# Governing Twin referee: proposed corrections and local Bockstein realization

Audit date: 2026-09-08. This note reviews the lead's proposed mathematics; it does not independently replay the missing 390-dimensional Manin-symbol computation or the canonical Kurihara value. Verdicts concern the exact statements below.

## 1. Canonical modular-symbol primitivity: conditional repair

**CONCUR**, with explicit hypotheses. Let $\phi:X_0(N)\to E$ be an optimal modular parametrization, $p$ odd, $p\nmid c_\phi$, and assume the Néron-normalized plus modular-symbol functional is integral over $\mathbb Z_p$ on the relative modular-symbol lattice under consideration. Then it is nonzero modulo $p$.

Reason: the induced quotient $J_0(N)\to E$ has connected kernel, hence is surjective on integral first homology. Abel–Jacobi identifies the first homology of $X_0(N)$ with that of $J_0(N)$. Surjectivity persists over $\mathbb Z_p$ and, because $2$ is invertible, on the plus summands for complex conjugation. A plus cycle mapping to a generator of $H_1(E,\mathbb Z_p)^+$ has normalized integral equal to a $p$-adic unit, since $\phi^*\omega_E=c_\phi\omega_f$. Changing from the least positive real period to the integral over all real components introduces only a factor of one or two. The restriction of the relative functional to absolute cycles is therefore primitive.

This corrects the invalid inference “nonzero modulo varying $p^k$ implies nonzero modulo $p$.” It requires no BSD statement, no Sha finiteness, and no Iwasawa main conjecture. Integrality and optimality are retained as hypotheses and must be sourced/checked in the actual application. This lemma alone does not verify the computed eigenspace or $\delta_{393427}=6$.

## 2. Norm quotient at an auxiliary good prime

**CONCUR.** Let $K/\mathbb Q_\ell$ be a local field, $\ell\ne p$, let $E/K$ have good reduction, and let $L/K$ be a totally ramified cyclic extension of degree $p$. Write $\widetilde E$ for the common residue curve. Then

$$
E(K)/N_{L/K}E(L)\simeq\widetilde E(k_K)/p\widetilde E(k_K).
$$

Proof: reduction commutes with the norm, and total ramification makes the residue norm multiplication by $p$. The kernel of reduction $E_1(K)$ is uniquely $p$-divisible, because multiplication by $p$ has invertible linear coefficient in the formal group. If $z\in E_1(K)$, choose $y\in E_1(K)$ with $py=z$; treating $y$ as an $L$-point gives $N(y)=z$. Conversely, if the reduction of $x\in E(K)$ belongs to $p\widetilde E(k_K)$, lift a preimage to $y\in E(K)$. Then $x-N(y)\in E_1(K)$, hence also lies in the norm image. This proves the stated kernel and quotient. No formal logarithm on all of $E(K)$ and no Sha input are needed.

In this setting $N E(L)=pE(K)$, since both have the same image under reduction and contain $E_1(K)$. Via Kummer and good reduction the quotient also identifies with $H^1_{\mathrm{ur}}(K,E[p])$.

## 3. Local first-order Bockstein

**CONCUR**, allowing an overall sign fixed by cocycle conventions. Let $V$ be an unramified $\mathbb F_p$-representation of $G_K$, where $K=\mathbb Q_\ell$, $\ell\ne p$ and $\ell\equiv1\pmod p$. Fix a nonzero tame additive character $\psi:G_K\to\mathbb F_p$. The deformation

$$
\rho_\varepsilon(g)=\rho(g)(1+\varepsilon\psi(g)),\qquad\varepsilon^2=0,
$$

gives a connecting morphism $\beta_\psi:H^1(K,V)\to H^2(K,V)$, namely cup product with $\psi$ (up to sign if inverse action is used). If $F$ denotes arithmetic Frobenius, then

$$
H^1_{\mathrm{ur}}(K,V)=V/(F-1)V,
\qquad
H^2(K,V)=V/(\ell^{-1}F-1)V,
$$

after choosing a basis of tame $p$-inertia. The second equality is Hochschild–Serre: $H^1(I_K,V)=V(-1)$, while the unramified quotient has $p$-cohomological dimension one. Since $\ell=1$ in $\mathbb F_p$, the quotients coincide. The cup product on the $H^1_{\mathrm{ur}}$ summand is multiplication by the nonzero value of $\psi$ on the chosen tame generator, hence is an isomorphism. This remains true for a nontrivial unipotent $F$.

