{"id":"2bdda5b3-6f52-4183-b524-92642dcf0aa5","arxiv_id":"2608.04234","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"The authors propose a joint kernel entropic Gromov-Wasserstein objective for multimodal alignment that provably achieves O(n^{-1/2}) sample complexity and is solvable by alternating Sinkhorn updates.","lead":"A new alignment method, JK-EGW, uses kernelized Gromov-Wasserstein optimal transport to map several frozen modality embeddings into one shared space with a user-chosen distribution, and it proves an n^{-1/2} sample-complexity rate. The accompanying alternating algorithm shows retrieval gains over existing baselines on small MS-COCO subsets.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"PSD-kernel assumption is unverified and likely violated in the experiments: label-agreement cross-modality blocks can be indefinite, so Theorem 5 and Pivoted Cholesky do not cover the reported configurations.","rationale":"The reader's weakest_assumption identifies the same load-bearing hinge: Assumption 1's PSD requirement. I agree with that assessment. The theory itself is plausible under Assumption 1, and the synthetic experiment in Section 5.2 uses a PSD kernel (RBF with cross-modality scaling 0.6), which is consistent with the η_R and n^{-1/2} predictions. The problem is not the conditional mathematics but the mismatch between Assumption 1 and the experimental kernels. The paper's own Section 5.1 provides a concrete violation: label-agreement similarity is indefinite, so the proof-of-concept experiment already runs outside the assumption. For Section 5.3, the kernel formula is omitted and W's PSD status is unchecked; without this, Pivoted Cholesky's nonnegative residual guarantee and the end-to-end error bound in Theorem 9 have no basis. A secondary issue is that the phrase 'η_R→0 yields O(n^{-1/2})' would require an explicit choice R=R_n with controlled constants, but even that does not change the conditional verdict. The concerns are addressable, so the conditional verdict remains appropriate.","tokens_in":38893,"tokens_out":10189,"duration_ms":96787,"concrete_test":"Compute λ_min(W) for the exact kernel matrices used in the MS-COCO 6K/9K experiments and for the MNIST Section 5.1 setup. If any eigenvalue is negative, Assumption 1 is violated and the theoretical guarantees do not cover those results; then replace the cross-modality block with a certified PSD construction (e.g., γ_{ij}K_base with Γ PSD and K_base RBF) and re-run Table 1 to see whether the retrieval conclusions survive.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Assumption 1 (Section 2.2) is the hinge of the theoretical edifice: it supplies the Mercer expansion with nonnegative eigenvalues, the residual bound |E_R|≤η_R used in Proposition 3, the lifting in Theorem 4, the empirical-process bounds in Proposition 6 and Theorem 5, and the Pivoted Cholesky guarantee in Algorithm 1 that the residual W−LL^T is PSD with ρ≤τ. The paper never verifies this assumption for its main experiments. Worse, the construction it does describe violates it: in Section 5.1 the cross-modality block is W^{i,j}_{k,k'}=1{same digit label}; such label-agreement blocks are indefinite. For example, with two samples per modality, within-RBF matrix [[1,a],[a,1]] for 0<a<1, and cross block all-ones J, the full block matrix has eigenvalue a−1<0. For Section 5.3 the cross-modality kernel is not specified, and no minimum-eigenvalue check is reported. If W is indefinite, the Mercer representation behind Theorem 5 and Corollary 7 does not exist, and the Pivoted Cholesky residual bound used in Proposition 12 and Algorithm 1 is invalid; the reported retrieval improvements are therefore not backed by the paper's theory.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes JK-EGW, a joint kernel entropic Gromov--Wasserstein formulation for aligning multiple modalities into a shared latent space with a prescribed reference distribution. The objective uses a global affinity kernel on the disjoint union of modality spaces, combines within- and cross-modality similarities, and is regularized by KL entropy. The theoretical core assumes the kernel is positive semidefinite with a uniformly controlled Mercer tail (Assumption 1), then proves a kernel-truncation error bound (Proposition 3), a variational lifting of the rank-truncated problem to a family of entropic optimal transport problems (Theorem 4), and a parametric O(n^{-1/2}) sample complexity bound for the full-kernel objective (Theorem 