{"id":"d4d89c42-0cdf-49e2-b5cc-7e4bceaaface","arxiv_id":"2506.19964","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":9,"one_line_summary":"A clause-based autoencoder architecture with Fowler-Nordheim annealing solves higher-order Ising problems directly, avoiding quadratization overhead and matching or beating prior Ising solvers on benchmarks.","lead":"This paper builds a higher-order neuromorphic Ising machine that optimizes objectives with three-or-more-spin interactions directly, without converting them to pairwise form. It reports better solution quality and competitive speed on MAX-CUT and MAX-SAT benchmarks, including a working FPGA implementation.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The asymptotic-convergence guarantee is proven only for the Bernoulli asynchronous model (Methods 4.3), while SI Algorithms 1 and 2—which produced all reported results—drop that noise and use rejection-free or colored parallel updates; detailed balance fails for these variants, so the headline…","rationale":"The paper's exact algebra around Eq. (2) and the autoencoder construction are a genuine contribution; I am not disputing that an asynchronous single-spin-flip implementation with Bernoulli-gated thresholds would implement simulated annealing and inherit the FN convergence theorem from [4]. The load-bearing gap is that the algorithms presented in the SI and used in every experiment are not that chain. Methods 4.6 explicitly says both variants use only N^E, and SI Algorithms 1–2 omit N^B. Rejection-free uniform selection among all threshold-passing spins changes the transition kernel: the move probability is an average of 1/|A| over random acceptance sets, not (1/N) alpha_i, and the two-spin ferromagnet calculation shows detailed balance is violated. Graph-colored simultaneous flips similarly use a product of independent single-flip acceptances rather than a single Metropolis acceptance for the joint move. Consequently, the FN log-cooling ground-state theorem cannot be invoked for the reported results unless a new proof is supplied. This is exactly the reader's weakest assumption, and my read agrees. The proposed check would settle whether the concern is real by directly measuring the transition ratios. I would keep the reader's CONDITIONAL verdict: the manuscript should either prove the guarantee for Algorithms 1–2, benchmark the Bernoulli-asynchronous version, or remove the convergence claim from the abstract; without one of these, the central claim is not supported.","tokens_in":31860,"tokens_out":14330,"duration_ms":168507,"concrete_test":"Run SI Algorithm 1 at fixed temperature beta on the two-spin ferromagnet E = -J s1 s2 with J = 1, beta = 1, and estimate P((+,+) -> (-,+)) and P((- ,+) -> (+,+)) from 10^6 independent trials. Detailed balance for the simulated-annealing chain requires the ratio P_forward / P_reverse to equal e^{-2 beta J} = e^{-2} ~ 0.1353; the rejection-free dynamics instead predicts 2e^{-2} - e^{-4} ~ 0.2524. If the measured ratio matches the latter, the implemented algorithm is not the SA chain of Methods 4.3, confirming that the convergence guarantee does not cover the reported solver.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Abstract, Section 3) is that the machine is functionally isomorphic to simulated annealing and therefore inherits the FN-annealer asymptotic ground-state guarantee. That isomorphism is established only for the asynchronous, Bernoulli-threshold model of Methods 4.3 (Eqs. 14–16). The versions that actually produced every reported number are different: SI Algorithm 1 drops the Bernoulli term N^B and, after evaluating all N thresholds in parallel, selects one accepting spin uniformly at random (\"rejection-free sampling\"); SI Algorithm 2 flips all accepting spins in a color group simultaneously. Section 3 even states that Algorithm 1 uses the Bernoulli asynchronous mechanism, but the pseudocode does not. These are not cosmetic differences. For rejection-free uniform selection, the transition probability from x to x^i is E_u[1_{i in A(x,u)} / |A(x,u)|], not (1/N) alpha_i(x). In a two-spin ferromagnet E = -J s1 s2 with J>0, the transition ratio from (+,+) to (-,+) over the reverse move is 2 e^{-2 beta J} - e^{-4 beta J}, not e^{-2 beta J}; detailed balance is violated and the stationary distribution is not Boltzmann. The colored variant has the same problem: it accepts multiple flips by a product of independent single-flip tests, which is not the Metropolis acceptance for the combined move. Hence the FN/log-cooling convergence theorem from [4] does not transfer to either implemented solver. Since all benchmark claims (Figs. 2–4) come from these variants, the headline guarantee is unsupported unless a new proof is supplied.