{"id":"d9ea1506-3640-4b63-8ee0-6f7bd0a66c71","arxiv_id":"2501.11886","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Itô's formula holds for planarly branched rough paths with Hölder roughness between 1/4 and 1/2.","lead":"This paper proves an Itô formula for planarly branched rough paths, a class of rough paths built on post-Lie Hopf algebras that includes more stochastic processes than ordinary or branched rough paths. The result gives a change-of-variable identity for smooth functions of such paths, with correction terms expressed as integrals against an extended bracket path.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Correction path X^{(ijk)} in (3.33) is not additive: its defining element is not primitive, so the fourth Young integral in Theorem 3.8 is ill-defined for α ≤ 1/3.","rationale":"The reader's weakest-assumption analysis already flagged the additivity of X^{(ijk)} as a concern, alongside the unproved extension lemma. I agree that this is load-bearing, but I regard it as fatal rather than patchable-in-place: (3.33) does not define a path because the defining element is not primitive, so the fourth integral in (3.34) has no meaning as a Young integral for the stated range α ∈ (1/4, 1/3]. The proof's appeal to α + 3α > 1 is insufficient, as it addresses only the regularity of the integrand and the nominal increment bound, not the Chen relation failure. Since Theorem 3.8 is the general-case Itô formula for the lower-roughness regime, the paper's central claim is not established. The error is concrete and can be settled by the proposed coproduct computation; if the computation shows primitivity, the concern would lapse, but the manuscript as written contains no such verification. A conditional acceptance would require a corrected definition of X^{(ijk)} (likely via a primitivization such as the log map) and a re-derivation of the resulting correction term, which is a substantive revision rather than a minor fix. Hence I recommend REJECT of the current version, while acknowledging the rest of the paper may be salvageable.","tokens_in":29456,"tokens_out":12935,"duration_ms":130796,"concrete_test":"Compute the coproduct Δ(E) for E defined in (3.33) and check whether the non-primitive (tensor-product) part vanishes. Concretely, evaluate the second-order increment δX^{(ijk)}_{0,1/2,1} for the canonical lift of a smooth path, e.g. X_t = (t, t^2) with d = 2, using explicit iterated integrals for all forests in hat HMKW. If δX^{(ijk)}_{0,1} − δX^{(ijk)}_{0,1/2} − δX^{(ijk)}_{1/2,1} is nonzero, the path defined by (3.33) is not additive and the Young integral in Theorem 3.8 is not well-defined. Also, re-run the sewing lemma estimate around (3.45): verify that the term involving δ^2 X^{(ijk)} has order |t−s|^{3α}, which violates the required 1+ε condition exactly when α ≤ 1/3.","verdict_should_be":"REJECT","load_bearing_attack":"Theorem 3.8 introduces X^{(ijk)} by the increment formula (3.33): δX^{(ijk)}_{s,t} = ⟨hat X_{s,t}, E⟩ with E = •i rr^j_k + rr^j_k •i − rrr^i_jk − rr^{(ij)}_k − rr^{(ji)}_k. The proof then declares that, since α + 3α > 1, the fourth integral in (3.34) is a Young integral. This is only valid if X^{(ijk)} is an actual path, i.e. if the increment is additive (Chen's relation). Additivity of ⟨hat X_{s,t}, E⟩ is equivalent to primitivity of E in the MKW Hopf algebra, because Chen's relation reads ⟨hat X_{s,t}, E⟩ = ⟨hat X_{s,u} ⊗ hat X_{u,t}, Δ(E)⟩. Unlike the element in (3.14), whose primitivity is verified directly after Theorem 3.6, no such check is given for E. Computing the coproduct under the standard MKW rules used in Lemma 3.2 gives cross terms: Δ(E) contains 2 rr^j_k ⊗ •i + •j ⊗ •i•k + •j ⊗ •k•i + •j•i ⊗ •k − •(ij) ⊗ •k − •(ji) ⊗ •k, in addition to the primitive part. These terms do not cancel (they involve distinct forests/trees). Hence δ^2 X^{(ijk)}_{s,u,t} = δX^{(ijk)}_{s,t} − δX^{(ijk)}_{s,u} − δX^{(ijk)}_{u,t} = ⟨hat X_{s,u} ⊗ hat X_{u,t}, cross terms⟩ is of order |t−s|^{3α}. The sewing lemma applied in (3.45) requires the increment Ξ_{s,t} = D2F(Y_s):(fi(Y_s), Dfj(Y_s):fk(Y_s)) δX^{(ijk)}_{s,t} to satisfy δΞ = O(|t−s|^{1+ε}). The non-additivity contributes an error of order |t−s|^{3α}; for α ≤ 1/3 this is not o(|t−s|), so the Young integral is not defined by the given argument. The condition α + 3α > 1 only bounds the product of the integrand's Hölder regularity and the claimed increment size; it does not repair the failure of additivity. Thus Theorem 3.8, which is the main result for the general case in the regime 1/4 < α ≤ 1/3, rests on an ill-defined object.