Paper 38 — Modular Universality: The Schubert Tower and the Matter–Gravity Split

We study the zero-divisor algebra ZD*(𝒜₄) of the sedenions 𝒜₄, equipped with its automorphism group G₈₄ = Aut(ZD*(𝒜₄)) of order 2688 = 2⁷·3·7, and identify three structural patterns connecting its representation theory, orbital geometry, and gravitational spectral action.

First (modular universality): the three-dimensional irreducible representation irr₃ of PSL(2,7) ≅ GL(3,𝔽₂) appears as the same 𝔽₂-module V₃ = irr₃|_{𝔽₂} in three independent constructions — spinorial boundary conditions on the Black Circle interior, the bosonic on-line sector decomposition under S₄ ⊂ PSL(2,7), and the group cohomology H²(PSL(2,7), (ℤ/2ℤ)⁴) of G₈₄. GAP 4.15.1 computations confirm the identification and establish the unexpected decomposition Sym²(V₃) ≅ V₃ ⊕ V₃' over 𝔽₂, in contrast to the characteristic-zero identity Sym²(irr₃) = irr₆.

Second (Schubert tower): the orbital stratification of G₈₄ on ZD*(𝒜₄) — with orbit sizes {2, 14, 84, 336} — has stabilizer chain PSL(2,7) ⊃ S₄ ⊃ V₄ ⊃ {e}, verified by GAP. This chain mirrors precisely the Hurwitz hierarchy of normed division algebras 𝒜₃ ⊃ 𝒜₂ ⊃ 𝒜₁ ⊃ 𝒜₀ by algebra dimension (8, 4, 2, 1), and under the IIB-II conjecture labels the four gauge sectors of the Standard Model plus gravity.

Third (matter-gravity split): the bosonic-pure coefficients of the one-loop induced Newton constant G_N⁻¹, derived from the sedenion scalar field theory S₈₄ via the Sakharov–Adler mechanism, belong entirely to the multiplicative semigroup {2ᵃ·3ᵇ·7ᶜ} — the prime signature of |G₈₄|. This structure is broken precisely by the fermionic correction ker(L_a) = 4, the algebraic matter content of the theory. As a logical cascade, the conjunction of conjectures IIB-II and IIB-IV selects α = 0 in the two-parameter spinorial boundary-condition family BC(α,β), partially resolving an open problem on the boundary conditions of the Black Circle interior.

Four supplementary GAP 4.15.1 scripts are deposited as ancillary files. (on Zenodo)