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a neuromorphic Ising machine for higher-order Hamiltonians of the form Eq. (1), organized as an autoencoder whose outer layer consists of clause neurons and whose latent layer consists of spin neurons. The central algebraic insight is that the energy change on flipping spin i can be expressed as 2∑_k H_{k,i} J_k T_{k,n-1} (Eqs. (7)-(10)), so that the Metropolis acceptance test of simulated annealing can be implemented by comparing this weighted clause sum against a noisy threshold. The noisy threshold is generated by an exponentially distributed random variable annealed with a Fowler-Nordheim schedule τ(t)=τ0/log(1+t/C) (Eq. (19)), plus (in the idealized model) a Bernoulli term to enforce asynchrony. The paper claims asymptotic convergence to the Ising ground state by exact functional isomorphism to simulated annealing, and claims resource complexity independent of interaction order. It reports benchmarks on MAX-CUT (Gset), MAX-3SAT (SATLIB), and MAX-5/7SAT instances, with CPU and FPGA implementations, comparing against the second-order NeuroSA machine and the dSBM simulator.","tokens_in":32172,"tokens_out":13248,"duration_ms":145009,"significance":"If the convergence and resource claims were established for the actual implementations, this would be a significant contribution: the algebraic reformulation in Eq. (2)/(10) is exact and clean, the mapping of the Metropolis accept/reject step to thresholded latent neurons (Eqs. (13)-(16)) is an elegant and genuinely useful rewrite for the asynchronous single-spin-flip model, and the paper ships extensive benchmark comparisons, including 1000-trial MAX-CUT runs, FPGA timing measurements, and comparisons to a commercial-style simulator (dSBM). The claim of avoiding quadratization overhead for higher-order interactions is of broad interest to the Ising-machine and neuromorphic-computing communities. However, the headline convergence guarantee is proven only for the idealized asynchronous Bernoulli model, while the solvers that produced all reported results use different update rules; this gap is central to the paper's reliability claims and must be addressed before the significance can be accepted.","major_comments":[{"comment":"The asymptotic convergence guarantee is established only for the asynchronous, Bernoulli-threshold update of Eqs. (14)-(16), which is an exact rewrite of Metropolis simulated annealing. The implementations that produced every reported figure — SI Algorithm 1 (uncolored) and SI Algorithm 2 (graph-colored) — drop the Bernoulli term N^B and instead perform parallel thresholding followed by either uniform random selection among all active neurons (Algorithm 1) or simultaneous flips of all active neurons in a color class (Algorithm 2). For these variants detailed balance fails. For example, for a two-spin ferromagnet E=-J s1 s2 with a=e^{-2βJ}, the rejection-free transition probability ratio is P((+,+)->(-,+))/P((- ,+)->(+,+)) = 2a - a^2, whereas the Boltzmann ratio is e^{-2βJ}; these are equal only in the trivial limits. Consequently the stationary distribution is not the Boltzmann distribution of Eq. (1), and the FN/log-cooling convergence theorem of [4] does not transfer to the implemented solvers. Section 3 states that the uncolored version \"uses the asynchronous update mechanism given by the noise threshold incorporating the Bernoulli random variable (N^B)\" and refers to Algorithm 1, but the pseudocode of SI Algorithm 1 contains no N^B term and no per-step single-spike constraint; Methods 4.6 explicitly says both variants use only the exponential noise N^E. This internal inconsistency is load-bearing, because the abstract's and Section 3's claims of asymptotic ground-state convergence and \"reliability\" rest on the isomorphism that the implemented algorithms do not satisfy.","section":null},{"comment":"For the graph-colored variant, even granting that same-color neurons are conditionally independent, the simultaneous acceptance of multiple flips via independent single-spin threshold tests is not the Metropolis acceptance probability for the joint multi-spin move. Since same-color spins have disjoint clause neighborhoods, the energy change of the joint move is the sum of the individual ΔE_i, but the proposed rule accepts with probability proportional to ∏_i min(1, e^{-ΔE_i/τ}), which is not equal to min(1, e^{-Σ_i ΔE_i/τ}) in