5). The empirical part develops a low-rank Pivoted Cholesky approximation and an alternating Sinkhorn algorithm (Algorithm 1), with an end-to-end error theorem (Theorem 9). Experiments include a digit multi-feature example, a synthetic finite-support rate study, and MS-COCO image--text alignment where JK-EGW is compared with CSA and STRUCTURE baselines.","tokens_in":39101,"tokens_out":6362,"duration_ms":59901,"significance":"If the theoretical claims hold under Assumption 1, the paper gives a clean separation between statistical error and kernel-truncation error for a kernelized multi-modal optimal transport problem, and the variational lifting is an elegant way to convert a quadratic GW-type problem into solvable EOT subproblems. The n^{-1/2} rate, the explicit role of the latent dimension and Mercer rank rather than raw feature dimension, and the synthetic validation of the rate are genuine strengths. However, the paper's own experiments appear to use cross-modality affinity blocks that are not guaranteed to be positive semidefinite, which is exactly what Assumption 1 requires. Because the stated theory does not cover the reported experimental configurations, the empirical claims are not currently supported by the manuscript's theorems. This is a load-bearing gap that requires a major revision.","major_comments":[{"comment":"Assumption 1 is load-bearing for Proposition 3, Theorem 4, Theorem 5, Corollary 7, Proposition 12, and the Pivoted Cholesky guarantee in Section 4.1. In Section 5.1 the cross-modality block is defined by Wi,j_{k,k'} = 1{same digit label}. Such a block matrix is not PSD in general: with n=2 per modality, taking the RBF within-block R=[[1,a],[a,1]] for 0<a<1 and the all-ones cross-block J, the block matrix [[R,J],[J,R]] has eigenvalue a-1<0 with eigenvector (1,1,-1,-1). Thus the Mercer decomposition and the positive-semidefinite residual property of Pivoted Cholesky do not apply to this configuration. In Section 5.3 the cross-modality kernel is not specified at all, and no minimum-eigenvalue check is reported. Consequently the reported retrieval improvements are not backed by the paper's theoretical guarantees. The authors should either construct and verify PSD empirical kernels (for example by projection onto the PSD cone) or clearly state that the experimental configurations lie outside the certified regime.","section":"Section 2.2 (Assumption 1) vs. Sections 5.1 and 5.3"},{"comment":"The proof of the rank-truncated sample complexity bound needs a uniform-in-A dual comparison. After Eq. (12), the text says 'By the same dual-comparison argument as in Corollary 2 of Mena and Niles-Weed (2019), uniformly over A in A_L, ...'. Lemma 13 provides regularity of optimal potentials for a fixed A, and the subsequent empirical-process bounds are uniform in the potential classes, but the manuscript does not prove that the dual comparison can be made simultaneously for all A in A_L. Since the supremum over A is essential for bounding the difference of the lifted objectives, this step is load-bearing. Please supply the missing uniform argument, or state and prove a uniform-in-A version of the potential regularity / dual comparison.","section":"Section B.1, Proposition 6 proof"},{"comment":"The Pivoted Cholesky error analysis explicitly relies on W being positive semidefinite: 'Under the positive-semidefinite kernel assumption, the Pivoted Cholesky factorization ... yields a positive-semidefinite residual W-LL^T ⪰ 0 at each truncation level.' If the empirical block matrix is indefinite, this residual need not be PSD and the bound rho_Rhat <= tau_Rhat cannot be guaranteed. Theorem 9 then invokes rho_Rhat <= tau_Rhat almost surely, but Algorithm 1 does not verify this condition. At minimum, the paper should add a PSD check or modify the algorithm to handle indefinite empirical kernels, and then restate the corresponding error guarantees.","section":"Section 4.1 and Proposition 12"}],"minor_comments":[{"comment":"The sentence 'JK-EGW produces the strongest alignment and retrieval results across all dimensions' is not supported by Appendix E: on the 9K subset, the purity of JK-EGW is below CSA and CSA+STRUCTURE at several latent dimensions. Please qualify the claim to retrieval and geometric coherence, or discuss the purity trade-off.","section":"Section 5.3 and Appendix E, Tables 1 and 3"},{"comment":"The text says 'the purple lines execute the sub-optimization for the transport plans and the maroon lines optimize A', but the printed algorithm contains no colored lines. Please remove the color references or add colors.","section":"Algorithm 1 and surrounding text"},{"comment":"The sentence 'Let x1,...,xM i.i.d.