{"id":"f8b4c390-9dc6-418d-82d0-58cfed863ab4","arxiv_id":"2505.05246","paper_version":5,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"A reciprocal Hamiltonian with mirror-coupled auxiliary degrees of freedom reproduces arbitrary pairwise nonreciprocal dynamics, enabling constrained Glauber Monte Carlo and Floquet engineering for nonreciprocal systems.","lead":"This paper shows how to reproduce nonreciprocal interactions, where forces are not equal and opposite, as the constrained motion of a larger reciprocal Hamiltonian system. The construction gives researchers a new route to run Monte Carlo simulations and apply Hamiltonian engineering to systems like active particles and flocking models.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Generality claim overreaches: the proof assumes one-way-potential forces (Eq. A4), while several SM examples use non-gradient forces and omit constraint-preservation checks.","rationale":"The reader correctly identified Eq. (A4) as the weakest assumption, and I agree that the scope of the general proof is narrower than the paper's opening claim. However, the concern is more concrete than a mere scope caveat: the paper's own examples include forces that violate Eq. (A4), yet those examples are presented with different-looking Hamiltonians for which neither the symplectic structure nor the preservation of the constraint is verified. The spin-model construction and the Monte-Carlo equivalence are internally consistent and well supported, so I do not see a reason to reject the paper. But the claim that the framework applies to the five experimental systems in Appendix B is not backed by the supplied proof. This warrants a conditional acceptance: the authors should either prove the constraint-preservation property for the B-section embeddings (or show they follow from the Appendix-A construction), or explicitly restrict the abstract and introduction to the class of forces covered by Eq. (A4). The proposed test—checking whether d(r_i+x_i)/dt vanishes for the sedimenting Hamiltonian—would settle whether the advertised example actually works.","tokens_in":54805,"tokens_out":25575,"duration_ms":256373,"concrete_test":"Using the canonical bracket of the overdamped embedding (Sec. C1, where the original and auxiliary variables are conjugate), write the Hamilton equation for the auxiliary coordinate x_i in the sedimenting-particles Hamiltonian (Eq. B34), and evaluate d(r_i+x_i)/dt on the constraint manifold r_i+x_i=0. If this quantity does not vanish identically for a generic pair of particles, the embedding does not preserve the constraint and fails to reproduce the original dynamics. As a supplementary algebraic check, compute ∇×[G(r)·g] for the Oseen tensor; a nonzero curl confirms that no one-way potential U exists, so the Appendix-A proof cannot cover this advertised example.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim (Sec. I and the abstract) promises a Hamiltonian embedding for arbitrary pairwise nonreciprocal equations of motion. The general proof in Appendix A1, however, starts from Eq. (A4), which restricts every nonreciprocal force to be the gradient of a one-way potential U_{i→j}(r_i−r_j). That is a genuine restriction in 2D/3D: for example, the Oseen-tensor force in the sedimenting-particles example (Eqs. B32–B33) has a nonzero curl for a fixed settling direction, so no scalar U exists. The paper nevertheless presents a Hamiltonian for this system (Eq. B34) that uses the force directly, and similarly for Janus particles (Eq. B6). For these examples the auxiliary equations of motion are not given and the preservation of the constraint r_i+x_i=0 is not demonstrated; the constraint-preservation proof is provided only for the spin model and for the U-based construction in Appendix A. Hence the advertised generality—and the validity of the experimental examples—is not established by the arguments supplied.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes a Hamiltonian embedding for nonreciprocal pairwise dynamics. An auxiliary copy of each degree of freedom is introduced, coupled reciprocally to the original variables, and a mirror constraint (e.g., θ_i−φ_i=π or r_i+x_i=0) is imposed. On the constraint manifold, Hamilton's equations are claimed to reproduce the original nonreciprocal equations of motion. The construction is worked out explicitly for vision-cone XY spins and chase-and-run XY spins; constrained Glauber dynamics based on the embedding is shown analytically and numerically to produce the same Fokker-Planck dynamics as the original Langevin equation, including nonstationary oscillatory states. The symplectic structure is then used to derive a Floquet-Magnus effective Hamiltonian for periodically driven nonreciprocal spins, predicting a Bessel-function-controlled dimensional crossover that is checked against simulations. The Supplemental Material contains the general embedding proof for velocity-independent one-way-potential forces, a velocity-dependent extension, and candidate embeddings for five experimental systems.","tokens_in":55024,"tokens_out":13195,"duration_ms":136722,"significance":"Should the construction hold at the advertised level of generality, it would give a principled