{"id":"c0c59759-0623-4906-bd61-a1d860eb90f3","arxiv_id":"2505.01084","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Realization functors from log prismatic F-crystals to étale, crystalline, and de Rham categories are constructed for semi-stable formal schemes with horizontal boundary divisors, and the étale realization is proved fully faithful (and bi-exact for n≤1).","lead":"This paper builds realization functors that translate coefficients in log prismatic cohomology into étale, crystalline, and de Rham coefficient objects on semi-stable p-adic formal schemes with horizontal boundary components. It extends prior work that only handled smooth or vertically semi-stable cases, and proves the étale realization is fully faithful and often exact.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 5.19 is the load-bearing step: its proof is delegated to the author's [Ino25, Prop 4.18], and the internal reduction may require an unstated flatness hypothesis when tensoring the exact sequence 0→A→B^(0)→B^(1).","rationale":"The paper's central claim is Theorem A and Theorem B: on semi-stable log formal schemes with horizontal boundary components, étale and crystalline realization functors exist, with T_ét fully faithful and bi-exact for framings with n≤1. I have read the full manuscript and cannot verify the deepest technical arguments; the most load-bearing premise is indeed Proposition 5.19, exactly as the reader identifies. My stress-test adds a sharper internal observation: in the proof of Proposition 5.19, the step from the exact sequence 0→A→B^(0)→B^(1) to an exact sequence after tensoring with Γ(Spec(A[1/f]),E) requires a flatness or splitness hypothesis that is not stated. If [Ino25, Proposition 2.26] supplies it, the concern does not land; if not, Theorem B lacks a complete proof. The paper has independent support in the special cases m=n=0 and m=0,n≥1 covered by [DLMS23, DLMS24, GR24, BS23], and the construction of the realization functors in Theorem A is detailed and largely self-contained outside this descent input. However, because the novel part of Theorem B depends on Proposition 5.19, the appropriate verdict remains CONDITIONAL rather than ACCEPT, and my read does not change the reader's verdict.","tokens_in":64193,"tokens_out":6555,"duration_ms":69194,"concrete_test":"Verify [Ino25, Proposition 2.26] and [Ino25, Proposition 4.18]. Specifically, check whether the sequence 0→A→B^(0)→B^(1) is split exact or whether B^(i) is flat over A; if neither is true or supplied, then the claim in Proposition 5.19 that tensoring with Γ(Spec(A[1/f]),E) preserves exactness is unjustified, and Proposition 5.19 fails as written. This check settles whether Theorems 6.27 and 6.32 have a gap.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 5.19 is not proved in this paper. The full-faithfulness half is reduced to [Ino25, Proposition 4.18] by replacing a prism with a flat cover and then claims that, because [Ino25, Proposition 2.26] gives an exact sequence 0→A→B^(0)→B^(1), tensoring with Γ(Spec(A[1/f]),E) preserves exactness. This is only automatic if the sequence is split exact or if B^(0) and B^(1) are flat over A; neither condition is stated or proved here. The bi-exactness half is even terser: for (A,I,MA)=(SR,(E),MSR), the proof concludes from faithful flatness of U(B^(0))→U(A,I) that the local functor is a bi-exact equivalence onto its essential image, but exactness of a sequence in the limit category does not formally follow from faithful flatness without checking that the sequence is already a complex over U(A,I) and that exactness can be tested on the cover. Since Proposition 5.19 is used in Theorem 6.27 for full faithfulness and in Theorem 6.32 for bi-exactness, any missing hypothesis in this descent statement would invalidate Theorem B. The delegation to [Ino25] is not by itself a flaw, but it is a load-bearing external input; moreover, Remark 1.7 says [Ino25] applies the current paper, so the two works should be checked for mutual dependence.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops realization functors for coefficient objects in log prismatic cohomology. On a semi-stable log formal scheme over O_K with possibly horizontal boundary components, it constructs an étale realization functor T_ét from analytic log prismatic F-crystals to Kummer étale Z_p-local systems and a crystalline realization functor T_fisoc to filtered F-isocrystals, and proves that they are associated in the sense of Definition 4.18(2) (Theorem A / Theorem 6.18). It further proves that T_ét is fully faithful (Theorem 6.27) and, when framings have n≤1 étale locally, bi-exact onto its essential image (Theorem 6.32). The main technical engine is a \"Kummer quasi-syntomic descent\" statement (Proposition 5.19), supported by a study of big log affinoid perfectoids and period sheaves with connections in Section 4 and of F-isocrystals with monodromy in Section 3.","tokens_in":64491,"tokens_out":16048,"duration_ms":170661,"significance":"If correct, this fills a genuine gap: the previous smooth and vertical semi-stable cases are extended to the mixed horizontal/vertical setting, and full faithfulness of the