{"id":"76780da5-a3a6-4e57-bb46-c16b5d6f36fa","arxiv_id":"2508.02250","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A Kuramoto oscillator network with one phase per cell can solve easy Sudoku puzzles in simulation and beats a Hopfield baseline, but fails on puzzles with many empty cells.","lead":"This paper encodes Sudoku puzzles as networks of coupled oscillators whose phases represent digits, and lets the oscillator dynamics settle toward a solution. In simulations, the oscillator network solves more test puzzles than a Hopfield-network baseline, at least when few cells are empty.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The printed dynamics are internally inconsistent: under Eq. (9), J=-1 unknown-unknown edges attract rather than repel, inverting the stated Sudoku mechanism.","rationale":"I agree with the reader's rejection. The central claim is that natural Kuramoto dynamics with the described weights solve Sudoku, and that claim collapses if the sign of the -1 couplings is inverted. I re-derived the two-oscillator phase difference under Eq. (9): for J=-1, dδ/dt = -2 sin δ, so the edge synchronizes rather than repels. This is not a matter of external consensus or implementation detail; the text is internally inconsistent with its own Background equation and with the stated purpose of the -1 coupling. The complex known-known coupling in Eq. (3) is a further sign of an undefined convention. Even if the authors' actual software produced the reported performance, the paper as written does not specify a reproducible model, so the central claim cannot be verified. The missing code and data are secondary but relevant. I therefore recommend no change to the reader's REJECT verdict.","tokens_in":7496,"tokens_out":5900,"duration_ms":72116,"concrete_test":"Check the minimal two-oscillator subsystem defined by Eq. (9): set J_12=J_21=-1, zero natural frequencies, initial phases θ1(0)=0 and θ2(0)=0.1, and integrate the ODE. If |θ1-θ2| monotonically decreases, the printed dynamics are attractive, directly contradicting the paper's claim that J=-1 repels and maximizes phase distance. To settle the paper-level claim, also re-run the 25-unknown 9×9 benchmark using an implementation that strictly follows Section III.B and Eq. (9); if the reported correct-recovery rate is not reproduced, the authors' actual solver must be using a sign convention different from the one printed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing failure is in Section III.B's sign and coupling conventions. The Kuramoto equation in Section II, Eq. (1), is dθ_i/dt = ω - Σ J_ij sin(θ_i - θ_j), but the dynamics actually used in Eqs. (4)-(9) are dθ_i/dt = -Σ J_ij sin(θ_j - θ_i), a global sign flip. This is not cosmetic: with J=-1 on an unknown-unknown edge, Eq. (9) gives dθ_i/dt = sin(θ_j - θ_i) = -sin δ and dθ_j/dt = sin(θ_i - θ_j) = sin δ for δ = θ_i - θ_j, so dδ/dt = -2 sin δ. The phase difference decays to zero; the edge synchronizes. Yet Section III.B states that J=-1 is chosen because 'the distance between the phases is maximized' and calls it repulsive. Thus the printed dynamics implement the opposite of the stated mechanism for exactly the edges that must force distinct digits. Equation (3) compounds the problem: a complex coupling e^{i(θ_i-θ_j)} multiplied by a real sine does not fit the real ODE in Eq. (9), and no phase-shifted-coupling convention is specified. Without a corrected sign convention or a released implementation, the reported ONN performance in Fig. 7 cannot be attributed to the described constraint-enforcing dynamics.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an oscillatory neural network (ONN) approach to Sudoku in which each cell is represented by a single Kuramoto oscillator, with the puzzle constraints encoded through row, column, and subgrid coupling subgraphs that are combined into one weight matrix. The authors claim that the natural phase dynamics converge to a valid solution and report simulations comparing the ONN against a self-implemented Hopfield-network baseline, concluding that the ONN outperforms the baseline for puzzles with up to roughly 20 unknown cells.","tokens_in":7721,"tokens_out":6231,"duration_ms":79143,"significance":"If the proposed mapping worked as described, the contribution would be interesting as a low-complexity, training-free analog constraint-satisfaction scheme: the ONN needs O(N_d^2) oscillators versus O(N_d^3) for the Hopfield formulation, and the use of one oscillator per cell with a direct phase encoding is conceptually appealing. The comparison is not circular: no parameters are fitted to the benchmark data, and the known-cell couplings encode the supplied clues rather than derived outputs. However, the paper as written has load-bearing inconsistencies in the equations of motion, an undefined complex coupling term, no explicit phase-to-digit readout rule, and an under-specified baseline, so the reported empirical claims cannot currently be evaluated or attributed to the stated mechanism.","major_comments":[{"comment":"The dynamics printed in Eqs. (4)-(9) are a sign-flipped