Jumping loci of the cylindrical commuting families: arithmetic ladders on the curve s² = (N−25)(N−1)

The three cylindrical families (“sheets” 1, 2, 3) of commuting charges of hep-th/0404195, whose densities are built from one boson and N bosons in O(N)-invariant pairs, live over the conic s² = (N−25)(N−1). Spin by spin from spin 5 through spin 19 we locate the points where, at a given spin, the centralizer of I₃ is enlarged, by two-prime residue scans and, through spin 17, an exact kernel computation at every flagged point other than the decoupling point at spin 17, and we compute the points predicted at spins 21–27. In the variable β² = (t−1)/(t+1) of a rational parametrization the conic takes the Coulomb-gas form N = 13 − 6(β² + β⁻²). On the branch t > 1 the detected locus consists of points β² = 1/q with q even, where N equals the Virasoro central charge c1,q = 13 − 6(q + q⁻¹) of the logarithmic (1,q) models: the decoupling point q = 2 on two of the three sheets, and the ladder points q = 4, 6, …, 16, where, through spin 27, sheet 3 jumps at spin 2q−3 and sheet 2 at spin (3q−2)/2 when that is odd. We prove that the classical jumping locus at each odd spin, a set of quartic symbols (the first-derivative parts of I₃ densities), is the line of symbols with no x²r² term together with an explicit two-index lattice, with points at infinity when 3 divides the spin; on sheet 3 it contains no point with odd q. A lifting rule — the decoupling point and the ladder points lift (the centralizer of I₃ is enlarged there too) at every spin at which they lie on the classical jumping locus, and no other classical point lifts — agrees with every recorded outcome; its positive predictions at spins 21–27, recorded before the computations, were all confirmed, and its null clauses have been tested through spin 19 and at two cells of spin 21. On sheet 3 the extra charge at c1,q has a vacuum eigenvalue that is a product of q−1 shifted copies of one quadratic, found at q = 4, 6 and confirmed at q = 8, …, 14 by predictions recorded in advance. Each half of the product is the eigenvalue of a singlet-algebra generator; at q = 4, 6, 8 the extra class is the product of two such generators, and we conjecture this at every even q. Every computational claim carries a status label. The certificates, run records and archived lab records cited in the paper are in integrable-boundary-certificates release v1.2.8 (doi:10.5281/zenodo.23956243).