Long-form mathematical articles: complete statements, hypotheses, and proofs behind the blog series. A blog post shows the result and the experiment; the article carries the full derivation, the fine print, and an honest ledger of what is proven, what is measured, and what remains open.

• 5 Sep 2026 Growth Vectors and Caustics of Magnetic Flux Lifts

  The mathematical companion to "The Geometry of Forbidden Directions": complete statements and hypotheses, proofs where they exist, and the caustic identification stated as the open conjecture it is. The flag computation behind Q = d + k + 2, the proven elliptic reduction behind the period-average law δ(ε) = 1 − (2/π)K(2ε), a computation showing the coefficients read the profile of the field and not just its gradient, the saddle-node corollary — and an explicit ledger separating theorem, measurement, and conjecture.