Paper 33 — Before the Zero-Divisors: Residues, Automorphisms and Gauge Coupling Constants in Cayley–Dickson Algebras

Paper 26 derived Newton's constant G_N from the failure of the Cayley–Dickson quadratic norm N to compose at level n=4 (sedenions), where zero-divisors first appear. Open Problem 26.7 asks: do the gauge coupling constants g₁², g₂², g₃² emerge from N at levels n=1,2,3 — the Hurwitz algebras where N composes perfectly — by the same Sakharov–Adler mechanism?

This paper builds the algebraic case. We compute all four residues of the ZD zeta function Z(x) = 336x⁴/[(1−x)(1−2x)(1−4x)(1−8x)] at its poles x = 2^{−k}, k=0,1,2,3, obtaining R₀=16, R₁=−7, R₂=7/8, R₃=−1/32, and derive a closed-form formula. The functional equation Z(x) = 64x⁴Z(1/8x) imposes the residue symmetry R_k = −2^{9−6k} R_{3−k}, which pairs (ℂ,ℍ) — the electroweak levels — and (ℝ,𝕆). The alternating-sum identity 1−7+14−8=0 is a corollary of the residue theorem on ℙ¹, not a numerical coincidence.

A new algebraic observation: the rescaled residues R_k · dim(A_k)⁴ = 16·(−1)^k · s_k yield (s₀,s₁,s₂,s₃) = (1,7,14,8), equal respectively to dim(ℝ), the number of points of the Fano plane PG(2,F₂), dim(G₂) = dim Aut(𝕆), and dim(𝕆). The identity s₂ = 2s₁ reflects the structural fact that dim(SO(7)/G₂) = 7 = |Fano|.

We also establish the spectral transition theorem: at n≤3, the Gram operator M_a = L_a^T L_a of any unit element has spectrum {1^{2^n}} (every direction is norm-preserving — no dark sector, no amplification); at n=4 the spectrum becomes {0⁴, 1⁸, 2⁴}. The dark sector V₀ = ker(L_a) is algebraically absent at all Hurwitz levels and first appears at n=4. This is the algebraic signature of the gauge-to-gravity transition in spectral language.

Under IIB-II-strong, we conjecture that the Sakharov–Adler loop at level k runs over a scalar field on Aut(A_k), with spectral invariant S_k = dim Aut(A_k) in the absence of fermionic corrections (which vanish because ker(L_a) = 0 at all Hurwitz levels). The complete derivation of g₁², g₂², g₃² requires identifying the matter field at each level (OP 33.1) and running from the Planck scale to observable energies (OP 33.2).

This paper is complementary to Furey's programme, which derives the gauge symmetry groups from the algebra of 𝕆; the present paper addresses the distinct question of the numerical coupling constants, which Furey's programme takes as experimental inputs.