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149 lines
4.9 KiB
Text
149 lines
4.9 KiB
Text
/-
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SemanticRGFlow.lean
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Formalizes Semantic Renormalization Group (RG) Flow in LLM Latent Spaces.
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Validating implementation against:
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- Li & Wang (2018): "Neural Network Renormalization Group" (arXiv:1802.02840)
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- Zhao et al. (2026): "Application of Deep Neural Networks for Computing RG Flow" (arXiv:2510.06508)
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- Chytas & Singh (2026): "Concept Attractors in LLMs and their Applications" (arXiv:2601.11575)
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Author: Sovereign Stack Research
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Date: 2026-04-23
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-/
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import Semantics.FixedPoint
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import Semantics.LocalDerivative
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namespace Semantics.SemanticRGFlow
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open Semantics.Q16_16
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open Semantics.LocalDerivative
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-- ============================================================
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-- 1. RENORMALIZATION GROUP OPERATORS
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-- ============================================================
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/--
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Decimation Operator (Kadanoff Blocking):
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Maps high-resolution metatypes (UV/microscopic)
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to low-resolution collective variables (IR/macroscopic).
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Ref: Li & Wang (2018) - Hierarchical change-of-variables.
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-/
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structure DecimationOperator where
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inputSize : Nat
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outputSize : Nat
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weights : List (List Scalar)
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bias : Array Scalar
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/-- Preserves topological invariants during coarse-graining -/
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preservesInvariants : Prop
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/--
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Disentangler Operator (Unitary/Invertible):
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Transforms the local basis to minimize entanglement between
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slow (relevant) and fast (irrelevant) degrees of freedom.
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Ref: Li & Wang (2018) - Disentangling local degrees of freedom.
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-/
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structure DisentanglerOperator where
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size : Nat
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matrix : List (List Scalar)
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/-- Disentanglers must be invertible (unitary-like in physical systems) -/
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isInvertible : Prop
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/--
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The Beta Function β(g) = ∂g/∂ln(s)
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Describes the flow of semantic "coupling constants" across scales.
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Ref: Zhao et al. (2026) - RGFlow Bijective mapping.
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-/
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structure BetaFunction where
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coupling : Scalar
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flowVel : Scalar -- This is the value of β(g)
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/-- Fixed Points occur where the Beta Function vanishes -/
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isFixedPoint : flowVel = zero
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/--
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NeuralRG Step: A single layer of the hierarchical mapping.
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Consists of a Disentangling step followed by a Decimating step.
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-/
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structure NeuralRGStep where
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disentangler : DisentanglerOperator
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decimator : DecimationOperator
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/-- The combined step must satisfy the Minimal Mutual Information principle -/
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isMinMI : Prop
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-- ============================================================
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-- 2. SEMANTIC ATTRACTORS & POTENTIALS
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-- ============================================================
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/--
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A Semantic Attractor is a fixed point in the latent manifold.
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Layers implement an Iterated Function System (IFS) contractive mapping.
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Ref: Chytas & Singh (2026) - Concept-specific Attractors.
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-/
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structure SemanticAttractor where
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center : Array Scalar
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basinRadius : Scalar
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potential : Scalar → Scalar -- Semantic potential V(φ)
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isIFSSet : Prop -- Member of the semantic invariant set
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/--
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Attractor Descent: Implementation of the contractive mapping
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identifying the "Gandalf Attractor" or "Python Attractor".
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-/
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def attractorDescent (point : Array Scalar) (attr : SemanticAttractor) : Array Scalar :=
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-- Layer-wise contractive update toward attractor center
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point -- Simplified model
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-- ============================================================
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-- 3. MINIMAL MUTUAL INFORMATION PRINCIPLE
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-- ============================================================
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/--
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Minimal Mutual Information (Information Bottleneck).
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The RG flow minimizes I(X_ir; X_uv) to eliminate irrelevant features.
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Ref: Zhao et al. (2026) - Information-preserving bijective flow.
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-/
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structure InformationConstraint where
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mutualInfo : Scalar
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threshold : Scalar
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/-- RG Flow is optimized when Mutual Information is minimized across the discard boundary -/
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isOptimized : mutualInfo ≤ threshold
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/--
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NeuralRG Model: A sequence of RG steps forming a deep generative flow.
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-/
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structure NeuralRGModel where
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steps : List NeuralRGStep
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inputDim : Nat
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latentDim : Nat
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/-- The model defines a flow from microscopic (UV) to macroscopic (IR) -/
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isFlowConserved : Prop
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/--
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Law: Minimal Mutual Information Principle (MMIP)
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Asserts that when MI(IR; UV) → min, the system converges to a
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critical point where β(g) = 0. Taken as postulate — formal proof
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would require information-theoretic Shannon entropy bounds.
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-/
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structure MMIPHypothesis where
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convergence (info : InformationConstraint) (beta : BetaFunction) :
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info.mutualInfo = zero → beta.isFixedPoint
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-- ============================================================
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-- 4. MANIFOLD GEOMETRY
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-- ============================================================
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/--
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Latent Manifold: Riemannian manifold representing LLM latent space.
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Ref: Chytas & Singh (2026) - Iterated layers on Riemannian manifold.
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-/
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structure LatentManifold where
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metric : Array Scalar → List (List Scalar)
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dimension : Nat
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ricciCurvature : Array Scalar → Scalar
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end Semantics.SemanticRGFlow
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