How fine can a hidden ripple be? A two-sided logarithmic law for the resolution gap of the arrow of time

The resolution gap g measures how far an observer who sees a pendulum's phase space through a Gaussian blur of width σ is from resolving the arrow of time near the hyperbolic top. A companion paper (paper E) showed that no σ-uniform gap exists, found the strip family whose smallest gap follows a logarithmic law, and conjectured that g ≥ c/√ln(1/σ) for all states. Here we prove the logarithmic law with the sharp constant on a larger family, the periodic windows: a Gaussian state whose density carries, inside a narrow window, any even periodic profile in the momentum-like direction, from a single cosine (the strip) to a comb of thin teeth. Under explicit assumptions collected in one place, we prove (1 − ε̄)(1 − 1.7 ln ln/ln) ≤ g_min √ln(1/σ) ≤ (1 + |ε_C|)(1 + 6.0 ln ln/ln) for every 10^−100000 ≤ σ ≤ 10^−24, where ε̄ ≤ 0.1 bounds paper E's remainder ε on the class and ε_C is the remainder of one explicit comb (numerically 4 × 10^−16 at σ = 10^−24, the smallest σ the exact evaluator reaches). Hence g_min √ln(1/σ) → 1: proved from below (up to 1 − ε̄), derived from above. The lower side combines an exact window inequality with an upward bootstrap and is certified on 421 cells covering the range of ln(1/σ) and all ε̄ ≤ 0.1 at once; the upper side follows an explicit comb sequence and is certified with exact exponents and interval arithmetic. At finite σ the comb is the strip seen through a lens 256 times sharper (a leading-order identity), and numerically it falls below the best Gaussian mixtures already at σ ≈ 1.4 × 10^−3. Towards general states we prove a decorrelation dichotomy and a three-case split; the remaining case is reduced to a single conjectured inequality. Every claim carries its status: proof, computer-assisted proof, derivation, numerical, or conjecture.Known issue (v1.0, found by the author on 2026-10-29): the window lemmas treat the Gaussian average over v as an exact period average. For states whose background v-width is only a few σ, a correction is needed. The narrow case is now closed by a new computer-assisted lemma; the remaining corrections are below 5 × 10^−9 relative, and the statements of the theorems are expected to be unchanged. A corrected version (v1.1) is in preparation; see https://github.com/neginyan/resolution-gap-two-sided. Code, certificates and verification: https://github.com/neginyan/resolution-gap-two-sided (doi:10.5281/zenodo.23746555).