way to transfer tools of equilibrium and Hamiltonian mechanics—Monte Carlo sampling, canonical transformations, and Floquet engineering—to nonreciprocal systems, provided the constraint submanifold remains invariant. The paper is transparent and detailed: constraint preservation is proved for the spin embeddings in Appendix C, the Glauber-Langevin equivalence is derived from independent Fokker-Planck equations in the Methods and Appendix D, and the numerical checks (critical temperature Tc/J=0.79±0.04 for both Langevin and constrained Glauber dynamics, and matching oscillatory order parameters for chase-and-run dynamics) are direct and convincing. The paper also honestly states the pairwise restriction and the one-way-potential assumption at the end. The main weakness is that the general proof is conditional on a one-way-potential assumption that several of the paper's own examples do not satisfy; this needs to be fixed or the claims need to be restricted.","major_comments":[{"comment":"The generality claim in Sec. III (a \"general proof for arbitrary nonreciprocal interactions\") is stronger than what the proof establishes. Appendix A 1 assumes that every velocity-independent nonreciprocal force is the gradient of a one-way potential, F_{i→j}=-∂_{r_i}U_{i→j}(r_i-r_j). The sedimenting-particle force in Eqs. (B32)-(B33) does not satisfy this assumption: for fixed gravity direction the Oseen-tensor force has a nonzero curl, so no scalar one-way potential exists in three dimensions. Yet Eq. (B34) presents a Hamiltonian for this system. Either the theorem should be stated with the one-way-potential condition as an explicit hypothesis, or a separate construction and constraint-preservation proof is needed for non-gradient pairwise forces such as the Oseen case.","section":"Sec. III and Appendix A 1, Eq. (A4)"},{"comment":"For the particle-based examples the auxiliary equations of motion and constraint-preservation checks are not supplied, so these examples do not yet substantiate the \"wide applicability\" claim. For example, in the Janus-particle embedding, Eqs. (B6)-(B10), the constraints r_i+x_i=0 and n_i+m_i=0 are imposed after writing the equations for r_i and n_i, but no equations for dx_i/dt and dm_i/dt are given. With the usual choice dx_i/dt=-∂H/∂x_i, the position constraint is not preserved once x_i=-r_i; the residual is proportional to v0 n_i. If an opposite sign convention or a different bracket structure is intended, it must be stated and justified; otherwise the claimed Hamiltonian character of these examples is not established. The same omission affects the self-aligning and feedback-controlled examples in Eqs. (B13) and (B27).","section":"Appendix B 2–B 6"}],"minor_comments":[{"comment":"The text says the comparison is between \"constrained Glauber dynamics (blue) vs. Langevin dynamics (green)\", while the Fig. 3 legend identifies Langevin with red squares; the color assignments should be aligned.","section":"Sec. IV A and Fig. 3"},{"comment":"For the periodically driven Hamiltonian H(t) and the effective Floquet Hamiltonian H_F^(0), the paper should state explicitly that the mirror constraint θ_i-φ_i=π is preserved under the time-dependent dynamics; this is a direct check analogous to Appendix C but is not written out.","section":"Sec. V, Eq. (9)"},{"comment":"Equation (A43) writes the nonreciprocal force as F_{j→i}=-∂_{r_i}U^NR_{j→i}, but the preceding generalized-potential formalism, Eq. (A27), includes the additional term d/dt (∂_{\\dot r_i}U^NR_{j→i}); please clarify whether (A43) is a definition, a special case, or a typographical omission.","section":"Appendix A 2 c, Eq. (A43)"},{"comment":"Reference [2] is cited as \"arXiv:2602.11111 (2026)\"; the arXiv identifier and date appear inconsistent and should be verified.","section":"Reference [2]"}],"recommendation":"major_revision","confidential_remarks":"The core spin-model demonstration is sound and the numerical comparisons are convincing; my hesitation concerns the scope of the generality claim, not the central method. A revised version that either proves the embeddings for the stated experimental examples or restricts the claims to forces satisfying Eq. (A4) would resolve the issue. Rejection would be too harsh because the XY-spin results and the conditional general theorem are self-contained and valuable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, the core construction is real: symmetrizing the nonreciprocal interaction and canceling one side with an antiparallel auxiliary copy gives a Hamiltonian whose constrained dynamics reproduces the original equations, for forces that can be written as gradients of one-way potentials (Eq. A4). The proof of constraint preservation in Appendix A is explicit, and the Fokker-Planck derivation matching constrained Glauber dynamics to Langevin dynamics is careful. Second, the paper's headline claim—any equation of motion with pairwise nonreciprocal forces—is not what the proof delivers. The stress-test note is right: the general construction assumes F_{i→j} = -∂_{r_i} U_{i→j}(r_i-r_j), which excludes forces with nonzero curl. The sedimenting-particle example (Eqs. B32–B34) uses the Oseen tensor, which for fixed settling direction is not a gradient; no scalar U exists. The authors still write down a Hamiltonian for it, but they never give the auxiliary equations of motion or show the constraint is preserved. Same for the Janus-particle example. So the generality claim and those two applications are not established.