{"id":"784934ed-7cfb-4b2d-b3d8-b2f54cb0a550","arxiv_id":"2505.05461","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Marked graph complexes B(g,n,r) stabilize sharply at n = ⌈3m/2⌉, where m = 3(g−1)+2(n−r), and stable top-degree multiplicities are given by Littlewood-Richardson coefficients.","lead":"The paper proves that certain graph complexes built from marked graphs are representation stable, with a sharp bound on when their irreducible decompositions stabilize. This gives new homology classes of these complexes and recovers, as a special case, Hersh and Reiner's stability theorem for configuration spaces in odd-dimensional Euclidean space.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified to the central claim; the reader's flagged Lemma 3.5 Step 1 is a standard Mackey-decomposition step.","rationale":"The Reader's CONDITIONAL verdict rests on an omitted verification in Lemma 3.5 Step 1 and on a perceived convention-sensitivity of Lemma 4.5. The first concern is not load-bearing: the asserted coset injection is immediate from J = I∩I^c, and the induced injection of representations is the identity double-coset summand in the Mackey decomposition, so the step is true as written. The second concern is not an internal gap, since Lemma 4.5 follows from the admissibility rules that are actually fixed in Definition 3.1. I checked the main induction of Proposition 4.11 case by case, including the base case g=1, the good-edge cutting argument, the marked-tadpole deletion, and the circuit subcases, and found no counterexample to the claimed bound. The sharpness witnesses θ_{g,ℓ}(p) do achieve the bound as stated. The paper would benefit from expanding the one-line citation in Lemma 3.5 Step 1, but this is an exposition issue, not a correctness risk. No load-bearing concern about the central claim was identified, so the Reader's verdict need not be changed.","tokens_in":26259,"tokens_out":61545,"duration_ms":621501,"concrete_test":"Re-derive the injection in Lemma 3.5 Step 1 without citing [Ser77]: write the Mackey decomposition of Res^{S_{n+2}}_{S_n} Ind^{S_{n+2}}_{I^c}X and isolate the identity double-coset summand; if it is not exactly Ind^{S_n}_J Res^{I^c}_J X, then the induction step in Proposition 4.11 Case 1 collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the central proof of Theorem 4.2 as internally sound. The only genuinely compressed argument, Lemma 3.5 Step 1, is exactly what the Reader flags: an omitted verification that an injection of cosets induces an injection of induced representations. That step is correct: with J = I_γ ∩ I_γ^c, the map S_n/J → S_{n+2}/I_γ^c is injective because the kernel is J, and the induced S_n-map is the identity double-coset summand in the Mackey decomposition. The citation to [Ser77] is heavier than needed, but the mathematics is standard. Lemma 4.5 is a direct counting consequence of the admissibility rules fixed in Definition 3.1, so it is not a convention-sensitivity gap. I also traced Proposition 4.11's induction cases, including the base g=1 case, the cutting case, the tadpole case, and the circuit analysis, and the sharpness witnesses θ_{g,ℓ}(p) do saturate the stated bound. I therefore find no load-bearing correctness risk in the proof of the sharp stability point.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a sharp representation stability result for the marked graph complexes B(g,n,r) of Payne and Willwacher: for fixed g and ℓ with m = 3(g−1)+2ℓ ≥ 0, the consistent sequence B(g,n,n−ℓ)⊗V_{1_n} is uniformly representation stable and stabilizes sharply at n = ⌈9(g−1)/2⌉ + 3ℓ = ⌈3m/2⌉. The proof decomposes each degree of the complex into direct sums of core sequences indexed by 'core graphs', bounds the stabilization point of each core sequence by an induction on genus, and exhibits explicit core graphs θ_{g,ℓ}(p) witnessing sharpness. The paper then draws consequences for homology multiplicities: a Littlewood–Richardson formula for the stable top-degree homology, vanishing statements for other degrees, and a computation of H_*(B(1,n,r)) as Whitehouse modules, recovering the sharp stability theorem of Hersh and Reiner for configuration spaces in odd-dimensional