version of the Kuramoto equation in Eq. (1). Eq. (1) has dθ_i/dt = ω - Σ J_ij sin(θ_i - θ_j), while Eq. (9) uses dθ_i/dt = -Σ J_ij sin(θ_j - θ_i). For a bidirectional unknown-unknown edge with J = -1, this gives dδ/dt = -2 sin δ, where δ = θ_i - θ_j, so the phase difference decays to zero and the oscillators synchronize. This is exactly opposite to the text's claim in Section III.B that J = -1 maximizes the distance between phases and is repulsive. The load-bearing repulsion that is supposed to force unknown cells to different digits is therefore not present in the printed dynamics. The sign convention must be corrected and the actual implemented equations stated before the reported results can be attributed to the described mechanism.","section":"III.B, Eqs. (1) and (9)"},{"comment":"Equation (3) defines J_ij^(known) as e^{i(θ_i-θ_j)}, which is complex-valued, but Eq. (9) is a real ODE in which J_ij multiplies sin(θ_j - θ_i). No real-part/imaginary-part convention, phase-shifted-coupling definition, or complex Kuramoto extension is specified. As written, the known-known edges, which are supposed to anchor the solution, do not have well-defined dynamics. This is a central issue because these edges are one of the three basic coupling cases on which the whole mapping rests.","section":"III.B, Eq. (3)"},{"comment":"The paper never specifies how the continuous oscillator phases are converted into discrete Sudoku digits after integration. Eq. (2) maps digits to phases, but the final phases can take arbitrary values in [0, 2π) (up to drift), and no nearest-digit rule, modulo-2π convention, or boundary handling is given. Without this readout rule, the 'correctly solved' percentages in Fig. 7 and the success/failure classification in the benchmark cannot be interpreted.","section":"III.A and IV"},{"comment":"The Hopfield-network baseline is not described beyond citing Refs. [13] and [14]. The implementation details of the HNN, the exact role of the seven puzzles per difficulty level, how the 100 independent runs are distributed across puzzles, and the precise definition of 'performance' are all under-specified. No code or data availability statement is provided. Since the central quantitative claim is that the ONN outperforms the HNN, this comparison needs enough detail to be reproduced or independently checked.","section":"III.C and Fig. 7"}],"minor_comments":[{"comment":"The text refers to 'the general Kuramoto dynamics given in Eq. (II)', but no Eq. (II) exists; the intended reference appears to be Eq. (1).","section":"III.B"},{"comment":"The y-axis of Fig. 6 lists both phase values in radians and Sudoku digits without explaining the correspondence; please add clear axis labels and a legend.","section":"Fig. 6"},{"comment":"The sentence 'one cell is directly represented by one oscillator, or neuron' is redundant and should be reworded for clarity.","section":"III.A"},{"comment":"The paper states that seven distinct puzzles were used for every difficulty level but also reports 100 independent runs; please clarify whether all runs use the same seven puzzles and how the random masking is applied across runs.","section":"III.C"},{"comment":"The natural frequency ω_i^(0) is said to be neglected in simulations. If all natural frequencies are equal, this is a harmless constant drift that can be removed by a rotating frame, but that transformation should be stated explicitly.","section":"II"},{"comment":"The conclusion states that the ONN 'correctly solv[es] 57.57 % of the runs' at 20 unknowns, but the results section does not report that number explicitly; please make the numbers in the text and figures consistent.","section":"IV and VI"}],"recommendation":"reject","confidential_remarks":"The sign inconsistency between Eq. (1) and Eq. (9), together with the undefined complex coupling in Eq. (3), is more than a typographical issue: it inverts the stated mechanism for the unknown-unknown edges. If the authors can supply the implemented code and a corrected, self-consistent formulation that reproduces the reported results, a resubmission could be considered, but as submitted the central claim is not supported by the printed equations."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, the mapping idea is genuinely new: one oscillator per Sudoku cell, digits placed on the unit circle, and row/column/subgrid constraints combined into a single adjacency matrix. That is a real departure from earlier multi-neuron HNN Sudoku solvers and worth a look. The complexity reduction (81 vs 729 neurons for 9x9) is real, and the authors are honest that performance collapses beyond 25 unknowns.\n\nSecond, the central dynamics as printed do the opposite of what the text claims. Eq. (9) is ˙θ_i = -Σ J_ij sin(θ_j - θ_i). For an unknown-unknown edge with J=-1, that gives dδ/dt = -2 sin δ for the phase difference δ, so the two oscillators synchronize. The paper says J=-1 is chosen to maximize phase distance and calls it repulsive. It isn't under their own sign convention. This is not cosmetic: those edges are exactly what must force distinct digits. Eq. (3) makes it worse by inserting a complex coupling e^{i(θ_i-θ_j)} into a real ODE without any explanation of how a complex weight acts in a real sine argument. The readout from phase to digit is also never specified, and no code or data are released.