Prime Number Studies: independent re-derivations, measurements and negative results around the division table

Notes, measurements and verified code from a self-directed study of the multiplicative structure of the integers, starting from an elementary object: the division table. Contains six papers and four structural notes, with figures and runnable verification scripts for every table and number printed. The results are independent re-derivations of known mathematics, and each paper names its prior art explicitly: the map characterising odd composites is the sieve of Sundaram (1934); the wheel pair-count is the truncated Hardy-Littlewood singular series (1923); the peak-matching construction is a truncated logarithmic derivative of the completed zeta function, so its numerics certify nothing about the critical line; and the single-column paper reaches the prime side of the logarithmic derivative of zeta through Ramanujan-Fourier lag analysis, gcd Gram kernels and Moebius inversion, closing three spectral routes for structural reasons via the Szego limit theorem and the bipartite symmetry of the operator it builds. The notes cover a measured structural difference against the matrix-valued von Mangoldt measure of Groskin (2026); the Li-Sekatskii shifted Li coefficients, where the shift is shown to have an exact optimal width; the phase of that same shift, which forces a strict n mod 4 cycle of detection and masking and meets Sekatskii's Theorem 3 at the far end of one rate curve, with two negative results; and the rational-angle locking of a von Mangoldt Toeplitz symbol, identified as the Hardy-Littlewood singular series multiplying a Lerch-Wood polylogarithm cusp, i.e. the major arcs of the circle method. The newest paper, A Numerical Study of the Division Table, is expository and is the natural entry point: it starts from the quotients n/d written as mixed numbers and reports at each step which classical object the sheet has become - the summatory divisor function, the harmonic numbers, the Mertens function through Redheffer's determinant, Fermat's method and the quadratic sieve, the Dirichlet divisor and Gauss circle problems and Voronoi's series, and, when the sheet is truncated rather than walked, Buchstab's function and the wheel sieve. Its ledger lists twenty-one items and decomposes all twenty-one. The sixth paper, A probe of the odd parity sector of the CvS Galerkin matrix, studies a sector of the Connes-van Suijlekom construction (A. Groskin, arXiv:2605.20224) that the published work lists as unprobed. It establishes an exact finite identification of that sector with a specific sine space, reports an unseeded sign-change scan that returns the first ten zeta ordinates at four prime cutoffs and all twenty-nine below height 100 at the smallest — the scan building its function from the eigenvector alone and calling for the ordinates only after its root list is frozen, with a random-vector control separating by some thirty orders of magnitude — and measures the Galerkin exponent against the archimedean cutoff. That last measurement is the section 11 test of the source paper, executed; the exponent proves nearly insensitive to the cutoff where the stated mechanism predicted proportionality. It also scans every near-null branch of both parity blocks at one cutoff, finding the number of roots a branch produces beyond the tested ordinates non-decreasing in the eigenvalue modulus across twelve cell-sector combinations, with the sign of the eigenvalue interleaved throughout and no separate sign effect detected; the negative eigenvalues of those blocks, a finite-cutoff artifact, are counted against the archimedean cutoff, and a parity-count pattern that held in five successive cells is reported together with the sixth that breaks it. The paper claims nothing about the limit as the cutoff grows, and nothing about the Riemann Hypothesis. No claim is made about the Riemann Hypothesis. The material is offered as an educational route from an elementary object to the explicit formula, together with documented negative results, controls, and three methodological rules the work exists to illustrate.