{"id":"05e62c33-1348-415a-9210-d0bcf174e11c","arxiv_id":"2608.04207","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The author establishes three equivalences of factorization module categories: the Iwahori-ramified versions of the Arkhipov-Bezrukavnikov, Bezrukavnikov, and Fundamental Local Equivalences.","lead":"This paper builds local building blocks for a tamely ramified version of the geometric Langlands correspondence, a recent major proof in mathematics. It lifts three known pointwise equivalences to a setting where points move and collide, a necessary upgrade for the global Iwahori-level statement.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unpublished fusability foundations in [BCG26] define the target of Theorem 0.2.8 and drive the pointwise reductions; until they appear, the three theorems are conditional, not independently verified.","rationale":"The reader's CONDITIONAL verdict is appropriate. My stress-test confirms the weakest assumption identified by the reader and sharpens it: the unpublished fusability theory enters not only in the final reduction of a constructed equivalence to its pointwise fiber, but already in the definition of the central spectral object Aff^spec_Ĝ (Definition 3.1.9 via Proposition 3.1.8 from [BCG26]) and in the constructions of AB and B via the QLisse lifting principle (2.1.7). Therefore a failure or modification of the unpublished foundations would affect the statements of Theorems 0.2.5, 0.2.8 and 0.2.11 themselves, not merely the proofs. I agree with the reader that this is the most load-bearing concern. The paper does contain substantial concrete material that would survive the appearance of the missing foundations: the six factorization module categories are explicitly defined, Theorem 1.5.6 gives a plausible approximation mechanism, and the pointwise character computation in Proposition 4.4.3 is a concrete adaption of [FG10]. The new notion of Iwahori-Hecke temperedness in Section 4.3 is argued internally, and the proof of Proposition 4.4.7 does not obviously circle back to the unproven pointwise equivalence because the temperedness of IKL(G)_{crit,x0} is deduced from the equivalence of the top arrow in the diagram of 4.4.1 together with conservativity of tensoring with QCoh(ň/Ȼ)_{x0} on tempered modules. The main unresolved risk remains the cluster of unpublished results in [BCG26] and [CF]; until those appear and are checked in the exact forms cited, the three theorems should be regarded as conditionally established rather than complete.","tokens_in":31829,"tokens_out":13523,"duration_ms":114698,"concrete_test":"When [BCG26] is available, verify the instance of Proposition 3.1.8 used in Definition 3.1.9: Fact_inertia(IndCoh((ň/Ȼ)_{x0} ×_{ĝ/Ĝ} (ň/Ȼ)_{x0})) should equal IndCoh^*(ň/Ȼ_{Ran_{x0}} ×_{(LS^mer_Ĝ)_{Ran_{x0}}} ň/Ȼ_{Ran_{x0}}) as a factorization Sph^spec_Ĝ-module, with convolution action as in 3.2.3. If extra hypotheses appear, re-check Theorems 0.2.8 and 0.2.11 under them, and confirm the fiber at x0 is IndCoh(ň/Ȼ ×_{ĝ/Ĝ} ň/Ȼ) with B_{x0} the classical Bezrukavnikov equivalence. Also apply the combinatorial criterion of Remark 2.1.6 to Whit!(FlG), since a failure there invalidates Theorem 0.2.5.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central reduction strategy—checking an equivalence of factorization module categories at the single point x0—is legitimate only if the unpublished theory of fusability from [BCG26] holds in the precise forms used here. The dependence is more structural than a mere verification step. Definition 3.1.9 defines the spectral affine Hecke category Aff^spec_Ĝ as Fact_inertia(IndCoh((ň/Ȼ)_{x0} ×_{ĝ/Ĝ} (ň/Ȼ)_{x0})), where Fact_inertia is the fully faithful functor whose essential image is characterized in Proposition 3.1.8, a result deferred to [BCG26]. Thus the target object in Theorem 0.2.8 is defined only conditional on an unpublished classification result; a change of hypotheses there changes the statement, not just the proof. The same unpublished input is used to justify that Whit!