{"id":"f06b1be0-9a50-41a3-938d-57acf2b5d037","arxiv_id":"2608.09726","paper_version":1,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A presentable six-functor formalism satisfying cohomological purity extends to Ind- and Pro-categories, defining motivic stable homotopy theory for ind-pro algebraic stacks such as the Hecke stack.","lead":"This paper extends abstract six-functor formalisms from schemes and stacks to filtered inductive and projective systems, and proves a functorial version of cohomological purity. The goal is a six-functor formalism for ind-pro objects like the Hecke stack, relevant to motivic Satake theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1.1 is the load-bearing step: Section 3.2 is explicitly only a sketch, and both functorial purity and the Pro-extension depend on it; the written proof does not yet establish it.","rationale":"The reader's weakest_assumption pinpoints Theorem 3.1.1, and my reading confirms that this is the correct load-bearing concern. The section titled 'Proof of Theorem 3.1.1' is an explicit sketch, and the text defers to [6] and [4] for the core lifting construction. I found no independent error in the categorical setup or in the examples; the issue is that the central theorem is not proved in the submitted text. Since functorial purity and the Pro-extension both reduce to Theorem 3.1.1, the paper's strongest claims are conditional on a missing technical proof. I would not change the reader's verdict: the appropriate status remains CONDITIONAL, with the request that Theorem 3.1.1 be proved in full, with particular attention to the removal of the admissibility hypothesis on E2. The proposed test isolates that non-admissible case and would tell whether the gap is purely expositional or reflects a false assertion.","tokens_in":25383,"tokens_out":6968,"duration_ms":63094,"concrete_test":"Specialize Theorem 3.1.1 to E1 = all morphisms and E2 = split monomorphisms, which is pullback-stable, contains isomorphisms, and is not admissible, and carry out the proof of Proposition 3.1.12 and the weak-contractibility argument in Section 3.2 for this class. Concretely, verify Lemma 3.1.8.3 for the inclusion of the cartesian subcomplex into Cart_n and the existence of the dotted lifting gcomm in diagram (60). If any step uses admissibility of E2—for example to show that tau_n(tau') preserves the exact or pullback squares defining Q—then Theorem 3.1.1 is false as stated, and the Pro/Ind extension theorems need a weaker hypothesis. If the proof succeeds for this non-admissible class, the conditional objection is answered, but a complete write-up is still required.","verdict_should_be":"UNCHANGED","load_bearing_attack":"All three central results—Theorem 3.0.1, Proposition 4.1.10, and Theorem 4.2.1(1)—reduce to Theorem 3.1.1, the extension along p^cart for non-admissible edges. Section 3.2 states that it only gives a sketch of the main points, and it then relies on unproved inputs: Lemma 3.1.5.3 (contractibility of Kart(tau)), Lemma 3.1.8.3 (the inclusion of the cartesian subcomplex into Cart_n is inner anodyne), and the assertion that the argument in [6, Thm B] goes through unchanged. The novelty of Theorem 3.1.1 is precisely that E2 need not be admissible and no truncation condition is required; Proposition 3.1.12(2)-(3) is where that difference matters, but the verification is deferred to [6]. If this lifting theorem fails, D^Sigma_! and PurD do not exist, and the Pro-extension in Theorem 4.2.1 collapses. This is a proof gap, not an observed contradiction; the surrounding framework is plausible and the examples are coherent.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims two theorems about abstract six-functor formalisms (6FFs). The first (Theorem 3.0.1 and Theorem 3.3.1) upgrades the pointwise cohomological purity equivalence f^# ≅ f^!