What I learned today

  Philosophy of GRM GRM is the attempt to discover the decomposition itself. The decisive question is therefore not merely “How do we solve this problem?” but “Which parts of the thing we are trying to solve actually belong to the problem?” Detailed Table of Contents Part I — The Problem of Representation 1. When the Map Becomes the Maze 1.1 The distinction between a problem and its representation 1.2 Intrinsic difficulty versus representation-induced difficulty 1.3 When a successful encoding becomes an inherited constraint 1.4 Mathematical work created by notation, coordinates, staging, and decomposition 1.5 Equivalent representations with unequal inferential closure 1.6 Why proof length does not measure structural complexity 1.7 The map–territory error in mathematical reasoning 1.8 Representation debt as accumulated artificial obligation 1.9 The first GRM question: what would remain if the representation disappeared? 1.10 From solving inside a representation to reconstructing the...

Use of AI in Mathematical Education Research-Oriented Table of Contents Part I — What Exactly Is Being Represented? 1. Mathematical knowledge as a carrier 1.1 Standard representation: curriculum as ordered topics; hidden assumptions: monotone progression, stable prerequisites, one dominant decomposition. 1.2 Knowledge graphs: nodes = concepts, edges = prerequisite/implication/dependence. 1.3 Learning spaces: admissible knowledge states K ⊆ 2 Q K\subseteq 2^Q , allowing multiple legal paths through the same domain. 1.4 Hypergraph representations: prerequisites may be conjunctive, disjunctive, compensatory, or context-dependent. 1.5 Concept lattices and Galois structures: concepts represented through shared attribute closure rather than curricular sequence. 1.6 Equivalence boundary: sequence ≃ \simeq DAG only when prerequisite relation is effectively total or near-total; DAG ≄ \not\simeq learning space when alternative acquisition paths matter. 1.7 Degenerate cases: one-concept domain...

  Semantic Cloud Research-Oriented Table of Contents 0. The Entire Primitive Alphabet 0.1 ḓ := THIS | NOT-THAT 0.2 Why distinction precedes object, identity, relation, context, space, probability and semantics 0.3 Orientation already implicit in THIS | NOT-THAT 0.4 Negation/complement: what exactly is NOT-THAT ? 0.5 Self-application: can distinction distinguish distinctions? 0.6 Closure problem: C l o s u r e ( ḓ ) Closure(ḓ) 0.7 Minimality criterion: p p is primitive only if p ∉ C l o s u r e ( ḓ ) p\notin Closure(ḓ) I. First-Order Consequences of Distinction 1.1 Difference from distinction 1.2 Boundary: THIS || NOT-THAT 1.3 Identity as absence of consequential distinction 1.4 Multiplicity from repeated distinction 1.5 Possibility domain U U as accumulated distinctions, not a primitive container 1.6 Inclusion, exclusion and complement 1.7 Symmetry versus oriented distinction II. Distinctions Acting on Distinctions 2.1 ḓ 1 → ḓ ( ḓ 1 ) ḓ_1\rightarrowḓ(ḓ_1) 2.2 Relations as distinc...

  Semantic Cloud Research TOC 0. Primitive Alphabet 0.1 Possibility domain U \mathcal U 0.2 Primitive distinction ḓ ( x , y ) : = x ∣ ¬ y ḓ(x,y):=x|\neg y 0.3 Context/perturbation C t C_t 0.4 Active distinctions D t D_t 0.5 Frozen invariants I t I_t 0.6 Refinement ⪯ \preceq , incompatibility ⊥ \perp , conditional independence ∥ \parallel 0.7 Directed transition a → b a\to b 0.8 Directional cost ω t ( a → b ) \omega_t(a\to b) 0.9 Propagation limiter m m 0.10 Surviving influence π t ( a → b ) \pi_t(a\to b) 0.11 Active relation A t = { ( a , b ) : π t ( a → b ) > 0 } A_t=\{(a,b):\pi_t(a\to b)>0\} 0.12 Epistemic type ϵ \epsilon 0.13 External discriminator Γ \Gamma 0.14 Witness W W , debt δ \delta , residue ρ \rho , counterkernel ϰ \varkappa 0.15 Persistent state Σ \Sigma 0.16 Terminal algebra Θ \Theta   I. Fundamental Laws 1.1 ¬ ḓ ⇒ \negḓ\Rightarrow no semantics 1.2 Meaning is distinction-relative, not token-intrinsic 1.3 Directionality: ω ( a → b ) ≠ ω ( b → a ) \omega(a\t...