The 731 Frog Calculus, Part 2: Two-Dimensional Frog Diagrams and the Ribbon-Leg Syntax
| Paper: 281 | Portfolio B — Mathematical Physics |
Abstract
Introduces 2D Frog Diagrams (Fano-Skeleton Diagrams): a strict visual syntax for the 731 Frog Calculus. Tree-Frogs (tetrahedra with exactly 4 legs) replace ZX spiders; 3-coloured Ribbon-Legs replace plain wires. The central compiler mnemonic: “Four legs good. More than four legs bad.” Develops Parenthesisation Bubbles, Poison Frogs (Associator Defect nodes carrying the $8/3$ Fano-Fisher cost), and five diagrammatic rewrite identities (Mirror Square, Malcev Resolution, Fano-Line Closure, No-Parallelisation, Moufang Echo). Companion to Part 1 (doi:10.5281/zenodo.19713350).
Key Results
- Ribbon-Leg syntax: each wire carries ordered Fano triple $(\alpha,\beta,\gamma)$ + orientation sign $\pm 1$ at the foot-joint
- Four-Leg Constraint: degree exactly 4 per node; fifth connection triggers Surgical Singularity
- Poison Frog: first-class diagram element for the $8/3$ associator cost; cannot be optimised away
- 5 rewrite rules visualised as diagrammatic identities, all verified to zero error on all 7 Fano lines
- Prior-art comparison: ZX-calculus, Penrose notation, spin foams, ribbon categories — all recovered as degenerate cases
Zenodo
Related Papers
- Paper 207 — Frog Calculus Part 1 (3D theory this syntax represents)
- Paper 258 — Origami ISA (machine code using these diagram opcodes)
- Paper 271 — G₂ Self-Duality (proves the five algebraic identities visualised here)
- Paper 221 — Fano-Fisher (establishes the $8/3$ eigenvalue cost of Poison Frogs)