Beyond few harmonics: the lower side of the logarithmic law of the resolution gap for windows with a weak ripple

Paper F proved the logarithmic law of the resolution gap of the arrow of time, with the sharp constant, for periodic windows with at most twenty harmonics or with damping exponent K ≥ 0.003. Here we extend the lower side to windows with any number of harmonics whose ripple is weak in the sup norm or poor in low-frequency energy. Three tools do the work: a remainder of the curvature step made relative to the ripple, a split sandwich in which a low-frequency block enters through its energy rather than its sup, and a mean-square bound of the curvature cross term that closes the edge of very wide focus. Cell-based computer-assisted proofs, with an explicit bound on floating-point rounding in every acceptance test, certify the full lower bound, remainders included. The main theorem gives g√ℓ ≥ (1 − ε̄)(λ/Υ)(1 + 12.1/ℓ + 2.5×10⁻⁴) for every state with τ ≥ 20 and ℓ = ln(1/σ) ∈ [55.26, 960.8] (σ from 10⁻²⁴ down to about 10⁻⁴¹⁷). In this range the coefficient one holds, up to the factor 1 − ε̄, for every window with η ≤ 1/ℓ, for a family of intermediate windows, and for seventeen thin-block windows certified one by one. A conjecture of our own (that ηe^{μ/2} stays bounded) is false; explicit counterexamples are given and covered by the theorem. The paper closes with the road to an unconditional theorem, from the nearest restriction to the farthest: states with a narrow focus (τ < 20), which reduce to one local inequality; the full bound beyond ℓ = 960.8 and the limit σ → 0; windows outside the covered families; the class constant ε̄; and the upper side. Every claim carries its status: proof, computer-assisted proof, derivation, numerical, or conjecture. Code, certificates and verification: https://github.com/neginyan/resolution-gap-weak-ripple (doi:10.5281/zenodo.23932577).