{"id":"c91b5f83-f920-4872-bc63-99439aedd74a","arxiv_id":"2412.16128","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves the p=2 principal-block case of a continuity conjecture for character rationality and conjectures a local criterion for the level-1 gap.","lead":"This paper proves that a new principal-block version of a recent conjecture about the continuity of 2-rationality of characters holds, and proposes a group-theoretic conjecture to explain when rationality jumps from level 2 to level 0. Generalist readers may care because it narrows the gap toward the McKay-Navarro conjecture, a major open problem in finite group theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 4.3 for the SL_{2n}(ε q) family is left as a one-line analogy to the E6 case; it never establishes that the 2-central elements of PGL_{2n}(ε q) form a cyclic group whose powers produce the needed lower rationality levels.","rationale":"The reader's weak-assumption analysis correctly spotlights Proposition 4.2's 'quick inspection' as a written gap, but I judge the SL_{2n} one-sentence analogy to be the more load-bearing concern. Proposition 4.2, if false, would affect only unipotent characters, a finite and well-tabulated family for each exceptional type; the assertion is concrete and readily checkable. The SL_{2n} case, by contrast, covers an infinite family of groups and dismisses the entire verification with 'arguing exactly as in the previous case'. That analogy requires the 2-central elements of PGL_{2n}(ε q) to have a very specific structure: a single cyclic group of order (q−ε)_2 whose powers generate all 2-central classes, with rationality levels determined by log_2 of the element order. The paper does not prove this structure; it cites tables for connectedness and a block-containment theorem, neither of which directly establishes the cyclic-power chain or the level formula. If that structure fails for some n, Theorem 4.3(b) has no proof for that family, and Theorem A is not established. The reader did mention the SL_{2n} deferral in the rationale, so we partially agree, but our weighting of which gap is most load-bearing differs. The appropriate verdict remains CONDITIONAL: the gaps are fillable but must be written out before the theorem is fully verified.","tokens_in":12942,"tokens_out":16928,"duration_ms":139599,"concrete_test":"For the smallest nontrivial case n=3, q=5 (q ≡ 1 mod 4, n not a 2-power), compute explicitly the 2-central semisimple classes of G* = PGL_6(5) and the 2-rationality levels of the corresponding semisimple characters in the principal 2-block of SL_6(5). Verify that the 2-central elements form a cyclic subgroup of order (5−1)_2 = 4, that the character in E(G,t) with |t| = 2^a has level a, and that for every A-invariant series E(G,t) the series E(G,t^k) are A-invariant with levels a − v_2(k). If the cyclic-power structure fails or the levels do not behave as log_2|t|, then the 'arguing exactly as in the previous case' step in Theorem 4.3 is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem A relies on Theorem 4.3, whose proof of Conjecture 3.1(b) for p=2 ends with the sentence: 'But then we can argue exactly as in the previous case' for G = SL_{2n}(ε q) with 4 | (q−ε) and n not a 2-power. The preceding E6 argument uses three specific facts: (1) the nontrivial elements of the centre Z of a Sylow 2-subgroup have connected centralizer of type D5, giving a unique semisimple character in each Lusztig series; (2) by [9, Thm 5.6] the 2-rationality level of every character in E(G,t) is log_2|t|; and (3) from A-invariance of one character in E(G,t), the classes t^k are A-stable, so the unique semisimple characters χ_{t^k} supply all levels β ≤ α. For SL_{2n}(ε q), the proof cites [8, Tab. 4.5.1] for connected centralizers and [3, Thm 21.14] for the principal-block containment, but it never shows that the 2-central elements form a single cyclic subgroup of order (q−ε)_2 whose powers exhaust all needed t's. If the 2-central classes are not of this cyclic-power form, the chain of characters at levels 2 ≤ β ≤ α is not constructed, and the continuity conclusion for this infinite family is unsupported. This is the least documented step in the proof of the main theorem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the p-part of character conductors of finite groups. Its main result, Theorem A, proves the principal-block form of Hung's continuity conjecture for p=2: if the principal 2-block B0(G) contains a p′-degree irreducible character whose 2-rationality level is α≥2, then B0(G) contains p′-degree irreducible characters of every level β with 2≤β≤α. The proof combines a reduction theorem (Theorem 3.3) that, assuming a conjectural statement for almost simple groups (Conjecture 3.1), reduces the problem to almost simple groups, with a verification for p=2 in Theorem 4.3 using Lusztig series and Jordan decomposition. The paper also formulates Conjecture B, a local group-theoretic criterion for the absence of p′-degree characters of p-rationality level 1, proves it for groups with a normal Sylow p-subgroup (Theorem 5.5), and derives applications to p-solvable, sporadic, symmetric, and