{"id":"64faf00d-4427-4310-b0f8-5e87e9afe0f8","arxiv_id":"2607.23455","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"All unramified motivic MMVs of depth ≤3 are classified, with two almost-extremal and one height-one infinite families proved unramified and general conjectures stated for higher depth.","lead":"The paper classifies exactly which motivic multiple mixed values of depth at most three can be written in terms of ordinary motivic multiple zeta values, and finds several infinite unramified families of higher depth. It advances the Galois-descent program that decides when level-two period numbers collapse to classical MZVs.","discovery_kind":"extension","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"The \"precisely three irregular values\" completeness claim rests on §5 subcase arguments that are partly sketched, assert non-cancellation of binomial coefficients without proof, and contain at least one degenerate edge case where the stated coefficient vanishes.","rationale":"I read the paper in good faith: the architecture (reduce unramifiedness to the Brown–Glanois descent criterion, then do explicit coaction computations) is standard and sound, the criterion is established in the literature, and the positive direction (Lemma 4.1, Theorems 6.2–6.3, 7.5) is proved in detail with explicit cut-by-cut computations. The load-bearing soft spot is narrower than the reader's flagged assumption: not the criterion itself, but the exhaustiveness of the §5 case analysis on which the \"precisely three\" completeness claim depends. Three of the four ramifiedness theorems are sketched, and the one fully written case already exhibits a degenerate edge (b=0) where the displayed coefficient formula vanishes and the stated positivity argument fails; the conclusion holds only through an unstated sign correction. This does not make the claim likely false — the edge case resolves in the authors' favor, and the overall pattern of cancellations is consistent — but it means the completeness half of Theorem 4.3 is verified at a lower standard than the rest of the paper. I agree partially with the reader: they located the right general area (reliance on the descent machinery and the §5 sketches) but misidentified the mechanism; the criterion is a theorem, the application is the risk. The reader's CONDITIONAL verdict, justified partly by the §5 sketches, already prices in this concern, so I recommend no change. A symbolic coaction computation over a finite weight range would settle completeness up to that bound and either close the concern or locate a counterexample.","tokens_in":37202,"tokens_out":5476,"duration_ms":82452,"concrete_test":"Implement the level-2 motivic coaction D_r on iterated integrals symbolically (Glanois's algorithm; cf. existing implementations for motivic Euler sums) and compute, for every irregular depth-3 pattern σ ∈ {(1,−1,−1),(1,1,−1),(−1,1,1),(−1,−1,1)} and all s with |s| ≤ 12, the images D_r M^m(s;σ) for all odd r < |s|, verifying each has a component outside L_r ⊗ H_{w−r}. Include the edge subcases b=0 of Theorem 5.1 Case III, e.g. M^m(2,1̌,2̌), M^m(3,1̌,2̌), M^m(2,1̌,3̌), and recompute Theorem 5.1 (III.1) with the corrected T^l(1)=−ζ̃^l(1) sign. Any irregular value outside the three listed passing the criterion refutes Theorem 4.3(5).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The strongest claim is Theorem 4.3's exact classification — in particular that ONLY three irregular-pattern triple MMVs are unramified. The unramifiedness direction is solid (Lemma 4.1 computes the coactions explicitly). The fragile direction is exclusion: Theorems 5.1–5.4 must show every other irregular MMV has some D_r landing outside L_r ⊗ H_{w-r}. Only Theorem 5.1 is proved in full; 5.2 is \"only sketched,\" and 5.3/5.4 repeatedly say \"we can show that C ≠ 0\" / \"clearly ramified\" without displaying the check. The completeness claim is only as strong as the weakest of these unshown non-cancellation arguments: if one coefficient vanishes with no compensating ramified term, an extra unramified value exists and the list is wrong.