étale realization is obtained without the detailed Breuil-Kisin ring computations used in earlier works. The paper is generally careful about functoriality and independence of framings, and it contains substantial local descriptions (Lemmas 4.6 and 4.12, Propositions 4.21 and 6.23) that are useful in themselves. The main caveat is that Proposition 5.19, the load-bearing descent input for Theorem B, is not proved self-containedly: it is reduced to results in the author's companion preprint [Ino25], whose relation to the present paper is only partially described.","major_comments":[{"comment":"The proof of the full faithfulness half of Proposition 5.19 is not carried out in this paper: it refers to [Ino25, Proposition 4.18] and [Ino25, Proposition 2.26] for the exact sequence 0→A→B^(0)→B^(1). Since Proposition 5.19 is then used verbatim in Theorem 6.27 and in Theorem 6.32, this is a load-bearing external input. Moreover, Remark 1.7 states that [Ino25] applies the results of the present paper, so there is a potential circularity between the two works. Please either give a complete proof of Proposition 5.19 or spell out the exact dependency graph and confirm that [Ino25, Prop. 2.26 and 4.18] do not use Proposition 5.19 or Theorem B.","section":"§5.2, Proposition 5.19; used in §6.5 and §6.6"},{"comment":"The deduction of exactness of 0→TdR(E1)→TdR(E2)→TdR(E3)→0 from exactness of the associated graded pieces is only sketched. One needs a lemma stating that, for finite (or at least bounded, separated, and exhaustive) filtrations with locally free graded pieces, exactness of the graded sequence implies exactness of the underlying filtered sequence, by induction over the filtration steps. Please supply this lemma or a precise reference, and state the boundedness of the Hodge filtrations in Definition 3.27 or in the construction of TdR.","section":"§6.6, Theorem 6.32"}],"minor_comments":[{"comment":"In the full faithfulness proof, the step \"Taking tensor products with Γ(Spec(A[1/f]),E)\" is valid because E|Spec(A[1/f]) is finite locally free and hence flat over A on each affine open Spec(A[1/f]); however, this justification is not recorded. Please add a sentence so that the reader does not need to supply a flatness hypothesis on B^(0) and B^(1).","section":"§5.2, proof of Proposition 5.19"},{"comment":"The bi-exactness half ends with the assertion that faithful flatness of U(B^(0))→U(A,I) proves the claim. Please expand this with the standard descent argument that exactness of the pullback sequence at the faithfully flat cover reflects exactness in Vect^{an,φ}(A,I).","section":"§5.2, proof of Proposition 5.19"},{"comment":"In the induction step for n≥2, the proof speaks of the \"vanishing locus of x_i\" for i=1,2. Since the x_i are invertible by Definition 1.1, these loci are empty; the intended variables are almost certainly the vertical coordinates z_i (or the corresponding t_j). Please correct the notation.","section":"§6.5, Theorem 6.27"},{"comment":"The parenthetical \"n≤1 (in other words, the special fiber of X is smooth)\" is inaccurate for n=1: in that case the special fiber has a normal-crossing divisor. The subsequent argument does not appear to use smoothness, so this is a presentation issue, but the wording should be fixed.","section":"§6.5, Theorem 6.27"},{"comment":"In the paragraph after the exact sequence of Tisoc(E_i)(S_R,M_{S_R}), the displayed sequence of Acrys(S)-modules repeats Tisoc(E1) in all three terms; it should read Tisoc(E1), Tisoc(E2), Tisoc(E3).","section":"§6.6, Theorem 6.32"}],"recommendation":"major_revision","confidential_remarks":"The main reservation is the reliance on the author's own preprint [Ino25] for Proposition 5.19, combined with the statement in Remark 1.7 that [Ino25] applies the present paper. I would ask the editor to verify that there is no circular dependence between the two works, ideally by requesting a self-contained proof of Proposition 5.19 or a precise dependency statement. The remaining issues are local and fixable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague —\n\nPunchline: this is a substantive generalization of realization functors for log prismatic F-crystals to semi-stable formal schemes with horizontal boundary components (arbitrary m,n). That's genuinely new: prior work covered smooth (m=n=0) and vertically semi-stable (m=0,n≥1). The paper also proves full faithfulness and bi-exactness of the étale realization (Theorem B), which is the expected result.\n\nWhat it does well: the constructions are detailed and the paper is honest about what it leaves open (essential image, Remark 1.6). The choice to follow the Fargues-functor method rather than Breuil-Kisin computations is sensible. The big log affinoid perfectoids and period sheaves with connections are set up carefully. This is not a routine extension; there's real work here.\n\nSoft spots: Proposition 5.19 is load-bearing for both full faithfulness and bi-exactness, and its proof is delegated to the author's own preprint [Ino25, Prop 4.18]. Delegation by itself would be fine, but Remark 1.7 says [Ino25] applies the current paper, so the referee must verify that [Ino25] does not itself depend on this paper's results. That is the main thing to check.