{"id":"48f48f91-8e81-4c78-a440-7748086575d3","arxiv_id":"2505.08329","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"For single particles, reducing Galilei or Poincaré symmetry to maximal proper subgroups such as very special relativity permits non-trivial velocity-dependent accelerations, which the paper derives explicitly.","lead":"This paper classifies which velocity-dependent forces are allowed for a single classical particle when spacetime symmetry is reduced to subgroups of the Galilei or Poincaré group, including very special relativity. It shows that some non-trivial interactions survive under these reduced symmetries, unlike the full relativistic case.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"VSR Poincaré solution in §3.2.1 is unproved and internally inconsistent: at β=1 it fails to reproduce the §3.2.2 'most special' solution unless A3 is corrected from (v3−β−1)(v3−β)^2F to (v3−β)^3F.","rationale":"The paper's Galilei analysis is explicit and verifiable, and the homogeneous-Poincaré no-go proof is complete. The only point on which the headline claim depends without supporting calculation is §3.2.1. The reader flagged this; stress-testing sharpens it to a concrete internal contradiction at β=1. This is load-bearing because §3.2.2's most-special solution is supposed to be a special case of the VSR family, and as printed it is not. The likely cause is a sign typo in A_3, and if so the physics claim may survive; therefore the right verdict is not rejection but a firm condition: provide the derivation and correct the displayed formula. This leaves the reader's CONDITIONAL verdict unchanged.","tokens_in":7192,"tokens_out":25786,"duration_ms":240104,"concrete_test":"Independently compute the anomaly conditions for L_1=K_1−βJ_2 and L_2=K_2+βJ_1 from eqs. (2.8)-(2.9), substitute the proposed ansatz, and solve the resulting PDE system. Minimal check: set β=1 and require the §3.2.1 family to contain the §3.2.2 solution; solve for F(s) from A_μ and test A_3. As printed the test fails. Stronger check: solve the full system for generic β; if only the free solution exists, the VSR interaction claim would not hold, while if the corrected family solves it, the issue is typographical and expository.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.2.1 states without derivation that the most general VSR-compatible acceleration is A_μ = v_μ (v_3−β)^2 F((v_3−β)/sqrt(1−v^2)), A_3 = (v_3−β−1)(v_3−β)^2 F(...). This displayed family is internally inconsistent with the β=1 limit computed in §3.2.2. Put B=v_3−1, q=sqrt(1−v^2), s=B/q. The §3.2.1 formula gives A_μ=v_μ B^2 F(s), A_3=(v_3−2)B^2 F(s). The 'most special' solution in §3.2.2 is A_μ=g v_μ q^3/B, A_3=g q^3. Matching A_μ forces F(s)=g/s^3=g q^3/B^3; then the printed A_3 becomes g q^3 (v_3−2)/(v_3−1), which equals g q^3 only if (v_3−2)/(v_3−1)=1, never true. Hence the β=1 solution explicitly given later is not contained in the purported general family. If the '−1' is a typo and A_3 should be (v_3−β)^3 F, the β=1 embedding works; but Section 3.2.1 still contains no derivation, no PDE analogue of (3.2)-(3.3), and no uniqueness proof. Because the paper's VSR existence claim for Poincaré is precisely this formula, the central claim rests on an unverified displayed solution.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies classical one-particle dynamics under subgroups of the Galilei and Poincaré groups using the world-line-condition (WLC) realization of symmetry generators on the tangent bundle of configuration space. After reproducing the standard no-interaction theorems for the full Galilei and Poincaré groups, it analyzes maximal proper subgroups: for Galilei, the static, very special (VSR), and anisotropic subgroups; for Poincaré, the VSR and 'most special' subgroups. The paper claims that VSR-type reductions admit nontrivial velocity-dependent accelerations for a single particle, while the homogeneous Galilei subgroup allows arbitrary accelerations and the homogeneous Lorentz subgroup forces vanishing acceleration. The main new quantitative claim is the general VSR Poincaré acceleration family in Section 3.2.1 and its beta=1 'most special' limit in Section 3.2.2.","tokens_in":7582,"tokens_out":5051,"duration_ms":47201,"significance":"The WLC framework is clean, and the Galilei calculations are explicit and checkable. If the VSR Poincaré result is correct, the paper establishes an interesting extension of the no-interaction theorems: reducing Poincaré symmetry to a maximal proper subgroup can restore interaction freedom for a single particle. The contrast between homogeneous Galilei (arbitrary acceleration) and homogeneous Lorentz (zero acceleration) is clearly argued. However, the VSR Poincaré central formula is the main new ingredient for relativistic VSR, and it is currently asserted without derivation and is internally inconsistent with the later 'most special' limit; the paper's significance is therefore conditional on repair of Section 