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