We study Hamming distances between consecutive canonical expansions in quadratic Ostrowski-type numeration systems, with the Zeckendorf and metallic-mean cases as the two concrete families developed here in full detail. The paper is organized around two concrete theories and an abstract transfer framework. In Part~I we treat the Zeckendorf case, (n) = (_ (n), _ (n+1) ), \and develop a complete concrete account of: local successor rewriting, an affine identity with the Zeckendorf sum-of-digits function, exact prefix sums, Fibonacci block self-similarity, exact finite-interval recurrences, superlevel counts, limiting laws, and a symbolic-dynamical interpretation via the Parry measure of the golden mean shift. In Part~II we extend the same program to the metallic-mean family attached to the quadratic irrationals\ₐ=a+a²+42, a 1. \For the canonical (a, 1) -recurrence numeration we prove a complete local successor rewriting rule, establish the exact affine formula\ a\, Cₐ (n) =a+1+sₐ (n) -sₐ (n+1), \obtain the global prefix-sum identity\₍=₀^N-1Cₐ (n) = (a+1) N-sₐ (N) a, \show that the mean Hamming cost tends to 1+1/a, and derive exact block-copy theorems on canonical metallic intervals. We further prove an exact recurrence and a closed formula for the metallic superlevel counts, identify the maximal-jump multiplicities, and deduce an explicit limiting distribution with rational probability generating function. The two concrete parts therefore furnish a complete Zeckendorf theory together with its first nonbinary quadratic extension. The final part extracts the common mechanism behind these two families into a quadratic-Ostrowski transfer framework. Local terminal rewriting, affine digit-statistic control, and canonical-interval decomposition appear there as structural inputs already verified in the concrete arguments. At the abstract level we prove a bridge theorem showing that, once the tail generating functions themselves are carried by a finite transfer package whose tail part is spectrally subcritical relative to block growth, rationality of the limiting law, exponential tails, exponential integrability, and moment convergence follow simultaneously from the same finite-state input. We also isolate a more native transfer criterion phrased directly in terms of successor-normalization templates and stationary canonical-block substitutions, so that the framework reaches beyond the Zeckendorf and metallic families to a broader class of purely periodic quadratic Ostrowski systems satisfying these finite-state hypotheses. Under a further transfer-primitivity assumption, one moreover obtains Perron--Frobenius type spectral control and exponential convergence for each fixed superlevel statistic.
Jianming Wang (Fri,) studied this question.