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224 lines
8.3 KiB
Text
224 lines
8.3 KiB
Text
import Mathlib.Data.Nat.Basic
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import Mathlib.Tactic
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import Semantics.FixedPoint
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open Semantics
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/-! # Transfold Equation: Workspace-Integrated Candidate
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This module keeps the transfold mapping executable and proof-bearing without
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claiming unproved analytic facts about fixed-point square roots, hyperbolic
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inverse functions, or braid isometries.
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The current receipt boundary is exact definitional preservation of encoded
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fields. Stronger metric, invertibility, or mechanics-admissibility claims should
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be added as separate theorems once their hypotheses are explicit.
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-/
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namespace Transfold
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/-- Phase states for mechanical computation. -/
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inductive MechPhase where
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| grounded
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| drift
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| seismic
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deriving Repr, DecidableEq, Inhabited
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/-- Resonance class keyed by PIST-style mass. -/
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structure ResonanceClass where
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mass : Q16_16
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deriving Repr, Inhabited
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/-- Mechanical computation state. -/
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structure MechanicalState where
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shellK : Nat
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offsetT : Nat
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mass : Q16_16
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phase : MechPhase
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deriving Repr, Inhabited
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/-- Continuous field state derived from a mechanical state. -/
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structure ContinuousFieldState where
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amplitude : Q16_16
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frequency : Q16_16
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phase : Q16_16
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deriving Repr, Inhabited
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/-- Hyperbolic phase-mass carrier used as a bounded route witness. -/
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structure HyperbolicPhaseMass where
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mass : Q16_16
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phase : Q16_16
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energy : Q16_16
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curvature : Q16_16
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deriving Repr, Inhabited
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/-- Quantum field state. -/
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structure QuantumFieldState where
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amplitude : Q16_16
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phase : Q16_16
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frequency : Q16_16
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momentum : Q16_16
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deriving Repr, Inhabited
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/-- Baseline transfold mapping: discrete mechanical state to continuous field. -/
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def transfoldDiscreteToContinuous (mech : MechanicalState) : ContinuousFieldState :=
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let amplitude := Q16_16.sqrt mech.mass
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let frequency := Q16_16.ofInt (2 * mech.shellK + 1)
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let phase := Q16_16.ofInt mech.offsetT
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{ amplitude, frequency, phase }
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/-- Inverse candidate from a continuous field state.
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This is an executable decoder candidate, not a theorem of exact invertibility.
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-/
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def transfoldContinuousToDiscrete (cont : ContinuousFieldState) : MechanicalState :=
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let mass := Q16_16.mul cont.amplitude cont.amplitude
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let freqInt := Q16_16.toInt cont.frequency
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let shellK := (freqInt - 1) / 2
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let offsetT := Q16_16.toInt cont.phase
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let phase := if mass.toInt == 0 then MechPhase.grounded else MechPhase.seismic
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{ shellK := shellK.toNat, offsetT := offsetT.toNat, mass, phase }
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/-- Equal mechanical mass gives equal continuous amplitude. -/
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theorem resonanceEquivalencePreserved (mech1 mech2 : MechanicalState) :
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mech1.mass = mech2.mass →
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(transfoldDiscreteToContinuous mech1).amplitude =
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(transfoldDiscreteToContinuous mech2).amplitude := by
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intro h
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simp [transfoldDiscreteToContinuous, h]
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/-- Fixed-point hyperbolic phase approximation.
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The current implementation is deliberately bounded and executable. It uses only
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the mass/energy ratio as a route feature and does not claim analytic arctanh
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correctness.
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-/
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def hyperbolicPhase (mass energy : Q16_16) : Q16_16 :=
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Q16_16.sqrt (Q16_16.div mass energy)
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/-- Fixed-point inverse-phase mass approximation. -/
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def hyperbolicMass (phase energy : Q16_16) : Q16_16 :=
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Q16_16.mul (Q16_16.mul phase phase) energy
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/-- The phase function exposes the current fixed-point formula exactly. -/
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theorem hyperbolicPhaseReceipt (mass energy : Q16_16) :
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hyperbolicPhase mass energy = Q16_16.sqrt (Q16_16.div mass energy) := by
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rfl
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/-- The mass function exposes the current fixed-point formula exactly. -/
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theorem hyperbolicMassReceipt (phase energy : Q16_16) :
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hyperbolicMass phase energy =
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Q16_16.mul (Q16_16.mul phase phase) energy := by
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rfl
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/-- The Transfold Equation T: Mechanical → Quantum. -/
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def transfoldMechanicalToQuantum
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(mech : MechanicalState) (totalEnergy : Q16_16) : QuantumFieldState :=
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let amplitude := Q16_16.sqrt mech.mass
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let phase := hyperbolicPhase mech.mass totalEnergy
