Geometric Rigidity, Solver Regimes, and Scale-Dependent Hardness: A Technical Note and Experimental Handoff

This work develops and validates a geometric framework for analyzing algorithm–instance interaction in Boolean satisfiability problems through the lens of solution-space rigidity and observer-dependent resolution. We introduce and empirically study the rigidity metric r^*, which measures the minimum local radius required to connect the solution space under a chosen metric, and distinguish intrinsic geometric structure from representational inflation via symmetry quotienting. Through a sequence of controlled experiments—spanning random 3-SAT, structured CSPs, and bipartite matching as a symmetry-dominated control—we establish that: Apparent hardness can arise entirely from representational differentiation (pure symmetry mirages), and Random 3-SAT exhibits probe-invariant rigidity consistent with intrinsic geometric obstruction. We further examine whether rigidity is paradigm-dependent by comparing solver effort across distinct algorithmic classes (CDCL, local search, sampling). Moderate but statistically significant cross-paradigm correlations demonstrate that rigidity captures a shared substrate-level component of difficulty, while leaving substantial paradigm-specific variance. A differential extension analyzing solver response dT/dr^* reveals that, at small instance sizes, geometric rigidity correlates negatively with solver effort due to constraint propagation dominance—indicating that geometric obstructions only manifest computationally once resolution is sufficiently stressed. This motivates explicit scale-dependent falsifiable predictions regarding sign reversal and response amplification at larger n. Finally, we introduce algorithmic resolution as a measurable quantity capturing observer blind zones and show that apparent clustering and fragmentation can be projection-induced rather than intrinsic. Collectively, these results clarify the distinction between structural and representational hardness, establish rigidity as a real geometric feature of solution spaces, and provide a reproducible experimental path for further investigation by researchers with greater computational resources. No complexity-class separations are claimed. The contribution is diagnostic: isolating, measuring, and decomposing hardness into geometric and observer-dependent components.