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 let k := Nat.sqrt n have ha : a (mirror n) = b n := by simp [a, mirror, k] omega have hb : b (mirror n) = a n := by simp [b, mirror, k] omega simp [hyperbolaIndex, ha, hb, Nat.mul_comm] /-- Theorem: Zero-mass iff square. -/ theorem zero_mass_iff_square (n : Nat) : hyperbolaIndex n = 0 ↔ (Nat.sqrt n)^2 = n := by simp [hyperbolaIndex, a, b] constructor · intro h cases Nat.eq_zero_or_pos (Nat.sqrt n + 1)^2 with | inl h_zero => -- Contradiction: (k+1)^2 is never zero for Nat have h_pos : (Nat.sqrt n + 1)^2 > 0 := Nat.pos_of_ne_zero (by intro h_z; injection h_z) exact False.elim (Nat.lt_irrefl 0 (h_pos.trans_le (Nat.zero_le _))) | inr h_pos => -- If a*b = 0 then a=0 or b=0. -- But b = (k+1)^2 - n > 0 because n < (k+1)^2 by sqrt properties. have hn : n < (Nat.sqrt n + 1)^2 := Nat.lt_succ_sqrt n have hb_pos : (Nat.sqrt n + 1)^2 - n > 0 := Nat.sub_pos_of_lt hn have ha_zero : n - (Nat.sqrt n)^2 = 0 := by exact Nat.eq_zero_of_mul_eq_zero_left h (Nat.ne_of_gt hb_pos) exact Nat.eq_of_sub_eq_zero ha_zero · intro h simp [h] /-! ## 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