embed_injective

"Conjecture 2.1.5: Tr(Frob, IndCoh_O(LS_restr)) is concentrated in degree 0 ... Ramanujan-Arthur over Q_ℓ (Conjecture 3.4.6)"

"The orbit decomposition of the centered symmetry group governs the within-orbit structure: worlds in the same orbit are conditionally identically distributed, and on orbits satisfying a richness condition and countable infinitude they are conditionally i.i.d. given a rigid orbit-specific directing measure."

"Developers retrospectively labeled their self-reported intentions, tool-supported actions, and emotions from screen recordings"

"incidence graph B(X,Y) … parity(g,f)-factor … discharging rules (i)–(iv)"

"two agents, without prior mathematical knowledge, can develop a shared symbolic protocol to solve a visually grounded task where the use of a numerical system facilitates extrapolation"

"A counterfactual parameter is normalization-free if and only if it is constant on equivalence classes; otherwise any point identification is created by the normalization rather than by the model."

"Information Gain Collapse—autoregressive generation makes each step highly dependent on prior outputs"

"The Maxwell count is fM(⟨r⟩)=6−5⟨r⟩/2... GRC fraction S∞=α[1−(u*)³]"

"Theorem 5.1. For every ℓ > 0 and every horizontal charge μ ... there is a finite positive measure ν on K_ℓ with μ = ∫ [γ] dν(γ)"

"k*=argmax σj/σj+1 ... gap ratio R(t)=σk*/σk*+1"

"a neural cellular automaton with only 321 learnable parameters achieves perfect length generalization"

"By theorem 1, any formula ϕ(¯x) with k free variables can be encoded as a finite set of types. ... M |= ϕ(¯a) iff tp(¯a) ∈ ι(ϕ)"

"the topological nature of gauging interfaces is obscured by the fact that the constrained Hilbert space depends on the location of the interface"

"Mt = (Mt−1 + ΔM) (mod 2π)... |eiθ| ≡ 1 regardless of sequence length"

"Corollary 1. A^{π_w→π}_s(r) = V^{π_w}(s;r) + [M^{π_w}_s(w)/M^{π_w}_w(w)] (V^π(w;r) − V^{π_w}(w;r)) − V^π(s;r)."

"Factorized credit assignment... phase-specific gradient weighting"

"Data augmentation via GPT-4o-mini back-translation to alternate languages effectively tripled the training corpus while preserving semantic content and class distribution ratios."

"Redundancy Ratio: Information density calculated as (1 - len_compressed / len_original)"

"four-namespace memory system (episodic, semantic, reflection, working) … anchor update strategies: EMA, Retrieval, Hybrid"

"Theorem 4.1 … Galois embedding problem … solvable iff K is the splitting field of the 3-division polynomial of an elliptic curve E/Q"

"Walsh functions... correspondence between Walsh series and diagonal operator compilation"

"if NN latent space geometry is known and possesses specific properties, label prediction complexity can be significantly reduced... by associating label prediction with the O(1) complexity closest cluster center search in a vector system used as target for latent space configuration (LSC)... finding indexes of several largest and lowest values in the embedding vector"

"letter alignment and inter-class decodability persist into the network and are gradually suppressed across layers... memory representations are initially overlapping, become more separated through the middle... target alignment... rises sharply near the end"

"the screening unit can also represent the absence of relevant context"

"the self-prior captures visual–proprioceptive associations, functioning as a probabilistic body schema"

"The Surrogate Verifier... operates in a completely independent LLM session... information isolation ensures that the verifier’s test generation is conditionally independent of the generator’s internal state"

"manifold complexity itself functions as an implicit supervisory signal—enabling single-neuron ANNs to distinguish correct mappings without pretraining"

"We observe that these decoded tokens meaningfully capture sound sources... meaningful audio information emerging in the last 5 layers"

"the signals governing update direction and update magnitude have asymmetric requirements: the directional signal can be sparse but must be reliable ... the magnitude signal ... benefits from being as dense as possible"

"Theorem 3.3. ... samples a panel uniformly from the set of all panels."