\n\nThe stress-test note about needing a flatness hypothesis when tensoring 0→A→B^(0)→B^(1) does not, as far as I can tell, land. The B^(i) are flat over A by construction (the log prism map is a strict flat cover), and E is a finite projective module over A[1/f], hence flat. The paper would benefit from saying so explicitly, but I don't see a hidden gap there. The bi-exactness half of Prop 5.19 is terse, though; the referee should ask for more detail.\n\nWho this is for: p-adic Hodge theorists working on compactified Shimura varieties or log prismatic coefficients. It deserves a serious referee. I would recommend conditional acceptance: the math looks right, but the dependence on [Ino25] must be sorted out before publication.","headline":"A serious paper that delivers the expected horizontal-boundary generalization of realization functors, but the referee should verify Prop 5.19 and its dependence on the author's companion preprint.","tokens_in":65095,"tokens_out":4006,"would_cite":true,"duration_ms":39165,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F30","14G22"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every log prismatic F-crystal on a semi-stable log formal scheme now has étale and crystalline realizations, and the étale one is faithful and, in the n≤1 case, preserves exactness.","keywords":["log prismatic cohomology","analytic log prismatic F-crystals","p-adic Hodge theory","realization functors","Kummer étale local systems","filtered F-isocrystals","semi-stable log formal schemes","Nygaard filtration"],"falsifier":"Exhibit a strict quasi-syntomic cover $(Y,M_Y)\\to (X_{\\infty,\\alpha},M_{X_{\\infty,\\alpha}})$ as in Setting 5.18 for which the restriction functor to the Čech nerve is not fully faithful or not bi-exact onto its essential image, or a counterexample to the companion proposition it invokes. Equivalently, in the $n\\leq 1$ case, exhibit two non-isomorphic analytic log prismatic F-crystals with isomorphic Kummer étale local systems, or an exact sequence of local systems whose preimage under the étale realization is not exact.","tokens_in":63920,"feed_emoji":"💎","tokens_out":12982,"duration_ms":125680,"temperature":0.7,"pith_summary":"Log prismatic cohomology packages the p-adic Hodge theory of varieties with semi-stable reduction into coefficient objects called log prismatic F-crystals. This paper establishes that, on semi-stable log formal schemes over a p-adic field whose boundary may contain horizontal components, every such crystal produces two classical coefficient objects: a Kummer étale $\\mathbb{Z}_p$-local system and a filtered F-isocrystal, and the two are associated in the sense of Definition 4.18(2). The étale realization is fully faithful, and when the local framings have at most one vertical boundary divisor ($n\\leq 1$) it is bi-exact: it preserves and reflects exact sequences. A companion paper applies the construction to toroidal compactifications of Shimura varieties. If the claims hold, log prismatic coefficients can be used interchangeably with étale and crystalline coefficients on semi-stable degenerations, extending results previously known in the smooth and vertically semi-stable cases.","feed_headline":"F-crystals on semi-stable log schemes now land in local systems","feed_subtitle":"New realizations extend p-adic comparison to degenerations with horizontal boundary components.","key_machinery":"The central object is the category of analytic log prismatic F-crystals, namely vector bundles on the log prismatic site equipped with a Frobenius-compatible isomorphism, defined on the locus away from $V(p,I)$. The argument's load-bearing mechanism is a descent statement the paper calls Kummer quasi-syntomic descent (Proposition 5.19): for a strict quasi-syntomic cover of the infinite-root log cover $X_{\\infty,\\alpha}$, restriction from the ambient category to the totalization over the Čech nerve is fully faithful and bi-exact onto its essential image. This descent lets the author reduce the proof of Theorem B, first to the known full faithfulness of the lattice realization functor on integral perfectoid algebras, and then to exact sequences of modules over the Breuil-Kisin log prisms associated with the local framings.","core_discovery":"On a semi-stable log formal scheme over $\\mathcal{O}_K$, with framings that may have both horizontal and vertical boundary coordinates, the paper constructs an étale realization functor from analytic log prismatic F-crystals to Kummer étale $\\mathbb{Z}_p$-local systems on the generic fiber, and a crystalline realization functor to filtered F-isocrystals, and proves the two realizations of the same crystal are associated in the sense of Definition 4.18(2). Theorem B strengthens this: the étale realization is fully faithful for all semi-stable log formal schemes, and when framings satisfy $n\\leq 1$ it is a bi-exact equivalence onto its essential image. The construction works with analytic log prismatic F-crystals, the larger coefficient category obtained by evaluating vector bundles on the complement of $V(p,I)$, and generalizes prior