{"id":"7711bbcd-b2af-4d95-9a1f-417d43766d25","arxiv_id":"2411.18955","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Primitive path homology is a new digraph invariant that coincides with GLMY path homology on asymmetric digraphs and differs on symmetric ones.","lead":"This mathematics paper defines a new invariant, called primitive path homology, for directed graphs, and proves it matches the existing path homology invariant on graphs that have no two-way arrows. On graphs with two-way arrows the two invariants differ, and the paper also builds variants for paths that start or end at a fixed vertex.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Theorem 3.7 is correct as stated, and the asymmetry hypothesis is used exactly where needed.","rationale":"I checked the proof of the central comparison theorem in detail. The key step is that in an asymmetric digraph an allowed elementary path cannot contain a consecutive fragment iji, so every term in its boundary is regular. Consequently the boundary in the quotient R is represented by the same linear combination as the boundary in Λ, and membership in Ω_n is equivalent to membership in Π_n. The natural isomorphism p_n is just the projection to the quotient, whose kernel meets A_n trivially because allowed paths are regular. Lemma 3.3, which supplies functoriality, is also correct: the only delicate case is when the image path is not allowed, but then a collapse pair f(i_k)=f(i_{k+1}) forces all other faces to remain irregular and the two faces omitting i_k and i_{k+1} cancel. The asymmetry hypothesis is not an unstated assumption; it is the theorem's scope, and the paper's Example 3.2 demonstrates that the theories genuinely differ without it. The reader's conditional verdict is reasonable because several later theorems are relegated to 'similar' proofs and Example 5.13 contains an unproved vanishing assertion. Those omissions do not affect the correctness of Theorem 3.7 itself, so I would not change the verdict. My agreement is partial because the reader's weakest_assumption points to asymmetry as the main risk, whereas I regard the main residual risk as the delegated proofs in the surrounding sections.","tokens_in":23468,"tokens_out":28833,"duration_ms":260318,"concrete_test":"Compute dim Π_n and dim Ω_n for all asymmetric digraphs on up to 5 vertices, including cases with repeated non-consecutive vertices and chords (e.g., 0→1→2→0 with the extra arrow 0→2), using a short enumeration of allowed paths; if any dimension differs for n ≤ 4, Theorem 3.7 is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim survives scrutiny. In an asymmetric digraph, every allowed path has no consecutive fragment iji, so every face appearing in ∂e_{i0...in} is regular; the quotient condition defining Ω_n therefore reduces exactly to the Λ-level condition defining Π_n, and the isomorphism p_n:Π_n→Ω_n is well-defined, injective, surjective, and commutes with the differential. Lemma 3.3's case analysis is sound: when the image path is not allowed, the collapsed pair cancels in f♯∂; when it is allowed, asymmetry of the target prevents iji fragments from appearing in the source path, so no face is accidentally zeroed. The reader's flagged asymmetry is a genuine scope limitation, but it is an explicit hypothesis of Theorem 3.7, not a hidden assumption, and Example 3.2 shows the distinction is real outside the category AD. The only soft spots lie outside the central claim: Theorem 7.1 and several results in Sections 5–6 are delegated with 'similar' proofs, and Example 5.13 leaves a vanishing computation as an exercise. These support the reader's CONDITIONAL verdict but do not undermine Theorem 3.7.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a primitive path homology theory for simple digraphs, defined by the submodule Π_n(G) = {w ∈ A_n(G) | ∂w ∈ A_{n-1}(G)} inside the free path module Λ_*(V_G), in contrast with the GLMY path homology Ω_*(G) defined in the quotient module R_*(V_G). The central theorem (Theorem 3.7) states that for every asymmetric digraph G, the chain complexes Π_*(G) and Ω_*(G) are naturally isomorphic, so the two homology theories coincide on the subcategory AD; Example 3.2 shows they differ in general, e.g., for the complete two-vertex digraph the primitive H_1 is Z while the path H_1 is 0. The paper then develops cluster digraphs, primitive (a,b)-cluster homology, primitive tail/head-fixed homology, establishes suspension and inverse-digraph results (Theorems 3.11, 5.12, 6.10), and states functoriality on AD (Theorem 7.1).","tokens_in":1669,"tokens_out":2602,"duration_ms":206575,"significance":"If the results are correct, the paper provides a genuinely new variant of path homology that agrees with the established GLMY theory on asymmetric digraphs but is distinct in the presence of two-way arrows. The main comparison theorem is a clean and useful statement, and the fixed-end and cluster constructions offer potential tools for studying paths with prescribed endpoints. The paper contains many worked examples that illustrate the behavior of the new invariants. However, the novelty is incremental relative to the existing GLMY framework, and