172 (right modules; ν_A = D Hom_A(−, A)). Proposition 2.3: let S be an algebra, V = S ⊕ W an S-module and B = End_S(V). If V is selforthogonal, then Ext^n_B(DB, B) ≅ Ext^n_S(ν_S V, V) for n ≥ 0; if V is also a cogenerator, then dom.dim B = ∞. Lemma 2.4: let V = S ⊕ W be a generator-cogenerator over S and B = End_S(V). If B is selfinjective, then V is injective; if S and B are selfinjective, then V is projective-injective. The Hochschild form of Tachikawa's conjecture is stated in Ringel's lecture notes "The homological conjectures". **Deduction.** Take S = A^op, which is symmetric because A is, so that left A-modules are right S-modules. Let V = S ⊕ M and B = End_S(V) = End_A(A ⊕ M), with composition ab = a ∘ b; this is the opposite of the release's Γ, and the hypothesis and conclusion of Tachikawa's first conjecture are invariant under A ↔ A^op (apply the duality D; self-injectivity is left-right symmetric). V is selforthogonal: Ext^i(M, M) = 0 by the release, Ext^i(A, −) = 0 since A is projective, and Ext^i(−, A) = 0 since A is injective over the self-injective algebra S. As S is symmetric, ν_S is isomorphic to the identity functor, so Proposition 2.3 gives Ext^n_B(DB, B) ≅ Ext^n_S(V, V) = 0 for all n > 0. V is a generator-cogenerator, since S is self-injective and DS ≅ S. If B were self-injective, Lemma 2.4 would make V projective-injective, so its summand M would be projective, a contradiction. For K = k only Theorem 1.1 of the release is used; for K/k the base change of the release gives A_K symmetric and M_K non-projective with Ext^i_{A_K}(M_K, M_K) = K ⊗\_k Ext^i_A(M, M) = 0 (flatness), and the same argument applies. For the Hochschild form: Hom_{A^e}(X, A ⊗ A) ≅ Hom_A(X ⊗\_A DA, A) naturally, because A ⊗ A ≅ Hom_k(DA, A) as bimodules, and P_• ⊗\_A DA is a projective resolution of DA when P_• is a projective bimodule resolution of A; hence Ext^n_{A^e}(A, A^e) ≅ Ext^n_A(DA, A). **Related.** Q. Tang (唐琪越), "A note on counterexamples to homological conjectures" ([Hexagon 2610.00143](https://hexagonmath.org/abs/2610.00143), 7 October 2026), extends the construction of family 199 to every field containing an element of infinite multiplicative order and obtains counterexamples over each such field to both Tachikawa conjectures and to further homological conjectures, the first Tachikawa conjecture through the same Proposition 2.3 of Enomoto–Marczinzik. That note precedes this entry and covers a larger class of base fields. **Scope.** The statement concerns finite-dimensional algebras over the fields K containing 𝔽₂(q, H₁, H₂), including an algebraic closure of k. ## 5. Wall's question, the Kapovich–Kleiner conjecture and the toral relative Cannon conjecture **Statement.** Conditional on Theorem 1.1 of the OpenAI paper "A Modulus Proof of Cannon's Conjecture" (family 246: every hyperbolic group with boundary homeomorphic to S² admits a proper cocompact isometric action on ℍ³ with finite kernel). - (a) (Wall's question, hyperbolic case.) Every word-hyperbolic group that is a PD(3) group over ℤ (the orientation character may be nontrivial) is the fundamental group of a closed hyperbolic 3-manifold, in particular of a closed aspherical 3-manifold; the manifold may be nonorientable. - (b) (Kapovich–Kleiner conjecture, torsion-free case.) Every torsion-free hyperbolic group whose Gromov boundary is a Sierpiński carpet is the fundamental group of a compact hyperbolic 3-manifold with nonempty totally geodesic boundary; equivalently, it acts properly and cocompactly by isometries on a convex subset of ℍ³ bounded by disjoint totally geodesic planes. - (c) (Toral relative Cannon conjecture, Groves–Manning–Sisto Conjecture 1.3.) Let (G, P) be a relatively hyperbolic pair with P a nonempty finite collection of free abelian peripheral subgroups (necessarily of rank 2) and Bowditch boundary homeomorphic to S². Then G is Kleinian, i.e. isomorphic to a discrete subgroup of Isom(ℍ³). **Release input.