independent_loop_count

formal source

 399    For D = 3: 3 · 2 / 2 = 3 independent loops.
 400    These 3 loops bound the 3 independent faces of Q₃.
 401    Each face contributes one topological charge (the flux through it). -/
 402def independent_loop_count (D : ℕ) : ℕ := D * (D - 1) / 2
 403
 404theorem three_independent_loops_D3 :
 405    independent_loop_count 3 = 3 := by native_decide
 406
 407/-- **THEOREM (Face Count = Loop Count for D = 3)**:
 408    The number of independent loops equals the number of face-pairs
 409    in D = 3. Each face of Q₃ corresponds to a loop, and each loop
 410    gives a topological charge. -/
 411theorem loops_eq_face_pairs_D3 :
 412    independent_loop_count 3 = ParticleGenerations.face_pairs 3 := by
 413  native_decide
 414
 415/-! ## Part 11: Summary Certificate -/
 416
 417/-- **F-013 CERTIFICATE: Winding Charges**
 418
 419    Conservation laws in RS are derived from winding numbers of lattice paths:
 420
 421    1. **INTEGER**: w_k(path) ∈ ℤ (counts net steps) → charge quantization
 422    2. **ADDITIVE**: w_k(p₁++p₂) = w_k(p₁) + w_k(p₂) → charges add
 423    3. **INVARIANT**: Cancelling pairs preserve winding → topological protection
 424    4. **INDEPENDENT**: D axes give D independent charges
 425    5. **D-SPECIFIC**: Charge count = dimension → D=3 gives 3 charges
 426    6. **UNIFIED**: 3 charges = 3 face-pairs = 3 colors = 3 generations
 427
 428    Conservation is NOT from symmetry (Noether). It is from TOPOLOGY:
 429    you cannot change a winding number by local deformations. -/
 430theorem winding_charges_certificate :
 431    -- 1. Winding numbers are additive
 432    (∀ (D : ℕ) (p₁ p₂ : LatticePath D) (k : Fin D),