{"id":"da6c784a-9393-41c2-b41e-9b5d152ee0d5","arxiv_id":"1908.06174","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For p=2, the support of patched modules meets every irreducible component of the potentially semi-stable deformation ring, yielding the Breuil-Mezard conjecture in the case where the residual representation is a twist of an extension of 1 by 1.","lead":"This paper proves a modularity lifting theorem for 2-adic Galois representations that removes a local restriction in prior proofs of the Fontaine-Mazur conjecture. It completes the Breuil-Mezard conjecture for 2-dimensional representations of the Galois group of Q2 in the previously open case where the residual representation is a twist of an extension of the trivial character by itself.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Exceptional-block support argument rests on Proposition 1.2.2, whose proof is explicitly left to the reader, and on an unverified p=2 adaptation of [Pas13, Lemma 10.35]; failure there breaks Prop 1.3.2 and hence finite generation of V(M_infty).","rationale":"The reader's weakest assumption is the correctness of the exceptional-block extension computations, centered on Proposition 1.2.2 and Lemma 1.2.7. My independent reading confirms that this is the most load-bearing unresolved point. Theorem 8.0.1 is the main result, and its proof splits into the ordinary locus, handled by Theorem 7.3.1 using a standard ordinary R=T argument, and the non-ordinary locus, handled by Theorem 6.3.7. The non-ordinary argument is a long chain of category-theoretic inputs, but the genuinely new and p=2-specific step is the equivalence of categories in Proposition 1.3.2 for the block {1,Sp}. Every later finiteness and faithfulness result (Proposition 6.3.2, Corollary 6.3.4, Proposition 6.3.5, Corollary 6.3.6) flows from it. The paper itself flags the weakness: Proposition 1.2.2 is stated with 'We leave the proof to the reader,' and Lemma 1.2.7 depends on a p>2 lemma of Paskunas together with an assertion that a Colmez proposition holds for p=2. These are not cosmetic gaps; they are exactly the places where the exceptional block differs from the previously treated cases. The rest of the paper appears coherent: the global patching follows [CEG+16] and [Paš16], and the ordinary part follows [All14b, Tho15, Sas19]. I found no deriving error in the main flow, and the reliance on unpublished preprints ([EP18], [Pyv18], [Tun18]) is a further risk, but the single most decisive unresolved step is the omitted proof of Proposition 1.2.2 and the associated verification of Lemma 1.2.7. Because the reader already assigned CONDITIONAL, and my analysis identifies the same concern without finding a definite error, the verdict should remain unchanged. The proposed concrete test would settle whether the concern lands: either the proof of Proposition 1.2.2 goes through and the injection in Lemma 1.2.7 is verified, in which case the main theorem is on much firmer ground, or one of the two fails, in which case the non-ordinary support claim is not established.","tokens_in":50523,"tokens_out":3683,"duration_ms":37044,"concrete_test":"Write out a complete proof of Proposition 1.2.2 for the exceptional block B={1,Sp}, following the arguments of [Pas13, Lemmas 10.26-10.29] and checking each hypothesis in the p=2, scalar-semisimplification setting. In parallel, recompute Lemma 1.2.6 and Lemma 1.2.7 by hand: enumerate all length <=2 objects of Mod^l.fin_{G/Z}(k) with Jordan-Holder factors in {1,Sp}, compute Ext^1_D(k)(T1,T1) using the explicit Iwahori-module extension of Lemma 1.2.3, and verify that ˇV induces an injection into Ext^1_{G_Q2}(1,1). If the injection is not injective, Lemma 1.3.1 and Proposition 1.3.2 fail. If Proposition 1.2.2 cannot be proven as stated, the non-ordinary support argument in Theorem 8.0.1 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 8.0.1 reduces the non-ordinary locus to Theorem 6.3.7, which uses Corollary 6.3.6, Proposition 6.3.5, and Proposition 6.3.2. Proposition 6.3.2 cites Proposition 1.3.3, whose proof relies on Lemma 1.3.1. Lemma 1.3.1 reduces to the exceptional block {1, Sp} of GL2(Q2), where the Hom claim is recorded and the Ext^1 injection is attributed to Lemma 1.2.7. Lemma 1.2.7 asserts that [Col10, Proposition VII.4.12] holds when p=2 and that the proof of [Pas13, Lemma 10.35] works verbatim after replacing Lemma 10.34 by Lemma 1.2.6. The chain also depends on Proposition 1.2.2, which supplies the duality