embed_injective

"ΔCC reduces to single-reference CC for M=1; size-extensive via cancellation of unlinked diagrams after C-condition"

"Lemma 4.1, 4.4, 4.5: close-coupling via acceptance probability and Gaussian proposals under non-smooth composite potential"

"sensk,b(C) = || sum_{j=0}^{k-1} C[:, jb+1] ||_2 for lower-triangular Toeplitz C"

"TV(q(x0|y)∥p(x0|y)) ≤ I(2ϵ + h(ϵ,M,λ,D)/(C−ϵ−h)) + Iϵ = Õ(I/√M + Iϵ)"

"Pert.(xt) = Var_Δ [P̃_Δ(xt)]"

"axion quality factor Qa ≡ (c/v)^2 ... coherence time τa ≈ Qa/νa ... stochastic lineshape"

"Theorem 3.2 (Instability via Policy Divergence) ... min L(Q) ≥ μ·E[∥πθ(s′)−πβ(s′)∥²]"

"value-revealing prints consistently outnumber assertions"

"support end component (SEC)... Reachf(S) ... maximal SECs"

"to leading order, RT mixing is governed by the competing dynamics between diffusion of ln ρ-bar and mass conservation"

"i*_C TQFT(C) ∩ i*_(ϕ,β) TQFT(Vect G) ≠ {0}"

"Λ_α(X_R) := (-1)^ℓ / (2^ℓ ℓ!) ∑ ... ∏ (1/d <x_rj(1), x_rj(2)> - bσ² I) ⋅ ∏ φ^(s_e-1)(τ)"

"Graph-Guided Policy Optimization ... pruning memory nodes associated with redundant actions ... critical path from the root to the answer node"

"query shifts amplify the hubness phenomenon"

"V^{(m)}_{LD} ≅ H^1(C;Z_2) ⊗_H S_m with S_m = ⊕_{ν=1}^m 2^{ν-1} (ker H_Z)⊗_H^{(ν-1)}"

"Restrict to maximal convex subset K* = conv(K_m) away from origin to bound α away from 0 (Corollary 6, Remark 10)"

"recursive error bound E(θk+1)≤(1−λ)E0+λ[(1−p̃k)ρE(θk)+p̃kEmax] (Eq. 7)"

"Posterior dynamics dm_t = P_t dŴ_t, dP_t = −P_t² dt independent of policy; entropy regularization orthogonal to learning."

"A novel Volterra-type (or mild) formulation allows to keep technical (rough-stochastic) consideration to a minimum."

"the degree of role confusion strongly predicts attack success even before generation"

"Corollary 1 … root among the divisors of its constant term … rational root theorem"

"unifying both domains through a shared deform map representation"

"Theorem 3.1. ... Q_n = G_Q ×1 C ×2 C ×3 C ×4 C ... mlrank(Q_n)=(4,4,4,4)"

"rigorous angular momentum conservation (AMC) in elastic, inelastic, and decay channels"

"metastable core shift structures can be obtained only when the core-shift times for the ABC stacking unit above is the same as that below or 1 more than that below... total numbers... 2^(csd/3)-1"

"Theorem 1 ... all maps in it are isomorphisms. ... The Gauss-Manin connection on H^i_dR(X/S, (V,∇/S)) corresponds to the action of Π(S/k) on H^i(π_geom(X/S),V) induced from the Lyndon-Hochschild-Serre sequence."

"We term the root cause as Contextual Inertia... over 70%–90% of multi-turn errors traced to propagation of previous responses"

"diffusion models generate novel samples by interpolating between training data and iFID measures the validity of these interpolated samples."

"transfer-matrix representation of the cumulant generating function"

"an online, entity-agnostic embedding model trained via contrastive learning for robust matching"