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136 lines
4.4 KiB
Text
136 lines
4.4 KiB
Text
/-
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BraidCross.lean - Braid Crossing and Strand Merge Operations
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Crossing topology: strands interact, merge, and generate residuals.
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The merge rule remains linear on phaseAcc; bracket is recomputed after.
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zᵢⱼ = zᵢ + zⱼ (linear merge)
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μᵢⱼ = X(μᵢ, μⱼ) (crossing slot operator)
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Bᵢⱼ = C(zᵢⱼ, μᵢⱼ) (bracket from merged state)
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Rᵢⱼ = Bᵢⱼ - (Bᵢ + Bⱼ) (interaction residual)
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-/
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import Semantics.DynamicCanal
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import Semantics.BraidBracket
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import Semantics.BraidStrand
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namespace Semantics.BraidCross
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open DynamicCanal
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open Semantics.BraidBracket
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open Semantics.BraidStrand
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/-- Crossing slot operator X(μᵢ, μⱼ)
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Combines transport slots from two strands into merged slot.
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Default: bitwise XOR of slot indices (creates unique crossing ID).
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-/
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def crossSlot (μᵢ μⱼ : Fix16) : Fix16 :=
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-- XOR the raw representations for unique crossing slot
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Fix16.mk (μᵢ.raw.xor μⱼ.raw)
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/-- BraidCross: merge two strands into a crossing
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This is THE fundamental merge operation. It:
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1. Linearly adds phase accumulations: zᵢⱼ = zᵢ + zⱼ
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2. Computes crossed slot: μᵢⱼ = X(μᵢ, μⱼ)
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3. Derives new bracket: Bᵢⱼ = C(zᵢⱼ, μᵢⱼ)
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4. Calculates residual: Rᵢⱼ = Bᵢⱼ - (Bᵢ + Bⱼ)
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Key: merge in linear space first, derive bracket afterward.
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-/
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def braidCross (sᵢ sⱼ : BraidStrand) : BraidStrand × BraidBracket :=
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-- Linear merge of phase accumulations
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let zᵢⱼ := PhaseVec.add sᵢ.phaseAcc sⱼ.phaseAcc
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-- Crossing slot operator
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let μᵢ := Fix16.mk sᵢ.slot
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let μⱼ := Fix16.mk sⱼ.slot
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let μᵢⱼ := crossSlot μᵢ μⱼ
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-- Derive new bracket from merged state (NOT from merging brackets)
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let Bᵢⱼ := BraidBracket.fromPhaseVec zᵢⱼ μᵢⱼ
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-- Calculate crossing residual
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let Rᵢⱼ := BraidBracket.crossingResidual Bᵢⱼ sᵢ.bracket sⱼ.bracket
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-- Construct merged strand
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let mergedStrand : BraidStrand :=
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{ phaseAcc := zᵢⱼ
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, parity := sᵢ.parity && sⱼ.parity
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, slot := sᵢ.slot.xor sⱼ.slot -- unique crossing slot
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, residue := Rᵢⱼ.kappa -- store residual magnitude
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, jitter := Fix16.add sᵢ.jitter sⱼ.jitter
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, bracket := Bᵢⱼ }
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(mergedStrand, Rᵢⱼ)
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/-- Concrete left-identity witness for the zero strand. -/
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theorem braidCrossZeroLeftWitness :
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(braidCross (BraidStrand.zero 0) (BraidStrand.zero 1)).1.phaseAcc = PhaseVec.zero := by
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native_decide
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/-- Concrete right-identity witness for the zero strand. -/
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theorem braidCrossZeroRightWitness :
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(braidCross (BraidStrand.zero 1) (BraidStrand.zero 0)).1.phaseAcc = PhaseVec.zero := by
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native_decide
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/-- Parallel crossing: merge multiple strands simultaneously
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z = Σᵢ zᵢ (linear sum over all strands)
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Then derive single bracket from total.
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-/
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def parallelCross (strands : List BraidStrand) : BraidStrand :=
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let totalPhase := strands.foldl (fun acc s => PhaseVec.add acc s.phaseAcc) PhaseVec.zero
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let totalSlot := strands.foldl (fun acc s => acc.xor s.slot) 0
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let totalJitter := strands.foldl (fun acc s => Fix16.add acc s.jitter) Fix16.zero
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let μ := Fix16.mk totalSlot
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let B := BraidBracket.fromPhaseVec totalPhase μ
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{ phaseAcc := totalPhase
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, parity := strands.all (fun s => s.parity)
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, slot := totalSlot
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, residue := Fix16.zero -- parallel merge has no pairwise residual
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, jitter := totalJitter
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, bracket := B }
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/-- Check if crossing is admissible (merged bracket valid) -/
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def crossingAdmissible (sᵢ sⱼ : BraidStrand) : Bool :=
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let (merged, residual) := braidCross sᵢ sⱼ
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merged.isAdmissible && residual.admissible
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/-- Total residual norm from a crossing -/
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def crossingResidualNorm (sᵢ sⱼ : BraidStrand) : Fix16 :=
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let (_, residual) := braidCross sᵢ sⱼ
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residual.kappa
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/-- Crossing history for AVMR audit trail -/
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structure CrossingHistory where
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leftSlot : UInt32
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rightSlot : UInt32
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mergedSlot : UInt32
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residual : BraidBracket
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timestamp : UInt64
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deriving Repr, DecidableEq
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namespace CrossingHistory
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def fromCross (sᵢ sⱼ : BraidStrand) (ts : UInt64) : CrossingHistory :=
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let (_, residual) := braidCross sᵢ sⱼ
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{ leftSlot := sᵢ.slot
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, rightSlot := sⱼ.slot
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, mergedSlot := sᵢ.slot.xor sⱼ.slot
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, residual := residual
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, timestamp := ts }
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end CrossingHistory
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#eval let s1 := BraidStrand.zero 1
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let s2 := BraidStrand.zero 2
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let (m, _) := braidCross s1 s2
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m.slot
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end Semantics.BraidCross
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