{"id":"5317de24-dc66-41b9-b0b6-23f77e9830fd","arxiv_id":"2608.09513","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New algorithms approximate multimarginal optimal transport with near-linear classical time and sublinear quantum time in the tensor dimension, plus matching query lower bounds.","lead":"This paper presents faster classical and quantum algorithms for approximating multimarginal optimal transport, which matches several probability distributions under one cost. The classical algorithm is near-linear in the storage size of the problem and in the inverse accuracy, while the quantum algorithms become sublinear in that size, with lower bounds that show why.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The quantum Sinkhorn bound in Corollary 5.2 hinges on the unverified LOGSUMEXP primitive of Lemma 5.1; if that primitive's true query cost has hidden dependence on 1/gamma or B, the claimed epsilon^{-5} and n^{(m+1)/2} exponents fail.","rationale":"I read the paper in good faith and checked the main mathematical chain. The classical result is solid: the packing reduction in Section 3.4 is exact, the deficit-completion step in Lemma 3.1 has correct marginals, and Theorem 3.3 correctly converts the multiplicative packing guarantee into additive MOT error. The quantum projected subgradient method also holds together: Proposition 4.2 gives a bounded optimal dual representative under full support, Lemma 4.1 supplies the needed Lipschitz constant and ball radius, and the max-finding subroutine in Lemma 4.2 is standard. The lower bound construction in Theorem 6.1 is a valid reduction from promised unstructured search, and the block-count arithmetic matches the stated Omega(n^m/(1+epsilon n)) and its square root. The quantum Sinkhorn analysis is the only place where the paper leans on an unverified external theorem. The internal lemmas in Section 5 are consistent: the noisy descent identity in Lemma 5.3 is correct, the range bound in Lemma 5.4 follows from the fiber-ratio argument, and Lemma 5.7's drift estimate is a standard supermartingale stopping-time bound. However, all of this is downstream of Lemma 5.1, and the paper does not supply a proof or a constant-checked statement of that primitive. This is not a demonstrated error, and the cited source is a legitimate doctoral thesis, so the reader's ACCEPT verdict stands. I would not move the verdict because the concern is a dependency caveat rather than a discovered flaw, but I would want the LOGSUMEXP query cost verified before relying on the exact epsilon-exponent in Corollary 5.2. Hence UNCHANGED, with partial agreement with the reader's weakest-assumption identification.","tokens_in":25405,"tokens_out":28518,"duration_ms":283229,"concrete_test":"Independently verify Lemma 5.1 against the actual statement of [Nie24, Thm 13.3.2], and stress-test its query complexity on the worst case where half the entries have a_j=gamma and w_j=-B and half have a_j=1 and w_j=B, for d=2^k, gamma=e^{-R}, and B=R with R large. Compute the number of coherent queries required to output log(sum a_j e^{w_j}) to additive delta. If the query count scales as Omega(1/(delta sqrt(gamma))) or Omega(e^{O(B)}), then Lemma 5.2 and Theorem 5.1 need an extra factor, and Corollary 5.2's epsilon^{-5} and n^{(m+1)/2} runtime should be weakened accordingly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The classical packing reduction and the quantum projected subgradient method are internally sound: the rank-one completion in Lemma 3.1 is exact, the dual geometry in Lemma 4.1 checks out, and the lower bound reduction in Theorem 6.1 is valid. The soft spot is the quantum multimarginal Sinkhorn branch, whose entire complexity rests on Lemma 5.1, a specialization of [Nie24, Thm 13.3.2] that is cited but not re-derived. The lemma claims that, with coherent access to a_j in [gamma,1] and w_j in [-B,B], one can estimate log((1/r) sum a_j e^{w_j}) to additive delta in O(sqrt(d)/delta * log^2(e/gamma) * polylog(B, etc.)) time. This is a strong claim: the sum can be exponentially large or small in B, and the minimum weight gamma can be exponentially small, yet the runtime is polylogarithmic in both. If the correct query complexity contains, for example, a factor 1/sqrt(gamma) or e^{O(B)}, then Lemma 5.2's per-fiber estimation cost changes, and with it the iteration cost in Theorem 5.1 and the final epsilon^{-5} and n^{(m+1)/2} exponents in Corollary 5.2. The paper does not provide a proof or even a statement of the cited theorem with all constants, so the quantum Sinkhorn result is conditional on that external primitive. This is a reliance on prior work rather than an internal inconsistency, but it is the most load-bearing unverified step in the paper.