\n\nThis is not hypothetical. In Theorem 5.1, Case (III.1) (w odd, b even, a,c>0), the S-part coefficient of X is 2^{b+1} − 2 + δ_{b≥c≥0}(b choose c), asserted \"clearly positive.\" For b=0 (which is allowed: b even, a,c>0, e.g. (a,b,c)=(1,0,1), i.e. M^m(2,1̌,2̌)) this coefficient is 0, and the t-part coefficient also vanishes, so the displayed formula gives D_{b+1}M^m = 0 and the subcase argument collapses. The ramified conclusion is rescued only because the reduction T^l(b+1) = (2^{b+1}−1)ζ̃^l(b+1) (Lemma 3.1(2), valid for k≥2) fails at b=0, where T^l(1) = log^l 2 = −ζ̃^l(1), flipping a sign so the true D_1 is −2log^l 2 ⊗ S^m(a+1,c+1) ≠ 0. The conclusion survives, but the written argument does not cover it — meaning the boundary cases of these coaction formulas are not being tracked reliably. Relatedly, Theorem 5.1 (III.1) cites Theorem 4.3(4) — a depth-3 statement, itself being proved via §5 — for a depth-2 S-value that is actually covered by Lemma 3.1(9): a citation slip inside the proof of the very theorem at issue, benign but symptomatic. Note the Brown–Glanois criterion itself (the reader's flagged assumption) is a published theorem (Brown Thm 3.3, Glanois Cor 2.4); the risk lies in its application here, not in the criterion.","agreement_with_reader":"partial"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The paper studies multiple mixed values (MMVs), level-two variants of MZVs defined by parity restrictions on summation indices, and asks which motivic MMVs are unramified, i.e., descend to the level-one motivic MZV algebra H_1 via the Brown–Glanois descent criterion (Theorem 2.1: D_1 vanishes and all odd coactions D_r land in L_r ⊗ H_{w−r}). Main results: (i) Theorem 4.3, a complete classification of unramified motivic MMVs in depth ≤ 3, including the assertion that exactly three irregular-parity values — M^m(1̌,1̌,3), M^m(1̌,2,2), M^m(1̌,4,4) — are unramified; (ii) two infinite unramified families of unbounded depth and near-extremal height (Theorems 6.2, 6.3), proved by induction with explicit cut enumeration; (iii) a height-one family V_j^m(d) in even depths with explicit MZV expressions (Theorems 7.5, 7.12), one member's closed form conditional on a clearly labeled conjecture (Conjecture 7.7); (iv) conjectures in all depths ≥ 4 based on numerical experiments. The unramifiedness direction of Theorem 4.3 is proved by explicit coaction computation (Lemma 4.1); the exclusion direction occupies §5 (Theorems 5.1–5.4).","tokens_in":37760,"tokens_out":10094,"duration_ms":50777,"significance":"If correct, this is the first complete motivic unramifiedness classification for all parity patterns in depth three, extending the authors' prior work on the regular t-/T-/S-families, and it identifies three sporadic irregular values plus three new infinite families with explicit reductions to MZVs. Strengths: the descent criterion reduces everything to finite, checkable linear algebra; the proofs of Lemma 4.1, Lemma 3.3, Theorems 6.2–6.3 and 7.5/7.12 are fully displayed with explicit cut diagrams; the conditional statement (Theorem 7.12 modulo Conjecture 7.7) is honestly flagged; and §8 formulates concrete, falsifiable conjectures. The results are unconditional at the motivic level, with analytic consequences conditional on the period conjecture in the standard way. The field is specialized but active, and completeness theorems of this kind are the natural benchmarks for it.","major_comments":[{"comment":"Theorem 5.1, Case (III.1): the boundary b=0 is not covered by the written argument, and the assertion that the S-part coefficient is 'clearly positive' is false there. For (a,b,c)=(1,0,1) (i.e. M^m(2,1̌,2̌), allowed: w=5 odd, b=0 even, a,c>0) the S-coefficient 2^{b+1}−2+δ_{b≥c≥0}C(b,c) equals 0, and the t-coefficient also vanishes (C(b+a,b−a)=C(1,−1)=0, δ_{a=b}=0), so the displayed formula gives D_{b+1}M^m=0 and the subcase argument collapses. The ramified conclusion survives only because the reduction T^l(b+1)=(2^{b+1}−1)ζ̃^l(b+1) (Lemma 3.1(2), stated for k≥2) fails at b=0, where T^l(1)=log^l 2=−ζ̃^l(1) (Lemma 3.1(1)); a direct computation gives D_1=−2log^l 