formula, projectivity, and essentiality properties of the quotient category D(O). Proposition 1.2.2 is stated as an 'easy variant' of [Pas13, Lemmas 10.26-10.29] and its proof is left to the reader; the cited results are for the p>2 or non-exceptional settings and do not directly cover the p=2 scalar-semisimplification block. If any part of Proposition 1.2.2 fails, the essentialness argument in Lemma 1.2.6 breaks, Lemma 1.2.7 loses its footing, and Proposition 1.3.2 collapses. That would remove the finite-generation of V(M_infty) and with it the non-ordinary support statement in Theorem 8.0.1. This is not a disagreement with established results; it is an omitted verification at a load-bearing step, explicitly flagged by the manuscript as 'We leave the proof to the reader.'","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proves, for p=2 and a totally real field F in which 2 splits completely, a support theorem for the patched modules M_infinity(sigma^o): the support of M_infinity(sigma^o) tensor_{Z_p} Q_p meets every irreducible component of the potentially semi-stable deformation ring R_infinity(sigma)[1/p] (Theorem 8.0.1). From this support theorem the author derives the Breuil-Mezard conjecture for two-dimensional representations of G_Q2, including the previously open case where the residual representation is a twist of an extension of the trivial character by itself (Corollary 8.0.2), and removes a local restriction in Paskunas' proof of the Fontaine-Mazur conjecture (Theorem 8.0.3). The strategy follows the author's earlier p>2 work: it combines the p-adic local Langlands correspondence via Colmez's Montreal functor with Taylor-Wiles patching, proves finite generation of V(M_infinity) using new representation-theoretic computations in the exceptional block {1,Sp} of GL_2(Q_2), and treats the ordinary locus by an ordinary R=T theorem.","tokens_in":50911,"tokens_out":6856,"duration_ms":67262,"significance":"If the exceptional-block computations are correct, this is a substantial contribution: it completes the Breuil-Mezard conjecture for GL_2(Q_2) in the scalar-reducible case, gives a new modularity lifting theorem for 2-adic potentially semi-stable representations, and improves Paskunas' Fontaine-Mazur result. The support theorem for patched modules is itself a useful structural result. The paper does not rely on circular reasoning: the main theorems are derived from external inputs such as p-adic local Langlands, patching machinery, and ordinary R=T theorems, and there is no data fitting or parameter normalization forcing the conclusion. However, the central argument rests on a chain of assertions that certain p>2 results work 'verbatim' in the p=2 exceptional block, and several load-bearing proofs are either omitted or deferred to preprints. The significance is therefore conditional on the correctness of those local computations, and the current manuscript does not yet provide enough detail for the reader to verify them.","major_comments":[{"comment":"This proposition is stated with the proof left to the reader and is described as an easy variant of [Pas13, Lemmas 10.26-10.29]. Those results are for settings outside the p=2 scalar-semisimplification block, which is exactly the block that the present paper must treat. Proposition 1.2.2 is used essentially in Lemma 1.2.6, Lemma 1.2.7, Lemma 1.3.1, Proposition 1.3.2, and ultimately in the finite-generation statement Proposition 6.3.2. Since the whole point of Sections 1.2-1.3 is to handle the exceptional block {1,Sp} of GL_2(Q_2), the proof cannot be omitted. I request a complete proof of parts (1)-(3) in the p=2 case, or a precise statement of the modifications needed in the cited arguments.","section":"Section 1.2.1, Proposition 1.2.2"},{"comment":"Lemma 1.2.7 asserts that [Col10, Proposition VII.4.12] holds when p=2 and that the proof of [Pas13, Lemma 10.35] works verbatim after replacing Lemma 10.34 by Lemma 1.2.6. No argument is supplied for the p=2 validity of [Col10, Proposition VII.4.12], and the manuscript does not identify any reason why the assumptions of that proposition remain satisfied. This injectivity statement is the unique input used to prove the Ext^1 assertion in Lemma 1.3.1 and the equivalence in Proposition 1.3.2. The proof should be written out rather than left as a 'works verbatim' assertion, because the exceptional block is precisely the case not covered by the cited results.","section":"Section 1.2.3, Lemma 1.2.7"},{"comment":"Proposition 