This paper asks whether the internal algebra used in the noncommutative-geometric formulation of the Standard Model, A_SM = ℂ ⊕ ℍ ⊕ M₃(ℂ), can be selected from Lorentzian Clifford and finite-spectral-triple data rather than assumed at the outset. Starting with a four-dimensional DeWitt superspace, the required KO-dimension-six reality structure selects the negative-trace Clifford branch of signature (6,4). Its compact spin factors generate the Pati-Salam algebra A_PS = M₄(ℂ) ⊕ ℍ_L ⊕ ℍ_R. An intrinsic carrier-nondegeneracy condition then excludes the unwanted order-zero representations without using dimension minimization. Finally, maximizing compatibility with the first-order condition over every nonzero projective Majorana line selects exactly the rank-one Segre locus: its equalizer has real dimension 24, compared with dimension 13 for every rank-two line. The resulting equalizer is A_SM. The Cartan-splitting space is also contractible, so the reference split is a gauge choice rather than an additional selection assumption. The reconstruction remains conditional on the presence of a nonzero Majorana coefficient. It does not derive that coefficient's magnitude, the number of fermion generations, Yukawa matrices, a global gauge-group quotient, compactification, or low-energy phenomenology. All algebraic claims are proved in the article, which is accompanied by six optional computational reproducibility certificates.