Binary-Valuation Identities and Lacunary Continued Fractions in Integer Sequences

We assemble exact identities linking binary valuations, digit blocks, and folded continued fractions. In addition to a modulo-four divisor-sum parity identity associated with OEIS A073089 and a generating-function proof of the A110037 difference relation, we give a disjoint classification and exact summatory count for integers with consecutive odd parts congruent to 3 modulo 4 (A106837). For the array A360967 we derive its full binary block structure, popcount, run count, and specified distances in the graph defined by fibbinary products. We also prove the exact index formula for every position in A081769 and establish a general weighted folding identity specialized to A123725. Known paperfolding results and an additive greedy sequence are credited to their recorded sources rather than counted as new original results. The mathematical statements are accompanied by their explicit index ranges and proofs. Licensing note: The manuscript is licensed under CC BY 4.0. The OEIS b-file snapshots included in the supplementary archive retain their original CC BY-SA 4.0 license and attribution. The independently authored verification code is made available under CC BY 4.0.