{"id":"60a24132-648a-42ce-be11-533c0ec7f38e","arxiv_id":"2605.30355","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.5,"correctness_risk":"low","formal_verification":"none","parameter_count":5,"one_line_summary":"Cosm finds certified optimal cuts on Gset G72/G77/G81 and reduces best-known times-to-target on G61/G70 from hundreds of hours to 36–303 s via switched circular dynamics.","lead":"Cosm is a new dynamical heuristic that uses circular variables, periodically switched conflict-free edge updates, and dual-window twists to solve large sparse Ising problems. It finds certified optima on the three largest Gset 2D spin glasses for the first time and cuts prior times-to-target from hundreds of hours to minutes on hard non-lattice instances.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the bounded-degree sparsity requirement (Sec. V-B) as the weakest assumption; it is stated explicitly and does not threaten the claims inside the sparse regime where Cosm is evaluated. Certified optima, ablation results, and scaling comparisons supply independent support. No stronger load-bearing flaw is present, so the ACCEPT verdict stands unchanged.","tokens_in":20348,"tokens_out":331,"duration_ms":14333,"concrete_test":"Re-implement Alg. 1 with the exact parameter tables (IX–XI) and greedy coloring of Table VIII; run 256 independent trials of 0.5 M sweeps on G61 and confirm that cut 5799 is reached with Ps ≥ 0.008 (the value underlying the 303 s TTT). If Ps falls materially below that figure the wall-clock claim requires re-examination.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims—first heuristic attainment of Gurobi-certified optima on G72/G77/G81 (cuts 7008/9940/14060) and multi-order TTT reductions on G61/G70—rest on posted bitstrings, ablation evidence (Table VII) that both SCS and DWT are essential, and an openly stated sparsity precondition (Sec. V-B) that matches the claimed regime. No hidden assumption fails inside that regime, no derivation gap appears, and hardware caveats are acknowledged. The moderate reproducibility gap (no public code) does not undermine the reported numerical advances.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper introduces Collective Switched Motion (Cosm), a dynamical-system heuristic for sparse Ising/Max-Cut/QUBO problems. Continuous circular variables interact via finite-magnitude sgn-based couplings applied sequentially over a proper edge coloring (Sequential Conflict-free Search, SCS), producing a periodically switched network; a Dual Window Twist (DWT) correlated perturbation is applied at regular intervals to promote cluster motion. After annealing, variables are binarized by testing multiple bisectors. On the three largest Gset 2-D toroidal spin glasses (G72, G77, G81; 10k–20k variables) Cosm reaches cuts 7008/9940/14060 that Gurobi certifies as optimal—the first heuristic attainment of these optima. On the largest bounded-degree non-lattice instances G61 and G70 it reaches the best-known cuts with wall-clock times-to-target of 303 s and 36 s. Ablation on G70 shows both SCS and DWT are essential; scaling exponents on tile-planted lattices are lower than those reported for seven prior dynamical solvers.","tokens_in":20550,"tokens_out":695,"duration_ms":7251,"significance":"If the numerical claims hold, Cosm supplies the first heuristic solutions that match Gurobi-certified optima on three long-standing large Gset spin-glass instances and multi-order-of-magnitude reductions in time-to-target on two hard non-lattice benchmarks. The combination of edge-colored switched dynamics, non-smooth finite-magnitude interactions, and a simple correlated perturbation is a concrete algorithmic contribution that is shown by ablation to be load-bearing. The work is carefully scoped to sparse bounded-degree graphs (explicitly stated in Sec. V-B), reports large trial counts, success probabilities, R99, STS and wall-clock TTT, and posts solution bitstrings for independent verification. These elements make the empirical advance reproducible and useful both for combinatorial optimization practice and for the design of future dynamical solvers.","major_comments":[],"minor_comments":[{"comment":"A short public reference implementation (or at least the precise C parameter files used for the Gset runs) would strengthen reproducibility; the manuscript already supplies bitstrings and detailed parameter tables, so this is a modest addition.","section":null},{"comment":"Fig. 9 caption and surrounding text should state more explicitly that the seven comparison exponents were extracted from the arXiv version of Hou et al. and are therefore approximate; the Cosm confidence interval is already shown.","section":null},{"comment":"In Sec. II-A the definition of the biased signum (Eq. 4) and the selected generalized gradient (Eq. 5) could be cross-referenced more clearly to the piecewise-linear potential of Eq. 2 so that readers immediately see the non-differentiable points.","section":null},{"comment":"Appendix B lists the number of edges per color for G61/G70; a one-sentence remark that the greedy coloring already attains the chromatic index would help readers who might otherwise re-run a more expensive coloring routine.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is a strong empirical algorithmic contribution that sits comfortably in the journal’s scope. The only residual concern is the absence of public code, which is common in this literature but still worth a gentle editorial nudge. No novelty or citation-pattern issues appear."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The headline results are solid. Cosm is the first heuristic to hit the Gurobi-certified optima on G72/G77/G81 (cuts 7008/9940/14060) and it drops wall-clock time-to-target for the best-known cuts on G61 and G70 from hundreds of hours to 303 s and 36 s. Those numbers are new, independently checkable via the posted bitstrings, and backed by large trial counts plus a clean ablation.\n\nWhat is actually new is the combination: circular variables, deterministic cyclic updates over edge-colored matchings (SCS), finite-magnitude sgn interactions that never go soft, and the dual-window-twist correlated perturbation. None of the usual SBM/OIM/CIM/Monte-Carlo baselines put those pieces together. The ablation on G70 shows both SCS and DWT are load-bearing; remove either and average cut collapses. Scaling on the tile-planted suite is also cleaner than the seven dynamical solvers Hou et al. compared, with a lower exponent under the same protocol.