Mathematical articles and manuscript companions behind the blog programs. Companion pages explain selected results and their hypotheses; their linked manuscripts contain the complete statements and proofs. Each page distinguishes proved results, numerical evidence, and open questions.

• 7 Oct 2026 Inverting the Exponential Map and Counting Geodesics on SE(2)

  An exact scalar inverse, finite rotating fibers, infinite winding families, and count formulas with explicit domains.

• 7 Oct 2026 All Conjugate Times of the Rotating Geodesics on SE(2)

  The exact bracket mechanism, its critical exception, and what the Lean development formalizes.

• 7 Oct 2026 The Conjugate Locus of Rotating Geodesics on SE(2)

  From the kernel of the exponential map to folds, cusp seams, and a conditional global incidence criterion.

• 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.