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This paper explains where the large environmental dispersion measured by a prior trilogy comes from by modelling hierarchy as a multiplicative amplifier of position: a cascade that turns tiny, often capability-irrelevant initial differences into large positional dispersion, so the dominance ratio R becomes an output of structure rather than a fact about people. It microfounds the feedback f(S) in Gould's deference model (linear at the bottom, saturating at the top), identifies the Kesten reset that creates the heavy tail with Boehm's reverse-dominance insurance, and pins the tail index in closed form, confirmed by a Clauset MLE. The calibrated amplifier reproduces R at all four aggregation levels and requires a near-random-walk between-environment process, with depth anchored independently from Maddison GDP divergence and income mobility so R is predicted rather than fitted. On the raw ICIJ Offshore Leaks officer network the predicted heavy tail holds (global alpha approximately 2.25), but no concentration metric tracks recording opacity across jurisdictions, a negative result that vindicates the demarcation paper since the quantity that would test the lever is hidden by construction. On 113,895 closed GitHub pull requests position multiplies the merge odds 3 to 7 times at fixed diff size; a within-author design bounds the position premium at 12.6 percentage points, and a promotion holding author and project fixed still finds acceptance rising 4.1 percentage points at unchanged proposal size, though even it cannot fully separate the badge from the competence that travels with it. The amplifier's gains are continuous and visible while its costs are rare, delayed and hidden, so hierarchy always appears beneficial and its net cost cannot be honestly settled.