Exact Heisenberg Determinant Limits and Tower-Dependent Homological Torsion

We compute the exact finite determinant limit for Kammeyer's element f=1-2a+2b in the integral group ring of the discrete Heisenberg group. Along the congruence tower modulo powers of any fixed odd prime p, the normalized Fuglede-Kadison determinant converges to 2^(1/(p+1)), whereas the infinite determinant is 2. An exact central-character factorization and an integral-lattice tail bound justify the ordinary limit without passing an unbounded logarithm through weak spectral convergence. Attaching a two-sphere and a three-cell to the Heisenberg nilmanifold realizes the same discrepancy in integral homology. The resulting finite connected three-dimensional CW complex has determinant-L2-acyclic universal cover and L2-torsion log 2, but its normalized alternating integral homology torsion tends to log 2/(p+1). This limit therefore depends on the residual tower of a single space and refutes the unrestricted finite-CW formulation of modified homological torsion approximation in Hughes and Luck, arXiv:2510.20959v2, Conjecture 1.2. The space is neither aspherical nor a closed manifold; no aspherical-manifold conjecture or original AIM 11.1 closure is claimed. Kammeyer's underlying counterexample and infinite determinant, Deninger's determinant formula, Luck's nilmanifold vanishing theorem, and Boschheidgen's representation-theoretic ingredients are explicitly credited. Two unchanged standard-library exact checkers support the displayed finite calculations; the general conclusions rest on the analytic proof, not finite sampling. This English preprint is AI-assisted, originating-researcher self-audited, and unrefereed. Novelty remains undetermined. No independent human review, proof-assistant verification, guaranteed indexing, or absolute-priority claim is made.