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops Itô-type change-of-variable formulas for planarly branched rough paths with Hölder roughness in (1/4,1/2]. It first recalls controlled planarly branched rough paths and rough integrals, then proves composition results and a Taylor expansion. In Section 3 it states two Itô formulas for F(X): Theorem 3.4 for 1/3<α≤1/2 and Theorem 3.6 for 1/4<α≤1/3, the latter containing an extra third-order Young correction. It then states analogous formulas for F(Y), where Y solves a rough differential equation driven by X: Theorem 3.7 for 1/3<α≤1/2 and Theorem 3.8 for 1/4<α≤1/3, with a fourth Young correction. The central claim is that smooth functions of planarly branched rough paths admit an explicit change-of-variable formula with correction terms encoded by the bracket extension and certain primitive elements.","tokens_in":29962,"tokens_out":25840,"duration_ms":260605,"significance":"If fully correct, this would be the first Itô formula for planarly branched rough paths and would extend the classical results of Friz--Hairer for rough paths and Kelly for branched rough paths to a setting covering more stochastic processes. The paper has useful and explicit ingredients: the Taylor-expansion skeleton is transparent, the composition results in Propositions 2.9, 2.10 and 2.12 are worked out in detail, and the simple-case formulas for F(X) are plausible. However, the validity of the general case for roughness α≤1/3 rests on the additivity of a correction path that is not established; the proof of Theorem 3.8 has a load-bearing gap, and the algebraic status of the bracket letters used in the primitivity checks also needs clarification. The paper's value is therefore contingent on repairing these points.","major_comments":[{"comment":"The correction path X^{(ijk)} is defined by δX^{(ijk)}_{s,t} = ⟨\\hat X_{s,t}, E⟩ with E = •i rr^j_k + rr^j_k •i − rrr^i_jk − rr^{(ij)}_k − rr^{(ji)}_k, and is then used as a Young integrator in (3.45). For this to define an actual path, E must be primitive in the extended MKW Hopf algebra so that Chen's relation gives additivity of the increments. No such verification is given. Under the coproduct rules used in the paper, Δ(E) contains cross terms such as rr^j_k ⊗ •i, •j ⊗ •i•k, •j ⊗ •k•i, •j•i ⊗ •k, −•(ij)⊗•k and −•(ji)⊗•k in addition to the primitive part. Hence δX^{(ijk)} is not additive, and the non-additivity is of order |t−s|^{3α}. Since α≤1/3 in this theorem, 3α≤1, so the sewing lemma invoked for the Young integral does not apply and the fourth integral in (3.34) is not defined by the given argument. This invalidates the proof of Theorem 3.8 as written.","section":"Section 3.2, Theorem 3.8, Eqs. (3.33)--(3.45)"},{"comment":"The verification that •k•j•i − r∨r r^i_jk − rr^{(ij)}_k is primitive implicitly identifies the new symbol •(ij) with the element •j•i − rr^i_j inside the extended Hopf algebra. Lemma 3.2, however, introduces (ij) as a new alphabet symbol and only prescribes its pairing with the bracket extension; it does not state an algebraic relation in \\hat H_MKW. Without such a relation, the displayed coproduct leaves a residual term •k⊗(•j•i − rr^i_j − •(ij)), and the element is not primitive. The additivity of \\tilde X in (3.14) and hence the Young integral in (3.15) therefore requires clarification. Please state explicitly whether •(ij) is a new independent primitive generator or a shorthand for •j•i − rr^i_j, and give a correct primitivity proof under that convention.","section":"Section 3.1, Theorem 3.6, primitivity check after Eq. (3.14)"},{"comment":"The bracket extension \\hat X is constructed by invoking Lemma 3.1, whose proof is only a citation to [11] and a sentence saying the result follows as in [23, Corollary 4.2.16]. Since every Itô formula in Section 3 uses the bracket extension with new components having prescribed Hölder estimates and Chen relations, this is a load-bearing input. A full proof, or a precise statement with a page/equation reference of the cited result as applied to the MKW Hopf algebra, should be included so that the existence of \\hat X is independently verifiable.","section":"Lemma 3.1 and Lemma 3.2"}],"minor_comments":[{"comment":"The third-order term in the final display of Theorem 3.8 is written as ∫ D^3F(Y_r):(f(Y_r),f(Y_r),f(Y_r))·d\\hat X^{(ijk)}_r, but it should be d\\tilde X^{(ijk)}_r, matching (3.32) and the proof around (3.44)--(3.46). This notational inconsistency should be corrected.","section":"Theorem 3.8, final display and Eq. (3.34)"},{"comment":"The paper uses the same symbol X for the driving path, the rough path, and in Theorem 3.8 for the correction path X^{(ijk)}; this makes statements such as (3.34) harder to read. Renaming the correction path, for example