{"id":"f92a7bb8-5965-4a8b-9b66-0332b29a3198","arxiv_id":"2502.05981","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"Any finite combinatorial problem with a known logical circuit can be encoded as a tensor network whose contraction defines an explicit, though generally inefficient, solution equation.","lead":"This paper presents MeLoCoToN, a recipe that turns any combinatorial problem described by a classical logic circuit into a tensor network whose contraction is claimed to be an exact explicit solution equation. The authors do not claim a speedup over existing algorithms, but they argue the tensor-network form gives a new mathematical perspective and, hypothetically, a fast contraction device would solve all NP-hard problems in polynomial time.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Optimization readout in Eq. (2.41) relies on a finite τ that the paper never specifies; for any fixed τ there exist instances where the Half Partial Trace selects a suboptimal state, so Theorem 1's 'explicit equation' is unsupported for optimization.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing gap: the optimization readout requires a sufficiently large tau, but no proof or constructive rule is given. My stress test sharpens this into a concrete failure of the proposed equation at any fixed finite tau. The example is not contrived in a way that depends on degenerate ties or non-unique optima; it has a unique optimum and a small positive cost gap. The failure arises because the Half Partial Trace uses unnormalized amplitudes, so a branch containing many near-optimal states can dominate a branch containing the single optimum. This is precisely the setting the paper's informal 'sufficiently large tau' paragraph is supposed to cover, and the paper does not cover it. Since the central claim Theorem 1 promises an exact explicit equation, an unspecified existential tau is insufficient; if instead tau is chosen before seeing the instance, the equation demonstrably returns suboptimal answers. The tensor-network catalog sections may retain independent value, and the inversion/CSP readout may be sound, but the universal optimization claim fails as stated. The reader's REJECT verdict therefore stands unchanged.","tokens_in":48954,"tokens_out":6651,"duration_ms":71447,"concrete_test":"Run the exact tensor contraction specified by Eqs. (2.37)-(2.41) on the two-variable instance C(10)=0, C(11)=100, C(00)=0.01, C(01)=0.02 with tau=1. If the returned state is 00 rather than the unique optimum 10, the finite-tau readout fails. Then compute the threshold tau* above which the readout returns 10 and check that tau* depends on the 0.01 gap; this demonstrates that no instance-independent explicit tau is provided in the manuscript.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The inversion and constraint-satisfaction readouts are count-based and survive degeneracy, but the optimization readout is not. Section 2.5.2 asserts that 'for a sufficiently large finite value of tau' the amplitude peak of the optimal state dominates, and Eq. (2.36) proves only the tau-to-infinity limit of the fully normalized state, not the unnormalized greedy decisions in Eq. (2.41). For any finite tau, contributions from many suboptimal states can outweigh the optimal state in the scalar Omega_n. Concretely, for two binary variables with costs C(10)=0, C(11)=100, C(00)=0.01, C(01)=0.02, at tau=1 the contraction behind Eq. (2.37) gives Omega_0 = 1 + e^{-100} - e^{-0.01} - e^{-0.02} < 0, so the equation returns x0=0 and then x1=0, the suboptimal state 00. Increasing tau repairs this instance, but the required threshold depends on the cost gap, the number of near-optimal states, and the degeneracy, i.e., on information that is the output of the problem. The paper gives no constructive rule for tau; the Conclusions list 'determination of the minimum tau value' as future work. Hence the claimed exact explicit solution equation is not established: either tau is left as an unspecified existential parameter, which is not explicit, or a concrete tau is inserted, in which case there are instances where the equation is wrong.