Adding an unramified character to $\psi$ does not change this restricted map: the extra cup product of two inflated unramified classes lies in a group of cohomological degree two for a quotient of cohomological dimension one. Scaling $\psi$ scales the map; the numerical normalization is not canonical without fixing that choice.

## 4. Actual global ambient-Galois realization

**CONCUR.** Let $F/\mathbb Q$ be the compositum of the unique degree-11 subfields of $\mathbb Q(\zeta_{397})$ and $\mathbb Q(\zeta_{991})$. Both are real because their degree is odd; their conductors are coprime, so

$$
G=\operatorname{Gal}(F/\mathbb Q)\simeq C_{11}\times C_{11}.
$$

Take $k=\mathbb F_{11}$, $A=k[G]$, augmentation ideal $J$, and $V=E[11]$ for $E=389.a1$. For $\Sigma$ containing $11,389,397,991$ and the archimedean place, the actual quotient character $G_{\mathbb Q,\Sigma}\to G$ gives the exact sequence

$$
0\to V\otimes J/J^2\to V\otimes A/J^2\to V\to0,
$$

using the tautological group-ring action (or its inverse with signs tracked). Its connecting map is an actual global Galois Bockstein

$$
\beta_\Sigma:H^1(G_{\mathbb Q,\Sigma},V)
\longrightarrow H^2(G_{\mathbb Q,\Sigma},V)\otimes J/J^2.
$$

Localization commutes with this connecting map. At 397, the 991-character is unramified and its cup product with an unramified Kummer class vanishes; at 991 the 397-character similarly contributes zero. Each own-prime character is nonzero tame and contributes the isomorphism in section 3. Thus, given the verified local rows $(1,2)$ and $(1,4)$ on $U=kP\oplus kQ$, and choosing the local cohomology basis $e_\ell=\beta_{\psi_\ell}(\operatorname{loc}_\ell P)$, the selected two-place localization of $\beta_\Sigma|_U$ has matrix

$$
\begin{pmatrix}X_{397}&2X_{397}\\X_{991}&4X_{991}\end{pmatrix}.
$$

Its determinant is

$$
2(e_{397}\wedge e_{991})\otimes X_{397}X_{991}.
$$

The word **selected** matters: this is a determinant of the two-place projection, not a determinant of the entire global Bockstein target, which may have other local components. Changing generators changes coordinates and cup-character normalizations compatibly. The invariant is the nonzero element in the specified tensor line, not an absolute scalar 2 in unspecified bases.

This adds real provenance from the actual cyclotomic extension. It does not yet identify a Selmer-complex local-condition cone, a BKS regulator, a Nekovář height, or a complex leading term. Those comparison edges remain open. It is reasonable to mark a precisely factored `LOCAL-BOC-REALIZATION` sub-obligation closed, but not `CANON-BocID` as a whole.

## 5. The residual norm kernel and Sha

**CONCUR.** Let

$$
0\to U\to S\xrightarrow{q}\Sha(E/\mathbb Q)[p]\to0
$$

be the Kummer exact sequence, with $U=E(\mathbb Q)/pE(\mathbb Q)$ and $S=\operatorname{Sel}_p(E/\mathbb Q)$. Let $\lambda:S\to W$ be the localization/norm map, and suppose $\lambda|_U:U\to W$ is an isomorphism. Then restriction of $q$ gives a canonical isomorphism

$$
\ker\lambda\xrightarrow{\sim}\Sha(E/\mathbb Q)[p].
$$

Proof: injectivity follows from $U\cap\ker\lambda=0$. Given a lift $s\in S$ of a Sha class, subtract the unique element $(\lambda|_U)^{-1}\lambda(s)$ of $U$; the result lies in $\ker\lambda$ and has the same image in Sha. Neither finiteness of full Sha nor any Selmer-dimension assumption is used.