the manuscript's reliance on 'similar' proofs and several incomplete arguments reduces its reliability in its current form.","major_comments":[{"comment":"The proof of the natural isomorphism Π_*(G) ≅ Ω_*(G) is missing a key justification. The text claims that 'Σ r_β e_β ∈ A_{n-1}(G) if and only if Σ r_β[e_β] ∈ η_{n-1}(A_{n-1}(G))', but this equivalence is not immediate. From p_{n-1}(∂w) = η_{n-1}(v) one obtains ∂w − v ∈ I_{n-1}, which does not by itself imply ∂w ∈ A_{n-1}. The missing step is the observation that in an asymmetric digraph every face of an allowed elementary path is regular, so the classes [e_β] form a basis of their span in R_{n-1}; this makes p_{n-1} injective on the span of the regular paths appearing in ∂w. Please add this argument to make the proof complete.","section":"Theorem 3.7, around Eq. (3.20)"},{"comment":"The claim that S_{n-1}^m(G)∩S_{n-1}^l(G) = 0 for m ≠ l is false in general. An elementary path can have two non-arrow 'skip' pairs at different positions and can arise both as ∂_m(p) and ∂_l(q) for different allowed paths p and q. For example, in the asymmetric graph with vertices 0,1,2,3,4,5 and arrows 0→1, 1→2, 2→3, 3→4, 0→5, 5→1, 2→5, 5→3, the path e_{0,1,2,3,4} equals ∂_1(e_{0,5,1,2,3,4}) and also ∂_3(e_{0,1,2,5,3,4}), and it satisfies both (0,2) ∉ E and (2,4) ∉ E. The proof of Proposition 3.9 therefore needs a different argument.","section":"Proposition 3.9, Eq. (3.31)"},{"comment":"The functoriality theorem for the cluster, tail, and head primitive homologies is stated with a one-line proof that says 'Similar to the proof of Proposition 3.5'. This is a load-bearing result for Sections 5 and 6, and the verification that f_♯ preserves the d-condition defining Θ^{[a,b]}_* (and the analogous conditions for Θ^{[a,·]}_* and Θ^{[·,b]}_*) requires checking the compatibility of the differential d with f_♯ and the boundary cases, especially n=1 and n=2. Please provide the details or a precise reduction to the already-proved cases.","section":"Theorem 7.1"}],"minor_comments":[{"comment":"The sentence 'It is an easy exercise for readers to check similarly above that H[a,b]_2(H) = 0 for i≥5' contains a typo and a delegated computation: it should likely read 'H[a,b]_i(H) = 0 for i≥5', and the vanishing claim should be either proved or explicitly marked as an exercise with enough details to be reproducible.","section":"Example 5.13"},{"comment":"The two homology theories are both denoted by H_* (or H_n) in the text, which makes sentences like 'the groups H_n(G) and H_n(G) are not isomorphic' in Example 3.2 unintelligible. Please use distinct notation, e.g., a different font or a superscript for the primitive theory, and apply it consistently.","section":"Notation throughout"},{"comment":"There is a typo in the reference 'By (3.1) and (2.3, for any digraph...'; the closing parenthesis should appear after (2.3).","section":"Section 3, Eq. (3.2)"},{"comment":"In the final line of the proof, '∂_m^n (w)' should be '∂_m(w)'. The superscript n appears to be a typographical error.","section":"Proposition 3.9 proof"},{"comment":"There are grammatical errors and missing words, e.g., 'Let a, b two different vertices of a digraph' should be 'Let a and b be two different vertices of a digraph'. A careful language edit would improve readability.","section":"Abstract and Introduction"}],"recommendation":"major_revision","confidential_remarks":"The main comparison theorem (Theorem 3.7) appears to be correct, but its proof needs a missing justification, and there is a false claim in the proof of Proposition 3.9. These issues are fixable without changing the paper's central contribution. The extensive use of 'similar' proofs and incomplete computations in examples should also be addressed before publication. I recommend major revision and a careful re-reading of the technical arguments."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis paper introduces primitive path homology Pi_* for simple digraphs and proves the central theorem: on asymmetric digraphs, Pi_* is naturally isomorphic to the GLMY path complex Omega_*. I checked the proof of Theorem 3.7, including Lemma 3.3, and the isomorphism holds; the asymmetry hypothesis does exactly what it is claimed to do, ruling out iji fragments. Example 3.2 shows the invariant is genuinely different on symmetric digraphs: for the complete two-vertex digraph, primitive H_1 is Z while path homology is 0. That is a real new object, not a notational variant.\n\nThe fixed-tail, fixed-head, and cluster (a,b) versions in Sections 5 and 6 are the other new pieces. They extend the cluster construction from [6], and the suspension theorem (5.12)—that H^{[a,b]}_n(SdG) = H_{n-2}(G)—is a clean, useful relation. The paper is plainly written, with precise definitions and internally consistent arguments.\n\nThe soft spots are about completeness. Many secondary results, including Theorem 7.1 and several propositions in Sections 5 and 6, are delegated to \"similar to\" proofs. Given how index-heavy path homology computations are, that means a referee has to redo a lot of work to verify them. Example 5.13 leaves the vanishing H^{[a,b]}_i(H)=0 for i≥5 as an \"easy exercise\"—that should be stated and proved or properly cited. None of this undermines Theorem 3.7, and none of it is a fitted-parameter circularity. The paper leans on the authors' own GLMY framework, which is natural given they are extending it, and the new complex is defined independently.