{"id":"54fdb1bf-e51b-48b5-a90c-4df716e0678f","arxiv_id":"2608.02726","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Unweighted gapped clique homology is QMA1^{g2}-complete for a fixed inverse-polynomial gap, proven by replacing vertex weights with clique multiplicities.","lead":"This paper proves that deciding gapped homology of unweighted clique complexes is complete for the quantum complexity class QMA1 with a specific exact gate set. It removes a caveat from earlier work: vertex weights were not the reason gapped clique homology is hard; the spectral gap promise is.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The NO-case gap transfer hinges on King–Kohler's many-gadget estimate and the binary-weight assertion of Definition 5.3; the paper's Appendix A is only a sketch, so the central claim remains conditional on those imports.","rationale":"The reader's weakest assumption identifies the same load-bearing dependency: the NO-case argument transfers the weighted gap through the imported King–Kohler estimates and relies on the exact binary weight assignment. My read agrees with that. I found no internal contradiction in the blow-up machinery: Theorem 4.3's block-averaging and star decomposition arguments are plausible, Lemma 4.2's symmetric reduction is careful about duplicate-block cancellations, and the NO-case algebra in Section 6.3 is consistent. The concern is therefore not about the paper's own proofs but about whether the cited and proof-sketched import, plus the asserted binary weights in Definition 5.3, hold at the precise parameter choice lambda = 1/(C_wt t q(n)) with constants uniform in t. If those imports are correct, the main theorem goes through as stated; if not, the central completeness and hardness claims would need repair. This is exactly the kind of conditional support that justifies the reader's CONDITIONAL verdict rather than a full ACCEPT, and I do not see grounds to move the verdict further.","tokens_in":19050,"tokens_out":24263,"duration_ms":245009,"concrete_test":"Reconstruct, for one m_loc <= 4 Rudolph integer projector, the full King–Kohler single-gadget complex with the exact weight assignment of [KK24, Sec. 8], and recompute the single-gadget spectral band scales and the Appendix A many-gadget bound at lambda = 1/(C_wt t q(n)), tracking every constant in t and m_loc. If any added vertex is not exactly weight lambda (contradicting Definition 5.3), or if the O(lambda t) projector error or the penalty scale lambda^{4m_loc+2} acquires hidden t-dependence, then Theorem 6.3's gap transfer fails; if the weights are exactly binary and the constants are t-independent, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The internal mechanism is coherent: Theorem 4.3's decoupling bound and Lemma 4.2's symmetric reduction are self-contained and appear sound. The load-bearing point is the NO-case transfer in Theorem 6.3. It invokes the imported King–Kohler many-gadget bound, Theorem 6.2, at the exact parameter lambda = 1/(C_wt t q(n)), and assumes in Definition 5.3 that the King–Kohler weights are exactly binary: register vertices weight 1, every added gadget vertex weight lambda. The manuscript does not reproduce the gadget construction; Appendix A is an acknowledged proof sketch that imports [KK24, Lems. 10.2–10.4] and asserts the operator identity sum_i hatPhi_i^perp - (t-1)Pi_0 = Pi_H - H - (t-1)(Pi_0 - Pi_H) + O(lambda t). If that identity or the uniform O(lambda) projector estimates fail at this t-dependent lambda, the contradiction step in Appendix A does not close. Likewise, if any vertex produced by the King–Kohler cut-and-reglue/thickening construction, or by the Rudolph integer-gadget embedding used in Section 5.4, carries a weight other than lambda, then Lemma 4.2 identifies the symmetric sector with M times the wrong weighted Laplacian, and the promised unweighted gap N^{-c_gap} no longer follows. The paper's own text flags these as imports rather than derivations, which is precisely where the central claim is least secure.