Σ_f to a natural transformation PurD between the functors D^# and D^Σ_!, where D^Σ_! is a functor on C^{E∩S} built via the multisimplicial and 2-tiled simplicial machinery of Liu–Zheng. The construction rests on a new lifting theorem (Theorem 3.1.1) that extends a functor along p^cart without assuming admissibility of the second edge class and without a truncation condition. The second (Theorem 1.1.3 / Theorem 4.2.1) extends a presentable 6FF arising from a Nagata setup to pro-systems and ind-systems of objects, producing Dpro on Corr(Pro^K_S(C))_{Adj^!_!(E),All} and Dind on Corr(Ind^K_{\\tilde P}(C))_{E,Adj^∗_!(All)}; the pro-extension uses functorial purity (Proposition 4.1.10) to identify colim^∗ with colim^! for arbitrary filtered indexing categories K, generalizing Yaylali's N-indexed result. These results are applied to Borel motivic stable homotopy theory, defining SH(−) for ind-pro algebraic stacks such as the Hecke stack (Theorem 5.2.4).","tokens_in":25585,"tokens_out":22550,"duration_ms":145190,"significance":"If the proofs are completed, the results would be valuable: functorial cohomological purity is a structural upgrade over pointwise purity, and the Ind/Pro extension of 6FFs with an arbitrary filtered indexing category is a useful generalization with concrete applications such as SH of the Hecke stack and DM of pro-algebraic stacks. The paper is honest about its limitations: Section 3.2 is explicitly a sketch, key combinatorial lemmas (Proposition 3.1.5(3), Lemma 3.1.8(3)) are quoted without proof, and the author refers to his own preprints [4]-[8] for the underlying compactification and partial-adjoint theorems. The architecture is coherent, and there is no observable circularity or parameter fitting. However, the written proofs do not yet establish the central claims, because the load-bearing lifting theorem and the main extension functors are not fully proved within the manuscript.","major_comments":[{"comment":"Theorem 3.1.1 is load-bearing for both Theorem 3.0.1 (existence of D^Σ_!) and, through Proposition 4.1.10 and the proof of Theorem 4.2.1(1), for the Pro-extension, yet its proof in Section 3.2 is explicitly 'just a sketch'. The sketch relies on inputs that are not proved in the manuscript: Proposition 3.1.5(3) (contractibility of Kart(τ)), Lemma 3.1.8(3) (inner anodicity of the inclusion ⊞^n_cart ↪ Cart_n), and the statement in the proof of Proposition 3.1.12 that parts (2)-(3) 'just exactly follows as the arguments in [6]'. Since the advertised novelty of Theorem 3.1.1 is precisely the removal of the admissibility of E2 and of the truncation condition, a blanket citation of [6, Thm B] does not cover the new content. The manuscript needs a complete proof of Theorem 3.1.1, or a precise reduction to numbered statements in [6]/[15] with a verification of their hypotheses in the non-admissible case.","section":"§3.1-3.2 (Theorem 3.1.1)"},{"comment":"The hypotheses of the extension theorems are weaker than what the proofs use. Theorem 1.1.3 and Theorem 4.2.1(1) assume pointwise cohomological purity (Definition 4.1.1(2): Pur_f and Σ_f equivalences for each f ∈ S), whereas Proposition 4.1.10 invokes Theorem 3.3.1, whose hypothesis is the global ambidexterity condition that Ex^#_! be an equivalence for all relevant pullback squares (Theorem 3.0.1(2)). Remark 4.1.2(2) only proves the implication ambidexterity ⇒ purity; the converse is asserted nowhere. Without a proof that pointwise purity implies the functorial/ambidexterity condition for the classes used (or a strengthening of the theorem statements to include ambidexterity), the functor-level identification D^#(X•) ≅ D^!(X•) and the colimit equivalence colim^∗ ≅ colim^! in Proposition 4.1.10 are not justified under the stated hypotheses.","section":"§4.1-4.2 (Prop. 4.1.10, Thm 4.2.1(1))"},{"comment":"These propositions are the technical input to the proof of Theorem 4.2.1 (via the maps α'_Pro and their duals), but each proof consists of the single sentence that it is 'just a variant' of [7, Theorem 5.2.3] or of the previous proposition. What needs to be checked is the assembly of a functor with mixed directions ((−)_!, (−)^*, (−)^*) into δ^*_3 tiled simplicial sets from the partial-adjoint and compactification