Consequently, invertibility of the two-row Mordell–Weil localization matrix by itself does not prove $\Sha[p]=0$. It isolates exactly the remaining Selmer kernel. This is a useful no-go against treating Mordell–Weil detection as full Selmer detection.

## 6. Arbitrary dual-number deformations do not determine an arithmetic Bockstein

**CONCUR** on the algebraic countermodel, with its domain stated precisely. For any linear map $B:U\to W$ and $R=k[\varepsilon]/\varepsilon^2$, form the two-term complex

$$
C_B=[U\otimes R\xrightarrow{\varepsilon B}W\otimes R]
$$

in degrees one and two. Every $C_B$ has the same reduction $[U\xrightarrow0 W]$ modulo $\varepsilon$, but its connecting map is $B$ (up to the chosen differential sign convention). In particular $B=0$ and an invertible $B$ have identical residual complexes and different Bocksteins.

Thus residual vector spaces, even together with a displayed invertible matrix, do not force a canonical arithmetic deformation. This does not refute section 4, where the actual global extension and tautological action have now been specified. It also does not prove that every arbitrary $C_B$ comes from a Selmer complex. It establishes only the exact insufficiency of the residual-data-only inference.

## 7. A literature-backed repair of canonical mod-11 primitivity

**CONCUR.** The following repairs the invalid use of Kim Corollary 1.6 to pass from nonzero quantities modulo varying powers of 11 to a nonzero canonical symbol modulo 11. It proves existence of a primitive canonical symbol; it does **not** reproduce the missing 390-dimensional eigenspace computation or certify the displayed value $\delta_\lambda=6$.

### 7.1 Exact elementary hypotheses for the curve

For $E:y^2+y=x^3+x^2-2x$, independently recomputed integer invariants are

$$
(b_2,b_4,b_6,b_8)=(4,-4,1,-3),\qquad
\Delta=389,\qquad c_4=112,\qquad j=1404928/389.
$$

The integral equation is minimal everywhere: its discriminant valuation is zero away from 389 and one at 389. At 389, $c_4$ is a unit, so the reduction is multiplicative of type $I_1$. Hence $N=389$ and $\operatorname{Tam}_E=c_{389}=1$. The nonintegral rational $j$ excludes complex multiplication.

Direct point enumeration gives $\#E(\mathbb F_2)=5$ and $\#E(\mathbb F_{11})=16$, hence $a_2=-2$ and $a_{11}=-4$. In particular 11 is a prime of good ordinary reduction. The Frobenius-at-2 polynomial modulo 11 is $T^2+2T+2$; its discriminant is $7$, whereas the squares modulo 11 are $\{0,1,3,4,5,9\}$. Thus $E[11]$ is irreducible as an $\mathbb F_{11}[G_\mathbb Q]$-module: a global invariant line would be invariant under this Frobenius and force a root in $\mathbb F_{11}$.

At the multiplicative prime 389, valuation one of the Tate parameter gives a nontrivial transvection in mod-11 inertia. The possible nonsplit quadratic twist is unramified and does not alter this inertia conclusion. Let $H$ be the residual image and $L$ the fixed line of this transvection. Irreducibility gives $g\in H$ with $gL\ne L$. In a basis of the two lines, the transvection and its conjugate are respectively nontrivial upper and lower unipotent matrices. Their powers give the complete upper and lower root groups, since the coefficient field is the prime field $\mathbb F_{11}$. These groups generate $\mathrm{SL}_2(\mathbb F_{11})$. The determinant is the surjective mod-11 cyclotomic character; therefore $H=\mathrm{GL}_2(\mathbb F_{11})$.