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a method, MeLoCoToN, which associates to any combinatorial problem—inversion, constraint satisfaction, or optimization—a tensor network whose contraction yields scalars Omega_n, and then defines the solution by nested Heaviside equations x_n = H(Omega_n(...)). The construction has four steps: choose variables, build a classical logical circuit (LSTC, LSVC, or LSMC), tensorize the circuit by Input-Output Indexing (Eq. 2.19), and read out variables iteratively via the Half Partial Trace. The paper presents tensor networks for a large list of problems, including QUBO/HOBO, integer sum, linear systems, k-colouring, TSP, knapsack, and several graph problems, and it describes approximation techniques under the name Motion Onion. It further claims that if a physical system could contract these tensor networks efficiently, every NP-hard problem would be solvable in polynomial time.","tokens_in":49296,"tokens_out":9288,"duration_ms":96292,"significance":"The circuit-to-tensor translation is coherent for finite circuits, and several of the example tensor definitions appear correct; the paper is also honest that no computational advantage over existing algorithms is claimed. If Theorem 1 were rigorously established, the work would offer a unified exact-formulation perspective on combinatorial problems and a conditional complexity statement. However, the central claim is weakened by three issues: the optimization readout depends on an unspecified sufficiently large tau or an unjustified limit interchange; Theorems 1-3 are asserted without formal proof; and the construction is essentially a transcription of the problem's truth table into tensor form, so the sense in which the resulting equation is 'explicit' needs to be made precise. These issues are load-bearing because the abstract and conclusions assert an exact explicit equation for every combinatorial problem.","major_comments":[{"comment":"The optimization readout is not justified, and the displayed limit is incorrect in the degenerate case. Eq. (2.36) claims that the normalized state tends to |X>; this holds only for a unique minimizer. With multiple optima, the limit is a uniform superposition over all minimizers. More importantly, Eq. (2.41) uses finite-tau unnormalized scalars Omega_n, and the paper only asserts that 'for a sufficiently large finite value of tau' the correct variable is selected, without a proof or a rule for choosing tau. This is load-bearing: for fixed tau, greedy Heaviside decisions can return suboptimal solutions. For example, with two binary variables and costs C(10)=0, C(11)=100, C(00)=0.01, C(01)=0.02, at tau=1 Eq. (2.37) gives Omega_0 = 1 + e^{-100} - e^{-0.01} - e^{-0.02} < 0, so x_0=0, and then x_1=0, returning x=(0,0) with cost 0.01 instead of the optimum (1,0) with cost 0. The Conclusions list 'determination of the minimum tau value' as future work, so the exactness claim for optimization is not established.","section":"Section 2.5.2, Eqs. (2.36)-(2.41)"},{"comment":"The main theorem is stated without a formal proof. The preceding text gives a construction, but no correctness statement, for example by induction on circuit size, shows that the contracted TLC equals the problem's indicator or cost tensor, nor that the nested Heaviside readout returns a solution for every input instance. The assertion in Section 2.3.1 that any known function can be implemented as an LSTC is also unproved. For finite circuits this is a standard fact, but the theorem's quantifier 'every combinatorial problem' requires a precise statement about the input representation and about the size of the circuit relative to the problem formulation.","section":"Section 2.6, Theorem 1"},{"comment":"The construction is essentially a transcription of the problem's defining relations into tensor elements via Input-Output Indexing. Because the tensor elements are defined directly from the same logical relations that define the problem, the contraction of the TLC is exactly the problem's truth table in tensor form. Correctness is therefore inherited by construction, and the claimed 'solution equation' is close to a notational restatement of exhaustive enumeration. The paper should define what counts as an 'explicit equation' and state why this transcription is not merely a disguised enumeration of all configurations. Without such a definition, the abstract's claim that the method proves the existence of an exact explicit equation for every combinatorial problem is not a substantive mathematical result.","section":"Section 2.4, Eq. (2.19)"},{"comment":"Several constrained formulations use a penalty weight lambda that is only described as 'large enough', with no finite instance-dependent bound. For example, Eq. (8.3) minimizes sum_i (C_{i,x_i} + lambda V_{x_i}) and states that lambda must be large enough to impose the maximum number of tasks performed, but no value or bound is given. Since the paper claims exact solutions, every free parameter used to enforce constraints needs a concrete, instance-dependent value that guarantees feasibility without changing the optimum. As written, the exactness claim for constrained optimization inherits the same unsupported-parameter problem as the tau readout.","section":"Section 8.1, Eq. (8.3); Sections 7 and 9"}],"minor_comments":[{"comment":"The manuscript contains numerous typographical errors, including 'sintetized', 'unassumingly expensive', and inconsistent capitalization of 'Half Partial Trace' and 'Humbucker'. A careful proofreading pass is needed.","section":"Front matter and