Research-Stack/.changes/2-Search-Space/PIST/PISTMachine.lean/concrete-history/1777674400569

{"type":"new","contents":"import Semantics.FixedPoint\nimport Mathlib.Data.Nat.Sqrt\n\nnamespace Semantics.PISTMachine\n\n/-! # PIST State Machine — Formal Core\nRevised and Neutralized Language Specification.\nAnchored to: ChatGPT-Making_It_Rigorous.md (Definitions 1-11)\n-/\n\n/-- Phase Sort: Energy bands for machine orchestration. -/\ninductive Phase\n  | grounded -- m(n) = 0 (Anchor/Square)\n  | drift    -- Low tension\n  | seismic  -- High tension\nderiving Repr, BEq, DecidableEq\n\n/-- Transfer Move Flags: Admissible transition events. -/\ninductive MoveFlag\n  | linearStep     -- n_{t+1} = n_t ± 1\n  | resonanceJump   -- mass preservation\n  | rejected        -- P_perp violation\n  | crystallized    -- m(n) hits 0\nderiving Repr, BEq, DecidableEq\n\n/-- State Vector: Formal machine configuration. -/\nstructure State where\n  n        : Nat     -- Active coordinate\n  phase    : Phase   -- Coarse energy class\n  friction : Nat     -- Loss register\n  mass     : Nat     -- Hyperbola Index m(n)\nderiving Repr, BEq, DecidableEq\n\n/-- Square Anchoring: Distance to lower square boundary. -/\ndef a (n : Nat) : Nat :=\n  let k := Nat.sqrt n\n  n - k^2\n\n/-- Square Anchoring: Distance to upper square boundary. -/\ndef b (n : Nat) : Nat :=\n  let k := Nat.sqrt n\n  (k + 1)^2 - n\n\n/-- Hyperbola Index: Symmetric square-gap tension. -/\ndef hyperbolaIndex (n : Nat) : Nat :=\n  (a n) * (b n)\n\n/-- Normalized Tension Ratio: ρ(n) ∈ [0, 1]. -/\ndef rho (n : Nat) : Float :=\n  let k := Nat.sqrt n\n  let maxMass := ((2 * k + 1)^2 : Nat).toFloat / 4.0\n  if maxMass == 0 then 0.0\n  else (hyperbolaIndex n).toFloat / maxMass\n\n/-- Phase Classifier: Maps mass to coarse energy bands. -/\ndef classifyPhase (n : Nat) (alpha : Float := 0.5) : Phase :=\n  let m := hyperbolaIndex n\n  if m == 0 then Phase.grounded\n  else if rho n < alpha then Phase.drift\n  else Phase.seismic\n\n/-- Mirror Involution: Symmetry-preserving resonance jump. -/\ndef mirror (n : Nat) : Nat :=\n  let k := Nat.sqrt n\n  (k + 1)^2 + k^2 - n\n\n/-- Lyapunov Functional: Scalar energy for strict descent. -/\ndef lambda (s : State) : Nat :=\n  s.mass + s.friction\n\n/-! # Theorems -/\n\n/-- Theorem: Mirror preserves mass. -/\ntheorem mirror_preserves_mass (n : Nat) : \n  hyperbolaIndex (mirror n) = hyperbolaIndex n := by\n  let k := Nat.sqrt n\n  have ha : a (mirror n) = b n := by\n    simp [a, mirror, k]\n    omega\n  have hb : b (mirror n) = a n := by\n    simp [b, mirror, k]\n    omega\n  simp [hyperbolaIndex, ha, hb, Nat.mul_comm]\n\n/-- Theorem: Zero-mass iff square. -/\ntheorem zero_mass_iff_square (n : Nat) :\n  hyperbolaIndex n = 0 ↔ (Nat.sqrt n)^2 = n := by\n  simp [hyperbolaIndex, a, b]\n  constructor\n  · intro h\n    cases Nat.eq_zero_or_pos (Nat.sqrt n + 1)^2 with\n    | inl h_zero => \n      -- Contradiction: (k+1)^2 is never zero for Nat\n      have h_pos : (Nat.sqrt n + 1)^2 > 0 := Nat.pos_of_ne_zero (by intro h_z; injection h_z)\n      exact False.elim (Nat.lt_irrefl 0 (h_pos.trans_le (Nat.zero_le _)))\n    | inr h_pos =>\n      -- If a*b = 0 then a=0 or b=0.\n      -- But b = (k+1)^2 - n > 0 because n < (k+1)^2 by sqrt properties.