1.3.2 states that the proof of [Pas13, Proposition 10.36] works verbatim with Lemma 10.35 replaced by Lemma 1.2.7, and Proposition 1.3.3 says that the proof of [Tun18, Proposition 2.8] works with Lemma 2.6 replaced by Lemma 1.3.1. Since [Tun18] is the author's own preprint and Proposition 1.3.3 is what converts the local equivalence into the finite generation of V(M_infinity) in Proposition 6.3.2, the manuscript should reproduce the relevant arguments or at least give precise lemma-by-lemma references with proofs of the modifications. As written, a reader cannot verify that the chain survives the p=2 exceptional case.","section":"Section 1.3, Propositions 1.3.2 and 1.3.3"},{"comment":"Proposition 6.3.2 says: 'Using Proposition 1.3.3, the proof of [Tun18, Proposition 3.4] works without any change.' This is the exact step where finite generation of V(M_infinity) is established, and it is load-bearing for Corollary 6.3.6 and for the non-ordinary support statement in Theorem 6.3.7. Given the dependence on the preceding unverified local statements, the proof of finite generation should be included in detail. If any part of Proposition 1.2.2 or Lemma 1.2.7 fails, then Proposition 6.3.2 collapses and with it the non-ordinary part of Theorem 8.0.1.","section":"Section 6.3, Proposition 6.3.2 and Theorem 6.3.7"}],"minor_comments":[{"comment":"The statement of Theorem 8.0.1 does not list the standing hypotheses under which M_infinity and R_infinity are constructed (for example, modularity of the residual representation, non-solvable image, and the conditions on the globalization). As written, the theorem appears to claim an unconditional statement about all such patched modules. Please state the full hypotheses in the theorem.","section":"Theorem 8.0.1"},{"comment":"In the construction of the nontrivial extension of I(Ind_B^G 1) by I(1), it would help to write the module as a quotient of a free module and explicitly verify the Hecke relations T^2=1 and (S+1)S=0. The current display is correct-looking but the verification is left implicit.","section":"Section 1.2.3, Lemma 1.2.3"},{"comment":"There are several typographical errors that should be corrected before publication, for example 'reduecd' (Section 3.2.7), 'sujective' (Proposition 4.6.5), and 'homomoprhism' (Sections 4.6 and 7.2).","section":"Throughout"},{"comment":"Several essential dependencies are arXiv preprints or unpublished notes, including [EP18], [Pyv18], [Tun18], [Pan19], and [Sas17]. The paper would benefit from a note on the publication status or version dates of these works, and from page/lemma-level references where 'works verbatim' is claimed.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The central obstacle is not a demonstrated error but an omitted verification at a load-bearing point. The author should be asked to provide complete proofs of Proposition 1.2.2, Lemma 1.2.7, and the finite-generation step Proposition 6.3.2, rather than relying on 'easy variant' and 'works verbatim' assertions in the p=2 exceptional block. The paper also relies heavily on the author's own unpublished preprint [Tun18]; the contribution relative to [Tun18], [Pan19], and [EP18] should be made explicit. If the local arguments are supplied, the result appears to be a valuable and correct contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is the natural p=2 sequel to your earlier p>2 paper [Tun18], and if the local exceptional-block computation goes through, it completes the Breuil-Mezard conjecture for G_Q2 and removes the local restriction in Paskunas' Fontaine-Mazur theorem. The overall strategy is clear and I did not find a deriving error. The paper should be sent to a referee, but the referee will need to fill a real gap.\n\nWhat's new: the p=2 case where the residual representation is a twist of an extension of 1 by 1. The genuinely new material is in Section 1.2.3: the extension computations in the block {1, Sp}, especially Lemma 1.2.3 and the table, plus the ordinary R=T argument in Section 7. The support theorem 8.0.1 is a coherent statement, and the reduction to Theorem 6.3.7 and the ordinary case is logical.