Encoding-Invariant Mode Geometry in Constraint Satisfaction Problems

This work presents an experimental study of encoding-invariant geometric structure in Boolean satisfiability (SAT) problems near the phase transition. Rather than focusing on mean runtime or satisfiability thresholds, the analysis examines the internal mode structure of solver-independent bias spaces constructed from partial resolutions of SAT instances. Using singular value decomposition (SVD) of bias vectors, we identify stable normalized mode ratios—including the leading mode ratio σ₂/σ₁ and a diagonal projection coefficient δ—that remain remarkably consistent across radically different logical encodings, including Tseitin expansion, XOR-chain encodings, and variable permutations. Across all tested encodings, the dominant ratios vary by less than 5%, indicating that they are not artifacts of solver heuristics or surface clause geometry. These normalized ratios correlate strongly with computational hardness (decision count and solve time) at the ensemble level, while individual instance correlations remain weak. This pattern suggests that the observed structure reflects intrinsic relational geometry of constraint satisfaction rather than instance-specific features. Additional experiments demonstrate that solver memory (e.g., CDCL heuristics) moderates variance but does not create the underlying geometric signatures. The results establish a falsifiable, representation-independent framework for probing computational hardness using geometric diagnostics. While no claims are made regarding complexity class separation or universality beyond the tested ensembles, the findings suggest that normalized mode geometry captures structural information that is invisible to mean-based analyses and may inform future algorithmic design and theoretical investigation. For questions, collaboration, or further discussion, readers are welcome to contact the author directly. The author is happy to provide assistance or insight freely.