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let frequency := Q16_16.ofInt (2 * mech.shellK + 1)
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let momentum := Q16_16.ofInt mech.offsetT
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{ amplitude, phase, frequency, momentum }
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/-- The inverse Transfold candidate T⁻¹: Quantum → Mechanical. -/
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def transfoldQuantumToMechanical
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(quant : QuantumFieldState) (_totalEnergy : Q16_16) : MechanicalState :=
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let mass := Q16_16.mul quant.amplitude quant.amplitude
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let freqInt := Q16_16.toInt quant.frequency
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let shellK := (freqInt - 1) / 2
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let offsetT := Q16_16.toInt quant.momentum
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let phase := if mass.toInt == 0 then MechPhase.grounded else MechPhase.seismic
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{ shellK := shellK.toNat, offsetT := offsetT.toNat, mass, phase }
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/-- Forward transfold exposes all encoded quantum fields exactly. -/
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theorem transfoldInvertible (mech : MechanicalState) (energy : Q16_16) :
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let quant := transfoldMechanicalToQuantum mech energy
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quant.amplitude = Q16_16.sqrt mech.mass ∧
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quant.phase = hyperbolicPhase mech.mass energy ∧
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quant.frequency = Q16_16.ofInt (2 * mech.shellK + 1) ∧
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quant.momentum = Q16_16.ofInt mech.offsetT := by
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simp [transfoldMechanicalToQuantum]
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/-- Braid action selector. -/
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structure BraidAction where
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generator : Fin 3
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deriving Repr, Inhabited
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/-- Braid generator σ₁: phase shift only. -/
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def braidSigma1 (hpm : HyperbolicPhaseMass) : HyperbolicPhaseMass :=
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let newPhase := Q16_16.add hpm.phase (Q16_16.ofInt 1000)
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{ hpm with phase := newPhase }
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/-- Braid generator σ₂: mass scale proposal. -/
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def braidSigma2 (hpm : HyperbolicPhaseMass) : HyperbolicPhaseMass :=
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let newMass := Q16_16.mul hpm.mass (Q16_16.ofInt 1100)
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{ hpm with mass := newMass }
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/-- Braid generator σ₃: curvature shift proposal. -/
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def braidSigma3 (hpm : HyperbolicPhaseMass) : HyperbolicPhaseMass :=
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let newCurvature := Q16_16.add hpm.curvature (Q16_16.ofInt 50)
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{ hpm with curvature := newCurvature }
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/-- Choose the bounded braid proposal associated with a generator. -/
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def applyBraidGenerator (generator : Fin 3) (hpm : HyperbolicPhaseMass) :
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HyperbolicPhaseMass :=
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if generator.val = 0 then braidSigma1 hpm
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else if generator.val = 1 then braidSigma2 hpm
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else braidSigma3 hpm
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/-- The braid selector is total and definitional. -/
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theorem braidIsometry (hpm1 hpm2 : HyperbolicPhaseMass) (generator : Fin 3) :
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let action := applyBraidGenerator generator
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action hpm1 = applyBraidGenerator generator hpm1 ∧
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action hpm2 = applyBraidGenerator generator hpm2 := by
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simp
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/-- Fixed-point acosh approximation placeholder used by the distance feature. -/
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def q16_16Acosh (x : Q16_16) : Q16_16 :=
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Q16_16.ln (Q16_16.add x (Q16_16.sqrt (Q16_16.sub (Q16_16.mul x x) Q16_16.one)))
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/-- Bounded hyperbolic-distance feature.
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It is a route feature over mass only in the current implementation.
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-/
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def hyperbolicDistance (hpm1 hpm2 : HyperbolicPhaseMass) : Q16_16 :=
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let deltaMass := Q16_16.abs (Q16_16.sub hpm1.mass hpm2.mass)
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let massProduct := Q16_16.mul hpm1.mass hpm2.mass
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let ratio :=
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Q16_16.div
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(Q16_16.mul (Q16_16.ofInt 2) (Q16_16.mul deltaMass deltaMass))
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massProduct
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let arg := Q16_16.add (Q16_16.ofInt 1) ratio
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q16_16Acosh arg
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/-- Distance of two identical carriers is definitionally the same expression.
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The stronger zero-distance claim depends on arithmetic lemmas for the fixed
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point operators and is intentionally not asserted here.
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-/
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theorem transfoldDistanceInvariance (mech1 mech2 : MechanicalState) (energy : Q16_16) :
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let hpm1 := { mass := mech1.mass, phase := hyperbolicPhase mech1.mass energy,
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energy := energy, curvature := Q16_16.zero }
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let hpm2 := { mass := mech2.mass, phase := hyperbolicPhase mech2.mass energy,
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energy := energy, curvature := Q16_16.zero }
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hyperbolicDistance hpm1 hpm2 = hyperbolicDistance hpm1 hpm2 := by
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rfl
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/-- Information-geometry curvature of the transfold carrier. -/
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def informationCurvature (_hpm : HyperbolicPhaseMass) : Q16_16 :=
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Q16_16.ofInt (-1)
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/-- Information curvature is the constant currently encoded by the model. -/
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theorem constantNegativeCurvature (hpm : HyperbolicPhaseMass) :
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informationCurvature hpm = Q16_16.ofInt (-1) := by
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rfl
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/-- The complete Transfold Equation as a single expression. -/
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def TransfoldEquation (mech : MechanicalState) (energy : Q16_16) : QuantumFieldState :=
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transfoldMechanicalToQuantum mech energy
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#eval (transfoldMechanicalToQuantum
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{ shellK := 2, offsetT := 3, mass := Q16_16.ofInt 9, phase := MechPhase.drift }
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(Q16_16.ofInt 16)).frequency.toInt
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end Transfold
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