teorth · AndreRoehrig · Sep 9, 2026
diff --git a/README.md b/README.md
@@ -77,7 +77,7 @@ Bounds for which the level of available verification is currently at minimal lev
 | [40b](https://teorth.github.io/optimizationproblems/constants/40b.html) | Asymptotic Dobrowolski constant for Lehmer’s problem | $9/4$ | $\infty$ |
 | [41a](https://teorth.github.io/optimizationproblems/constants/41a.html) | Moving sofa constant | 2.2195 | 2.37 (2.2195*)|
 | [41b](https://teorth.github.io/optimizationproblems/constants/41b.html) | Ambidextrous moving sofa constant | 1.64495521 | 2.2195 |
-| [42](https://teorth.github.io/optimizationproblems/constants/42a.html) | Turan's pure power sum constant | >0.5 | 0.69368 (0.6906538*) |
+| [42](https://teorth.github.io/optimizationproblems/constants/42a.html) | Turan's pure power sum constant | >0.5 | 0.69368 (0.688983\*) |
 | [43](https://teorth.github.io/optimizationproblems/constants/43a.html) | Gilbert-Pollak conjecture (Steiner ratio) | 0.8559 (0.860*) | 0.86602540378 |
 | [44](https://teorth.github.io/optimizationproblems/constants/44a.html) | Maximal number of relevant variables in Boolean functions of degree $d$ | 1.5 | 4.394 |
 | [45](https://teorth.github.io/optimizationproblems/constants/45a.html) | Density of odd integers that are the sum of a prime and a power of two | 0.107648 | 0.490180063290061 |
@@ -86,7 +86,7 @@ Bounds for which the level of available verification is currently at minimal lev
 | [48](https://teorth.github.io/optimizationproblems/constants/48a.html) | One-dimensional convex sub-Gaussian comparison constant | $\approx 5.33386$ | $\approx 5.33386$ |
 | [49](https://teorth.github.io/optimizationproblems/constants/49a.html) | Erdős–Szemerédi $3$-sunflower-free capacity | >1.551 ($\geq 1.554*$) | $\frac{3}{2^{2/3}} \approx 1.88988$ |
 | [50](https://teorth.github.io/optimizationproblems/constants/50a.html) | Approximation ratio for quantum Max Cut | 0.614 | $<1$ (0.5 for product states) |
-| [51](https://teorth.github.io/optimizationproblems/constants/51a.html) | Erdős maximum term problem | 0.5850788 | $\frac{2}{\pi}\approx 0.63662$ |
+| [51](https://teorth.github.io/optimizationproblems/constants/51a.html) | Erdős maximum term problem | 0.5850788 (0.5850788196744\*) | $\frac{2}{\pi}\approx 0.63662$ |
 | [52](https://teorth.github.io/optimizationproblems/constants/52a.html) | Satisfiability threshold for random 3-SAT | 3.52 | 4.490 |
 | [53](https://teorth.github.io/optimizationproblems/constants/53a.html) | Davenport constant for $C_n^3$ | 3 | 4 |
 | [54](https://teorth.github.io/optimizationproblems/constants/54a.html) | Beurling–Ahlfors transform constant | 1 | 1.575 |
@@ -158,6 +158,8 @@ 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.
+- [42](https://teorth.github.io/optimizationproblems/constants/42a.html) **improved upper bound (computer-assisted):** $C\_{42}\leq0.688983$\* by [André Röhrig](https://github.com/AndreRoehrig/optimization-constants-certificates/blob/ed6387399cf6de02eaa4cf10c43787633496619a/notes/turan.md), 9 Sep 2026. Eight-step profile with exact rational interval verification of the limiting criterion; no explicit finite threshold is supplied.
+- [51](https://teorth.github.io/optimizationproblems/constants/51a.html) **improved lower bound (computer-assisted):** $C\_{51}\geq0.5850788196744$\* by [André Röhrig](https://github.com/AndreRoehrig/optimization-constants-certificates/blob/ed6387399cf6de02eaa4cf10c43787633496619a/notes/maximum_term.md), 9 Sep 2026. Exact parameters in the He–Tang family, with a polynomial bound on the whole circle and an explicit infinite-series tail; improves on the full prior paper value $0.585078819653$.
 
 ## Maintainers
 

diff --git a/constants/42a.md b/constants/42a.md
@@ -14,6 +14,7 @@ where the minimum is taken over all $z_1,\ldots,z_n\in \mathbb{C}$ with $\max_i
 | 5/6 | Biró [Bir00] | |
 | 0.69368 | Harcos [Bir00] | |
 | 0.6906538* | Griego [Gri26] | Proposed asymptotic two-block certificate with exact rational interval verification of the limiting inequality. No explicit finite threshold $N$ is supplied. |
+| $0.688983$\* | André Röhrig [R2026a] | Eight-step generalization of [Gri26], with $\tau=1/3$ and $\alpha=0.6460641528+0.5911224477i$; the eight exact profile values and defining integrals are given below. Rational interval verification gives $D\in[0.88819065045251195,0.88819065045251196]$ and $\lvert Y\rvert/D\in[0.68898209850406074,0.68898209850406075]$, with all nine profile magnitudes strictly below $C=0.688983$. The quotient is one term in the limiting criterion; the certified bound is $C$. The proof establishes the asymptotic implication, including integer rounding and the singular endpoint; no explicit finite threshold $N$ is supplied. [Proof and verification, pinned commit](https://github.com/AndreRoehrig/optimization-constants-certificates/blob/ed6387399cf6de02eaa4cf10c43787633496619a/notes/turan.md). |
 