Euclidean space.","tokens_in":26464,"tokens_out":34607,"duration_ms":314118,"significance":"If the main theorem is correct, this is a substantial contribution to representation stability and to the structural understanding of the marked graph complexes appearing in the cohomology of moduli spaces of curves. The paper is notable for giving a sharp, not merely eventual, stability bound, and for proving that the bound is witnessed by explicit graphs. The passage from chain-level stability to homology multiplicities via Theorem 2.9 is a useful general mechanism, and the genus-1 computation is self-contained and independently recovers a known result, which strengthens confidence in the framework. The explicit stable multiplicity formula and the applications to weight 11 and weight 15 cohomology of moduli spaces give the results concrete computational content. I found no load-bearing correctness error in the central derivation; the main theorem is supported by a detailed chain-level decomposition and an explicit sharpness witness.","major_comments":[],"minor_comments":[{"comment":"The proof of Step 1 in Lemma 3.5 is compressed: the injectivity of the coset map S_n/J → S_{n+2}/I^c and the resulting injection (3.1) are dismissed with a reference to [Ser77, Prop. 7.3]. Since this is the one place in the genus induction where a Mackey-decomposition argument is invoked, please spell out the double-coset argument, noting that the kernel of the induced coset map is J = I^c ∩ S_n.","section":"Lemma 3.5"},{"comment":"In the equality part of Lemma 4.5, the symbol t is used before it is defined: after introducing p and y, please define t as the number of neutral vertices connected to the distinguished vertex by three edges, so that the statement 'the genus is 2t+q+p+1' is immediately readable.","section":"Lemma 4.5"},{"comment":"The estimates in Proposition 4.11 are written with decimals such as '4.5(g−2) + 3(n+2−r) + .5'; restating all bounds with ceilings, e.g. ⌈9(g−2)/2⌉, would make the induction and the final rounding easier to verify.","section":"Section 4.3"},{"comment":"The notation V_{n−m} in the definition of X◦V_{n−m} can be confused with the convention V_λ for irreducibles; please state explicitly that V_n denotes the trivial representation of S_n, in contrast with V_{1_n}, which denotes the sign representation.","section":"Section 2.1"},{"comment":"In the paragraph after Theorem 6.2, the phrase 'of dimensions n,k, the Stirling number of the first kind' appears to be a typo for the unsigned Stirling number typically written s(n,k); please correct the notation.","section":"Section 6"},{"comment":"Corollary 4.6 is stated without proof; it follows immediately from Lemma 4.5 by taking the core of γ and letting t be the number of marked legs, but adding this one-sentence justification would improve readability.","section":"Corollary 4.6"}],"recommendation":"minor_revision","confidential_remarks":"This is a well-written and carefully argued paper, squarely within the scope of math.AT. The central theorem is internally sound, and the sharpness witnesses are concrete and checkable. The only substantive request is to expand the standard but unstated Mackey-decomposition verification in Lemma 3.5; all other issues are local presentation fixes. I recommend minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main result is a sharp representation stability bound for the marked graph complexes B(g,n,r): for fixed g and ℓ, the sequence B(g,n,n−ℓ)⊗sgn stabilizes exactly at n = ceil(3m/2) with m = 3(g−1)+2ℓ. I read the proof of Theorem 4.2 as internally sound. The sharpness witnesses θ_{g,ℓ}(p) are explicit and do saturate the bound, and the induction on genus is a genuine piece of work, not a repackaging. The paper also gives a formula for stable top-degree multiplicities in terms of Littlewood-Richardson coefficients, and the genus-one computation recovers Hersh and Reiner's configuration-space bound as a consequence. That is a real extension of [PW24] and [CLPW24], which only computed low-excess examples, and the novelty is substantive even though the machinery is standard.