** Family 246, "A Modulus Proof of Cannon's Conjecture" (OpenAI, September 23, 2026), Theorem 1.1: "Let G be a hyperbolic group with boundary of G homeomorphic to S². There is a homomorphism ρ : G → Isom(ℍ³) for which the action on ℍ³ is proper and cocompact. Its kernel is finite." A finite kernel is allowed and the image may contain orientation-reversing isometries. Corollary 8.1: a torsion-free hyperbolic group with boundary S² is the fundamental group of a closed hyperbolic 3-manifold (possibly nonorientable). The Lean formalization of the release covers Theorem 1.1; Corollary 8.1 is outside its scope. **Bridge.** - (a) M. Bestvina, "Local homology properties of boundaries of groups", Michigan Math. J. 43 (1996) 123–139: Example 1.2(i) (a torsion-free hyperbolic group has a 𝒵-structure whose boundary is the Gromov boundary, by Bestvina–Mess, J. Amer. Math. Soc. 4 (1991) 469–481); Theorem 2.8 (if G is a Poincaré duality group of dimension n over L whose orientation character has finite image and Z is a boundary of G, then Z is a homology (n − 1)-sphere over L); Remark 2.9 (homology manifolds of dimension ≤ 2 are manifolds, by Wilder, so Z ≅ S^(n−1) for n ≤ 3). For torsion-free hyperbolic groups, the equivalence of Wall's question with the Cannon conjecture is due to Bestvina, building on Bestvina–Mess. Wall's question is Question G2, p. 391, of Wall, Homological group theory (1979): is every PD(3) group the fundamental group of a closed aspherical 3-manifold? - (b) M. Kapovich and B. Kleiner, "Hyperbolic groups with low-dimensional boundary", Ann. Sci. École Norm. Sup. (4) 33 (2000) 647–669 (journal numbering): Theorem 5 (for a hyperbolic group G with Sierpiński carpet boundary, the double Ĝ of G along the collection of peripheral subgroups is hyperbolic, contains G as a quasiconvex subgroup, and has boundary S²), Corollary 12 (Ĝ is torsion-free if G is) and Corollary 13: "Let G be a torsion-free hyperbolic group with Sierpinski carpet boundary. Suppose either (A) Cannon's conjecture is true, or (B) every 3-dimensional Poincaré duality group with a nontrivial splitting is the fundamental group of a closed 3-manifold. Then G is the fundamental group of a compact hyperbolic 3-manifold with totally geodesic boundary." - (c) D. Groves, J. F. Manning, A. Sisto, "Boundaries of Dehn fillings", Geom. Topol. 23 (2019) 2929–3002: Conjecture 1.3 (relative Cannon conjecture) and Corollary 1.4 (the Cannon conjecture implies the relative Cannon conjecture). **Deduction.** - (a) Let G be hyperbolic and PD(3) over ℤ. Poincaré duality groups have finite cohomological dimension, so G is torsion-free. By Example 1.2(i) the Gromov boundary Z is a boundary of G, and the orientation character takes values in {±1}. Theorem 2.8 with n = 3 and L = ℤ shows that Z is a homology 2-sphere, and Remark 2.9 gives Z ≅ S². Theorem 1.1 of the release provides ρ with a proper cocompact action and finite kernel. The kernel is a finite subgroup of the torsion-free group G, hence trivial, and point stabilizers of the proper action are finite, hence trivial. So ℍ³/ρ(G) is a closed hyperbolic 3-manifold with fundamental group G, as in Corollary 8.1 of the release. - (b) Let G be torsion-free and hyperbolic with carpet boundary. By Theorem 5 and Corollary 12 of Kapovich–Kleiner, Ĝ is torsion-free and hyperbolic with boundary S². Corollary 8.1 of the release gives Ĝ = π₁(M) with M a closed hyperbolic 3-manifold; this is the step "if (A) holds, then Ĝ is the fundamental group of a closed hyperbolic 3-manifold" in the proof of Corollary 13. The remainder of that proof (Mostow rigidity for the reflection group acting on Ĝ by conjugation, convex cocompactness of G, convex hull bounded by totally geodesic planes) does not use Cannon's conjecture again, and Mostow rigidity applies to lattices in the full group Isom(ℍ³). - (c) Groves–Manning–Sisto, Corollary 1.4, states that the Cannon conjecture implies the relative Cannon conjecture; the release theorem is the Cannon conjecture. **Scope.** Part (a) concerns word-hyperbolic PD(3) groups over ℤ, part (b) is for torsion-free groups, and part (c) concerns pairs (G, P) with P a nonempty finite collection of free abelian peripheral subgroups of rank 2. ## 6. C∗-simplicity of Thompson's group T and further consequences of the nonamenability of F **Statement.