(FlG) and Aff_G are fusable (Propositions 2.1.9 and 3.1.10, with only sketches), that forming the QLisse subcategory and lifting functors from it is allowed (2.1.7), and that factorization restriction along Rep(Ĝ)→Sph^spec_Ĝ is conservative (Proposition 3.1.3, using the unpublished functor of [CF]). Consequently Theorems 0.2.5, 0.2.8 and 0.2.11 are not presently established as self-contained statements: each depends at a load-bearing point on results that the manuscript itself locates in forthcoming or unpublished work. This is a correctness risk, not merely a presentation issue, because the pointwise equivalences in Sections 2.3, 3.3 and 4.4 are only promoted to equivalences of pairs through these exact unpublished mechanisms.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs three tamely ramified local equivalences of factorization module categories at a fixed point x0 of a smooth projective curve X. The first, AB, is claimed to be a factorization version of the Arkhipov–Bezukravnikov equivalence between (Whit!(G), Whit!(FlG)) and (Rep(Ǧ), QCoh(ň/B̌)). The second, B, is claimed to be a monoidal equivalence between (SphG, AffG) and (Sph^{spec}_{Ǧ}, Aff^{spec}_{Ǧ}). The third, IFLE, is claimed to be an Iwahori-ramified factorization version of the Fundamental Local Equivalence between (KL(G)_crit, IKL(G)_crit) and (IndCoh*(Op^{mon-free}_{Ǧ}), IndCoh*(Op^{mer}_{Ǧ} ×_{LS^{mer}_{Ǧ}} ň/B̌)). The central technical strategy is to reduce each equivalence of factorization module categories to a pointwise equivalence at the single point x0, using the theory of fusability for factorization Rep(Ǧ)-module categories and a factorization restriction 2-functor. The paper also proves an approximation theorem for pairs of factorization categories and module categories, used in constructing B and IFLE.","tokens_in":32197,"tokens_out":3969,"duration_ms":37913,"significance":"If all three theorems are established, the paper would provide the expected spectral descriptions for the Iwahori-ramified local categories that appear in the tamely ramified geometric Langlands program, and would yield compatibility with the relevant Hecke actions. The paper is well-structured and the overall strategy is natural and ambitious: it upgrades known pointwise equivalences to factorization module categorical ones, and it gives explicit constructions of the six pair objects involved. Credit is due for the detailed setup of the factorization module categories, for the explicit reduction strategy, and for engaging with the recent literature on the geometric Langlands conjecture. However, the central claims are currently conditional on substantial unpublished work: the fusability theory of [BCG26] and the factorization restriction functor of [CF]. Because these inputs are used not merely as citations but as load-bearing components in the definitions and reduction steps, the theorems are not presently established as self-contained statements.","major_comments":[{"comment":"The proof of Theorem 0.2.5 depends at a load-bearing point on the unpublished theory of fusability in [BCG26]. Definition 2.1.5 defines a fusable factorization Rep(Ǧ)-module category as one in the essential image of Fact^{restr}, whose full faithfulness is only asserted in Proposition 2.1.3 as a result of [BCG26]. The reduction in §2.3.1 uses the claim from §2.1.7 that an equivalence between fusable categories can be checked at x0, and Proposition 2.1.9 asserting that Whit!(FlG) is fusable is proven only by a two-sentence sketch. If any hypothesis in the unpublished theory fails or requires modification, the equivalence AB may not follow from the pointwise result. This is a correctness risk, not a presentation issue, because the statement of Theorem 0.2.5 itself is conditional on a not-yet-available classification result.","section":"§2.1 and Theorem 0.2.5"},{"comment":"Theorem 0.2.8 relies on two pieces of unpublished work. First, the conservativity of factorization restriction FactRes along Rep(Ǧ)→Sph^{spec}_{Ǧ} (Proposition 3.1.3) is asserted for the 2-functor FactRes of [CF], which is not available to the reader; the proof's cube diagram uses implicit generation claims that are not fully justified. Second, the target object Aff^{spec}_{Ǧ} is defined in Definition 3.1.9 as Fact_{inertia}(IndCoh((ň/B̌)_{x0} ×_{ǧ/Ǧ} (ň/B̌)_{x0})), where Fact_{inertia} is a fully faithful 2-functor whose essential image is characterized only in Proposition 