\n\nThis is a solid contribution for people working in path homology or applied algebraic topology, not a field-changer. It deserves a serious referee; I would send it to review, with a request to fill in the delegated proofs and the Example 5.13 computation.\n\nBest,","headline":"Primitive path homology is a genuinely new digraph invariant that provably matches GLMY path homology on asymmetric digraphs; worth refereeing despite several deferred proofs.","tokens_in":24229,"tokens_out":2513,"would_cite":false,"duration_ms":22722,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55N35","05C20","05C38","05C25","55U15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper introduces a primitive path homology theory for digraphs and proves that it coincides with the established path homology exactly on asymmetric digraphs, with a two-vertex example showing the two theories differ in general.","keywords":["primitive path homology","path homology","digraph","asymmetric digraph","chain complex","cluster path","fixed tail vertex","fixed head vertex"],"falsifier":"Compute the first primitive path homology group of the complete digraph on two vertices with arrows in both directions: the paper predicts $\\mathbb{Z}$, while the standard path homology group is 0. For the main theorem, search for any asymmetric digraph $G$ and any $n \\geq 0$ for which the natural map $\\Pi_n(G) \\to \\Omega_n(G)$ is not an isomorphism, since Theorem 3.7 asserts no such example exists.","tokens_in":23227,"feed_emoji":"➡️","tokens_out":8864,"duration_ms":68941,"temperature":0.7,"pith_summary":"This paper introduces a primitive path homology theory for finite simple directed graphs, defined by a leaner boundary condition than the usual path homology: a chain is kept only if its alternating face sum remains inside the allowed paths of the graph. Its main theorem states that for asymmetric digraphs, those with no pair of opposite arrows, the primitive chain complex is naturally isomorphic to the ordinary path chain complex, so the two homology theories agree. The agreement is sharp: in the complete two-vertex digraph, where both arrows are present, primitive homology is $\\mathbb{Z}$ in degree 1 while path homology is 0. The construction also yields homology theories of paths with a fixed tail vertex, a fixed head vertex, or both, and these theories are functorial only on asymmetric digraphs; an example shows the naturality diagram can fail when the target graph has two-way arrows.","feed_headline":"Primitive homology equals path homology when arrows are one-way","feed_subtitle":"A simpler boundary condition reproduces the standard theory only when no arrow has a reverse arrow.","key_machinery":"The primitive chain complex $\\Pi_*(G) = \\{w \\in A_n(G) : \\partial w \\in A_{n-1}(G)\\}$ uses the ordinary alternating face differential $\\partial$ on the free module of allowed elementary paths of length $n$. The load-bearing fact is that asymmetry of the digraph rules out the repeated-consecutive-vertex fragment $iji$ inside allowed paths; this makes the projection from the free path module to its quotient by irregular paths injective on allowed paths, so the condition defining $\\Pi$ coincides with the condition defining the standard path complex $\\Omega$. The proof of Theorem 3.7 is carried by a commutative diagram comparing the two complexes through this projection, and Proposition 3.9 shows that membership in $\\Pi_n$ is equivalent to each individual face $\\partial_m(w)$ staying allowed, which is what makes the tail/head/cluster decompositions possible.","core_discovery":"The central claim is Theorem 3.7: for every asymmetric digraph $G$, the primitive chain complex $\\Pi_*(G)$ is naturally isomorphic to the path chain complex $\\Omega_*(G)$, and therefore the primitive path homology $H_*(G)$ equals the path homology $\\mathrm{H}_*(G)$. The isomorphism is the identity on allowed elementary paths in each dimension; it is natural because, with no opposite arrows present, no allowed path contains a fragment $iji$, so the difference between the two boundary conditions disappears. For general digraphs the theories diverge, as shown in Example 3.2 where the complete two-vertex digraph has primitive $H_1 = \\mathbb{Z}$ but path $\\mathrm{H}_1 = 0$. The paper also proves functoriality of the primitive theory on the subcategory of asymmetric digraphs, obtains a direct-sum decomposition of each $\\Pi_n(G)$ into tail-fixed, head-fixed, and $(a,b)$-cluster summands, and relates primitive $(a,b)$-cluster homology of a directed suspension to primitive homology shifted by two dimensions. One structural limitation is noted in Remark 4.8: the $(a,b)$-cluster submodules do not themselves form a chain complex, so cluster homology must be built through a separate complex.","pith_inferences":["If asymmetry is exactly where the two theories agree, the difference