** Conditional on Theorem 1.1 of the OpenAI paper "Thompson's group F is nonamenable" (family 248: Thompson's group F is not amenable). Here F is the group of increasing piecewise linear homeomorphisms of [0,1] with finitely many pieces, dyadic breakpoints and slopes in 2^ℤ, and T is the group of orientation-preserving piecewise linear homeomorphisms of ℝ/ℤ with finitely many breakpoints, all dyadic, slopes in 2^ℤ, mapping dyadic rationals to dyadic rationals (Cannon–Floyd–Parry convention); F is the stabilizer of the point 0 in T. - (1) T is C∗-simple, i.e. its reduced group C∗-algebra C∗\_r(T) is simple (the Haagerup–Olesen question), and C∗\_r(T) has a unique tracial state. - (2) F, Aut(F) and the abstract commensurator group Comm(F) are C∗-simple. More generally, every countable subgroup of Homeo(S¹) that contains the standard copy of F (S¹ = ℝ/ℤ) is C∗-simple, and every countable subgroup of Homeo(ℝ) containing the standard copy of F in PL₂(ℝ) is C∗-simple. - (3) T acts freely (all point stabilizers trivial) on its Furstenberg boundary. - (4) For every field K the group algebra K[F] is a domain but not an Ore domain. - (5) Myhill's property fails over F: there is a cellular automaton with finite alphabet carried by F that has a Garden of Eden but no mutually erasable patterns (it is pre-injective but not surjective). **Release input.** Family 248, "Thompson's group F is nonamenable" (OpenAI, September 23, 2026), Theorem 1.1: "Thompson's group F is not amenable." Conventions: F as above, regarded as a discrete group; a group is amenable if the bounded real functions on it admit a positive normalized left-invariant mean. The Lean formalization of the release covers Theorem 1.1. **Bridge.** - A. Le Boudec and N. Matte Bon, "Subgroup dynamics and C∗-simplicity of groups of homeomorphisms", Ann. Sci. École Norm. Sup. (4) 51 (2018), no. 3, 557–602 (arXiv:1605.01651). Theorem 4.1: if F is non-amenable, every countable subgroup of Homeo(S¹) containing F is C∗-simple. Corollary 4.2: F non-amenable ⇔ F is C∗-simple ⇔ T is C∗-simple. Theorem 4.3: the same for countable subgroups of Homeo(ℝ) containing F. Corollary 4.4: F non-amenable ⇔ Aut(F) is C∗-simple ⇔ Comm(F) is C∗-simple. Theorem 4.11: the point stabilizers for the action of T on its Furstenberg boundary are either as in their case (i), in which case F is amenable, or trivial, in which case F is non-amenable. The implication (iii) ⇒ (i) of Corollary 4.2 is due to Haagerup and Olesen. - E. Breuillard, M. Kalantar, M. Kennedy, N. Ozawa, "C∗-simplicity and the unique trace property for discrete groups", Publ. Math. IHÉS 126 (2017) 35–71, Theorem 1.3: a discrete group has the unique trace property if and only if its amenable radical is trivial; in particular every C∗-simple group has the unique trace property. - L. Bartholdi, with an appendix by D. Kielak, "Amenability of groups is characterized by Myhill's theorem", J. Eur. Math. Soc. 21 (2019), no. 10, 3191–3197 (numbering of arXiv:1605.09133). Theorem 1.1: a non-amenable group carries a cellular automaton with Gardens of Eden but no mutually erasable patterns. Corollary 1.2 gives the equivalences with amenability. Theorem A.1 (Kielak): if KG has no zero divisors, then G is amenable if and only if KG is an Ore domain (see also Guba, Comm. Algebra (2021), arXiv:2101.01848). **Deduction.** Write R for Theorem 1.1 of the release. (1) By Corollary 4.2, (i) ⇒ (iii), R gives that T is C∗-simple; by Theorem 1.3 of Breuillard–Kalantar–Kennedy–Ozawa, C∗\_r(T) has a unique tracial state. (2) Corollary 4.2, (i) ⇒ (ii), gives F; Corollary 4.4, (i) ⇒ (ii), (iii), gives Aut(F) and Comm(F); Theorems 4.1 and 4.3 give the overgroups. (3) In Theorem 4.11, case (i) would make F amenable, so by R the stabilizers are trivial. (4) F is bi-orderable (compare two elements at the left end of the first interval where they differ), hence K[F] has no zero divisors for every field K; Theorem A.1 in the direction "Ore domain ⇒ amenable", together with R, shows that K[F] is not an Ore domain. (5) Bartholdi's Theorem 1.1 and Corollary 1.2, (1) ⇔ (3), with R. Hypotheses match: the groups F and T of Le Boudec–Matte Bon are the standard Thompson groups of the definitions above; the opposite multiplication convention is immaterial, since G and G^op are isomorphic by inversion and amenability is invariant; non-amenability has the same meaning on both sides (discrete group, left-invariant mean on ℓ^∞); the countability required in Theorems 4.1 and 4.3 holds for T, Aut(F) and Comm(F); and the direction (i) ⇒ (ii), (iii) of Corollary 4.2 is the one used. **Related.