# Equatorial cycles and projective periods

The main formal chapter proves the exact affine/projective tube comparison, center factor two, positive normal coefficient one, affine scalar-fiber splitting and logarithm argument. The learner has four examples and eight fully solved exercises totaling 100 points. The accompanying analytic reading retains every mathematical expression and solved exercise in the original authored tangent/contour/polynomial-continuation proof body. Its all-integer angular Fourier identity is supplied by the complete linked angular-distribution chapter. The positive-polar tube analytic-wavefront theorem, analytic differential decrease and ordinary-wavefront inclusion are supplied by the complete analytic-boundary chapter. The homogeneous Euler bound and sphere-conormal exclusion are proved in Dilation equations and tangential analytic singularities, Theorem 4.1 and Corollary 4.2. General smooth and analytic wavefront operations are proved in Pulling back and combining analytic singularities, Sections 4–8. Gaussian packets and homogeneous Fourier directions, Theorem 8.1, proves exact ordinary and analytic Fourier-wavefront interchange away from both origins for arbitrary complex degree, finite Euler chains and arbitrary origin extensions. [Derivative sequences and microlocal regularity](../../AN02-L190.html#complete-proof), Sections 1–8, supplies the exact admissible derivative-sequence convention, bounded localization and base regularity, all analytic-map operation bounds, homogeneous Fourier interchange away from both origins, and the uniform class estimates needed for limits. Ordinary normal continuity and hypocontinuity retain their stated hypotheses; stronger class estimates require the stated uniform constants.

Run with Python 3, NumPy, mpmath and Matplotlib:

    python -B -X utf8 affine-cycle-check-and-figures.py

The code generates both PNG/SVG figures and exact geometry data. One drawing uses the actual negatively oriented affine filling disc for F=tau, with fixed imaginary coordinate -7/20 and its exact projective image. The other distinguishes center multiplicity, the relative-disc normal map and the forgotten scalar-fiber model. Every diagram specifies whether it is an exact coordinate plot or a map/class diagram.

The 5713 finite identities supplement the complete proof. They verify parity factors, homogeneous pole degrees, affine nonvanishing, projective normal determinants, positive radial scales and the canonical/scalar logarithmic periods. They are not a numerical proof of the general theorem.

Original material is CC0 1.0. The DejaVu font retains its own terms. Linked historical works are not reproduced.