arXiv:2611.05420v1 Announce Type: new Abstract: Bloch's identity $\int_\gamma\eta(f,g)=D_E((f)\diamond(g))$, relating the regulator of a $K_2$-symbol on an elliptic curve $E/\mathbb{Q}$ to the elliptic dilogarithm of the diamond product of the divisors, is transmitted in the Mahler-measure literature with $\gamma$ generating the anti-invariant subgroup $H_1(E,\mathbb{Z})^-$ and factor $1$ (e.g. Lalin-Ramamonjisoa, Thm. 6). We observe that, when $E(\mathbb{R})$ has one connected component ($\Delta<0$), this reading is incompatible with the published intersection-product normalization (Brunault, Proposition 19): the anti-invariant subgroup then has index $2$ in the regulator quotient $H_1(E,\mathbb{Z})/H_1(E,\mathbb{Z})^+$. The correct statement -- factor $1$ for the quotient generator, hence factor $\iota(E)\in\{1,2\}$ for the subgroup generator, with $\iota(E)=1\iff\Delta>0$ -- follows from Brunault's formula by an elementary index computation. We document the phenomenon twice: on 11.a3 by proved values (a Siegel-unit anchor from [Zheng, arXiv:2609.26119]; the diamond side by Brunault and Bertin), and on 20.a4 by high-precision computation of both integrals against the $q$-series for $D_E$ (subgroup/quotient ratio $2$ to $13$ digits). We also show that Touafek's Theorem 2 (involving a $\Delta<0$ curve) mixes two independent normalizations (subgroup versus quotient, and a $(2\pi)^{-1}$ prefactor versus the bare diamond value), and we identify the two-part reinterpretation making its printed formulas mutually compatible. In the cases we have checked, no Mahler-measure or $D_E$-identity is affected: the main theorem of Lalin-Ramamonjisoa (conductor $17$, $\Delta>0$) and Bertin's proved exotic relations stand untouched, and Touafek's final $D_E$-relation is not contradicted and is numerically supported, though its printed derivation retains the gap described in the text. A systematic audit is beyond the scope of this note.
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