** Statement (1) and the C∗-simplicity of F in (2) are also recorded, from family 248 and Le Boudec–Matte Bon, as entry GR-5 of the catalogue [atimics/math2, docs/COROLLARIES.md](https://github.com/atimics/math2/blob/fcaf37abb2a57c1aaace218546c6388c673838f1/docs/COROLLARIES.md) (8 October 2026, 01:15 UTC), which precedes this entry. **Scope.** The statement concerns the groups F, T, Aut(F), Comm(F) and the countable overgroups listed in (2), the group algebras K[F] over arbitrary fields K, and cellular automata with finite alphabet on F. ## 7. Group von Neumann algebras of surface groups **Statement.** Conditional on Theorems 1.1 and 1.2 of the OpenAI paper "An isomorphism of the free group factors" (family 287: L(F_n) ≅ L(F_{n+1}) trace-preservingly for n ≥ 3, and L(F₂) ≅ L(F₃)) and on Shlyakhtenko, "On the II₁ factors of Fuchsian groups", arXiv:2609.11074: for every integer g ≥ 2, the group von Neumann algebra L(π₁(Σ_g)) of the fundamental group of the closed orientable surface of genus g is isomorphic, as a tracial von Neumann algebra, to L(F_{2g−1}), hence to L(F_n) for every 2 ≤ n ≤ ∞, in particular to L(F₂). Consequently all surface group factors of genus g ≥ 2 are isomorphic to each other and to L(F₂). **Release input.** Family 287, "An isomorphism of the free group factors" (OpenAI, September 23, 2026). Conventions: L(G) = {λ(g)}″ on ℓ²(G) with canonical trace τ(x) = ⟨xδ_e, δ_e⟩; F_n is the free group on n generators. Theorem 1.1: for every integer n ≥ 3 there is a unital trace-preserving normal ∗-isomorphism L(F_n) ≅ L(F_{n+1}). Theorem 1.2: there exists a unital normal trace-preserving ∗-isomorphism L(F₂) → L(F₃). Corollary 6.1: for all 1 < r, s ≤ ∞ there is a unital normal trace-preserving ∗-isomorphism L(F_r) ≅ L(F_s). The Lean formalization of the release covers the isomorphism of all interpolated free group factors L(F_r) ≅ L(F_s), 1 < r, s ≤ ∞, which includes the integer ranks used here. **Bridge.** D. Shlyakhtenko, "On the II₁ factors of Fuchsian groups", arXiv:2609.11074. Theorem 1: if t + s > 0, or g ≥ 2, or g ≥ 1 and k ≥ 2, then L(Γ(g; m₁, …, m_k : t, s)) ≅ L(F_r) with r = 2g − 1 + s + t + Σ_j (1 − 1/m_j), where Γ(g; m₁, …, m_k : t, s) is the standard presentation of a finitely generated non-elementary discrete subgroup of PSL₂(ℝ) and L(F_r) is the interpolated free group factor for non-integer r (the case t + s > 0 is credited to Dykema). Corollary 1: for the fundamental group of a surface of genus g ≥ 2, L(π₁(Σ_g)) ≅ L(F_{2g−1}). The identification of the factors L(Γ) of Fuchsian groups Γ with interpolated free group factors goes back to a problem of de la Harpe and Voiculescu (Astérisque 232 (1995), 155–158, as cited in Shlyakhtenko's introduction). **Deduction.** Let g ≥ 2 and Λ = π₁(Σ_g), a cocompact torsion-free Fuchsian group (k = s = t = 0). Corollary 1 of Shlyakhtenko (case g ≥ 2 of Theorem 1) gives L(Λ) ≅ L(F_{2g−1}) with 2g − 1 ≥ 3 an integer. Theorems 1.1 and 1.2 of the release give L(F₂) ≅ L(F₃) and L(F_n) ≅ L(F_{n+1}) for n ≥ 3, so by chaining L(F_m) ≅ L(F₂) for every integer m ≥ 2, and by Corollary 6.1 also for m = ∞. Composing the trace-preserving normal ∗-isomorphisms gives L(π₁(Σ_g)) ≅ L(F₂); for II₁ factors every ∗-isomorphism preserves the unique trace. Hypotheses match: the direction used is "a surface group factor is a free group factor, and free group factors are all isomorphic"; the ranks 2g − 1 are integers inside the range of the release; the isomorphism notion on both sides is ∗-isomorphism of tracial von Neumann algebras with the canonical trace, and L(F_r) for integer r is the usual group von Neumann algebra in both sources. **Scope.