theorems, including the behavior of the tilings □_S and the (3,4)-direction compatibility. These are exactly the points where the extension to Ind/Pro categories could fail, so the reduction to [7] should be carried out explicitly, or the relevant numbered statements in [6]/[7] should be cited with their hypotheses verified.","section":"§4.1 (Propositions 4.1.4 and 4.1.5)"},{"comment":"The construction of Pur'_D, which is the core of the natural transformation PurD, is not verified. The proof asserts that the simplices defined by the case split in equations (90)-(92) 'glue together to form ∆^m × ∆^1 → δ^*_2 C_{S,E}', but no check of the compatibility of those squares with the tiling and two-marked simplicial structure is given, and the naturality of the resulting PurD (needed for the functor-level claim in Proposition 4.1.10) is asserted rather than proved. In addition, the case split contains an evident typo: both (b) and (c) are labeled 'a>b', yet case (c) contains 'b=i+1, a=i', which is only possible for a<b. A complete and corrected verification of the gluing and naturality is required.","section":"§3.3 (Theorem 3.3.1)"}],"minor_comments":[{"comment":"Theorem 4.2.1(2) writes the morphism class of the ind-extension as Adj^∗_∗(All), whereas Definitions 1.1.1(2)(b), 4.1.8(2), Theorem 1.1.3(2) and Notation 5.2.3 use Adj^∗_!(All); the notation should be made uniform.","section":"§4.2"},{"comment":"The proof of Theorem 4.2.3 says 'one checks' that the corresponding pairs on Pro-categories are again nice geometric/exceptional pairs; this check should be included or sketched, since the applications in Section 5 depend on it. Also, the statement of part (1) writes the nice geometric pair as (C,S,E) ⊂ (C,S',E'), whereas Definition 2.4.1(1) allows a different ambient category (C',S',E'); the application to Sch ⊂ AlgSt requires the general form.","section":"§4.2 (Theorem 4.2.3)"},{"comment":"Example 4.1.9(1) describes cohomological purity as 'f^! ≅ f^∗', which is not Definition 4.1.1(2), where purity is f^# ≅ f^!Σ_f with Σ_f an equivalence; moreover the displayed chain of isomorphisms 'fk! p! ∼= fk′! p! p! ∼= ...' is garbled as printed. Both the terminology and the display need correction.","section":"§4.1 (Example 4.1.9)"},{"comment":"Several cross-references are wrong or missing: Theorem 1.2.1 is labeled '(??)'; the proof of Theorem 4.2.1(1)(a) refers to a non-existent 'Theorem 4.1.3'; and Theorem 3.1.1 refers to 'Theorem 3.0.2' where Notation 3.0.2 is meant. These should be fixed.","section":"§1.1, §3.2, §4.2"},{"comment":"The definition of Σ_f is not uniform: Eq. (13) and Eq. (46) define Σ_f as pr1_#δ_{f!}, while Definition 4.1.1(2)(b) writes Σ_f := pr2_#∆_{f!}; the convention should be made consistent, and the directions of the projections in diagram (49) should be aligned with the chosen definition.","section":"§1.2, §4.1"},{"comment":"The theorem does not say which of the classes E1 and E2 corresponds to S and to E in the application to Theorem 3.0.1; since the hypotheses differ (E1 admissible, E2 merely weakly stable), a remark identifying the classes and verifying the hypotheses, in particular the admissibility of E1, is needed.","section":"§3.1 (Theorem 3.1.1)"},{"comment":"Theorem 3.0.1 and Theorem 3.3.1 are stated with target Cat_∞, while Proposition 4.1.10 needs equivalences in Fun(K^op, Pr^L) and colimits in Pr^L_cl; the compatibility of the constructed functors and of PurD with the Pr^L-structure should be stated explicitly.","section":"§3 and §4.1"},{"comment":"There are numerous typos and small inconsistencies, e.g., 'multiplicaltion map' in Remark 2.3.4, 'induced age' in Notation 2.1.5, 'Algst' versus 'AlgSt', and the unnumbered Theorem 1.2.1; I recommend a careful editorial pass.","section":"throughout"}],"recommendation":"major_revision","confidential_remarks":"The manuscript depends for its main results on the author's own unpublished preprints ([4], [6], [7]) for foundational statements: Theorem 2.2.5 is [7, Theorem 5.2.5], the compactification theorem is [6, Thm B], and the partial