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

Marko Lela

TL;DR

IECZ-III develops a compact, size-aware blueprint for passing structure through gadget lifts by isolating two low-order invariants—the cumulative mod-$q$ Fourier mass up to degree $k$ and the noise stability $\mathrm{Stab}_\rho$—into a reusable profile tied to the gadget’s affine interface. The framework preserves the profile exactly under coordinate permutations ($\Delta=1$) and permits degree expansion by at most a factor $\Delta$ under bounded fan-in, with all overheads explicitly tracked in a fixed $O(\log N)$ header within a balanced window $m=(1+\gamma)n$. It proves a distributional erasure-complexity (EC) lower bound in the window (via a switching-path argument and a row-distance hypothesis) and derives an “echo”: correlations against size-aware $\mathrm{AC}^0{+}\log$ and a logarithmic-degree lower bound in polynomial calculus, all connected through a reproducible, prefix-free accounting scheme. The work also offers a transferable template for hardness transfers that preserves measurable structure, supported by empirical diagnostics such as pattern-context gaps (PCG) and low-order mass/stability proxies, demonstrating alignment between analytic invariants and compression-based signals. Overall, IECZ-III provides a concrete, reusable interface for size-budgeted gadget lifts, enabling rigorous, quantifiable hardness transfers while keeping all overheads transparent and verifiable.

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 $\mathrm{Stab}_ρ$ -- are treated as a reusable "profile" tied to the gadget's affine interface. Under coordinate permutations ($Δ=1$) the profile is preserved exactly; under bounded fan-in the degree budget relaxes by at most $+Δk$ (i.e., $k \mapsto k + Δk$), with all overheads tracked explicitly. In a balanced window $m=(1+γ)n$ 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.

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

TL;DR

IECZ-III develops a compact, size-aware blueprint for passing structure through gadget lifts by isolating two low-order invariants—the cumulative mod- Fourier mass up to degree and the noise stability —into a reusable profile tied to the gadget’s affine interface. The framework preserves the profile exactly under coordinate permutations () and permits degree expansion by at most a factor under bounded fan-in, with all overheads explicitly tracked in a fixed header within a balanced window . It proves a distributional erasure-complexity (EC) lower bound in the window (via a switching-path argument and a row-distance hypothesis) and derives an “echo”: correlations against size-aware and a logarithmic-degree lower bound in polynomial calculus, all connected through a reproducible, prefix-free accounting scheme. The work also offers a transferable template for hardness transfers that preserves measurable structure, supported by empirical diagnostics such as pattern-context gaps (PCG) and low-order mass/stability proxies, demonstrating alignment between analytic invariants and compression-based signals. Overall, IECZ-III provides a concrete, reusable interface for size-budgeted gadget lifts, enabling rigorous, quantifiable hardness transfers while keeping all overheads transparent and verifiable.

Abstract

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

Paper Structure

This paper contains 144 sections, 28 theorems, 60 equations, 2 figures, 2 tables.

Table of Contents

  1. Scope and guiding principle.
  2. Introduction
  3. Problem framing: balanced window without worst-case claims
  4. Contributions
  5. Conventions and notation
  6. Motivation: structure profiles and contextual compression
  7. Roadmap
  8. Limitations and scope
  9. Invariants and the size-aware API
  10. Windowed API (balanced 3XOR → 3SAT)
  11. Constants and loss table (size/meta exposed)
  12. Controlled overheads and amortization
  13. How the API is used downstream
  14. Auxiliary notes and cross-references
  15. Normalization conventions
  16. ...and 129 more sections

Key Result

Theorem 2.1

There is a window-preserving map $R:\mathsf{XOR}\to\mathsf{SAT}$ with a prefix-free header of length $H=O(\log N)$ such that for $\nu=R_\#\mu$ the following hold. PROF (multimode profile). Deterministic push-forwards are $\operatorname{TV}$-1-Lipschitz, so empirical fluctuations are preserved up to $o(1)$ on the window. Under output-mixing normalization ($\Delta=1$) the affine relabeling is a coo

Figures (2)

  • Figure 1: EC proxy (average backtrack pops) vs. $n$ on balanced 3XOR$\to$3SAT. Parameters as in §\ref{['subsec:D5-ec']}.
  • Figure 2: $k$-gram predictor on restriction-path parity histories: average log-loss (bits) vs. $k$ for $n\in\{64,96,128\}$ ($\varepsilon=0.05$, $p_{\mathrm{hist}}=0.06$, $d\ge 3$, $L=32\cdot n$). Error bars: s.e.m.; sharp drop and plateau match the echo narrative.

Theorems & Definitions (64)

  • Theorem 2.1: A — API, windowed form
  • Theorem 2.2: A$'$ — Block product stability
  • Theorem 2.3: A$"$ — Gadget stability under small branching
  • Remark 2.4: Normalization for EC
  • Remark 3.1: Scope of \ref{['assm:row-dist']}
  • Lemma 3.2: Surviving parity
  • proof
  • Lemma 3.3: Width reduction
  • proof
  • Remark 3.4: Switching calibration and explicit $c_\star$
  • ...and 54 more