\n\nThe reader's main concern is Lemma 3.5 Step 1, an omitted verification that a coset injection induces an injection of induced representations. The authors defer to a citation of Serre. I agree with the stress-test: this is a standard Mackey-decomposition fact, and the step is correct. It is compressed, and a referee could reasonably ask for one sentence explaining the double-coset argument, but it is not a load-bearing gap. The other flagged point, Lemma 4.5, depends on the admissibility rules as fixed in Definition 3.1. That is a convention within the model, not a hidden assumption; the counting bound follows directly from those rules. I do not see a circularity issue: the sharpness witnesses are independent of the upper bound, and the Hersh–Reiner recovery is a benchmark, not an input.\n\nIf I have a complaint, it is that the paper is dense and some sign/convention choices are easy to trip over. That is a presentation issue, not a correctness issue. The citation pattern looks honest: [War24] is cited for the genus-one precursor, but the current proof is self-contained in the B(g,n,r) language and goes beyond it.\n\nWho is this for? Specialists in representation stability and graph complexes, especially people working on moduli space cohomology. It is not a paradigm shift, but it is a solid, important-in-the-subfield result with a clear new bound and computable stable multiplicities. It deserves a serious referee and, I expect, acceptance after minor revision.\n\nRecommendation: send it to peer review; if I were the editor I would not desk-reject.","headline":"A solid, genuinely new sharp stability theorem for marked graph complexes; the reader's conditional verdict is too cautious about one standard Mackey step.","tokens_in":26987,"tokens_out":1409,"would_cite":true,"duration_ms":16211,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55R80","05E10","20C30","18G85"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the marked graph complexes B(g,n,n−ℓ) are representation stable and stabilize sharply at n = ⌈3m/2⌉, with the stable top-degree homology multiplicities computed by Littlewood–Richardson sums.","keywords":["representation stability","graph complexes","marked graphs","symmetric group representations","moduli spaces of curves","Whitehouse modules","configuration spaces","Littlewood-Richardson coefficients"],"falsifier":"For $g=2$, $\\ell=1$ (so $m=5$ and the claimed sharp point is $n=8$), compute the full irreducible decompositions of $B(2,7,6)\\otimes \\mathrm{sgn}_7$ and $B(2,8,7)\\otimes \\mathrm{sgn}_8$. If the decomposition at $n=8$ matches the stability rule derived from $n=7$---each partition of $8$ either has zero multiplicity or is $(8-|\\lambda|,\\lambda)$ with the same multiplicity as at $n=7$---then the sequence is already stable at $7$ and the 'sharply at $8$' claim fails.","tokens_in":26065,"feed_emoji":"📐","tokens_out":16262,"duration_ms":139412,"temperature":0.7,"pith_summary":"This paper proves a sharp representation stability theorem for the marked graph complexes $B(g,n,r)$, the chain complexes whose homology computes pieces of the compactly supported cohomology of moduli spaces of marked curves. Fixing the genus $g$ and the excess $\\ell$, so that $r = n-\\ell$ and $m = 3(g-1)+2\\ell$, the sequence of complexes formed by adding one marked leg, tensored with the sign representation, is representation stable and stabilizes exactly at $n = \\lceil 3m/2\\rceil$. From that point on, the irreducible decomposition of every degree of the complex is a formal consequence of the decomposition at the stabilization point. The paper also computes the resulting stable multiplicities in the top degree as an explicit sum of Littlewood–Richardson coefficients, and shows that in genus $1$ the theorem recovers the sharp stability bound for configuration spaces of points in odd-dimensional Euclidean space.","feed_headline":"Marked graph complexes stabilize sharply at n = ceil(3m/2)","feed_subtitle":"Once n reaches this point, all irreducible multiplicities are determined, and the bound is provably tight.","key_machinery":"The central object is the marked graph complex $B(g,n,r)$: its degree-$i$ piece is spanned by isomorphism classes of connected graphs with a distinguished vertex, $n$ labeled legs, and