adjoint theorem is [7, Theorem 3.2.1]. While self-citation of preprints is common in this area, the volume of deferred material makes it difficult to assess the correctness of the current paper independently. If the editor proceeds, it would be appropriate to ask the author to supply, in a version of the file intended for referees, the statements of the cited theorems from [4]-[8] (or precise numbered pointers), and to indicate which parts of those preprints are already accepted. There is no indication of a novelty disclosure problem: the new claims are extensions of the cited work rather than restatements, but the evidentiary base needs to be accessible for a fair review."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the genuinely new content is the extension of a presentable six-functor formalism to Pro^K_S(C) and Ind^K_tildeP(C) for arbitrary filtered K, and to Ind-Pro combinations, with the Hecke stack as a concrete target. Second, the main theorem is not actually proven in the written paper: Theorem 3.1.1, the extension along p^cart for non-admissible edges, is the load-bearing result, and Section 3.2 says explicitly that it only gives a sketch.\n\nThe paper does well at laying out the architecture. The definitions of Adj^!_!(E) and Adj^*_!(All) are motivated by real examples, and the payoff—SH(Hecke stack) and a spectral Satake category—is substantial. The Pro part genuinely generalizes Yaylali's N-indexed rational motives to arbitrary filtered indexing categories, and the Ind-Pro combination is new. The paper is also honest about where proofs are deferred, which is more than many preprints do.\n\nThe soft spot is exactly where the stress-test note puts it. Theorem 3.1.1 is the hinge for both functorial purity (Theorem 3.0.1) and the Pro-extension (Theorem 4.2.1(1)). Its proof relies on unproved lemmas 3.1.5.3 and 3.1.8.3, and on the assertion that the argument of [6, Thm B] goes through unchanged. But the novelty of Theorem 3.1.1 is precisely that E2 need not be admissible and no truncation condition is required; those are the differences that need checking, and they are deferred. So the written proof does not establish the main results. This is a proof gap, not an observed contradiction: the surrounding framework is coherent and there is no sign of a circular argument.\n\nThe heavy reliance on the author's own preprints is worth noting. Self-citation is not itself a flaw, but here the main cited compactification theorem is itself from a preprint, and the adaptation is asserted rather than demonstrated. Minor issue: Theorem 1.2.1 is cross-referenced as '??'.\n\nWho should read this: people working on six-functor formalisms, motivic homotopy of stacks, and spectral Satake categories. If Theorem 3.1.1 is eventually proven, this will be a useful tool. It deserves a serious referee: the referee should ask for a complete proof of Theorem 3.1.1, including the quoted lemmas, and a careful check of the non-admissible case. I would accept it for peer review, but I would not cite it as a proven result until the gap closes.","headline":"A plausible and useful extension of six-functor formalisms to Ind/Pro categories, but the written proof leaves the load-bearing Theorem 3.1.1 as a sketch, so the main results are not yet established.","tokens_in":26157,"tokens_out":3809,"would_cite":false,"duration_ms":33253,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F42","14A20","14F08"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that abstract six-functor formalisms extend to ind- and pro-categories of geometric setups under cohomological purity, and that cohomological purity is itself functorial.","keywords":["six-functor formalism","ind-categories","pro-categories","cohomological purity","motivic stable homotopy theory","Hecke stack","multisimplicial sets","algebraic stacks"],"falsifier":"Take $D = SH$ on schemes, let $S$ be the class of smooth morphisms, and compute the two sides of the functorial purity natural transformation on a composable pair $X_0 \\to X_1 \\to X_2$: the triangle comparing $D^\\Sigma_!