From Three to Fifteen Grade Reciprocity, Phase-Spatial-Chirality, and the A₃ Completion Problem

3 → 3|1 → 3|1|3 → 3|1|3|8 A generator count is not yet a generator genealogy. sector genealogy → Lie closure → physical realization Why do sector dimensions resembling 3|1|3|8 appear in the symmetry architecture of mathematical physics? Recognizing these dimensions after familiar Lie groups are known is different from explaining how typed precursor sectors with those dimensions might arise before their Lie brackets are assumed. This paper investigates the latter problem as a generator genealogy. A Grade Reciprocity criterion, dim V = dim Λ²V, uniquely selects the nontrivial finite-dimensional rank dim V = 3. Adjoining a primitive one-dimensional phase carrier produces a four-dimensional phase-spatial carrier V_PS = V₃ ⊕ P₁, whose second exterior power decomposes into two real triplets. Thus a second three-dimensional relational sector appears without assuming SU(2) or another three-generator Lie algebra. Spin-cover structure then motivates a finer interpretation of the second triplet, producing the intermediate Phase-Spatial-Chirality architecture PSC₇ = 3_E|1_P|3_S. Separately, if the reciprocal six-dimensional relational space naturally acquires a compatible complex structure J and positive Hermitian form, its stabilizer yields 𝔲(3) = 𝔲(1) ⊕ 𝔰𝔲(3), giving a conditional 1+8 decomposition. Before identifying primitive phase with the resulting central one-dimensional action, the complete typed inventory is sixteen-dimensional. Reduction to PSI₁₅ = 3_E|1_P|3_S|8_T requires a separate Phase Descent result. Three mathematical completion problems are isolated. B_J asks whether the required complex structure follows naturally from phase-spatial geometry. B_P asks whether primitive phase descends to the central 𝔲(1) action rather than supplying an independent generator. B_A asks whether the resulting fifteen-dimensional typed architecture admits an independently derived Lie bracket whose complexification has Dynkin type A₃. The dimensional correspondence between the internal 1|3|8 sectors and the Standard Model gauge-generator inventory is treated as a candidate physical correspondence rather than a derivation. The principal result is the forward typed genealogy and the explicit localization of the remaining proof obligations. The manuscript distinguishes sector genealogy (𝒢), Lie closure (𝒜), and physical realization (𝒫), and applies three methodological rules throughout: Anti-Circularity; Type Before Identification; and Provenance Before Correspondence. Keywords: closure mathematics; Grade Reciprocity; generator genealogy; phase-spatial geometry; Phase-Spatial-Chirality; exterior algebra; spin geometry; Lie algebra; A₃; gauge symmetry; mathematical physics.