realizations from the smooth and vertically semi-stable cases to horizontal boundary components. The de Rham realization is also described through the Nygaard filtration on the Breuil-Kisin log prisms attached to a framing.","pith_inferences":["An immediate next problem is to identify the essential image of the étale realization; the paper's naive expectation is the category of semi-stable local systems, which would turn the bi-exact embedding into a full Tannakian equivalence.","The bi-exactness theorem stops at $n\\leq 1$; whether multi-dimensional vertical degenerations with $n\\geq 2$ destroy bi-exactness is left open and could be tested with multi-variable Breuil-Kisin log prisms.","Because Proposition 5.19 is proved by reduction to a companion preprint, an independent proof of that descent statement would be the fastest way to check the robustness of Theorem B."],"forward_implications":["A log prismatic F-crystal on a semi-stable log formal scheme is determined up to isomorphism by its Kummer étale local system, because the étale realization functor is fully faithful.","The étale and crystalline realizations of the same crystal are associated in the sense of Definition 4.18(2), so passage between the two coefficient worlds loses no comparison information.","When the local framings have at most one vertical boundary direction, exact sequences of crystals and exact sequences of local systems correspond bi-exactly, giving a sharp embedding theorem.","The de Rham realization is computed from the Nygaard filtration on a Breuil-Kisin log prism, so the filtered vector bundle and Griffiths transversality are explicit from the crystal itself.","The construction supplies the log prismatic realization used in a companion work on toroidal compactifications of Shimura varieties of Hodge type (Remark 1.7)."],"supporting_citations":[{"why":"Supplies the equivalence between Laurent F-crystals and Kummer étale local systems on the log diamond, which is the base of the étale realization functor.","marker":"[KY23]"},{"why":"Constructs the smooth-case realization functors and the lattice realization argument that Theorem B adapts to the semi-stable setting.","marker":"[GR24]"},{"why":"Treats the vertically semi-stable case and supplies the log prismatic F-crystal formalism and Breuil-Kisin log prisms used in Sections 5 and 6.","marker":"[DLMS24]"},{"why":"Its Proposition 4.18 is the input to which Proposition 5.19's Kummer quasi-syntomic descent proof is reduced, so Theorem B depends on it.","marker":"[Ino25]"},{"why":"Supplies the prismatic F-crystal description of crystalline Galois representations and the full faithfulness method the paper follows.","marker":"[BS23]"},{"why":"Gives bi-exactness and Nygaard-filtration results in the smooth case, which Sections 6.4 and 6.6 generalize.","marker":"[IKY24]"},{"why":"Introduces log prisms and the log prismatic site that defines the category of log prismatic F-crystals.","marker":"[Kos22]"},{"why":"Provides foundations of prismatic cohomology, including quasi-syntomic descent for prismatic crystals, used throughout.","marker":"[BS22]"},{"why":"The original comparison setup behind the notion of association recorded in Definition 4.18(2).","marker":"[Fal89]"}],"fun_headline_variants":["Log prismatic F-crystals realize into étale local systems","Fully faithful étale realization for log F-crystals","Étale realization for log F-crystals now covers horizontal boundaries","New realizations connect log F-crystals to étale local systems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is a descent theorem, Proposition 5.19, for analytic log prismatic F-crystals over strict quasi-syntomic covers; its proof is reduced to a proposition in a companion preprint, so if that comparison fails, the full faithfulness and bi-exactness of the étale realization would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Log prismatic F-crystals realize into étale local systems","Fully faithful étale realization for log F-crystals","Étale realization for log F-crystals now covers horizontal boundaries","New realizations connect log F-crystals to étale local systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001232,"raw_usage":{"total_tokens":4992,"prompt_tokens":806,"completion_tokens":4186,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":422,"completion_tokens_details":{"reasoning_tokens":4123}},"tokens_in":422,"tokens_out":4186,"duration_ms":27834,"temperature":1.0,"reasoning_tokens":4123,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:27:39.981265+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a strict quasi-syntomic cover $(Y,M_Y)\\to (X_{\\infty,\\alpha},M_{X_{\\infty,\\alpha}})$ as in Setting 5.18 for which the restriction functor to the Čech nerve is not fully faithful or not bi-exact onto its essential image, or a counterexample to the companion proposition it invokes. Equivalently, in the $n\\leq 1$ case, exhibit two non-isomorphic analytic log prismatic F-crystals with isomorphic Kummer étale local systems, or an exact sequence of local systems whose preimage under the étale realization is not exact.","supporting_citations":[],"review_version":1}