Blog Notes

These notes complement the theory and calculator pages with longer-form mathematical exposition. Each post starts from motivation, moves into formal statements and worked constructions, and ends by connecting back to the interactive tools in the repository.

The updated solver is now the computational companion for the posts below: it covers both the paper-backed composita workflow and a local analytic fixed-point workflow, with rendered formulas, residual checks, and graph-based verification.

• The role of fixed points in functional iteration

  Why fixed points are the first real test for solvability, how multipliers constrain local half-iterates, and why maps such as \(x^2-2\) and \(x^2+1\) lead to radically different real behaviors.
  2026-06-01
• Solving \(A^{2^n}(x) = F(x)\) with the Composita recurrence

  A formal derivation of the triangular coefficient system, the logic behind repeated square-root extraction, and the reason the composita table gives a practical static-site implementation path.
  2026-06-01
• Comparing numerical methods for half-iterates

  A comparison of truncated series, local conjugacy, and direct recomposition checks, with explicit discussion of where the analytic generalization of \(f(f(x)) = F(x)\) is local, stable, or structurally obstructed.
  2026-06-01

All three posts refer back to the curated references on the bibliography page, and the figures remain lightweight so the site stays responsive on static hosting.