Congruence Structures and Diagonal Inversion in Formal Generating Functions

We study formal generating functions determined by compositional iteration, inversion, integral exponential substitution and coefficientwise reciprocal-diagonal conditions. Several common algebraic frameworks yield all-index congruences for a group of thirty-three OEIS sequences. The tools include compositional congruences over finite rings, Frobenius reductions, integral reversion recurrences and labelled prime-cycle orbit formulas. For the integral exponential family we obtain residue formulas modulo three, four, five and six; in particular the full five-class formula establishes the residue exclusion recorded for A396804. For the two reciprocal-diagonal sequences A393170 and A393173, we prove the asymptotic constants (e-1)²/e³ and (1-e⁻²)², respectively, using exact half-truncation identities, uniform normalization bounds and an explicit three-term convolution inverse. The constants and source equations are credited to their OEIS contributors. Further exact structural identities include an infinite dyadic lower-bound family and power-of-two digit-support classifications. The results are illustrated by shared small, medium and arbitrarily large parameter constructions, distinguishing all-index proofs from finite source-data checks. No claim of historical priority is made for previously recorded patterns or techniques. 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.