\n      have hn : n < (Nat.sqrt n + 1)^2 := Nat.lt_succ_sqrt n\n      have hb_pos : (Nat.sqrt n + 1)^2 - n > 0 := Nat.sub_pos_of_lt hn\n      have ha_zero : n - (Nat.sqrt n)^2 = 0 := by\n        exact Nat.eq_zero_of_mul_eq_zero_left h (Nat.ne_of_gt hb_pos)\n      exact Nat.eq_of_sub_eq_zero ha_zero\n  · intro h\n    simp [h]\n\n/-! ## MNLOG-001 Mass Number Valuations for PISTMachine Theorems\n\n    Doctrine: Logic can have a mass-number value only after we say which reality is weighing it.\n    These valuations are field-local under the PIST machine reality contract.\n-/\n\n/-- Reality contract for PIST machine theorems -/\nstructure PISTRealityField where\n  domain      := \"PIST state machine\"\n  contract    := \"hyperbola index preservation and square boundary invariants\"\n  validator   := \"algebraic proof (omega tactics)\"\n\n/-- Residual model for PIST machine theorems -/\nstructure PISTResidualModel where\n  uncertainty  : Nat  -- Unresolved edge cases\n  assumptions  : Nat  -- Axiomatic dependencies (sqrt properties)\n  cost         : Nat  -- Proof complexity\n\n/-- Projection rule for PIST machine theorems -/\nstructure PISTProjectionRule where\n  name     := \"linear projection\"\n  scaling  := 256  -- Q8_8 approximation\n\n/-- Logical mass structure for PIST theorems -/\nstructure PISTLogicalMass where\n  field          : PISTRealityField\n  admissible     : Nat  -- Proof strength, invariant preservation\n  residual       : PISTResidualModel\n  projection     : PISTProjectionRule\n\n/-- Compute mass number for PIST theorem -/\ndef PISTLogicalMass.massNumber (lm : PISTLogicalMass) : Q0_16 :=\n  let totalResidual := lm.residual.uncertainty + lm.residual.assumptions + lm.residual.cost\n  let denom := 1 + totalResidual\n  let maxVal : Nat := 32767\n  if denom = 0 then Q0_16.zero\n  else\n    let scaled := if lm.admissible ≥ maxVal then maxVal else lm.admissible\n    let denomScaled := if denom ≥ maxVal then maxVal else denom\n    let result := scaled * lm.projection.scaling / denomScaled\n    ⟨result.toUInt16⟩\n\n/-- Mass number for mirror_preserves_mass theorem -/\ndef mirrorPreservesMassMass : PISTLogicalMass :=\n  {\n    field := { domain := \"PIST state machine\", contract := \"hyperbola index preservation\", validator := \"algebraic proof\" },\n    admissible := 80,  -- Strong invariant: mass preservation is core property\n    residual := { uncertainty := 2, assumptions := 3, cost := 5 },  -- Moderate proof complexity\n    projection := { name := \"linear projection\", scaling := 256 }\n  }\n\n/-- Mass number for zero_mass_iff_square theorem -/\ndef zeroMassIffSquareMass : PISTLogicalMass :=\n  {\n    field := { domain := \"PIST state machine\", contract := \"square boundary invariants\", validator := \"algebraic proof\" },\n    admissible := 75,  -- Strong invariant: characterizes grounded phase\n    residual := { uncertainty := 3, assumptions := 3, cost := 7 },  -- Higher proof complexity\n    projection := { name := \"linear projection\", scaling := 256 }\n  }\n\n/-- Demonstrate MNLOG-001: PIST theorems have field-local numerical valuations -/\n#eval! mirrorPreservesMassMass.massNumber\n-- Note: This valuation means \"high admissibility under algebraic proof validator\"\n-- It does NOT mean \"this theorem is universally true\". Truth is proven by the theorem itself.\n\n#eval! zeroMassIffSquareMass.massNumber\n-- Note: This valuation means \"moderate admissibility with higher proof cost\"\n-- Truth still requires the formal proof provided in the theorem.\n\nend Semantics.PISTMachine\n","mtime":1777674400569}