3.2.1.","major_comments":[{"comment":"The displayed general solution in Section 3.2.1 is not consistent with the beta=1 solution given in Section 3.2.2. Let B=v3-1 and q=sqrt(1-v^2). Matching the A_mu components of the Section 3.2.2 solution forces F(s)=g/s^3 with s=B/q; substituting into the printed A3 in Section 3.2.1 gives A3=g q^3 (v3-2)/(v3-1), which equals g q^3 only if v3-2=v3-1. Thus the 'most special' solution is not contained in the purported general family, and the two central equations of the paper contradict each other. This must be corrected and the corrected family verified.","section":"§3.2.1 and §3.2.2"},{"comment":"No derivation is given for the claim that the displayed family is the most general VSR Poincaré-compatible acceleration. The Galilei case in Section 3.1.2 is justified by explicit anomaly equations, but the Poincaré case jumps from the algebra to the final formula. Since this family is the paper's central existence result for VSR one-particle dynamics, a derivation, or at least a complete verification of the anomaly-cancellation conditions and a uniqueness proof, is necessary; as written, the reader cannot check the claim.","section":"§3.2.1"}],"minor_comments":[{"comment":"There are several typographical errors, including 'analize' in the abstract, 'Ponciaré' in Section 2, and 'thís' in Section 2.1; a careful proofreading pass is needed.","section":"Abstract and §2"},{"comment":"The displayed solution reads 'F (vec v^2 + (v3-beta))2)', which should presumably be F(vec v^2 + (v3-beta)^2); the parentheses and exponent are misplaced.","section":"§3.1.2"},{"comment":"The argument of F and the denominators involve sqrt(1-v^2) and v3-1; the domain of validity (e.g., v3 != 1, 1-v^2 > 0) should be stated explicitly.","section":"§3.2.1 and §3.2.2"},{"comment":"The notation A_mu with mu=1,2 followed by A3 mixes Greek and spatial indices; using A_i throughout would avoid confusion, and 'suplemented' should be 'supplemented'.","section":"§3.2.2"}],"recommendation":"major_revision","confidential_remarks":"The reader's conditional verdict is appropriate. The Galilei part is solid and publishable, but the Poincaré VSR section is the headline result and it currently fails an internal consistency check. I would ask the authors to redo Section 3.2.1 carefully; if the corrected family is verified with a full derivation, the paper can likely be accepted after revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: the paper is a genuine extension of the classic no-interaction theorems to reduced symmetry groups, and the Galilei-side classification is solid. The Poincaré VSR part, which is the headline, is not proven: the general solution in §3.2.1 is asserted without derivation, and it is internally inconsistent with the β=1 solution in §3.2.2 unless you correct a sign. So the central VSR existence claim currently rests on an unverified displayed formula.\n\nWhat is new: the WLC framework and VSR concept are prior work, but the classification of one-particle dynamics under the maximal proper subgroups of Galilei (static, very special, anisotropic) and of Poincaré (VSR, most special), plus the homogeneous subgroup results, is new. The Galilei very special case is worked out carefully: equations (3.2)–(3.3), the reduction to f′ = 0, and the general solution via W. That part is clean and convincing. The β→0 static limit and the β→∞ anisotropic limit check out. The homogeneous Poincaré result—Lorentz algebra forces zero acceleration—is a nice capstone, and the contrast with arbitrary Galilei homogeneous accelerations is physically sensible.\n\nSoft spots: the Poincaré VSR §3.2.1 gives a family of accelerations without showing the differential equations or a uniqueness argument. More seriously, the displayed A3 contains (v3−β−1), and at β=1 it does not reproduce the most special solution of §3.2.2. The stress-test note does the arithmetic: matching A_μ forces F = g q^3/B^3, which makes A3 = g q^3 (v3−2)/(v3−1), never equal to g q^3. If the −1 is a typo and the factor should be (v3−β), the β=1 limit works, but the derivation is still missing. Since the paper's existence claim for VSR-compatible Poincaré interactions is exactly this formula, this needs to be fixed before the result can be trusted.\n\nThe citation pattern is fine; the framework citations to earlier work are appropriate. No curve-fitting or circularity concerns.