\n\nSoft spots are real but proportionate. The method is explicitly limited to sparse, bounded-degree graphs; once degree rises the intra-sweep rotations become large and the circular representation breaks. That is stated in Sec. V-B, not hidden. No public code is a moderate reproducibility gap, though the pseudocode, coloring heuristic, and parameter tables are detailed enough for re-implementation. Free parameters exist (α0, DWT period/ratio, bisectors) but they are not fitted post-hoc to invent the optima. Citation pattern is normal for the subfield; self-cites are earlier arXiv notes that simply document the same solutions.\n\nThis is for people who care about classical dynamical solvers for sparse Ising/Max-Cut and for anyone who needs strong classical baselines before claiming quantum or hardware advantage. The math is elementary, the data are careful, and the central claims hold inside the stated regime. I would send it to peer review without hesitation and would cite the Gset numbers myself.","headline":"Cosm delivers the first heuristic optima on the three largest Gset lattices (Gurobi-certified) and multi-order TTT cuts on G61/G70; the algorithmic mix is real and the sparsity limit is stated honestly.","tokens_in":21129,"tokens_out":521,"would_cite":true,"duration_ms":5698,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"A switched dynamical heuristic finds certified optima on the largest Gset spin-glass Ising problems and cuts time-to-target from hours to minutes on sparse graphs.","keywords":["Ising optimization","Max-Cut","switched dynamical systems","circular variables","sparse graphs","collective computation","edge coloring","dual window twist"],"falsifier":"Run Cosm on a sequence of the same lattice family while systematically raising average degree (or on dense Erdős–Rényi graphs of comparable size) and measure whether success probability and scaling exponent degrade sharply once typical intra-sweep angular displacement approaches 180 degrees.","tokens_in":21257,"feed_emoji":"⚡","tokens_out":614,"duration_ms":6289,"temperature":0.7,"pith_summary":"The paper introduces Cosm, a dynamical-system heuristic that treats Ising variables as continuous angles on a circle and updates them through a periodic sequence of conflict-free edge matchings together with occasional coordinated twists of opposing phase windows. The switching dynamics keep interactions finite-magnitude and non-smooth so that collective fluctuations never freeze, while the dual-window twists encourage whole clusters of variables to rotate together toward lower-energy configurations. On the three largest Gset lattice spin glasses (10 000–20 000 variables) Cosm reaches the optimal cuts for the first time; an exact solver later certifies those cuts. On two large irregular bounded-degree graphs it reaches the previous best-known solutions in tens to hundreds of seconds rather than hundreds of hours. On a family of tile-planted lattices with controlled hardness its median time-to-solution scales more gently than seven earlier dynamical solvers. The authors argue that the combination of local switched interactions and global cluster motion is what lets the method traverse the rugged energy landscapes that defeat ordinary gradient or annealing approaches on sparse graphs.","feed_headline":"Heuristic finds certified optima on largest Gset spin glasses","feed_subtitle":"Switched circular dynamics cut times-to-target from hours to minutes on sparse Ising graphs","key_machinery":"Sequential Conflict-free Search (SCS): the edges are colored into matchings and the system is advanced one matching at a time, producing a periodically switched interaction network whose finite-magnitude signum updates sustain structured intra-sweep fluctuations and prevent force cancellation.","core_discovery":"Cosm’s periodically switched, edge-partitioned dynamics plus dual-window twist perturbations enable a continuous circular-variable system to discover certified optimal solutions on the three largest Gset 2-D spin-glass instances and to reach best-known cuts on large bounded-degree non-lattice graphs orders of magnitude faster than prior heuristics.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Cosm finds certified optima on largest Gset spin glasses","Switched dynamics hit optima on 20k-var Ising graphs first","Cosm cuts sparse Ising times-to-target from hours to minutes","Collective switched motion solves huge Gset instances","Periodic edge switching yields Gset optima and faster cuts"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The method works only while the graph stays sparse and bounded-degree so that no variable rotates by roughly 180 degrees or more inside a single full sweep of color classes; denser graphs break that bound and the circular representation collapses.","fun_headline_variants_meta":{"raw":{"variants":["Cosm finds certified optima on largest Gset spin glasses","Switched dynamics hit optima on 20k-var Ising graphs first","Cosm cuts sparse Ising times-to-target from hours to minutes","Collective switched motion solves huge Gset instances","Periodic edge switching yields Gset optima and faster cuts"]},"model":"grok-4.5","effort":"low","cost_usd":0.004776,"raw_usage":{"total_tokens":1366,"prompt_tokens":759,"num_sources_used":0,"completion_tokens":69,"cost_in_usd_ticks":47760000,"prompt_tokens_details":{"text_tokens":759,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":538,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":759,"tokens_out":69,"duration_ms":4306,"temperature":1.0,"reasoning_tokens":538,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T19:43:14.463246+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Run Cosm on a sequence of the same lattice family while systematically raising average degree (or on dense Erdős–Rényi graphs of comparable size) and measure whether success probability and scaling exponent degrade sharply once typical intra-sweep angular displacement approaches 180 degrees.","supporting_citations":[],"review_version":1}