** The statement concerns the fundamental groups of closed orientable surfaces of genus g ≥ 2. ## 8. Rank r Donaldson–Thomas invariants of Calabi–Yau threefolds from rank 0 **Statement.** Conditional on Corollary 1.3 (equivalently, the uncorrected large-volume tail (1.4) of Theorem 1.2) of the OpenAI paper "Prescribed large-volume charges on threefolds with trivial canonical bundle" (family 055). Let X be a smooth projective complex threefold with trivial canonical bundle and O_X(1) an ample line bundle, with H = c₁(O_X(1)). Then the weak Bogomolov–Gieseker condition (wBG) of Feyzbakhsh–Thomas, Section 3.1, holds for (X, O_X(1)). Hence, by Theorem 1 of Feyzbakhsh–Thomas (Duke Math. J. 173 (2024) 2063–2116, "Rank r DT theory from rank 0"), for every numerical K-theory class v of rank ≥ 1, the Joyce–Song generalised Donaldson–Thomas invariant J(v) ∈ ℚ counting H-Gieseker semistable sheaves of class v is given by J(v) = F(J(α₁), J(α₂), …), a universal polynomial in the invariants J(α_i) of classes α_i of rank 0 and dimension 2, whose coefficients depend only on H^∗(X, ℚ) as a graded ring with pairing, ch(v), the Chern classes of X and H. There are countably many terms and only finitely many are nonzero. If h¹(O_X) > 0, then J(v) = 0 for rank ≥ 1, as noted by Feyzbakhsh–Thomas. **Release input.** Family 055, second paper: "Prescribed large-volume charges on threefolds with trivial canonical bundle" (OpenAI, September 24, 2026). Theorem 1.2: for a smooth projective complex threefold X, an ample integral divisor H and B₀ ∈ N¹(X)\_ℚ, there are constants k ∈ ℚ_{≥0} and R_∗ > 0, depending only on (X, H, B₀), such that finite-slope ν_{a,b}-semistable objects E in B_{H,B₀+bH} with ν_{a,b}(E) = 0 satisfy ch₃^(B₀+bH)(E) ≤ (a²/6 + k) H² ch₁^(B₀+bH)(E), and the stronger bound with k = 0 holds whenever a > R_∗, for every b ∈ ℝ. Corollary 1.3: for every polarized smooth projective threefold (X, H) there is R_∗ > 0 such that, if α > f(β) := (β² + R_∗²)/2, then Q_{α,β}(v_H(E)) ≥ 0 for every nonzero semistable object E in B_{H,βH}, for the slope (v₂ − βv₁ + (β² − α)v₀)/(v₁ − βv₀) (+∞ at zero denominator), where Q_{α,β}(v) = α(v₁² − 2v₀v₂) + β(3v₀v₃ − v₁v₂) + 2v₂² − 3v₁v₃ and v_H(E) = (H³ch₀, H²ch₁, H ch₂, ch₃)(E). No Lean formalization is listed for this family. **Bridge.** S. Feyzbakhsh and R. P. Thomas (FT), "Rank r DT theory from rank 0", Duke Math. J. 173 (2024), no. 11, 2063–2116 (arXiv:2103.02915). Theorem 1: if (X, O_X(1)) is a Calabi–Yau 3-fold satisfying the weak Bogomolov–Gieseker condition (wBG), then for fixed v of rank ≥ 1, J(v) = F(J(α₁), J(α₂), …) is a universal polynomial in invariants J(α_i) with all α_i of rank 0 and dimension 2, with coefficients depending only on H^∗(X, ℚ) as a graded ring with pairing, ch(v), the Chern classes of X and H. Condition (wBG), Section 3.1: for v of rank r > 0 and n ≫ 0 (n = n(r, p₁, p₂, q) as in FT), put v_n = v − [O_X(−n)] and let l_{v,n} be the lowest line through Π(v_n) meeting the boundary of U = {w > b²/2} in two points with b-values b₁ < b₂ < μ_H(v) such that b₁ ≤ −n + 1/(4r²H³) and b₂ ≥ μ_H(v) − 1/(4r²H³); then there is no ν_{b,w}-semistable object of class v_n for (b, w) ∈ l_{v,n} ∩ U. Conjecture 1.4 (BG): B_{b,w}(E) = (2w − b²)Δ_H(E) + 4(ch₂^(bH)·H)² − 6(ch₁^(bH)·H²) ch₃^(bH) ≥ 0 for ν_{b,w}-semistable E and (b, w) ∈ U; Remark 1.5: ½B_{b,w}(E) = (C₁² − 2C₀C₂)w + (3C₀C₃ − C₁C₂)b + (2C₂² − 3C₁C₃) with C_i = ch_i(E)·H^(3−i); Lemma 2.2: there are no semistable objects of class v_n at points of U below the line l_f(v_n) on which B vanishes. **Deduction.