3.1.8, a result deferred to [BCG26]. Thus a change of hypotheses in the unpublished theory would change the statement of Theorem 0.2.8, not just its proof. The reduction to the pointwise equivalence B_{x0} in §3.3.1 inherits this dependence.","section":"§3.1, Proposition 3.1.3, Definition 3.1.9, Theorem 0.2.8"},{"comment":"Theorem 1.5.6 is an approximation result that is used to construct the morphisms B in §3.2.3 and IFLE in §4.1.2, yet its proof is only a sketch. The H={e} case is declared \"obvious\" without argument, and the general case is reduced to it via a Cech nerve and an identification of totalization categories that is stated without proof. Since Theorem 1.5.6 is load-bearing for the existence of the central morphisms, the proof needs to be written out in detail, at least for the base case H={e} and for the compatibility of the totalization identifications with pullbacks.","section":"§1.5, Theorem 1.5.6"},{"comment":"The pointwise Iwahori Fundamental Local Equivalence is proven in §4.4, but the proof contains a step that is only sketched: in Proposition 4.4.3, the map from O_{Op^{λ-nilp}} to the cohomology H^∞/2(n((t)),n[[t]], M_λ⊗Ψ_0) is said to be injective \"because O is irreducible as a module over N*\", and the character computation is said to follow from [FG10] \"essentially verbatim\" with the finite Weyl module replaced by the finite Verma module. Since the comparison of characters is the core of the essential surjectivity argument, this step should be either proven in detail or accompanied by a precise reference to a result that covers the Verma case.","section":"§4.2.2 and §4.4.3"}],"minor_comments":[{"comment":"The text contains a typo: \"greatful\" should be \"grateful\".","section":"Acknowledgements"},{"comment":"The word \"automorphsim\" is a typo for \"automorphism\".","section":"§2.3.2"},{"comment":"In the proof, \"we agian use\" should be \"we again use\".","section":"Proposition 3.1.10"},{"comment":"The remark about unital versus non-unital Ran space is useful, but the terminology \"Ran without any decorations\" is informal; a fixed notation such as Ran^un would be clearer in later sections where unitality is important.","section":"§1.1.4"},{"comment":"The reference [Ras] is cited as an available PDF without a publication year; if there is a published or arXiv version, it should be cited accordingly.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"This is a serious contribution from a strong group, but the dependence on unpublished work ([BCG26] and [CF]) is more than incidental: it is woven into the definitions and the reduction strategy. If the editors have information that these works are imminent and stable, a conditional acceptance may be appropriate; otherwise the manuscript should be returned with a request to either make the dependencies available, prove the needed statements here, or clearly mark the main theorems as conditional. The gap in Theorem 1.5.6 is also worth attention, as it is used in constructing two of the three equivalences."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Taeuk Nam's paper does something genuinely new: it lifts the three known pointwise Iwahori-ramified equivalences (Arkhipov-Bezrukavnikov, Bezrukavnikov, Fundamental Local Equivalence) to equivalences of factorization module categories at a fixed point x0. That is exactly the local input the tamely ramified global program needs after the unramified five-paper proof, and I do not see these module-category statements in the cited literature. The six pairs are constructed carefully, and the overall architecture—use fusability to reduce an equivalence of factorization modules to the fiber at x0, then invoke pointwise results—is coherent. I also want to credit the author for not hiding the dependencies: sections point explicitly to [BCG26] and [CF], and there are no fitted parameters or circular arguments that I can see.\n\nThe soft spots are real, though, and they are load-bearing. The fusability theory from [BCG26] is not a decoration. It is what defines 'fusable' (Definition 2.1.5), supports the conservativity reduction (Propositions 2.1.3, 3.1.8), and is used to prove Whit!