Paper 26 — Induced Gravity from the Sedenion Scalar Field Theory S84: Newton Constant as the Cayley–Dickson Norm

Paper 26 of the Cayley–Dickson Zero-Divisor Series.

We apply the one-loop Sakharov–Adler induced gravity mechanism to S₈₄, the quartic scalar field theory on the 84 signed zero-divisors of the sedenion algebra A₄. Integrating out the84 real scalars and 4 Majorana fermions in a curved spacetime background via the Frolov–Fursaev supertrace formula, we derive Newton's gravitational constant G_N as an explicit function of the Cayley–Dickson quadratic norm N evaluated on the sedenionic condensate C₀ (Theorem 4.2):

1/G_N = (16λN(C₀)/12π) · [164·ln(Λ²/16λN(C₀)) − 220·ln2 − 42·ln3]

The effective spectral invariant S_eff = 164 = |GL(3,F₂)| − 4 encodes the Fano-incidence combinatorics of A₄ after the Majorana fermion correction reduces the bosonic value S = 168 = |GL(3,F₂)| by 4. A key structural result (Lemma 3.1) shows that the 14 massless Level-2 modes are gravitationally decoupled: the algebraic factor (2+μ₂) = 0 that protects their masslessness under PSL(2,7) simultaneously makes them gravitationally neutral — one mechanism, two physical consequences.

A complementary geometric derivation (Route B) uses the Riemannian structure ZD(S) ≅ V₂(ℝ⁷) = G₂/SU(2) (Reggiani 2024): the heat-kernel a₁ coefficient at the physical metric r = 2/3 gives G_N⁻¹|_B = √(6π)/128 ≈ 0.0339 M_Pl². Both routes confirm G_N⁻¹ ∝ N(C₀), providing independent algebraic and geometric evidence for the IIB-II conjecture. A self-consistency equation (Theorem 4.6) expresses G_N implicitly as a function of the Black Circle mass M and coupling λ, making the Planck scale an algebraic output rather than an input.

Companion paper: Two-Loop Induced Gravity and Branch Selection for The Black Circle Self-Consistency Equation https://doi.org/10.5281/zenodo.20703988