Project T Volume XXXVII : The Dimensional Selection Why K = 3 Is the Unique Physical Dimension: Commutation Deficit, Quadratic Surd Boundary, and the Five Selection Criteria

We prove that K = 3 is the unique physical dimension selected by the algebraic structure of the symmetric group chain. Five independent criteria converge on a single answer: (i) the commutation deficit C(K) = K(K−3)/2 has exactly one nontrivial zero at K = 3; (ii) S3 is the only symmetric group with an integer spectral gap (λ1 = 1); (iii) the palindromic pattern [1, 2, 2, 1] is the simplest nontrivial palindrome; (iv) S3 is the root of the number field tower Q ⊂ Q(√2) ⊂ Q(√2, √5) ⊂ Q(√2, √3, √5); and (v) S3 maximizes spectral efficiency at 67%. Furthermore, the quadratic surd boundary at n = 6 defines the maximal algebraically constructible chain S3–S6, beyond which spectral gaps involve cubic irrationals. The case is closed with zero free parameters.