Recursive Cube–Quadrant Deduction: A Geometric Probability Theory of Search Narrowing

This paper develops Recursive Cube–Quadrant Deduction (RCQD) as a geometric–probabilistic theory of search. A problem is represented as a search cube: a bounded volumetric region of candidate configurations. At each stage the cube is partitioned into four congruent subregions ("quadrants"). A probability (or plausibility) measure ranks the four quadrants; the three lower-ranked quadrants are carved away and attention concentrates on the single most plausible quadrant. The same operation is applied recursively until the retained region is small enough that a terminal locator can identify the solution inside it. The contribution is not an empirical claim that every exponential search collapses to polynomial time, nor a proof that P = NP. It is a precise theoretical framework: axioms for the cube model, a formal visualization of recursive carving, principles governing when single-quadrant focus is licensed, complexity theorems conditional on those principles, and an explicit account of failure modes when the probability ranking is wrong. Under ideal ranking, depth is logarithmic in volume and work is linear in depth. Under imperfect ranking, backtracking restores completeness at a cost that re-enters the exponential regime in the worst case. The theory therefore clarifies both the power and the limits of recursive geometric narrowing.