### 7.2 The Manin hypothesis can be supplied without assuming the database label is optimal

Castella–Sano fix a modular parametrization $\varphi:X_0(N)\to E$ and assume $11\nmid c_\varphi$. Their PDF, p. 2, cites Mazur, Corollary 4.1, for this property for the optimal strong Weil parametrization when $11^2\nmid N$. Kim, §1.4.1, likewise treats the Manin hypothesis as vacuous at a semistable prime. For a fully explicit bridge to this particular equation, let $E_0$ be its optimal isogenous curve. A minimum-degree rational isogeny $E\to E_0$ has degree prime to 11: otherwise its kernel intersects $E[11]$ nontrivially; irreducibility forces the intersection to be all of $E[11]$, and factoring through $[11]$ gives a smaller degree. The dual isogeny $E_0\to E$ also has degree prime to 11. At good reduction at 11, both isogenies act integrally on Néron differentials and their differential multipliers multiply to a degree prime to 11. Each multiplier is therefore an 11-adic unit. Composing the optimal parametrization with this isogeny gives the required $\varphi$ with $11\nmid c_\varphi$.

This argument needs only modularity and the standard optimal quotient and Néron differential facts; it does not require the unprovided numerical period normalization or the disputed Kurihara certificate.

### 7.3 Exact theorem application and its scope

Primary source: Francesc Castella and Takamichi Sano, *On refined nonvanishing conjectures by Kurihara and Kolyvagin*, [arXiv:2601.14504v1, PDF pp. 1–3](https://arxiv.org/pdf/2601.14504v1), [HTML §§1.1.1–1.1.4](https://arxiv.org/html/2601.14504v1). The arXiv v1 identifier is dated 20 January 2026; the PDF prints 22 January 2026. The dynamically rendered HTML currently prints a different internal date; the theorem text checked here agrees with the fixed v1 PDF. Local original and extracted text: `work/twin_sources/cs_v1.pdf`, `cs_v1.txt`, `cs_v1.html`.

The standing hypotheses in §1.1.1 are that $E/\mathbb Q$ is non-CM and the mod-$p$ representation is surjective. Page 2 fixes the positive total real Néron period, the parametrization, the Manin-unit hypothesis, and the $p$-integral canonical modular symbols. Conjecture A asserts that, under these hypotheses and $p\nmid\operatorname{Tam}_E$, some canonical $\overline\delta_n$ is nonzero modulo $p$. Conjecture B, for $p>3$, states $M_\infty(\delta)=\operatorname{ord}_p(\operatorname{Tam}_E)$. **Theorem B, p. 3, proves Conjecture B (and hence Conjecture A) for good ordinary $p$ under the stated surjectivity and Manin hypotheses.** Its other branch, good supersingular with squarefree conductor, is unnecessary here. There is no analytic-rank, Sha-finiteness, or local-$p$-torsion hypothesis in this theorem's ordinary branch.

All these hypotheses are supplied above for $(E,p)=(389.a1,11)$. Since $\operatorname{Tam}_E=1$, the theorem gives $M_\infty=0$. The proof of Corollary A on the same page explicitly identifies this with existence of a $p$-indivisible $\delta_n$. Thus some canonical $\overline\delta_n\ne0$, so the canonical plus modular-symbol functional is nonzero modulo 11. A finite linear combination of zero symbols could not give this nonzero quantity. This uses a theorem giving **primitivity**, which the earlier nonvanishing-modulo-$I_n$ argument lacked.

If the missing finite computation really establishes that the relevant plus Hecke eigenspace on the stated integral lattice has dimension one modulo 11, and produces $\lambda$ with $\overline\delta_\lambda=6$, then both $\lambda$ and the canonical functional lie in this same one-dimensional eigenspace and are nonzero. Their proportionality scalar is therefore nonzero modulo 11 (an 11-adic unit when lifted in the claimed integral proportionality). This makes the **normalization bridge valid conditional on those finite data**. The original eigenspace dimension, matching lattice/Hecke conventions, and $\delta_\lambda=6$ have not been replayed, so the particular finite Kurihara certificate remains **UNVERIFIED**. The theorem supplies no identified conductor $n$ and no numerical value 6.

Kim comparison source: [arXiv:2203.12159v6, §1.4.1 and Corollary 1.6](https://arxiv.org/html/2203.12159v6). The source's positive-period convention agrees with Castella–Sano's real integral definition (and any explicitly tracked factor 2 in another plus convention is an 11-adic unit). The old inference should be replaced rather than retained as an alternative proof.