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Corollary 38 ([Domination by the minimizer]). For the minimizing ray w and every v∈ Q , v(f)b(v)≤2nw(f)b(w) (f∈R). For rational heights the same statement holds on the central slice with all symbols carrying a superscript 0 .

Corollary 37 ([Near-minimizers for a height]). Fix an allowed height b and normalize its common rays by b=1 . If wi,vi have functional values within relative o(1) of the minimum, then wi=(1+o(1))vi on all regular functions, with an …

Theorem 5.1 ([Exact common-slice minimizer]). For every allowed height b , the homogeneous functional b(v)(v)1/n on centred common lc rays has a unique minimizing ray. Its A=1 representative depends continuously on the coefficients of b …

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Normalize both rays by b=1 and interpolate them with weights 1/2 . The resulting centred u has b(u)≤1 , so minimality gives (u)≥(w) . For w(f)>0 , gives 1≤2^n(w(f))/(w(f)+v(f)). Thus v(f)≤(2^n-1)w(f) , which implies the stated bound. If f is a unit both values are zero. The central assertion follows from the same argument and Theorem 5.1 ([Exact common-slice minimizer]).

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