general. This violates detailed balance for the joint move as well. No convergence analysis is provided for the colored variant, yet Fig. 2 and Fig. 3 explicitly attribute improved time-to-solution and solution quality to graph coloring. The empirical results therefore come from a heuristic whose stationary distribution, if it exists, is not shown to be the target Boltzmann distribution; the paper's claim that the FN annealer's optimality transfers to the colored machine is unsupported.","section":null},{"comment":"The claim that the architecture \"maintains resource complexity independent of interaction order\" (Abstract) and that \"the resource complexity always scales as O(n^2) irrespective of the order of interaction\" (Section 3) is not supported by the model or by the manuscript's own MAX-SAT mapping. The number of encoder connections is the number of nonzeros of H, S = Σ_k |C_k|. For the MAX-SAT expansion in SI Section S4, each clause of width p contributes Σ_{ℓ=1}^{p} ℓ C(p,ℓ) = p 2^{p-1} nonzero interactions, so S = M_orig p 2^{p-1}, which grows exponentially in the original clause width p. Thus resource complexity is not independent of interaction order unless one fixes the clause set and per-clause order, which is not the regime described in the abstract or Fig. 1b. The comparison in Fig. 1b is also based on a single quadratization scheme (Chancellor's transformation, Methods 4.1), so the broad statement that quadratization in general leads to exponential overhead is not established. This overstatement affects a headline contribution and should be corrected or carefully qualified.","section":null}],"minor_comments":[{"comment":"Equation (10) gives only the sign equivalence for ΔE_{i,n}; substituting ΔE = 2Σ into the Metropolis condition Eq. (12) yields Σ_k H_{k,i} J_k T_{k,n-1} < -(τ_n/2) log(u/B+ε), not < -τ_n N^E as written in Eq. (13). The factor of two can be absorbed into the hyperparameters, but as written the derivation is not exact and the claim of functional isomorphism is slightly overstated.","section":null},{"comment":"The statement that N^E = log(u/B+ε) \"follows an exponential distribution\" is imprecise: if u is uniform and ε=0, B=1, then -log u is Exp(1), but the shifted and scaled quantity with general B and ε is not exactly exponential. Since the mean of N^E is later treated as a free hyperparameter (Methods 4.4), this is a presentation issue rather than a technical flaw.","section":null},{"comment":"The sentence \"the resource complexity always scales as O(n^2) irrespective of the order of interaction\" is also internally in tension with the following sentences giving O(n×q) for the encoder and O(m×p) for the decoder; the O(n^2) worst case should be stated as a dense-bound assumption, and the dependence on the number of clauses and their orders should be stated explicitly.","section":null},{"comment":"The time axis in Fig. 2b is labeled \"a.u.\" and the inset reports an offset of approximately 5×10^5; the text should state explicitly whether these are CPU iterations, FPGA clock cycles, or wall-clock time, and whether the comparison in Fig. 2c uses the same time units as the dSBM data.","section":null},{"comment":"There are several typographical errors, e.g., \"T able S1\" in SI, \"MAX-SA T\" in the Section 2.3 heading and Fig. 3 caption, and \"the chunk\" phrasing in Section 4.6; a careful proofread is needed.","section":null}],"recommendation":"major_revision","confidential_remarks":"The decisive issue is the mismatch between the theoretically analyzed asynchronous Bernoulli model and the rejection-free/colored algorithms used in all experiments. The paper would be salvageable either by proving detailed balance (or a controlled approximation) for the implemented variants, or by repositioning the contributions as empirical heuristics and removing the inherited asymptotic convergence claim. The resource-complexity overstatement should also be corrected, as it is a headline claim. The reliance on [4] for the convergence guarantee is legitimate in itself, but the manuscript needs to state precisely which dynamics inherit that guarantee."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The exact derivation in Eq. (10) is the real content here, and it checks out: flipping a spin changes the energy by a weighted sum of clause outputs, so you can navigate higher-order Ising landscapes without quadratization overhead. The autoencoder framing with toggle-neuron decoders is a clean way to implement that update, and the resource argument—hardware cost proportional to clause-variable incidence rather than interaction order—is genuinely useful for the neuromorphic Ising community. The FPGA numbers (microsecond TTS on MAX-3SAT up to 250 variables) are suggestive and internally consistent, and the authors are appropriately careful to compare against NeuroSA with the same annealing schedule.