Three Laws for Stepwise Qubit Reset: Exact Dissipation, an Iterated Logarithm at Fixed Step Number, and the log* Step Count

A two-level system is reset to error eps by N quenches of its energy gap, each followed by full equilibration with a bath at temperature T. For every dissipation budget, the number of steps required is log*(1/eps) up to an additive constant, and log*(1/eps)+1 steps keep the dissipation below 0.7 kT at every eps. At fixed N the minimal dissipation diverges as 1/2 times the N-fold iterated logarithm of 1/eps, with an explicit constant. For all N and eps it is at least L^2/(2N), where the thermodynamic length L = arcsin(1 - 2 eps) stays below pi/2, and this is sharp as N grows at fixed eps, so the limits do not commute. At eps = 1e-6 the minimal dissipation is 6.2 kT for one step, 1.3 kT for two and 0.68 kT for three. The three laws of thermodynamics for this protocol are written in the visibility V = 1 - 2p, with gamma(V) = (1 - V^2)^(-1/2) and rapidity eta = artanh V. The entropy produced in a step is exactly ln(gamma_2/gamma_1) - V_1 (eta_2 - eta_1). It reduces to (1/2) gamma^2(V) (Delta V)^2 for a small step and to ln gamma(V) for a one-step reset, so no floor of the form (1/2) I(V) per event holds; that floor, stated in earlier preprints of the author, is withdrawn here. The exact step formula and the bound L^2/(2N) are machine-checked in Lean 4 (github.com/bharathsrivats/fisher-lorentz-formalization, tag v2.1.0). Related prior work is credited in the paper, and no claim of priority is made.