at least $r$ marked flags at the distinguished vertex, with edges in degree $1$ and marked flags in degree $-1$. The argument is carried by the core-graph decomposition. Forgetting the marked legs of any graph leaves a core graph; each core graph $\\xi$ of type $(g,k,u)$ determines a consistent sequence $\\mathbf{p}A_\\xi$ of $S_n$-modules, and Lemma 2.7 says that sequence stabilizes sharply at $k+\\rho_\\xi$, where $\\rho_\\xi$ is the maximum number of rows of an irreducible summand. The key counting lemma bounds $k \\le 3(g-1)+2(k-u)$, with equality only for the explicit graphs $\\theta_{g,\\ell}(p)$; combining this with induction on genus over edge-cutting and vertex-removal steps yields $k+\\rho_\\xi \\le \\lceil 3m/2\\rceil$. The same graphs $\\theta_{g,\\ell}(0)$ and $\\theta_{g,\\ell}(1)$ have $\\rho_\\xi$ exactly large enough to make the bound sharp.","core_discovery":"The central claim is that the consistent sequence $\\cdots \\to B(g,n,n-\\ell)\\otimes V^{1_n} \\to B(g,n+1,n+1-\\ell)\\otimes V^{1_{n+1}} \\to \\cdots$ is representation stable and stabilizes sharply at $n = \\lceil 3m/2\\rceil = \\lceil 9(g-1)/2\\rceil + 3\\ell$, where $m = 3(g-1)+2\\ell$. 'Sharply' means the sequence is stable at that value of $n$ but not at $n-1$. The proof decomposes each degree of $B(g,n,n-\\ell)$ into summands indexed by core graphs---graphs with no marked legs---and bounds the stability point of each summand by counting flags and applying induction on genus. Two explicit families of core graphs, $\\theta_{g,\\ell}(0)$ and $\\theta_{g,\\ell}(1)$, are shown to stabilize exactly at the bound, forcing sharpness. On homology, the sharp bound gives vanishing and stability of certain multiplicities: partitions with fewer than $\\lceil m/2\\rceil$ rows contribute nothing, and in the top degree the stable multiplicities are given by a sum of Littlewood–Richardson coefficients. In genus $1$ the full homology $H_*(B(1,n,r))$ is identified with the Whitehouse modules (a family of $S_n$-representations with Stirling-number dimensions), and the theorem recovers the sharp $n=3i$ bound of [HR17] for the cohomology of configuration spaces in $\\mathbb{R}^3$.","pith_inferences":["Editorial inference: the sharpness witnesses $\\theta_{g,\\ell}(0)$ and $\\theta_{g,\\ell}(1)$ are the most symmetric core graphs saturating the counting bound, which suggests that the $\\lceil 3m/2\\rceil$ threshold is a feature of the admissible-marking conventions (trivalent neutral vertices, no neutral tadpoles, no double-marked tadpoles); altering those conventions should move the threshold predict","Editorial inference: the same core-graph decomposition should extend to variants with several distinguished vertices, the setting the paper's outlook ties to commutative graph homology; a natural conjecture is that the multi-rooted complex stabilizes at the same $\\lceil 3m/2\\rceil$ with $m$ measured by total genus and excess, with the present stable multiplicities serving as genus-$1$ building blo","Editorial inference: the identification of $B(1,n,r)$ with the Whitehouse modules gives a purely combinatorial witness for the sharp $n=3\\ell$ configuration-space bound of [HR17], so one could test whether the topological genus-$2$ constructions discussed in the outlook can be realized by the same graph-complex machinery, with the top-degree stable classes here predicting their irreducible content","Editorial inference: if the computed stable multiplicities are correct, the equivariant spectral sequence to commutative graph homology cannot kill these classes by differentials landing inside the stable range, so the nonzero classes in $H^m(B(g,n,n-\\ell))$ should survive to the $E^\\infty$ page unless paired with classes outside the stable range."],"forward_implications":["For fixed $g$ and $\\ell$, once $n \\ge \\lceil 3m/2\\rceil$, the entire $S_n$-character of $B(g,n,n-\\ell)\\otimes V^{1_n}$ in every degree is determined by the character at the stabilization point.","Corollary 1.2 gives explicit homology constraints: in any degree, the multiplicity of $(\\lambda,1^{n-N})$ in $H_i(B(g,n,n-\\ell))$ is zero when $N > \\lceil 3m/2\\rceil$, and is independent of $n$ when $N = \\lceil 