Reconstruction cores in cyclic homometry: Finite intrinsic groups, spectral blocks, and local trades

How many finite point sets share exactly the same differences, counted with multiplicity? This working paper studies homometry in cyclic groups and finitely generated abelian groups, identifying reconstructions up to translation and inversion. Principal results The cyclic seven-point maximum is S(7) = 8. Every seven-point autocorrelation fibre contains at most eight inequivalent reconstructions, and an eight-class example attains the bound. The proof combines companion classification theorems, analytical reductions, and exact finite certificates. It is not an independent computation-free proof. Prime-order fibres admit a sharper analytical bound. For every prime p ≥ 11 except 302891, a seven-point fibre has at most four classes. The exceptional prime concerns this sharper bound; it does not affect S(7) = 8. The optimal consecutive Fourier-intensity bandwidth is ⌊k²/4⌋. For k ≥ 2, this many intensities makes every anchored complex fibre finite. Explicit families show that fewer cannot suffice uniformly. The theorem establishes finiteness, rather than unique reconstruction. Methods and scope Integral relation lattices preserve the torsion essential to cyclic reconstruction. Group-ring identities, character moments, Gram matrices, and algebraic counting bounds reveal the structure of complete fibres. Exact certificates resolve the remaining finite seven-point sectors, with independent implementations and unrestricted decoding checks supporting their verification. For general cardinality, the paper proves exponential counting bounds in generic and several structured regimes. The optimal bandwidth theorem also yields a general bound of exp(O(k log k)). Open questions The uniform bound S(k) ≤ exp(Ck) remains open. Its principal obstruction is counting unrestricted torsion placements, especially with heavy layers and spectral zeros. Fully analytical replacements for the seven-point certificates and the exceptional prime-order four-class case also remain unresolved. The project includes the manuscript, editable LaTeX, verification code and records, dependency documentation, and a separate corrections-and-issues account.