Z^{(ijk)}, would improve clarity.","section":"General notation"}],"recommendation":"major_revision","confidential_remarks":"The paper contains a promising framework and several correct-looking intermediate results, but the proof of Theorem 3.8 is not valid as written: the correction path X^{(ijk)} is used as a Young integrator without establishing additivity, and the non-additivity is too large for the sewing lemma in the range α≤1/3. The related primitivity check in Theorem 3.6 also relies on an unstated algebraic identification of the bracket letters. I would ask the authors to either prove the additivity with a correct primitive element, or revise the claims and state precisely for which range of α the general Itô formula is established. The other formulas, especially Theorem 3.4 and Theorem 3.7, appear to be on much firmer ground."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves Itô formulas for planarly branched rough paths, which is a real and sensible next step after Kelly's branched case. The simple-case results and the α>1/3 general case are executed cleanly: the bracket extension is explicit, the Taylor expansions are careful, and the algebraic bookkeeping is mostly transparent. It is genuinely new to bring the MKW Hopf algebra into the Itô formula picture.\n\nThe soft spot is Theorem 3.8. The stress-test about X^{(ijk)} lands. The element E = •i rr^j_k + rr^j_k •i − rrr^i_jk − rr^{(ij)}_k − rr^{(ji)}_k is not primitive. I computed the coproduct using the paper's own rules and found cross terms like 2 rr^j_k ⊗ •i, •j ⊗ •i•k, and •j•i ⊗ •k, none of which cancel. So Chen's relation fails: δX^{(ijk)}_{s,t} is not the increment of any path, and the Young integral in (3.45) has no rigorous meaning. The error contributes at order |t−s|^{3α}, which is not o(|t−s|) when α ≤ 1/3, so the sewing lemma cannot rescue it. That means the main general result for the regime 1/4 < α ≤ 1/3 is unsupported. There is also a typo in the final display of Theorem 3.8 (d\\hat X instead of d\\tilde X for the third integral).\n\nTwo minor points: Lemma 3.1 is imported by citation and not proved; given that Lemma 3.2 does provide an explicit construction of the bracket components, the gap may be patchable, but it should be addressed. And the abstract claims an Itô formula for planarly branched rough paths with roughness 1/4 < α ≤ 1/2; as it stands, the general F(Y) result only holds for α > 1/3.\n\nThe paper is worth engaging. Theorems 3.4, 3.6, and 3.7 look solid, and the overall strategy is sound. But Theorem 3.8 needs either a different correction term that is actually primitive, or a genuine proof that the non-primitive contributions cancel in the sum; without that, the main low-regularity result is not established.\n\nI would send this to a serious referee, because the topic is timely and the flaw is specific and potentially repairable. For my own work, I would not cite Theorem 3.8 in its current form, but I would cite the paper for the α>1/3 results once the low-regularity gap is fixed.","headline":"A mostly careful extension of the Itô formula to planarly branched rough paths, but Theorem 3.8 has a genuine gap: the correction path X^{(ijk)} is not additive, so the fourth Young integral in the low-regularity general case is ill-defined.","tokens_in":30539,"tokens_out":5700,"would_cite":false,"duration_ms":56861,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60L20","60L50","60H99","34K50","37H10","05C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves an Itô change-of-variable formula for planarly branched rough paths with Hölder roughness $1/4<\\alpha\\le 1/2$.","keywords":["Itô formula","planarly branched rough path","controlled planarly branched rough path","bracket extension","rough differential equation","Young integral","Hopf algebra of planar rooted forests","roughness 1/4"],"falsifier":"Take a planarly branched rough path and compute the bracket-extension component $\\langle \\hat X,\\bullet_{(ij)}\\rangle=\\langle X,\\bullet_j\\bullet_i-\\mathrm{tree}_{ij}\\rangle$ on three consecutive intervals; verify whether the increments satisfy $\\hat X_{s,u}+\\hat X_{u,t}=\\hat X_{s,t}$. A failure would contradict Lemma 3.2. Independently, compute $\\delta X^{(ijk)}_{s,u}+\\delta X^{(ijk)}_{u,t}-\\delta X^{(ijk)}_{s,t}$ from (3.33); a nonzero defect would invalidate the Young-integral term in Theorem 3.8.","tokens_in":29234,"feed_emoji":"🧮","tokens_out":7374,"duration_ms":68822,"temperature":0.7,"pith_summary":"The paper proves that smooth functions of planarly branched rough paths satisfy an Itô (change-of-variable) formula for Hölder roughness $1/4<\\alpha\\le 1/2$. Planarly branched rough paths generalize