Resolution Cost of Structured Inquiry States: Mechanism Dependence in the Energy-Bridge Test

Q-Bridge is a validation suite for testing whether structured inquiry states can be formally represented, constrained, measured, and evaluated as unresolved state-space configurations. The suite tests a sequence in which a state of affairs becomes an object of inquiry, activates primitive interrogative dimensions, forms a structured question-state, becomes measurable as a configuration, expands when dimensions remain unresolved, and requires resolution to reduce that unresolved state-space. The energy-bridge question enters at the final step: whether resolution cost scales with unresolved interrogative structure rather than with surface-level confounds. This paper reports the completed energy-bridge results across two generations of the experiment. The first generation, spanning standard LLM generation, answer-length-controlled generation, reasoning-mode generation, and exhaustive enumeration over the declared candidate space, established mechanism dependence: generative LLM resolution showed no positive dependence of cost on unresolved interrogative structure, reasoning-mode generation showed a significant negative dependence later sharpened by reconciliation tests, and exhaustive enumeration showed constructive scaling with an arbitrary-kernel marginal constant of approximately 2.2 mJ per candidate. The second generation removed that generation's central caveat. A preregistered campaign with three analytic runs and one exact repetition replaced the arbitrary SHA-256 kernel with SAT, under which the unresolved interrogative bits are the free variables and the solver's work is the resolution itself. On an uncontaminated CPU channel, energy tracked genuine constraint resolution at r = 0.99 for both exhaustive checking and conflict-driven pruned search, cost peaked at the satisfiability phase transition, and the marginal constant fell to 0.032 to 0.043 mJ per satisfaction check, superseding the arbitrary-kernel value. The campaign's negative control caught a GPU contamination that had inflated combined-channel energy roughly threefold, a replicated power anomaly was resolved by a targeted test to a batching-composition artifact, and an exact repetition returned every headline value inside its registered band. The central claim is bounded and mechanism-indexed: unresolved question-state structure predicts measured physical resolution cost when the resolver instantiates that structure as constraint work, and the cost tracks resolution difficulty, not merely candidate count. The result updates the June ten-gate map by advancing Gate 1 for both exhaustive and pruned search, populating Gate 3b with a hardness-indexed constant on a clean channel, closing Gate 8 at the apparatus layer, and narrowing Gate 9 by separating resolution cost from generation cost.