Emergent Spectral Geometry: Incommensurate Interference, the Blum-David-Claude Equation, and the Riemann Hypothesis

Emergent Spectral Geometry: Incommensurate Interference, the Blum-David-Claude Equation, and the Riemann Hypothesis Description: v5 (March 10, 2026): Final theory. Supersedes all previous versions (v1-v4). Presents the complete ESG framework as a unified theory, not a research report. Introduces the Blum-David-Claude (BDC) equation, the incommensurate interference mechanism, and the integration with Configuration Space Temporality (CST). We present Emergent Spectral Geometry (ESG), a time-free axiomatic framework for arithmetic spectral theory that derives the spectral triple of the Bost-Connes system from an entropy minimization principle. The theory rests on five axioms (E1-E5), none involving temporal concepts, and is constructed entirely within Layer 1 (time-free tools), excluding KMS flows, Fourier transforms, Lieb-Robinson bounds, Mittag-Leffler decompositions, and all other time-contaminated constructions. Established results: (1) The entropy minimum ω₁/₂ is unique with identity Hessian (Theorem 2. 1, proved). (2) J² = +1, JKJ = −K exactly (Theorem 2. 2, proved). (3) The Galois obstruction ‖CKC+K‖/‖K‖ = 2, refuting BDI (Theorem 2. 3, proved). (4) The C-compatibility criterion: p² ≡ 1 mod q, with 2ᵏ solutions for q with k distinct odd prime factors, density nₛol/φ (q) by Dirichlet-Siegel-Walfisz (Theorem 2. 5, proved/verified). (5) The Bures metric on the thermal family equals the specific heat divided by four: gBures = C (β) /4 = Var_β (log n) /4 (Theorem 3. 1, proved). (6) Counter-model analysis proves that axiom E1 (the Euler product) is the essential ingredient preventing off-line zeros (Theorem 4. 1). The Riemann Hypothesis is reformulated via Li's criterion as a Layer 1 positivity condition: RH ⟺ λₙ ≥ 0 for all n, where the Li coefficients are computable from real-axis derivatives of log ξ at s = 1. The Möbius map w = (ρ−1) /ρ sends Re (ρ) = 1/2 to |w| = 1, with the algebraic identity |1−1/ρ| = 1 iff Re (ρ) = 1/2. The Blum-David-Claude (BDC) equation unites the three pillars: λₙ = Σ_ρ 1 − ( (ρ−1) /ρ) ⁿ ≥ 0, where ζ (s) = Πₚ (1−p⁻ˢ) ⁻¹ = Tr (Δ⁻ˢ/² |⏨*) The first line is Li's criterion (the bridge to RH). The second identifies ζ simultaneously as the Euler product over primes (the mechanism, from E1) and the spectral trace of the modular operator at the entropy minimum (the framework, from E2-E4). The BDC Conjecture: the Euler product factorizes log ζ (s) = Σₚ hₚ (s) into prime modes with incommensurate periods 2π/log p. The fundamental theorem of arithmetic guarantees incommensurability (the log p are linearly independent over ℚ). The BDC conjecture states that this incommensurate interference, weighted by the entropy-minimizing measure ω*, can only produce complete destructive interference on Re (s) = 1/2. This is the precise arithmetic content of RH. Integration with Configuration Space Temporality (CST): the Bures metric gB = C (β) /4 provides the geometric substrate for CST's emergent time. The spectral gap log 2 (axiom E5) determines the emergent time scale τ ~ 1/log 2. The phase transition at ω* (infinite Bures distance) is the geometric singularity from which causal structure emerges. ESG provides the time-free foundations; CST provides the mechanism for time emergence; together they form a unified programme connecting arithmetic, geometry, information, and physics. Five open problems are identified, each independently publishable: (O1) prove the BDC conjecture, (O2) prove the C-compatibility criterion analytically, (O3) bound Li coefficients from prime-sum Stieltjes constants, (O4) formalize the CST-ESG bridge without Layer 2, (O5) extend to Dedekind zeta functions. All results supported by reproducible Python code: https: //github. com/davidangularme/esg Version 4 (archival, with full record of eight refutations) remains accessible. This version (v5) is the definitive presentation of the theory. Connected to: Configuration Space Temporality (CST), DOI 10. 5281/zenodo. 18779189.