# Are all relations among F_q[t] multizeta values consequences of Thakur's shuffle relations? Date: 2026-10-23, America/New_York. Written by an AI agent (claude-opus-5-5, Claude Code fork) for Taylor. Saved to disk by the parent agent because the fork harness blocked the fork's own write. AI-generated research; no human or formal review. The fork's notification said it was still waiting on its own background work, so this report may be superseded. ## Verdict The statement as posed is **false**, and it is not a published Thakur conjecture. The corrected statement is a **theorem**: > Every $\bar K$-algebraic relation among Thakur's MZVs follows from > (i) the $\mathbb F_p$ q-shuffle product formula and (ii) **one** weight-$q$ > $K$-linear relation, the fundamental binary relation > $$R_1:\ S_d(q)+D_1S_{d+1}(1,q-1)=0\ \ \forall d,\quad\text{i.e.}\quad \zeta(q)+D_1\zeta(1,q-1)=0,\qquad D_1=\theta^q-\theta,$$ > propagated by Todd's power-sum operators $B_U$, $C_V$ and $B_U\circ C_V$ > (or equally $B^*_U$ in place of $B_U$). This combines three results: - Chang, Compositio 2014, Thm 2.2.1: the $\bar K$-algebra of MZVs is graded by weight and defined over $K$. - Chang–Chen–Mishiba (CCM), arXiv 2205.09929, Forum Math. Pi 2023, Thm 5.4.2: the $B^*$ version (refereed). - Hu, arXiv 2609.08263 v2, 2026-10-05: the full Todd relation conjecture (Thm 1.5), **unrefereed**. ## Refutations and corrected expectations **N1 (proved here, for every $q$).** It was natural to expect that shuffle relations generate all relations, perhaps together with Frobenius and Carlitz's Eulerian evaluations $\zeta(w)\in K\tilde\pi^w$ for $(q-1)\mid w$. In fact $R_1$ is not in the ideal $I$ generated by these. Proof: 1. All product and Frobenius generators have coefficients in $\mathbb F_p$. 2. For $q\ge3$, every Eulerian generator has weight at least $2(q-1)>q$, so none reaches weight $q$. For $q=2$, the only one in weight 2 is $X_1^2-X_2$, since $\zeta(1)^2=\zeta(2)$ in characteristic 2. 3. Hence $I_q\cap V_q=\bar K\otimes(\mathbb F_p\text{-rational relations of weight }q)$. 4. $R_q$ is one-dimensional over $\bar K$, by $\dim\mathcal Z_q=2^{q-1}-1$ (CCM/IKLNDP) and Chang's theorem. So $R_q=\bar K\cdot(X_{(q)}+D_1X_{(1,q-1)})$. 5. This line contains no nonzero $\mathbb F_p$-rational vector, because $D_1\notin\mathbb F_p$. So $I\cap V_q=0\ne R_q$. The proof needs neither associativity of the product nor the formal Frobenius identity. **N2 (mis-attribution).** Thakur (IMRN 2009) says the opposite of the attributed conjecture: - p. 2320: "we do not know a simple description of all identities even conjecturally". - p. 2319: he separates "shuffle-type relations with $\mathbb F_p$-coefficients" from "Euler-type relations with $\mathbb F_p(t)$-coefficients". The only published all-relations conjecture is Todd's Conj. 5.1 (preprint 2017; JNT 187, 2018). It uses a single weight-$q$ relation plus operators, not shuffles alone. **N3 (two meanings of "Eulerian").** The prompt glossed "Eulerian" as the condition $(q-1)\mid w$, i.e. Carlitz's evaluation $\zeta(w)/\tilde\pi^w\in K$. Thakur's "Euler-type" relations are something else: $K$-linear relations within one weight, such as $\zeta(m,m(q-1))=\zeta(mq)/\ell_1^m$ (Thakur 2009, Thm 5(2)). - Under the first reading, the conjecture is false (N1). - Under the second, it is true but circular until the generators are specified. Todd and Hu specify them: $R_1$ alone. **N4 (computational).** It was natural to expect that $\mathbb F_p$-rational linear relations never occur, so that $\mathbb F_p$-coefficients would mark "shuffle-type". In fact, for $q=2$ there are $\mathbb F_2$-rational relations in weights 5, 6 and 7, of dimensions 1, 4 and 13. For example, in weight 5: $$\zeta(1,1,3)+\zeta(1,2,2)+\zeta(2,1,2)+\zeta(3,1,1)+\zeta(4,1)=0.$$ - The partial sums over $d\le D$ have valuations $8,16,\dots,2048$ for $D\le9$. - One weight-7 relation is *fixed*: it holds exactly at every partial sum $d\le D$. It is $(2,1,1,3)+(2,1,2,2)+(2,1,4)+(2,2,1,2)+(2,2,3)+(2,3,1,1)+(2,3,2)+(2,4,1)+(3,1,3)+(3,2,2)$. - None were found for $q=3$ ($w\le6$), $q=4$ ($w\le5$) or $q=5$ ($w\le5$). So $\mathbb F_p$-rationality does not identify shuffle-derived relations. **N5 (the fork's own error, corrected).** A first truncated-series run gave false results for $q=5$, $w=5$: relation rank 3, and an apparent relation "$\zeta(1,1,1,1,1)=0$". There were two causes: - The expansions are sparse modulo $q-1$, because $\theta\mapsto\lambda\theta$ scales $S_d(k)$ by $\lambda^{-dk}$. - Deep MZVs have huge valuation; for example $v(\zeta(1^5))=780$ when $q=5$, so at low precision they look like zero. The precision is now $N\ge\max_s v(\zeta(s))+3(q-1)\cdot\#\text{unknowns}$, and with that the $\mathbb F_5$ artifact is gone. The $q=5$, $w=5$ rank was not re-run because it exceeded the time budget. Carry-forward: there were no prior dump inputs (0 carried, 0 excluded). ## Theorem C (assembled from published results) Let: - $\mathcal P=\bar K[X_s]$, with evaluation $X_s\mapsto\zeta(s)$; - $J$ be the ideal generated by $X_\emptyset-1$ and $X_sX_t-X_{s*_\zeta t}$; - $\mathcal R_w$ be the $K$-span of $B_U(R_1)$, $C_V(R_1)$ and $B_U C_V(R_1)$ in weight $w$. Then $\ker(\mathrm{ev})=J+\sum_w\bar K\mathcal R_w$. The same holds with $B^*$ in place of $B$. Proof: 1. Reduce any polynomial modulo $J$ to $c_0+\sum_w\ell_w$, using CCM Prop 2.3.5(1). 2. Chang's Thm 2.2.1(1) gives $c_0=0$ and $\mathrm{ev}(\ell_w)=0$ for each $w$. 3. Chang's Thm 2.2.1(2) puts each $\ell_w$ in $\bar K\otimes R_w^K$. 4. $R_w^K=\mathcal R_w$, by Hu Thm 1.5, or by CCM Thm 5.4.2 for the $B^*$ version. For $w