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript gives classical and quantum approximation algorithms for discrete multimarginal optimal transport (MOT). The classical result reduces MOT to a positive packing LP on the marginal-incidence matrix, solves the packing problem with the Allen-Zhu--Orecchia solver, and exactly restores marginals by a rank-one completion, yielding an explicit coupling in O(m^2 n^m (||C||_∞/ε) polylog) time. The first quantum algorithm applies projected subgradient descent to the MOT dual, using quantum maximum finding to evaluate the nonsmooth dual objective, and estimates the MOT value in O(m^3 n^{m/2+1} (||C||_∞/ε)^2 polylog) time. The second quantum algorithm randomizes multimarginal Sinkhorn updates, using a quantum LOGSUMEXP primitive to estimate fiber log-sums without materializing the tensor; it returns scaling potentials for an approximate entropic coupling in O(m^5 n^{(m+1)/2} ...) time, and after the standard entropic-to-unregularized reduction the paper claims an implicit unregularized approximation in O(m^8 n^{(m+1)/2} (||C||_∞/ε)^5 polylog) time. Finally, query lower bounds of Ω(n^m/(1+εn)) for randomized classical algorithms and Ω(sqrt(n^m/(1+εn))) for quantum algorithms are proved by reduction from promised unstructured search.","tokens_in":25676,"tokens_out":10302,"duration_ms":98905,"significance":"The classical packing reduction is the paper's strongest contribution: Theorem 3.2 gives an exact equality between MOT and a packing LP, Lemma 3.1's rank-one completion is elementary and correct, and Corollary 3.1 improves on previous near-linear-in-N methods in its combined dimension and accuracy dependence. The QPSM value algorithm is also internally coherent: the dual geometry in Lemma 4.1 supplies the needed ball radius and Lipschitz constant, and the runtime follows from standard projected-subgradient theory. The lower-bound construction in Section 6 is clean and correctly interpolates between the n^{m-1} and n^m regimes. The quantum Sinkhorn branch is more conditional: its exponents rest on an unproved LOGSUMEXP specialization, and the unregularized reduction is largely delegated to [LHCJ22]. If Lemma 5.1 is supplied with a complete proof, the claimed quantum speedups are significant; as written, that part should be treated as a conditional result.","major_comments":[{"comment":"The claimed complexity of the quantum multimarginal Sinkhorn method (Theorem 5.1 and Corollary 5.2) is carried by the LOGSUMEXP primitive, but the lemma is presented only as a 'specialized' version of [Nie24, Theorem 13.3.2] with no proof and no statement of the cited theorem's hypotheses and constants. The asserted runtime is polylogarithmic in B and 1/γ, which is a strong claim because the target log((1/r)∑ a_j e^{w_j}) ranges over an interval of length O(B) and the weights can be as small as γ. If the true query complexity of the primitive contains a factor 1/√γ, e^{O(B)}, or any similar dependence, then the per-fiber estimation cost in Lemma 5.2 changes and with it the ε^{-5} and n^{(m+1)/2} exponents in Corollary 5.2. Since this is a load-bearing step for one of the two quantum speedups, the manuscript should either prove Lemma 5.1 in full or state the exact theorem from [Nie24], including all constants and coherence assumptions, and verify that the rescaled Gibbs tensor and the potentials used in Section 5.2 satisfy those hypotheses.","section":"Section 5, Lemma 5.1"},{"comment":"The proof delegates three load-bearing steps to [LHCJ22]: the entropic-bias bound η m log n, the rounding bound O(||C||_∞ δ) for a coupling with total marginal error δ, and the