alternating groups via the McKay–Navarro conjecture.","tokens_in":13266,"tokens_out":14936,"duration_ms":127495,"significance":"The main theorem is a genuine refinement of Hung's continuity result, since it localizes the phenomenon to the principal block and the reduction is effective for p=2. The proof strategy is transparent: external results of Hung, Malle, Navarro–Tiep, and others are used as hypotheses rather than as circular inputs, and I see no circularity in the main argument. Theorem 3.3 is carefully structured and is a reusable reduction statement. Section 5 gives a substantive normal-Sylow proof of a new local conjecture, and the use of semi-inertia subgroups and coprime actions is elegant. If the compressed points in Section 4 are filled in, the paper will be a solid contribution to the area and relevant to the Alperin–McKay–Navarro program.","major_comments":[{"comment":"The treatment of G=SL_{2n}(εq) is too compressed to be checked as written. After quoting [8, Tab. 4.5.1] and [3, Thm 21.14], the proof ends with 'we can argue exactly as in the previous case', but the previous case used the explicit cyclic group Z of order (q−ε)_2 and then formed the powers t^k. For PGL_{2n}(εq) this step is not automatic from connectedness of centralizers: one must say that if t is the given 2-central element of order 2^α, then each t^{2^{α−β}} is again 2-central and its centralizer is among the connected Levi subgroups, so [9, Thm 5.6] supplies rationality level β and the unique semisimple character in E(G,t^{2^{α−β}}) is A-invariant. As written, the chain of characters realizing levels 2≤β≤α is not actually constructed for this infinite family. Please expand this paragraph explicitly.","section":"§4, Theorem 4.3 (final paragraph)"},{"comment":"The proof of the claim that every unipotent character of odd degree is rational is reduced to 'a quick inspection shows that all irrational unipotent characters have even degree'. This is load-bearing, because Theorem 4.3 uses it to discard all unipotent characters from the p=2 argument. The finite check over the exceptional families should be made explicit, for example by citing the relevant character-degree and character-field tables in [6] and explaining why every row with irrational field has even degree. Without this, the statement is an assertion rather than a verified step.","section":"§4, Proposition 4.2"}],"minor_comments":[{"comment":"The sentence 'By hypothesis, we can assume that P⊴G' is not justified by Conjecture 2.1 as stated. Conjecture 2.1 gives a bijection between Irr_{p′}(G) and Irr_{p′}(N_G(P)); the proof should transfer the given character to N_G(P), construct characters there, and transfer back. As written the lemma is incomplete, although the intended reduction is clear.","section":"§2, Lemma 2.7"},{"comment":"The symbol G is used for both the finite group and the Galois group Gal(Q_p/Q), which makes the proof hard to follow (for example, 'Write G = Gal(Q_p/Q)' while G is also the group under study). Please use a different notation such as Γ or 𝒢 for the Galois group.","section":"§5, proof of Theorem 5.5"},{"comment":"The odd-degree condition for the semisimple characters χ_t is implicit: it follows because t is 2-central, so a Sylow 2-subgroup is contained in C_{G^*}(t) and the degree is odd. Stating this explicitly would make the argument easier to verify and would prepare the same fact for the SL_{2n} case.","section":"§4, Theorem 4.3 (E6 paragraph)"}],"recommendation":"major_revision","confidential_remarks":"The mathematical core appears sound, but the Section 4 proof has two places where infinite-family verification is asserted rather than demonstrated. Both points are fixable within the manuscript's scope, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: Theorem A is real progress. It proves Hung's principal-block continuity conjecture for p=2, and Conjecture B gives a plausible local criterion for the gap at level 1. The reduction theorem 3.3 is well-structured, and the normal-Sylow proof in Section 5 is the strongest part. I'd send this to a good referee; it will need a revision to fill two compressed spots, but none of it smells wrong.\n\nWhat's new: principal block version for p=2 (Theorem A) and the two equivalent local conditions in Conjecture B. Both are genuine extensions, not repackaged. The use of Navarro-Tiep, Malle, Hung is standard tool-style citation, with no circularity: the target statements are not used as inputs.\n\nSoft spots: The proof of Theorem 4.3 for SL_{2n}(εq) is a one-line analogy to E6. In the E6 case you need the cyclic center of a Sylow 2-subgroup to produce t's of every order 2^β; for SL_{2n}, the analogous cyclic structure of 2-central elements is not stated, only cited to GLS tables. The missing sentence is small but load-bearing for the infinite family. Likewise, Proposition 4.2's 'quick inspection' for exceptional unipotent characters is a finite check that should be one table or a reference. These are fillable gaps, not contradictions. My stress-test note about the SL2n case doesn't change my overall confidence; the argument 'argue exactly as before' is a compression, not an error.