both ordinary rough paths and branched rough paths by taking values in the Hopf algebra of planar rooted forests, so the formula covers a wider class of driving signals. The identity expresses $\\delta F(X)_{s,t}$ as a first-order rough integral plus a correction integral against a bracket extension of $X$; when $\\alpha\\le 1/3$, a third-order Young integral appears. The same structure is proved for $F(Y)$ where $Y$ solves a rough differential equation driven by $X$. A sympathetic reader would care because it gives a concrete change-of-variable calculus for the largest known combinatorially defined class of rough paths in this roughness range.","feed_headline":"Itô formula proved for planarly branched rough paths","feed_subtitle":"Smooth functions of these paths obey an explicit change-of-variable identity down to roughness 1/4.","key_machinery":"The central object is a planarly branched rough path: a two-parameter path taking values in the graded dual of the Hopf algebra of planar rooted forests, satisfying Chen's identity and Hölder estimates. The argument is carried by the bracket extension $\\hat X$, which adds extra letters $(ij)$ to the alphabet with prescribed level-two components, and by the observation that certain linear combinations of forest elements—such as $\\bullet_j\\bullet_i$ minus the corresponding rooted tree—are prime, meaning their coproduct is $h\\otimes 1+1\\otimes h$, so their increments can serve as integrands. Controlled planarly branched rough paths, whose components approximately transform by $\\langle X_{s,t}\\star\\tau,Y_s\\rangle$, provide the integrands, and Young integrals handle the third-order terms when $\\alpha\\le 1/3$.","core_discovery":"On its own terms, the paper establishes four theorems. For $\\alpha\\in(1/3,1/2]$ it proves $$\\delta F(X)_{s,t}=\\int_s^t DF(X_r):dX_r+\\int_s^t $D^{2}$F(X_r):d\\hat X_r,$$ where $\\hat X$ is the bracket extension of $X$, whose level-two component is $\\langle\\hat X_{s,t},\\bullet_{(ij)}\\rangle=\\langle X_{s,t},\\bullet_j\\bullet_i-\\mathrm{tree}_{ij}\\rangle$. For $\\alpha\\in(1/4,1/3]$ it proves the analogous identity with an additional third-order Young integral $\\int D^3F(X_r):d\\tilde X_r$. Theorem 3.7 and Theorem 3.8 give the corresponding formulas for $\\delta F(Y)_{s,t}$ when $Y$ solves the rough differential equation $dY=f(Y)\\cdot dX$, with correction terms evaluated on $f(Y)$ and its derivatives, including a mixed second-order Young integral in the low-regularity case. The proofs proceed by Taylor expansion, re-expressing the remainder as increments of the bracket extension and then identifying the remainder as a rough or Young integral.","pith_inferences":["The paper's approach suggests that the same bracket-extension recipe should yield Itô formulas for other controlled planarly branched rough paths, provided composition and RDE existence are established for $\\alpha\\le 1/4$; the authors explicitly leave those two difficulties open.","A reader should check whether the correction path $X^{(ijk)}$ defined in (3.33) has additive increments. If it does not, the fourth term in Theorem 3.8 would need reinterpretation, for instance as a rough integral over a larger alphabet rather than a Young integral.","Because planarly branched rough paths sit between branched and geometric rough paths, the formula may provide the right interpolation for stochastic processes with roughness just above $1/4$, such as certain fractional Brownian motions, though the paper does not discuss specific processes."],"forward_implications":["For $\\alpha\\in(1/3,1/2]$, every smooth function of a planarly branched rough path satisfies the two-term Itô identity, with the bracket correction explicit in terms of $\\hat X$.","For $\\alpha\\in(1/4,1/3]$, the formula gains a third-order Young integral for $F(X)$ and, for $F(Y)$, an additional mixed second-order Young integral driven by the correction path $X^{(ijk)}$.","The same extension result implies that any $X$-controlled planarly branched rough path is automatically controlled by the bracket extension $\\hat X$, so integrals against the enlarged alphabet are available.","If the proof of the extension lemma is correct, the framework includes the previously known branched and ordinary rough-path Itô formulas as special cases at the corresponding truncations.","The formulas hold for the RDE solution $Y$, not only for the driving path $X$, which is what is needed for applications to stochastic differential equations."],"supporting_citations":[{"why":"Provides the controlled planarly branched rough path framework and the rough-integral estimates used throughout the proofs.","marker":"[15]"},{"why":"Defines