\n\nSoft spots: Proposition 1.2.2 is load-bearing and its proof is explicitly left to the reader. It states a duality formula, projectivity, and essentiality for D(O), as an 'easy variant' of Paskunas' Lemmas 10.26-10.29. The stress-test chain is accurate: Lemma 1.2.6 uses it, Lemma 1.2.7 uses it together with [Col10, VII.4.12] and a 'works verbatim' adaptation of [Pas13, Lemma 10.35], and all of that feeds Proposition 1.3.2, Proposition 1.3.3, and the finite generation of V(M_infty) in Proposition 6.3.2. If any of those fail, the non-ordinary support statement loses its footing. I do not see a contradiction in the exceptional block computations, but the proof is not actually supplied. Also, several central dependencies are unpublished preprints: [EP18], [Pyv18], and your own [Tun18]. That is not a flaw by itself, but it makes the chain hard for a referee to check.\n\nThe citation pattern looks fine. There is no data-fitting or normalization forcing the result; the argument derives the support statement from p-adic local Langlands, patching, and ordinary R=T.\n\nWho this is for: specialists in p-adic Langlands and modularity lifting. If the omitted proof gets written down, this is a genuine completion of the local story. Recommendation: engage with it; send to a serious referee and require the proof of Proposition 1.2.2 and a verification of the p=2 adaptations before acceptance. My own verdict would be conditional until those are supplied.","headline":"Serious 2-adic Breuil-Mezard paper with a plausible global argument, but the load-bearing local computation is explicitly left to the reader; it deserves refereeing, not desk rejection.","tokens_in":51463,"tokens_out":2110,"would_cite":false,"duration_ms":21245,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F80","11F33","11S37"],"pacs":[],"model":"deepseek-v4-flash","headline":"The patched module is supported on every component of the 2-adic potentially semi-stable deformation ring, proving Breuil-Mézard for Q2 in the final open case and strengthening Fontaine-Mazur.","keywords":["p-adic local Langlands","Breuil-Mézard conjecture","deformation rings","modularity lifting","Taylor-Wiles patching","GL_2(Q_2)","Colmez Montreal functor","potentially semi-stable representations"],"falsifier":"Check the claimed extension table for the exceptional block $\\{1,\\mathrm{Sp}\\}$: compute $\\dim_k \\mathrm{Ext}^1_{G/Z}(\\mathrm{Sp},1)$, $\\dim_k \\mathrm{Ext}^1_{G/Z}(1,\\mathrm{Sp})$ and $\\dim_k\\mathrm{Ext}^1_{\\mathcal{D}(k)}(T_1,T_1)$ directly from the Iwahori-Hecke algebra presentation $T^2=1$, $(S+1)S=0$, and test whether they are $1$, $3$ and $3$ respectively. A single mismatch would invalidate Proposition 1.3.2 and the finite generation of $\\check{V}(\\widetilde{M}_\\infty)$ in Proposition 6.3.2, breaking Theorem 8.0.1. Alternatively, run the patched-module specialization at a closed point of an irreducible component of $R_\\infty(\\sigma)[1/p]$: the theorem predicts $M_\\infty(\\sigma^\\circ)\\otimes_{R_\\infty,y}E_y\\neq 0$, so a zero specialization would be a direct counterexample.","tokens_in":50310,"feed_emoji":"🧩","tokens_out":10170,"duration_ms":87276,"temperature":0.7,"pith_summary":"The paper proves a modularity statement for $p=2$: for a totally real field $F$ in which $2$ splits completely, the patched module attached to algebraic quaternionic forms is supported on every irreducible component of the potentially semi-stable deformation ring (Theorem 8.0.1). The route goes through the $p$-adic local Langlands correspondence for $\\mathrm{GL}_2(\\mathbb{Q}_2)$ and an ordinary $R=\\mathbb{T}$ theorem, avoiding any restriction on the residual representation at $2$. If correct, this gives a new proof of the Breuil-Mézard conjecture for two-dimensional Galois representations of $\\mathbb{Q}_2$, including the previously open case where the residual representation is a twist of a nontrivial extension of the trivial character by itself. It also removes the local restriction in the earlier proof of the Fontaine-Mazur conjecture, so global potentially semi-stable lifts with distinct Hodge-Tate weights are automorphic under the stated residual assumptions. The reader should care because the $p=2$ case has been the obstacle: the exceptional block $\\{1,\\mathrm{Sp}\\}$ of $\\mathrm{GL}_2(\\mathbb{Q}_2)$ requires new extension computations that the paper carries out.","feed_headline":"Patching hits every 2-adic deformation component","feed_subtitle":"Over the 2-adics, a patched-module proof finishes Breuil-Mézard and widens Fontaine-Mazur.","key_machinery":"The argument is carried by three pieces. First, the patched module $M_\\infty$: a module over a completed local ring $R_\\infty$, built from completed cohomology of quaternionic forms by Taylor-Wiles-Kisin patching, carrying a commuting