/- # HyperKitty Core Definitions ## SNAPKITTYWEST Research Institute ## Formal Verification Suite for Deterministic Routing **Author:** Ahmad Ali Parr **Affiliation:** SNAPKITTYWEST, Bel Esprit D'Accord Irrevocable Trust **Repository:** https://github.com/SNAPKITTYWEST/hyperkitty **Date:** August 2026 **Version:** 1.0.0 - Gold Standard This module defines all canonical types and constants for the HyperKitty system. All definitions are constructive and fully computable. -/ -- ============ GLYPH: The Six Routing Primitives ============ /-! Glyph: The six canonical routing primitives from paper Section 2.1. These correspond to the six dimensions of the QLG sphere and the six states of the QRA deterministic finite automaton. -/ inductive Glyph where | Pi -- Propositio: send proposition (0x01) | Gamma -- Guard: receive guard check (0x03) | Delta -- Transition: execute state transition (0x04) | Omega -- Conclusio: absorbing terminal (0x0A) | Lambda-- Locality: identity element (0xFF) | Psi -- Negative transition (0x0B) deriving DecidableEq, Repr -- Enumeration matching paper Section 2.1 @[simp] def Glyph.idx : Glyph → Fin 6 | .Pi => 0 | .Gamma => 1 | .Delta => 2 | .Omega => 3 | .Lambda => 4 | .Psi => 5 @[simp] def Glyph.ofIdx : Fin 6 → Glyph | 0 => .Pi | 1 => .Gamma | 2 => .Delta | 3 => .Omega | 4 => .Lambda | 5 => .Psi @[simp] theorem Glyph.idx_ofIdx (i : Fin 6) : (Glyph.ofIdx i).idx = i := by fin_cases i <;> rfl @[simp] theorem Glyph.ofIdx_idx (g : Glyph) : Glyph.ofIdx g.idx = g := by cases g <;> rfl -- ============ QRA ROUTING TENSOR ============ /-! Q: The 6×6 QRA routing tensor from paper Section 3.1. This is the transition matrix for the deterministic 6-state automaton. Q[i][j] tells us what state to transition to from state i with previous state j. Row meanings: 0 = Pi row 1 = Gamma row 2 = Delta row 3 = Omega row (absorber: always stays in Omega) 4 = Lambda row (identity: returns the previous state) 5 = Psi row Key properties: - Lambda row is identity: Q[4][j] = j for all j - Omega row is absorber: Q[3][j] = 3 for all j - All other rows deterministically route based on paper Table 1 -/ -- Paper Table 1 (Zenodo PDF, Ahmad Parr 2026): -- prev: Pi(0) Ga(1) De(2) Om(3) La(4) Ps(5) -- Pi(0): 2 2 3 3 2 2 -- Ga(1): 2 3 3 3 2 3 -- De(2): 3 3 3 3 2 3 -- Om(3): 3 3 3 3 3 3 [absorber] -- La(4): 0 1 2 3 4 5 [identity] -- Ps(5): 2 3 3 3 2 3 def Q : Fin 6 → Fin 6 → Fin 6 | 4, j => j -- Lambda row: identity | 3, _ => 3 -- Omega row: absorber | 0, j => if j = 2 ∨ j = 3 then 3 else 2 -- Pi: Omega when prev∈{Delta,Omega}, else Delta | 1, j => if j = 0 ∨ j = 4 then 2 else 3 -- Gamma: Delta when prev∈{Pi,Lambda}, else Omega | 2, j => if j = 4 then 2 else 3 -- Delta: Delta when prev=Lambda, else Omega | 5, j => if j = 0 ∨ j = 4 then 2 else 3 -- Psi: Delta when prev∈{Pi,Lambda}, else Omega | _, _ => 3 /-! Glyph.next: Compute the next state in QRA evolution. Given current state curr and previous state prev, compute the next state by looking up Q[curr.idx][prev.idx]. -/ def Glyph.next (curr prev : Glyph) : Glyph := Glyph.ofIdx (Q curr.idx prev.idx) -- ============ LEDGER: Symbolic Ledger Algebra ============ /-! Ledger: A balanced ledger from paper Section 2.2. Structure components: s : ℤ - ledger size δ : ℤ - debit (outflow) ι : ℤ - credit (inflow) ω : ℤ - domain identifier Invariant: δ + ι = 0 (always balanced) -/ structure Ledger where s : ℤ -- size δ : ℤ -- debit ι : ℤ -- credit ω : ℤ -- domain deriving Repr -- Balance axiom R(λ) = δ + ι = 0 from paper def Ledger.balance (λ : Ledger) : Prop := λ.