Words / CONCEPT

Quantified finite-prefix convolution correction and higher-arity evil-odious monotonicity

Ordered k-tuple representation counts over the evil and odious supports evolve in arity by discrete convolution against binary-valued indicators satisfying adjacent-pair complementation a(2q) + a(2q + 1) = 1. A general signed forward-difference identity splits the convolution increment above any finite cutoff R into a nonnegative boundary term, a finite prefix whose negative variation is bounded by V_R(f), and a tail bounded below by floor((n - R + 1) / 2), forcing strict increase for all n >= R + 2 V_R(f) + 1 and, under weak-composition binomial bounds on f, for all n >= R + 2 choose(R + k - 1, k) + 1. Propagating from the sixfold thresholds r_6(n) < r_6(n + 1) for n >= 37 and s_6(n) < s_6(n + 1) for n >= 5 yields explicit quantitative strict monotonicity across every arity k >= 6, together with sharp odious strict monotonicity s_k(n) < s_k(n + 1) for all k >= 6 and n >= k - 1 via the shifted odious bijection s_k(n) = q_k(n - k).