Research-Stack/3-Mathematical-Models/unified_9pattern_samples.json

{
  "coupling": [
    "z = n pn (z 2 ), 2 (1 − z 2 )n+1 where z = sin(σ/2)",
    "n=85,695, I=0",
    "ki=1 Buexi ≥ 1 − α, and report this set ⋆",
    "xr = 0 imply that ωxr+1 = 0",
    "out = Cer , where \u0002 \u0003\u0001 r(z) = Φ−1 CΩ∗ ln |w|2 (z) φ As Ω ∈ QD0 (h), Theorems 2",
    "I = Z ∩ [−ea , ea ]",
    "K = ω ′ |K , K ∈ K(W), ω ′ ∈ Ω, and we have ′ ′ η−1 γ (C(K, ω )) = C(K, ηγ−1 (ω )) (4",
    "A = (ax )x∈X is called α-intersubjective if X ∀B = (bx,x′ )x,x′ ∈X ∈ JM(A, A), bx,x ≥ α1S",
    "e = (1 − e−8β )# 1 F1 \u0010 1 F1 −4β 2 1; γ2 + 1; e 4 |z| \u0010 1; γ2 + 1; |z|4 2 \u0011 \u0011 #",
    "t=0 t=0 PT −1 ⋆ Since there is only one phase, the comparator is θmE throughout, so the left side equals t=0 ηt Lfs t − PT −1 \u0001 ⋆ 1 fs ⋆ 2 η L",
    "M = (S, A, O, E, Tϕ , Rϕ , γ) as follows: • Latent state st = [zt , ht ]: The agent models the latent state at time t as the combination of the current latent code zt , which encodes the most recent sparse observations, and a recurrent history ht , which maintains a memory of the field’s evolution",
    "j = 0 z i,j − zi,j−1 − yn+1−j,i = 0 ∀i ∈ [ℓ], (6",
    "Xn = k X ··· i1 =1 k X A1,n (i1 ,",
    "j ≤ c, vρ,j = 0",
    "g = c = 1) 0"
  ],
  "scaling": [
    "d ≈ N regime of a noiseless gradient flow dXt = (yJ − J⊤ JXt ) dt",
    "c = Th,c Σs /T since A is assumed to be independent of T while D ∝ T (see also Ref",
    "F = τ |F and Σ′τ (A)/Στ (A) ∼ = Σ′τ ′ (D)/Στ ′ (D) ([Jan07, 1 Lemma 2])",
    "Hmod ≈ 0",
    "R<x> := PolynomialRing(K); f := 5^3*x^12 - 22*5^2*x^10 - 33*5^2*x^8 + 44*5*x^6 - 33*5*x^4 - 22*x^2 + 1; H := HyperellipticCurve(f); GroupName(GeometricAutomorphismGroup(H)); As a result, we verify that Aut(H) ∼ = C2 × A5 , and therefore H is of Type 12",
    "N ≈ −0",
    "j ≈ 90) 106 50 10 Relative Error (vs",
    "g = αtj g∞ kf zt−1 j 7→ [a(zf )], where α ∈ Z∞ , g ∈ G(F ), g∞ ∈ G∞ , kf ∈ Kf , z = z∞ zf ∈ ZA , is a well-defined group homomorphism, and induces an isomorphism e[j] /Z∞ Γ[j] ∼ Z∞ Γ = ClF [2]",
    "k = E2,J,J = u ,△ def M ′ ′′ E2−ℓ,k ⊗E ∧k Hom(zJu , E), k′ +k′′ =k which is non-zero for some (−ℓ, k) satisfying k−ℓ = #Ju \\J −#J if and only if ℓ = #J, k ′ = #Ju \\J and k ′′ = 0, in which case we have −#J,#Ju \\J+ℓ(u) E2,u,△ −#J,#Ju \\J = E2 −#J,#Ju \\J ∼ ⊆ E2,J,Ju −#J,#Ju \\J+ℓ(u) = E2,u",
    "r≥0 We introduce a second generating series in the ramification exponent k: X D(x, y) := Dp,k (x)y k",
    "I ≈ I1 + I2 := dr2 dr1 (r2 − r1 )a |J1 (r1 , r2 )| + |J2 (r1 , r2 )| , 0 0 51 (5",
    "by ≈ Orb(C), is the set of all images of C under the elements of SO(3) (Abud and Sartori, 1983; Olive, Kolev and ≈ ≈ Auffray, 2017): n o Orb(C) = C ∈ Ela | C = g ⋆ C, g ∈ SO(3)",