~rho gives a dataset of {x1_k,...,xM_k}' is confusing: if each modality has its own distribution mu_i, the samples should be i.i.d. from the appropriate marginal or from rho conditional on each modality component. Please clarify the sampling model.","section":"Section 3.1"},{"comment":"The text in Section 5.2 reports n in {50,100,200,400,800,1600}, but the horizontal axis of Figure 4 is labeled 2^6 through 2^10, i.e. {64,128,256,512,1024}. Please make the sample sizes in the figure consistent with the text.","section":"Figure 4"},{"comment":"There are several typos and grammatical issues: 'Deparment' in the affiliation, 'allowing us to take the advantage' in the abstract, and 'instantiates as' in Section 5.3. A copyedit would improve readability.","section":"Abstract and author affiliations"}],"recommendation":"major_revision","confidential_remarks":"The theoretical machinery is serious and the lifting idea is a solid contribution, but the mismatch between Assumption 1 and the reported experiments is substantial. The paper could become acceptable after a major revision if the experiments are either brought under the PSD assumption or explicitly presented as outside the certified regime. I see no indication of circularity or novelty concerns; the main issue is the scope of the claims relative to the verified assumptions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Do read this one. The theoretical core is genuinely new and mostly sound, but the experiments are not covered by the paper's own assumptions, and one claim in the main text is flatly contradicted by the paper's tables.\n\nWhat's actually new: the rank-R kernel EGW objective, the variational lifting to a linear EOT problem with an auxiliary matrix, and the n^{-1/2} sample complexity with an explicit kernel-truncation term. The proof technique, building regularity of EOT potentials in the Mercer feature space and getting dimension dependence R∨dz instead of the raw input dimension, is a real contribution. The alternating Sinkhorn algorithm with closed-form A-update is practical, and the synthetic experiments do check the rate and the truncation behavior. That part of the paper deserves a serious referee.\n\nNow the soft spots, in order of importance. First, Assumption 1 -- the global affinity kernel K must be positive semidefinite with Mercer-tail control -- is the hinge of the whole theory. The paper never verifies it for the main experiments. Worse, the MNIST experiment in Section 5.1 defines cross-modality blocks by label agreement, and those blocks can make the full block matrix indefinite even when within-modality blocks are RBF. The stress-test example is correct: with two points per modality, within-block [[1,a],[a,1]] and cross-block all-ones, the matrix has eigenvalue a-1 < 0. So the Mercer expansion, Proposition 3, Theorem 5, and the Pivoted Cholesky PSD residual guarantee do not apply to that experiment. For the image-text experiment the cross-modality kernel is not specified, so there is no way to check PSD at all. This is a load-bearing gap between theory and practice, not a cosmetic one.\n\nSecond, the text says JK-EGW \"achieves the highest purity in both datasets.\" Table 1 shows JK-EGW purity on MS-COCO 9k is 0.633, below CSA's 0.673 and CSA+STRUCTURE's 0.689. The Appendix table confirms JK-EGW purity is below CSA at every tested dimension on 9k. That sentence is simply wrong and should be fixed.\n\nThird, reproducibility: no code, no error bars, and the image-text section omits key implementation details (the affinity kernel, the entropic regularization ε, and the GMM parameters for ν). A reader cannot reproduce the reported retrieval numbers.