2⊗S^m(a+1,c+1)≠0. The same gap affects Case (III.2) when a=b=0 (c even ≥2): the text claims 'δ_{a=b}=0 since b is odd' — b is even in Case III, and δ_{a=b}=1 at a=b=0 — and the displayed X vanishes identically. Please add the b=0 bo","section":"§5, Theorem 5.1, Cases (III.1)–(III.2)"},{"comment":"The exclusion half of Theorem 4.3 (the 'precisely three irregular values' claim) rests on Theorems 5.2–5.4, where the key non-vanishing/non-cancellation steps are asserted rather than shown. Theorem 5.2 is 'only sketched'; Case (III) ends with 'it can be shown that D_{b+1}M^m is ramified' with no argument, and Case (IV) lists five subcases with 'we can show'. Theorem 5.3 Case (V) says 'By consider the coaction D_{a+b+1}... we can show that it is always ramified', and Case (VI) asserts C≠0 in five subcases without display. Theorem 5.4 (III.2) contains the incomplete sentence 'But we can show that D_{a+c}X by the same idea as used before.' Given that the one fully written proof (Theorem 5.1) harbors the genuine b=0 gap above, these omitted checks are load-bearing for the completeness claim and should be written out in full, at least in an appendix.","section":"§5, Theorems 5.2–5.4"},{"comment":"Case (III.1) restricts to a=0, b=c≥5 and declares D_{b+4}M^m(1̌,b′,b′) 'clearly ramified'. This requires two unstated inputs: (i) T^m(1,b−2) is ramified (weight b−1 even, so via Lemma 3.1(9) or [19, Thm. 1.2]), and (ii) the two displayed terms cannot cancel since their left factors are proportional via T^l(b+4)=(2^{b+4}−1)ζ̃^l(b+4) while exactly one right factor is ramified. More importantly, the threshold b=c≥5 is unexplained: for b=c=3 (the exception M^m(1̌,4,4)) the cut D_{b+4} exists (r=7≤w−1=8) but the displayed formula must fail, since Lemma 4.1(iii) computes D_7=T^l(7)⊗ζ̃^m(2) with no second term. Please explain why the cut formula degenerates at b=c=3 and justify the restriction; this is another instance of small-parameter degeneration of the coaction formulas that the reader needs to see handled.","section":"§5, Theorem 5.4, Case (III.1)"}],"minor_comments":[{"comment":"Remark 4.2: 'Since D_3M^m(1̌,1̌,3)=0, we now can lift (4.1)' contradicts the line above, which computes D_3=−2T^l(3)⊗ζ^m(2)≠0. Presumably D_1=0 is meant.","section":"§4, Remark 4.2"},{"comment":"Typos: §1.4 'conjecture of Kaneko and Tsumura. Tsumura.'; §7 title 'unrmaified'; Lemma 3.3 'Turing to'; Theorem 4.3 proof 'mixed values of with irregular'; Theorem 5.2 'the proceeding proof'; Theorem 5.4 'Case (IV))'; missing superscript on ζ̃(b+4) in Theorem 5.4 (III.1).","section":"passim"},{"comment":"The index ranges for the unramified V_j(d) are stated inconsistently: §7 intro and Theorem 7.1 say j=∅,2,…,d−1 (with a log 2 correction needed at j=1,d), while Theorem 7.5's display is given 'for all 1≤j≤d'. Please state once, precisely, which V_j^m(d) are unramified and which require the correction term 2(δ_{j=1}−δ_{j=d})T^m(d)log^m 2. In Lemma 7.4 the 'In particular' cases overlap (j=n=1 satisfies n≥1).","section":"§7, Theorems 7.1/7.5, Lemma 7.4"},{"comment":"In the proof of Theorem 7.5, cut ④ cites 'Cor. 7.4' (presumably Cor. 7.2) and 'Lemma (7.4)'; please check cross-references. It would help the reader to state explicitly at the start of §5 that non-admissible (c′=1) MMVs are included via shuffle regularization, since Cases (II) of Theorems 5.1–5.4 concern c=0.","section":"§7, Theorem 7.5; §5"},{"comment":"The numerical evidence underlying Conjectures 8.2–8.6 is described as 'extensive' but not documented. A brief description (weight/depth ranges searched, use of [1], and the motivic vs. analytic status of the checks) would strengthen §8 without lengthening it much.","section":"§8"}],"recommendation":"major_revision","confidential_remarks":"The paper is a natural continuation of the authors' earlier work [19] and fits the journal's scope. My concern is confined to §5: the completeness direction of the flagship Theorem 4.3 depends on four exclusion theorems of which only one is fully proved, and that one contains a genuine (though repairable) boundary-case error at b=0. The repair appears routine — a direct D_1 computation — but until the sketched non-cancellation arguments in Theorems 5.2–5.4 are written out and independently checked, the 'precisely three' list should be treated as not fully established. Everything else (Sections 6–7) is carefully argued and I expect no difficulties there."