** The condition (wBG) follows from Corollary 1.3 as follows; the rest is Theorem 1 of FT. - Step 0 (dictionary). Put α = w and β = b. By FT Remark 1.5, Q_{α,β}(v_H(E)) = ½B_{b,w}(E). The slope of the release equals ν_{b,w} − β, where ν_{b,w} = (v₂ − wv₀)/(v₁ − bv₀) is the slope of FT, a constant shift on a fixed heart, so the two notions of semistability agree; the heart B_{H,bH} is FT's A_b; the shift E[k] changes v to −v, and Q is even. - Step 1. For v of rank r ≥ 1 and n large, the chord l_{v,n} ∩ U has b-endpoints b₁ ≤ −n + ε and b₂ ≥ μ_H(v) − ε, with ε = 1/(4r²H³). At the midpoint P = (b, w) of the chord, a² := 2w − b² = (b − b₁)(b₂ − b) ≥ (n + μ_H(v) − 2ε)²/4. Hence for n ≥ n₀(R_∗, μ_H(v)) the point P satisfies α = w > (b² + R_∗²)/2. - Step 2. The point P lies below l_f (FT: l_{v,n} lies below l_f), so B_P(v_n) < 0 (FT Lemma 2.2). Corollary 1.3 gives B_P(E) ≥ 0 for every ν_P-semistable E, hence there is no ν_P-semistable object of class v_n. - Step 3. The ν-semistability of objects of class v_n is constant along the line l_{v,n} through Π(v_n) (wall-and-chamber structure, FT Proposition 1.3 and the use of l_{v,n} in the quintic verification of Section 3.1). So there is no ν-semistable object of class v_n on all of l_{v,n} ∩ U, which is (wBG). The constant R_∗ depends only on (X, H), not on v or n, so enlarging n costs nothing. - Step 4. FT Theorem 1 applies. Hypotheses match: the Calabi–Yau 3-folds of FT are smooth projective threefolds with trivial canonical bundle (FT note that h¹(O_X) > 0 forces J(v) = 0 and assume h¹ = 0 without loss of generality); Theorem 1.2 and Corollary 1.3 of the release need no canonical-bundle hypothesis, hold for every ample integral H and all real b, and include objects of rank 0 (the zero-denominator case). **Scope.** The statement concerns H-Gieseker semistable sheaves for the fixed polarisation, numerical classes v of rank ≥ 1, and invariants J(α_i) of classes of rank 0 and dimension 2, with the universal polynomial of FT Theorem 1. ## 9. Ground configurations of the planar disordered Ising ferromagnet **Statement.** Conditional on Theorem 1.1 of the OpenAI paper "No bigeodesics in planar first-passage percolation" (family 212). Let (J_e) be i.i.d. couplings on the nearest-neighbour edges of ℤ² with common law G on [0, ∞) such that G has no atoms and E[min{J₁, J₂, J₃, J₄}²] < ∞ for i.i.d. J_i ∼ G. Consider the ferromagnetic Ising Hamiltonian H(σ) = − Σ_{⟨x,y⟩} J_{xy} σ_x σ_y on {+1, −1}^(ℤ²), and call σ a ground configuration if H(σ) ≤ H(σ′) for every σ′ that differs from σ on a finite set (only energy differences are used). Then almost surely the only ground configurations are σ ≡ +1 and σ ≡ −1. The coupling law may be singular continuous, bounded or of infinite mean; no shape regularity is used. **Release input.** Family 212, "No bigeodesics in planar first-passage percolation" (OpenAI, September 24, 2026), Theorem 1.1: "Suppose that G has no atoms and that, for independent X₁, X₂, X₃, X₄ with law G, E[min{X₁, X₂, X₃, X₄}²] < ∞. Then, with probability one, there is no sequence (v_i)\_{i∈ℤ} of pairwise distinct vertices with consecutive vertices adjacent and Σ_{k=i}^{j−1} t_{{v_k,v_{k+1}}} = T(v_i, v_j) for every i < j." Model: the weights t_e on the unoriented nearest-neighbour edges of ℤ² are independent with common Borel law G on [0, ∞), and T(x, y) is the infimum of the costs of finite nearest-neighbour walks. The Lean formalization of family 212 covers a different paper of the family (the differentiability of the planar first-passage norm); Theorem 1.1 of this paper is outside its scope. **Bridge.** Damron–Hanson, "Bigeodesics in first-passage percolation", Comm. Math. Phys. 349 (2017) 753–776 (arXiv:1512.00804), Section 2.3, after C. Newman, Topics in disordered systems, Lectures in Math. ETH Zürich, Birkhäuser, 1997: on the dual lattice (ℤ²)\_∗ = ℤ² + (1/2, 1/2) with i.i.d. positive couplings J_{x,y}, set t_e = J_{x,y} for the primal edge e dual to the bond {x, y}; there are non-constant ground states of the spin model if and only if the induced first-passage percolation model has bigeodesics, and any non-constant ground state has a two-sided circuitless infinite path of primal edges dual to bonds with σ_x = −σ_y (an interface), which is a bigeodesic. The same statement is in Auffinger–Damron–Hanson, "50 years of first-passage percolation", AMS University Lecture Series 68 (2017) (arXiv:1511.03262), Section 4, after Theorem 4.25 (equivalence with Question 31). The definition of a ground configuration by finite modifications follows Bassan–Gilboa–Peled, "Non-constant ground configurations in the disordered ferromagnet", arXiv:2309.06437. **Deduction.