marginal perturbation of [LHCJ22, Algorithm 3]. The first two are cited as Theorems 4.4-4.5, but the algorithm here produces noisy potentials from randomized coordinate updates rather than the exact or greedy Sinkhorn iterates analyzed in that paper. Please state the relevant results with their hypotheses and confirm that the noise level from Theorem 5.1 (α_upd, α_cert) is compatible with the cited rounding and perturbation bounds; otherwise the ε dependence in Corollary 5.2 is not established.","section":"Section 5, Corollary 5.2"},{"comment":"The proof asserts that a fixed-point precision B = polylog(m,n,1+κ_C) makes the accumulated arithmetic error at most ε/8, and that the final estimate eF satisfies |eF - F(ȳ)| ≤ ε/8, but no derivation is given for the accumulation over T = O(mMκ_C^2) iterations and over the quantum max-finding comparisons. Since the final guarantee is for the real MOT value and the certificate A^T ŷ ≤ c is stated for the represented LP, the manuscript should give the explicit precision bookkeeping, or a short lemma, showing that the ε/8 slacks are not violated by the projection, averaging, and final-shift steps.","section":"Section 4, Theorem 4.1"}],"minor_comments":[{"comment":"The displayed quantum Sinkhorn runtime contains an unmatched closing parenthesis: O(m^8 n^{(m+1)/2} ε^{-5} polylog(...)))$; remove the extra ')'.","section":"Abstract and Section 1.1"},{"comment":"The entry 'QSinkhorneO' is missing a space, and the eO notation is used in the table before it is explicitly defined in the text; please define it in the caption or before the table.","section":"Table 1"},{"comment":"The factor log^2(e/γ) should be defined precisely; if 'e' denotes the base of the natural logarithm, write it as (1+ln(1/γ))^2 or specify the convention so that it is not confused with the accuracy parameter ε.","section":"Section 5.2, Lemma 5.1"},{"comment":"The sentence 'For more details; the entropy-regularized multimgarginal Sinkhorn guarantee...' contains a semicolon splice and the typo 'multimgarginal'; please correct both.","section":"Section 1.1"},{"comment":"The paragraph introducing O_C describes both a direct tensor-entry oracle and a structured implementation via a coherent evaluator for c; please clarify which oracle is assumed for each theorem, since the lower bounds of Section 6 apply only to the unstructured query model.","section":"Section 4, cost-entry model"}],"recommendation":"major_revision","confidential_remarks":"The main unresolved issue is the external LOGSUMEXP theorem [Nie24, Theorem 13.3.2] used as the key primitive for the quantum Sinkhorn branch. If the authors can supply an exact statement and proof of that primitive, or otherwise justify the claimed polylogarithmic dependence on B and 1/γ, I would be comfortable with the revised version. No other concerns about citation patterns or scope arose."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper is the real thing on the classical side: it gives the first near-linear-in-n^m and linear-in-1/epsilon algorithm for discrete MOT, via a packing reduction and an exact rank-one completion. The reduction is simple enough to verify by hand, and the completion lemma is correct. The lower bounds are elementary but valid, and they calibrate the quantum results nicely. If you work on OT complexity, the classical result alone is worth your time.\n\nThe quantum projected subgradient method is also internally consistent: the dual geometry checks out, the Lipschitz bound is fine, and the algorithm is honestly described as estimating the value, not returning a coupling. The quantum Sinkhorn branch is where I get cautious. Everything rests on Lemma 5.1, a specialization of a theorem cited from Nieuwboer's thesis, which estimates log-sum-exp with polylog dependence on 1/gamma and B. That is a strong claim. If the true query cost has hidden factors like 1/sqrt(gamma) or e^B, the epsilon^{-5} and n^{(m+1)/2} exponents in Corollary 5.2 collapse. The paper does not re-derive the primitive or even state the constants. So the quantum Sinkhorn result is conditional in a way the rest of the paper is not.