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The new content is the full irregular-parity classification at depth 3 (Theorem 4.3), the three exceptional unramified triples in Lemma 4.1, the two almost-extremal families (6.2–6.3), and the height-one family in §7. Prior papers only handled regular patterns (t, T, S). That is genuine progress inside motivic level-two periods.\n\nWhat works: the unramified direction is clean. Lemma 4.1 computes the coactions explicitly; the induction proofs for the unbounded families use cut diagrams and stay inside the Brown–Glanois criterion; the V_j^m (j ≠ 0) lift is careful and the period map closes it. The machinery is standard and applied correctly for the positive results.\n\nThe soft spot is the exclusion half of Theorem 4.3. Theorems 5.2–5.4 are partly sketched; several non-cancellation claims are asserted (“C ≠ 0”, “clearly ramified”) without the binomial check written out. The stress-test edge case in 5.1(III.1) with b = 0 is real: the displayed coefficient vanishes and the argument as written does not cover it, though the conclusion is rescued by the k = 1 case of Lemma 3.1(2). So the list is probably still correct, but a referee will have to re-check the boundary cases. One formula in §7 is conditional on a labelled analytic conjecture; the higher-depth conjectures are numerical only. Neither is hidden.\n\nCitation pattern is normal for this circle (Brown, Glanois, their own [19], Murakami, Charlton). No free parameters, no circular reduction to earlier irregular results.\n\nThis is for people already working on motivic MZVs / Euler sums / Galois descent. A serious editor should send it to referees; the main theorems deserve that time even if §5 needs tightening. I would engage with the proved low-depth statements and the two families.","headline":"Solid depth-3 classification and three new infinite families; the exclusion half of Theorem 4.3 is real work but the write-up of §5 is thinner than the claim needs.","tokens_in":39047,"tokens_out":521,"would_cite":true,"duration_ms":18357,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M32","11G99","14E18","18M25"],"pacs":[],"model":"grok-4.5","headline":"Motivic multiple mixed values of depth under four that reduce to ordinary multiple zeta values are completely classified, together with several infinite families of any depth.","keywords":["motivic multiple zeta values","motivic Euler sums","multiple mixed values","unramifiedness","Galois descent","motivic coaction","level-two variants"],"falsifier":"Compute the full set of odd coactions D_r for any claimed unramified triple mixed value outside the list of Theorem 4.3 (or for a member of one of the infinite families) and check whether every image lands in the predicted level-one subspace; a single nonzero component outside that subspace falsifies the claim.","tokens_in":38585,"feed_emoji":"∫","tokens_out":979,"duration_ms":24163,"temperature":0.7,"pith_summary":"Multiple mixed values are level-two cousins of multiple zeta values obtained by forcing fixed parity patterns on the summation indices. The analytic question of which of them actually lie in the span of ordinary multiple zeta values is still out of reach, but the same question can be answered completely on the motivic side. Using the Brown–Glanois descent criterion, the paper lists every unramified motivic triple mixed value and exhibits three infinite families that remain unramified in arbitrarily large depth. The resulting classification confirms earlier special-case conjectures and supplies an explicit dictionary between those values and ordinary motivic multiple zeta values. The work therefore turns a transcendental-number-theory problem into a finite linear-algebra check that can be carried out for every low-depth parity pattern.","feed_headline":"Every