Emergent Spectral Geometry I: Arithmetic Symmetry Classification of the Bost-Connes System

We investigate the symmetry classification of the Bost-Connes system at critical inverse temperature β = 1 from the perspective of Emergent Spectral Geometry (ESG), a framework in which the spectral triple of noncommutative geometry is derived from a thermodynamic variational principle rather than axiomatically prescribed. The paper establishes five main results: (1) The Haar barycenter state ω₁/₂ is the unique minimum of the Kullback-Leibler divergence on the KMS₁ Choquet simplex, with identity Hessian (Theorem 3. 2). (2) The Tomita-Takesaki modular conjugation J satisfies J² = +1 and JKJ = −K exactly, providing KO-dimension signs ε = +1, ε' = −1 (Theorem 5. 1). (3) No diagonal operator (including arithmetic parity via the Liouville function) can serve as a chiral grading for the modular Hamiltonian (Theorem 6. 1). (4) The Galois conjugation σ₋₁ does not anticommute with K: the relative error ‖CKC + K‖/‖K‖ = 2 exactly, for all truncations q = 5 through 2003 (Theorem 7. 1). This definitively refutes the BDI classification proposed in earlier work. (5) As a constructive alternative, we introduce the adelic Hamiltonian Kₐd = (K − CKC) /2, which anticommutes with both J and C simultaneously (Theorem 8. 2), and discover the p-adic defect tower: the null space of the defect matrix Dₚ = CKₚC + Kₚ, indexed by primes, provides a growing family of compatible Hamiltonians. The C-compatibility condition for a prime p is p² ≡ 1 mod q (Theorem 9. 2). The null space dimension grows with q, producing an invariant in (ℤ₂) N that goes beyond the standard Altland-Zirnbauer tenfold classification. All results are supported by reproducible Python code (included as supplementary material). Each claim is explicitly marked as proved, numerically verified, or open. This is the first paper in the ESG programme. It connects to the Configuration Space Temporality (CST) framework deposited at DOI 10. 5281/zenodo. 18779189.