(X_0) \\to D^\\Sigma_!(X_2)$ with $D^\\#(X_0) \\to D^\\#(X_2)$ must commute after identifying each object via $f^\\# \\cong f^!\\Sigma_f$. If, for an explicit pair such as $\\mathbb{A}^1 \\to \\mathbb{A}^2$ followed by the structure map $\\mathbb{A}^2 \\to \\mathrm{pt}$, the triangle fails to commute in $SH$, then Theorem 3.3.1 is false; and since Theorem 4.2.1 reduces to the same lifting theorem, that computation would also falsify the pro-extension.","tokens_in":25129,"feed_emoji":"🧮","tokens_out":12873,"duration_ms":98484,"temperature":0.7,"pith_summary":"This paper claims that an abstract six-functor formalism--a symmetric monoidal package of pullback, pushforward, and exceptional functors satisfying base change and projection formulas--can be extended from a geometric category to categories of projective and inductive systems of its objects, under a cohomological purity hypothesis on the transition maps. The extension theorems (Theorem 4.2.1) construct presentable six-functor formalisms on $\\mathrm{Pro}^K_S(C)$ and on $\\mathrm{Ind}^K_{\\tilde P}(C)$ from any presentable formalism coming from a Nagata setup. The paper also upgrades the pointwise purity isomorphism $f^\\# \\cong f^!\\Sigma_f$ to a natural transformation between functors (Theorem 3.3.1), so purity can be chosen compatibly across compositions. If these results are correct, motivic stable homotopy theory $SH(-)$ becomes a six-functor formalism on ind-pro algebraic stacks such as the Hecke stack, yielding a spectral Satake category, and rational motives on pro-algebraic stacks generalize from $\\mathbb{N}$-indexed systems to arbitrary filtered indexing categories.","feed_headline":"Six-functor formalisms extend to ind-pro algebraic stacks","feed_subtitle":"Motivic stable homotopy theory gains a six-functor formalism for the Hecke stack.","key_machinery":"The load-bearing object is the multisimplicial set formalism of [15]: marked and tiled simplicial sets, the poset $\\mathrm{Cart}_n$ of up-sets of $[n]\\times[n]$, and right Kan extensions along the map $\\sigma_n: [n]\\times[n] \\to \\mathrm{Cart}_n$. Inside this language, Theorem 3.1.1 is the technical engine: a lifting theorem along $p^{\\mathrm{cart}}$ for non-admissible edges, which takes a functor defined on the tiled subcomplex $\\delta^*_2 C^{Q,\\mathrm{cart}}_{E_1,E_2}$ (squares satisfying the $\\mathrm{Ex}^{\\#!}$ equivalence) and extends it to $\\delta^*_2 C^R_{E_1,E_2}$ (squares whose decomposition has one pullback in $Q$). This theorem upgrades the pointwise purity isomorphism to the natural transformation $\\mathrm{Pur}_D$ and supplies the coherence needed to assemble the colimit and limit formulas for pro- and ind-objects. The construction of $\\mathrm{Pur}_D$ in Section 3.3 then defines the natural transformation by gluing the relevant simplices in the tiled simplicial set.","core_discovery":"On its own terms, the paper establishes Theorem 1.1.3 (= Theorem 4.2.1): if $D: \\mathrm{Corr}(C)_{E,\\mathrm{all}} \\to \\mathrm{Pr}^L_{\\mathrm{cl}}$ is a presentable six-functor formalism arising from a Nagata setup and $S \\subset E \\cap HL$ is a weakly stable class satisfying cohomological purity, then $D$ extends to a presentable six-functor formalism $D_{\\mathrm{pro}}: \\mathrm{Corr}(\\mathrm{Pro}^K_S(C))_{\\mathrm{Adj}^!_!(E),\\mathrm{All}} \\to \\mathrm{Pr}^L_{\\mathrm{cl}}$, with $D_{\\mathrm{pro}}(X) \\cong \\mathrm{colim}^*_{k} D(X_k)$; and for $\\tilde P \\subset P$ a pullback-stable subclass, $D$ extends to $D_{\\mathrm{ind}}: \\mathrm{Corr}(\\mathrm{Ind}^K_{\\tilde P}(C))_{E,\\mathrm{Adj}^*_!(\\mathrm{All})} \\to \\mathrm{Pr}^L_{\\mathrm{cl}}$, with $D_{\\mathrm{ind}}(Y) \\cong \\mathrm{lim}^!_{k} D(Y_k)$. The paper also proves functorial cohomological purity: the equivalence $f^\\# \\cong f^!\\Sigma_f$, one morphism at a time, is upgraded to a natural transformation $\\mathrm{Pur}_D$ between the functors $D^\\#$ and $D^\\Sigma_!$, and this functoriality is what makes the colimit formula $\\mathrm{colim}^* \\cong \\mathrm{colim}^!