Niven's Theorem on the Cosine Ruler: Why Its Dial and the Tessellation Grids Share Only the Vesica Piscis

Two instruments of this series measure the same pair of overlapping rings. The cosine ruler reads their center distance on a dial of fourteen settings, d = 2R·cos(6k°), and selects electron shells from it. The tessellation grids read the same two rings musically. Each ring carries an inscribed triangle cut into finer and finer equal rows, and at certain distances the two grids' lines fall exactly onto one another. This short paper proves the two instruments can agree at exactly one place. Every grid alignment solves a linear equation with whole-number fractions, so alignments happen only at distances that are exact fractions of R, while Niven's theorem (1956) makes every setting of the cosine dial irrational except one, k = 10, where cos(60°) = 1/2 and d = R exactly. That distance is the Vesica Piscis. And there the agreement is total. The two triangles close into the vesica diamond, sharing an edge, and their slanted grid families coincide line for line at every level tested (1 through 12, difference exactly zero). Atomic structure and musical harmony, measured on the Fibonacci wheel, meet at one and only one distance, and a seventy-year-old theorem guarantees no other meeting point exists. Paper 7 of 10 in the music series of The Strassner Series: Fibonacci Geometry and Harmony. The program that checks every number in the paper is printed in its appendix.