Stopped Channels and Offset Alignment: De-Conditionalizing the Real-Channel Hypotheses for Mixed-Base Tetration

The mixed-base phase law established in Paper I and its circle-diffeomorphism companion in Paper II are conditional on two hypotheses concerning the regular tetration branch: real-channel positivity and a peeling-growth condition. This paper removes most of that conditionality. First, the growth hypothesis is shown to hold unconditionally for every admissible tower base above the classical threshold e^(1/e), with an explicit positive lower bound for the growth gap. Second, positivity of the finite logarithmic chain is proved unconditionally down to an explicit stop level, called the Stoppregel. A self-contained downward induction produces the required channel rather than assuming it, and the evaluation and limit arguments from Papers I and II require only this stopped chain. Third, for the limiting objects, the correct channel condition is shown to be offset-aligned: positivity is required only beyond an explicitly computable shift determined by the mean-shift hub. The corresponding unshifted interpretation is false for sufficiently asymmetric base pairs, including the pairs (10, 2) and (e, 1.5), whereas the offset-aligned formulation holds with explicit margins. Numerical certification is reported across all 45 measured base pairs. Together with the base differentiability of the Kneser family established by the response-kernel program, these results reduce the remaining conditional core of the series to a single mode-certifiable inequality for each pair of bases.