3m/2\\rceil$.","Theorem 1.3 computes top-degree stable homology: the nonzero stable multiplicities are indexed by partitions $\\lambda$ of $\\lceil 3m/2\\rceil$ with $\\lceil m/2\\rceil$ rows, and their values are Littlewood–Richardson sums independent of $n$.","In genus $1$, the isomorphism $H_i(B(1,n,r)) \\cong W_{n,r-1}$ for $i=2(n-r)$ and $0$ otherwise identifies the homology with Whitehouse modules, and the sharp chain-level bound descends to the sharp $n=3\\ell$ bound for $H_{2\\ell}(C(\\mathbb{R}^3,n))$ of [HR17].","Because the differentials in the spectral sequence to commutative graph homology are equivariant, the stable multiplicities and vanishing statements constrain which classes can appear on the $E^1$ page and how they can pair, informing higher-genus computations."],"supporting_citations":[{"why":"It supplies the definition of the marked graph complexes B(g,n,r), the admissible-marking conventions, the edge-contraction differential, and the weight-11 cohomology motivation.","marker":"[PW24]"},{"why":"It provides the definitions of consistent sequences, uniform representation stability, and multiplicity stability used throughout the paper.","marker":"[CF13]"},{"why":"It proves the sharp n=3i stability bound for configuration-space cohomology in R^3 that the genus-1 case recovers as a corollary.","marker":"[HR17]"},{"why":"It introduces the Whitehouse modules W_{n,k}, which are identified with H_*(B(1,n,r)) in Theorem 6.2.","marker":"[Whi97]"},{"why":"It gives the topological interpretation of Whitehouse modules as configuration-space cohomology tensored with the sign representation, used to match the genus-1 computation.","marker":"[ER19]"},{"why":"It computes the homology of the Stirling complexes later shown quasi-isomorphic to B(1,n,r), providing the precursor for the genus-1 character computation.","marker":"[War24]"},{"why":"It supplies the Young-diagrammatic formulas for induced representations from hyperoctahedral and Pieri-type inductions used in the sharpness argument and in Theorem 5.4.","marker":"[KT87]"},{"why":"It provides low-excess weight-11 homology computations (for example H_4(B(7,4,11))) used to confirm the stable multiplicity formulas against known cases.","marker":"[Bur25]"}],"fun_headline_variants":["Sharp stability bound for marked graph complexes at n=ceil(3m/2)","Marked graph complexes stabilize sharply, generalizing Hersh-Reiner","Core graphs prove sharp stability threshold for graph complexes","Exact stability point for marked graph complexes: n=ceil(9(g-1)/2)+3l"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The sharp bound rests on Lemma 4.5, the counting claim that every core graph of type $(g,n,r)$ has $n \\le 3(g-1)+2(n-r)$, with equality only for the explicit family $\\theta_{g,\\ell}(p)$; if the admissibility rules for marked graphs changed, the count---and hence the stability point---could change.","fun_headline_variants_meta":{"raw":{"variants":["Sharp stability bound for marked graph complexes at n=ceil(3m/2)","Marked graph complexes stabilize sharply, generalizing Hersh-Reiner","Core graphs prove sharp stability threshold for graph complexes","Exact stability point for marked graph complexes: n=ceil(9(g-1)/2)+3l"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000782,"raw_usage":{"total_tokens":3441,"prompt_tokens":920,"completion_tokens":2521,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":536,"completion_tokens_details":{"reasoning_tokens":2438}},"tokens_in":536,"tokens_out":2521,"duration_ms":18567,"temperature":1.0,"reasoning_tokens":2438,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:03:22.871640+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $g=2$, $\\ell=1$ (so $m=5$ and the claimed sharp point is $n=8$), compute the full irreducible decompositions of $B(2,7,6)\\otimes \\mathrm{sgn}_7$ and $B(2,8,7)\\otimes \\mathrm{sgn}_8$. If the decomposition at $n=8$ matches the stability rule derived from $n=7$---each partition of $8$ either has zero multiplicity or is $(8-|\\lambda|,\\lambda)$ with the same multiplicity as at $n=7$---then the sequence is already stable at $7$ and the 'sharply at $8$' claim fails.","supporting_citations":[],"review_version":1}