Section 2"},{"comment":"Several subsections are placeholders stating 'Subsubsection not available due to paper pending publication', and references such as '[pending to publish]' appear throughout. These must be either filled in or removed before the paper can be considered a complete manuscript.","section":"Sections 5.1.1, 5.1.2, 10.2, 10.3"},{"comment":"The statement that the equation 'can be obtained in a polynomial time with respect to the time needed to formulate it' is not a well-defined complexity measure. The paper should specify the input size and the computational model, and prove the claim with respect to that measure.","section":"Section 2.6, Theorem 2"},{"comment":"In the definition of the Plus Vector with local imaginary time evolution, the notation '+_i = e^{-tau C_i}' uses C_i without defining it; the symbol should be connected to the problem's cost function or removed.","section":"Appendix D.1"},{"comment":"The same tensor names, such as delta and DOT, are reused for tensors of different arities in different problems. A summary table of index conventions for each problem would improve readability and reduce ambiguity.","section":"Equations (4.3)-(4.4) and (5.1)-(5.4)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is far from ready in its present form: it contains placeholder sections, informal theorem statements, and an optimization readout whose exactness rests on an unspecified parameter. The core idea is simple and partly correct, but the authors should be asked to provide a formal correctness theorem, a precise definition of 'explicit equation', and a constructive choice of tau (and lambda where used). If these cannot be supplied, the central claim should be weakened accordingly. The paper might fit a venue that welcomes exploratory formalization papers, but for a standard journal in this field the current version does not meet the rigor bar."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the catalog of explicit tensor networks for dozens of combinatorial problems is a real contribution, but the headline theorem is not established for the optimization case because the readout equation depends on an unspecified finite tau and fails on concrete small instances.\n\nWhat's good: the circuit-to-tensor translation in Section 2.4 is coherent, and Input-Output Indexing is a clean way to turn logical circuits into sparse tensors. The paper gives explicit tensor definitions for QUBO, T-QUDO, linear systems, k-colouring, partition, shortest path, TSP, knapsack, ILP, and more. That is a useful reference even if the networks are exponentially large. The inversion and CSP readouts are count-based, so they survive degeneracy, and the projection-vector handling of multiple solutions is sensible. The paper is also honest in the abstract and conclusions that the equations are not efficient.\n\nThe soft spots: Theorem 1 is asserted, not proved. For optimization, the Half Partial Trace readout in Eq. (2.41) requires a 'sufficiently large finite value of tau' that is never specified. The stress-test example is correct: two binary variables with costs C(10)=0, C(11)=100, C(00)=0.01, C(01)=0.02, at tau=1 the amplitude sum for x0=0 exceeds that for x0=1, so the equation returns the suboptimal state 00. This is not a pathological corner case; the required tau depends on the cost landscape, which is exactly what the equation is supposed to return. The paper lists 'determination of the minimum tau value' as future work, which concedes the point. For optimization, the claimed exact explicit equation is either tautological (the tensor network encodes the truth table) or under-specified (tau as an existential parameter). The reader's circularity concern is fair: the tensor elements are defined directly from the problem's logical relations, so the contraction is a restatement of the problem.\n\nThe paper is also incomplete: several subsections are missing because they refer to unpublished work, including prime factorization and SHA-3 inversion. That makes a full evaluation harder.\n\nBottom line: readers who want a catalogue of tensor-network formulations will find value here; the theorem as stated does not hold for optimization. I would send it to a serious referee but expect major revision: either fix the tau issue or weaken the theorem to the inversion and CSP cases. I would cite the catalog, not the theorem.","headline":"A genuinely useful catalog of tensor-network encodings, but the optimization readout's unspecified tau sinks the 'explicit equation' theorem.","tokens_in":49769,"tokens_out":3472,"would_cite":true,"duration_ms":34609,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Every finite combinatorial problem has an exact explicit equation, built by tensorizing its logic circuit and reading off variables with nested Heaviside steps.","keywords":["tensor networks","combinatorial