planarly branched rough paths and the rough differential equations on homogeneous spaces that this paper extends.","marker":"[9]"},{"why":"Cited for the geometric embedding that underpins Lemma 3.1, the existence of the bracket extension.","marker":"[11]"},{"why":"Supplies the branched-rough-path Itô formula and the extension technique adapted to the planar setting.","marker":"[23]"},{"why":"Gives the rough-path Itô formula that this paper generalizes to planarly branched rough paths.","marker":"[13]"},{"why":"Introduces branched rough paths and controlled branched paths, the model for the planar version.","marker":"[19]"},{"why":"Defines the Hopf algebra of planar rooted forests that underlies the whole construction.","marker":"[28]"},{"why":"Provides the Young integral used for the third-order correction terms.","marker":"[30]"},{"why":"Gives the sewing lemma used to identify the Young integrals in the low-regularity case.","marker":"[3]"}],"fun_headline_variants":["Itô formula now holds for planarly branched rough paths","Planarly branched rough paths satisfy Itô formula from roughness 1/4","Itô formula extends to planarly branched rough paths down to 1/4","Itô formula for planarly branched rough paths at roughness 1/4","Planarly branched paths: Itô formula proved for roughness above 1/4"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that every planarly branched rough path admits a bracket extension over an enlarged alphabet with the prescribed level-two components (Lemma 3.1); the paper cites this to a survey rather than proving it, and the correction path $X^{(ijk)}$ used as a Young integrator in Theorem 3.8 is not shown to have additive increments.","fun_headline_variants_meta":{"raw":{"variants":["Itô formula now holds for planarly branched rough paths","Planarly branched rough paths satisfy Itô formula from roughness 1/4","Itô formula extends to planarly branched rough paths down to 1/4","Itô formula for planarly branched rough paths at roughness 1/4","Planarly branched paths: Itô formula proved for roughness above 1/4"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001222,"raw_usage":{"total_tokens":5007,"prompt_tokens":909,"completion_tokens":4098,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":525,"completion_tokens_details":{"reasoning_tokens":3999}},"tokens_in":525,"tokens_out":4098,"duration_ms":27860,"temperature":1.0,"reasoning_tokens":3999,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T17:47:05.680324+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a planarly branched rough path and compute the bracket-extension component $\\langle \\hat X,\\bullet_{(ij)}\\rangle=\\langle X,\\bullet_j\\bullet_i-\\mathrm{tree}_{ij}\\rangle$ on three consecutive intervals; verify whether the increments satisfy $\\hat X_{s,u}+\\hat X_{u,t}=\\hat X_{s,t}$. A failure would contradict Lemma 3.2. Independently, compute $\\delta X^{(ijk)}_{s,u}+\\delta X^{(ijk)}_{u,t}-\\delta X^{(ijk)}_{s,t}$ from (3.33); a nonzero defect would invalidate the Young-integral term in Theorem 3.8.","supporting_citations":[{"cited_title":"Rough differential equations and planarly branched universal limit theorem","cited_arxiv_id":"2412.16479","evidence_quote":"Provides the controlled planarly branched rough path framework and the rough-integral estimates used throughout the proofs."},{"cited_title":"Curry, K","cited_arxiv_id":null,"evidence_quote":"Defines planarly branched rough paths and the rough differential equations on homogeneous spaces that this paper extends."},{"cited_title":"Ebrahimi-Fard and L","cited_arxiv_id":null,"evidence_quote":"Cited for the geometric embedding that underpins Lemma 3.1, the existence of the bracket extension."},{"cited_title":"Kelly, Itˆ o corrections in stochastic equations, Ph.D","cited_arxiv_id":null,"evidence_quote":"Supplies the branched-rough-path Itô formula and the extension technique adapted to the planar setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the rough-path Itô formula that this paper generalizes to planarly branched rough paths."},{"cited_title":"Gubinelli, Ramification of rough paths, J","cited_arxiv_id":null,"evidence_quote":"Introduces branched rough paths and controlled branched paths, the model for the planar version."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Hopf algebra of planar rooted forests that underlies the whole construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Young integral used for the third-order correction terms."},{"cited_title":"Broux and L","cited_arxiv_id":null,"evidence_quote":"Gives the sewing lemma used to identify the Young integrals in the low-regularity case."}],"review_version":1}