mirror of
https://github.com/allaunthefox/Research-Stack.git
synced 2026-07-31 03:05:21 +00:00
109 lines
3.7 KiB
Text
109 lines
3.7 KiB
Text
import Mathlib.Data.List.Basic
|
||
import Mathlib.Logic.Equiv.Basic
|
||
import Semantics.BraidedField
|
||
|
||
/-!
|
||
# Braided Field Sets & Virtual Information Paths
|
||
|
||
This module formalizes the topology of virtual information paths
|
||
created by braiding polaron-polariton quasiparticles (anyons).
|
||
Instead of tracking particle locations in Cartesian space, information
|
||
is stored in the topological equivalence class of the braid sequence.
|
||
|
||
This is a finite braid-word scaffold. It does not prove physical topological
|
||
protection, material feasibility, or local-noise immunity.
|
||
-/
|
||
|
||
namespace Semantics.BraidedField
|
||
|
||
/--
|
||
A single generator `σ_i` representing the virtual operation of
|
||
swapping adjacent quasiparticles `i` and `i+1` in a 2D manifold.
|
||
-/
|
||
structure Swap (n : Nat) where
|
||
i : Nat
|
||
valid : i + 1 < n
|
||
deriving Repr, DecidableEq
|
||
|
||
/--
|
||
A Virtual Information Path is a temporal sequence of swaps.
|
||
In formal topology, this is known as a braid word.
|
||
This acts as an encoding system where data is written into spacetime history.
|
||
-/
|
||
abbrev VirtualPath (n : Nat) := List (Swap n)
|
||
|
||
/--
|
||
Topological equivalence of virtual paths.
|
||
|
||
Two paths are treated as the same candidate path if they can be deformed into
|
||
one another via the Artin braid relations. This is a structural equivalence
|
||
relation for the virtual-path model, not a physical immunity claim.
|
||
-/
|
||
inductive TopologicallyEquivalent {n : Nat} : VirtualPath n → VirtualPath n → Prop where
|
||
| refl (p : VirtualPath n) : TopologicallyEquivalent p p
|
||
|
||
| symm {p q : VirtualPath n} :
|
||
TopologicallyEquivalent p q → TopologicallyEquivalent q p
|
||
|
||
| trans {p q r : VirtualPath n} :
|
||
TopologicallyEquivalent p q → TopologicallyEquivalent q r → TopologicallyEquivalent p r
|
||
|
||
/--
|
||
Far-commutation: Swapping particles on opposite sides of the field
|
||
are independent operations. (σ_i σ_j = σ_j σ_i if |i - j| ≥ 2)
|
||
-/
|
||
| commute (i j : Swap n) (left right : VirtualPath n)
|
||
(h_dist : i.i + 1 < j.i ∨ j.i + 1 < i.i) :
|
||
TopologicallyEquivalent (left ++ [i, j] ++ right) (left ++ [j, i] ++ right)
|
||
|
||
/--
|
||
The Yang-Baxter Equation / braid relation.
|
||
The core braid-word equivalence of the virtual-path encoding system.
|
||
(σ_i σ_{i+1} σ_i = σ_{i+1} σ_i σ_{i+1})
|
||
-/
|
||
| braid_rel (i j : Swap n) (left right : VirtualPath n)
|
||
(h_adj : j.i = i.i + 1) :
|
||
TopologicallyEquivalent (left ++ [i, j, i] ++ right) (left ++ [j, i, j] ++ right)
|
||
|
||
/--
|
||
A basic theorem demonstrating that the relation is explicitly symmetric.
|
||
-/
|
||
theorem path_integrity_preserved {n : Nat} (p1 p2 : VirtualPath n)
|
||
(h : TopologicallyEquivalent p1 p2) :
|
||
TopologicallyEquivalent p2 p1 := by
|
||
exact TopologicallyEquivalent.symm h
|
||
|
||
namespace Examples
|
||
|
||
def s0 : Swap 4 := ⟨0, by decide⟩
|
||
def s1 : Swap 4 := ⟨1, by decide⟩
|
||
def s2 : Swap 4 := ⟨2, by decide⟩
|
||
|
||
def farLeft : VirtualPath 4 := [s0, s2]
|
||
def farRight : VirtualPath 4 := [s2, s0]
|
||
|
||
def yangBaxterLeft : VirtualPath 4 := [s0, s1, s0]
|
||
def yangBaxterRight : VirtualPath 4 := [s1, s0, s1]
|
||
|
||
def pathLength {n : Nat} (p : VirtualPath n) : Nat :=
|
||
p.length
|
||
|
||
def generatorIndexSum {n : Nat} (p : VirtualPath n) : Nat :=
|
||
p.foldl (fun acc swap => acc + swap.i) 0
|
||
|
||
theorem far_commute_example : TopologicallyEquivalent farLeft farRight := by
|
||
exact TopologicallyEquivalent.commute s0 s2 [] [] (by decide)
|
||
|
||
theorem yang_baxter_example : TopologicallyEquivalent yangBaxterLeft yangBaxterRight := by
|
||
exact TopologicallyEquivalent.braid_rel s0 s1 [] [] (by decide)
|
||
|
||
theorem far_commute_symmetric : TopologicallyEquivalent farRight farLeft := by
|
||
exact path_integrity_preserved farLeft farRight far_commute_example
|
||
|
||
#eval pathLength farLeft
|
||
#eval generatorIndexSum yangBaxterLeft
|
||
#eval generatorIndexSum yangBaxterRight
|
||
|
||
end Examples
|
||
|
||
end Semantics.BraidedField
|