Integral Square Roots and Digital Congruences of Formal Generating Functions

Conjectures recorded in the On-Line Encyclopedia of Integer Sequences concern integral square roots of formal generating functions and their congruence patterns. We prove the integrality of the square root associated with an odd-square Stieltjes fraction (A193550), and show that its coefficients reduce to Catalan numbers modulo $4$. For self-referential continued fractions, we establish the complete modulo-$8$ and ternary-digit congruences conjectured for A338636. The same formal-reduction method proves the three conjectures for A338632, the binary parity conjectures for A338633 and A338634, and the Fibbinary support conjecture for A338635. For the square-root sequence A261594 we establish the conjectured ternary-digit characterization modulo $3$ and a binary-digit classification modulo $4$, yielding a complete classification modulo $12$. The latter conclusions extend to an integer-parameter binomial-transform family, for which divisibility of the parameter by $4$ is necessary and sufficient for the formal square root to have integral coefficients. A separate family of binomial-square series yields two previously recorded source/root pairs as further applications of the same classical root criterion. The resulting eleven conjectural claims admit all-index arguments, including the initial positive index in the A338635 classification. The formal-root congruence principle itself is due to Heninger, Rains and Sloane; our use concerns these explicit generating functions and their digital properties. 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.