    "t = 0, so the initial profile must contain oscillations of exponentially large amplitude in order for them to survive up to t̃ = 0",
    "B ≈ 625 MeV [9]",
    "ds ≈ e−τ = e−τ = e−τ M X (−1)k hk X k=0 0≤m1 <···<mk <Nm M X X (−1)k hk k=0 0≤m1 <···<mk <Nm M X X k=0 (−1)k hk Up (t)[Ap (mk h) + Amax I] · · · [Ap (m1 h) + Amax I] Up (t) 1 Y Up† (mj h)L̃(mj h)Up (mj h) j=k 1 h Y 0≤m1 <···<mk <Nm j=k i Up (mj h, mj+1 h) · L̃(mj h) · Up (m1 h)"
  ],
  "inequality_constraint": [
    "r > cArt,F (K) + 1",
    "g ≥ 3 using Fuchsian systems, and by numerical experiments based on our construction",
    "m ≥ 2",
    "ip ≤ m",
    "t ≥ T0 : Load ξ for φ and ξ1 for child node φ1",
    "h > or include ALL necessary headers",
    "n≥0 Y π0 Pin+ (Mn+1 ) , (370) n≥0 where the two arrows are (Pn ) 7→ (gn∗ Pn ) and (Pn ) 7→ (Pn+1 )",
    "t≥ θ log θ + δ, (θ − 1)(α − cθ,u ) we have n PG θ,t,u (Aη ) ≥ θ log θ α − ξ − cθ,u (1 + ε) − (θ−1)t (1 + θ)(α + ξ) − cθ,u (1 + ε)",
    "m ≥ 2 samples (non-isolation)",
    "d ≥ 2",
    "index ≤ 1, since we can construct an obvious splitting",
    "f ≤ (1 + 2η)d − 2 such that F (M1 , f ) holds is bounded from above by 74 IRINA D̄ANKOVIĆ, MAARTEN MARKERING, JASON MILLER, AND YIZHENG YUAN (p′ )ηd−4 ≤ (p′ )−4 · R−αd",
    "rv > τ and vertex v is not coherently factual",
    "t ≥ 0",
    "X = L U 2 U1 + X \u0013 Ψ1 (n)Ψ1 (n + lq2 ) , 0<|l|<L n,n+lq2 ∈I which deduces the final result"
  ],
  "assignment_boundary": [
    "optim = optimizer self",
    "N = 2n (for n qubits) [13]",
    "id=pOq9vDIYev",
    "hk = n − jk + 1 and define y1 ,",
    "X = lim K",
    "T = 0 and is also Hamiltonian dynamics",
    "dxdt = 0, Rn (4",
    "M = δ (D) (x − x′ ) (3",
    "t = (t1 ,",
    "IB = (3",
    "b = (b1 ,",
    "AT = log log T + log W",
    "x = 0} to the order made explicit in (3",
    "x = C",
    "pl / t = 1 (a) and 4 (b)"
  ],
  "mass": [
    "t = k1 one has θ(t) = 0 and hence τ (Φ1/k ) = k",
    "CAKK = 0, the value of CAKK computed on the connected configuration can be interpreted as the difference relative to the disconnected configuration",
    "S = θ−1 2 , for which the loop weight is 2S + 1 = θ; see [20, Section 3]",
    "zi = −zi + 2ai Γ/α2 + 32 zi = 12 zi + 2ai Γ/α2",
    "Q /Q = 0, 1,0 E22,0 = Coker(d1,0 1 ) = Q/Im(d1 ) = Q/Q = 0",
    "kcrit = p π/2/τ ≈ 1",
    "r≤(ℓ+ 1 )3/2 + 1(ℓ+ 1 )3/2 <r≤(ℓ+ 1 )2 + (ℓ + 21 )3 1r>(ℓ+ 1 )2",
    "k = 0 into (61) yields the reduced system:  1/2 e A0 ψ = −2ia · [αm,∗ ϕ♭,m ]♭∈{L,R},m∈Z0 ; (63) [α1/2 ϕ b m,∗ ♭,m ψ]♭∈{L,R},m∈Z0 = (I2N0 + M )a",
    "d = \\frac {-d_{\\mathcal {L}}(\\bm {m}_i,\\bm {p}_e)+b_d}{s_d}, \\qquad \\bm m_i^a = \\frac {-\\text {ext}(\\bm {m}_i,\\bm {p}_e)+b_a}{s_a}, \\label {eq:scale_shift} (19) (23) where bd ,ba are bias terms that shift the distance and angle values into a suitable range for sigmoid activation, and sd ,sa are scaling parameters controlling sensitivity",
