The Web of Bonds: How the DICTONIC Bonds Hand Over

In short. DICTO's two loops of space-filling pairs, the golden loop of paper #48 and the Delian loop of paper #51, each slide one pair of solids through a whole family of shapes. Draw one line, a bond, between every two pieces that share a face, and the lines make a web. We show that each loop is really one fixed crowd of points of two kinds, the piece centres, and that the only thing that moves is one number, a weight given to one kind against the other. Every piece is the region nearer (in this weighted sense) to its own point than to any other. Then the web is exactly the web of the classical weighted Delaunay construction. As the weight slides, the bond between unlike pieces never breaks. The two like bonds hand over: in the golden loop gradually, both alive all the way and equal at Kelvin's cell; in the Delian loop all at once, at the octet, where every rhombus of a hidden rhombic dodecahedron trades its long diagonal for its short one. Read physically, the weight is a ball radius squared: the golden loop is the caesium chloride structure, the Delian loop the fluorite structure, and the range where the balls can touch is bounded exactly by Pauling's classical radius-ratio limits (the same contact condition as Pauling's). Main theorem. For every x in [0, ∞] the golden network is the power diagram of the body-centred cubic sites with weight difference (φ²/4)(x − 1)/(x + 1); for every t in [0, 1] the Delian network is the power diagram of the fluorite sites with weight difference (3/8)(2t − 1). The web is the 1-skeleton of the dual regular subdivision, with exact bond areas. At the ends, at Kelvin's cell and at the octet these are classical Voronoi or Delaunay tessellations; the theorem gives the whole family between them. In the golden loop all three bond types live on the whole open loop, with constant face lattice; the two like bonds have area ratio x² and are equal only at x = 1. In the Delian loop the unlike bond lives exactly on (0, 1), oct–oct exactly on [0, 1/3), tet–tet exactly on (1/3, 1]; t = 1/3, the octet, is the only parameter at which both pieces exist and only one bond type is present. Theorem 2 (the balls): the weight is a squared radius; with unlike neighbours touching, ρ = (x + 2)/(2x + 1) and ρ = t/(1 − t), and the touching windows end exactly at Pauling's limits √3 − 1 and √(3/2) − 1 (Pauling's own contact condition, rederived). All geometric results are exact (371 + 61 checks, none failed; the 24 physical checks of the scale-anomaly study are estimates). The ideas are DICTO's: the DICTONIC SOLIDS and the DICTONIC BONDS and their names, and the web of bonds, a network of bonds that "should create a web of lattices that change with the rhythm of the loop" (DICTO, 25 October 2026); computation, exact proofs and drafting with Claude (Anthropic). More at dictospheres.com. Licence CC BY-SA 4.0. Scripts, exact checks and data of the three studies attached (commit 28cfdde).