Square Synchronization Tools for Landau's Fourth Problem (Extended Edition with Oppermann Verification)

Description This dataset provides five HTML tools for verifying square synchronization, the second-order generation rule governing the distribution of odd primes and odd composite numbers inside square intervals. This extended edition adds a new tool for Oppermann's conjecture, completing the unified framework connecting Goldbach, Legendre, twin primes, Oppermann, and Landau's fourth problem. Natural numbers possess a first-order generation rule in which sums of consecutive odd numbers generate perfect squares. Square synchronization is the second-order generation rule, expressed by: (a + b)^2 = a^2 + b^2 + 2ab Here: a = number of odd primes b = number of odd composite numbers Inside each square interval n^2 to (n+1)^2, the second-order rule enforces the inequality: a > a^2 / (a + b) The monotonic increase of a^2 / (a + b) forms the structural core of Landau's fourth problem. This dataset provides tools to verify this structure across all four classical conjectures and Oppermann's conjecture. Included Tools GoldbachViewer.html For even numbers 2n, this tool computes: a = odd primes b = odd composites Pp = prime-prime pairs Cp = composite-composite pairs Mp = prime-composite heterogeneous pairs It verifies the synchronized oscillation: DeltaP( Pp - a^2 / (2a + 2b) ) = DeltaC( Cp - b^2 / (2a + 2b) ) LegendreVerification.html For each square interval n^2 to (n+1)^2, it verifies: a^2 + b^2 + 2ab = (n+1)^2 LegendreViewer.html For cumulative counts at (n+1)^2, it visualizes: generation buffer: 2ab / (a + b) monotonic increase of a^2 / (a + b) inequality: a > a^2 / (a + b) TwinPrimeViewer.html For intervals xn^2 to x(n+1)^2, it compares: Pp = observed twin prime count C2 * a^2 / (a + b) = expected value using the twin prime constant C2 This shows the growth of twin prime density under square synchronization. OppermannViewer.html (NEW) This new tool extends square synchronization to Oppermann's conjecture by analyzing the two-step structure: n^2 -> n^2 + n It computes: a, b = cumulative counts a^2 / (a + b) b^2 / (a + b) ab / (a + b) Cp = composite-composite pairs Mp = heterogeneous pairs (Mp = a - 2Cp) Pp = prime-prime pairs The viewer shows that the two-step acceleration from n^2 to n^2 + n produces the density jump required by Oppermann's conjecture, completing the unified structure of Landau's fourth problem. Citation Hamaji, Shinsuke (2026). Square Synchronization Tools for Landau's Fourth Problem (Extended Edition with Oppermann Verification). Zenodo. https://zenodo.org/records/21085783