\n\nRecommendation: worth a serious referee, but only after the authors supply the missing derivation and correct the VSR formulas. For now, treat the Galilei side as reliable and the Poincaré VSR side as conjectural. If you're teaching or writing on VSR, cite the Galilei classification, not the Poincaré formula.","headline":"The Galilei-side classification is solid and checkable; the Poincaré VSR general solution is unproved and internally inconsistent at β=1, so the paper's central VSR existence claim needs a derivation and a fix before it can be trusted.","tokens_in":8059,"tokens_out":2376,"would_cite":false,"duration_ms":22221,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83A05","70S10","70H40"],"pacs":[],"model":"deepseek-v4-flash","headline":"Reducing Poincaré or Galilei symmetry to maximal proper subgroups, especially the very special relativity subgroups, permits non-trivial velocity-dependent accelerations for a single classical particle, so the no-interaction theorem no…","keywords":["no-interaction theorems","very special relativity","world line conditions","Lorentz invariance violation","classical particle dynamics","symmetry subgroups","Galilei group","Poincaré group"],"falsifier":"Substitute the claimed VSR Poincaré accelerations into the anomaly conditions $X_i A_l + v_l A_i=0$ and into the commutator anomalies for $K_1-\\beta J_2$ and $K_2+\\beta J_1$; if the equations admit solutions other than the stated one-function family, or if the displayed family fails to close the algebra for generic $F$, then the central claim is false.","tokens_in":7000,"feed_emoji":"⚛️","tokens_out":10195,"duration_ms":97661,"temperature":0.7,"pith_summary":"The paper asks whether the classical no-interaction theorem, which says that point particles cannot interact while preserving full Galilei or Poincaré symmetry, survives when the symmetry is reduced to maximal proper subgroups. It establishes that it does not: in very special relativity (VSR) subgroups, a single particle can have velocity-dependent accelerations, with explicit families parameterized by arbitrary functions. The result matters because VSR theories are a candidate description of Lorentz-invariance violation, and it shows that such theories can support non-trivial particle dynamics without invoking additional degrees of freedom. The same world-line-condition analysis also reveals an asymmetry between homogeneous Galilean and homogeneous Lorentz symmetries: acceleration is unconstrained in the former and forced to zero in the latter.","feed_headline":"Single particles can accelerate when relativity is 'very special'","feed_subtitle":"Trimming the symmetry group to maximal subgroups evades the no-interaction theorem and leaves room for Lorentz-violating forces.","key_machinery":"The central machinery is the world-line-condition (WLC) realization of space-time symmetries: the symmetry generators are written as vector fields on the tangent bundle of configuration space, with the time-translation generator $H$ containing the unknown acceleration $A^{(a)}(v,x)$. Requiring these vector fields to close under the Galilei or Poincaré algebra turns each Lie bracket into a differential equation for the acceleration, and the anomalous terms in those brackets are what force the no-interaction conclusions. For reduced symmetry, the paper studies maximal proper subgroups, especially the very special relativity subgroups generated by translations, time translation, a pair of boost-rotation combinations, and one rotation, and derives the differential equations that the remaining freedom must satisfy.","core_discovery":"On the paper's own terms, the central discovery is that the vanishing interaction forced by the no-interaction theorem is an artifact of using the full Galilei or Poincaré algebra. Replacing these by maximal proper subgroups, especially the one-parameter VSR families, leaves enough room for non-trivial single-particle accelerations. Concretely, for the VSR Poincaré subgroup the paper finds a general acceleration family of the form $A_\\mu = v_\\mu (v_3-\\beta)^2 F\\!\\left(\\frac{v_3-\\beta}{\\sqrt{1-\\vec v^{\\,2}-v_3^2}}\\right)$ for $\\mu=1,2$ and $A_3=(v_3-\\beta-1)(v_3-\\beta)^2 F\\!\\left(\\frac{v_3-\\beta}{\\sqrt{1-\\vec v^{\\,2}-v_3^2}}\\right)$, with $F$ an arbitrary function; for the most-special eight-dimensional subgroup the only solution is $A_\\mu = g\\, v_\\mu \\frac{(1-\\vec v^{\\,2}-v_3^2)^{3/2}}{v_3-1}$ and $A_3=g(1-\\vec v^{\\,2}-v_3^2)^{3/2}$. In the Galilean version, VSR allows accelerations that are gradients of an arbitrary function of $\\vec v^{\\,2}+(v_3-\\beta)^2$, while the anisotropic subgroup yields constant acceleration along the $x_3$ axis. The same world-line-condition machinery shows that for homogeneous Galilei subgroups the acceleration is unconstrained, whereas for the homogeneous Lorentz subgroup it must vanish.","pith_inferences":["Editorial inference: The arbitrary function $F$ in the VSR Poincaré family means that VSR phenomenology would need to constrain a whole function rather than a single Lorentz-violating coefficient, a qualitatively different task for experiments.","Editorial inference: Because the allowed accelerations single out the $x_3$ axis through $v_3-\\beta$, the framework predicts direction-dependent dynamics; one could search for axis-dependent anomalous accelerations in datasets already used to test Lorentz-invariance violation.","Editorial inference: The paper does not ask whether the admissible accelerations come from a Lagrangian or Hamiltonian; a natural next check is whether these velocity-dependent