    "rn = 2(1 − γ∗ )n/2 , n ∈ N",
    "x<p≤x log p x log x ≪ ≪ x, p X X and √ λ2 (p) log p x<p≤x \u0012 \u00133/5 x ≪ x",
    "c > 0 so that we actually have Q < q 3/2",
    "z = iy (y ∈ R), Z ∞ ℜF (z) + ℜF (2σ − 1 − z) = ℜ e−iyu (1 + e−(2σ−1)u )f (u)du Z0 ∞ =ℜ e−iyu/η w(u)du (14) (15) (16) 0 = ℜW (iy/η) ≥ 0",
    "n ≥ 1, ∀σ ∈ Gal(K/Q) , where ZK denotes the ring of integers of K",
    "V = A ⊔ B with |A| = is limited by the number of ground configurations rather |B| = N/2, we restrict to those automorphisms that prethan by the Hilbert space dimension"
  ],
  "gradient": [
    "i=1 ∂ci =d(ιvξ Θ) − ιξ dL + de ci = k X ∂J[vξ ] i=1 ∂ci e dci = (A",
    "D = C⟨zi , ∂i ; i = 1,",
    "wi =\u0010 max(Ui ); (iii) the global best score bi−1 = Si−1 \u0011 max j=1 Uj ; and (iv) the improvement ∆i = Termination mechanism",
    "IIC = sup ζ(y, θ)h(t∗ )r−6 Re L [τ1 ,τ2 ]⊆[0,τ ], {τ1 ≤t∗ ≤τ2 } h:[τ1 ,τ2 ]→R, ∥h∥C 1 ≤1 \u0002 \u0003o × ∂t∗ O(r−1 )ψ̄ (0;j̄) + O(r−4 )|χϕ̃ + ψ|2 ∂t2∗ ψ̄ (0;j̄) dvolg ≲ 1 Z X A=0 \u0001≤A 2 ♯ (0;j̄) \u00011−A 1 1 ∂t∗ (r Φ1 ) · ∂t∗ V (rψ) L L \u0010 \u0011 × ψ (0;j̄) + r−3 |χϕ̃ + ψ|2 · ∂t2∗ ψ (0;j̄) sin θdydθdφdt∗ r−6 {0≤t∗ ≤τ }∩suppζ + Boundary terms + Boundary terms′ , (7",
    "v = ∆v, H(0) v = hv",
    "T = 0, with condition ∆h = 0 in common, the somewhat basic problem is whether ∆h = 0 is necessary for the validity of T = 0 in Eq",
    "m = 3 we put Gm = 21+2m + to denote the stabiliser of ∆m in Aut(Λm )) is a normal subgroup of index 2 in√the real Clifford group ⟨Gm , h⟩ ≤ GLN (R) (see for instance [1], [9])",
    "m > 0 is defined as the symmetric second-order tensor field on M whose components in an (arbitrary) coordinate representation are 1 Tµν (x) := ∂µ ϕ(x)∂ν ϕ(x) − gµν (∂α (x)ϕ∂ α (x)ϕ + m2 ϕ(x)2 )",
    "t = e−Kt HN eKt + (i∂t e−Kt )eKt e−Kt ΦN,t",
    "n = ∂0 ϵ − (s∂1 ϵ − ϵ∂1 s) + ϵ1 ∂1 n − n∂1 ϵ1 , δϵH s = ∂0 ϵ1 + (ϵ1 ∂1 s − s∂1 ϵ1 ) + ϵ∂1 n − n∂1 ϵ , δϵH σ = −λn ϵπ ρ + ϵ1 ∂1 σ + 2∂1 ϵ1 , δϵH ρ = −λn ϵπ σ + ϵ1 ∂1 ρ , ϵ δϵH ∆ = λn ∆2 π ∆ + ϵ1 ∂1 ∆ , ρ ϵ δϵH ψ = λn ∆2 π ψ + ϵ1 ∂1 ψ , ρ δϵH π σ = − 1 ∂1 (ϵ∂1 ρ) + ∂1 (ϵ1 π σ ) , λn δϵH π ρ = − \u0011 2 ϵλn \u0010 1 ∂1 (ϵ∂1 σ) − ∂12 ϵ + 2 ∆2 (π ψ )2 + (π ∆ )2 λn λn 2ρ − δϵH π ∆ ϵ (∂1 ∆)2 + (∂1 ψ)2 + ∂1 (ϵ1 π ρ ) , 2λn ∆2 \u0011 ρ ϵλn \u0010 ψ 2 2 2 ∆ 2 = ϵ (∂ ∆) + (∂ ψ) − ∆ (π ) + (π ) 1 1 λn ∆3 ρ \u0012 1 ρ ∂1 ϵ∂1 ∆ + ∂1 (ϵ1 π ∆ ) , λn ∆2 \u0012 + \u0013 \u0013 8 δϵH π ψ = 1 ρ ∂1 ϵ∂1 ψ + ∂1 (ϵ1 π ψ )",
    "V ≥ 0, such a bound is not a priori given by the energy law, because the energy is not monotonic due to the spatial variations of a: indeed, for V ≥ 0, we have formally Z 1 ∂t E[u(t)] = (non-positive terms) − ∆a|u|2σ2 +2",
    "n = (∂x2 −∂t2 )n",
    "k = σ+ ∂+ + σ− ∂− , the action becomes: A0 = m Z δCµ = ωµν Cν",
    "HN = −∆ + N X j=1 αj δ(|x| − Rj ), (1",
