/-
MasterEquation.lean - Anisotropically Frustrated Torsional Gradient Flow

This module formalizes the "Minimal Compact System" (Section 7) as the 
authoritative governing evolution for the Sovereign Informatic Manifold.

Equations (Discrete Time):
1. Phase Flow:   ϕ_{t+1} = ϕ_t + Δt [ ∇_i(M^ij ∇_j μ) - σ (∂I_lock/∂ϕ) ]
2. Local Potential: μ = δF/δϕ
3. Embedding Flow: X^A_{t+1} = X^A_t + Δt [ -Λ^AB(X^B - X_0^B) - δF/δX^A + τ T^A ]

One-line interpretation: The fabric folds back into n-space, snagging on 
anisotropic frustration, storing stress as torsional geometry.
-/

import Semantics.DynamicCanal
import Semantics.BraidBracket
import Semantics.ManifoldFlow

namespace Semantics.MasterEquation

open DynamicCanal
open Semantics.BraidBracket
open Semantics.ManifoldFlow

-- =============================================================================
-- THE MINIMAL COMPACT SYSTEM (Section 7)
-- =============================================================================

/-- 
Executes one step of the "n-space foldback-lock" equation.
Encapsulates the hyperfluid phase evolution and embedding dynamics.
-/
def foldbackLockStep 
  (p : ManifoldPoint) 
  (dt : Fix16) 
  (prevX : PhaseVec) 
  : ManifoldPoint :=
  let nextPhi := flowPhi p dt
  let nextX   := flowEmbedding p dt prevX
  { p with phi := nextPhi, x_pos := nextX }

/-- 
Master Equation: Recursive Manifold Evolution
Σ̂_{t+1} = FoldbackLockStep(Σ_t, Δt, Σ_{t-1})
-/
def masterEquation
  (curr : ManifoldPoint)
  (prev : ManifoldPoint)
  (dt : Fix16)
  : ManifoldPoint :=
  foldbackLockStep curr dt prev.x_pos

-- =============================================================================
-- LEGACY MAPPING (Braid Compatibility)
-- =============================================================================
-- Keeping these as shims for the SLUG-3 decoder which operates on segments of 
-- the manifold generated by these flows.

structure CMYK where
  c : Fix16
  m : Fix16
  y : Fix16
  k : Fix16
  deriving Repr, DecidableEq, BEq

def CMYK.zero : CMYK := ⟨Fix16.zero, Fix16.zero, Fix16.zero, Fix16.zero⟩

end Semantics.MasterEquation