(FlG) and Aff_G are fusable (Propositions 2.1.9, 3.1.10, both sketches). Theorem 0.2.8's spectral target Aff^spec_G is itself defined in 3.1.9 via Fact_inertia, whose essential image is characterized in the unpublished Proposition 3.1.8. So a change of hypotheses there changes the statement, not just the argument. Similarly, Proposition 3.1.3 uses the unpublished factorization restriction functor of [CF]. And Theorem 1.5.6, which is used to construct B and IFLE, is only sketched: the proof says the general case follows from H={e} and pullbacks, without giving the higher-categorical details. I would not call any of these fatal—they are consistent with the program and likely fixable—but as submitted, Theorems 0.2.5, 0.2.8, and 0.2.11 are conditional on sources the reader cannot check.\n\nOne smaller point: Section 2.3.2 admits that ABx0 may not identify with ABclassical. That does not threaten Theorem 0.2.5 since both are equivalences, but it could matter for compatibilities later and should be resolved before publication.\n\nWho is this for? Geometric Langlands people working on the tamely ramified case. A serious referee should engage with it—the strategy deserves referee time—but the report should insist that the author either quote the needed statements from [BCG26] and [CF], or make the paper explicitly conditional and give the exact hypotheses. I would not accept the current arXiv version as final.","headline":"The three factorization-module equivalences are a real step for tamely ramified geometric Langlands, but right now their proofs hinge on unpublished [BCG26] and [CF], so the paper reads as conditional rather than self-contained.","tokens_in":32725,"tokens_out":3031,"would_cite":true,"duration_ms":28087,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14D24","14F10","17B67"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the Arkhipov–Bezukravnikov, Bezrukavnikov, and Fundamental Local Equivalences of geometric Langlands can each be upgraded to an equivalence of factorization module categories with Iwahori ramification at a fixed…","keywords":["geometric Langlands","Iwahori ramification","factorization module categories","Arkhipov–Bezukravnikov equivalence","Bezrukavnikov equivalence","Fundamental Local Equivalence","affine Hecke category","Kazhdan–Lusztig category"],"falsifier":"A decisive check would be to compute the fiber of $\\mathrm{IFLE}$ on a Verma module $M_{\\check{\\lambda}}$: the proof identifies its image with the structure sheaf $O_{\\mathrm{Op}^{\\check{\\lambda}-\\mathrm{nilp}}_{\\check{G},x_0}}$ and its character with $\\prod_{n>0}(1-q)^{-\\ell}$; any mismatch in these characters would falsify Theorem 0.2.11. More broadly, once the cited fusability theory appears, testing its reduction principle on a category in the essential image of $\\mathrm{Fact}^{\\mathrm{restr}}$ whose fiber at $x_0$ is an equivalence but which is not an equivalence on a stratum with extra moving points would falsify the common promotion argument.","tokens_in":31533,"feed_emoji":"📐","tokens_out":11326,"duration_ms":94357,"temperature":0.7,"pith_summary":"This paper's central claim is that three local equivalences at the heart of geometric Langlands—the Arkhipov–Bezukravnikov, Bezrukavnikov, and Fundamental Local Equivalences—survive in factorized form when $G$ is tamely ramified at a fixed point $x_0$. Each is upgraded from a pointwise equivalence at $x_0$ to an equivalence of factorization module categories, a structure that keeps the equivalences compatible as the other points move, collide, and vary in families. The paper constructs these as equivalences of pairs $(\\mathcal{C},\\mathcal{M})$ in $\\mathrm{FactModCat}_{x_0}$: the first component is the unramified factorization category and the second is a factorization module category encoding Iwahori level structure at $x_0$. If the construction is correct, it supplies the local spectral descriptions needed for a tamely ramified, Iwahori-level geometric Langlands correspondence.","feed_headline":"Iwahori ramification gets three factorized Langlands equivalences","feed_subtitle":"All three local spectral equivalences now hold at Iwahori level, with Hecke compatibility as points collide.","key_machinery":"The load-bearing object is the category $\\mathrm{FactModCat}_{x_0}$ of pairs $(\\mathcal{C},\\mathcal{M})$: $\\mathcal{C}$ a factorization category living over the Ran space, and $\\mathcal{M}$ a factorization $\\mathcal{C}$-module category over the Ran space $\\mathrm{Ran}_{x_0}$ of finite subsets that must contain the fixed point $x_0$. The Iwahori subgroup is not factorizable, so $x_0$ is held fixed while other points move and collide with it; factorization module categories are exactly the device that encodes this. The argument runs on three mechanisms: fusability, a condition on factorization $\\mathrm{Rep}(\\check{G})$-module categories that lets a functor be shown an equivalence by checking its fiber at $x_0$; conservativity of factorization restriction along $\\mathrm{Rep}(\\check{G}) \\to \\mathrm{Sph}^{\\mathrm{spec}}_{\\check{G}}$; and Iwahori–Hecke temperedness, a condition on $\\mathrm{Aff}^{\\mathrm{spec}}_{\\check{G},x_0}$-module categories that makes tensoring with the Whittaker affine-flag category conservative. These three levers reduce each theorem to a pointwise equivalence that can be proven by adapted classical arguments.","core_discovery":"The central discovery is that ramification does not break the factorization pattern: the three equivalences hold not only fiberwise but as equivalences of pairs in $\\mathrm{FactModCat}_{x_0}$. Concretely, $\\mathrm{AB}$ identifies the Whittaker-invariant $D$-modules on the affine flag variety, $\\mathrm{Whit}^!(\\mathrm{Fl}_G)$, with quasi-coherent sheaves on the stack $\\check{\\mathfrak{n}}/\\check{B}$; $\\mathrm{B}$ identifies the affine Hecke category $\\mathrm{Aff}_G$ with its spectral counterpart $\\mathrm{Aff}^{\\mathrm{spec}}_{\\check{G}}$; and $\\mathrm{IFLE}$ identifies the Iwahori Kazhdan–Lusztig category at critical level, $\\mathrm{IKL}(G)_{\\mathrm{crit}}$, with $\\mathrm{IndCoh}_*$ of the meromorphic-opers stack fibered over $\\check{\\mathfrak{n}}/\\check{B}$. Theorems 0.2.5, 0.2.8, and 0.2.11 state that these morphisms of pairs are equivalences in $\\mathrm{FactModCat}_{x_0}$, with $\\mathrm{B}$ monoidal and with $\\mathrm{IFLE}$ equivariant with respect to $\\mathrm{B}$. The pointwise fibers at $x_0$ are the classical Arkhipov–Bezukravnikov, Bezrukavnikov, and Iwahori Fundamental Local Equivalences, and the additional content is that these fibers can be promoted to statements about all of $\\mathrm{Ran}_{x_0}$.","pith_inferences":["The same 'check at $x_0$ and invoke fusability' strategy would plausibly extend to other parahoric level structures at a fixed point, provided the relevant module categories can be shown fusable and tempered; the paper itself confines the argument to Iwahori level.","A natural next step, which the paper does not take, is to feed these local equivalences into the compact-generation result for $D$-modules on $\\mathrm{Bun}^I_G$ to construct the global tamely ramified Langlands functor; the paper stops at the local statements.","Because the fusability theory is cited as forthcoming, the most direct stress test of the arguments is to isolate the pointwise-to-global promotion step: if the unpublished criterion required hypotheses not verified for $\\mathrm{Whit}^!(\\mathrm{Fl}_G)$ or $\\mathrm{Aff}_G$, the reductions in all three theorems would need repair."],"forward_implications":["If the three equivalences hold, the local Iwahori-ramified spectral description at a fixed point $x_0$ is fixed: $\\mathrm{IKL}(G)_{\\mathrm{crit},x_0}$ is equivalent to $\\mathrm{IndCoh}_*(\\mathrm{Op}^{\\mathrm{mer}}_{\\check{G}} \\times_{\\mathrm{LS}^{\\mathrm{mer}}_{\\check{G}}} \\check{\\mathfrak{n}}/\\check{B})$, in agreement with the expected right-hand side for Iwahori level structure.","The equivalence $\\mathrm{B}$ transfers every module category over the geometric affine Hecke category $\\mathrm{Aff}_G$ to a module category over its spectral counterpart, so actions used in local geometric Langlands become interchangeable.","Because each statement is an equivalence of pairs in $\\mathrm{FactModCat}_{x_0}$, the equivalences remain compatible as mobile points collide with the fixed point $x_0$, not merely at a single fiber; this is the structure needed to globalize to an Iwahori-level Langlands functor.","The compatibility between $\\mathrm{B}$ and $\\mathrm{IFLE}$ supplies the