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

更新後 Description(全文) Square Synchronization Tools for Landau's Fourth Problem (Extended Edition with Oppermann Verification and FLT Supplementary Viewer) Authors/Creators Hamaji, Shinsuke Description This dataset provides six 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 and an FLT supplementary viewer, completing the unified framework connecting Goldbach, Legendre, twin primes, Oppermann, Landau's fourth problem, and the pair-structure mechanism behind Fermat's Last Theorem. 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 the four classical conjectures, Oppermann's conjecture, and the FLT pair-structure. 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 This 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. 2-Type Binomial Square Synchronization Viewer.html (NEW, FLT supplementary tool) This viewer illustrates the two-type synchronization behavior of the binomial square (a + b)^2 = a^2 + b^2 + 2ab, where the two types correspond to odd primes and odd composites. It shows how these two types of contributions synchronize around the mixed term 2ab, and why this synchronization is unique to the 2-term quadratic case. By contrast, the cubic expansion (a + b)^3 = a^3 + b^3 + 3a^2b + 3ab^2 contains multiple mixed terms and cannot produce the same 2-dimensional synchronization. This structural breakdown explains why the pair-structure mechanism applies only to the 2-term square case and not to the 2-term cube case, providing a supplementary structural viewer for the FLT manuscript. Citation Hamaji, Shinsuke (2026). Square Synchronization Tools for Landau's Fourth Problem (Extended Edition with Oppermann Verification and FLT Supplementary Viewer). Zenodo. https://zenodo.org/records/21223132