Collision Geometry of Finite Radix Value Sets: Restricted Product Regions, Dyadic Divisor Fibers, and Hyperbolic Kernels

Abstract

We study the finite value sets of two quadratic state forms arising in a binary--ternary native-radix comparison. The negative sectors admit exact factorizations into restricted multiplication-table regions. After removing the repeated binary zero state, the binary and ternary negative input domains have the same cardinality, so distinct-value comparison is exactly equivalent to a comparison of collision defects. On the ternary side we prove a dyadic odd-kernel decomposition, an ordered two-block theorem for divisor fibers, factor-$2$ localization inside each orientation, and a dyadic singleton cutoff. We then connect the problem with Ford's divisor-union functional, derive an exact consecutive-gap formula and a strict-concavity kernel, and introduce a hyperbolic core functional that separates radial and divisor geometry. Exact radial kernels yield a positive pure-block contribution. A finite exact computation gives a global two-center lower bound, and a one-dimensional curvature argument gives positivity for every center cluster of multiplicative diameter at most $2$. Finally, squarefree cores admit a primitive-quotient hypercube decomposition which isolates the remaining cross-orientation arithmetic. We sharpen this by proving a cross-edge blocker theorem: every consecutive augmented cross-copy edge forces multiplicative isolation in the residual hypercube and a divisor-free interval around the inserted prime on the primitive side; outside an explicit terminal condition this produces a genuine large consecutive-divisor gap. The full ternary-versus-binary extremality statement is not proved here; the purpose of the paper is to establish the structural theory and to identify the remaining analytic obstruction precisely.