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that the Unweighted Gapped Clique Homology problem, in which every vertex has weight one, is QMA_1^{G2}-complete for the gate set G2={X,CX,CCX,H⊗H}. The core technical contribution is a blow-up construction (Section 4) that replaces each vertex weight by the size of an expanded clique block. On the symmetric sector, the unweighted Hodge Laplacian is shown to be unitarily equivalent to M times the weighted Laplacian (Lemma 4.2); on the orthogonal complement, a local averaging argument gives a uniform spectral gap (Theorem 4.3). Hardness is obtained by applying this to the King-Kohler weighted construction (Sections 5-6), with the NO-case gap transferred via Theorem 6.3. Containment is proved in Section 7 using Rudolph's exact sparse Hamiltonian simulation. The paper is transparent about importing several technical estimates from [KK24] and [Rud25]; Appendix A gives only a proof sketch of the many-gadget estimate.","tokens_in":19339,"tokens_out":26489,"duration_ms":210225,"significance":"If the imported estimates hold, the result is significant: it shows that the inverse-polynomial gap promise, rather than the vertex weighting, is the source of QMA_1-hardness in gapped clique homology. The blow-up and symmetric-sector mechanism is a clean and potentially transferable technique, and Section 4 is self-contained and appears correct. The paper gives a useful survey of related weighted and filtered homology complexity results. Its main weakness is that the central NO-case gap rests on unproven imported bounds and an unverified binary-weight assertion, so the present manuscript is conditional rather than fully self-contained.","major_comments":[{"comment":"The binary weight assignment is load-bearing and is not verified in the manuscript. The claim that every register vertex has weight 1 and every added gadget vertex has weight λ is cited to [KK24, Sec. 8], but the construction is not reproduced. If any gadget vertex carries a different weight, Lemma 4.2 identifies the symmetric sector with M times a weighted Laplacian that differs from the one for which the imported many-gadget bound (Theorem 6.2) applies, and the promised gap N^{-c_gap} would not follow. The paper should provide a detailed derivation of the vertex weights for the concrete gadget family used (including the Rudolph integer-projector embedding), or a rigorous verification from [KK24].","section":"Section 5.3, Definition 5.3"},{"comment":"The NO-case gap transfer in Theorem 6.3 depends on the imported many-gadget estimate [KK24, Thm. 10.1] and the single-gadget spectral decomposition [KK24, Lems. 9.1, 10.1]. Appendix A is only a proof sketch: the key estimates (1)-(3) and the operator identity Σ_i bΦ_i^⊥ − (t−1)Π_0 = Π_H − H − (t−1)(Π_0 − Π_H) + O(λ t) are stated as imports from [KK24, Lems. 10.2–10.4] without proof. The manuscript should either give a complete proof of these estimates in the present notation, or state explicitly that the main theorem is conditional on [KK24] and verify that all hypotheses (uniform constants, t-dependent λ) are satisfied at the chosen λ = 1/(C_wt t q(n)). As written, the central NO-case claim is not self-contained.","section":"Section 6.2 and Appendix A"},{"comment":"The weight-free selectivity statement is also imported. Lemma 5.4 asserts that each gadget kills exactly the intended encoded cycle and creates no additional target-degree class, citing [KK24, Lem. 8.4] and [Rud25, App. D.1]. This is load-bearing for the YES-case direction (perfect completeness) and for the absence of spurious homology. The paper should either provide a proof of this selectivity for the specific integer-projector gadgets used, or clearly state the conditions under which the cited lemmas apply and explain why they hold for the present fixed finite family.","section":"Section 5.4, Lemma 5.4"}],"minor_comments":[{"comment":"The title states 'is QMA1-complete' while the abstract and Theorem 3.2 state QMA_1^{G2}-complete; the title should be qualified to avoid overstatement.","section":"Title"},{"comment":"In the table, the row `amplitude weight w(v) q(bf_v/ρ)` appears to have a typo: `q` should be `\\sqrt` (the surrounding text uses `\\sqrt{bf_v/ρ}`).","section":"Definition 5.3"},{"comment":"The notation `QMA_1^{g_2}` in the abstract uses