    "SPST = 2 4κ ∂M This contribution vanishes provided that we impose the boundary condition ξ3 |∂M = 0"
  ],
  "chain": [
    "EN =2 (n, r) ϕ and EN =2 (0, t) at r+t+1 = ϕ",
    "k = −ηDk mk and combining these intermediate bounds yields: η ∥θk+1 − θk ∥ = η∥Dk mk ∥ ≤ η∥Dk ∥ ∥mk ∥ ≤ (1 + 2Cα )G = ηM",
    "i=1 Define the natural projection πQ : Σ → Q: Σ ∋ w = (w1 w2",
    "g = gcd(qr, k) and ρ = qr g , κ = g",
    "K = 1, for which we set N = 2k = 4t with n−1 t = 1, 2,",
    "I = T a Iˆ = a a a ̃ ←→ a a (663) ˆ for clarity)",
    "m=1 m=1 m=1 If we take the principal branch of the powers and use Newton’s expansion, we get, for any ρ ∈ (0, 1), p ∞ Y X (1 − ρα j+ℓ−1 eiΨ j+ℓ−1 (θm ) )sm = Ck ρk αkj+ℓ−1 m=1 k=0 22 T",
    "a + b = 1, then Nq (F) = (q − 1)2 ; q−1 (iv) If n = 2m = pr −1 with r < h such that r ∣ h and either a, b ∈ F pr or 2r ∣ h, a ∈ F pr and b ∈ F p2r r with b p = −b, then Nq (F) = n2 r (p − 2) + 2n; 2 13 (v) If n = 2m = 2(q−1) r pr −1 with r < h, r ∣ h, and a, b ∈ F p , then Nq (F) = { m2 (pr − 3) + 4m, m2 (pr − 1) + 2m, if a is a square in Fq ; if a is not a square in Fq (vi) In all remaining cases, an upper bound for Nq (F) is given by: 10(mn − m − n − gcd(m, n)) + (q + 5)2n α(4m − 11) + (n − α)(2m − 6) − 5 5 β(4n − 11) + (m − β)(2n − 6) −",
    "BiHS = Tr[A† B], kAkHS = hA, AiHS , A, B ∈ B(H), 2n 2 which induces an orthonormal operator basis {bi }i=1 ⊂ B(H) satisfying hbi , bj iHS = δi,j",
    "m=1 1 βm Hl+1,m (xl+1 ) if l is odd; x l+1 odd G(1, xl+1 ) = Pm (l+1)  l m=1 βm Hl+1,m (xl+1 ) if l is even",
    "L = 16 L = 18 L = 20 L = 22 L = 24 L = 26 L = 28 L = 30 0",
    "Q = 0, the RN coefficient simplifies to f (r) = 1 − 2M , r and the transformed even matrix loses its r−4 charge contribution",
    "W = W, t ∈ R, ΛLW (t) = LΛW (t)L−1 , jW (W ◦ ) = (W ′ )◦ , jLW = LjW L−1 , L ∈ P(d + 1)",
    "Kf = Q(a), a2 + 2a − 1 = 0, Aut(Kf ) = {id, σ} and Kg = Q",
    "p ≥ 2 the conditions \u0012 \u0013 \u0012 \u0013 p−ν ν X X ℓ−ν ν − 1 ℓ ℓ−p+ν p − ν − 1 (−1) w βp−ℓ−1 = (−1) wℓ βp−ℓ−1 , ℓ−1 ℓ−1 ℓ=1 ℓ=1 \u0004p\u0005 for 1 ≤ ν ≤ 2"
  ],
  "entropy": [
    "d = 2 can be derived retracing the same arguments, considering the regularized version of g defined as 1 ζ0ε (x) := − 2π log(|x| + ε) (ε > 0)",
    "Ln = − ln ε̃n , C1 the following hold: L0 ≥ 300 ln C1 , L1 ≥ max{50 ln(2C2 ), 2}, (7",
    "t = O Γ−1 log(1/ϵ)",
    "a = γ log (2n ) for γ > 0 Z 1 − 1 x,Σ−1 x d2 xe 2 ( k1 ,k2 ) J =p 2 (2π) det Σk2 −k1 A \u0014 \u0010 \u0013 \u0011 \u0015−1 \u0012 x2 +x2 Z ∞ Z ∞ i−1/2 h s12 2 1 2 − s12 x x \u0001 − 1− 1 2 2 s 2 s 11 11 (2π)−1 = 1 − ss12 dx1 dx2 e",
    "IT = O( T log(1/δ)) into both architectures recovers exactly the bounds stated in Theorem 2",
    "s≤ log(|A| − 1) − log 18",
    "nj=j (Ej ,Êj )) w(s) w (s) + Ej 0 j=1 (b) δ satisfies the jump conditions δ+ (z) = δ− (z) + log(1 − r3 (z)r4 (z)), z ∈ (Êj0 , +∞) \\ (∪nj=j0 [Ej , Êj ]), δ+ (z) + δ− (z) = iδj , z ∈ (Ej , Êj ), j = 1,",