\n\nNone of this kills the theory. The theorems are stated under explicit assumptions and the proofs look plausible, modulo the uniform-in-A dual-comparison step that is asserted rather than fully shown. But the empirical section overclaims and does not satisfy the paper's own conditions. I'd send this to peer review with a request for major revision: verify or repair the experimental kernels, correct the purity claim, and release code and details. The theory alone justifies referee time.","headline":"Solid OT theory with a load-bearing PSD-kernel assumption that the experiments neither verify nor satisfy, plus an overclaimed purity result.","tokens_in":39740,"tokens_out":3960,"would_cite":false,"duration_ms":39618,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49Q22","62G20","46E22"],"pacs":[],"model":"deepseek-v4-flash","headline":"Kernelized Gromov–Wasserstein transport aligns several modalities onto a user-chosen latent distribution, converges at the parametric $n^{-1/2}$ sampling rate for its empirical objective, and improves MS-COCO image–text retrieval over…","keywords":["multimodal alignment","kernel Gromov–Wasserstein optimal transport","entropic optimal transport","finite-rank Mercer approximation","sample complexity","variational lifting","representation learning","Pivoted Cholesky"],"falsifier":"On the exact MS-COCO configurations of Section 5.3, construct the full empirical block kernel matrix $W$ (within-modality similarity blocks and the cross-modality affinity blocks as actually defined for those experiments) and compute its smallest eigenvalue and its Mercer tail $\\eta_R$ at the rank used by Algorithm 1; a negative eigenvalue of size comparable to the diagonal scale settles that Assumption 1 fails for the evaluated kernel, so Theorems 5 and 9 do not formally apply to the retrieval numbers, and a slow decay of $\\eta_R$ would inflate the truncation term far beyond the claimed rate.","tokens_in":38596,"feed_emoji":"🎯","tokens_out":21676,"duration_ms":169075,"temperature":0.7,"pith_summary":"The paper sets out to show that several frozen single-modality encoders can be aligned into one shared latent space, with the geometry and marginal distribution of that space prescribed by the user, even when cross-modal paired data are scarce. It introduces a joint kernel entropic Gromov–Wasserstein (JK-EGW) objective, a transport framework that aligns distributions through their pairwise relational structure rather than raw coordinates, in which a global affinity kernel scores pairs of observations both within and across modalities and each modality is coupled by an optimal transport plan to a common reference measure $\\nu$. The core theoretical claim is that despite the nonconvexity of the quadratic program, the empirical objective converges to its population counterpart at the parametric rate $n^{-1/2}$, plus a kernel-truncation term that decays with the Mercer tail of the kernel. The core practical claim is that the objective is solvable by an alternating scheme whose transport-plan step is a batch of entropic optimal transport problems, and that the resulting shared representations improve bidirectional image–text retrieval on MS-COCO over three recent baselines. A sympathetic reader would care because scarce paired data is the practical bottleneck in multimodal learning, and this result promises a provably convergent, computationally tractable way to control the geometry of the shared space without training a heavy joint encoder.","feed_headline":"Images and text align at the optimal sampling rate","feed_subtitle":"Kernelized optimal transport maps each modality onto a chosen latent distribution, beating three baselines on MS-COCO.","key_machinery":"The load-bearing object is the variational lifting of the rank-truncated objective (Theorem 4), which converts the nonconvex quadratic interaction into a linear entropic optimal transport problem. For a fixed truncation rank $R$, the rank-$R$ JK-EGW objective equals the infimum over an auxiliary matrix $A \\in \\mathbb{R}^{R\\times d_z}$ of $2\\|A\\|_F^2 + \\sum_i \\mathrm{OT}^i_{A,m_\\Phi,\\epsilon}(\\mu_i,\\nu)$, where each $\\mathrm{OT}^i$ is a standard entropic transport problem with cost $c^i_{A,m_\\Phi}(x,z) = 2\\Phi_R(x)^\\top m_\\Phi \\|z\\|^2 - 4\\langle A,\\Phi_R(x)z^\\top\\rangle_F$; here $\\Phi_R(x)$ stacks the first $R$ scaled Mercer eigenfunctions (the Mercer decomposition being the spectral expansion of a positive-semidefinite kernel into orthogonal eigenfunctions) and $m_\\Phi = \\sum_j \\int \\Phi_R \\, d\\mu_j$ is the averaged feature moment. The identity that does the work is $-2\\|\\sum_i V_i\\|_F^2 = \\inf_A (2\\|A\\|_F^2 - 4\\langle A, \\sum_i V_i\\rangle_F)$ with $V_i = \\int \\Phi_R(x) z^\\top d\\pi_i$, which moves the quadratic dependence on the couplings into a bounded auxiliary variable $A$. This reduction carries both halves of the paper: statistically it lets the proof borrow linear-EOT sample-complexity and potential-regularity machinery, yielding the $n^{-1/2}$ rate without tail conditions on the modality measures; computationally it yields Algorithm 1, an alternating scheme in which the transport-plan