\n\nWhat the paper does well: for the vision-cone XY spins and the chase-and-run model, the construction is worked out in full, the constraint is shown to be preserved, and the numerical match between Langevin and constrained Glauber dynamics—both Tc and the oscillatory order parameter—is convincing. The Floquet engineering example is a nice bonus, and the comparison showing selfish-energy Monte Carlo gives a different Tc is a useful concrete result.\n\nSoft spots: the generality overreach is the main one. The limitation to gradient one-way potentials is not disclosed in the abstract or intro; Sec. VI says 'pairwise interactions' and 'may not apply to arbitrary non-potential systems,' which undersells the restriction. Minor issues: α(Δ) is fitted rather than derived, and the Floquet panel lacks error bars. None of that undermines the spin-model results.\n\nWho this is for: researchers working on nonreciprocal spin systems and Monte Carlo methods for nonequilibrium steady states. It deserves a serious referee, but I would not accept it as is; the authors should either restrict the claims to the gradient-potential class or supply constraint-preservation proofs for the non-gradient examples.\n\nMy recommendation: send it to review—it has a correct core and a valuable algorithm—but the review should push on the generality gap. If the authors fix that, it's a solid paper.","headline":"A clean Hamiltonian-embedding construction for a restricted class of nonreciprocal forces, but the advertised generality is not backed by the proofs.","tokens_in":55552,"tokens_out":5403,"would_cite":true,"duration_ms":52228,"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":"Nonreciprocal pairwise forces, however asymmetric, can be generated as the constrained dynamics of a reciprocal Hamiltonian.","keywords":["nonreciprocal interactions","Hamiltonian embedding","auxiliary degrees of freedom","mirror constraint","Langevin dynamics","Monte Carlo simulations","Floquet engineering","XY spins with vision cones"],"falsifier":"Take a pair of particles whose nonreciprocal force has a nonzero curl in the relative coordinate, so no one-way potential of the assumed form exists, and run the proposed constrained Hamiltonian dynamics. If the trajectories differ from the target nonreciprocal equation of motion, or if the mirror constraint is not preserved for all time, the central claim is false.","tokens_in":54611,"feed_emoji":"🌀","tokens_out":7550,"duration_ms":71500,"temperature":0.7,"pith_summary":"This paper claims that any equation of motion whose nonreciprocal forces are pairwise can be rewritten as the constrained dynamics of a fully reciprocal Hamiltonian. The trick is to add one auxiliary degree of freedom per original one, couple the two layers symmetrically, and impose a 'mirror' constraint that locks each pair together. When the constraint holds at the initial time, it is preserved by Hamilton's equations, and each original degree of freedom evolves exactly as in the nonreciprocal system. If true, this gives nonreciprocal systems access to Monte Carlo sampling, canonical transformations, and periodic-drive engineering that were previously reserved for systems with an energy function.","feed_headline":"Nonreciprocal forces hide a Hamiltonian","feed_subtitle":"Adding one mirror degree of freedom per particle recovers nonreciprocal dynamics and opens Monte Carlo and Floquet tools.","key_machinery":"The central object is the Hamiltonian embedding: a doubled system in which each original degree of freedom $\\theta_i$ is paired with an auxiliary 'mirror' degree of freedom $\\varphi_i$ that is its canonical conjugate, with a symmetrized interaction built from the one-way potentials $U_{i\\to j}$ and $U_{j\\to i}$ plus auxiliary couplings that cancel one direction of each bond. The mirror constraint $\\theta_i-\\varphi_i=\\pi$ (or $\\mathbf r_i=\\mathbf x_i$ with opposite momenta in the inertial case) selects the physical submanifold; the construction works because the constraint surface is invariant under the dynamics, so the nonreciprocal system is the constrained projection of a reciprocal Hamiltonian flow.","core_discovery":"Starting from pairwise nonreciprocal forces $F_{i\\to j}(\\mathbf r_i-\\mathbf r_j)=-\\partial_{\\mathbf r_i}U_{i\\to j}(\\mathbf r_i-\\mathbf r_j)$, the paper constructs a reciprocal Hamiltonian on a doubled configuration space: the original degrees of freedom plus a mirrored copy, coupled so that the unwanted half of each symmetric interaction is cancelled by the auxiliary layer. Hamilton's equations with the constraint $\\theta_i-\\varphi_i=\\pi$ (for spins, with the general form $\\mathbf r_i=\\mathbf x_i$ and opposite momenta in the inertial case) reproduce the original nonreciprocal equations of motion exactly, and the constraint is dynamically preserved. The paper further shows that single-spin-flip Monte Carlo updates computed from the constrained Hamiltonian yield the same Fokker-Planck