optimization","constraint satisfaction","inversion problems","explicit solution equation","imaginary time evolution","QUBO","Heaviside step function"],"falsifier":"Run the Half Partial Trace iteration, with increasing values of $\\tau$, on a small optimization instance whose optimum is known (for example a 4-variable QUBO with a unique minimizer) and record whether the sign of each $\\Omega_n$ eventually matches the optimal assignment for every instance; any instance for which no finite $\\tau$ yields the optimum would falsify the optimization version of Theorem 1.","tokens_in":48708,"feed_emoji":"🧩","tokens_out":6947,"duration_ms":59022,"temperature":0.7,"pith_summary":"The paper's central claim is that every finite combinatorial problem—whether it asks to invert a function, satisfy constraints, or optimize a cost—has an exact explicit equation that returns its solution or solutions. The construction starts from a classical logic circuit for the problem, translates each gate into a tensor, and contracts the resulting tensor network while summing over all variables except one at a time. Each variable's value is then given by the sign of a scalar, expressed as a nested Heaviside step function, Eq. (2.41). The paper stresses that this equation need not be computable in a reasonable time and is not claimed to beat state-of-the-art complexity, but it does give every problem a closed-form-like analytic handle. A side consequence is the conditional statement that if some physical device could contract these networks in polynomial time, every NP-hard problem would be solvable in polynomial time.","feed_headline":"Every combinatorial problem gets an exact solution equation","feed_subtitle":"The equation is exact yet not fast: a polynomial-time contraction would make every NP-hard problem tractable.","key_machinery":"The load-bearing object is the Tensor Logical Circuit (TLC) together with the Half Partial Trace readout. A TLC is a tensor network obtained from the classical logical circuit of the problem by replacing each operator with a sparse tensor whose nonzero entries enforce the input-output relation and multiply the state's amplitude; equality of indexes encodes the circuit wiring. The readout contracts the TLC with Plus Vectors (all-ones vectors) on all variables except the one being determined, and with a Minus Vector $(-1,1)$ on that variable, producing a scalar $\\Omega_n$ whose sign decides the bit through the Heaviside function. Earlier determined values enter as projection vectors, which makes the expression for the $n$-th variable a nested composition $x_n = H(\\Omega_n(H(\\Omega_0), H(\\Omega_1(H(\\Omega_0))),\\dots))$. For optimization, the amplitude factor $e^{-\\tau C(\\vec x)}$ is the engine that makes low-cost states dominant, with the intended limit $\\tau\\to\\infty$ isolating the optimum.","core_discovery":"The central discovery is Theorem 1: given any combinatorial problem—inversion, constraint satisfaction, or optimization—there is an exact explicit equation for its solution(s). The equation is produced by (1) rewriting the problem in chosen variables, (2) building a logical circuit (LSTC for inversion, LSVC for constraints, LSMC for optimization) whose operators carry only the necessary internal signals, (3) tensorizing it into a Tensor Logical Circuit, and (4) contracting with the 'Half Partial Trace': impose the known output, or for optimization apply imaginary-time weight $e^{-\\tau C(\\vec x)}$, then sum over all variables except one using Plus Vectors and read that variable from a Minus Vector $(-1,1)$. Each decision is a Heaviside step $x_n = H(\\Omega_n(...))$, and because earlier decisions are fed back as projection vectors, the full solution is a nesting of Heaviside functions inside tensor-network contractions. The paper further claims (Theorem 2) that this equation can be written down in time polynomial in the problem's formulation, and (Theorem 3) that infinitely many equivalent equations exist.","pith_inferences":["Editorial inference: if the claimed construction is correct, it reframes the P versus NP question as a question about the contraction cost of a specific family of tensor networks, giving complexity theorists a concrete combinatorial object to bound.","Editorial inference: a natural testable next step is to search for polynomial-time contractible subfamilies (for example, low-treewidth or chain-structured TLCs) where the Half Partial Trace readout provably returns the optimum, bypassing the unresolved general choice of $\\tau$.","Editorial inference: the paper's explicit equations could be used to generate certificates, because evaluating the nested Heaviside composition at a candidate solution gives a direct check, though checking sign consistency would itself cost as much as contraction.","Editorial inference: since the TLC is a positive tensor network, Monte-Carlo