{"id":"61341825-7b34-40b2-8237-029bcc848874","arxiv_id":"2508.19863","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New formal series and formal integral spaces give tridendriform and dendriform lifts of Multiple Zeta Values, leading to tree-indexed zeta values that respect associative products.","lead":"The paper builds formal versions of Multiple Zeta Values as series and integrals, and shows these formal objects carry finer algebraic structures (tridendriform and dendriform) that explain the shuffle identities for the classical numbers. It then uses the universal property of Schroeder trees to create tree-indexed zeta values that respect associative products, linking them to arborified and Shintani zeta values.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Definition 4.1's strict inequality d(v) > |Ang(v)| makes the map ι ill-defined on convergent trees and contradicts Theorem 4.8; replacing it with '≥' is the minimal correction needed for Corollaries 4.12–4.13 to hold.","rationale":"The reader's weakest assumption correctly identifies the strict inequality in Definition 4.1 as the main obstacle. My independent reading of the paper confirms that this inequality makes the map ι not well-defined on the generator i(1) and on convergent angle-decorated trees such as a root with two angles decorated 1. Consequently, Theorem 4.10's identification Ψ = flat∘ι and Corollary 4.12's formula relating ζTri to ζT are not well-defined as stated. The quoted Theorem 4.8 from [8] also cannot hold for the restricted domain defined by '>', because it would exclude binary vertices decorated 1 and hence fail to hit all convergent words. Replacing '>' with '≥' is the natural fix: the image of ι then lies in Tree(N*), the right-comb trees generating arbitrary words are admitted, and the surjectivity argument in Lemma 4.4 still works since any d(v) ≥ |Ang(v)| can be partitioned into |Ang(v)| positive integers. The rest of the algebraic framework—formal series, tridendriform and dendriform morphisms, universal properties of Schroeder trees—is coherent and internally consistent; the Shintani section's omitted proof is a secondary concern that does not affect the main algebra-morphism claims. Therefore, the appropriate verdict remains CONDITIONAL, exactly as the reader proposed, and no adjustment is needed.","tokens_in":32710,"tokens_out":13043,"duration_ms":135495,"concrete_test":"Let t be the angle-decorated Schroeder tree with one internal vertex (the root), two leaves, and both angles decorated by 1. Check: (a) by Definition 4.11, t ∈ Tridend(N*)conv, since the root does not have a unique angle decorated by 1; (b) under Definition 4.1 as written, ι(t) has d(root)=2 and |Ang(root)|=2, so ι(t) ∉ Tree(N*), making the right-hand side ζT(ι(t)) of Corollary 4.12 undefined; (c) with Definition 4.1 changed to d(v) ≥ |Ang(v)|, ι(t) becomes admissible, and the same fix admits the generator i(1) and all right-comb binary trees needed for flat(Tree(N*)conv) = W_conv(N*). This single example shows the strict inequality is not merely a typo but the exact obstruction; verifying that the rest of §4 remains valid under the non-strict inequality would settle the concern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central construction relating tridendriform zeta values to arborified zeta values depends on the map ι : Tridend(N*) → Tree(N*) of Definition 4.2. However, Definition 4.1 requires d(v) > |Ang(v)| for every internal vertex, while ι sends a vertex with angle decorations D(a) to d(v) = Σ D(a). Since each D(a) is a positive integer, the natural image satisfies only d(v) ≥ |Ang(v)|, with equality when all angles at v are decorated by 1. Under the strict inequality, ι is not defined on such trees. This is not a harmless technicality: take t ∈ Tridend(N*) with a single internal vertex (the root) having two leaves and both angles decorated by 1. By Definition 4.11, t is convergent, because its root does not have a unique angle decorated by 1. But ι(t) has root decoration 2 and |Ang(root)| = 2, violating Definition 4.1, so the expression ζT(ι(t)) in Corollary 4.12 is undefined. The same problem occurs for the generator i(1) ∈ Tridend(N*), whose image under ι is a binary vertex decorated 1 (equal to its number of angles); this makes the map Ψ = flat∘ι in Theorem 4.10 ill-defined on generators. Moreover, the strict inequality makes the quoted Theorem 4.8 false for the stated domain: the right-comb binary tree with a single vertex decorated 1 (which flattens to the word \"1\") is