forces satisfy the inverse-problem conditions of the calculus of variations, which would decide whether a variational description exists.","Editorial inference: The homogeneous-subgroup contrast suggests a classification programme: for any subalgebra of the Poincaré or Galilei algebra, determine whether the anomaly equations admit non-zero solutions; the VSR, anisotropic, and homogeneous cases are the first entries of such a classification."],"forward_implications":["For VSR Poincaré and VSR Galilei one-particle systems, non-trivial velocity-dependent accelerations exist, so the no-interaction theorem does not apply to these Lorentz-violating symmetry reductions.","For the most-special Poincaré subgroup, the allowed acceleration is unique up to a constant and has a fixed functional form in $v_3$ and $v^2$, so a VSR-compatible force law is fully determined up to that constant.","In the anisotropic Galilean case the only allowed acceleration is constant along the $x_3$ axis, reproducing uniform-force motion such as parabolic trajectories.","For homogeneous, translation-free subgroups, Galilei invariance imposes no restriction on the acceleration while Lorentz invariance forces it to vanish, showing that the time-shifting character of Poincaré boosts is decisive.","The multiparticle Galilean analysis admits interactions through functions of relative positions and velocities, whereas the authors expect the full Poincaré non-interaction theorem to persist for multiparticle systems."],"supporting_citations":[{"why":"Supplies the original no-interaction theorem that defines the baseline being generalized.","marker":"[1]"},{"why":"Provides the Lagrangian-formulation proof of no-interaction, cited as an alternative route the WLC approach parallels.","marker":"[9]"},{"why":"Further develops the no-interaction result in the same geometric and Lagrangian style.","marker":"[10]"},{"why":"Gives the world-line-condition realization of symmetry generators as vector fields, which is the paper's main machinery.","marker":"[11]"},{"why":"Defines the very special relativity subgroups whose one-particle dynamics are the focus of the paper.","marker":"[13]"},{"why":"Provides the Galilei no-interaction result that the Galilean part of the analysis extends.","marker":"[15]"}],"fun_headline_variants":["No-interaction theorem dodged in very special relativity","VSR subgroups let single particles feel forces","Maximal subgroups rescue particle interactions","Acceleration allowed when Poincaré is pared down","Very special relativity opens door to single-particle forces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument's load-bearing premise is that the formula displayed in Section 3.2.1 is the general solution of the VSR Poincaré anomaly equations; the derivation is summarized as \"after some work,\" so if that calculation is wrong or incomplete, the paper's main positive VSR result is unsupported.","fun_headline_variants_meta":{"raw":{"variants":["No-interaction theorem dodged in very special relativity","VSR subgroups let single particles feel forces","Maximal subgroups rescue particle interactions","Acceleration allowed when Poincaré is pared down","Very special relativity opens door to single-particle forces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000141,"raw_usage":{"total_tokens":1167,"prompt_tokens":952,"completion_tokens":215,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":568,"completion_tokens_details":{"reasoning_tokens":146}},"tokens_in":568,"tokens_out":215,"duration_ms":2842,"temperature":1.0,"reasoning_tokens":146,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:58:56.303533+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Substitute the claimed VSR Poincaré accelerations into the anomaly conditions $X_i A_l + v_l A_i=0$ and into the commutator anomalies for $K_1-\\beta J_2$ and $K_2+\\beta J_1$; if the equations admit solutions other than the stated one-function family, or if the displayed family fails to close the algebra for generic $F$, then the central claim is false.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the original no-interaction theorem that defines the baseline being generalized."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Lagrangian-formulation proof of no-interaction, cited as an alternative route the WLC approach parallels."},{"cited_title":"Marmo, N","cited_arxiv_id":null,"evidence_quote":"Further develops the no-interaction result in the same geometric and Lagrangian style."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the world-line-condition realization of symmetry generators as vector fields, which is the paper's main machinery."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the very special relativity subgroups whose one-particle dynamics are the focus of the paper."},{"cited_title":"Sen Gupta Nuov","cited_arxiv_id":null,"evidence_quote":"Provides the Galilei no-interaction result that the Galilean part of the analysis extends."}],"review_version":1}