Extremal stopping-time coefficients for the Collatz map

Let T be the shortcut Collatz map, B the set of positive integers whose orbit reaches 1, τ(n) the number of steps needed, and μ = 2/log(4/3) = 6.95211…, the typical value of τ(n)/log n on B. We prove that for every c < 7.626913 there is e > 0 such that every sufficiently large interval [X, 2X) contains at least Xe integers n ∈ B with τ(n) ≥ c log n; near 7.626913 one may take e close to 0.8895. Previously Applegate and Lagarias proved the coefficient 6.14316, and Amano (in a technical report) the coefficient μ, each for infinitely many n and each by a large computer search of inverse trees. Our proof has three parts. A transfer theorem shows that if, for Haar-almost every 3-adic unit, some inverse path eventually has at least a fraction θ of odd steps, then every coefficient below c(θ) = (1 + μθ log 3)/log 2 occurs in every large octave. A zero-one law shows that the best such θ is an almost-sure constant Θ∞, with Θ∞ ≤ θ* = 0.6090897…, the root of H(θ) = θ log 3. Finally, a drift certificate for an adaptive window policy on the 28,697,814 unit residues modulo 316, verified in exact integer arithmetic, gives Θ∞ ≥ 0.56124. Without any window certificate, absolute continuity of a tilted 3-adic Syracuse law, imported from a companion paper where it rests on one exact scalar inequality, gives Θ∞ ≥ 0.550024 and every c < 7.503325. The arithmetic inputs are the Gaussian law for τ on B and the density-one coalescence of consecutive integers. A kernel-checked formalization in the author's Lean development, not yet public, proves that some coefficient strictly above μ occurs infinitely often. The method cannot exceed c(θ*) = 8.1541626…; stochastic models predict 1/(log 2 − θ* log 3) = 41.677647…. The supplement (extremal-supplement.zip, with the W = 16 certificate in three parts certificate-W16.npz.part-aa/ab/ac; see certificate-W16-REASSEMBLE.txt) contains the window certificates for W = 4, 6, 12, 14 and 16, the exact verifiers with two negative controls, the scalar certificates, their outputs and a SHA-256 manifest. Companion papers: Brownian motion in Collatz orbits; Tilted 3-adic Syracuse laws; The stochastic 3x+1 model is a theorem: Brownian structure of Collatz orbits.