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

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 ${\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{b}}_{\rho }$ – are treated as a reusable “profile” tied to the gadget’s affine interface. Under coordinate permutations ($\mathrm{\Delta }=1$) the profile is preserved exactly; under bounded fan-in the degree budget relaxes by at most $+\mathrm{\Delta }k$ (i.e., $k\mapsto k+\mathrm{\Delta }k$), 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 ${\mathrm{A}\mathrm{C}}^0+\log$ 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.