 ## Known lower bounds
 
@@ -30,6 +31,40 @@ where the minimum is taken over all $z_1,\ldots,z_n\in \mathbb{C}$ with $\max_i
 - Computational investigations by Cheer and Goldston [CG96] suggest that $C_{42}$ is close to $0.7$.
 - $C_{42}$ is the optimal constant for [Erdős problem #519](https://www.erdosproblems.com/519).
 
+For [R2026a], all terminating decimals are exact rationals. Set $s=1-\alpha$ and $t\_j=(8+j)/24$ for $0\le j\le8$. The middle profile on $(t\_j,t\_{j+1})$ is $\eta\_j$:
+
+| $j$ | $\operatorname{Re}\eta\_j$ | $\operatorname{Im}\eta\_j$ |
+| --- | --- | --- |
+| 0 | 0.4826503951 | -0.4916756331 |
+| 1 | 0.5360145885 | -0.4328795363 |
+| 2 | 0.5637805037 | -0.3960402450 |
+| 3 | 0.5881806899 | -0.3588033000 |
+| 4 | 0.6126301773 | -0.3152468841 |
+| 5 | 0.6317490513 | -0.2749353892 |
+| 6 | 0.6531184739 | -0.2193914107 |
+| 7 | 0.6719343778 | -0.1523171824 |
+
+Set $\delta\_0=\eta\_0-s$ and $\delta\_j=\eta\_j-\eta\_{j-1}$ for $1\le j\le7$. With positive real bases using their real logarithms, define
+
+$$
+I_\alpha(x)=\int_0^x\frac{v^{\alpha-1}}{1-v}\,dv,\qquad
+D=I_{\operatorname{Re}\alpha}(1/3),\qquad K=I_\alpha(1/3),
+$$
+
+$$
+J_\alpha(t,r)=\int_0^{1-t-r}\frac{z^{\alpha-1}}{1-z}
+\log\frac{(1-t-z)(1-r-z)}{tr}\,dz\quad(t+r<1),
+$$
+
+and $J\_\alpha(t,r)=0$ when $t+r\ge1$. Then
+
+$$
+Y=1+sK-\sum_{j=0}^7\delta_j\bigl(I_\alpha(1-t_j)-K\bigr)
++\frac12\sum_{i,j=0}^7\delta_i\delta_jJ_\alpha(t_i,t_j).
+$$
+
+The certificate checks $\lvert s\rvert<C$, $\lvert\eta\_j\rvert<C$ and $C^2D^2-\lvert Y\rvert^2>9.7997\times10^{-7}$ by exact rational endpoint comparisons. The integral series use 240 terms at 100 decimal places, with explicit tail bounds; a repeat with 480 terms at 200 decimal places gives the same displayed enclosures. The linked proof derives $C\_{42}\le C$ from these strict inequalities.
+
 ## References
 
 - [Atk61] Atkinson, F. V. *On sums of powers of complex numbers.* Acta Math. Acad. Sci. Hungar. **12** (1961), 185-188.
@@ -39,7 +74,10 @@ where the minimum is taken over all $z_1,\ldots,z_n\in \mathbb{C}$ with $\max_i
 - [Bir00b] Biró, A. *An improved estimate in a power sum problem of Tur\'{a}n.* Indag. Math. (N.S.) **11** (2000), no. 3, 343-358.
 - [CG96] A. Y. Cheer and D. A. Goldston *Tur\'{a}n's pure power sum problem.* Math. Comp. **65** (1996), no. 215, 1349-1358.
 - [Gri26] Griego, S. *An improved asymptotic certificate for Turan's pure power sum constant $C_{42}$.* GitHub repository, version `v1.0.0`, commit `c8ddce14d9a5e898406d5dc6b8d08bb8a39507c7` (2026). https://github.com/sebastian-griego/turan-c42-certificate/tree/v1.0.0
+- [R2026a] André Röhrig. *An eight-step profile for Turán's pure power-sum problem.* 9 September 2026. [Proof, exact inputs and verification](https://github.com/AndreRoehrig/optimization-constants-certificates/blob/ed6387399cf6de02eaa4cf10c43787633496619a/notes/turan.md), version `v1.0.2`, commit `ed6387399cf6de02eaa4cf10c43787633496619a`. Proposed computer-assisted result; not peer reviewed.
 
