Eight citations to the companion paper point to the wrong items
Where the issue occurs
Section 7.1, proof of Lemma 7.2 (Choice of a point) · p. 60
These are the spanning denominators of [23, Lemma 6.4] and the Frobenius separation resultants of [23, Equation (8.11) and Lemma 8.6].
Should cite [23, Equation (8.10) and Lemma 8.7]. In [23], Equation (8.11) is the inertia identity and Lemma 8.6 concerns Hodge–Newton contacts.
Section 7.2, proof of Lemma 7.4 (The two p-operators) · p. 62
by the matched unramified theta calculation of [23, Lemma 4.2 and Equation (4.10)].
The spherical parameter is Equation (4.11) in Lemma 4.2 of [23]; Equation (4.10) there defines the moving-prime pivot test.
Section 7.3, proof of Lemma 7.5 (Bounded cuts of unary moments) · p. 63
For these unary theta functions use the ordinary Hilbert modular Igusa tower with the fixed tame Schwartz level and its multiplicative differential trivializations, as in [23, Lemma 4.4].
Item 4.4 of [23] is a convention for frames; the unary-moment result is Lemma 4.5 and its proof.
Section 7.3, proof of Lemma 7.5 (Bounded cuts of unary moments) · p. 64
These algebraicity and comparison facts are the finite-level facts proved in the unary-moment argument of [23, Lemma 4.4]; they require finiteness of the chosen list, not a uniform bound for its depth.
Should cite the proof of [23, Lemma 4.5]; item 4.4 of [23] is a convention.
Section 7.3, proof of Proposition 7.6 (Integral theta sections) · p. 66
These are the mixed-expansion identities of [23, Lemma 4.5]; the argument above checks the only changing input, the increasing p-depth.
Should cite [23, Section 4.5 and Lemma 4.6], where the mixed Fourier–Jacobi expansions are treated; Lemma 4.5 of [23] is the unary-moment result.
Section 7.5, paragraph on the local functional equations after Equation (7.3) · p. 68
These are the normalizations in [23, Equations (5.8) and (5.10)].
The two functional equations are Equations (5.9) and (5.11) of [23]; Equation (5.10) may be kept for the explicit gamma factor.
Section 7.5, paragraph on the global Siegel–Weil/Rallis identity · p. 68
The global identity is the equal-rank, anisotropic binary Siegel–Weil/Rallis identity [33, 10] combined with the two Waldspurger identities [31], with the specific normalization of [23, Equation (5.6)].
The normalization is Equation (5.7) of [23], the Rallis–Waldspurger period identity; Equation (5.6) there is a cusp-constant formula.
Section 8.3, proof of Lemma 8.3 (Uniform cusp congruence) · p. 76
The differential scalings have lower bounds λ′₃ + λ′₄ for j = 2 and λ′₄ for j = 3, exactly as in [23, Lemma 7.3].
Should cite [23, Lemma 7.4 and Equation (7.2)]; Lemma 7.3 of [23] concerns cusp base change.
Issue & evidence
Several numbered citations to [23], The Selmer converse for elliptic curves at every prime, do not match its October 7, 2026 version. Item 4.4 is a convention, the unary-moment result is Lemma 4.5, and the mixed-expansion result is Lemma 4.6. The spherical-parameter, separation and period-formula citations also contain incorrect equation numbers. The intended passages occur nearby, so this finding concerns citation accuracy and does not establish a mathematical obstruction.
The citation in the proof of Lemma 8.3 has the wrong lemma number. In the October 7, 2026 version of [23], The Selmer converse for elliptic curves at every prime, Lemma 7.3 concerns cusp base change. The stated differential-scaling bounds occur in the proof of Lemma 7.4, pp. 50–51, and the normalized operators are defined in equation (7.2).
Proposed repair by OSVerify
Correct the [23] locators as follows: the citation in the proof of Lemma 7.2 to Equation (8.10) and Lemma 8.7; the citation in the proof of Lemma 7.4 to Lemma 4.2 and Equation (4.11); the two citations in the proof of Lemma 7.5 to the proof of Lemma 4.5, retaining Convention 4.4 when referring to frames; the citation in the proof of Proposition 7.6 to Section 4.5 and Lemma 4.6; the citation after Equation (7.3) to Equations (5.9) and (5.11), with (5.10) optionally retained for the explicit gamma scalar; and the Siegel–Weil/Rallis normalization in Section 7.5 to Equation (5.7) and its Section 5.4 discussion. These changes correct the references to the intended inputs; they do not replace the candidate's separate arguments extending the fixed-level constructions.
Replace the reference to [23, Lemma 7.3] in the proof of Lemma 8.3 with [23, Lemma 7.4] and equation (7.2). Those passages contain the stated scaling bounds and normalization exponents, so this citation correction requires no change to the displayed operator formulas.
Related context
Background References, entry [23]
The companion paper The Selmer converse for elliptic curves at every prime (OpenAI, October 7, 2026), cited as [C] in the report. All corrected locators refer to this version.