Exact Resolution of Geometric Reasoning Problems via Discrete Search in SE(5)

Algebraic Completeness, Phase-Based Counting, and Isometric Identification Abstract We demonstrate that a well-defined class of geometric reasoning problems — those reducible to compositions of rigid isometries with semantic coupling in ℝ⁵ — admits exact solutions within a finite, enumerable lattice. The central result is an algebraic completeness theorem: for any problem in the affine isometric class PaffPaff, a solution exists in the discrete lattice SE(5)ΛSE(5)Λ and exhaustive search finds it exactly. Approximation is unnecessary by construction within the defined class. We formalize the representation space ℝ⁵ = (x,y,z,t,c)(x,y,z,t,c), decompose its transformation group SE(5) into 10 independent rotation planes via Givens factorization, characterize precisely the problem class and its boundaries, and prove injectivity of the encoding map ϕϕ with an explicit construction. We show that discrete counting operations emerge naturally as spatial frequency phenomena from helix projections in the (x,z)(x,z) plane — requiring no separate logical module. We further formalize the Z-buffer projection operator as a tropical linear projection in the Max-Plus semiring, establishing its mathematical role beyond graphics hardware into tropical geometry. The Abstract and Reasoning Corpus (ARC) serves as one concrete illustration among possible applications. It is used because its transformation structure falls within the covered class — not because it defines the framework. We conclude with a structural discussion of connections to active inference, proposing that both frameworks independently converge on the same insight: efficient cognition operates by identifying invariants that make problems tractable, then searching within the constrained space.