Iwahori-level counterpart of the standard compatibility between Bezrukavnikov's equivalence and the Fundamental Local Equivalence in the unramified setting."],"supporting_citations":[{"why":"Supplies the unramified factorization categories and equivalences (geometric Casselman–Shalika, Satake, and the Fundamental Local Equivalence) that form the first components of all three pairs.","marker":"[ABC+24a]"},{"why":"Constructs the semi-infinite version of the Arkhipov–Bezukravnikov equivalence and proves the pointwise equivalence used in Theorem 0.2.5.","marker":"[CCF+24]"},{"why":"Defines fusability and supplies the reduction principle that an equivalence between fusable factorization $\\mathrm{Rep}(\\check{G})$-module categories can be checked at the fiber over $x_0$; all three theorems rely on this.","marker":"[BCG26]"},{"why":"Defines the Fact functor, restricted local systems, and the full faithfulness result used to characterize fusable module categories in Proposition 2.1.3.","marker":"[Bog24]"},{"why":"Defines the factorization restriction 2-functor whose conservativity along $\\mathrm{Rep}(\\check{G}) \\to \\mathrm{Sph}^{\\mathrm{spec}}_{\\check{G}}$ underlies the reduction for Theorem 0.2.8.","marker":"[CF]"},{"why":"Provides the Betti analogue of pointwise Bezrukavnikov's equivalence that is adapted in Section 3.3 to prove the pointwise fiber of $\\mathrm{B}$.","marker":"[DT25]"},{"why":"Supplies the character computations and the comparison with the structure sheaves of nilpotent opers used to show that IFLE sends Verma modules to the expected objects.","marker":"[FG10]"},{"why":"Gives the pointwise identification $\\mathrm{Whit}^!(\\mathrm{Fl}_G)_{x_0} \\simeq \\mathrm{Whit}^!(\\mathrm{LG})^{\\infty/2}_{x_0}$ used in the pointwise chain for Theorem 0.2.5.","marker":"[Ras16]"},{"why":"Constructs the functor $\\Gamma: D$-mod$_{\\mathrm{crit}}(\\mathrm{Fl}_G) \\to \\mathrm{IKL}(G)_{\\mathrm{crit},x_0}$ and establishes the bounded cohomological amplitude used in the proof of Iwahori–Hecke temperedness.","marker":"[FG09]"}],"fun_headline_variants":["Three Langlands equivalences hold under Iwahori ramification","Factorization beats ramification for three local equivalences","Iwahori level yields factorization of three spectral equivalences","From point to Ran: Iwahori equivalences factorize","Triple Iwahori equivalence preserves factorization"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is an unpublished 'fusability' criterion—that an equivalence of the relevant factorization module categories can be checked at the single fixed point $x_0$—and if this criterion fails, or demands conditions not proven here, the three theorems do not follow from the paper's arguments.","fun_headline_variants_meta":{"raw":{"variants":["Three Langlands equivalences hold under Iwahori ramification","Factorization beats ramification for three local equivalences","Iwahori level yields factorization of three spectral equivalences","From point to Ran: Iwahori equivalences factorize","Triple Iwahori equivalence preserves factorization"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000812,"raw_usage":{"total_tokens":3545,"prompt_tokens":913,"completion_tokens":2632,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":529,"completion_tokens_details":{"reasoning_tokens":2554}},"tokens_in":529,"tokens_out":2632,"duration_ms":17810,"temperature":1.0,"reasoning_tokens":2554,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:41:33.579905+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A decisive check would be to compute the fiber of $\\mathrm{IFLE}$ on a Verma module $M_{\\check{\\lambda}}$: the proof identifies its image with the structure sheaf $O_{\\mathrm{Op}^{\\check{\\lambda}-\\mathrm{nilp}}_{\\check{G},x_0}}$ and its character with $\\prod_{n>0}(1-q)^{-\\ell}$; any mismatch in these characters would falsify Theorem 0.2.11. More broadly, once the cited fusability theory appears, testing its reduction principle on a category in the essential image of $\\mathrm{Fact}^{\\mathrm{restr}}$ whose fiber at $x_0$ is an equivalence but which is not an equivalence on a stratum with extra moving points would falsify the common promotion argument.","supporting_citations":[],"review_version":2}