\n\nSo the performance curves in Fig. 7 cannot be reproduced, and the mechanism claimed to produce them is, as written, inverted. The HNN baseline is self-implemented, and details of its update rule are missing, which weakens the comparison but is a lesser problem.\n\nThe paper deserves a serious referee rather than a desk reject because the mapping idea is creative and the flaw, while load-bearing, is identifiable and potentially fixable. I would engage with the authors and ask for a corrected sign convention, a real readout rule, and a released implementation. But I would not cite the results until the dynamics actually match the claim. The paper is for people working on oscillator-based analog computing who want to see whether this problem class can be mapped at all; they should read it critically and only after the authors fix the math.","headline":"A genuinely new oscillator mapping for Sudoku, but the printed dynamics invert the claimed repulsion between unknown cells, so the reported results cannot be attributed to the described mechanism.","tokens_in":8289,"tokens_out":2284,"would_cite":false,"duration_ms":28511,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34C15","68T07","90C27"],"pacs":[],"model":"deepseek-v4-flash","headline":"An oscillatory neural network with one oscillator per Sudoku cell solves puzzles through phase synchronization and outperforms a Hopfield baseline on puzzles with up to 20 unknowns.","keywords":["Sudoku","oscillatory neural networks","Kuramoto model","constraint satisfaction","phase encoding","Hopfield neural networks","combinatorial optimization","non-von Neumann computing"],"falsifier":"Run the two-oscillator case of Eq. (9) with $J=-1$ and initial phases $\\theta_1=0$, $\\theta_2=\\pi/2$; the difference $\\delta=\\theta_1-\\theta_2$ obeys $d\\delta/dt=-2\\sin\\delta$, so $\\delta$ decays to $0$ rather than to $\\pi$. If the implemented solver drives this pair into phase lock instead of anti-alignment, the printed $J=-1$ coupling does not produce the claimed repulsion, and the mechanism would have to rely on a different sign convention or readout.","tokens_in":1611,"feed_emoji":"🧩","tokens_out":4481,"duration_ms":138534,"temperature":0.7,"pith_summary":"This paper proposes encoding a Sudoku puzzle directly into the phase dynamics of a coupled-oscillator network: one oscillator per cell, with each digit represented by a phase evenly spaced on the unit circle. The row, column, and subgrid constraints are written into a single coupling matrix, and the unmodified Kuramoto equation is run until the phases lock, with the locked phases read out as the solved grid. The authors report that this mapping solves 9×9 puzzles with 5–20 unknown cells with high success, outperforming a Hopfield-network baseline, and still recovers some correct assignments at 25–30 unknowns before both methods fail at 35 unknowns. If correct, the result matters because it gives a training-free, analog route from a discrete constraint-satisfaction problem to hardware that computes with oscillator phases.","feed_headline":"Phases of coupled oscillators crack Sudoku, beating Hopfield nets","feed_subtitle":"One oscillator per cell encodes digits as phases; Kuramoto dynamics solve puzzles with up to 30 blanks.","key_machinery":"The central object is the complex Kuramoto coupling matrix $J$ built from the three constraint subgraphs of a Sudoku: rows, columns, and subgrids. The phase mapping $\\theta_{\\mathrm{digit}} = 2\\pi(n-1)/N_d$ places the $N_d$ digits on the unit circle, and the update law is $\\dot{\\theta}_i = -\\sum_{j\\in\\Omega_i} J_{ij}\\sin(\\theta_j-\\theta_i)$, where $\\Omega_i$ is the set of row, column, and subgrid neighbors of cell $i$. The coupling values carry the argument: known-known edges use $J_{ij} = e^{i(\\theta_i-\\theta_j)}$, unknown-unknown edges use $J_{ij}=-1$ to push phases apart, and known-unknown edges are unidirectional $-1$ couplings that let known cells steer unknown ones. The machinery turns the discrete constraint-satisfaction puzzle into a continuous phase-locking problem.","core_discovery":"The central claim is that Sudoku can be solved by the natural dynamics of an oscillatory neural network, without logic operations or learning. Each cell is one oscillator; digits are phases via $\\theta_{\\mathrm{digit}} = 2\\pi(n-1)/N_d$; and the weight matrix is assembled from three edge types: Hermitian couplings between known cells, repulsive $-1$ couplings between unknown cells, and directed $-1$ couplings from known to unknown cells. The total dynamics $\\dot{\\theta}_i = -\\sum_{j\\in\\Omega_i} J_{ij}\\sin(\\theta_j-\\theta_i)$ combine row, column, and subgrid constraints, and random initial phases evolve to a locked state that is read as a digit assignment. In the paper's simulations of 9×9 puzzles, the ONN achieves near-100% performance at 