    "SANNA = c5 [llog t]nt= n3 −1 + c6 Z +∞ (log s)3 ds s2 log(n3 −1) Combining (83), (84), (85), and Eq",
    "op ≈ C∥Σ∥op reff + log(1/δ)",
    "t = c∈C X c∈C max x1 ,x2 ∈X n o , fk (x1 , c) − fk (x2 , c) (10) without a central server, and \u0010 n o\u0011 p IT = O min DK T log(1/δ), DT , where we let Xk,t (c) ≜ {xt : (xt , yt , c) ∈ Dk,t } and    arg max fk (x, c), if Xk,t (c) ̸= ∅, best xk,t (c) = x∈Xk,t (c)  arg min fk (x, c), otherwise",
    "T > 0 and Λ > 1, we introduce the self-similar scaling: τ =− log(T − t) , Λ y= x (T − t) 1 Λ = eτ x, τ0 = − log T , Λ (2",
    "Rn ≤ 1 ln Wn∗ − ln (π(λ∗n )|λ∗n |)",
    "t = π, and reaches its global maximum SLmax = 2/3 (the maximum entropy of a maximally mixed qutrit) at the two values χt = 2π/3 and χt = 4π/3 within a period, where both (1 + 2 cos χt) and (1 + 2 cos 2χt) vanish simultaneously",
    "k ≥ l (pk − k) log l + pk log log l − k log (log l)(1 + log l \u0010 \u0011 log k ≥ l (pk − k) log l + pk log log l − k log log l − k − k log k − pk + 0",
    "dG = −SdT + V dP + µdQ, gravity thermodynamics orbit radius r rh horizon radius celestial coordinate β G free energy celestial coordinate α T temperature entropy slope F -S spin a P pressure deformation parameter η Q charge volume conjugate quantity A V conjugate quantity Θ µ chemical potential self-intersection point phase transition point (34) where T , P , and Q represent the temperature, pressure, and charge of the black hole system, respectively"
  ],
  "feedback": [
    "k=0 The radii iterate to ϵK (ω, m) = R K−1 Y k=0 2 sup x∈A(θ mk ω) \u0001 σjΛ∗ +1 Dϕ(m, θmk ω, x)",
    "n = 1 case, the control (1",
    "Y = (Yn )n∈Z be a stationary, irreducible, and aperiodic Markov chain with countable state space B, transition matrix P , and unique stationary distribution π",
    "P = □ − 1, the exact Klein–Gordon operator, but the arguments use only the principal form of P (and dynamical things like global hyperbolicity), so they apply for more general metrics that are asymptotically Minkowski or for operators P which differ from the Klein–Gordon operator □g − 1 of an asymptotically Minkowski metric g by lower-order terms",
    "n=2 X 2≤k≤2n+1 We used the almost analyticity of g̃ to control the remainder",
    "s = n (full cache), our bound gives only L ≥ ⌈log n/(Hmp)⌉, weaker than Ω(log k) but holding unconditionally on the controller",
    "k = ±ℓ: pseudopoles ωj,±,ℓ and nearby true QNMs ωj,±,ℓ , stable labeling, and microlocal control of the corresponding resolvent singularities after equatorial localization",
    "L = s=1 Ls , then the variables among different clusters can be decomposed in (26), where different clusters can be updated in parallel based on the clustered augmented Lagrangian function",
