Square Synchronization Tools for Landau's Fourth Problem

This dataset provides four HTML tools for verifying square synchronization (second-order generation rule) in Landau’s fourth problem. This dataset contains four HTML tools designed to verify square synchronization in natural numbers, a structure generated by the second-order generation rule (a+b)^2=a^2+b^2+2ab and observed within Landau’s fourth problem. Natural numbers possess a first-order generation rule in which sums of consecutive odd numbers generate perfect squares (1+3=2^2, 1+3+5=3^2, ...). In addition, the square of a sum, (a+b)^2 = a^2 + b^2 + 2ab, acts as a second-order generation rule, producing synchronized growth between: a: the number of odd primes b: the number of odd composite numbers within each square interval n^2 to (n+1)^2. This second-order rule implies the key inequality: a > a^2/(a+b) and this value increases monotonically, forming the core of square synchronization. The tools in this dataset allow verification of this structure across: Goldbach’s conjecture Legendre’s conjecture the twin prime conjecture and Landau’s fourth problem as a whole. Included Tools 1. GoldbachViewer.html For even numbers 2n, this tool computes: a: number of odd primes b: number of odd composites Pp: number of prime–prime pairs Cp: number of composite–composite pairs It verifies the synchronized oscillation: DeltaP{ Pp - a^2/(2a+2b) } = DeltaC{ Cp - b^2/(2a+2b) }. 2. LegendleVerification.html For each square interval n^2 to (n+1)^2, it verifies: a^2 + b^2 + 2ab = (n+1)^2. 3. LegendleViewer.html For the cumulative counts at (n+1)^2, it visualizes: the generation buffer: 2ab/(a+b) the monotonic increase of: a^2/(a+b) and confirms the inequality: a > a^2/(a+b). 4. 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 to show the growth of twin prime density under square synchronization. Citation Hamaji, Shinsuke (2026). Square Synchronization Tools for Landau’s Fourth Problem. Zenodo. DOI 10.5281/zenodo.20847081