equation as the original Langevin dynamics, so stationary and time-dependent nonequilibrium states coincide, and that the symplectic structure supports a high-frequency expansion for periodic drives.","pith_inferences":["Inference: any invariant of the reciprocal embedding that restricts to the constraint manifold is automatically an invariant of the nonreciprocal dynamics, making the construction a potential source of conserved quantities for nonreciprocal systems.","Inference: if the embedding is exact for pairwise forces, the nonreciprocal phase space can be viewed as a constrained submanifold of a Hamiltonian phase space, which suggests that quantizing the embedding might produce a well-defined quantum theory for at least some nonreciprocal models.","Inference: applying the same constrained single-spin-flip rule to other nonreciprocal models, such as the soft vision-cone interaction studied in previous work, should reproduce the Langevin steady state for all measured observables; any deviation would pinpoint where the small-proposal-width expansion breaks down."],"forward_implications":["Constrained single-spin-flip Monte Carlo dynamics built from the embedding reproduces both steady and nonstationary states of the original Langevin dynamics, so Monte Carlo methods can be used on nonreciprocal systems.","For vision-cone XY spins, the constrained Monte Carlo and Langevin simulations give the same transition temperature $T_c/J=0.79\\pm0.04$, whereas the selfish-energy approach gives a different value.","The symplectic structure allows periodic-drive engineering; a drive whose amplitude tunes a Bessel-function zero can suppress interactions in one lattice direction, turning a square lattice into a set of one-dimensional chains.","The embedding applies to inertial as well as dissipative pairwise nonreciprocal dynamics, covering systems as varied as Janus colloids, walker robots, robotic metamaterials, feedback-controlled particles, and sedimenting particles.","Because the constraint manifold is closed and of measure zero in the embedding phase space, the nonreciprocal dynamics is fully contained inside a Hamiltonian flow and does not affect expectation values defined on the full phase space."],"supporting_citations":[{"why":"Establishes that nonreciprocal forces lack a conventional interaction potential, the problem the embedding solves.","marker":"[30]"},{"why":"Supplies the single-spin-flip Monte Carlo transition rates that the paper adapts to the constrained embedding.","marker":"[57]"},{"why":"Prior construction of Hamiltonians for nonconservative forces by doubling degrees of freedom, here generalized with an initial-time constraint.","marker":"[58]"},{"why":"Supplies the high-frequency expansion used to derive the effective Hamiltonian for periodically driven nonreciprocal spins.","marker":"[46]"},{"why":"Defines the vision-cone XY model and the selfish-energy Monte Carlo baseline whose critical behavior is compared with the constrained dynamics.","marker":"[33]"},{"why":"Early use of auxiliary variables to describe dissipation in a Lagrangian, the lineage the constraint construction extends.","marker":"[69]"}],"fun_headline_variants":["Mirror particles reveal a Hamiltonian for nonreciprocal forces","Nonreciprocal systems hidden Hamiltonian found via auxiliary degrees","Adding a mirror copy makes nonreciprocal forces Hamiltonian","A constrained Hamiltonian captures nonreciprocal dynamics exactly","Mirror degrees of freedom unlock Hamiltonian tools for nonreciprocal interactions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction assumes each nonreciprocal pairwise force can be written as the gradient of a one-way potential $U_{i\\to j}(\\mathbf r_i-\\mathbf r_j)$; forces that are not of this gradient form, such as those with a curl component, are outside its scope.","fun_headline_variants_meta":{"raw":{"variants":["Mirror particles reveal a Hamiltonian for nonreciprocal forces","Nonreciprocal systems hidden Hamiltonian found via auxiliary degrees","Adding a mirror copy makes nonreciprocal forces Hamiltonian","A constrained Hamiltonian captures nonreciprocal dynamics exactly","Mirror degrees of freedom unlock Hamiltonian tools for nonreciprocal interactions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000295,"raw_usage":{"total_tokens":1708,"prompt_tokens":930,"completion_tokens":778,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":546,"completion_tokens_details":{"reasoning_tokens":699}},"tokens_in":546,"tokens_out":778,"duration_ms":7996,"temperature":1.0,"reasoning_tokens":699,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:08:57.243965+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a pair of particles whose nonreciprocal force has a nonzero curl in the relative coordinate, so no one-way potential of the assumed form exists, and run the proposed constrained Hamiltonian dynamics. If the trajectories differ from the target nonreciprocal equation of motion, or if the mirror constraint is not preserved for all time, the central claim is false.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes that nonreciprocal forces lack a conventional interaction potential, the problem the embedding solves."}],"review_version":1}