excluded from Tree(N*), so flat(Tree(N*)conv) cannot equal W_conv(N*). Replacing '>' by '≥' in Definition 4.1 solves all these issues simultaneously: the image of ι is contained in Tree(N*), the generator i(1) is admissible, and the right-comb trees that suffice to realize every word in W_conv(N*) are admitted. Since Corollaries 4.12 and 4.13—which assert that arborified zeta values are algebra morphisms for the quasi-shuffle of Schroeder trees—rely on these maps, the central claim is not fully established without this correction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a framework of formal series (Def. 2.1) and formal Chen integrals (Def. 5.1) carrying tridendriform and dendriform algebra structures. Using freeness of Schroeder and planar binary trees, it defines tridendriform and dendriform zeta values (Defs. 3.6, 6.5), and shows that the classical MZV maps factor as ζ = ev∘ζ_FS and ζ_int = ev∘ζ_FI with ζ_FS and ζ_FI being algebra morphisms (Props. 2.10, 5.10). It then relates these maps to arborified zeta values via a flattening map and the map ι (Thm. 4.10, Cor. 4.12, Cor. 7.6), obtaining new morphism properties for arborified zeta values with respect to associative products on trees (Cors. 4.13, 7.7). Finally it gives a Shintani zeta representation of dendriform zeta values (Thm. 8.4).","tokens_in":33239,"tokens_out":18301,"duration_ms":218732,"significance":"If the domain issues are fixed, the paper gives a clean algebraic explanation of the shuffle/quasi-shuffle property of MZVs by lifting evaluation to formal objects. The universal-property arguments are elegant, and the resulting associative tree products for arborified zeta values are genuinely new and explicitly computable. Strengths include explicit decompositions ζ = ev∘ζ_FS and ζ_int = ev∘ζ_FI, the construction of new formal series and integral spaces, and concrete relations expressing arborified zeta values as rational linear combinations of MZVs. These are falsifiable and testable claims, and the framework is likely to be useful for further generalizations.","major_comments":[{"comment":"Definition 4.1's strict inequality d(v)>|Ang(v)| makes Tree(N*) too small for the map ι of Definition 4.2. For a vertex all of whose angle decorations are 1, d(v)=|Ang(v)|, so ι(t) is not in Tree(N*). Already the generator i(1) is mapped to a binary vertex decorated 1, which Definition 4.1 excludes; consequently flat∘ι is not defined on all of Tridend(N*) and the universal-property proof of Theorem 4.10 breaks. Moreover Theorem 4.8 is false for this domain: the word 21 ∈ W_conv(N*) is flat(B+_2(|, B+_1(|,|))), but the binary child B+_1(|,|) violates the strict inequality. Corollaries 4.12–4.13 also use ζT(ι(t)) for convergent trees whose root has two angles decorated 1, giving d(v)=|Ang(v)|. Remove the inequality (or impose only d(v)≥1) and let convergence be defined by root≠1 as in Theorem 4.8; this restores ι, Lemma 4.4 and the corollaries.","section":"Definitions 4.1–4.2; Lemma 4.4; Theorem 4.8; Corollary 4.12"},{"comment":"The main Shintani result is stated without proof: the text says 'We will omit the proof since it is an adaptation of [9, Theorem 1.19]'. This is a load-bearing claim for the abstract and Section 8. The segment/matrix construction in Definitions 8.2–8.3 is intricate, and the claimed invariance under x-decorated bifurcated vertices is not demonstrated. At least a proof sketch, or a precise explanation of why the proof of [9, Theorem 1.19] carries over unchanged, should be included. Also, Remark 8.1 ('permuting rows... does not change the value') is only true if the word ω_t is permuted correspondingly; as written it is misleading.","section":"Theorem 8.4"}],"minor_comments":[{"comment":"In the second displayed identity, the product bound 'n∏' should presumably be 'd∏'; as written it is a typo.","section":"Proposition 2.2"},{"comment":"The notation w_{i,j} is used without definition; it should be defined explicitly, e.g. as the sum of the two letters.","section":"Lemma 1.11"},{"comment":"The definition of B(t) has ambiguous precedence: 'v ∈ νy(t) or v ∈ νx(t) and v is a bifurcated vertex'. Parentheses would clarify the intended logical grouping.","section":"Definition 8.2"},{"comment":"The vector space K is used in Definition 5.1 without specifying the field; the earlier notation KX suggests it is the base field, but the field itself is never named.","section":"Section 5.1"}],"recommendation":"major_revision","confidential_remarks":"The domain mismatch in Definition 4.1 is