Three Exact Theorems

3 Three Exact Theorems

Theorem 1 (Universal Boson Norm). For any n-site equal superposition with arbitrary phases,

∘ ------ ∥B ∥ = n-−-1. n

Proof. ρ_ij = (1∕n)e^{i(ϕ_i−ϕ_j)}, so |B_ij|² = 1∕n² for all i≠j. There are n(n − 1) off-diagonal pairs, giving ∥B∥² = n(n − 1)∕n² = (n − 1)∕n. □ □

Theorem 2 (Universal Fermion Norm). For any n-site equal superposition with arbitrary phases,

∥F ∥ = √1-. n

Proof. F_k = |ψ_k|² = 1∕n for all k, so ∥F∥² = ∑ _k1∕n² = 1∕n. □ □

Theorem 3 (Fermion-Boson Conservation Law). For any pure state ψ,

∥F ∥2 + ∥B∥2 = 1.

Proof. For any pure state, ρ² = ρ, so ∥ρ∥² = Tr(ρ²) = Tr(ρ) = 1. Since F and B occupy orthogonal subspaces, ∥ρ∥² = ∥F∥² + ∥B∥² = 1. □ □

Remark (Connection to Paper VII). For ψ(𝜃) = cos𝜃e_i + isin𝜃e_j, ∥B(𝜃)∥² = sin²(2𝜃)∕2 (Paper VII, exact). Then ∥F(𝜃)∥² = cos⁴𝜃 + sin⁴𝜃, and

∥F∥2 + ∥B ∥2 = cos4𝜃 + sin4 𝜃 + 2 sin2 𝜃cos2𝜃 = (cos2𝜃 + sin2𝜃)2 = 1.

Theorem 3 subsumes the Paper VII result as a special case.

Corollary 1 (Pauli exclusion). For a localised fermion ψ = e_j: ∥F∥ = 1, ∥B∥ = 0. No compression residual is produced. Double occupancy has zero off-diagonal content and is informationally invisible.

Table 1 summarises the theorems for small n.

√--
n — Table 1: Universal fermion and boson norms. All rows satisfy ∥F∥² + ∥B∥² = 1 exactly.

∘ ---------
(n− 1)∕n — Table 1: Universal fermion and boson norms. All rows satisfy ∥F∥² + ∥B∥² = 1 exactly.

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n	∥F∥ = 1∕	∥B∥ =	∥F∥²	∥B∥²
1	1	0	1	0
2	1∕	1∕	1∕2	1∕2
3	1∕		1∕3	2∕3
4	1∕2	∕2	1∕4	3∕4