Refactor matrix multiplication exponent entries in README and constants documentation

Author: damek

README.md

@@ -34,7 +34,8 @@ We are arbitrarily numbering the constants as $C_{1}$, $C_{2}$, etc., mostly bas
 | [13a](https://teorth.github.io/optimizationproblems/constants/13a.html) | Moser's convex worm cover constant | 0.232239 | 0.2617993878 |
 | [13b](https://teorth.github.io/optimizationproblems/constants/13b.html) | Lebesgue's convex universal cover constant | 0.832 | 0.8440935944 |
 | [14](https://teorth.github.io/optimizationproblems/constants/14a.html) | Smallest $n$ for which the value of $BB(n)$ is undecidable | 6 | 432 |
-| [15](https://teorth.github.io/optimizationproblems/constants/15a.html) | Matrix multiplication exponent | 2 | 2.371339 |
+| [15a](https://teorth.github.io/optimizationproblems/constants/15a.html) | Matrix multiplication exponent | 2 | 2.371339 |
+| [15b](https://teorth.github.io/optimizationproblems/constants/15b.html) | Dual matrix multiplication exponent | >0.321334 | 1 |
 | [16](https://teorth.github.io/optimizationproblems/constants/16a.html) | Brezis–Gallouet–Wainger remainder constant on the 2D torus | $\frac{\beta + \pi}{\pi} \approx 1.82283$ | $\approx 2.15627$ |
 | [17](https://teorth.github.io/optimizationproblems/constants/17a.html) | Exponential growth constant of diagonal Ramsey numbers | $\sqrt{2} \approx 1.4142$ | 3.7992027396 |
 | [18](https://teorth.github.io/optimizationproblems/constants/18a.html) | Marton's conjecture constant (PFR) | 1 | 9 |

constants/15a.md

@@ -2,7 +2,7 @@
 
 ## Description of constant
 
-We define $C_{15}$ to be the matrix multiplication exponent $\omega$, the smallest real number such that two $n \times n$ matrices over a field can be multiplied using $O(n^{\omega + o(1)})$ arithmetic operations.
+We define $C_{15a}$ to be the matrix multiplication exponent $\omega$, the smallest real number such that two $n \times n$ matrices over a field can be multiplied using $O(n^{\omega + o(1)})$ arithmetic operations.
 
