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Record four 2026 lower bounds for the Shannon capacity of C₇ #174

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6 changes: 5 additions & 1 deletion README.md
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Expand Up @@ -30,7 +30,7 @@ Bounds for which the level of available verification is currently at minimal lev
| [7a](https://teorth.github.io/optimizationproblems/constants/7a.html) | Irrationality measure of $\pi$ | 2 | 7.103205334137 |
| [7b](https://teorth.github.io/optimizationproblems/constants/7b.html) | Irrationality measure of $\Gamma(1/4)$ | 2 | $10^{143}$ |
| [8](https://teorth.github.io/optimizationproblems/constants/8a.html) | Classical zero-free region constant | 0.755106 | 4.896 |
| [9](https://teorth.github.io/optimizationproblems/constants/9a.html) | Shannon capacity of the 7-cycle | 3.2578 | 3.3177 |
| [9](https://teorth.github.io/optimizationproblems/constants/9a.html) | Shannon capacity of the 7-cycle | 3.25883262 | 3.3177 |
| [10a](https://teorth.github.io/optimizationproblems/constants/10a.html) | The real Grothendieck constant | $\frac{6\pi}{11}\approx 1.71360$ | $\frac{\pi}{2\log(1+\sqrt{2})} - 10^{-4} \approx 1.78211$ |
| [10b](https://teorth.github.io/optimizationproblems/constants/10b.html) | The complex Grothendieck constant | 1.338 | 1.40491 |
| [10c](https://teorth.github.io/optimizationproblems/constants/10c.html) | Spencer discrepancy constant (“six standard deviations suffice”) | 1.697749 | 3.674235 (3.65*) |
Expand Down Expand Up @@ -158,6 +158,10 @@ Bounds for which the level of available verification is currently at minimal lev
- [15a](https://teorth.github.io/optimizationproblems/constants/15a.html) **improved upper bound:** $C_{15a} \leq 2.371177$ by [E. Dupont, M. Eisenberger, B. Kozlovskii, A. Mehrabian, F. J. R. Ruiz, A. See, R. Zhou, J. Alman, V. Vassilevska Williams, M. Balog](https://arxiv.org/abs/2608.16884), 17 Aug 2026.
- [43](https://teorth.github.io/optimizationproblems/constants/43a.html) **improved lower bound (unverified):** $C_{43} \geq 0.860*$ (exact $43/50$; certificate-layer result conditional on the lemma set of [KHSHGW2026](https://arxiv.org/abs/2601.22365)) by [J. Savva](https://doi.org/10.5281/zenodo.22223485), 1 Sep 2026.
- [88a](https://teorth.github.io/optimizationproblems/constants/88a.html) **improved upper bound:** $C_{88a} \leq 186$ via $\mathrm{DHL}[40,2]$, by [OpenAI](https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/short_gaps.pdf), 30 Aug 2026, with a Lean 4 formalization conditional on three declared axioms.
- [9](https://teorth.github.io/optimizationproblems/constants/9a.html) **improved lower bound:** $C_{9} \ge 134753^{1/10} \approx 3.258020$ by [N. Itty, C. D. Rosin, C. Carstensen, D. Reichman](https://arxiv.org/abs/2607.21517), 23 Jul 2026.
- [9](https://teorth.github.io/optimizationproblems/constants/9a.html) **improved lower bound:** $C_{9} \ge 3.258789153908\ldots$ by [Y. Gao](https://arxiv.org/abs/2607.27869), 30 Jul 2026.
- [9](https://teorth.github.io/optimizationproblems/constants/9a.html) **improved lower bound:** $C_{9} \ge 3.258805369885\ldots$ by [P. Buys, S. Polak, J. Zuiddam](https://arxiv.org/abs/2607.29681), 31 Jul 2026, with a Lean 4 formalization.
- [9](https://teorth.github.io/optimizationproblems/constants/9a.html) **improved lower bound:** $C_{9} \ge 3.25883262\ldots$ by [R. Tandon](https://arxiv.org/abs/2608.30273), 31 Aug 2026.