** Dualise: the Ising model lives on the dual lattice (ℤ²)\_∗ ≅ ℤ². Set t_e := J_{e_∗} for the dual bond e_∗ crossing the primal edge e. Then (t_e) are i.i.d. with law G, so Theorem 1.1 applies to the induced first-passage percolation (G has no atoms, hence J > 0 almost surely, and the moment hypothesis is the same). The constant configurations are ground configurations since J ≥ 0. The direction needed is: a non-constant ground configuration yields a bigeodesic (Damron–Hanson Section 2.3, after Newman). Let Γ be the set of primal edges dual to bonds with σ_x ≠ σ_y; every primal vertex has even degree in Γ. - (i) Γ has no finite cycle: if a simple cycle c were in Γ, flipping the finite set S of dual sites enclosed by c changes only the bonds crossing c, all of which are unsatisfied, so ΔH = −2 Σ_{e∈c} J_e < 0, contradicting the ground configuration property. - (ii) A component of Γ is therefore an acyclic graph with all degrees even, hence infinite, and it contains a bi-infinite simple path π. - (iii) Every finite sub-path P of π is a geodesic: for any other path P′ between its endpoints, each excursion e′ of P′ away from P, closed by the segment s of P, forms a simple cycle c. The ground configuration inequality gives Σ_{c∖Γ} J ≥ Σ_{c∩Γ} J; since s ⊂ Γ and c∖Γ lies in e′, the cost of e′ is at least the cost of s. Replacing the excursions by segments gives a walk inside the path graph P, of cost at least that of P. Hence the cost of P equals the passage time between its endpoints, and π is a bigeodesic with pairwise distinct vertices. Since Theorem 1.1 excludes all bigeodesics simultaneously on one event of probability one, the only ground configurations on that event are the two constant ones. No ergodicity or limit-shape input is used beyond Theorem 1.1. **Scope.** The statement concerns i.i.d. couplings on ℤ² with the hypotheses of Theorem 1.1, and ground configurations in the sense of finite modifications. ## 10. The weak recovery threshold of the four-community stochastic block model **Statement.** Conditional on Theorem 1.1 of the OpenAI paper "The Reconstruction Threshold for the Ferromagnetic Four-State Potts Model" (family 229). Fix a ≥ b > 0 and let G_n be the symmetric 4-community sparse stochastic block model: labels σ₁, …, σ_n are i.i.d. uniform on {1, 2, 3, 4}; each pair of vertices is joined independently with probability a/n if the labels agree and b/n otherwise; let d = (a + 3b)/4 be the mean degree. For an estimator σ̂\_n (a function of G_n and independent randomness) let ov(σ, σ̂) = max over permutations π of S₄ of (1/n) #{v : σ̂\_v = π(σ_v)}. - (i) If (a − b)² ≤ 4(a + 3b) and d > 1, then for every sequence of estimators lim_n E ov(σ, σ̂\_n) = 1/4; weak recovery is information-theoretically impossible, including at equality. - (ii) If (a − b)² > 4(a + 3b), there exist ε > 0 and an algorithm running in time O(n log n) with P(ov ≥ 1/4 + ε) → 1. Hence, for a ≥ b > 0 and d > 1, weak recovery is possible exactly when (a − b)² > 4(a + 3b), the Kesten–Stigum threshold. **Release input.** Family 229, "The Reconstruction Threshold for the Ferromagnetic Four-State Potts Model" (OpenAI, October 5, 2026). Model: the root spin is uniform on {1, 2, 3, 4}; each edge transmits through P_λ(j|i) = λ·1{i = j} + (1 − λ)/4 with λ ∈ [0,1]; the tree is the d-ary tree or the Galton–Watson tree with independent Poisson(d) offspring; the observer sees the rooted tree T and the spins σ_{L_n} at level n; a_n(d, λ) = E[½ Σ_{i=1}^{4} |Pr(σ_ρ = i | T, σ_{L_n}) − 1/4|], the expectation being over the tree and the spins and not conditioned on survival. Theorem 1.1: if dλ² ≤ 1, then lim a_n(d, λ) = 0 on the rooted d-ary tree for every integer d ≥ 2 and on the Poisson(d) Galton–Watson tree for every real d > 0; the assertion includes dλ² = 1. The proof is computer-assisted (exact-arithmetic verification of polynomial inequalities). The Lean formalization of family 229 covers the three-state supercritical reconstruction theorem of a different paper of the family; the four-state non-reconstruction theorem is outside its scope. **Bridge.