The Arithmetic and Cube-Path Geometry of A392243

This paper gives a unified study of OEIS A392243, a(n)=∑i=1ni(−1)⌈i⌉a(n)=\sum_{i=1}^{n} i(-1)^{\lceil\sqrt{i}\rceil}, organized around the exact identity 2(−1)ka(n)=k2(k+1)2−n(n+1)2(-1)^k a(n)=k^2(k+1)^2-n(n+1), where k=⌊n⌋k=\lfloor\sqrt n\rfloor. The sequence’s square-block structure leads to a common pronic coordinate governing several apparently different phenomena: a critical modulo-threshold recurrence whose canonical orbit reproduces ∣a(n)∣|a(n)| after a one-step clock shift, square- and pronic-centered reflection laws for polynomially weighted variants, exact fourth-power block sums, forced sixth powers, and an infinite family of nonforced Fibonacci sixth powers arising from the norm equation f2−fh−h2=±1f^2-fh-h^2=\pm1. The paper then broadens from A392243 to seam-perfect alternating-block systems whose block peaks are cubes. In this wider setting it develops a negative-Pell foundation, an exact cube-edge transition equation, a global characterization of the canonical triangular path, a formal Fibonacci subcanonical boundary, the three-regime canonical wall, cumulative offset and Nicomachus defect identities, fixed-label Diophantine finiteness, and quantitative restrictions on noncanonical paths. Any hypothetical counterexample to canonical cube-path uniqueness must produce at least one of two surviving mechanisms: a finite noncanonical return to a later triangular seam (Branch R) or an infinite all-off-diagonal tail (Branch N). Branch N is forced into unbounded label and normalized-depth escape, while infinite families of Branch R returns must escape in width, terminal boundary ratio, and label; additional pointwise inequalities constrain any individual return. Neither Branch R nor Branch N is eliminated, so uniqueness of the anchored infinite cube path remains open. Likewise, the conjecture that the forced and Fibonacci constructions exhaust all sixth-power occurrences for bases m≥2m\ge2 remains open. Finite computations are reported only as scoped evidence and enter no universal proof, while results depending on Siegel’s theorem on integral points are identified explicitly. This flagship paper consolidates, corrects where necessary, and extends the author’s earlier investigations of A392243, and is intended as the preferred unified citation for the results reproduced here. Earlier deposits remain available for provenance, computational certificates, and specialist detail.