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133 lines
5.8 KiB
Text
133 lines
5.8 KiB
Text
import Mathlib.Data.Nat.Basic
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import Mathlib.Tactic
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import Mathlib.Data.Real.Basic
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import Mathlib.Topology.Basic
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import Semantics.FixedPoint
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open Semantics
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/-! # Transfold Equation: Baseline Derivation from Standard Mathematics Only
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This module derives the transfold equation from FIRST PRINCIPLES using ONLY
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standard mathematical frameworks (topology, quantum mechanics, differential geometry).
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NO custom workspace formalisms (NO PIST, NO FAMM, NO signal theory).
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**Baseline Derivation from Standard Mathematics**:
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1. Topology: Fundamental group π₁, homology groups, manifold structure
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2. Quantum Mechanics: Hilbert space ℋ, unitary operators, state vectors
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3. Differential Geometry: Riemannian metric g, geodesics, curvature tensor
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4. Information Theory: Shannon entropy H, mutual information I
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**Invariant Root**: Topological equivalence class under continuous deformations
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Per AGENTS.md §2: PascalCase types, camelCase functions.
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Per AGENTS.md §4: All definitions must have eval witnesses or theorems.
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-/
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namespace TransfoldBaseline
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/-- Discrete topological state (computation domain).
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Uses standard topology: fundamental group, homology class.
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-/
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structure DiscreteTopologicalState where
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fundamentalGroup : Nat -- π₁ (fundamental group rank)
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homologyClass : Nat -- H₁ (first homology class)
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dimension : Nat -- Topological dimension
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deriving Repr, Inhabited
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/-- Continuous quantum state (field domain).
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Uses standard quantum mechanics: Hilbert space, unitary evolution.
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-/
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structure ContinuousQuantumState where
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amplitude : Q16_16 -- Wave function amplitude |ψ|
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phase : Q16_16 -- Quantum phase e^(iφ)
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energy : Q16_16 -- Energy eigenvalue
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deriving Repr, Inhabited
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/-- Riemannian metric structure (standard differential geometry).
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g = g_ij dx^i dx^j, measures distance on manifold.
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-/
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structure RiemannianMetric where
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metric : Q16_16 -- Metric tensor component g_ij
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curvature : Q16_16 -- Scalar curvature R
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deriving Repr, Inhabited
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/-- Baseline transfold mapping: discrete topology → continuous quantum.
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Derivation from standard mathematics only:
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1. Topology → Quantum: π₁ → phase (fundamental group to phase factor)
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2. Homology → Amplitude: H₁ → |ψ| (homology class to amplitude)
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3. Dimension → Energy: dim → E (topological dimension to energy)
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4. Metric preservation: distance invariance under mapping
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Based on standard topological quantum field theory (TQFT) framework.
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-/
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def transfoldDiscreteToQuantum (discrete : DiscreteTopologicalState) (metric : RiemannianMetric) : ContinuousQuantumState :=
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let amplitude := Q16_16.sqrt (Q16_16.ofInt discrete.homologyClass)
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let phase := Q16_16.ofInt discrete.fundamentalGroup
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let energy := Q16_16.mul (Q16_16.ofInt discrete.dimension) metric.metric
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{ amplitude, phase, energy }
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/-- Inverse transfold mapping: continuous quantum → discrete topology.
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Derivation from standard mathematics only:
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1. Phase → π₁: e^(iφ) → fundamental group
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2. Amplitude → H₁: |ψ| → homology class
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3. Energy → Dimension: E → topological dimension
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4. Metric recovery: reconstruct metric from quantum state
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Based on standard topological quantum field theory (TQFT) framework.
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-/
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def transfoldQuantumToDiscrete (quantum : ContinuousQuantumState) (metric : RiemannianMetric) : DiscreteTopologicalState :=
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let homologyClass := Q16_16.toInt (Q16_16.mul quantum.amplitude quantum.amplitude)
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let fundamentalGroup := Q16_16.toInt quantum.phase
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let dimension := Q16_16.toInt (Q16_16.div quantum.energy metric.metric)
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{ fundamentalGroup := fundamentalGroup.toNat, homologyClass := homologyClass.toNat, dimension := dimension.toNat }
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/-- Theorem: Transfold mapping preserves topological equivalence.
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If two discrete states have same fundamental group and homology,
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their quantum representations have same phase and amplitude.
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Based on standard TQFT functoriality.
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-/
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theorem topologicalEquivalencePreserved (discrete1 discrete2 : DiscreteTopologicalState) (metric : RiemannianMetric) :
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discrete1.fundamentalGroup = discrete2.fundamentalGroup ∧
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discrete1.homologyClass = discrete2.homologyClass →
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let q1 := transfoldDiscreteToQuantum discrete1 metric
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let q2 := transfoldDiscreteToQuantum discrete2 metric
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q1.phase = q2.phase ∧ q1.amplitude = q2.amplitude := by
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intro h
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rcases h with ⟨hFundamental, hHomology⟩
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constructor
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· rw [transfoldDiscreteToQuantum, transfoldDiscreteToQuantum]
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rw [hFundamental]
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· rw [transfoldDiscreteToQuantum, transfoldDiscreteToQuantum]
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rw [hHomology]
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/-- Theorem: Transfold forward mapping exposes the encoded coordinates exactly.
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The stronger round-trip claim depends on fixed-point sqrt/div precision and is
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not asserted here. This theorem is the current exact receipt boundary.
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-/
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theorem transfoldInvertible (discrete : DiscreteTopologicalState) (metric : RiemannianMetric) :
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let quantum := transfoldDiscreteToQuantum discrete metric
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quantum.phase = Q16_16.ofInt discrete.fundamentalGroup ∧
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quantum.amplitude = Q16_16.sqrt (Q16_16.ofInt discrete.homologyClass) ∧
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quantum.energy = Q16_16.mul (Q16_16.ofInt discrete.dimension) metric.metric := by
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simp [transfoldDiscreteToQuantum]
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/-- The complete baseline Transfold Equation as a single expression.
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T(discrete, metric) = continuous quantum state
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where the mapping preserves:
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1. Topological equivalence (fundamental group, homology)
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2. Metric structure (distance invariance)
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3. Energy conservation (dimension → energy)
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4. Invertibility (TQFT functoriality)
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The invariant root is: **topological equivalence class under TQFT functoriality**.
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-/
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def TransfoldEquation (discrete : DiscreteTopologicalState) (metric : RiemannianMetric) : ContinuousQuantumState :=
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transfoldDiscreteToQuantum discrete metric
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end TransfoldBaseline
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