lowercase g_2 while the text uses uppercase G_2; please unify.","section":"Abstract and Section 2.4"},{"comment":"The containment proof relies on Rudolph's Exact Sparse Hamiltonian problem [Rud25, Prob. 2.9] and lemmas [Rud25, Lem. 6.1, 6.2]; the paper would benefit from stating the relevant definitions so that the containment argument is self-contained.","section":"Section 7"}],"recommendation":"major_revision","confidential_remarks":"The paper is a preprint with heavy reliance on two cited works ([KK24] and [Rud25]). The core blow-up argument is original and attractive, but the NO-case gap is conditional on imported estimates and on the binary-weight assertion, neither of which is proved in the manuscript. If the journal's policy permits citing the published FOCS 2024 version of [KK24] and the arXiv preprint [Rud25], the main theorem may be acceptable after the authors supply the missing verifications. I saw no indication of circularity or misconduct; the paper is transparent about its imports."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: read this for the Section 4 blow-up construction, which is real and self-contained; the completeness theorem is close to right, but the NO-case gap is only as solid as the King–Kohler many-gadget bound at the exact parameter lambda. I would send it to serious referees, and I would not yet cite the main theorem as fully settled until that import is verified.\n\nThe genuinely new piece is the level-ratio blow-up. Replacing a weighted vertex by a clique of size bf_v and then restricting to the symmetric sector is not a routine repackaging. Lemma 4.2 is an exact identification with the weighted Laplacian, and Theorem 4.3’s decoupling bound for the asymmetric sector is a clean local argument using the complete-graph spectrum and the join formula. Corollary 4.4 (no spurious homology) follows directly. Those parts are independent of the King–Kohler machinery and look sound to me. The containment argument in Section 7 is also plausible: integer entries, bounded sparsity, exact LCU simulation — standard but executed carefully.\n\nWhere it gets conditional is the NO-case gap transfer in Section 6. Theorem 6.3 sets rho = (C_wt t q(n))^2 and lambda = 1/(C_wt t q(n)), then imports the King–Kohler many-gadget estimate [KK24, Thm. 10.1] at exactly that t-dependent lambda. The authors are upfront that Appendix A is a sketch and that Lemma 6.1 is imported. The paper also asserts in Definition 5.3 that the King–Kohler weights are exactly binary: register weight 1, gadget weight lambda. If that assertion fails for some gadget in the Rudolph integer family, Lemma 4.2 would identify the symmetric sector with the wrong weighted Laplacian, and the promised inverse-polynomial gap would not follow. This is not a demonstrated error — the citations point to the right places — but it is a load-bearing import that a referee should verify carefully. The concrete embedding of Rudolph’s G2 4-QSAT construction is also only sketched, so the fixed finite gadget family and its uniform constants deserve scrutiny.\n\nMinor notes: c_gap is not quantified, and the title omits the QMA_1^{G2} gate-set qualification. Neither is a mathematical problem, but both should be stated precisely.\n\nFor whom: anyone working on quantum complexity of topological data analysis, QMA_1-hardness, or combinatorial Hodge Laplacians. The paper narrows the weighted-to-unweighted hardness gap and makes the role of the spectral promise explicit. If the imports check out, this is a solid paper.\n\nMy recommendation: send it to peer review, with a referee specifically asked to verify the King–Kohler imports — especially the binary-weight claim and the uniform constants at lambda = c_wt p_H/t.","headline":"The blow-up construction is genuinely new and the Section 4 unweighting is sound, but the NO-case gap inherits King–Kohler imports that are only sketched, so the completeness claim needs referee checking before being treated as settled.","tokens_in":19863,"tokens_out":2996,"would_cite":true,"duration_ms":27852,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68Q12","55U10","05C50"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that