central and must be fixed before publication; once corrected, the main structural results appear plausible and well-motivated. The omission of the proof of Theorem 8.4 should be addressed, either by a proof or a detailed appendix, because the Shintani application is one of the paper's selling points. The paper fits the journal's scope and the algebraic framework is original enough to warrant revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid structural paper, and it deserves a serious referee. What is genuinely new and useful: the paper inserts formal series and formal Chen integrals as intermediate objects between Multiple Zeta Values and their arborified versions. Showing that ζ and ζ_int factor as ev∘ζ_FS and ev∘ζ_FI, with ζ_FS a tridendriform morphism and ζ_FI a dendriform morphism, is a clean explanatory result. Applying the universal property of Schroeder and binary trees then gives arborified zeta values that are algebra morphisms for associative quasi-shuffle and shuffle products (Corollaries 4.13 and 7.7). The freeness arguments are standard and mostly coherent, and the paper is honest about not attacking a long-standing conjecture.\n\nThe main soft spot is real. Definition 4.1 requires d(v) > |Ang(v)| at every internal vertex, but the map ι : Tridend(N*) → Tree(N*) sends a vertex with angle decorations D(a) to d(v)=ΣD(a), which only guarantees d(v) ≥ |Ang(v)|. Take a root with two angles both decorated 1: Definition 4.11 declares the tree convergent, but ι sends it to a root decorated 2 with two angles, violating Definition 4.1. The generator i(1) has the same problem. So Lemma 4.4, Proposition 4.5, and Corollary 4.12 are not fully well-defined as written. The stress-test note is right: changing '>' to '≥' in Definition 4.1 fixes all of this and is consistent with Theorem 4.8 and the intended convergence criterion. This is a genuine gap, but a one-line fix, and the surrounding structure is otherwise sound.\n\nSecond soft spot: Theorem 8.4, the Shintani zeta representation, is stated without proof, with only a remark that it adapts [9, Theorem 1.19]. Since the matrix A_t in Definition 8.3 is explicitly rebuilt according to the new decoration rules, and this is the only bridge to Shintani values, the claimed adaptation deserves at least a sketch. As written it is a citation with commentary, not a proof.\n\nSmaller issues: there are some notation typos (for instance in Definition 2.9), and certain convergence claims are stated more tersely than ideal, but nothing else blocks the main argument. The citation practice is fair: prior work on arborified zeta values and free algebras is properly credited.\n\nWho this is for: people working on MZVs, arborified zeta values, and combinatorial Hopf algebras. It is not a breakthrough paper, but it is a well-motivated structural one. My recommendation: send it to a capable referee, with an explicit request to verify the ≥ fix and to ask for a proof sketch of Theorem 8.4.","headline":"Worth reading and worth refereeing: a clean structural contribution to MZV/tree combinatorics, with one real definitional bug in the vertex-decoration bridge and one unproved theorem in the Shintani section; both look fixable.","tokens_in":33785,"tokens_out":4042,"would_cite":true,"duration_ms":46971,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16W99","16S10","11E45","05C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Multiple Zeta Values are shown to factor through tridendriform and dendriform algebra morphisms on Schroeder trees, so arborified zeta values are multiplicative for associative tree products.","keywords":["Multiple Zeta Values","tridendriform algebras","dendriform algebras","Schroeder trees","shuffle products","quasi-shuffle products","arborified zeta values","Shintani zeta values"],"falsifier":"Take the decorated Schroeder tree whose root has two angles both labelled 1. The paper's map ι sends it to a vertex decorated by 2, but Definition 4.1 admits only decorations strictly larger than the number of angles, so ι(t) is not an element of Tree(N*) as written. Checking whether the intended non-strict inequality was meant settles whether the equality ev∘ζ_Tri(t)=ζ_T(ι(t)) can be evaluated as stated.","tokens_in":1713,"feed_emoji":"🌳","tokens_out":2211,"duration_ms":110812,"temperature":0.7,"pith_summary":"Multiple Zeta Values are known to be multiplicative for two word products: the quasi-shuffle for the series representation and the shuffle for the integral representation. This paper claims those