Hadamard Designs and Incidence Repair for Kaplansky's Direct-Finiteness Problem

We give a finite incidence construction for one-sided inverses in group algebras of torsion-free groups in every odd prime characteristic. Projective lines supply the required intersection congruences; Hadamard blocks provide regular auxiliary incidences; and unions of concurrent lines correct inverse-letter balances exactly over the integers. The repair-only union rows have fixed multiplicities, leaving a replicated bulk of lines and Hadamard-decorated lines whose reduced-turn kernel admits a uniform spectral bound. If p ≥ 5 and a real Hadamard matrix of order 4p exists, a plane of order p³ gives p⁶ + p³ + 4p signed letters; Du–Jiang's existence theorem gives at most p⁶ + p³ + 2¹⁰p^(23/20) for p ≥ 2²³. In characteristic three we specify a 6654-letter input with root degree 93 and maximum graph degree 325. The signed-matching and bounded-path methods of OpenAI and Ivanov, together with Mghirbi–Douglas's cone criterion, realize these data in finitely presented torsion-free word-hyperbolic groups, with a finite classifying complex of dimension at most two and witnesses ab = 1, ac = 0, c ≠ 0. The contribution is a quantitative refinement of the finite input to the qualitative group-algebra result of Mghirbi–Douglas.