action of $\\mathrm{GL}_2(\\mathbb{Q}_2)$. Second, Colmez's Montreal functor $\\check{V}$, an exact contravariant functor from certain $\\mathrm{GL}_2(\\mathbb{Q}_2)$-representations to $\\mathrm{Gal}(\\overline{\\mathbb{Q}}_2/\\mathbb{Q}_2)$-modules; the key new input is that $\\check{V}(\\widetilde{M}_\\infty)$ is finitely generated over $\\widetilde{R}_\\infty$ (Proposition 6.3.2), so Nakayama's lemma makes specializations nonzero. Third, the exceptional-block computations for $\\{1,\\mathrm{Sp}\\}$: the author computes extension groups such as $\\mathrm{Ext}^1_{\\mathcal{D}(k)}(T_1,T_1)\\simeq k^3$ and proves the injectivity statement of Lemma 1.2.7, which are exactly what is needed to make the finiteness and equivalence arguments work when $p=2$ and the residual representation has scalar semi-simplification. The ordinary components are then covered by an ordinary $R=\\mathbb{T}$ theorem, and the irreducible/non-ordinary components by the $\\check{V}$-based argument, so together the support theorem follows.","core_discovery":"The paper's central claim is Theorem 8.0.1: with $p=2$ and $F$ a totally real field in which $p$ splits completely, the support of $M_\\infty(\\sigma^\\circ)\\otimes_{\\mathbb{Z}_p}\\mathbb{Q}_p$ meets every irreducible component of $R_\\infty(\\sigma)[1/p]$ for every locally algebraic type $\\sigma$. Here $R_\\infty(\\sigma)$ is the patched, fixed-type potentially semi-stable deformation ring and $M_\\infty(\\sigma^\\circ)$ the corresponding patched module of algebraic quaternionic modular forms. The author proves this by combining a finite-generation statement for $\\check{V}(\\widetilde{M}_\\infty)$, the image of the patched module under Colmez's Montreal functor, with an ordinary $R=\\mathbb{T}$ theorem for the reducible locus. From the support theorem, Corollary 8.0.2 derives the Breuil-Mézard conjecture, in both multiplicity and cycle form, for all continuous two-dimensional representations of $\\mathrm{Gal}(\\overline{\\mathbb{Q}}_2/\\mathbb{Q}_2)$, including the previously open case in which the residual representation is a twist of an extension of the trivial character by itself. Theorem 8.0.3 then removes the local restriction in the earlier proof of the Fontaine-Mazur conjecture: global potentially semi-stable representations with distinct Hodge-Tate weights are modular whenever the residual representation is modular, totally odd and has non-solvable image.","pith_inferences":["The decisive local input is the computation of extensions in the block $\\{1,\\mathrm{Sp}\\}$: the $3$-dimensionality of $\\mathrm{Ext}^1_{\\mathcal{D}(k)}(T_1,T_1)$ and the injectivity statement for $\\check{V}$ are what make finite generation of $\\check{V}(\\widetilde{M}_\\infty)$ possible. Similar block calculations for other groups or for ramified extensions of $\\mathbb{Q}_2$ would likely carry the su","Because the non-ordinary locus is handled entirely by $\\check{V}$ and the ordinary locus by an ordinary $R=\\mathbb{T}$ theorem, the paper suggests a template: separate components by whether the associated local Galois representation is reducible, then use local-global compatibility in the irreducible case and Hida-theoretic methods in the reducible case. A testable extension is to verify the same ","The density statement for $n_y=1$ implies that, away from a thin set, the $p$-adic local Langlands correspondence attaches a single representation to each point; this could be used to compute the cycles in the geometric Breuil-Mézard conjecture directly rather than through multiplicities.","The faithfulness result the paper imports is essential for passing from finite generation of $\\check{V}(M_\\infty)$ to nonzero specializations; a natural stress test is to check whether the support theorem survives for residual representations where that faithfulness may fail."],"forward_implications":["The Breuil-Mézard conjecture, in both multiplicity and cycle forms, holds for every continuous two-dimensional representation of $\\mathrm{Gal}(\\overline{\\mathbb{Q}}_2/\\mathbb{Q}_2)$; the previously open case of a twist of an extension of $1$ by $1$ is included.","A modularity lifting theorem holds for $p=2$ over totally real fields in which $2$ splits completely, for potentially semi-stable lifts with distinct Hodge-Tate weights, assuming only that the residual representation is modular, totally odd, and has non-solvable image.","The local restriction in the earlier proof of the Fontaine-Mazur conjecture is