two facts are shadows of finer structures: the series representation factors through a tridendriform algebra of formal series and the integral representation through a dendriform algebra of formal integrals, with the classical zeta maps themselves being algebra morphisms before evaluation. Using the freeness of these algebras on Schroeder trees, the authors extend the construction to tree-valued arguments and obtain arborified zeta values that remain multiplicative for associative tree products. The payoff is a systematic way to write tree-indexed zeta values, and a family of Shintani zeta values, as rational linear combinations of classical Multiple Zeta Values.","feed_headline":"Multizeta values are evaluations of tree-algebra morphisms","feed_subtitle":"Formal series and integrals make multiple zeta values morphisms of tree algebras.","key_machinery":"The engine is the universal property of free tridendriform and dendriform algebras: angle-decorated Schroeder trees—planar rooted trees whose internal vertices have at least two children—freely generate the tridendriform category, while {x,y}-decorated binary trees freely generate the dendriform category. Specifying the value on one-node trees determines a unique algebra morphism into the spaces of formal series and formal integrals, giving ζ_FS and ζ_FI and hence ζ_Tri and ζ_Dend. The maps ι and flat connect the tree spaces back to words, and the evaluation map ev turns convergent formal objects into real numbers.","core_discovery":"The central claim is that ζ = ev ∘ ζ_FS and ζ_int = ev ∘ ζ_FI, where ζ_FS is a tridendriform algebra morphism from words to formal series and ζ_FI is a dendriform algebra morphism from words to formal integrals. Applying the universal properties of Schroeder trees and binary trees yields tridendriform zeta values ζ_Tri and dendriform zeta values ζ_Dend, which are algebra morphisms for associative quasi-shuffle and shuffle products of trees. As a consequence, arborified zeta values ζ_T and ζ_T_int are algebra morphisms for these associative products, and every convergent tridendriform or dendriform zeta value is an explicit rational combination of usual Multiple Zeta Values.","pith_inferences":["The strict inequality in Definition 4.1 appears to be a domain-definition slip: a vertex whose angles are all decorated by 1 has decoration equal to the number of angles, so relaxing the condition to 'greater than or equal' makes the map ι total and is consistent with the paper's own convergence theorem.","The construction is insensitive to the particular monoid structure on the decorations, so the same formal-series recipe could build zeta-like tree-indexed functions for other commutative semigroups.","Because the new tree products are associative but not commutative, iterating them should produce identities among arborified zeta values that have no word-shuffle analogue, potentially yielding new relations among classical Multiple Zeta Values.","The paper notes the lack of a tree-level analogue of Kontsevitch's map; if such a map exists, it would close the diagram between ζ_Tri and ζ_Dend and likely reveal a single tree-indexed object unifying both zeta families."],"forward_implications":["Arborified zeta values ζ_T and ζ_T_int are multiplicative for associative, noncommutative tree products, not just for word shuffles.","Every convergent tridendriform or dendriform zeta value can be written explicitly as a rational linear combination of classical Multiple Zeta Values.","The classical multiplicativity of Multiple Zeta Values for quasi-shuffles and shuffles is recovered as the shadow of finer tridendriform and dendriform morphisms that are lost only at the final evaluation.","A family of Shintani zeta values can be expressed as linear combinations of Multiple Zeta Values with rational coefficients.","The formal series and formal integral spaces inherit standard analytic-bookkeeping properties before any convergence issue is addressed, separating algebra from analysis."],"supporting_citations":[{"why":"Supplies the integral representation of Multiple Zeta Values and the equality ζ = ζ_int ∘ s that the paper lifts to formal objects.","marker":"[35]"},{"why":"Gives the quasi-shuffle product on words and the statement that ζ is an algebra morphism for it.","marker":"[20]"},{"why":"Provides the free tridendriform algebra structure on Schroeder trees and its universal property, used to define ζ_Tri.","marker":"[36]"},{"why":"Gives the explicit tridendriform products on words and Schroeder