    "et = max(0, θd,t − Td,t ) is the temperature tracking error, and Kp , Ki , and Kd are controller gains",
    "h ≤ e h s hmax ≥ Cκ(ν, ρ)h while h2 2pe+1 γ 2h b δ,Rmax gn,b 2pe = νρh e (5) Scale-free adaptive planning for deterministic dynamics & discounted rewards In the case γ 2 κ ≥ 1 we can simply solve the following equations",
    "q = 0, while the behavior at the cusp 1 is controlled by the growth condition in (40)",
    "k ≥ 2, Pk ∈ {Ak , Bk } and Ck satisfy the recurrence Pk = −4Pk−1 − 8Pk−2 + 5 · 2k−1 , Ck = −4Ck−1 − 8Ck−2 , with initial values B0 = 0, A0 = A1 = B1 = C0 = 1, and C1 = −4",
    "z = wCT (fmeta ) · zCT + wHE (fmeta ) · zHE where – zCT = logits from CT model – zHE = logits from histopathology model – wCT , wHE = dynamic modality weights derived from clinical metadata The final classification probabilities are obtained using the softmax function: ŷ = sof tmax(z) This fusion mechanism enables the system to adaptively prioritize radiological or pathological information depending on patient-specific context",
    "pt = 1, the update vector reduces to a simple mini-batch stochastic gradient gt of size Nt",
    "k > 0 controls the sharpness of the gate"
  ],
  "unknown": [
    "0.017 Coding (Graph) GPT Qwen +16.51 +12.17 +39.1 +69.1 +0.42 +0.18",
    "/M tokens) BMBE 01 02",
    "0.33 -77.9%",
    "40.18 . stock-based compensation cost is measured at the date of grant based on the calculated fair value of the award and is generally recognized on a straight-line basis over the vesting period of the equity grant . the compensation cost is determined based on awards ultimately expected to vest ; therefore , we have reduced the cost for estimated forfeitures based on historical forfeiture rates . forfeitures are estimated at the time of grant and revised , if necessary , in subsequent periods to reflect actual forfeitures . there were no stock-based compensation costs capitalized as the amounts were not material . during the year ended december 31 , 2017 , we issued 2.1 million rsus and 1.6 million stock options under the lti plan . these rsus and stock options generally vest in equal amounts over a three-year vesting period provided that the employee has remained continuously employed by the company through such vesting date . stock based compensation expense was",
    "0.15/",
    "0.15/",
    "[...] dand - 47 9",
    "b + 7 [...] 6",
    "b + 7 [...] 7 )",
    "7.4k on Anthropic claude-opus-4-6 (",
    ",5! 𝑣4 𝑥6% + 𝜆73889: log 𝑃;,3 𝑣4 𝑧#\" log 𝑃",
    "p ≠ ℓ is a k-explosive prime if V appears in Gℓ (Fpk )",
    "0.130 Coding (Tree) GPT Qwen +20.73 +5.93 +13.0 +23.5 +1.59 +0.25",
    "I≠∅ (127) As aforementioned, the action on Majorana operators is that of the group of signed permutations Bn ≡ (Z2 )2n ⋊ S2n , i",
    "b +for 7"
  ]
}