 ## Known upper bounds
 

constants/15b.md

@@ -0,0 +1,83 @@
+# Dual matrix multiplication exponent
+
+## Description of constant
+
+In algebraic complexity theory, for each real $k \ge 0$, let $\omega(k)$ denote the exponent for multiplying an $n \times n^k$ matrix by an $n^k \times n$ matrix. We define
+$$
+\alpha := \sup\{k \ge 0 : \omega(k) = 2\}.
+$$
+Thus $\alpha$ is the largest aspect-ratio exponent for which rectangular matrix multiplication can still be performed in essentially quadratic arithmetic time.
+<a href="#LGU2018-def-omega">[LGU2018-def-omega]</a> <a href="#LGU2018-def-alpha">[LGU2018-def-alpha]</a>
+
+We define
+$$
+C_{15b} := \alpha.
+$$
+
+The current best established range is
+$$
+0.321334 < C_{15b} \le 1.
+$$
+<a href="#WXXZ2024-alpha-0321334">[WXXZ2024-alpha-0321334]</a> <a href="#CLLZ2025-ub-1">[CLLZ2025-ub-1]</a>
+
+## Known upper bounds
+
+| Bound | Reference | Comments |
+| ----- | --------- | -------- |
+| $1$ | [[CLLZ2025](#CLLZ2025)] | Current best general upper bound. <a href="#CLLZ2025-ub-1">[CLLZ2025-ub-1]</a> |
+
+## Known lower bounds
+
+| Bound | Reference | Comments |
+| ----- | --------- | -------- |
+| $0$ |  | Trivial: $\omega(0)=2$ since multiplying an $n\times 1$ matrix by a $1\times n$ matrix is an outer product. |
+| $0.172$ | [[LGU2018](#LGU2018)] | Coppersmith’s 1982 theorem yields $\omega(0.172)=2$. <a href="#LGU2018-alpha-0172">[LGU2018-alpha-0172]</a> |
+| $0.29462$ | [[LG2012](#LG2012)] | Previous record due to Coppersmith (1997), as recorded in Le Gall’s 2012 paper. <a href="#LG2012-alpha-029462-030298">[LG2012-alpha-029462-030298]</a> |
+| $0.30298$ | [[LG2012](#LG2012)] | Le Gall’s 2012 improvement. <a href="#LG2012-alpha-029462-030298">[LG2012-alpha-029462-030298]</a> |
+| $0.31389$ | [[LGU2018](#LGU2018)] | Improvement from the fourth-power Coppersmith–Winograd analysis. <a href="#LGU2018-alpha-031389">[LGU2018-alpha-031389]</a> |
+| $0.321334$ | [[WXXZ2024](#WXXZ2024)] | Current best published lower bound. <a href="#WXXZ2024-alpha-0321334">[WXXZ2024-alpha-0321334]</a> |
+
+## Additional comments and links
+
+- **Algorithmic relevance.** Better bounds for rectangular matrix multiplication improve algorithms for all-pairs shortest paths and sparse matrix multiplication.
+  <a href="#LG2012-applications">[LG2012-applications]</a>
+
+- [Wikipedia page on computational complexity of matrix multiplication](https://en.wikipedia.org/wiki/Computational_complexity_of_matrix_multiplication)
+
+## References
+
+- <a id="CLLZ2025"></a>**[CLLZ2025]** Christandl, Matthias; Le Gall, François; Lysikov, Vladimir; Zuiddam, Jeroen. *Barriers for rectangular matrix multiplication.* Computational Complexity **34** (2025), article 4. DOI: [10.1007/s00037-025-00264-9](https://doi.org/10.1007/s00037-025-00264-9). Preprint: [arXiv:2003.03019 PDF](https://arxiv.org/pdf/2003.03019.pdf). [Google Scholar](https://scholar.google.com/scholar?q=Matthias+Christandl+Francois+Le+Gall+Vladimir+Lysikov+Jeroen+Zuiddam+Barriers+for+rectangular+matrix+multiplication)
+	- <a id="CLLZ2025-ub-1"></a>**[CLLZ2025-ub-1]**
+	  **loc:** arXiv PDF p.4, Figure 2 caption
+	  **quote:** “The red points are the best upper bounds on $\alpha$, namely $1$.”
+
+- <a id="LG2012"></a>**[LG2012]** Le Gall, François. *Faster Algorithms for Rectangular Matrix Multiplication.* In: *2012 IEEE 53rd Annual Symposium on Foundations of Computer Science (FOCS 2012)*, 514–523. DOI: [10.1109/FOCS.2012.80](https://doi.org/10.1109/FOCS.2012.80). Preprint: [arXiv:1204.1111 PDF](https://arxiv.org/pdf/1204.1111.pdf). [Google Scholar](https://scholar.google.com/scholar?q=Francois+Le+Gall+Faster+Algorithms+for+Rectangular+Matrix+Multiplication+FOCS+2012)
+	- <a id="LG2012-alpha-029462-030298"></a>**[LG2012-alpha-029462-030298]**
+	  **loc:** arXiv PDF p.1, Abstract
+	  **quote:** “In this paper we show that $\alpha>0.30298$, which improves the previous record $\alpha>0.29462$ by Coppersmith (Journal of Complexity, 1997).”
+	- <a id="LG2012-applications"></a>**[LG2012-applications]**
+	  **loc:** arXiv PDF p.1, Abstract
+	  **quote:** “For example, we directly obtain a $O(n^{2.5302})$-time algorithm for the all-pairs shortest paths problem over directed graphs with small integer weights, improving over the $O(n^{2.575})$-time algorithm by Zwick (JACM 2002), and also improve the time complexity of sparse square matrix multiplication.”
+
+- <a id="LGU2018"></a>**[LGU2018]** Le Gall, François; Urrutia, Florent. *Improved Rectangular Matrix Multiplication using Powers of the Coppersmith-Winograd Tensor.* In: *Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2018)*, 1029–1046. DOI: [10.1137/1.9781611975031.67](https://doi.org/10.1137/1.9781611975031.67). Preprint: [arXiv:1708.05622 PDF](https://arxiv.org/pdf/1708.05622.pdf). [Google Scholar](https://scholar.google.com/scholar?q=Francois+Le+Gall+Florent+Urrutia+Improved+Rectangular+Matrix+Multiplication+using+Powers+of+the+Coppersmith-Winograd+Tensor)
+	- <a id="LGU2018-def-omega"></a>**[LGU2018-def-omega]**
+	  **loc:** arXiv PDF p.3, §1.1 “Background”
+	  **quote:** “In analogy to the square case, the exponent of rectangular matrix multiplication, denoted $\omega(k)$, is defined as the minimum value such that this product can be computed using $O(n^{\omega(k)+\epsilon})$ arithmetic operations for any constant $\epsilon>0$.”
+	- <a id="LGU2018-def-alpha"></a>**[LGU2018-def-alpha]**
+	  **loc:** arXiv PDF p.3, §1.1 “Background”
+	  **quote:** “$\alpha = \sup\{k \mid \omega(k)=2\}$.”
+	- <a id="LGU2018-alpha-0172"></a>**[LGU2018-alpha-0172]**
+	  **loc:** arXiv PDF p.3, §1.1 “Background”
+	  **quote:** “Coppersmith [8] showed in 1982 that $\omega(0.172)=2$.”
+	- <a id="LGU2018-alpha-031389"></a>**[LGU2018-alpha-031389]**
+	  **loc:** arXiv PDF p.4, §1.2 “Our results”
+	  **quote:** “$\alpha \ge 0.31389$”
+
+- <a id="WXXZ2024"></a>**[WXXZ2024]** Vassilevska Williams, Virginia; Xu, Yinzhan; Xu, Zixuan; Zhou, Renfei. *New Bounds for Matrix Multiplication: from Alpha to Omega.* In: *Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2024)*, 3792–3835. DOI: [10.1137/1.9781611977912.134](https://doi.org/10.1137/1.9781611977912.134). Preprint: [arXiv:2307.07970 PDF](https://arxiv.org/pdf/2307.07970.pdf). [Google Scholar](https://scholar.google.com/scholar?q=Virginia+Vassilevska+Williams+Yinzhan+Xu+Zixuan+Xu+Renfei+Zhou+New+Bounds+for+Matrix+Multiplication+from+Alpha+to+Omega)
+	- <a id="WXXZ2024-alpha-0321334"></a>**[WXXZ2024-alpha-0321334]**
+	  **loc:** arXiv PDF p.3, Introduction, paragraph beginning “In particular”
+	  **quote:** “In particular, we show that $\alpha >0.321334$ (improving upon the previous bound $0.31389$)”
+
+## Contribution notes
+
+Prepared with assistance from ChatGPT 5.2 Pro.