step is a batch of Sinkhorn-style entropic solves and the $A$-step is the closed-form update $A = L^\\top \\hat\\Pi \\hat Z$ applied to the Pivoted-Cholesky factor $L$ (a greedy low-rank factorization) of the empirical kernel matrix.","core_discovery":"The discovery is that a kernelized Gromov–Wasserstein alignment objective, though it is a nonconvex quadratic program over the couplings, becomes statistically and computationally tame once the global affinity kernel $K$ admits a finite-rank Mercer truncation. Writing the kernel as $K(x,x') = \\sum_{r\\ge 1} \\lambda_r \\varphi'_r(x)\\varphi'_r(x')$ and keeping the first $R$ scaled eigenfunctions $\\Phi_R(x)$, the quadratic interaction among couplings depends on each coupling only through the feature-latent moment $\\int \\Phi_R(x) z^\\top d\\pi_i$, so the rank-$R$ objective lifts (Theorem 4) to an infimum over an auxiliary matrix $A$ of $2\\|A\\|_F^2$ plus a sum of linear entropic optimal transport problems with cost $c^i_{A,m_\\Phi}(x,z) = 2\\Phi_R(x)^\\top m_\\Phi \\|z\\|^2 - 4\\langle A, \\Phi_R(x) z^\\top\\rangle_F$, where $m_\\Phi$ is the averaged feature moment. That lifting routes the statistics through linear-EOT empirical-process bounds: Theorem 5 bounds the two-sample deviation $\\mathbb{E}|S_\\epsilon(\\{\\mu_i\\},\\nu) - S_\\epsilon(\\{\\hat\\mu_i^n\\},\\hat\\nu_n)|$ by a constant times $n^{-1/2}$ plus $\\sigma\\eta_R$, with constants depending on the number of modalities, the latent dimension, the truncation rank, and the ratio $\\sigma/\\epsilon$ but not on the ambient feature dimension, and Theorem 9 extends the same parametric rate to the fully computable objective with a Pivoted-Cholesky low-rank kernel factor. Empirically, on a 6K and a 9K MS-COCO image–text benchmark with a 20-component Gaussian-mixture reference distribution in 32 latent dimensions, JK-EGW is reported as the only method among those compared with positive silhouette scores and as the best on bidirectional R@1 and R@5 retrieval, ahead of the CSA, CSA+STRUCTURE, and MLP+STRUCTURE baselines.","pith_inferences":["A check the paper does not run: the MS-COCO experiments of Section 5.3 do not specify how the cross-modality affinity blocks are built, and the numerals experiment uses label agreement, which is not guaranteed to be positive semidefinite; computing the smallest eigenvalue of the empirical block kernel $W$ on those configurations would settle whether Assumption 1 covers the reported numbers.","Because the lifted objective is linear in the aggregate feature-latent moment $A$, the reference distribution $\\nu$ could itself be optimized inside the same variational form, over a parametric family such as Gaussian mixtures with learnable means, turning the latent geometry from a fixed design choice into part of the alignment task; the authors list adapting $\\nu$ as future work but do not pursu","The $n^{-1/2}$ bound is independent of the raw feature dimension, which makes JK-EGW a natural candidate for aligning very high-dimensional frozen encoder embeddings; whether that advantage materializes depends on the practical rank $\\hat R$ of the empirical kernel, since the factorization cost scales with $\\hat R$ rather than the ambient dimension.","The paper's validation of the rate (Section 5.2) carries its own stated caveat that optimization error is assumed negligible against statistical error, so a clean experimental confirmation of Theorem 5 would need a certified optimality gap rather than a converged objective value."],"forward_implications":["If the rank-$R$ kernel truncation has a fast-decaying spectrum (periodized RBF kernels give $\\eta_R$ decaying faster than any polynomial in $R$), the effective statistical complexity of JK-EGW is governed by the latent dimension $d_z$ and the truncation rank $R$, not by the raw feature dimension of the modalities.","For fixed regularization $\\epsilon$, the two-sample error decomposes into a statistical term of order $n^{-1/2}$ and a kernel-truncation term of order $\\sigma\\eta_R$; choosing $R$ so that $\\eta_R \\ll n^{-1/2}$ makes the empirical objective match the known parametric rate up to constants.","Because the user chooses the reference measure $\\nu$, the practitioner can impose a target geometry on the shared space, such as the Gaussian mixture with means along orthogonal directions used in the experiments, and retain the same $n^{-1/2}$ guarantee.","The alternating algorithm costs $O(Mn^2)$ per iteration for the entropic transport updates, with the