** - E. Mossel, A. Sly, Y. Sohn, "Exact phase transitions for stochastic block models and reconstruction on trees", arXiv:2212.03362; STOC 2023; Ann. Probab. (2025), doi 10.1214/24-AOP1723. Setting: G(n, q, a/n, b/n), labels i.i.d. uniform, d = (a + (q − 1)b)/q, λ = (a − b)/(a + (q − 1)b), broadcast channel P(σ_u = j | σ_v = i) = λ + (1 − λ)/q if i = j and (1 − λ)/q otherwise, which for q = 4 is the channel of the release. Definition 1.4: detection is possible if limsup_n (1/n) E max_{Γ∈S_q} #{u : σ̂\_u = Γ(σ_u)} ≥ 1/q + ε for some ε > 0, for any G-measurable estimator. Definition 1.9: reconstruction is solvable on a tree T if limsup_n d_TV(σ^i(n), σ^j(n)) > 0 for some i, j; otherwise there is non-reconstruction. Theorem 2.1: if there is non-reconstruction on the Poisson(d) Galton–Watson tree for the broadcast process with second eigenvalue λ, then detection is information-theoretically impossible on G(n, q, a/n, b/n) for (a, b) corresponding to (d, λ). - E. Abbe and C. Sandon, "Detection in the stochastic block model with multiple clusters: proof of the achievability conjectures, acyclic BP, and the information-computation gap" (arXiv:1512.09080; Comm. Pure Appl. Math. 71 (2018) 1334–1406, "Proof of the achievability conjectures for the general stochastic block model"). Corollary 1: acyclic belief propagation solves detection in SBM(n, k, a, b) if (a − b)²/(k(a + (k − 1)b)) > 1, and can be run in time O(n log n). Lemma 1: for uniform community sizes, an algorithm solving detection also achieves agreement at least 1/k + ε/k − o(1) with high probability. **Deduction.** For q = 4, d = (a + 3b)/4 and λ = (a − b)/(a + 3b), so dλ² = (a − b)²/(4(a + 3b)); for a ≥ b > 0, λ ∈ [0, 1), the ferromagnetic channel of the release, with mean d > 0. - (i) If (a − b)² ≤ 4(a + 3b), then dλ² ≤ 1 and Theorem 1.1 gives a_n → 0 on Poisson(d) trees. The passage to the fixed-tree hypothesis of Mossel–Sly–Sohn follows Section 7 of the release's three-state paper "The exact reconstruction threshold for the three-state symmetric channel" (Corollary 1.2). For a fixed tree T let A_n(T) = E[½ Σ_i |P(σ_ρ = i | T, σ_{L_n}) − 1/4| | T]; by information degradation A_n(T) is non-increasing in n; the offspring beyond depth n are independent of the root and of the depth-n spins given T_{≤n}, so E A_n(T) = a_n, hence E lim A_n = 0 and lim A_n(T) = 0 almost surely. By Bayes' rule A_n(T) = ¼ Σ_i d_TV(μ_i^T, μ^T), with μ_i^T the law of the level-n spins given root i and μ^T their average, so all pairwise d_TV(μ_i^T, μ_j^T) → 0 for almost every T, which is non-reconstruction in the sense of Mossel–Sly–Sohn. Their Theorem 2.1 then gives limsup (1/n) E max_Γ agreement ≤ 1/4; the quantity is always at least 1/4 (average over the 24 permutations), so the limit is 1/4. Randomized estimators reduce to deterministic ones by fixing the seed. - (ii) If (a − b)² > 4(a + 3b), Corollary 1 of Abbe–Sandon with k = 4 gives an O(n log n) detection algorithm, and Lemma 1 turns it into overlap at least 1/4 + ε/4 − o(1) with high probability. Here dλ² > 1 forces d > 1. - Hypotheses match: the block models agree (i.i.d. uniform labels, edge probabilities a/n and b/n); the channels agree; the release covers every real Poisson mean d > 0 and λ ∈ [0, 1], and λ ≥ 0 is exactly a ≥ b. Mossel–Sly–Sohn require non-reconstruction with the observer knowing T, which is the observer of the release, since the spins at level n depend only on T_{≤n}. **Scope.** The statement concerns the assortative case a ≥ b > 0, with part (i) for mean degree d > 1. ## 11. The quantum Fourier transform and fanout in constant depth **Statement.** Conditional on Theorem 1 of the OpenAI paper "Product-projection localization and the QAC⁰ parity lower bound" (family 274: Parity is not in QAC⁰, measured-output model) and on Theorem 1.1 of Gretta–Joshi, arXiv:2608.06703 (which rests on Gretta–Gupta–Joshi, arXiv:2604.02793 and arXiv:2604.15298): the quantum Fourier transform cannot be approximated in QAC⁰, even to inverse-polynomial fidelity. Precisely, fix an integer d ≥ 0 and a real c ≥ 1; for each n let q_n be any modulus with 2^n ≤ q_n < 2^(n+1), where QFT_q|x⟩ = q^(−1/2) Σ_{y