Emergent Spectral Geometry II: A Time-Free Framework for Arithmetic Spectral Theory

Voici la description corrigée à coller dans Zenodo pour ESG II : v2 (March 10, 2026): Corrections to Pillar III (σ_KMS scaling is log(q)/q, not φ(q)) and entropy monotonicity (|ζ(σ+iγ)|² minimized at σ=1/2 only at zeros, not universally). Five axioms, time-free reformulation, and two solid pillars unchanged. We construct Emergent Spectral Geometry (ESG) as a time-free framework for arithmetic spectral theory. A systematic audit reveals that every failed attempt to connect the spectral gap of the Bost-Connes system to the zeros of the Riemann zeta function passed through time-dependent constructions (KMS flows, Fourier duality, Lieb-Robinson bounds), creating a hidden circularity in a programme where time is supposed to emerge rather than be assumed. We identify a clean separation into Layer 1 (time-free: entropy functional V, modular operator Δ, conjugation J, spectral trace) and Layer 2 (time-contaminated: KMS analyticity, Fourier transform, gradient flows). We strip the framework to Layer 1 and reformulate the Riemann Hypothesis without invoking time at any stage: Tr(Δ-s/2) = 0 implies Re(s) = 1/2 where Δ is the modular operator of the Bost-Connes system at the unique entropy-minimizing state ω₁/₂. The framework rests on five time-free axioms and two established pillars: (1) ω₁/₂ is the unique minimum of the KL divergence V with identity Hessian (proved); (2) the spectral gap is log 2, an arithmetic fact requiring no protection (the integers are discrete and 2 is the smallest prime). A third pillar — a time-free mechanism confining zeros to the critical line — remains an open problem. Two candidates were tested and found insufficient: σ_KMS scaling (corrected from φ(q) to log(q)/q due to normalization error) and entropy monotonicity of |ζ(σ+iγ)|² (minimized at σ=1/2 only at zeros, not universally). We introduce the entropy route to the zeros: the deformed state ω_s = n⁻ˢ/ζ(s) becomes ill-defined exactly at the zeros of ζ(s), where the entropy cost V(s) diverges. Numerical computation confirms that |ζ(σ+iγ)|² achieves its minimum at σ=1/2 precisely at the zeros — a geometric restatement of RH, not an independent proof strategy. This paper also corrects an error in ESG I regarding the NC2 bridge: the constant log ζ(2) is a density-of-states correction, not an operator shift. The true spectral gap is log 2 ≈ 0.693 (not log ζ(2) ≈ 0.498), and it is inherent in the spectrum {log n} rather than created by a constant shift. The framework provides a time-free home for RH with self-correcting diagnostic capacity. The proof requires a new idea — the third pillar — which has not yet been found. Companion code available at https://github.com/davidangularme/esg