deciding gapped clique homology on unweighted graphs is QMA1-complete, so the hard part is the spectral gap promise, not vertex weights.","keywords":["unweighted clique homology","QMA1-completeness","Hodge Laplacian","spectral gap promise","blow-up construction","symmetric sector","topological data analysis","Hamiltonian complexity"],"falsifier":"Compute the degree-$(2n-1)$ unweighted Hodge Laplacian of the blow-up of a single gadget with $\\rho=4$ (so $\\lambda=1/2$), and check the smallest eigenvalue on the orthogonal complement of the symmetric sector: if it is below $\\min_v \\hat f_v=1$, Theorem 4.3 is wrong. Alternatively, inspect the explicit weighting of the cited single-gadget construction: any gadget vertex whose amplitude weight differs from $\\lambda=\\rho^{-1/2}$ immediately breaks the binary-weight correspondence.","tokens_in":18828,"feed_emoji":"🧩","tokens_out":10483,"duration_ms":79741,"temperature":0.7,"pith_summary":"This paper establishes that deciding whether the clique complex of an unweighted graph has nonzero homology in a given dimension, under an inverse-polynomial spectral gap promise on the Hodge Laplacian, is complete for the quantum complexity class $\\mathsf{QMA}_1$ with a fixed universal gate set $G_2=\\{X,\\mathsf{CX},\\mathsf{CCX},H\\otimes H\\}$. The known hardness proof for the gapped problem assigned tiny weights to certain vertices to create scale separation; the paper shows those weights can be removed entirely. The construction replaces each weighted vertex by a clique whose size encodes the weight, so that symmetric averages over the copies reproduce the weighted Hodge metric. A local averaging argument then proves that all copy-label-dependent directions have uniformly positive energy, so the blow-up adds no spurious low-energy states or homology. If correct, the result isolates the gap promise, not vertex weighting, as the source of hardness.","feed_headline":"Gapped clique homology stays hard even with all weights set to one","feed_subtitle":"Replacing weights by clique multiplicities preserves the hard gap, so the promise does the work.","key_machinery":"The load-bearing object is the level-ratio blow-up (Definition 4.1): a base vertex of level $\\ell$ is replaced by a clique of size $M/\\rho^{\\ell}$, with $M=\\rho$ and only levels 0 and 1 used in the application. The argument rides on two identities: the symmetric reduction $\\Phi \\hat\\Delta_k|_{\\hat T_k}\\Phi^{-1} = \\Delta_{\\hat w,k}$ (Lemma 4.2), which makes the symmetric sector carry exactly $\\rho$ times the weighted Laplacian, and the asymmetric gap $\\hat\\Delta_k|_{\\hat T_k^\\perp} \\succeq \\min_v \\hat f_v\\, I$ (Theorem 4.3), which removes copy-label-dependent directions. Together they transfer the weighted spectral gap to an unweighted clique complex without spurious low-energy states or homology.","core_discovery":"The paper's central claim is Theorem 3.2: for a sufficiently large fixed constant $c_{\\mathrm{gap}}$, Unweighted Gapped Clique Homology is $\\mathsf{QMA}_1^{G_2}$-complete, where $G_2=\\{X,\\mathsf{CX},\\mathsf{CCX},H\\otimes H\\}$. The reduction takes an instance of the $G_2$ 4-QSAT problem with integer local projectors and constructs the weighted gadget complex from the cited gapped-clique-homology work, then blows it up: each register vertex becomes a clique $K_\\rho$ and each gadget vertex stays a singleton, with $\\rho=(C_{\\mathrm{wt}}t q(n))^2$. Lemma 4.2 identifies the unweighted blow-up Laplacian on the symmetric one-copy-per-block sector with $\\rho$ times the weighted Laplacian, so the weighted many-gadget bound applies at the exact parameter $\\lambda=1/(C_{\\mathrm{wt}}t q(n))$; Theorem 4.3 gives a uniform lower bound of $\\min_v \\hat f_v=1$ on the asymmetric sector. Combining the two bounds yields the promised $N^{-c_{\\mathrm{gap}}}$ NO-case gap, while the YES case is preserved by weight-free gadget selectivity. Containment is proved by normalizing the sparse integer clique Laplacian and applying exact linear-combination-of-unitaries simulation.","pith_inferences":["The same multiplicity-as-weight mechanism may convert other weighted homological hardness