removed: the earlier theorem required the residual representation at $2$ not to be a twist of an extension of a character by itself, and this paper drops that condition.","Every irreducible component of the potentially semi-stable deformation ring is automorphic, so the patched eigenvariety contains points of every such component, giving geometric control beyond the numerical Breuil-Mézard equality.","The author notes the same patching method applies to $p>2$ without change, so the support theorem is not special to $p=2$ except for the block computations carried out here."],"supporting_citations":[{"why":"Supplies the Montreal functor $\\check{V}$ and its basic values, including the exceptional-block facts used in Lemma 1.2.7.","marker":"[Col10]"},{"why":"Provides the block decomposition and the template for Lemmas 1.2.6 and 1.2.7, whose $p=2$ variants carry the finiteness argument.","marker":"[Paˇ s13]"},{"why":"Establishes the $p=2$ patching and modularity lifting framework that this paper extends by removing the local restriction.","marker":"[Paˇ s16]"},{"why":"Original strategy showing that patched ring isomorphisms imply the Breuil-Mézard conjecture.","marker":"[Kis09a]"},{"why":"The Taylor-Wiles-Kisin patching construction with $\\mathrm{GL}_2(\\mathbb{Q}_p)$-action and local-global compatibility used for $M_\\infty$.","marker":"[CEG+16]"},{"why":"Faithfulness of the $R_\\infty$-action on $M_\\infty$, used to pass from finite generation of $\\check{V}(M_\\infty)$ to nonzero specializations.","marker":"[EP18]"},{"why":"States the Breuil-Mézard conjecture whose $p=2$ case is the main target.","marker":"[BM02]"},{"why":"Reformulates Breuil-Mézard as cycle equalities and provides the equivalence with patched-module support used in Theorem 5.3.3.","marker":"[EG14]"},{"why":"The author's earlier proof for $p>2$; the present paper follows its strategy with the necessary $p=2$ extension computations.","marker":"[Tun18]"},{"why":"Ordinary $R=\\mathbb{T}$ machinery used for the ordinary and partially ordinary components.","marker":"[Ger10]"}],"fun_headline_variants":["2-adic Breuil-Mézard: patched modules cover all components","Patched modules hit every 2-adic deformation component","New proof for 2-adic Breuil-Mézard via patched support","Removing the local restriction in Fontaine-Mazur via patching"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper's load-bearing premise is that the new local extension computations in the exceptional block $\\{1,\\mathrm{Sp}\\}$ of $\\mathrm{GL}_2(\\mathbb{Q}_2)$ are correct—especially Proposition 1.2.2, which is stated as an easy variant of earlier results with the proof left to the reader, and Lemma 1.2.7, which depends on two quoted local facts; if either fails, the finite generation of $\\check{V}(\\widetilde{M}_\\infty)$ and hence the support theorem collapse through Lemma 1.3.1 and Proposition 6.3.2.","fun_headline_variants_meta":{"raw":{"variants":["2-adic Breuil-Mézard: patched modules cover all components","Patched modules hit every 2-adic deformation component","New proof for 2-adic Breuil-Mézard via patched support","Removing the local restriction in Fontaine-Mazur via patching"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000238,"raw_usage":{"total_tokens":1524,"prompt_tokens":973,"completion_tokens":551,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":589,"completion_tokens_details":{"reasoning_tokens":473}},"tokens_in":589,"tokens_out":551,"duration_ms":5157,"temperature":1.0,"reasoning_tokens":473,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:53:54.689766+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the claimed extension table for the exceptional block $\\{1,\\mathrm{Sp}\\}$: compute $\\dim_k \\mathrm{Ext}^1_{G/Z}(\\mathrm{Sp},1)$, $\\dim_k \\mathrm{Ext}^1_{G/Z}(1,\\mathrm{Sp})$ and $\\dim_k\\mathrm{Ext}^1_{\\mathcal{D}(k)}(T_1,T_1)$ directly from the Iwahori-Hecke algebra presentation $T^2=1$, $(S+1)S=0$, and test whether they are $1$, $3$ and $3$ respectively. A single mismatch would invalidate Proposition 1.3.2 and the finite generation of $\\check{V}(\\widetilde{M}_\\infty)$ in Proposition 6.3.2, breaking Theorem 8.0.1. Alternatively, run the patched-module specialization at a closed point of an irreducible component of $R_\\infty(\\sigma)[1/p]$: the theorem predicts $M_\\infty(\\sigma^\\circ)\\otimes_{R_\\infty,y}E_y\\neq 0$, so a zero specialization would be a direct counterexample.","supporting_citations":[],"review_version":1}