Effective Local-to-Global Obstructions for Fixed-Excess Induced Saturation in Square-Grid Quotients

Preprint, version 1. We study induced saturation by even cycles in finite Cayley graphs obtained as quotients of the square grid. For an induced path whose addition closes a cycle, its complement defines a two-defect marking of the grid; marked–marked edges form an energy whose budget is determined by the cycle excess. We prove an effective local-to-global theorem for this defect model. If a radius-2‖δ‖₁ + 12K + 8 window admits a two-defect marking of energy at most K, then the infinite grid admits one with the same budget. Combining effective shell splicing with diagonal-cut confinement and a mod-six charge law yields an explicit piecewise upper bound on the ℓ₁-systole of every simple square-grid quotient that is addition-saturated for an even cycle at fixed positive even excess. Under a bounded successive-minima ratio, this gives an explicit order bound and hence finiteness. We also establish well-rounded and Gaussian specializations, show why a systolic bound alone cannot control quotient order, and record a certified primitive-Gaussian classification through excess 16 as an application. This record contains the preprint manuscript and its Supporting Information. The computer-assisted classification is logically separated from the symbolic general theory.