Algebraic‑Geometric Duality in Spin(9): Clifford Spectra, Heat Kernels, and Cartan Depth Hierarchy

Abstract

This manuscript constitutes Part I of a three‑part framework for the algebraic‑geometric foundations of fermion flavor structure, developed without phenomenological input. We examine the real Clifford algebra Cl(9), the associated Spin(9)=B4 root system, and the spectral geometry of the round compact three‑sphere S3. The main mathematical results are:1.chiral spinor decomposition of Cl(9), with normalization constant Cnorm=32 fixed by Bott periodicity;2.complete branching rule Spin(9)⊃B2×D2, with Coxeter invariants for B4 and its subalgebras;3.a rigorous definition of the flavor 3‑chain (Weyl 3‑node orbit) and an integer‑valued Cartan depth recursion for quark and lepton weights;4.an effective branching Casimir operator C2eff that satisfies a factorization identity separating the three lepton generations via Cartan coordinates.All derivations use only standard Lie algebra theory and S3 heat kernel analysis. No symmetry breaking, extra scalars, or phenomenological fitting is introduced. This work provides the geometric backbone for two companion papers: Part II constructs LO and NLO flavor phenomenology for all 35 Standard Model observables; Part III establishes neutrino–dark energy homology. Numerical predictions for mixing angles, masses, and cosmology are deferred to those subsequent articles.