The Global Web of Bonds: How Bonds Hand Over Across Crystal Networks

In short. Paper #52 showed that DICTO's two loops of space-filling pairs are one fixed crowd of points of two kinds, with one number, a weight, slid between the kinds; the bonds (one line for each face two pieces share) form a web. In the golden loop the two like bonds hand over gradually; in the Delian loop they hand over all at once, at the octet, where only bonds between unlike pieces are left. DICTO proposed that these branches turn up elsewhere in slightly different guises. We took eight more crystal arrangements of two kinds of points (three in the plane, five in space) and one of three kinds, and slid the weight through every value at which both kinds still have a piece. For each we found, exactly, every point where the pattern changes, which bonds exist in between, and whether there is a moment with only unlike bonds. The proposal holds: the hand-over with a single switch recurs in the square lattice with its centres, in the honeycomb with its hexagon centres (the switch is the trihexagonal tiling) and in rock salt; the gradual hand-over recurs in zinc blende, in the holes of hexagonal close packing, and in the A15 and C15 alloy structures. In every two-kind network the bond between unlike pieces never breaks, and we prove that it never can. Much of this sits inside known theory, and we say where. Main theorem. For each of the ten two-kind networks of Table 1, with weight difference λ = w A − w B : both kinds have cells exactly on the open range of Table 2; the face lattice changes exactly at the listed rational flips and is constant between them; every bond type lives on one interval, given exactly in Table 2, and the unlike bond lives on the whole open range; a parameter at which both kinds have cells and only unlike bonds exist occurs exactly at λ = 0 (square + centres), −1/2 (honeycomb + centres), 0 (rock salt) and −1/8 (fluorite, the octet), and nowhere in the other six. For the three-kind perovskite the coexistence region is the open rectangle −2 < λ < 2, −1 < μ < 1, cut by three straight walls into five chambers, and no point of it has only unlike bonds. All results are exact (648 exact and float checks, none failed). The named tilings at the stops follow from a classical rule (Aurenhammer; Ash and Bolker); the A15 walk up to its first flip is Weaire and Phelan's (with Kashyap and Neuhoff, as reported by Sloane and Vaishampayan); the switch points are walls in the sense of Gelfand, Kapranov and Zelevinsky. Figure 1. The ten walks of the Main theorem, each lane spread over its own coexistence range (ends at left and right, in the network's units, Table 1). Gold: the unlike bond, alive on the whole open range in every lane. Blue: the like bond of the kind whose weight rises with λ; green: the other like bond. Black ticks: the flips, with their exact λ. Red diamonds: the switch points, where only unlike bonds exist. Where blue and green overlap, the hand-over is gradual; where they meet at a diamond, it has a single switch. The golden loop of paper #48 is exactly the overlap (−1, 1) of the CsCl lane, with its ends; the Delian loop of paper #51 is the whole fluorite lane. Kagome + centres has only one like bond (the centre–centre bond appears only at the end λ = −1, as a dot). 1. Where it started Two loops of DICTO's showed two ways in which bonds can hand over. DICTO proposed that the same branches appear in other networks. We walked the weight through ten networks to find out. The ideas are DICTO's: the DICTONIC SOLIDS and the DICTONIC BONDS and their names, the web of bonds, and DICTO's idea, proposed on 25 October 2026, that "the dictonic cycle branches could show up elsewhere in geometric networks in slightly different guises"; computation, exact proofs and drafting with Claude (Anthropic). In short. Paper #52 showed that DICTO's two loops of space-filling pairs are one fixed crowd of points of two kinds, with one number, a weight, slid between the kinds; the bonds (one line for each face two pieces share) form a web. In the golden loop the two like bonds hand over gradually; in the Delian loop they hand over all at once, at the octet, where only bonds between unlike pieces are left. DICTO proposed that these branches turn up elsewhere in slightly different guises. We took eight more crystal arrangements of two kinds of points (three in the plane, five in space) and one of three kinds, and slid the weight through every value at which both kinds still have a piece. For each we found, exactly, every point where the pattern changes, which bonds exist in between, and whether there is a moment with only unlike bonds. The proposal holds: the hand-over with a single switch recurs in the square lattice with its centres, in the honeycomb with its hexagon centres (the switch is the trihexagonal tiling) and in rock salt; the gradual hand-over recurs in zinc blende, in the holes of hexagonal close packing, and in the A15 and C15 alloy structures. In every two-kind network the bond between unlike pieces never breaks, and we prove that it never can. Much of this sits inside known theory, and we say where. Main theorem. For each of the ten two-kind networks of Table 1, with weight difference λ = w A − w B : both kinds have cells exactly on the open range of Table 2; the face lattice changes exactly at the listed rational flips and is constant between them; every bond type lives on one interval, given exactly in Table 2, and the unlike bond lives on the whole open range; a parameter at which both kinds have cells and only unlike bonds exist occurs exactly at λ = 0 (square + centres), −1/2 (honeycomb + centres), 0 (rock salt) and −1/8 (fluorite, the octet), and nowhere in the other six. For the three-kind perovskite the coexistence region is the open rectangle −2 < λ < 2, −1 < μ < 1, cut by three straight walls into five chambers, and no point of it has only unlike bonds. All results are exact (648 exact and float checks, none failed). The named tilings at the stops follow from a classical rule (Aurenhammer; Ash and Bolker); the A15 walk up to its first flip is Weaire and Phelan's (with Kashyap and Neuhoff, as reported by Sloane and Vaishampayan); the switch points are walls in the sense of Gelfand, Kapranov and Zelevinsky. Figure 1. The ten walks of the Main theorem, each lane spread over its own coexistence range (ends at left and right, in the network's units, Table 1). Gold: the unlike bond, alive on the whole open range in every lane. Blue: the like bond of the kind whose weight rises with λ; green: the other like bond. Black ticks: the flips, with their exact λ. Red diamonds: the switch points, where only unlike bonds exist. Where blue and green overlap, the hand-over is gradual; where they meet at a diamond, it has a single switch. The golden loop of paper #48 is exactly the overlap (−1, 1) of the CsCl lane, with its ends; the Delian loop of paper #51 is the whole fluorite lane. Kagome + centres has only one like bond (the centre–centre bond appears only at the end λ = −1, as a dot). 1. Where it started Two loops of DICTO's showed two ways in which bonds can hand over. DICTO proposed that the same branches appear in other networks. We walked the weight through ten networks to find out. Exact study, scripts and data attached (study commit 5307e7b). Licence CC BY-SA 4.0. More at dictospheres.com.