$ valid for pro-objects indexed by an arbitrary filtered $K$. As an application, $SH(-)$ (and similarly $DM(-)$) extends to ind-pro algebraic stacks with the loop-group transition maps, so the Hecke stack receives a spectral Satake category $SH(Hk_G)$.","pith_inferences":["Inference: The functorial purity natural transformation should also provide a uniform replacement for the $\\mathbb{N}$-indexing trick in other contexts: any six-functor formalism whose relevant class $S$ is cohomologically pure should satisfy continuity-like $\\mathrm{colim}^*$-versus-$\\mathrm{colim}^!$ identities on arbitrary filtered diagrams, without needing a total order on the index category.","Inference: Because the lifting theorem is stated for any $\\infty$-category $D$ and any pullback-stable class $E_2$, the same proof strategy could yield functorial purity for formalisms valued in other targets, such as derived categories of sheaves with supports, as long as the $\\mathrm{Ex}^{\\#!}$ ambidexterity square is an equivalence.","Inference: A concrete stress test would be to compute the natural transformation $\\mathrm{Pur}_D$ on a two-simplex of smooth maps in $SH$ and verify the associativity and composition triangle; the paper does not spell out this explicit triangle, and its validity would exercise exactly the gluing identities that Section 3.3 sketches."],"forward_implications":["$SH(-)$ and $DM(-)$ become six-functor formalisms on the category of $K$-ind-pro algebraic stacks with smooth and closed transition maps, so in particular the Hecke stack $Hk_G$ has a presentable stable symmetric monoidal category $SH(Hk_G)$.","The pro-extension generalizes the earlier rational-motive theory for pro-algebraic stacks from $\\mathbb{N}$-indexed projective systems to arbitrary filtered $\\infty$-categories $K$, via the equivalence $\\mathrm{colim}^* \\cong \\mathrm{colim}^!$ supplied by functorial purity.","For any pro-object $X_\\bullet$ in $\\mathrm{Pro}^K_S(C)$, the identity $D^\\#(X_\\bullet) \\cong D^!(X_\\bullet)$ holds functorially, so pullback and exceptional pullback computations agree on such pro-systems.","The extension procedures are compatible with nice geometric pairs and exceptional pairs, so the formalism can be iterated, for example from schemes to algebraic stacks and then to ind-pro stacks.","For any object of this ind-pro category, $SH(X)$ is presentable and stable, because it is computed as filtered colimits of presentable stable categories."],"supporting_citations":[{"why":"Supplies the multisimplicial set formalism and the enhanced six-functor framework that both main theorems are built on.","marker":"[15]"},{"why":"Provides the $\\infty$-categorical compactification theorem, called Theorem B, whose proof Theorem 3.1.1 revisits and adapts for non-admissible edges.","marker":"[6]"},{"why":"Gives the construction of presentable six-functor formalisms from Nagata setups via partial adjoints, used throughout Sections 3 and 4.","marker":"[7]"},{"why":"Supplies the theorem for constructing functors using the category of simplices, invoked at the end of the proof of Theorem 3.1.1.","marker":"[4]"},{"why":"Establishes the prior rational-motive formalism on pro-algebraic stacks indexed by $\\mathbb{N}$, which the pro-extension generalizes.","marker":"[27]"},{"why":"Builds the motivic stable homotopy theory of algebraic stacks that is the paper's main example formalism.","marker":"[5]"},{"why":"Provides generalized cohomology theories for algebraic stacks, also used to get $SH(-)$ on stacks.","marker":"[13]"},{"why":"Develops non-representable six-functor formalisms, needed to handle non-representable maps in the application to $SH$ on stacks.","marker":"[8]"},{"why":"Introduces the loop groups, affine Grassmannian, and Hecke stack whose ind-pro structure motivates and supports the application section.","marker":"[22]"},{"why":"Gives the upper-shriek extension for prestacks that the ind-extension formula $\\mathrm{lim}^!