\n\nOne thing I disagree with in the reader's take: I'd weight the gap slightly smaller. The structure of 2-central elements in PGL_{2n} is standard, and the citation to [8, Tab 4.5.1] plus [3, Thm 21.14] nearly does the job. It still needs to be written out.\n\nBottom line: worth a referee's time. The main theorem is likely correct; Conjecture B is well-motivated and proven in the normal Sylow case. I would ask the authors to expand the two compressed steps before publication.","headline":"A genuine principal-block refinement of Hung's p=2 theorem plus a new local criterion for the rationality gap; the Lie-type checks are compressed but the argument holds up.","tokens_in":13813,"tokens_out":4180,"would_cite":true,"duration_ms":33424,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20C15","20C33"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves Hung's continuity conjecture for p=2 inside the principal 2-block, and proposes a local criterion for the level-1 gap.","keywords":["p-rationality","character conductors","principal block","characters of p'-degree","finite groups of Lie type","unipotent characters","McKay–Navarro conjecture","Galois automorphisms"],"falsifier":"Compute directly the unipotent character tables of $E_7(q)$ or $E_8(q)$ with $q$ odd and check whether any odd-degree unipotent character has non-rational values; such a character would refute Proposition 4.2 and with it the proof of Theorem A.","tokens_in":12728,"feed_emoji":"🧮","tokens_out":10868,"duration_ms":81372,"temperature":0.7,"pith_summary":"The paper proves that the continuity of 2-rationality is already visible inside the principal 2-block: if that block contains a character of odd degree whose conductor has 2-part $2^a$, then it contains odd-degree characters with conductor 2-part $2^b$ for every integer $b$ from 2 up to $a$. This confirms a conjecture of Hung for $p=2$, and it does so by reducing the conjecture for all primes to two explicit problems about almost simple groups, then solving those problems for $p=2$. The same reduction shows the Alperin–McKay–Navarro conjecture would imply the full statement. In a separate thread, the paper proposes a local group-theoretic criterion that explains why continuity stops at level 1: it predicts, in terms of the action of $N_G(P)/\\Phi(P)$ on $P/\\Phi(P)$, exactly when no $p'$-degree character has $p$-rationality level 1, and verifies this criterion for groups with a normal Sylow $p$-subgroup.","feed_headline":"Odd-degree characters fill every 2-rationality level","feed_subtitle":"Principal 2-blocks are shown to contain odd-degree characters at every conductor level 2^b up to the maximum.","key_machinery":"The key objects are the $p$-rationality level of a character, the exponent $\\alpha$ such that $c(\\chi)_p=p^\\alpha$, and the principal block $B_0(G)$. The proofs use the Galois automorphism $\\sigma_\\alpha$, which fixes $p$-power order roots and sends $p'$-roots to their $1+p^\\alpha$-th power, and the semi-inertia subgroup of a character under the Galois action. For the $p=2$ verification, the load-bearing structure is the decomposition of $\\mathrm{Irr}(B_0(G))$ into Lusztig series $\\mathcal{E}(G,t)$ indexed by 2-elements $t$ in the dual group, together with Jordan decomposition, which forces odd-degree characters to correspond to 2-central $t$, and Proposition 4.2, which says odd-degree unipotent characters are rational valued.","core_discovery":"The central result is Theorem A: Conjecture 1.1 holds for $p=2$. That is, if $B_0(G)$ contains an irreducible character $\\chi$ of odd degree with $c(\\chi)_2=2^a$, then for every $2\\le b\\le a$ there is an odd-degree $\\psi\\in \\mathrm{Irr}(B_0(G))$ with $c(\\psi)_2=2^b$. The route is a general reduction, Theorem 3.3, showing that the conjecture follows for any $p$ once an explicit statement about almost simple groups (Conjecture 3.1) is verified; the authors verify it for $p=2$ in Theorem 4.3, using the description of the principal 2-block of a group of Lie type as a union of Lusztig series and the fact that odd-degree unipotent characters are rational. The paper also formulates Conjecture B, a necessary-and-sufficient local condition on $N_G(P)/\\Phi(P)$ acting on $P/\\Phi(P)$ for the absence of characters of $p'$-degree with $p$-rationality level 1, and proves it for groups with a normal Sylow $p$-subgroup.","pith_inferences":["The dependence on $P/\\Phi(P)$ in Conjecture B suggests the level-1 gap is governed by the minimal number of generators of a Sylow subgroup; one could test this by computing, in families not yet covered, whether the condition can be simplified to an inequality involving that number.","The $p=2$ proof for Lie-type groups points to a possible odd-prime analogue: when $p$ divides $q-\\varepsilon$, the role of 2-central elements might be played by elements of order $|t|_p$, and the semisimple characters in the corresponding Lusztig series would supply the required levels.","The paper's $D_{24}$ example shows that level-1 phenomena are not visible in the principal block; this suggests that any general principal-block version of the gap must pass to $G/O_{p'}(G)$ or to the defect group, as Conjecture 