between primitive and path homology on general digraphs could be interpreted as a measure of two-cycle content, a reading the paper does not pursue.","The fixed-tail and fixed-head complexes are natural candidates for relative homology groups of a vertex pair; a long exact sequence relating $\\Pi_*$, $\\Theta^{[a,\\cdot]}_*$, and $\\Theta^{[\\cdot,b]}_*$ would be a plausible next step not taken here.","Because the boundary condition is purely combinatorial, primitive path homology may be noticeably easier to compute algorithmically than path homology; benchmarking on random asymmetric digraphs would test this expectation.","In characteristic 2 the alternating signs that produce the two-vertex counterexample vanish, so testing whether the isomorphism of Theorem 3.7 survives there is a concrete check of how essential the signs are."],"forward_implications":["On every asymmetric digraph, primitive path homology equals the established path homology, so calculations on such graphs can use the simpler primitive boundary condition.","Primitive $(a,b)$-cluster homology of the directed suspension $\\mathrm{Sd}\\,G$ is isomorphic to the primitive homology of $G$ shifted down by two dimensions.","The tail-fixed, head-fixed, and cluster decompositions give a direct-sum splitting of $\\Pi_n(G)$, allowing primitive homology computations to be organized by tail, head, or both.","The tail-fixed and head-fixed primitive homology theories are functorial on asymmetric digraphs and satisfy the cone and suspension shift relations of Theorem 6.10.","Since a digraph homotopy equivalent to a point via path homology can still have nonzero primitive $H_1$, primitive path homology is not homotopy invariant in the same sense as path homology."],"supporting_citations":[{"why":"Defines the path chain complex $\\Omega_*$ that Theorem 3.7 identifies with the primitive complex.","marker":"[8]"},{"why":"Establishes homotopy invariance of path homology, used in the two-vertex comparison of Example 3.2.","marker":"[9]"},{"why":"Supplies the inverse-digraph involution and suspension arguments adapted in Theorems 3.11, 5.10, and 6.7.","marker":"[10]"},{"why":"Provides the cluster-path decomposition and Lemma 2.2 used for the direct-sum splitting of $\\Pi_n$ and the cluster complex $\\Theta^{[a,b]}_*$.","marker":"[6]"},{"why":"Contains Lemma 2.3 used in Proposition 3.5 to prove functoriality of the primitive theory on asymmetric digraphs.","marker":"[7]"},{"why":"Introduces the path homology framework that the primitive theory is defined against.","marker":"[5]"}],"fun_headline_variants":["Primitive path homology equals path homology for one-way digraphs","Two path homologies agree exactly when digraph is asymmetric","New primitive path homology matches old for one-way digraphs","For one-way digraphs, primitive path homology = standard","One-way digraphs make primitive homology match path homology"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire comparison rests on the digraph having no pair of opposite arrows; if even one two-way arrow appears, the primitive and path complexes may disagree and the induced maps on them need not commute.","fun_headline_variants_meta":{"raw":{"variants":["Primitive path homology equals path homology for one-way digraphs","Two path homologies agree exactly when digraph is asymmetric","New primitive path homology matches old for one-way digraphs","For one-way digraphs, primitive path homology = standard","One-way digraphs make primitive homology match path homology"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001145,"raw_usage":{"total_tokens":4735,"prompt_tokens":915,"completion_tokens":3820,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":531,"completion_tokens_details":{"reasoning_tokens":3735}},"tokens_in":531,"tokens_out":3820,"duration_ms":22405,"temperature":1.0,"reasoning_tokens":3735,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T10:42:22.816722+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the first primitive path homology group of the complete digraph on two vertices with arrows in both directions: the paper predicts $\\mathbb{Z}$, while the standard path homology group is 0. For the main theorem, search for any asymmetric digraph $G$ and any $n \\geq 0$ for which the natural map $\\Pi_n(G) \\to \\Omega_n(G)$ is not an isomorphism, since Theorem 3.7 asserts no such example exists.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes homotopy invariance of path homology, used in the two-vertex comparison of Example 3.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the inverse-digraph involution and suspension arguments adapted in Theorems 3.11, 5.10, and 6.7."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the cluster-path decomposition and Lemma 2.2 used for the direct-sum splitting of $\\Pi_n$ and the cluster complex $\\Theta^{[a,b]}_*$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Contains Lemma 2.3 used in Proposition 3.5 to prove functoriality of the primitive theory on asymmetric digraphs."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the path homology framework that the primitive theory is defined against."}],"review_version":1}