Square and Pronic Reflection Laws for Weighted Square-Block Sums

For p≥0, define Sp(n)=∑i=1nip(−1)⌈i⌉. This note places the OEIS sequences A392243 (p=1) and A392677 (p=2) inside a common all-weight framework. For every weight p, a unique polynomial Qp, called the pronic potential, converts the signed square-shell partial sum into a difference between Qp(k(k+1)) and the Faulhaber polynomial Fp(n). The construction is an application of classical Faulhaber–Euler–Salié polynomiality to the square-block sign pattern. The pronic potential yields two coupled reflection laws: reflection across a square produces a symmetric Faulhaber second difference, while reflection about the half-step between the consecutive pronic-centered indices K−1 and K produces a baseline-minus-increment formula. The resulting factorizations show that p=1 is uniquely exceptional within the family: its square-reflection defect is independent of the shell and its pronic baseline vanishes identically. Consequences include exact fourth-power and other block sums for A392243, weight-two reflection formulas for A392677, a classical Pell criterion for square defects, and the exact coincidence locus ∣S1(n)∣=∣S2(n)∣. The classical Faulhaber, Euler, Bernoulli, Salié, and Pell mechanisms are treated as prior; the contribution is the sequence-specific all-weight assembly and coupled square/pronic reflection structure.