5.7 does."],"forward_implications":["Hung's continuity conjecture holds for $p=2$ for every finite group, so the principal 2-block cannot hide gaps in the ladder of 2-rationality levels.","Any counterexample to Conjecture 1.1 for an odd prime must occur in an almost simple group, reducing the full conjecture to the verification of Conjecture 3.1.","The results are consistent with the Alperin–McKay–Navarro conjecture: both the continuity in the principal block and the level-1 gap criterion are consequences of that conjecture.","Conjecture B gives a computable local criterion for the level-1 gap, and Theorem 5.5 confirms it for groups with a normal Sylow $p$-subgroup, covering $p$-solvable, sporadic, symmetric, alternating, and defining-characteristic Lie-type groups."],"supporting_citations":[{"why":"States Hung's conjecture and proves the p=2 case without the principal-block restriction, providing the baseline and motivation.","marker":"[9]"},{"why":"Gives the p=2 case of Conjecture 3.1(a) and the exponent bound for Sylow 2-subgroups used in Theorem 4.3.","marker":"[13]"},{"why":"Shows the largest p-rationality level is governed by exp(P/P') and is attained in the principal block, used in the reduction.","marker":"[20]"},{"why":"Provides the Broué–Michel description of the principal 2-block as a union of Lusztig series over 2-elements.","marker":"[3]"},{"why":"Supplies the character theory of finite reductive groups, including character fields, Jordan decomposition, and semisimple characters.","marker":"[6]"},{"why":"Establishes that the relevant semisimple characters lie in the principal 2-block for the E6 and SL2n cases.","marker":"[5]"},{"why":"The McKay–Navarro conjecture, used to derive the continuity and the level-1 gap criterion.","marker":"[16]"},{"why":"Provides canonical extensions, Clifford–Gallagher theory, and Brauer's lemma on character tables used in the block arguments.","marker":"[17]"}],"fun_headline_variants":["Odd-degree characters in principal 2-blocks span all 2-rationality levels","Principal 2-blocks hold odd-degree characters at every conductor level","Odd-degree characters realize all 2-rationality levels in principal blocks","Each 2-rationality level appears via odd-degree characters in B0"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the $p=2$ simple-group check relies on the claim, justified in the text only by a quick inspection, that every irrational unipotent character of a finite group of Lie type in odd characteristic has even degree; if an odd-degree unipotent character were irrational, the chain leading to Theorem A would break.","fun_headline_variants_meta":{"raw":{"variants":["Odd-degree characters in principal 2-blocks span all 2-rationality levels","Principal 2-blocks hold odd-degree characters at every conductor level","Odd-degree characters realize all 2-rationality levels in principal blocks","Each 2-rationality level appears via odd-degree characters in B0"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000961,"raw_usage":{"total_tokens":4044,"prompt_tokens":847,"completion_tokens":3197,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":463,"completion_tokens_details":{"reasoning_tokens":3115}},"tokens_in":463,"tokens_out":3197,"duration_ms":21786,"temperature":1.0,"reasoning_tokens":3115,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:45:14.110195+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute directly the unipotent character tables of $E_7(q)$ or $E_8(q)$ with $q$ odd and check whether any odd-degree unipotent character has non-rational values; such a character would refute Proposition 4.2 and with it the proof of Theorem A.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States Hung's conjecture and proves the p=2 case without the principal-block restriction, providing the baseline and motivation."},{"cited_title":"Malle , The Navarro–Tiep Galois conjecture for p = 2","cited_arxiv_id":null,"evidence_quote":"Gives the p=2 case of Conjecture 3.1(a) and the exponent bound for Sylow 2-subgroups used in Theorem 4.3."},{"cited_title":"Navarro and P","cited_arxiv_id":null,"evidence_quote":"Shows the largest p-rationality level is governed by exp(P/P') and is attained in the principal block, used in the reduction."},{"cited_title":"Geck and G","cited_arxiv_id":null,"evidence_quote":"Supplies the character theory of finite reductive groups, including character fields, Jordan decomposition, and semisimple characters."},{"cited_title":"Enguehard , Sur les l-blocs unipotents des groupes r´ eductifs finis quand l est mauvais","cited_arxiv_id":null,"evidence_quote":"Establishes that the relevant semisimple characters lie in the principal 2-block for the E6 and SL2n cases."},{"cited_title":"Navarro, The McKay conjecture and Galois automorphisms","cited_arxiv_id":null,"evidence_quote":"The McKay–Navarro conjecture, used to derive the continuity and the level-1 gap criterion."},{"cited_title":"Navarro , Character Theory and the McKay Conjecture","cited_arxiv_id":null,"evidence_quote":"Provides canonical extensions, Clifford–Gallagher theory, and Brauer's lemma on character tables used in the block arguments."}],"review_version":1}