Uniform Block Counting for Champernowne Concatenations

This paper gives a source-level proof audit and a self-contained reconstruction of the classical normality and discrepancy arguments for Champernowne concatenations. For an integer base b≥2, let Cb=0.(1)b(2)b(3)b⋯ denote the number obtained by concatenating the base-b representations of the positive integers. The normality of Cb in its generating base is known. The purpose of the present paper is not to establish this result for the first time, but to examine in detail how the relevant counting arguments work and to make their uniformity requirements explicit. The paper first revisits Champernowne’s original 1933 proof and finds that the essential arbitrary-prefix and insertion arguments are already present, although in a compressed form. It then compares the first arXiv version and the published version of the recent discrepancy proof by Becher and Graus. A specific issue is identified in the published treatment of occurrences crossing a concatenation boundary t∣(t+1). An intermediate implication there effectively transfers a prescribed prefix of t+1 to a prefix of t, which is not valid in the presence of carries. The simple example 19∣20,B=92 isolates this failure. The discrepancy theorem itself is not contradicted: the required boundary estimate remains correct. A carry-safe replacement is proved by combining a residue-class condition for the suffix of t with an interval condition for the prefix of t+1. This yields the required boundary estimate in every fixed base b≥2. The replacement is then embedded in a complete uniform block-counting argument, including arbitrary endpoints and explicit dependence on the block length k. The resulting estimate is occ(Cb,B,N)=b−kN+Ob(kbn−k)+Ob(kbk)+Ob(kn), where n=n(N) is the digit length of the integer containing the N-th digit of the concatenation. From this, the known upper discrepancy bound D(Cb,N)=Ob(logN1) is recovered, and hence normality in the generating base. The contribution is therefore twofold: a precise diagnosis and repair of one carry-sensitive step in a published proof, and a self-contained verification that the repaired estimate remains uniform throughout the complete upper-discrepancy argument. Artificial intelligence tools were used during the preparation and checking of the manuscript, including for proof auditing, comparison of formulations, and referee-style review. The mathematical claims and final responsibility for the manuscript remain with the author.