\n\nThe load-bearing problem is the convergence guarantee. Methods 4.3 proves isomorphism to simulated annealing for the asynchronous, Bernoulli-threshold model. But SI Algorithm 1 drops the Bernoulli noise and uses rejection-free selection, and SI Algorithm 2 uses graph-colored parallel updates. The stress-test note is right: in a two-spin ferromagnet, the rejection-free transition ratio is 2e^{-2βJ} - e^{-4βJ}, not e^{-2βJ}, so detailed balance fails and the stationary distribution is not Boltzmann. The paper itself concedes the departure in Methods 4.6—\"rejection-free sampling boosts acceptance probability compared to the original simulated-annealing framework\"—but never proves what the new stationary distribution is. Section 3 says dense graphs need the uncolored version, but even that version is rejection-free, not Metropolis. So the abstract's claim that asymptotic convergence is \"ensured\" does not cover the implemented solvers. That's not a cosmetic gap; it means the reported results come from a heuristic that may or may not converge to the ground state.\n\nOther soft spots are minor by comparison. The MAX-CUT TTS picks the best of three annealing steps per graph and tunes A per graph, which is fitting. The MAX-SAT comparison has no external higher-order or non-Ising baseline, so \"SOTA\" is only with respect to NeuroSA. No code or data is released, which makes the FPGA results hard to verify. And there are mechanical issues—an unrelated EPOS-LHC citation, an imprecise statement that N^E follows an exponential distribution (it's a shifted log-uniform), and a sign inconsistency between the pseudocode and the derivation. All fixable.\n\nWho should read this: anyone working on higher-order Ising hardware or neuromorphic optimization. The resource-scaling idea is worth taking seriously even if the guarantee needs repair. I'd send it to peer review, but ask the authors to either prove convergence for the rejection-free and colored variants or explicitly reframe the machine as an empirically strong heuristic. And require code/data.\n\nRecommendation: accept for peer review with major revision.","headline":"The delta-E rewrite and autoencoder framing are sound and worth building on, but the convergence guarantee doesn't cover the solvers that produced the benchmarks—so the headline claim needs either a new proof or a haircut.","tokens_in":32912,"tokens_out":2584,"would_cite":true,"duration_ms":29532,"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":"Operating on Ising clauses rather than on spins lets a neuromorphic architecture anneal arbitrary-order Hamiltonians with hardware cost that does not grow with interaction order, and the Fowler–Nordheim schedule preserves the asymptotic…","keywords":["Neuromorphic Computing","Ising Machines","Higher-order Ising","Fowler-Nordheim Annealing","Autoencoder","MAX-SAT","MAX-CUT","Graph Coloring"],"falsifier":"Take a planted 3R-3X instance with a known ground state, run the graph-colored implementation of SI Algorithm 2 with the Bernoulli threshold term removed, as in the FPGA results, and check whether it always reaches the planted ground state under the Fowler–Nordheim schedule. If any instance gets stuck in a local minimum that the asynchronous Bernoulli-threshold dynamics escapes, the claim that the implemented variants inherit the asymptotic convergence guarantee is false.","tokens_in":31502,"feed_emoji":"🧠","tokens_out":10289,"duration_ms":94897,"temperature":0.7,"pith_summary":"This paper tries to establish that a neuromorphic Ising machine can solve higher-order combinatorial optimization problems—MAX-SAT with three-, five-, and seven-body clauses—without first quadratizing them into pairwise interactions. The architecture makes the dynamic variables the clauses themselves, $\\prod_i s_i^{H_{k,i}}$, while sampling the energy landscape in the lower-dimensional spin space through an autoencoder, so hardware resources scale with the clause–variable incidence structure rather than with interaction order or auxiliary-variable overhead. The same Fowler–Nordheim annealer that guarantees asymptotic convergence for pairwise Ising machines is shown to transfer to this setting, because flipping a spin lowers the energy exactly when the weighted clause sum feeding that spin is negative. If