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The orphan theorem: degenerate perpendicular orbits in rational right triangles

Primarily AI-generated textHuman understanding: some partsmath.DS — Dynamical Systemsmath.GT — Geometric Topologymath.NT — Number Theory

Contributed by Joshua Boorstein ↗

Version 1 / Nov 07, 2026 / CC BY 4.0

Abstract

Let TT be a right triangle with acute angle α\alpha, legs L1L_1, L2L_2 and hypotenuse HH, and suppose TT is rational: 2α/π=P/Q2\alpha/\pi = P/Q in lowest terms. Shoot a beam perpendicular to the leg L1L_1 from each of its points. The leg is partitioned into branches — maximal intervals of launch heights sharing a combinatorial type — and we prove that the partition is finite, with at most 2Q+12Q + 1 branches, which cover the leg but for finitely many points. The first-return map JJ of the beam to L1L_1 is an involution, and on each branch it does one of two things: it carries the branch onto a distinct branch of equal width by a reflection, or it fixes the branch pointwise. A pointwise-fixed branch — a whole interval of trajectories that hit a side head-on and retrace — is an orphan. Theorem A. There is no orphan branch when PP is even and at most one when PP is odd, with no hypothesis on the direction; when PP is odd and QQ is a prime power, or twice or four times a prime, the orphan exists. Non-existence is a parity count on the velocity set of the unfolded beam. Uniqueness is an Euler count on the genus-zero quotient of the unfolded surface, which admits minimal components. Existence is arithmetic: on a folded disc, an absent orphan forces 11 to be a signed sum of doubled heights of images of the right-angle vertex, which is impossible at the centres named. Running the direction count and the Euler count for each side shows that at most two of the three sides carry a retracing cell, at most one each, the two ends of one band of double normals. On the unfolded surface — the holonomy double cover of a genus-zero quadratic differential Q(P−2,Q−P−2,−1Q)Q(P-2, Q-P-2, -1^Q) — the images of the right-angle vertex are Weierstrass points, the leg is the quotient by the hyperelliptic involution of a geodesic segment through one of them, and the first-return map is read in that quotient.

Author comments: First version. Companion papers in preparation: "The necklace of a rational right triangle" and "No arrivals". Code: https://doi.org/10.5281/zenodo.24037744

Provenance statement

The text, the proofs and the code were drafted by an AI system (Claude, Anthropic) working under the author's direction over an extended research project, and checked by independent AI review passes and by the exact computations deposited at https://doi.org/10.5281/zenodo.24037744 (directories lib/ and orphan/). The author has read parts of the paper in detail and takes responsibility for its content.

Tools used

Anthropic
Claude Opus
Python
SymPy/NumPy (exact cyclotomic arithmetic and certificates)

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Version history

  1. v1 · 60 pagesSubmitted by Joshua BoorsteinFirst version. Companion papers in preparation: "The necklace of a rational right triangle" and "No arrivals". Code: https://doi.org/10.5281/zenodo.24037744CurrentNov 07, 2026