or approximate contraction methods may be adapted to sample from its distribution, connecting the exact equation to randomized algorithms."],"forward_implications":["Every finite well-formulated combinatorial problem acquires a closed-form-like equation, so problems without a known analytic solution still have an explicit mathematical expression for their answer.","The same recipe covers inversion, constraint satisfaction, and optimization uniformly, suggesting a common analytic language for problems usually treated by separate algorithms.","Because the equation's construction time is polynomial in the problem formulation, any future method that contracts these tensor networks in polynomial time would imply polynomial-time algorithms for all NP-hard problems.","The framework yields concrete equations for many named problems (QUBO, TSP, knapsack, integer programming, k-colouring, maximum flow, and others), which can be studied analytically or approximated by tensor-network compression.","The equations also give a new target for approximation: compressing the TLC with Matrix Product States or removing constraint layers can turn the exact form into an approximate solver."],"supporting_citations":[{"why":"It supplies the quantum-inspired imaginary-time-plus-postselection template that the present method generalizes and simplifies.","marker":"[30]"},{"why":"It provides the tensor-network formalism used throughout to represent logic circuits as contracted index sums.","marker":"[27]"},{"why":"It defines the QUBO formulation used as the base example and main extension target.","marker":"[25]"},{"why":"It gives the tensor-train decomposition used to split high-valence logical operators into low-rank chains.","marker":"[45]"},{"why":"It is the earlier tensor-network treatment of the TSP whose filter-layer and iteration ideas the method systematizes.","marker":"[37]"},{"why":"It is the earlier polynomial-time tridiagonal QUBO and QUDO solver that motivates the chain-shaped networks and efficient contraction.","marker":"[39]"},{"why":"It is the classical shortest-path baseline that the tensor-network formulation reproduces.","marker":"[8]"}],"fun_headline_variants":["Every combinatorial problem has an explicit exact solution equation","Tensor networks give an exact equation for any combinatorial problem","Exact equation for all combinatorial problems, built from tensor networks","If tensor networks contract fast, every NP-hard problem gets solved","One explicit equation solves any combinatorial problem exactly"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"For optimization problems the readout assumes that a single finite damping parameter $\\tau$ can be made large enough that each partial-trace scalar $\\Omega_n$ keeps the sign of an optimal assignment; the paper states this informally in Section 2.5.2 and gives no proof or constructive rule for choosing $\\tau$.","fun_headline_variants_meta":{"raw":{"variants":["Every combinatorial problem has an explicit exact solution equation","Tensor networks give an exact equation for any combinatorial problem","Exact equation for all combinatorial problems, built from tensor networks","If tensor networks contract fast, every NP-hard problem gets solved","One explicit equation solves any combinatorial problem exactly"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000281,"raw_usage":{"total_tokens":1652,"prompt_tokens":923,"completion_tokens":729,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":539,"completion_tokens_details":{"reasoning_tokens":652}},"tokens_in":539,"tokens_out":729,"duration_ms":7400,"temperature":1.0,"reasoning_tokens":652,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T17:10:00.227176+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the Half Partial Trace iteration, with increasing values of $\\tau$, on a small optimization instance whose optimum is known (for example a 4-variable QUBO with a unique minimizer) and record whether the sign of each $\\Omega_n$ eventually matches the optimal assignment for every instance; any instance for which no finite $\\tau$ yields the optimum would falsify the optimization version of Theorem 1.","supporting_citations":[{"cited_title":"A quantum-inspired tensor network algorithm for constrained combinatorial optimization problems","cited_arxiv_id":null,"evidence_quote":"It supplies the quantum-inspired imaginary-time-plus-postselection template that the present method generalizes and simplifies."},{"cited_title":"Tensor Network Based HOBO Solver","cited_arxiv_id":"2407.16106","evidence_quote":"It is the earlier polynomial-time tridiagonal QUBO and QUDO solver that motivates the chain-shaped networks and efficient contraction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It is the classical shortest-path baseline that the tensor-network formulation reproduces."}],"review_version":1}