$ mirrors.","marker":"[23]"}],"fun_headline_variants":["Six-functor formalism reaches ind-pro stacks","Ind-pro stacks get six-functor formalism","Motivic homotopy for Hecke stack via six functors","Functorial purity extends six-functor formalisms"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that a functor known on the well-behaved squares of a diagram extends to all squares even when the smooth-like morphisms are not required to be admissible, and both headline results reduce to that extension, whose proof is only sketched and defers key lemmas to earlier work.","fun_headline_variants_meta":{"raw":{"variants":["Six-functor formalism reaches ind-pro stacks","Ind-pro stacks get six-functor formalism","Motivic homotopy for Hecke stack via six functors","Functorial purity extends six-functor formalisms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000203,"raw_usage":{"total_tokens":1439,"prompt_tokens":1055,"completion_tokens":384,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":671,"completion_tokens_details":{"reasoning_tokens":318}},"tokens_in":671,"tokens_out":384,"duration_ms":2658,"temperature":1.0,"reasoning_tokens":318,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:54:02.645676+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $D = SH$ on schemes, let $S$ be the class of smooth morphisms, and compute the two sides of the functorial purity natural transformation on a composable pair $X_0 \\to X_1 \\to X_2$: the triangle comparing $D^\\Sigma_!(X_0) \\to D^\\Sigma_!(X_2)$ with $D^\\#(X_0) \\to D^\\#(X_2)$ must commute after identifying each object via $f^\\# \\cong f^!\\Sigma_f$. If, for an explicit pair such as $\\mathbb{A}^1 \\to \\mathbb{A}^2$ followed by the structure map $\\mathbb{A}^2 \\to \\mathrm{pt}$, the triangle fails to commute in $SH$, then Theorem 3.3.1 is false; and since Theorem 4.2.1 reduces to the same lifting theorem, that computation would also falsify the pro-extension.","supporting_citations":[{"cited_title":"Liu and W","cited_arxiv_id":null,"evidence_quote":"Supplies the multisimplicial set formalism and the enhanced six-functor framework that both main theorems are built on."},{"cited_title":"Six-Functor Formalisms II : The $\\infty$-categorical compactification","cited_arxiv_id":"2412.03231","evidence_quote":"Provides the $\\infty$-categorical compactification theorem, called Theorem B, whose proof Theorem 3.1.1 revisits and adapts for non-admissible edges."},{"cited_title":"Six-Functor Formalisms III: The construction and extension of 6FFs","cited_arxiv_id":"2412.20548","evidence_quote":"Gives the construction of presentable six-functor formalisms from Nagata setups via partial adjoints, used throughout Sections 3 and 4."},{"cited_title":"Six-Functor Formalisms I : Constructing functors using category of simplices","cited_arxiv_id":"2304.11742","evidence_quote":"Supplies the theorem for constructing functors using the category of simplices, invoked at the end of the proof of Theorem 3.1.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the prior rational-motive formalism on pro-algebraic stacks indexed by $\\mathbb{N}$, which the pro-extension generalizes."},{"cited_title":"Chowdhury","cited_arxiv_id":null,"evidence_quote":"Builds the motivic stable homotopy theory of algebraic stacks that is the paper's main example formalism."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides generalized cohomology theories for algebraic stacks, also used to get $SH(-)$ on stacks."},{"cited_title":"Non-representable six-functor formalisms","cited_arxiv_id":"2409.20382","evidence_quote":"Develops non-representable six-functor formalisms, needed to handle non-representable maps in the application to $SH$ on stacks."},{"cited_title":"RICHARZ and J","cited_arxiv_id":null,"evidence_quote":"Introduces the loop groups, affine Grassmannian, and Hecke stack whose ind-pro structure motivates and supports the application section."},{"cited_title":"Richarz and J","cited_arxiv_id":null,"evidence_quote":"Gives the upper-shriek extension for prestacks that the ind-extension formula $\\mathrm{lim}^!$ mirrors."}],"review_version":1}