δ + λ.ι = 0 /-! Ledger.mkBalanced: Constructor that enforces balance invariant. Creates a balanced ledger by accepting debit δ and automatically computing credit as ι = -δ, ensuring δ + ι = 0. -/ def Ledger.mkBalanced (s δ ω : ℤ) : Ledger := {s := s, δ := δ, ι := -δ, ω := ω} @[simp] theorem Ledger.balance_mkBalanced (s δ ω : ℤ) : (Ledger.mkBalanced s δ ω).balance := by simp [Ledger.balance] omega /-! Ledger.comp: Composition of two balanced ledgers. Two ledgers can be composed only if they have matching domain (ω). The result is a new ledger with combined size and summed debit/credit. -/ def Ledger.comp (λ₁ λ₂ : Ledger) : Option Ledger := if h : λ₁.ω = λ₂.ω then some { s := λ₁.s + λ₂.s δ := λ₁.δ + λ₂.δ ι := λ₁.ι + λ₂.ι ω := λ₁.ω } else none -- ============ VEC3: Quadratic Ledger Geometry ============ /-! Vec3: Three-dimensional integer vectors for QLG. The canonical QLG surface is the unit integer sphere: x² + y² + z² = K where K = 1 Only 6 integer solutions exist on the unit sphere: (±1, 0, 0), (0, ±1, 0), (0, 0, ±1) -/ structure Vec3 where x : ℤ y : ℤ z : ℤ deriving Repr -- Canonical QLG: unit integer sphere x² + y² + z² = 1 def QLG.canonical (v : Vec3) : Prop := v.x^2 + v.y^2 + v.z^2 = 1 def QLG.K : ℤ := 1 /-! Vec3.ofGlyph: Bijection from glyphs to canonical QLG points. Maps each glyph to its unique point on the unit sphere: Pi ↔ (1, 0, 0) Gamma ↔ (-1, 0, 0) Delta ↔ (0, 1, 0) Psi ↔ (0, -1, 0) Lambda ↔ (0, 0, 1) Omega ↔ (0, 0, -1) -/ def Vec3.ofGlyph : Glyph → Vec3 | .Pi => {x:=1,y:=0,z:=0} | .Gamma => {x:=-1,y:=0,z:=0} | .Delta => {x:=0,y:=1,z:=0} | .Psi => {x:=0,y:=-1,z:=0} | .Lambda => {x:=0,y:=0,z:=1} | .Omega => {x:=0,y:=0,z:=-1} /-! Glyph.ofVec3: Inverse bijection from QLG points to glyphs. Converts a vector to its corresponding glyph, or returns none if the vector is not a canonical QLG point. -/ def Glyph.ofVec3 : Vec3 → Option Glyph | {x:=1,y:=0,z:=0} => some .Pi | {x:=-1,y:=0,z:=0} => some .Gamma | {x:=0,y:=1,z:=0} => some .Delta | {x:=0,y:=-1,z:=0} => some .Psi | {x:=0,y:=0,z:=1} => some .Lambda | {x:=0,y:=0,z:=-1} => some .Omega | _ => none -- ============ SPIN FACTOR ALGEBRA ============ /-! SpinFactor: Parameterized algebra structure (α, v) where α ∈ ℤ, v ∈ ℤⁿ. The spin factor product x ∘ y is defined as: x = (α, v), y = (β, w) x ∘ y = (α*β + ⟨v, w⟩, α*w + β*v) This is commutative, idempotent, and has exactly 2 primitive idempotents. -/ structure SpinFactor where scalar : ℤ vector : List ℤ deriving Repr /-! SpinFactor.mul: The spin factor product operation. Implements x ∘ y commutative product. For clarity, we compute: - Scalar part: α*β + dot(v, w) - Vector part: α*w + β*v -/ def SpinFactor.mul (x y : SpinFactor) : SpinFactor := let α := x.scalar let β := y.scalar let dot := List.zipWith (· * ·) x.vector y.vector |> List.sum let scalar_part := α * β + dot let vector_part := List.map (· * β) x.vector ++ List.map (· * α) y.vector {scalar := scalar_part, vector := vector_part} -- Commutativity property (proven separately in Jordan.lean) def SpinFactor.commutative (x y : SpinFactor) : Prop := x.mul y = y.mul x