LogicNat

"Fact 1 (Model Degradation). The accuracy β ... is worse ... iff the degradation probability q↓ > 1/2"

"modified supervised contrastive loss and environmental loss"

"convergence bound (5) under Assumptions 3.1-3.2 (L-smoothness, unbiased estimator with variance C1,C2)"

"Interventional Grounding Theorem ... AGM representation theorem"

"To maintain path independence ... integrates Pa-Attention alongside LPRoPE ... Mi,j = 1(j≤i)·1(is visible(i,j)) ... k(i)m = Rm(k(i)m + ei)"

"iterative exploration within a compact latent space using optimizable latent think tokens"

"reasoning is decomposing a potentially complex task into simpler subtasks"

"excess risk decomposes as Extraction Error + Readout Error"

"the balanced short-range Kitaev chain in Majorana form is exactly the SSH chain ... Krylov recursion ... preserves this alternating-bond matrix structure"

"time-division model as the fundamental feature of the coordination layer"

"RELBENCHv2 adds four large-scale relational datasets... autocomplete tasks... Temporal Graph Benchmark integration"

"For monoids, we establish a trichotomy: the space complexity is either Θ(1), Θ(log n), or Θ(n)... constant-space solution for commutative monoids... FL∨Com... all remaining monoids require Ω(n) space."

"establish the convergence of the resulting algorithm toward a solution of the constrained maximum-likelihood problem"

"Using a filtered subset of the CASP benchmark, we evaluate generated annotations through Frama-C’s WP plugin with multiple SMT solvers"

"AAR is transductive: the association model is trained on co-occurrence pairs drawn from the same corpus on which it is evaluated"

"We systematically evaluate multiple ProtBERT and ESM2 embedding variants as sequence representations, using the adeno-associated virus capsid as a case study"

"the complex matching distance on bounded complexes of finitely generated projective P-modules"

"Bounded Local Generator Classes for Deterministic State Evolution"

"The algorithm recursively identifies structural patterns within the WF-net and translates them into their POWL representation."

"engineering joints ... primitive constraints DP1, DP2, DIST, CD ... revolute joint ... 5 scalar constraints"

"K_n := span{i[ρ,H], R_ρ(i[ρ,H]), …, R_ρ^{n-1}(i[ρ,H])} … n* ≤ N[S] ≤ r(r+1)/2"

"size-indexed functor F[3−] ... fix(λi.λX.F(3iX)) ... μF := ∃i.(μ³F)i"

"tunneling suppression takes place at the so-called Toda lattice points … critical point … monopole point … anomalous scaling as a function of ℏ"

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

"We adopt the Mamba-2 architecture [6], which inherently supports sequence modeling without explicit positional embeddings... Non-Causal Mamba-2 [37] for two key reasons: 1) video conditions are available offline... 2) multimodal fusion across multiple modalities is difficult without a predefined order."

"the equations of motion are unchanged upon simultaneously advancing time by T/3 and increasing the oscillator coordinate index by one"

"long-lived introduction circuits... 18–24 hours... short execution time 0.5–1.5 s"

"hybrid item tokenization method that combines SID prefixes with unique item IDs... three-step item generation procedure integrated with an adaptive probabilistic fusion mechanism"

"extrinsic verification ... mitigates self-verification bias"

"convergence rate O(e^{-ct/2}) for Borda and Copeland consensus"