low-depth unramified mixed zeta value is now listed","feed_subtitle":"Motivic descent turns a hard analytic question into a finite linear-algebra check for all depths under four","key_machinery":"The Brown–Glanois descent criterion: a weight-w motivic Euler sum lies in the level-one Hopf algebra if and only if its D_1 coaction vanishes and every odd-r coaction lands inside L_r ⊗ H_{w-r}. All classification proofs reduce to verifying this finite-dimensional linear-algebra condition on the motivic coproduct.","core_discovery":"Every unramified motivic multiple mixed value of depth at most three is one of the ordinary motivic multiple zeta values, one of the previously known regular-parity families (multiple t-, T- or S-values), or exactly one of the three irregular-parity exceptions M^m(1̌,1̌,3), M^m(1̌,2,2) and M^m(1̌,4,4). In addition, three infinite families of unbounded depth—two of almost extremal height and one of height one—are proved unramified, and explicit expressions in terms of motivic multiple zeta values are given for all but one of them.","pith_inferences":["The same parity-pattern analysis should extend, with only notational changes, to the alternating multiple mixed values already introduced by the authors, producing a parallel unramified list at level two with signs.","If the depth-four and depth-five conjectures hold, the only unramified wide MMVs that are not ordinary MZVs or multiple t-values will be a handful of sporadic examples, suggesting a strong rigidity once width exceeds three.","The explicit coaction formulae for the height-one family give a practical algorithm that computer-algebra systems can use to decide unramifiedness for any fixed depth and parity signature."],"forward_implications":["Under the period conjecture the analytic triple mixed values that equal linear combinations of ordinary MZVs are precisely the ones listed in Theorem 4.3.","The three infinite families give infinitely many new explicit identities expressing irregular-parity Euler sums as rational combinations of MZVs.","The same coaction technique yields a complete dictionary for the height-one family V_j(d) in every even depth.","The conjectures of Section 8 supply a concrete checklist that can be tested numerically for every depth greater than three."],"fun_headline_variants":["All unramified motivic MMVs of depth <4 now classified","Depth-three unramified MMVs reduce to MZVs plus three exceptions","Unramified motivic mixed values listed completely below depth four","Three irregular exceptions finish the depth-three unramified list","Infinite unramified MMV families proven at almost extremal height"],"cache_read_input_tokens":32896,"weakest_assumption_plain":"The descent criterion is taken as both necessary and sufficient; if it misses a hidden ramification, the lists of unramified values are incomplete.","fun_headline_variants_meta":{"raw":{"variants":["All unramified motivic MMVs of depth <4 now classified","Depth-three unramified MMVs reduce to MZVs plus three exceptions","Unramified motivic mixed values listed completely below depth four","Three irregular exceptions finish the depth-three unramified list","Infinite unramified MMV families proven at almost extremal height"]},"model":"grok-4.5","effort":"low","cost_usd":0.00475,"raw_usage":{"total_tokens":1362,"prompt_tokens":801,"num_sources_used":0,"completion_tokens":75,"cost_in_usd_ticks":47504000,"prompt_tokens_details":{"text_tokens":801,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":486,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":801,"tokens_out":75,"duration_ms":9222,"temperature":1.0,"reasoning_tokens":486,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T21:43:01.019303+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Compute the full set of odd coactions D_r for any claimed unramified triple mixed value outside the list of Theorem 4.3 (or for a member of one of the infinite families) and check whether every image lands in the predicted level-one subspace; a single nonzero component outside that subspace falsifies the claim.","supporting_citations":[],"review_version":1}