\n\nThat said, this is reliance on a cited theorem, not an internal error, and it is standard practice in theory papers to import quantum primitives. The fix is straightforward: ask the authors to either prove Lemma 5.1 in an appendix or give a precise statement with all constants from the thesis. The classical result and the QPSM do not share this fragility.\n\nWho is this for? Complexity theorists working on OT and Sinkhorn, and quantum algorithm people interested in LP-type problems. The paper deserves a serious referee: the central classical claim is strong and likely correct, and the quantum parts are worth refereeing even if the Sinkhorn branch needs tightening.\n\nRecommendation: send it to review. If you are the editor, ask for the LOGSUMEXP proof or an exact statement with all dependencies before accepting. The classical result is publishable as is.","headline":"The classical packing reduction is the real contribution and is sound; the quantum Sinkhorn branch is conditional on an unproved log-sum-exp primitive, but the paper as a whole deserves refereeing.","tokens_in":26318,"tokens_out":1503,"would_cite":true,"duration_ms":14132,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Discrete multimarginal optimal transport admits a classical $O(m^2 n^m \\varepsilon^{-1}\\mathrm{polylog})$-time algorithm returning an exactly feasible coupling, and quantum algorithms estimate the value or return implicit couplings in…","keywords":["multimarginal optimal transport","positive packing linear program","quantum Sinkhorn","projected subgradient method","entropic regularization","query complexity","quantum speedup","Wasserstein barycenters"],"falsifier":"Run the quantum LOGSUMEXP subroutine of Lemma 5.1 on a single fiber of size $d=n^{m-1}$ with known answer, with array entries stored in the specified quantum-readable form, and count the actual coherent queries and fixed-point operations needed to reach accuracy $\\delta$; if the count grows like $d$ rather than like $\\sqrt{d}/\\delta$, then Lemma 5.2 and Theorem 5.1 are refuted. For the classical claim, one can take a small random MOT instance with known optimum, run the packing solver with $\\delta=\\varepsilon/\\Delta$, and verify both the completion's exact feasibility and that the observed runtime tracks $m^2 n^m \\kappa_C$.","tokens_in":25151,"feed_emoji":"⚛️","tokens_out":13135,"duration_ms":97180,"temperature":0.7,"pith_summary":"The paper studies discrete multimarginal optimal transport (MOT), where $m$ probability distributions supported on $n$ points are coupled to minimize a general cost tensor, a problem that underlies Wasserstein barycenters, matching in teams, and density-functional theory. Its central claim is a classical randomized algorithm that returns an exactly feasible coupling whose transportation cost is within an additive $\\varepsilon$ of the MOT value in time $O(m^2 n^m \\varepsilon^{-1}\\mathrm{polylog}(m,n,\\varepsilon^{-1}))$, the first bound that is simultaneously linear in the tensor dimension $n^m$ and in the inverse accuracy. It further claims two quantum algorithms: a projected subgradient method that estimates the MOT value without producing a coupling in $\\tilde O(m^3 n^{m/2+1}\\varepsilon^{-2})$ time, and a quantum multimarginal Sinkhorn method that outputs an implicit Gibbs-potential description of an $\\varepsilon$-optimal coupling in $\\tilde O(m^8 n^{(m+1)/2}\\varepsilon^{-5})$ time. Matching-style lower bounds are also claimed: $\\Omega(n^m/(1+\\varepsilon n))$ classical cost queries and $\\Omega(\\sqrt{n^m/(1+\\varepsilon n)})$ quantum cost queries for any $\\varepsilon<1/2$. If these claims hold, general MOT is solvable at the cost of writing down the coupling, and quantum algorithms genuinely reduce the dimensional dependence whenever cost entries are accessible coherently.","feed_headline":"Near-linear MOT algorithm, plus quantum speedups","feed_subtitle":"A coupling within ε of optimal in O(m²nᵐ/ε) time; quantum paths beat the tensor dimension.","key_machinery":"The argument is carried by three machines. First, the packing reduction with rank-one completion: all marginal deficits $d_k$ of a feasible packing share the same mass, so $D=\\tau^{1-m}d_1\\otimes\\cdots\\otimes d_m$ closes every marginal at