Logarithmic Shell Geometry, Discrete Scale Invariance, and the Convexity of the Riemann Chi Function

We develop a unified analytic and spectral framework arising from the logarithmic embedding θ(n) = log n (mod 2π), which induces an exact discrete scale invariance on the positive integers. This structure naturally partitions the integers into exponentially spaced logarithmic shells Sₖ = {n ∈ ℕ : e2πk ≤ n < e2π(k+1)}. We establish precise asymptotics for the weighted shell sums K(p,k) = Σn∈Sₖ n⁻ᵖ, proving K(p,k) = C_p · e−(p−1)2πk + O(e−2πpk), with explicit renormalisation coefficient C_p = (1 − e−(p−1)2π) / (p−1). The shell decomposition is lifted to the Hilbert space ℓ²(ℕ) through spectral projection operators, yielding a renormalisation hierarchy governed by the eigenvalue Λ_p = e−(p−1)2π. In the second part we investigate the Riemann chi function χ(s) = ζ(s)ζ(1−s) and its canonical factorisation χ(s) = (2π)^s π⁻¹ sin(πs/2) Γ(1−s). We prove that χ(s) is strictly convex on (0,1), from which every equation χ(s) = cs (c > 0) possesses a unique solution r* ∈ (0,1). The framework unites discrete scale invariance, shell geometry, operator theory, and analytic properties of zeta-type functions into a single coherent spectral structure, independent of any physical interpretation.