# Independent Twin audit: finite Kurihara certificate

**Verdict: CONCUR.** The independent verifier accepts the finite certificate for $E:y^2+y=x^3+x^2-2x$, $p=11$, $n=397\cdot991=393427$. This note records the Twin's completed replay and mutation audit; the root's subsequent fresh-build gate is a separate check.

## Recomputed result

The fixed generator order is $(1,0),(1,1),\ldots,(1,388),(0,1)$. The normalization is that the first nonzero coordinate equals 1. Its index is 5. Independent computation gives

$$\operatorname{rank}R=325,\quad \dim M=65,\quad
\dim M_f^\vee=2,\quad\dim(M_f^\vee)^+=1.$$

All seven Hecke eigenvalues at $2,3,5,7,13,17,19$ agree with independently counted curve traces. Every one of the **392040** unit-indexed CSV rows was recomputed, including its symbol value and logarithms. The unreduced integer sum is **43605160**, yielding

$$\boxed{\delta_{393427}^{(\lambda)}=5\in\mathbb F_{11}^{\times}.}$$

The archived literal **6** remains **NOT_REPRODUCED_NORMALIZATION_UNSPECIFIED**. The archive supplies no vector or precise normalization rule permitting identification with this vector. The verified replacement is the explicit normalized value 5 and its invariant nonvanishing. No scalar was selected to force 6. The canonical Néron-normalized value is not asserted to equal 5: the established bridge gives $5u$ for an unspecified $u\in\mathbb F_{11}^{\times}$.

## Independence and adverse checks

The verifier imports neither the producer nor NumPy. Its methods differ as follows:

| Object | Producer | Independent verifier |
|---|---|---|
| Linear algebra | NumPy Gauss–Jordan | Scalar sparse elimination and backward substitution |
| Hecke action | Cusp paths and ordinary continued fractions | Positive Merel matrices; exact formal-cusp verification of condition $C_q$ |
| Full symbol sum | Ordinary continued fractions | Balanced Farey parents obtained by modular inverses |
| Logarithms | Full primitive-root walk | Projection to the subgroup of order 11 |
| Point counts | Quadratic-discriminant characters | Multiplicity table for $y^2+y$ |

The verifier checks every saved pivot, RREF, kernel, intermediate constraint matrix, and basis. All seven Hecke operators agree on the entire 65-dimensional quotient, not only on the chosen eigenvector. It also verifies descent through all relations and the $T_2,T_3$ commutator. All 390 generator paths are recovered using Farey parents on the full quotient.

Nine corrupted copies were rejected **after their binding hashes were recomputed**. They alter the vector, rank, RREF, Hecke operator, log table, final residue, an internally consistent CSV symbol, or the normalization. The failure locations are mathematical checks, not hash mismatches. An explicitly declared scale-2 vector with all CSV data recomputed was accepted through the test-only mathematical-equivalence API and gives residue 10. The default fixed-normalization protocol rejects this alternative. The real producer certificate was unchanged.

Evidence: `results/independent_verification.json`, `results/independent_negative_checks.json`. Commands executed successfully:

```text
python3 src/verify_independent.py --results results --report results/independent_verification.json
python3 -m unittest discover -s tests -p test_independent.py -v
```

## Conventions and the bounded theorem consequence

Stein's [Chapter 3](https://wstein.org/books/modform/modform/weight_two.html) supplies the bottom-row Manin presentation and cusp Hecke action. [Chapter 8](https://wstein.org/books/modform/modform/_sources/modular_symbols.txt), `eqn:tnmanin` and `prop:heilbronn`, supplies the independent Merel formula. **Stein's star involution includes an extra minus sign.** Our plus condition is instead geometric conjugation, $\lambda(c,d)=\lambda(-c,d)$, appropriate to the real part of $2\pi i\int f$. This matches [Kim v6, §§1.4.1–1.4.3](https://arxiv.org/html/2203.12159v6).

**CONCUR on the proposed compact consequence chain, using the previously established canonical-primitivity and residual/Manin inputs.** The primitive canonical plus functional belongs to the now verified one-dimensional eigenspace, hence its conductor-$n$ value is $5u\ne0$. Here $I_n=(11)$, since $v_{11}(396)=v_{11}(990)=1$. The previously replayed localization determinant 2 proves independence of $P,Q$ modulo 11; good reduction and $\#E(\mathbb F_2)=5$ rule out rational 11-torsion, giving Mordell–Weil rank at least 2.

[Kim v6, Theorem 1.8(1),(3)](https://arxiv.org/html/2203.12159v6) applies without assuming Sha finiteness. Its corank formula and the unit witness give rank = Selmer corank = collection order = 2 and $\partial^{(2)}=\partial^{(\infty)}=0$. Thus the $11^\infty$-Selmer group is divisible of corank 2. The Kummer injection of $E(\mathbb Q)\otimes\mathbb Q_{11}/\mathbb Z_{11}$ has divisible quotient of corank zero, so that quotient is zero. Consequently $\Sha[11^\infty]=0$, and the full mod-11 Selmer group equals $E(\mathbb Q)/11=\langle P,Q\rangle$. The norm-kernel dimensions $2,1,1,0$ follow from the two rows.

This uses Theorem 1.8(1),(3), **not** its statements (4)–(6), which separately assume Sha finiteness. It establishes no Mazur–Tate initial-form identity, Selmer regulator comparison, or complex leading-term formula. `CANON-BocID` and `CPLX-GPR` remain open.