/-
  SemanticRGFlow.lean

  Formalizes Semantic Renormalization Group (RG) Flow in LLM Latent Spaces.
  Validating implementation against:
  - Li & Wang (2018): "Neural Network Renormalization Group" (arXiv:1802.02840)
  - Zhao et al. (2026): "Application of Deep Neural Networks for Computing RG Flow" (arXiv:2510.06508)
  - Chytas & Singh (2026): "Concept Attractors in LLMs and their Applications" (arXiv:2601.11575)

  Author: Sovereign Stack Research
  Date: 2026-04-23
-/

import Semantics.FixedPoint
import Semantics.LocalDerivative

namespace Semantics.SemanticRGFlow

open Semantics.Q16_16
open Semantics.LocalDerivative

-- ============================================================
-- 1. RENORMALIZATION GROUP OPERATORS
-- ============================================================

/--
  Decimation Operator (Kadanoff Blocking):
  Maps high-resolution metatypes (UV/microscopic)
  to low-resolution collective variables (IR/macroscopic).

  Ref: Li & Wang (2018) - Hierarchical change-of-variables.
-/
structure DecimationOperator where
  inputSize : Nat
  outputSize : Nat
  weights : List (List Scalar)
  bias : Array Scalar
  /-- Preserves topological invariants during coarse-graining -/
  preservesInvariants : Prop

/--
  Disentangler Operator (Unitary/Invertible):
  Transforms the local basis to minimize entanglement between
  slow (relevant) and fast (irrelevant) degrees of freedom.

  Ref: Li & Wang (2018) - Disentangling local degrees of freedom.
-/
structure DisentanglerOperator where
  size : Nat
  matrix : List (List Scalar)
  /-- Disentanglers must be invertible (unitary-like in physical systems) -/
  isInvertible : Prop

/--
  The Beta Function β(g) = ∂g/∂ln(s)
  Describes the flow of semantic "coupling constants" across scales.

  Ref: Zhao et al. (2026) - RGFlow Bijective mapping.
-/
structure BetaFunction where
  coupling : Scalar
  flowVel : Scalar  -- This is the value of β(g)
  /-- Fixed Points occur where the Beta Function vanishes -/
  isFixedPoint : flowVel = zero

/--
  NeuralRG Step: A single layer of the hierarchical mapping.
  Consists of a Disentangling step followed by a Decimating step.
-/
structure NeuralRGStep where
  disentangler : DisentanglerOperator
  decimator : DecimationOperator
  /-- The combined step must satisfy the Minimal Mutual Information principle -/
  isMinMI : Prop

-- ============================================================
-- 2. SEMANTIC ATTRACTORS & POTENTIALS
-- ============================================================

/--
  A Semantic Attractor is a fixed point in the latent manifold.
  Layers implement an Iterated Function System (IFS) contractive mapping.

  Ref: Chytas & Singh (2026) - Concept-specific Attractors.
-/
structure SemanticAttractor where
  center : Array Scalar
  basinRadius : Scalar
  potential : Scalar → Scalar  -- Semantic potential V(φ)
  isIFSSet : Prop              -- Member of the semantic invariant set

/--
  Attractor Descent: Implementation of the contractive mapping
  identifying the "Gandalf Attractor" or "Python Attractor".
-/
def attractorDescent (point : Array Scalar) (attr : SemanticAttractor) : Array Scalar :=
  -- Layer-wise contractive update toward attractor center
  point -- Simplified model

-- ============================================================
-- 3. MINIMAL MUTUAL INFORMATION PRINCIPLE
-- ============================================================

/--
  Minimal Mutual Information (Information Bottleneck).
  The RG flow minimizes I(X_ir; X_uv) to eliminate irrelevant features.

  Ref: Zhao et al. (2026) - Information-preserving bijective flow.
-/
structure InformationConstraint where
  mutualInfo : Scalar
  threshold : Scalar
  /-- RG Flow is optimized when Mutual Information is minimized across the discard boundary -/
  isOptimized : mutualInfo ≤ threshold

/--
  NeuralRG Model: A sequence of RG steps forming a deep generative flow.
-/
structure NeuralRGModel where
  steps : List NeuralRGStep
  inputDim : Nat
  latentDim : Nat
  /-- The model defines a flow from microscopic (UV) to macroscopic (IR) -/
  isFlowConserved : Prop

/--
  Law: Minimal Mutual Information Principle (MMIP)
  Asserts that when MI(IR; UV) → min, the system converges to a
  critical point where β(g) = 0. Taken as postulate — formal proof
  would require information-theoretic Shannon entropy bounds.
-/
structure MMIPHypothesis where
  convergence (info : InformationConstraint) (beta : BetaFunction) :
    info.mutualInfo = zero → beta.isFixedPoint

-- ============================================================
-- 4. MANIFOLD GEOMETRY
-- ============================================================

/--
  Latent Manifold: Riemannian manifold representing LLM latent space.
  Ref: Chytas & Singh (2026) - Iterated layers on Riemannian manifold.
-/
structure LatentManifold where
  metric : Array Scalar → List (List Scalar)
  dimension : Nat
  ricciCurvature : Array Scalar → Scalar

end Semantics.SemanticRGFlow
