mirror of
https://github.com/allaunthefox/Research-Stack.git
synced 2026-07-31 03:05:21 +00:00
176 lines
6.4 KiB
Text
176 lines
6.4 KiB
Text
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
|