IECZ-III: Hardcore Condensation Lift with Size-Aware Invariants

Abstract

This paper develops a compact, size-aware blueprint for transferring structure through gadget lifts. Two low-order invariants -- cumulative mod-$q$ Fourier mass up to degree $k$ and noise stability ${\mathrm{Stab}}_{\rho }$ -- are treated as a reusable "profile" tied to the gadget's affine interface. Under coordinate permutations ($\Delta =1$) the profile is preserved exactly; under bounded fan-in the degree budget relaxes by at most $+\Delta k$ (i.e., $k\mapsto k+\Delta k$), with all overheads tracked explicitly. In a balanced window $m=(1+\gamma )n$ the framework yields a distributional lower bound for a monotone cost (Erasure Complexity, EC) and an "echo" to correlation against size-aware ${\mathrm{AC}}^0+\log$ and to logarithmic degree in the polynomial-calculus setting. The accounting keeps total-variation non-expansion and a single $O(\log N)$ prefix-free header visible end to end, avoiding hidden slack.

Third paper in the IECZ series (IECZ-III). Standalone; minimal reproducibility scripts included (make repro_all). Single-directory LaTeX sources for arXiv