## Maintainers

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13 changes: 11 additions & 2 deletions constants/9a.md
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Expand Up @@ -37,7 +37,11 @@ is the strong graph product.
| $343^{1/5} \approx 3.2141$ | [BMRRST1971] |
| $108^{1/4} \approx 3.2237$ | [VZ2002] |
| $350^{1/5} \approx 3.2271$ | [MO2017] | |
| $367^{1/5} \approx 3.2578$ | [PS2018] | |
| $367^{1/5} \approx 3.2578$ | [PS2018] | Independent set of size $367$ in ${\mathcal C}\_{7}^{\boxtimes 5}$ |
| $134753^{1/10} \approx 3.258020$ | [IRCR2026] | Independent set of size $134753$ in ${\mathcal C}\_{7}^{\boxtimes 10}$; constructions at https://github.com/nathanielitty/lower-bounds-for-shannon-capacity |
| $3.258789153908\ldots$ | [Gao2026] | Recursive product of the size-$367$ gadget; independent set of size $M_{40}$ in ${\mathcal C}\_{7}^{\boxtimes 200}$. Verification code: commit `b13031ba76e3` of https://github.com/xyz2606/recursive_construction_of_the_Shannon_capacity_of_C_7 |
| $3.258805369885\ldots$ | [BPZ2026] | Valid-tuple product in ${\mathcal C}\_{7}^{\boxtimes 200}$. Lean 4 formalization: commit `aa21eeb12b75` of https://github.com/spectra-research/shannon-capacity-lean |
| $3.25883262\ldots$ | [Tan2026] | Heterogeneous recursion; independent set in ${\mathcal C}\_{7}^{\boxtimes 500}$. Certificates: https://github.com/tandonravi/C7-Shannon-Capacity-Heterogeneous-Recursion |


## Additional comments and links
Expand All @@ -49,6 +53,7 @@ channel whose confusability graph is $G$.
known for $\Theta({\mathcal C}\_{7})$.
- It is possible that $\Theta({\mathcal C}\_{2k+1})=\vartheta({\mathcal C}\_{2k+1})$ for all $k$,
but this is currently open beyond $k=2$.
- [BPZ2026] also records improved lower bounds for larger odd cycles (not split out as separate constants): $\Theta({\mathcal C}\_{11})\ge 5.294502522149\ldots$, $\Theta({\mathcal C}\_{13})\ge 6.302455083464\ldots$, $\Theta({\mathcal C}\_{15})\ge 7.301600534487\ldots$, $\Theta({\mathcal C}\_{19})\ge 9.357192705918\ldots$, $\Theta({\mathcal C}\_{21})\ge 10.342455853338\ldots$, $\Theta({\mathcal C}\_{23})\ge 11.328224257774\ldots$.

## References

Expand All @@ -62,7 +67,11 @@ RI (1971), 97–108.
Codes and Cryptography, 84 (2017), 13–22.
- [VZ2002] A. Vesel, J. Zerovnik. Improved lower bound on the Shannon capacity of $C\_7$. Information Processing
Letters, 81 (2002), 277–282.
- [IRCR2026] N. Itty, C. D. Rosin, C. Carstensen, D. Reichman. Improved lower bounds for the Shannon capacity of odd cycles. 2026. [arXiv:2607.21517](https://arxiv.org/abs/2607.21517)
- [Gao2026] Y. Gao. A recursive construction improving the lower bound on the Shannon capacity of $C_7$. 2026. [arXiv:2607.27869](https://arxiv.org/abs/2607.27869)
- [BPZ2026] P. Buys, S. Polak, J. Zuiddam. Lean-verified lower bounds for the Shannon capacity of odd cycles. 2026. [arXiv:2607.29681](https://arxiv.org/abs/2607.29681)
- [Tan2026] R. Tandon. Strengthening recursive constructions for zero-error Shannon capacity. 2026. [arXiv:2608.30273](https://arxiv.org/abs/2608.30273)

## Contribution notes

ChatGPT DeepResearch was used to prepare an initial version of this page.
ChatGPT DeepResearch was used to prepare an initial version of this page. The July–August 2026 lower-bound cascade was added from the cited arXiv texts (abstracts, theorems, and construction sizes checked against the papers).