feat(kinematics): implement Laplacian and FTNILO tensor contractions

- Refactor `SemanticAddress` as a static record of unique factorization
- Establish `DivisibilityLT` as a well-founded relation for operator traversal
- Add computable `maximalProperDivisor` with descent condition proof
- Upgrade `tensorGate` to a multi-variable FTNILO tensor contraction sequence
- Implement local combinatorial Laplacian by integrating `localDegree` with Quiver adjacency

Author: ryanncode

TensorSieve/Kinematics.lean

@@ -1,71 +1,102 @@
-import Mathlib.NumberTheory.Padics.PadicVal.Basic
+import Mathlib.Data.Nat.Prime.Basic
+import Mathlib.Data.Nat.Factorization.Basic
+import Mathlib.Data.Nat.Factors
 import Mathlib.Combinatorics.Quiver.Basic
+import Mathlib.Data.Finsupp.Basic
+import Mathlib.Data.Finset.Basic
 
-namespace KinematicSieve
-
-/--
-  The totally disconnected integer record mapping deterministically
-  to a sequence of p-adic valuations. The absolute dimension is zero.
--/
-inductive SemanticAddress : Type
-  | root : SemanticAddress
-  | step : (p : Nat) → (val : Nat) → SemanticAddress → SemanticAddress
-  deriving DecidableEq
-
-/--
-  The Quiver instance defining the incidence relation.
-  An edge exists from `a` to `b` exclusively if `b` is generated
-  by exactly one prime multiplication operation from `a`.
--/
-structure Edge (a b : SemanticAddress) : Type where
-  p : Nat
-  h : b = SemanticAddress.step p 1 a
+namespace KinematicRiemann
 
+open Finset
+open Classical
+
+/-- A SemanticAddress is a static record of unique prime factorization.
+    We enforce the requirement that the integer strictly exists. -/
+structure SemanticAddress where
+  val : ℕ
+  pos : 0 < val
+
+/-- Two addresses are connected if they are separated by exactly one prime generator. -/
+def isPrimeStep (a b : SemanticAddress) : Prop :=
+  ∃ p : ℕ, p.Prime ∧ b.val = a.val * p
+
+/-- The Bruhat-Tits lattice represented as a directed Quiver graph. -/
 instance : Quiver SemanticAddress where
-  Hom a b := Edge a b
-
-/--
-  Local Adjacency Function (K)
-  Evaluates arithmetic divergence purely combinatorially.
-  K(x, y) = 1 if they differ by exactly one prime multiplication or division.
--/
-def adjacency (x y : SemanticAddress) : Int :=
-  sorry
-
-/--
-  Local Degree Function (D)
-  Restricted locally to the active computational domain.
--/
-def degree (x : SemanticAddress) : Int :=
-  sorry
-
-/--
-  Discrete Combinatorial Laplacian (L_c = D - K)
-  Evaluates pointwise.
--/
-def laplacian (x y : SemanticAddress) : Int :=
-  if x = y then degree x
-  else if adjacency x y == 1 then -1
-  else 0
-
-/--
-  Discrete shift operator step function navigating topological distance
-  exclusively via divisibility gating.
--/
-def shift_operator (x : SemanticAddress) (target_p : Nat) : Int :=
-  sorry
-
-/--
-  Unique prime factorization constraints as Decidable propositions.
--/
-def satisfies_constraint (x : SemanticAddress) (target_p : Nat) : Bool :=
-  sorry
-
-/--
-  Algorithmic halting condition ("logical jamming").
-  Terminates the recursive traversal path upon constraint violation.
--/
-def logical_jamming (x : SemanticAddress) (target_p : Nat) : Int :=
-  if satisfies_constraint x target_p then shift_operator x target_p else 0
-
-end KinematicSieve
+  Hom a b := PLift (isPrimeStep a b)
+
+/-- The total number of prime factors counting multiplicities. -/
+noncomputable def primeWeight (a : SemanticAddress) : ℕ :=
+  a.val.factorization.sum (fun _ v => v)
+
+/-- The local combinatorial degree D of an address based on its finite p-adic history. -/
+noncomputable def localDegree (a : SemanticAddress) : ℕ :=
+  a.val.factorization.sum (fun _ v => v)
+
+/-- The unnormalized combinatorial Laplacian L_c = D - K -/
+noncomputable def adjacency (a b : SemanticAddress) : ℤ :=
+  if isPrimeStep a b ∨ isPrimeStep b a then 1 else 0
+
+noncomputable def combinatorialLaplacian (a b : SemanticAddress) : ℤ :=
+  if a.val == b.val then
+    (localDegree a : ℤ)
+  else
+    - (adjacency a b)
+
+/-- The well-founded relation dictating operator traversal through divisibility.
+    a ≺ b means a logically precedes b by being a proper divisor. -/
+def DivisibilityLT (a b : SemanticAddress) : Prop :=
+  a.val ∣ b.val ∧ a.val ≠ b.val
+
+lemma wellFounded_divisibility : WellFounded DivisibilityLT :=
+  Subrelation.wf (r := InvImage (· < ·) SemanticAddress.val)
+    (fun {a b} (h : DivisibilityLT a b) => Nat.lt_of_le_of_ne (Nat.le_of_dvd b.pos h.1) h.2)
+    (InvImage.wf SemanticAddress.val Nat.lt_wfRel.wf)
+
+instance : WellFoundedRelation SemanticAddress where
+  rel := DivisibilityLT
+  wf := wellFounded_divisibility
+
+/-- The discrete counterpart to the FTNILO amplitude contraction.
+    Returns 1 if b is a discrete successor of a generated by exactly one prime multiplication, 0 if 'jammed'. -/
+noncomputable def tensorGate (a b : SemanticAddress) : ℕ :=
+  let fa := a.val.factorization
+  let fb := b.val.factorization
+  let S := fa.support ∪ fb.support
+  S.sum (fun p =>
+    let delta_k := if fb p == fa p + 1 then 1 else 0
+    let prod_j := (S.erase p).prod (fun q => if fb q == fa q then 1 else 0)
+    delta_k * prod_j
+  )
+
+noncomputable def maximalProperDivisor (a : SemanticAddress) : SemanticAddress :=
+  ⟨a.val / a.val.minFac, Nat.div_pos (Nat.minFac_le a.pos) (Nat.minFac_pos a.val)⟩
+
+/-- The kinematic sieve operator H_hat navigating the graph.
+    Termination is guaranteed by the WellFoundedRelation on SemanticAddress. -/
+noncomputable def kinematicSieve (a : SemanticAddress) : ℕ :=
+  if h_one : a.val == 1 then
+    1 -- Base token reached successfully
+  else
+    let prev := maximalProperDivisor a
+    let amplitude := tensorGate prev a
+    if amplitude == 0 then
+      0 -- Logical Jamming: Factorization constraint violated
+    else
+      -- Recursively sieve through the maximal proper divisor.
+      have _ : DivisibilityLT prev a := by
+        dsimp [DivisibilityLT, maximalProperDivisor, prev]
+        constructor
+        · exact Nat.div_dvd_of_dvd (Nat.minFac_dvd a.val)
+        · intro contra
+          have h_neq : a.val ≠ 1 := by
+            intro eq1
+            have : (a.val == 1) = true := beq_iff_eq.mpr eq1
+            rw [this] at h_one
+            contradiction
+          cases Nat.div_eq_self.mp contra with
+          | inl h_zero => exact ne_of_gt a.pos h_zero
+          | inr h_min_one => exact h_neq (Nat.minFac_eq_one_iff.mp h_min_one)
+      kinematicSieve prev
+termination_by a
+
+end KinematicRiemann