A two-to-one executable map between Tunnell representation sets under the Tunnell balance
Let B(n) and A(n) be the integer solution sets of 2x^2 + y^2 + 8z^2 = n and 2u^2 + v^2 + 32w^2 = n, respectively. For every odd positive squarefree n satisfying |B(n)| = 2|A(n)|, we construct a deterministic computable map Phi_n : B(n) -> A(n) that sends exactly two source points to each target point. The construction combines an elementary even-coordinate map, three explicit quarter-turn correspondences, and a unique stable matching of the remaining antipodal orbits, with preferences determined by primitive midpoint directions. The map, its two-to-one property, the branch formulas, and the…
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