Exact algebraic degrees and finite digit rules for normalized Stieltjes fractions

AI-assisted research note. Global novelty and priority have not been established, and the manuscript has not been independently reviewed or proof-assistant verified. We study the normalized Stieltjes series A_w(x) = F_w(x/A_w(x)^2). Formal stabilization gives existence and uniqueness over commutative rings, and transfer matrices give a polynomial equation for periodic weights. Over a field K, when the first zero weight occurs at index r and all preceding weights are nonzero, the note proves that [K(x,A_w):K(x)] = r and that the resulting monic equation is irreducible. The proof uses continuants and the standard rational-map degree lemma. The degree statement gives exact degrees (p+1)/2 for odd weights 2j-1 modulo odd primes p, and p for the positive-power weights j^m for every prime p and every integer m >= 1. The square-weight identification with OEIS A158119 uses its recorded continued-fraction characterization; its alternative bisection definition is not independently rederived here. Classical Furstenberg diagonal and Cartier-operator methods, credited to the existing literature, produce finite digit rules reading base-p digits least significant first. A complete 13-state rule for OEIS A338632 modulo 5 is included. The manuscript does not claim a new general automatic-sequence algorithm. The PDF contains full arguments. The source archive includes the LaTeX manuscript, a standard-library verification program, polynomial state certificates, and exact check results: 30 family/prime cases and 40 first-zero blocks through degree 48; 16 composite-modulus periodic cases; three complete machines compared with integer coefficients through degree 256; and an independent modulo-5 recurrence comparison through degree 4096. All checks pass. These finite checks supplement the general proofs rather than establish infinite results by extrapolation. ChatGPT generated the mathematical arguments, English manuscript, and verification program at Peggy Pei-Yu Chao's request. She initiated the investigation and authorized this public dissemination. The work is an application and synthesis of established tools. Novelty of this exact degree formulation and application remains unresolved; this is an unreviewed preprint, not an accepted journal article or a claimed solution of a major open conjecture.