5–10 unknowns, stays above 90% at 15 unknowns, and still solves or partially solves some puzzles at 25–30 unknowns, while the Hopfield baseline falls to 0% at 25 unknowns. The authors state that the ONN mapping outperforms the established HNN methodology and supports a proof of concept for phase-based non-von Neumann constraint solving.","pith_inferences":["Because the phase mapping is not binary, the same construction should transfer to other multi-valued constraint-satisfaction problems, such as graph coloring or Latin-square completion, where each node must take one of $N$ labels and constrained neighbors must differ.","A practical variant would rerun the Kuramoto dynamics with different random initial phases until a globally consistent readout appears, using the paper's proposed ratio $r = N_{\\mathrm{unknowns}}/N$ to decide how many repeats are worth attempting.","In physical hardware, locked-phase spreads caused by oscillator noise and coupling mismatch would be the main failure mode; measuring the distribution of readout phases as a function of $r$ would test how robust the analog solver is beyond the idealized simulation."],"forward_implications":["Standard 9×9 Sudoku needs only 81 oscillators and an 81×81 coupling matrix, compared with 729 neurons and a 729×729 matrix in the Hopfield formulation.","Puzzles with up to about 20 unknown cells can be solved by free oscillator dynamics, with no backtracking, SAT search, or training.","The ONN continues to produce partially correct assignments at 25–30 unknowns in cases where the Hopfield baseline solves nothing, indicating that phase-based constraint encoding retains useful information under sparse clues.","The construction scales naturally from the 4×4 illustration to any $N_d$-valued Sudoku, since the phase mapping and constraint subgraphs are defined for a general number of digits.","For highly ambiguous puzzles, the authors propose augmenting the dynamics with a checker module that repeats solves according to the ratio of unknowns to cells; this is an explicit extension the paper offers rather than a demonstrated result."],"supporting_citations":[{"why":"Provides the Kuramoto model whose phase dynamics form the solver.","marker":"[17]"},{"why":"Introduces the Hopfield-network approach to Sudoku used as the baseline.","marker":"[13]"},{"why":"Supplies the HNN formulation and the benchmark puzzle used for direct comparison.","marker":"[14]"},{"why":"Supplies the six public Sudoku puzzles used in the evaluation.","marker":"[21]"}],"fun_headline_variants":["Oscillator phases solve Sudoku without logic","Coupled oscillators beat Hopfield nets on Sudoku","Kuramoto dynamics outperform Hopfield on Sudoku","Sudoku solved by phase-locked oscillators","Oscillatory neural nets crack Sudoku"],"cache_read_input_tokens":10368,"weakest_assumption_plain":"The load-bearing premise is that the printed coupling signs and phase readout behave as intended, so unknown-unknown edges push phases apart, known-known edges anchor fixed phases, and known-unknown edges steer free cells toward the correct digit.","fun_headline_variants_meta":{"raw":{"variants":["Oscillator phases solve Sudoku without logic","Coupled oscillators beat Hopfield nets on Sudoku","Kuramoto dynamics outperform Hopfield on Sudoku","Sudoku solved by phase-locked oscillators","Oscillatory neural nets crack Sudoku"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000626,"raw_usage":{"total_tokens":2907,"prompt_tokens":967,"completion_tokens":1940,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":583,"completion_tokens_details":{"reasoning_tokens":1866}},"tokens_in":583,"tokens_out":1940,"duration_ms":17977,"temperature":1.0,"reasoning_tokens":1866,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T05:04:15.387457+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the two-oscillator case of Eq. (9) with $J=-1$ and initial phases $\\theta_1=0$, $\\theta_2=\\pi/2$; the difference $\\delta=\\theta_1-\\theta_2$ obeys $d\\delta/dt=-2\\sin\\delta$, so $\\delta$ decays to $0$ rather than to $\\pi$. If the implemented solver drives this pair into phase lock instead of anti-alignment, the printed $J=-1$ coupling does not produce the claimed repulsion, and the mechanism would have to rely on a different sign convention or readout.","supporting_citations":[{"cited_title":"Mori and Y","cited_arxiv_id":null,"evidence_quote":"Provides the Kuramoto model whose phase dynamics form the solver."},{"cited_title":"Searching for memories, Sudoku, implicit check-bits, and the iterative use of not-always-correct rapid neural computation","cited_arxiv_id":"q-bio/0609006","evidence_quote":"Introduces the Hopfield-network approach to Sudoku used as the baseline."},{"cited_title":"Solving sudoku puzzles by using hopfield neural networks,","cited_arxiv_id":null,"evidence_quote":"Supplies the HNN formulation and the benchmark puzzle used for direct comparison."},{"cited_title":"3 million sudoku puzzles with ratings,","cited_arxiv_id":null,"evidence_quote":"Supplies the six public Sudoku puzzles used in the evaluation."}],"review_version":1}