the claim holds, quadratization ceases to be the bottleneck for higher-order Ising hardware, and even pairwise problems like MAX-CUT can be solved at time-to-solution competitive with specialized solvers.","feed_headline":"An autoencoder lets Ising machines handle any interaction order","feed_subtitle":"Higher-order clauses are annealed directly, so hardware cost no longer explodes with interaction order.","key_machinery":"The load-bearing object is the clause–spin autoencoder defined by the binary incidence matrix $H\\in\\{0,1\\}^{M\\times N}$: the encoder maps the $M$ clause outputs $T_k$ to $N$ latent spin neurons through the weighted sum $\\sum_k H_{k,i} J_k T_k$, and the decoder reconstructs each clause from the parity of latent-neuron spikes. The identity carrying the argument is $\\Delta E_i<0 \\iff \\sum_k H_{k,i} J_k T_{k,n-1}<0$, which turns the energy comparison of simulated annealing into a thresholded weighted sum. The Fowler–Nordheim annealer supplies the time-varying noisy thresholds $\\mu_{i,n}=\\tau_n N^E_n + A N^B_{i,n}$ with $\\tau_n\\sim 1/\\log n$, reproducing the optimal annealing schedule and, in the idealized asynchronous model, preserving asymptotic convergence. Decoder toggle neurons implement clause flips, and graph coloring of latent neurons allows conditionally independent spins to update in parallel.","core_discovery":"The central discovery is that the general higher-order Ising Hamiltonian $E(s)=-\\sum_{k=1}^{M} J_k \\prod_{i=1}^{N} s_i^{H_{k,i}}$ admits an autoencoder implementation whose outer layer stores the clause outputs $T_k=\\prod_i s_i^{H_{k,i}}$ and whose latent layer stores the spins $s_i$. Flipping spin $i$ lowers the energy if and only if $\\sum_{k=1}^{M} H_{k,i} J_k T_{k,n-1} < 0$; the encoder computes that weighted clause sum at each latent neuron, the noisy threshold implements the simulated-annealing acceptance step, and the decoder toggles a clause whenever the parity of contributing spin flips is odd. Because clauses are manipulated directly, a $p$-body interaction costs one row of the interconnection matrix instead of $O(n^p)$ pairwise couplings, and the auxiliary variables of quadratization disappear. With the Fowler–Nordheim temperature schedule $\\tau(t)=\\tau_0/\\log(1+t/C)$, the paper argues the system asymptotically converges to the Ising ground state. On MAX-CUT and MAX-3SAT benchmarks, the higher-order machine yields better solutions in fewer iterations than the same annealer applied to a quadratized second-order model; graph coloring and an FPGA implementation further improve time-to-solution.","pith_inferences":["If the resource claim holds at scale, the real scaling ceiling shifts to clause count and clause density; a clean test would hold the clause count fixed while raising the interaction order and measure whether encoder and decoder resources stay flat.","The convergence guarantee is proven for the asynchronous single-spin-flip model, while the reported runs use rejection-free and graph-colored parallel updates; a reader should treat the guarantee as covering the idealized dynamics until the variants are proven equivalent.","Because a clause output is a parity product of spins, the same encoder–decoder factorization could serve parity-check decoding problems beyond SAT, such as LDPC decoding, where higher-order parity constraints are native and quadratization is also unnatural.","A physical Fowler–Nordheim tunneling device is the natural next noise source; the paper notes digital emulation was chosen for 16-bit precision, so a device-based implementation would test whether the annealer's hardware promise survives outside simulation."],"forward_implications":["Any-order MAX-SAT clauses can be embedded directly, so the search space is not inflated by auxiliary variables; on the paper's benchmarks this yields near-perfect satisfiability at sizes where the quadratized second-order solver's success rate collapses.","For sparse problem encodings the hardware footprint becomes $O(nq)$ for the encoder and $O(mp)$ for the decoder instead of $O(n^p)$ couplings or quadratization's exponential auxiliary-variable overhead.","Graph coloring massively parallelizes updates of conditionally independent latent neurons, accelerating high-temperature convergence; sufficiently dense graphs must fall back to the uncolored asynchronous variant to preserve ergodicity.","An FPGA implementation with 16-bit noise thresholds reaches 99% clause satisfaction on size-250 MAX-3SAT instances in median times under a microsecond, a 2500–3000x speedup over the CPU emulation at the same