Locality Causes Tractability? An Intervention Study on the Variable-Interaction Radius in Geometric Random Satisfiability

A popular intuition holds that constraint instances are easy because their “causes act locally”: variables interacting in small neighborhoods should let solvers scale. But observation cannot separate locality from the other regularities instances carry. We turn it into a falsifiable, interventional claim and test it on conflict-driven clause-learning (CDCL) Boolean satisfiability (SAT) solvers: with variable count fixed, density matched to each radius's satisfiability threshold, and clause-content statistics invariant by construction, we vary the radius from which clause variables are drawn. A locality-kernel generator sweeps nine radii on a 2-D torus with preregistered threshold-matched offsets, 30 paired seeds and two solvers; a degree-preserving swap destroys spatial locality while holding every literal occurrence count fixed. Three findings. First, the operational satisfiability boundary is not radius-invariant: below density 2.5 at radius 0.06, near the classical ${\approx}4.3$ region by radius 0.22. Second, at matched density, difficulty spans 3.4 orders of magnitude across radius among decided instances ($F(8,216)=2808.5$; partial $\eta^2\approx0.99$). Third, the swap raises difficulty by $16$–$68\times$ ($p\le7\times10^{-10}$) and flips 53 of the 82 resolved pairs, all toward satisfiability; within the nine status-stable sat–sat pairs the elevation is $\times10.7$. Exploratory mediation over four collinear mediators yields a ${\sim}94\%$ community-restructuring share (a consistency check, not a mechanism fraction). Under these controls, a degree-preserving structure-destroying intervention raises difficulty at matched density --- consistent with spatial locality contributing to the causal channel. Claims are scoped to $n=400$, direct conjunctive-normal-form (CNF) encodings, and CDCL-family solvers.

Deposit contents. The package consists of three files: a README describing the package, the changes relative to the previously deposited version (v7), and the licenses; the paper PDF (Springer-formatted version: 25 pages, unstructured abstract, Statements and Declarations back matter, and appendices A–D); and a single tarball containing the complete research repository — the LaTeX sources of the journal, compact, and Springer-formatted versions of the paper, the locality-kernel instance generator, the solver and analysis pipeline, all raw result databases and summary JSON files, the preregistration chain with its numbered amendments and the prespecification-vs-implementation audit, the experiment log, the full report in English and Chinese, literature notes, the AI-use disclosure, the test suite, and a bundled third-party probSAT solver. The snapshot is byte-identical to the repository state tagged v8.0.0; the complete commit history is available at https://github.com/essnt/correlation-vs-hardness.