one-time Pivoted-Cholesky factorization costing $O(N\\hat R^2)$ arithmetic operations and $O(N\\hat R)$ memory for $N = Mn$, making the method applicable when $N$ is in the tens of thousands.","On the tested MS-COCO subsets, JK-EGW attains the highest purity, the only positive silhouette score, and the best R@1 and R@5 in both retrieval directions, with the retrieval advantage persisting across latent dimensions $d \\in \\{32, 48, 64, 128\\}$."],"supporting_citations":[{"why":"Defines Gromov–Wasserstein distances, the relational-structure framework whose multiple-modality extension the paper builds.","marker":"(Mémoli, 2011)"},{"why":"Introduces the quadratic Gromov–Wasserstein formulation on kernel and distance matrices that the JK-EGW objective generalizes and entropically regularizes.","marker":"(Peyré et al., 2016)"},{"why":"Supplies the parametric $n^{-1/2}$ sample-complexity bound for entropic optimal transport and the potential-regularity machinery the proof of Proposition 6 adapts to kernel costs.","marker":"(Mena and Niles-Weed, 2019)"},{"why":"The nearest precursor: duality and sample complexity for entropic Gromov–Wasserstein with Euclidean costs, whose auxiliary-matrix lifting Theorem 4 extends from raw coordinates to Mercer feature maps.","marker":"(Zhang et al., 2024)"},{"why":"The Sinkhorn algorithm used to solve the entropic optimal transport subproblems inside Algorithm 1.","marker":"(Cuturi, 2013)"},{"why":"Provides the Pivoted-Cholesky low-rank factorization, with its residual bounds and linear-in-$N$ complexity, used to approximate the empirical kernel matrix.","marker":"(Harbrecht et al., 2012)"},{"why":"The spectral decomposition of positive-semidefinite kernels that underlies Assumption 1's Mercer-truncation structure.","marker":"(Mercer, 1909)"},{"why":"Supplies the MS-COCO dataset on which the image–text retrieval experiments are evaluated.","marker":"(Lin et al., 2014)"},{"why":"The CSA baseline, the closed-form alignment method that JK-EGW outperforms in retrieval.","marker":"(Li et al., 2025)"},{"why":"The STRUCTURE baseline and its geometry-preserving regularization, used as a comparison and as a component of two baseline variants.","marker":"(Gröger et al., 2025)"}],"fun_headline_variants":["Kernelized optimal transport aligns modalities at n^{-1/2} rate","Multimodal alignment achieves sample-optimal complexity","Entropic Gromov-Wasserstein improves multimodal retrieval","Scarce paired data? One kernel transport aligns it","Cross-modal embedding via kernelized optimal transport"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Every theoretical guarantee in the paper rests on a single premise: the global affinity kernel that scores similarities within and across modalities is symmetric positive semidefinite, has a uniformly bounded diagonal, and has a Mercer tail $\\eta_R = \\sup_x \\sum_{r>R} \\lambda_r \\varphi'_r(x)^2$ that vanishes as the truncation rank grows, and the experimental configurations with label-agreement cross-modality blocks are never checked against this premise.","fun_headline_variants_meta":{"raw":{"variants":["Kernelized optimal transport aligns modalities at n^{-1/2} rate","Multimodal alignment achieves sample-optimal complexity","Entropic Gromov-Wasserstein improves multimodal retrieval","Scarce paired data? One kernel transport aligns it","Cross-modal embedding via kernelized optimal transport"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000883,"raw_usage":{"total_tokens":3962,"prompt_tokens":1244,"completion_tokens":2718,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":860,"completion_tokens_details":{"reasoning_tokens":2638}},"tokens_in":860,"tokens_out":2718,"duration_ms":18739,"temperature":1.0,"reasoning_tokens":2638,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:41:45.740570+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On the exact MS-COCO configurations of Section 5.3, construct the full empirical block kernel matrix $W$ (within-modality similarity blocks and the cross-modality affinity blocks as actually defined for those experiments) and compute its smallest eigenvalue and its Mercer tail $\\eta_R$ at the rank used by Algorithm 1; a negative eigenvalue of size comparable to the diagonal scale settles that Assumption 1 fails for the evaluated kernel, so Theorems 5 and 9 do not formally apply to the retrieval numbers, and a slow decay of $\\eta_R$ would inflate the truncation term far beyond the claimed rate.","supporting_citations":[],"review_version":1}