precision.","On the second-order MAX-CUT benchmark the same machine matches or beats the reported state-of-the-art cuts on G4, G11, and G15 with competitive time-to-solution."],"supporting_citations":[{"why":"Supplies the Fowler–Nordheim annealer, the asynchronous neuromorphic Ising framework, and the quadratized second-order baseline used throughout the comparisons.","marker":"[4]"},{"why":"Contributes the greedy graph-coloring heuristic used to parallelize updates of conditionally independent latent neurons.","marker":"[12]"},{"why":"Provides the auxiliary-variable gadget used to quadratize MAX-3SAT clauses for the second-order comparison.","marker":"[28]"},{"why":"Supplies the Gset MAX-CUT graphs (G4, G11, G15) used for the cut-quality and time-to-solution comparisons.","marker":"[38]"},{"why":"Defines the time-to-solution formula and provides the simulated-bifurcation baseline for the MAX-CUT comparison.","marker":"[39]"},{"why":"Supplies the satisfiable and unsatisfiable MAX-3SAT benchmark instances used in solution-quality and time-to-solution evaluations.","marker":"[45]"},{"why":"Gives the Chancellor mapping used to estimate quadratization costs and to construct the quadratized 3R-3X Hamiltonian.","marker":"[55]"},{"why":"Is the simulated-annealing acceptance criterion from which the latent-neuron thresholding and the convergence argument are derived.","marker":"[56]"},{"why":"Supplies the rejection-free sampling strategy that the uncolored implementation uses to preserve ergodicity on synchronous hardware.","marker":"[57]"}],"fun_headline_variants":["Any-order Ising solved directly: no quadratization needed","Autoencoder + Fowler-Nordheim: Ising scales to any order","Higher-order Ising without cost explosion via autoencoder","Ising machines handle any order with autoencoder annealer","No quadratization: Autoencoder annealer scales Ising to any order"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The convergence guarantee is derived for asynchronous single-spin-flip updates with Bernoulli noise, but the benchmark results come from rejection-free parallel selection and graph-colored simultaneous updates; if those variants do not preserve the same sampling statistics, the reported performance is a heuristic rather than a consequence of the guarantee.","fun_headline_variants_meta":{"raw":{"variants":["Any-order Ising solved directly: no quadratization needed","Autoencoder + Fowler-Nordheim: Ising scales to any order","Higher-order Ising without cost explosion via autoencoder","Ising machines handle any order with autoencoder annealer","No quadratization: Autoencoder annealer scales Ising to any order"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000905,"raw_usage":{"total_tokens":3981,"prompt_tokens":1123,"completion_tokens":2858,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":739,"completion_tokens_details":{"reasoning_tokens":2770}},"tokens_in":739,"tokens_out":2858,"duration_ms":20323,"temperature":1.0,"reasoning_tokens":2770,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:24:27.765016+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a planted 3R-3X instance with a known ground state, run the graph-colored implementation of SI Algorithm 2 with the Bernoulli threshold term removed, as in the FPGA results, and check whether it always reaches the planted ground state under the Fowler–Nordheim schedule. If any instance gets stuck in a local minimum that the asynchronous Bernoulli-threshold dynamics escapes, the claim that the implemented variants inherit the asymptotic convergence guarantee is false.","supporting_citations":[{"cited_title":"Nature Communications16(1), 3086 (2025) https://doi.org/10.1038/ s41467-025-58231-5","cited_arxiv_id":null,"evidence_quote":"Supplies the Fowler–Nordheim annealer, the asynchronous neuromorphic Ising framework, and the quadratized second-order baseline used throughout the comparisons."},{"cited_title":"In: 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’05), vol","cited_arxiv_id":null,"evidence_quote":"Provides the auxiliary-variable gadget used to quadratize MAX-3SAT clauses for the second-order comparison."},{"cited_title":"https://web.stanford.edu/∼yyye/yyye/Gset/ (2003)","cited_arxiv_id":null,"evidence_quote":"Supplies the Gset MAX-CUT graphs (G4, G11, G15) used for the cut-quality and time-to-solution comparisons."},{"cited_title":"283–292 (2000)","cited_arxiv_id":null,"evidence_quote":"Supplies the satisfiable and unsatisfiable MAX-3SAT benchmark instances used in solution-quality and time-to-solution evaluations."}],"review_version":2}