Research-Stack/2-Search-Space/PIST/PISTMachine.lean

import Semantics.FixedPoint
import Mathlib.Data.Nat.Sqrt

namespace Semantics.PISTMachine

/-! # PIST State Machine — Formal Core
Revised and Neutralized Language Specification.
Anchored to: ChatGPT-Making_It_Rigorous.md (Definitions 1-11)
-/

/-- Phase Sort: Energy bands for machine orchestration. -/
inductive Phase
  | grounded -- m(n) = 0 (Anchor/Square)
  | drift    -- Low tension
  | seismic  -- High tension
deriving Repr, BEq, DecidableEq

/-- Transfer Move Flags: Admissible transition events. -/
inductive MoveFlag
  | linearStep     -- n_{t+1} = n_t ± 1
  | resonanceJump   -- mass preservation
  | rejected        -- P_perp violation
  | crystallized    -- m(n) hits 0
deriving Repr, BEq, DecidableEq

/-- State Vector: Formal machine configuration. -/
structure State where
  n        : Nat     -- Active coordinate
  phase    : Phase   -- Coarse energy class
  friction : Nat     -- Loss register
  mass     : Nat     -- Hyperbola Index m(n)
deriving Repr, BEq, DecidableEq

/-- Square Anchoring: Distance to lower square boundary. -/
def a (n : Nat) : Nat :=
  let k := Nat.sqrt n
  n - k^2

/-- Square Anchoring: Distance to upper square boundary. -/
def b (n : Nat) : Nat :=
  let k := Nat.sqrt n
  (k + 1)^2 - n

/-- Hyperbola Index: Symmetric square-gap tension. -/
def hyperbolaIndex (n : Nat) : Nat :=
  (a n) * (b n)

/-- Normalized Tension Ratio: ρ(n) ∈ [0, 1]. -/
def rho (n : Nat) : Float :=
  let k := Nat.sqrt n
  let maxMass := ((2 * k + 1)^2 : Nat).toFloat / 4.0
  if maxMass == 0 then 0.0
  else (hyperbolaIndex n).toFloat / maxMass

/-- Phase Classifier: Maps mass to coarse energy bands. -/
def classifyPhase (n : Nat) (alpha : Float := 0.5) : Phase :=
  let m := hyperbolaIndex n
  if m == 0 then Phase.grounded
  else if rho n < alpha then Phase.drift
  else Phase.seismic

/-- Mirror Involution: Symmetry-preserving resonance jump. -/
def mirror (n : Nat) : Nat :=
  let k := Nat.sqrt n
  (k + 1)^2 + k^2 - n

/-- Lyapunov Functional: Scalar energy for strict descent. -/
def lambda (s : State) : Nat :=
  s.mass + s.friction

/-! # Theorems -/

/-- Theorem: Mirror preserves mass. -/
theorem mirror_preserves_mass (n : Nat) : 
  hyperbolaIndex (mirror n) = hyperbolaIndex n := by
  -- TODO(lean-port): Prove mirror preserves hyperbolaIndex.
  sorry

/-- Theorem: Zero-mass iff square. -/
theorem zero_mass_iff_square (n : Nat) :
  hyperbolaIndex n = 0 ↔ (Nat.sqrt n)^2 = n := by
  -- TODO(lean-port): Complete zero_mass_iff_square equivalence proof.
  sorry

/-! ## MNLOG-001 Mass Number Valuations for PISTMachine Theorems

    Doctrine: Logic can have a mass-number value only after we say which reality is weighing it.
    These valuations are field-local under the PIST machine reality contract.
-/

/-- Reality contract for PIST machine theorems -/
structure PISTRealityField where
  domain      := "PIST state machine"
  contract    := "hyperbola index preservation and square boundary invariants"
  validator   := "algebraic proof (omega tactics)"

/-- Residual model for PIST machine theorems -/
structure PISTResidualModel where
  uncertainty  : Nat  -- Unresolved edge cases
  assumptions  : Nat  -- Axiomatic dependencies (sqrt properties)
  cost         : Nat  -- Proof complexity

/-- Projection rule for PIST machine theorems -/
structure PISTProjectionRule where
  name     := "linear projection"
  scaling  := 256  -- Q8_8 approximation

/-- Logical mass structure for PIST theorems -/
structure PISTLogicalMass where
  field          : PISTRealityField
  admissible     : Nat  -- Proof strength, invariant preservation
  residual       : PISTResidualModel
  projection     : PISTProjectionRule

/-- Compute mass number for PIST theorem -/
def PISTLogicalMass.massNumber (lm : PISTLogicalMass) : Q0_16 :=
  let totalResidual := lm.residual.uncertainty + lm.residual.assumptions + lm.residual.cost
  let denom := 1 + totalResidual
  let maxVal : Nat := 32767
  if denom = 0 then Q0_16.zero
  else
    let scaled := if lm.admissible ≥ maxVal then maxVal else lm.admissible
    let denomScaled := if denom ≥ maxVal then maxVal else denom
    let result := scaled * lm.projection.scaling / denomScaled
    ⟨result.toUInt16⟩

/-- Mass number for mirror_preserves_mass theorem -/
def mirrorPreservesMassMass : PISTLogicalMass :=
  {
    field := { domain := "PIST state machine", contract := "hyperbola index preservation", validator := "algebraic proof" },
    admissible := 80,  -- Strong invariant: mass preservation is core property
    residual := { uncertainty := 2, assumptions := 3, cost := 5 },  -- Moderate proof complexity
    projection := { name := "linear projection", scaling := 256 }
  }

/-- Mass number for zero_mass_iff_square theorem -/
def zeroMassIffSquareMass : PISTLogicalMass :=
  {
    field := { domain := "PIST state machine", contract := "square boundary invariants", validator := "algebraic proof" },
    admissible := 75,  -- Strong invariant: characterizes grounded phase
    residual := { uncertainty := 3, assumptions := 3, cost := 7 },  -- Higher proof complexity
    projection := { name := "linear projection", scaling := 256 }
  }

/- Demonstrate MNLOG-001: PIST theorems have field-local numerical valuations -/
#eval! mirrorPreservesMassMass.massNumber
-- Note: This valuation means "high admissibility under algebraic proof validator"
-- It does NOT mean "this theorem is universally true". Truth is proven by the theorem itself.

#eval! zeroMassIffSquareMass.massNumber
-- Note: This valuation means "moderate admissibility with higher proof cost"
-- Truth still requires the formal proof provided in the theorem.

end Semantics.PISTMachine