Paper Referee

Global quantum geometric Langlands at irrational level

OpenAI preprintReviewed version: October 4, 2026

Original paper

Findings and severity labels come from OSVerify’s internal review and do not necessarily establish that the paper’s main theorem is false.

Major issues 2

The absolute pairing fails to preserve direct sums

Where the issue occurs

Section 4.3, Proposition 4.3 (Whittaker duality), immediately after Equation (4.3) · p. 28

In a pole bound this formula uses the bounded-zero star-extension convention of Proposition 4.2. It then extends continuously to the full pole model.

The issue is the asserted continuous extension of this absolute pairing over the full stated parameter class.

Issue

The absolute pairing (4.3) is asserted to be continuous for the stated parameter class, which includes non-quasi-compact spaces.

Counterexample

Take S to be a countable disjoint union of points with constant marking, H = SL₂, X = ℙ¹, and s = 0. The zero-label stratum is 𝔸¹; the normalized pairing of opposite exponential objects is ℂ.

Let W carry the positive object on every component, and Vₙ the negative object only on component n. All lie in the zero-pole bound. Ordinary global sections give:

B(W, ⨁ₙ Vₙ) = ∏ₙ ℂ

⨁ₙ B(W, Vₙ) = ⨁ₙ ℂ

The canonical comparison from the direct sum to the product is not an isomorphism. Increasing pole bounds does not remove the failure. This tests the stated generality at s = 0; it is not a counterexample at irrational level, and does not challenge the relative pairing or the pairing on Xᴵ.

Proposed repair by OSVerify

For the stated non-quasi-compact parameter generality, distinguish ordinary absolute de Rham sections from a continuous categorical evaluation and justify the construction supporting the asserted duality. Check agreement with (4.3) on quasi-compact parameters and compatibility with the claimed closed-image transposition identities. In the disjoint-union example, continuity requires the direct sum of the componentwise evaluations; that observation alone does not construct the evaluation for general S. The current extension in pole bounds does not supply this parameter-direction argument.

Related context

Background Section 2.1, conventions for functors
Specifies that functors are continuous unless explicitly stated otherwise.
Background Section 4.1, opening description of the parameter scheme S
Gives the parameter generality tested by the counterexample; this is an assumption, not a second erroneous assertion.
Background Section 2.3 and Section 4.2, quasi-compactness and bounded-zero opens
Supplies the safe-pushforward and quasi-compactness conditions used to compare the claimed pairing with its bounded construction.
Background Section 4.3, proof of Proposition 4.3, passage to increasing pole bounds
Explains the extension argument being challenged: increasing pole bounds does not by itself address the parameter-direction failure.
Background Section 8.5, proof of Lemma 8.5, description of the zero-pole bound
Identifies the zero-label model used in the counterexample. This location is supporting evidence, not another reported error.

Constant fibres do not imply a global scalar identification

Where the issue occurs

Section 6.1, proof of Proposition 6.1, final paragraph of the diagonal calculation · p. 45

Its cohomology is nonzero and varies constantly along the diagonal. The composite therefore acts as the identity tensored with a nonzero complex of vector spaces, and is conservative.

The disputed step is the inference from constant fibre cohomology to a global scalar identity. The report does not disprove the required conservativity.

Issue

The global scalar-identity assertion in the finite-averaging argument is too strong: constant fibre cohomology does not eliminate a torsor’s derived extension data.

Counterexample

After the stated cancellations, the diagonal coefficient is E = p_*ω_H×N, where p(h′, n) = h′B⁻, up to fixed normalization shifts. For H = SL₂, the constant-sheaf counterpart is Rπ_*ℂ_SL₂ on ℙ¹ for π : SL₂ → ℙ¹. Here B⁻ is the opposite Borel subgroup and N the unipotent subgroup in the averaging construction.

With complex coefficients and dimensions summed over all degrees, a constant derived coefficient with fibre H^*(B⁻) would give total dimension four. The actual pushforward instead has the cohomology of SL₂, of total dimension two:

dim H^*(ℙ¹) · dim H^*(B⁻) = 2 · 2 = 4

dim H^*(SL₂) = 2

The claimed global scalar identification fails. The report proposes a repair of conservativity using the actual coefficient and descent.

Proposed repair by OSVerify

Keep E=p_*ω_H×N, with p(h',n)=h'B⁻. Trivialize H→H/B⁻ on translates of the big cell. On each covering open U, the coefficient becomes ω_U⊗V, where V=Γ_dR(B⁻×N,ω) is a nonzero finite complex. Over ℂ, V splits into shifted finite-dimensional vector spaces, so tensoring with V detects zero for arbitrary input complexes. Descent therefore makes tensoring with E conservative globally. The same calculation on products with spectator charts preserves spectators. Together with the preceding diagonal-support calculation, this proves conservativity of the composite and hence of finite averaging, supplying the conservativity needed downstream without asserting a global splitting.

Related context

Background Section 6.1, proof of Proposition 6.1, finite-averaging kernel and diagonal calculation
Defines the composite and the coefficient whose fibre cohomology is being computed.
Downstream use Section 6.6, proof of Theorem 6.7, invocation of Proposition 6.1
Uses conservativity of the finite-averaging functor. It is included to explain what the proposed repair must preserve, not as a separate error in Theorem 6.7.

Minor issues 1

Three references point to the wrong equation

Where the issue occurs

Section 10.5, paragraph defining the exponential period, before Proposition 10.6 · p. 92

let ψ₀ be the zero-defect function in (4.1), pulled back to this stack.

The reference should identify the definition of ψ_D in Section 4.2, specialized to D = 0.

Section 10.5, proof of Proposition 10.6, paragraph beginning “Enlarge l once more” · p. 93

Pulling this presentation to the stratum identifies normalized averaging with the conjugated pull after star image along the stratum map q_D in (4.1).

Here the intended references are the definition of q_D and the projector formula (4.2).

Section 10.5, proof of Proposition 10.6, later in the same paragraph · p. 93

The averaging correspondence and the Whittaker stratum projector in (4.1) are thus !-base changes of the same relative quotient presentation.

The projector displayed in (4.2) is the intended target of this third reference.

Issue & evidence

The references to (4.1) in the three passages quoted above are inaccurate: that equation displays finite-loop averaging, whereas these passages refer to the zero-defect phase, the parameter map q_D, and the stratum projector. The intended definitions and projector are present elsewhere in the submission, so this is a nonblocking cross-reference defect.

Proposed repair by OSVerify

Correct the reference in the first quoted passage to the definition of psi_D in Section 4.2, specialized to D=0. In the second quoted passage, refer to the definition of q_D in Section 4.2 and the formula (4.2). In the third quoted passage, replace (4.1) with (4.2). Add labels to the definitions if needed; the associated shift calculation is already given immediately after Equation (4.2).

Related context

Background Section 4.2, definitions of q_D and ψ_D
These are the correct definition targets for the first two citations.
Background Section 4.2, proof of Proposition 4.2, Equations (4.1) and (4.2)
Equation (4.1) is finite-loop averaging; (4.2) is the stratum projector. The formulas are not themselves the errors reported here.