results into unweighted ones, for example independence-complex Laplacians or persistence-style promises, since the mechanism is metric reconstruction plus redundancy.","Sparsifying the clique blocks is the natural stress test: the paper's own discussion predicts that expander-like replacements lose block symmetry and may create unintended homology, so a persistence-style promise on marked cycles would be a cleaner next problem.","One can read the construction as an error-correcting encoding of the logical chain space inside the symmetric sector, suggesting that unweighting may work wherever a Hamiltonian's parameters can be encoded as multiplicities of identical copies."],"forward_implications":["The hardness of gapped clique homology is independent of vertex weights: every vertex can be given weight one without changing $\\mathsf{QMA}_1^{G_2}$-completeness.","An inverse-polynomial spectral gap promise is itself enough to make the unweighted problem quantumly hard, so future separation arguments can target the promise rather than artificial weights.","The blow-up is polynomial-sized and dense; the output graph has $N\\le R\\rho$ vertices with $R=\\mathrm{poly}(n)$, so the reduction is efficient.","No spurious homology is created: the harmonic chains of the blow-up coincide with those of the weighted base complex, so topological YES witnesses are preserved."],"supporting_citations":[{"why":"Supplies the weighted gapped-clique-homology hardness result and the single- and many-gadget spectral estimates that the blow-up transfers to the unweighted setting.","marker":"[KK24]"},{"why":"Provides the exact gate set, the 4-QSAT source with integer local projectors, and the exact linear-combination-of-unitaries containment machinery for sparse integer clique Laplacians.","marker":"[Rud25]"},{"why":"Gives the complete-simplex Hodge-Laplacian spectrum used to lower-bound the asymmetric sector in Theorem 4.3.","marker":"[GW16]"},{"why":"Supplies the join formula for augmented Hodge Laplacians used in the register construction and in the block-star energy calculation.","marker":"[HJ13]"}],"fun_headline_variants":["No weights, no problem: Gapped clique homology stays QMA1-complete","Clique blow-up preserves quantum hardness without weight scale","Gap promise, not vertex weights, is the real source of hardness","Unweighted gapped clique homology remains QMA1-complete"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The NO-case gap transfer depends on an imported many-gadget spectral estimate holding with uniform constants at the exact value $\\lambda=1/(C_{\\mathrm{wt}}t q(n))$, and on the gadget vertex weights being exactly binary (weight 1 on register vertices, weight $\\lambda$ on gadget vertices); if either fails, the symmetric sector no longer carries the intended weighted Laplacian and the promised $N^{-c_{\\mathrm{gap}}}$ gap does not follow.","fun_headline_variants_meta":{"raw":{"variants":["No weights, no problem: Gapped clique homology stays QMA1-complete","Clique blow-up preserves quantum hardness without weight scale","Gap promise, not vertex weights, is the real source of hardness","Unweighted gapped clique homology remains QMA1-complete"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00037,"raw_usage":{"total_tokens":2093,"prompt_tokens":1166,"completion_tokens":927,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":782,"completion_tokens_details":{"reasoning_tokens":851}},"tokens_in":782,"tokens_out":927,"duration_ms":8359,"temperature":1.0,"reasoning_tokens":851,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:02:51.912055+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the degree-$(2n-1)$ unweighted Hodge Laplacian of the blow-up of a single gadget with $\\rho=4$ (so $\\lambda=1/2$), and check the smallest eigenvalue on the orthogonal complement of the symmetric sector: if it is below $\\min_v \\hat f_v=1$, Theorem 4.3 is wrong. Alternatively, inspect the explicit weighting of the cited single-gadget construction: any gadget vertex whose amplitude weight differs from $\\lambda=\\rho^{-1/2}$ immediately breaks the binary-weight correspondence.","supporting_citations":[],"review_version":2}