import Semantics.FixedPoint
import Semantics.ManifoldStructures

namespace Semantics.VirtualWarpMetric

open Semantics

/--
Virtual Warp Metric (Layer 7)
dI² = -dτ² + (dH - v_eff * f * Ω * dτ)²

This metric governs the information-theoretic displacement of entropy
across the manifold's virtual surface.
-/
structure VirtualWarpParameters where
  kappa : Q16_16             -- κ : Steepness / coupling intensity
  opcodeEfficacy : Q16_16    -- Ω : Performance tier of the instruction
  localVelocity : Q16_16     -- v_local : Native hardware throughput
  coherence : Q16_16          -- φ : Phase coherence with past anchors
  properTime : Q16_16        -- dτ : Execution clock cycles
  entropyDisplacement : Q16_16 -- dH : Target entropy change
  deriving Repr, DecidableEq

/--
Effective compression velocity: v_eff = v_local / (1 - φ)
Encapsulates the 'boost' achieved by high coherence.
-/
def effectiveVelocity (params : VirtualWarpParameters) : Q16_16 :=
  let one := Q16_16.one
  let denom := Q16_16.sub one params.coherence
  -- Avoid division by zero: clamp denominator to at least epsilon
  let safeDenom := if Q16_16.le denom Q16_16.zero then Q16_16.epsilon else denom
  Q16_16.div params.localVelocity safeDenom

/--
Warp coupling constant: f_warp = f(x_i) * Ω
Typically derived from the SSS constant Φ_sss.
-/
def warpCoupling (params : VirtualWarpParameters) (sssConstant : Q16_16) : Q16_16 :=
  -- f = sigmoid(-κ * Φ_sss)
  -- Approximation: sigmoid(x) ≈ 0.5 + 0.5 * x for small x
  let inner := Q16_16.mul params.kappa sssConstant
  let ramp := Q16_16.sub (Q16_16.ofNat 1 / Q16_16.ofNat 2) inner
  Q16_16.mul ramp params.opcodeEfficacy

/--
Calculates the Virtual Warp Metric value: dI²
-/
def calculateVirtualWarpMetric (params : VirtualWarpParameters) (sssConstant : Q16_16) : Q16_16 :=
  let dt := params.properTime
  let dh := params.entropyDisplacement
  let veff := effectiveVelocity params
  let f_omega := warpCoupling params sssConstant
  
  -- dI² = (dH - v_eff * f * Ω * dτ)² - dτ²
  let velocityWork := Q16_16.mul veff (Q16_16.mul f_omega dt)
  let spaceTermInner := Q16_16.abs (Q16_16.sub dh velocityWork)
  let spaceTerm := Q16_16.mul spaceTermInner spaceTermInner
  let timeTerm := Q16_16.mul dt dt
  
  Q16_16.sub spaceTerm timeTerm

/--
Invariant: The metric is stable if the SSS constant and opcode efficacy
yield a positive coupling (sustained bubble).
-/
def isVirtualWarpStable (params : VirtualWarpParameters) (sssConstant : Q16_16) : Bool :=
  Q16_16.ge (warpCoupling params sssConstant) Q16_16.zero

-- =============================================================================
-- VERIFICATION
-- =============================================================================

#eval calculateVirtualWarpMetric 
  { kappa := Q16_16.one
  , opcodeEfficacy := Q16_16.one
  , localVelocity := Q16_16.one
  , coherence := Q16_16.ofNat 0 -- 0 coherence
  , properTime := Q16_16.epsilon
  , entropyDisplacement := Q16_16.one 
  } Q16_16.zero

end Semantics.VirtualWarpMetric