trees, including the quasi-shuffle action used throughout.","marker":"[4]"},{"why":"Provides arborified zeta values' factorisation ζ_T = ζ ∘ flat and the convergence criterion used in corollary 4.12.","marker":"[8]"},{"why":"Supplies the rooted-forest generalisation and the matrix/segment construction used to express dendriform zeta values as Shintani zeta values.","marker":"[9]"},{"why":"Introduces arborified zeta values as tree-indexed sums and integrals, the objects the paper connects to its new zeta maps.","marker":"[27]"},{"why":"Independently introduces forest-indexed integral zeta values whose properties are reused in section 7.","marker":"[34]"},{"why":"Establishes the free dendriform algebra structure on planar binary trees and its universal property, used to define ζ_Dend.","marker":"[29, 24]"},{"why":"Provides Chen's iterated-path-integral lemma underlying the formal integral shuffle product.","marker":"[7]"}],"fun_headline_variants":["Tree zeta values act as algebra morphisms on multizeta","Schroeder trees give new zeta values as algebra maps","Arborified zeta values generalize as morphisms","New tree-based zeta values extend multizeta algebraically"],"cache_read_input_tokens":35200,"weakest_assumption_plain":"The construction assumes that summing angle decorations always gives a vertex decoration allowed by Definition 4.1, but that definition requires the decoration to be strictly larger than the number of angles, so a vertex with two angles both labelled 1 violates it and the map ι is not defined on some trees the paper later calls convergent.","fun_headline_variants_meta":{"raw":{"variants":["Tree zeta values act as algebra morphisms on multizeta","Schroeder trees give new zeta values as algebra maps","Arborified zeta values generalize as morphisms","New tree-based zeta values extend multizeta algebraically"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000139,"raw_usage":{"total_tokens":944,"prompt_tokens":643,"completion_tokens":301,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":387,"completion_tokens_details":{"reasoning_tokens":231}},"tokens_in":387,"tokens_out":301,"duration_ms":3777,"temperature":1.0,"reasoning_tokens":231,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T15:25:03.431079+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the decorated Schroeder tree whose root has two angles both labelled 1. The paper's map ι sends it to a vertex decorated by 2, but Definition 4.1 admits only decorations strictly larger than the number of angles, so ι(t) is not an element of Tree(N*) as written. Checking whether the intended non-strict inequality was meant settles whether the equality ev∘ζ_Tri(t)=ζ_T(ι(t)) can be evaluated as stated.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the integral representation of Multiple Zeta Values and the equality ζ = ζ_int ∘ s that the paper lifts to formal objects."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the quasi-shuffle product on words and the statement that ζ is an algebra morphism for it."},{"cited_title":"Free ( tri) dendriform family algebras","cited_arxiv_id":null,"evidence_quote":"Provides the free tridendriform algebra structure on Schroeder trees and its universal property, used to define ζ_Tri."},{"cited_title":"Tridendriform structures","cited_arxiv_id":null,"evidence_quote":"Gives the explicit tridendriform products on words and Schroeder trees, including the quasi-shuffle action used throughout."},{"cited_title":"Double shuffle relations for arborified z eta values","cited_arxiv_id":null,"evidence_quote":"Provides arborified zeta values' factorisation ζ_T = ζ ∘ flat and the convergence criterion used in corollary 4.12."},{"cited_title":"Generalisations of m ultiple zeta values to rooted forests","cited_arxiv_id":null,"evidence_quote":"Supplies the rooted-forest generalisation and the matrix/segment construction used to express dendriform zeta values as Shintani zeta values."},{"cited_title":"Arborified multiple zeta values, 20 16","cited_arxiv_id":null,"evidence_quote":"Introduces arborified zeta values as tree-indexed sums and integrals, the objects the paper connects to its new zeta maps."},{"cited_title":"Yamamoto","cited_arxiv_id":null,"evidence_quote":"Independently introduces forest-indexed integral zeta values whose properties are reused in section 7."},{"cited_title":"Iterated path integrals","cited_arxiv_id":null,"evidence_quote":"Provides Chen's iterated-path-integral lemma underlying the formal integral shuffle product."}],"review_version":1}