once, turning a multiplicative packing approximation into an additive MOT approximation at no extra cost. Second, quantum maximum finding inside projected subgradient descent: the dual objective $F(y)=\\max_i(\\sum_k y_{k,i_k}-C_i)-b^\\top y$ has subgradients of the form $a_i-b$ with $\\|a_i-b\\|_2\\le 2\\sqrt{m}$, and finding the maximizing index over $N=n^m$ entries in $O(\\sqrt{N})$ queries is what makes the value estimate sublinear. Third, quantum log-sum-exp fiber estimation in a randomized Sinkhorn scheme: each block update needs $\\log q_{k,a}$, which is a log-sum over $n^{m-1}$ entries minus a log-sum over all $n^m$ entries; the quantum LOGSUMEXP primitive estimates both in roughly $\\sqrt{n^{m-1}}/\\alpha$ and $\\sqrt{n^m}/\\alpha$ coherent evaluations, and the noisy descent analysis, with a potential-range bound $R_K=\\log(K_{\\max}/K_{\\min})+\\log(1/\\mu_{\\min})$, keeps the iteration count at $O(m^2\\bar R_K/\\delta)$. The lower bounds are a fourth piece: partitioning $[n]^m$ into shifted matchings and blocks of size about $\\varepsilon n$ reduces promised search on $\\Omega(n^m/(1+\\varepsilon n))$ items to MOT.","core_discovery":"On its own terms, the paper's discovery is that discrete MOT is exactly a positive packing LP, not merely an LP that resembles one. The primal cost $\\min_{X\\in U(\\mu)}\\langle C,X\\rangle$ is replaced by $\\max\\{\\langle U\\mathbf{1}-C,X\\rangle : \\mathrm{Proj}_k(X)\\le \\mu_k\\}$, and the loss from using a partial feasible packing is repaired exactly: the deficit vectors $d_k=\\mu_k-\\mathrm{Proj}_k(X)$ all have equal total mass $\\tau$, so $D=\\tau^{1-m}d_1\\otimes\\cdots\\otimes d_m$ is a nonnegative rank-one tensor whose marginals are exactly the deficits, and $X+D$ is a true coupling. A $(1-\\delta)$-optimal packing therefore rounds to a coupling of cost at most $\\mathrm{MOT}(C,\\mu)+\\delta\\Delta$, yielding the classical near-linear runtime. For the value, the dual is written as the nonsmooth convex function $F(y)=\\max_{i\\in[n]^m}(\\sum_k y_{k,i_k}-C_i)-b^\\top y$; quantum maximum finding over the $n^m$ indices supplies subgradients in $O(\\sqrt{N})$ queries, and projected subgradient descent on a ball of radius $\\|C\\|_\\infty\\sqrt{mn}$ gives the sublinear value estimate. For the entropy-regularized problem, the block update $\\beta_{k,a}\\leftarrow\\beta_{k,a}+\\log\\mu_{k,a}-\\log q_{k,a}$ requires only log-marginals, and a quantum log-sum-exp estimator computes those over fibers of size $n^{m-1}$ in $O(m n^{(m-1)/2}\\alpha^{-1})$ time, with a noisy-coordinate-descent analysis showing convergence to residual $\\delta$; standard entropic-to-unregularized rounding converts this into the implicit-coupling guarantee.","pith_inferences":["Beyond the paper's explicit theorems, the packing reduction is exact, so any future improvement to general positive packing solvers would transfer directly to MOT without modifying the rounding argument.","The $O(\\sqrt{N})$ subgradient evaluation hints that quantum speedups could transfer to other linear programs with sparse columns and entry-query cost access, a direction the paper does not pursue.","The lower-bound construction places fine-precision MOT at the level of unstructured search, suggesting that practical quantum advantage will require exploiting cost structure such as low rank, metric, or geometric form rather than relying on the generic oracle.","The large $m^8$ factor in quantum Sinkhorn versus $m^2$ classically should be tested at small $m$ before claiming an end-to-end quantum advantage in practice."],"forward_implications":["For general MOT with $m$ marginals on $n$ points, an exactly feasible coupling with cost at most $\\mathrm{MOT}(C,\\mu)+\\varepsilon$ can be computed in $O(m^2 n^m \\kappa_C \\log(mn\\kappa_C)\\log\\kappa_C)$ arithmetic operations with high probability, where $\\kappa_C=\\max\\{1,\\|C\\|_\\infty/\\varepsilon\\}$.","The quantum projected subgradient method estimates the MOT value without a coupling in $\\tilde O(m^3 n^{m/2+1}\\kappa_C^2)$ time, which is sublinear in $n^m$ for $m\\ge 3$.","The quantum multimarginal Sinkhorn method outputs scaling potentials for a Gibbs tensor whose rounding has cost at most $\\mathrm{MOT}(C,\\mu)+\\varepsilon$, in $\\tilde O(m^8 n^{(m+1)/2}\\kappa_C^5)$ time, with an additional $O(n^m)$ classical cost if the explicit coupling tensor must be materialized.","Any randomized classical algorithm needs $\\Omega(n^m/(1+\\varepsilon n))$ cost queries and any quantum algorithm needs $\\Omega(\\sqrt{n^m/(1+\\varepsilon n)})$ cost queries, so at constant accuracy the $n^{m-1}$ dimensional dependence is unavoidable classically.","For bimarginal MOT, $m=2$, the quantum Sinkhorn specialization gives $\\tilde O(n^{3/2}(\\|C\\|_\\infty/\\varepsilon)^5+n^2)$, a modest quantum speedup in $n$ over the classical near-linear-in-$n^2$ baseline."],"supporting_citations":[{"why":"Supplies the nearly linear-time packing LP solver invoked as Theorem 3.1; the classical MOT runtime inherits its guarantee.","marker":"[AZO19]"},{"why":"Establishes the packing-based approach for bimarginal OT that this paper generalizes to $m$ marginals, and provides the baseline the classical result improves.","marker":"[BJKS24]"},{"why":"Provides the multimarginal Sinkhorn framework, greedy block updates, potential-range induction, and the rounding theorems used in Corollary 5.2.","marker":"[LHCJ22]"},{"why":"Supplies the quantum LOGSUMEXP primitive specialized in Lemma 5.1; the quantum Sinkhorn runtime bound depends on its stated accuracy and failure guarantees.","marker":"[Nie24]"},{"why":"Quantum minimum/maximum finding used in Lemma 4.2 to locate the maximizing tensor index in $O(\\sqrt{N})$ queries, underpinning the QPSM speedup.","marker":"[DH96]"},{"why":"Provides the projected-subgradient convergence theorem used to bound the QPSM iteration count.","marker":"[Bub15]"},{"why":"Cited in Theorem 6.1 for the quantum query lower bound on promised unstructured search.","marker":"[BBBV97]"},{"why":"Establishes optimality of Grover's search, converting the promised-search reduction into the stated quantum query lower bound.","marker":"[Zal99]"},{"why":"Provides the near-linear-time entropic rounding reference used in converting entropic MOT outputs to unregularized MOT guarantees.","marker":"[AWR17]"}],"fun_headline_variants":["MOT in near-linear time, quantum too","Exact rounding to coupling speeds MOT","Quantum speedups for multimarginal transport","Linear-time coupling via packing rounding","MOT fast: classical linear, quantum sublinear"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The quantum speedups assume that cost values and current iterate arrays can be read in quantum superposition with polylogarithmic overhead per entry, and that the quantum log-sum-exp subroutine truly costs $O(\\sqrt{d}/\\delta)$ coherent evaluations; if either fails, the sublinear quantum claims collapse, while the classical packing algorithm does not depend on this.","fun_headline_variants_meta":{"raw":{"variants":["MOT in near-linear time, quantum too","Exact rounding to coupling speeds MOT","Quantum speedups for multimarginal transport","Linear-time coupling via packing rounding","MOT fast: classical linear, quantum sublinear"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000486,"raw_usage":{"total_tokens":2578,"prompt_tokens":1305,"completion_tokens":1273,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":921,"completion_tokens_details":{"reasoning_tokens":1207}},"tokens_in":921,"tokens_out":1273,"duration_ms":565627,"temperature":1.0,"reasoning_tokens":1207,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T15:52:39.075748+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the quantum LOGSUMEXP subroutine of Lemma 5.1 on a single fiber of size $d=n^{m-1}$ with known answer, with array entries stored in the specified quantum-readable form, and count the actual coherent queries and fixed-point operations needed to reach accuracy $\\delta$; if the count grows like $d$ rather than like $\\sqrt{d}/\\delta$, then Lemma 5.2 and Theorem 5.1 are refuted. For the classical claim, one can take a small random MOT instance with known optimum, run the packing solver with $\\delta=\\varepsilon/\\Delta$, and verify both the completion's exact feasibility and that the observed runtime tracks $m^2 n^m \\kappa_C$.","supporting_citations":[],"review_version":1}