mirror of
https://github.com/allaunthefox/Research-Stack.git
synced 2026-07-31 03:05:21 +00:00
110 lines
3.5 KiB
Text
110 lines
3.5 KiB
Text
/-
|
|
PhiShellEncoding.lean — φ-Shell Encoding for Topology/Manifold Routing
|
|
|
|
Extraction-friendly φ-shell routing surface.
|
|
|
|
The shell spacing uses a rational golden-ratio approximation for executable
|
|
metadata, while the verified invariants stay structural: receipts preserve
|
|
endpoints, paths are nonempty, and capacities/radii are monotone by definition.
|
|
-/
|
|
|
|
import Std
|
|
import Mathlib.Data.Nat.Basic
|
|
import Mathlib.Data.List.Basic
|
|
|
|
namespace Semantics.PhiShellEncoding
|
|
|
|
def phiNum : Nat := 1618
|
|
def phiDen : Nat := 1000
|
|
|
|
structure PhiShell where
|
|
level : Nat
|
|
radius : Nat
|
|
capacity : Nat
|
|
deriving Repr, Inhabited, DecidableEq
|
|
|
|
def computePhiShellRadius (level : Nat) (baseRadius : Nat) : Nat :=
|
|
baseRadius * (level + 1)
|
|
|
|
def computePhiShellCapacity (level : Nat) : Nat :=
|
|
level + 1
|
|
|
|
structure PhiShellAddress where
|
|
shellLevel : Nat
|
|
shellIndex : Nat
|
|
deriving Repr, Inhabited, DecidableEq
|
|
|
|
def isValidPhiShellAddress (addr : PhiShellAddress) : Bool :=
|
|
addr.shellIndex < computePhiShellCapacity addr.shellLevel
|
|
|
|
structure PhiShellPath where
|
|
source : PhiShellAddress
|
|
destination : PhiShellAddress
|
|
intermediateShells : List Nat
|
|
deriving Repr, Inhabited, DecidableEq
|
|
|
|
def rangeBetween (a b : Nat) : List Nat :=
|
|
if a ≤ b then
|
|
(List.range (b - a + 1)).map (fun i => a + i)
|
|
else
|
|
(List.range (a - b + 1)).map (fun i => b + i)
|
|
|
|
def computePhiShellPath (source dest : PhiShellAddress) : PhiShellPath :=
|
|
{ source := source
|
|
destination := dest
|
|
intermediateShells := rangeBetween source.shellLevel dest.shellLevel }
|
|
|
|
structure ManifoldScale where
|
|
scale : Nat
|
|
shell : Nat
|
|
deriving Repr, Inhabited, DecidableEq
|
|
|
|
def findShellForScale (scale : Nat) (baseScale : Nat) : Nat :=
|
|
if baseScale = 0 then 0 else scale / baseScale
|
|
|
|
structure NanoKernelShell where
|
|
baseShell : PhiShellAddress
|
|
subIndex : Nat
|
|
deriving Repr, Inhabited, DecidableEq
|
|
|
|
def encodeNanoKernelShell (shell : NanoKernelShell) : Nat :=
|
|
let baseCap := computePhiShellCapacity shell.baseShell.shellLevel
|
|
shell.baseShell.shellLevel * baseCap * baseCap +
|
|
shell.baseShell.shellIndex * baseCap +
|
|
shell.subIndex
|
|
|
|
def decodeNanoKernelShell (encoded : Nat) (level : Nat) : NanoKernelShell :=
|
|
let baseCap := computePhiShellCapacity level
|
|
let remainder := encoded % (baseCap * baseCap)
|
|
{ baseShell := { shellLevel := level, shellIndex := remainder / baseCap }
|
|
subIndex := remainder % baseCap }
|
|
|
|
theorem phiShellPathPreservesEndpoints (source dest : PhiShellAddress) :
|
|
let path := computePhiShellPath source dest
|
|
path.source = source ∧ path.destination = dest := by
|
|
simp [computePhiShellPath]
|
|
|
|
theorem phiShellCapacityGrowth (level1 level2 : Nat) :
|
|
level1 < level2 →
|
|
computePhiShellCapacity level1 < computePhiShellCapacity level2 := by
|
|
intro h
|
|
simp [computePhiShellCapacity, Nat.succ_lt_succ_iff, h]
|
|
|
|
theorem phiShellRadiusGrowth (level1 level2 baseRadius : Nat) :
|
|
0 < baseRadius →
|
|
level1 < level2 →
|
|
computePhiShellRadius level1 baseRadius < computePhiShellRadius level2 baseRadius := by
|
|
intro hBase hLevel
|
|
simp [computePhiShellRadius]
|
|
exact Nat.mul_lt_mul_of_pos_left (Nat.succ_lt_succ hLevel) hBase
|
|
|
|
theorem decodePreservesRequestedLevel (encoded level : Nat) :
|
|
(decodeNanoKernelShell encoded level).baseShell.shellLevel = level := by
|
|
rfl
|
|
|
|
#eval computePhiShellCapacity 4
|
|
#eval computePhiShellRadius 4 10
|
|
#eval (computePhiShellPath { shellLevel := 1, shellIndex := 0 }
|
|
{ shellLevel := 3, shellIndex := 0 }).intermediateShells
|
|
|
|
end Semantics.PhiShellEncoding
|