/-
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 Mathlib.Data.Nat.Basic
import Mathlib.Data.Int.Basic
import Mathlib.Tactic
import Semantics.DynamicCanal
import Semantics.BraidStrand
import Semantics.BraidBracket
import Semantics.FixedPoint
import Semantics.ManifoldFlow
import Semantics.VirtualWarpMetric

namespace Semantics.MasterEquation

open Semantics
open Semantics.Q16_16
open DynamicCanal
open Semantics.BraidBracket
open Semantics.ManifoldFlow
open Semantics.VirtualWarpMetric

/-- 
Executes one step of the "n-space foldback-lock" equation.
Encapsulates the hyperfluid phase evolution and embedding dynamics.
-/
def foldbackLockStep
  (p : ManifoldPoint)
  (dt : Q16_16)
  (prevX : PhaseVec)
  : ManifoldPoint :=
  let nextPhi := flowPhi p dt
  let nextX   := flowEmbedding p dt prevX
  { p with phi := nextPhi, x_pos := nextX }

/-- 
Mechanical Cycle Step 1: Expand
Project current state into the proposed manifold neighborhood.
-/
def expand (p : ManifoldPoint) : ManifoldPoint := p -- Placeholder for expansion logic

/-- 
Mechanical Cycle Step 2: Score
Calculate the metabolic and torsional cost (including Virtual Warp Metric penalty).
-/
def score (_p : ManifoldPoint) (params : VirtualWarpParameters) (sss : Q16_16) : Q16_16 :=
  calculateVirtualWarpMetric params sss

/-- 
Mechanical Cycle Step 3: Stabilize
Apply the foldback-lock torsion correction to minimize displacement.
-/
def stabilize (p : ManifoldPoint) (dt : Q16_16) (prevX : PhaseVec) : ManifoldPoint :=
  foldbackLockStep p dt prevX

/-- 
Mechanical Cycle Step 4: Prune
Deselect non-coherent branches of the manifold.
-/
def prune (p : ManifoldPoint) : ManifoldPoint := p

/-- 
Mechanical Cycle Step 5: Gossip
Synchronize state across the Triumvirate nodes (Warden, Builder, Judge).
-/
def gossip (p : ManifoldPoint) : ManifoldPoint := p

/-- 
Mechanical Cycle Step 6: MLGRU
Final recursive update of the engram state.
-/
def mlgru (p : ManifoldPoint) : ManifoldPoint := p

/-- 
The Mechanical Cycle (Layer 8)
S_{t+1} = MLGRU(Gossip(Prune(Stabilize(Score(Expand(S_t))))))
-/
def primaryMechanicalCycle
  (curr : ManifoldPoint)
  (prev : ManifoldPoint)
  (dt : Q16_16)
  (warpParams : VirtualWarpParameters)
  (sss : Q16_16)
  : ManifoldPoint :=
  let s1 := expand curr
  let _  := score s1 warpParams sss -- Verification Step
  let s3 := stabilize s1 dt prev.x_pos
  let s4 := prune s3
  let s5 := gossip s4
  mlgru s5

/-- 
Master Equation: Recursive Manifold Evolution
Anchored to the Mechanical Cycle.
-/
def masterEquation
  (curr : ManifoldPoint)
  (prev : ManifoldPoint)
  (dt : Q16_16)
  : ManifoldPoint :=
  -- Default warp params for standard execution
  let defaultParams : VirtualWarpParameters := 
    { kappa := Q16_16.one
    , opcodeEfficacy := Q16_16.one
    , localVelocity := Q16_16.one
    , coherence := Q16_16.zero
    , properTime := dt
    , entropyDisplacement := Q16_16.zero }
  primaryMechanicalCycle curr prev dt defaultParams Q16_16.zero

-- =============================================================================
-- 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 : Q16_16
  m : Q16_16
  y : Q16_16
  k : Q16_16
  deriving Repr, DecidableEq, BEq

def CMYK.zero : CMYK := ⟨Q16_16.zero, Q16_16.zero, Q16_16.zero, Q16_16.zero⟩

end Semantics.MasterEquation