The Area Lattice of a Regular Polygon: Rational Fractions and Signed Cell Decompositions

Draw every diagonal of a regular n-gon of area A, and let f_1, ..., f_m be the areas of the resulting cells. We determine, for every n >= 3, the largest positive integer a(n) such that A/a(n) is an integer combination of these areas: it is 1 when n is prime, 36 when n = 6, n when n is odd and composite or a power of two, and 2n in all remaining cases. The main strengthening is that the optimum always admits a signed decomposition into distinct cells, with every coefficient in {-1, 0, 1}. The constructions use disjoint angular wedges, exact trigonometric identities, and sums of roots of unity. Cyclotomic prime-ideal valuations establish optimality. For odd prime powers, we further determine the entire normalized area lattice as a fractional ideal in the maximal real cyclotomic field. All general proofs are computer-free. Whether the optimum can always be realized by a union of cells without negative coefficients remains open.