Lower bounds for the Shannon capacity of C₁₁
the Shannon capacity of the eleven-cycle
- steps
- 6
- by AI
- 2
- since
- 2026
The line is the frontier over time, climbing as the bound goes up: higher is better here. Filled dots are steps that moved it; muted dots are results that did not. Orange dots are catalog entries, results with AI in the loop. Hollow dots are candidates under review and never move the line.Grey dots along the bottom edge are results from before the quantity had a number, placed at the worst end because they have no value on this axis. Dots that would overlap are nudged sideways a few pixels. Hover a dot for its value and attribution.
About this frontier
The Shannon capacity of a graph measures how many distinct messages can be sent with zero error through a noisy channel whose confusability is described by that graph. Lovász settled the five-cycle in 1979 with the theta function; every longer odd cycle has stayed open since, and progress comes as explicit independent sets in large strong powers, each giving a better lower bound. This frontier tracks the best certified lower bound for . The exact capacity is not known, and no upper bound here is claimed to approach it.
Every step, newest first
| Date | Value | Who | Model | Status | Source |
|---|---|---|---|---|---|
| 9 Sept 2026 | best R10, dimension 213, and the current record. R9 reached 5.295524149098711 the same day on the same construction and is not given its own step. R9 and R10 change only the terminal code and import every R6 child construction unchanged, so they inherit R6's native-evaluation trust dependencies. | Matthew Protti with Astra 6 Pro and Codex GPT-6 Astra | Astra 6 Pro; Codex GPT-6 Astra Extra-High | AI step lean-checked | entry |
| 9 Sept 2026 | R6, dimension 213. Uses avoidance information to get stronger concrete constructions from ordinary 58-letter certificates. Build carries disclosed native-evaluation dependencies. | Matthew Protti with Astra 6 Pro and Codex GPT-6 Astra | – | AI step | source ↗ |
| 8 Sept 2026 | R5, dimension 213. Changes the recursive assembly and the finite substitution and terminal tables while keeping the seeds and the seven-family system. Build carries disclosed native-evaluation dependencies. | Matthew Protti with Astra 6 Pro and Codex GPT-6 Astra | – | AI step | source ↗ |
| 8 Sept 2026 | R3, dimension 207. A single replacement at BPZ node x27; the gain in the constructed root is about 1.6e-7. | Matthew Protti with Astra 6 Pro and Codex GPT-6 Astra | – | AI step | source ↗ |
| 10 Aug 2026 | The same group's updated construction in dimension 207, pinned at commit aa21eeb. This is the baseline every later step is compared against, by exact integer comparisons, using cross-powers when dimensions differ. | Buys, Polak and Zuiddam (updated certificate) | – | historical | source ↗ |
| 31 Jul 2026 | The published odd-cycles record paper, which set records for seven odd cycles at once with model assistance and formal verification. | Buys, Polak, Zuiddam, Gao, Itty, Rosin, Carstensen and Reichman | ChatGPT 5.6 Sol Pro, ChatGPT 5.6 Sol, Claude Opus 5 | AI step lean-verified | entry |
Steps transcribed from the version-history table in the README at v0.5.0, cross-checked against the release tags. The July row is the published odd-cycles record; the August row is the same group's updated certificate, the baseline later steps compare against by exact integer comparisons, using cross-powers when dimensions differ. Trust caveat from the source itself: the R5 and R6 builds retain disclosed native-evaluation dependencies, and R9/R10 import the R6 constructions unchanged, so the top of this staircase is not kernel-only. No independent expert endorsement is claimed for any step.