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 (σKMS 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: σKMS 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 ωₛ = 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