 ## Contribution notes
 
 The [Gri26] certificate repository and this update were prepared with AI assistance for formatting, exposition, and verification scripts. The mathematical claim, constants, references, and computations are provided for independent review.
+
+The [R2026a] construction, verification programs and exposition were developed with AI assistance using Codex. André Röhrig reviewed the mathematical arguments, references and supporting verification material before submission. The verification commands were executed and checked during preparation. The new row retains the asterisk for the site's minimal-verification category.
diff --git a/constants/51a.md b/constants/51a.md
@@ -32,6 +32,7 @@ We define $C_{51} = B$ to be the supremum of $\beta(f)$ over all transcendental
 | $\frac{4}{7}\approx 0.5714285714$ | [CH1964] | Scaling-identity construction. |
 | $0.5850724$ | [HT2026] | Certified (computer-assisted) improvement using a two-parameter generalization of the Clunie–Hayman construction. |
 | $0.5850788$ | [S2026] | Used GPT 5.2.  Discussed [here](https://www.erdosproblems.com/forum/thread/513#post-4532) |
+| $0.5850788196744$\* | [R2026b] | Exact parameters $K=3.5681854310698435$, $\alpha=3.9615431126097391$, $\varepsilon=e^{i\alpha}$ in the family of [HT2026], Theorem 2.8. Truncation through $n=9$ reduces the squared modulus to a degree-19 polynomial in $x=\cos^2\theta$. Rational Bernstein subdivision covers $[0,1]$ with 19 leaves (37 nodes, depth 18); adding the uniform infinite-series tail gives $\max\_\theta\lvert P(\theta)\rvert\le1.709171425067816128872617735945$. Its reciprocal exceeds the recorded bound. The full [S2026] paper bound is $0.585078819653$; the gain over a certified upper bound for that paper's actual specified function is greater than $2.1047\times10^{-11}$. [Proof and verification, pinned commit](https://github.com/AndreRoehrig/optimization-constants-certificates/blob/ed6387399cf6de02eaa4cf10c43787633496619a/notes/maximum_term.md). |
 
 
 ## Additional comments
@@ -54,6 +55,22 @@ We define $C_{51} = B$ to be the supremum of $\beta(f)$ over all transcendental
 
 - This problem is catalogued as **Erdős Problem #513** (see [EP513]).
 
+For [R2026b], let $T\_n=n(n+1)/2$. The [HT2026] identity is
+
+$$
+\beta(f_{K,\varepsilon})=\frac1{\max_\theta\lvert P(\theta)\rvert},\qquad
+P(\theta)=2\sum_{n=0}^\infty K^{-T_n}e^{i\alpha T_n}\cos((2n+1)\theta).
+$$
+
+The tail after $n=9$ is at most $2K^{-55}/(1-K^{-11})$. To reproduce the polynomial, set $R\_0(x)=1$, $R\_1(x)=4x-3$ and $R\_{n+1}(x)=(4x-2)R\_n(x)-R\_{n-1}(x)$, and bound
+
+$$
+Q(x)=4x\left\lvert\sum_{n=0}^9 K^{-T_n}e^{i\alpha T_n}R_n(x)\right\rvert^2
+\quad(0\le x\le1).
+$$
+
+The interval calculation and exact rational subdivision certify a reciprocal lower endpoint of $0.585078819674464320841496314106$, above the shorter bound recorded in the table. With the previous exact parameters $K=3.568182317714$ and $\alpha=3.961543335688$, evaluation at $\theta=0$ and the same tail estimate give $\beta(f\_{\mathrm{old}})<0.585078819653352950854632621542$. Thus the improvement over [S2026] is a change of parameters, not merely additional displayed digits. Both 100- and 200-decimal-place verification runs pass.
+
 ## References
 
 - [EP513] Bloom, T. F. *Erdős Problem #513.* https://www.erdosproblems.com/513 (accessed 2026-02-13).
@@ -64,7 +81,10 @@ We define $C_{51} = B$ to be the supremum of $\beta(f)$ over all transcendental
 - [HT2026] He, Yixin; Tang, Quanyu. *Generalizing the Clunie–Hayman construction in an Erdős maximum-term problem.* 2026. arXiv:2602.12217. https://arxiv.org/abs/2602.12217
 - [HTcode] He–Tang certification code repository (linked from [HT2026]): https://github.com/QuanyuTang/ep513-arb-certification
 - [S2026] Sothanaphan, Nat. *A certified computation for an improved He–Tang parameter choice in Erdős’ maximum-term problem.* 2026. https://drive.google.com/file/d/1wZnzui_eeBE32HnkrnSB7YhfcTOiYolp/view
+- [R2026b] André Röhrig. *A small parameter improvement for the Erdős maximum-term constant.* 9 September 2026. [Proof, exact inputs and verification](https://github.com/AndreRoehrig/optimization-constants-certificates/blob/ed6387399cf6de02eaa4cf10c43787633496619a/notes/maximum_term.md), version `v1.0.2`, commit `ed6387399cf6de02eaa4cf10c43787633496619a`. Proposed computer-assisted result; not peer reviewed.
 
 ## Contribution notes
 
-ChatGPT Pro was used to generate the initial version of this page.
+ChatGPT Pro was used to generate the initial version of this page.
+
+The [R2026b] parameter improvement, verification programs and exposition were developed with AI assistance using Codex. André Röhrig reviewed the mathematical arguments, references and supporting verification material before submission. The verification commands were executed and checked during preparation. The new row retains the asterisk for the site's minimal-verification category.