{"id":"d90da85a-6338-4a31-8539-d863640e373b","arxiv_id":"1908.08232","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A[G]-equivalence generalizes Mather's right-left equivalence to G-structures, and it coincides with G-congruence exactly when the G-preserving infinitesimal vector fields are linear.","lead":"This mathematics paper introduces a general way to compare smooth maps when the target space is equipped with a geometric structure such as a metric or volume form. The same framework covers several known classification schemes and shows when they reduce to ordinary rigid congruence.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 7.1's proof is incomplete: the identification T_1 Diff[G]=θ[G]_0 is shown in only one direction, and the 'same method' proof of Diff_0[G]=G contains a non-sequitur; both steps are needed to conclude A0[G]-equivalence equals G-congruence.","rationale":"The reader's verdict is CONDITIONAL and we agree. Our pass found that the reverse-inclusion gap they flagged is real but repairable via the Lie-group ODE argument: any η with Dη∈g integrates to a flow with Jacobian in G. A more specific flaw is the 'same method' argument: the inference from linearity of η to dh_t/dt=0 is invalid, since η is linear in the target coordinate φ_{t0}(y), not in the source coordinate y. The correct route through the right logarithmic derivative and the linear ODE dφ_t/dt=B(t)φ_t repairs the proof. Both gaps affect the written proof of the central equivalence but not the likely truth of the statement, so no change to the CONDITIONAL verdict is needed.","tokens_in":25436,"tokens_out":25505,"duration_ms":242110,"concrete_test":"Re-derive the proof of Theorem 7.1 (2)=>(3) using the right logarithmic derivative v_t = dφ_t/dt∘φ_t^{-1} for an arbitrary path in Diff[G]: show Dv_t∈g from d/dt Jφ_t = Dv_t(φ_t)Jφ_t and T_A G=gA, then under condition (2) conclude v_t(y)=B(t)y, so dφ_t/dt=B(t)φ_t. Apply this to G=SO(3) with a nonconstant B(t)∈so(3) and verify that all solutions of the ODE are rotations, confirming Diff_0[SO(3)]=SO(3). If this ODE argument cannot be completed for some connected G satisfying condition (2), the proof of Theorem 7.1 is not secure.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central equivalence (Cor. 7.2) rests on Theorem 7.1's conclusion Diff_0[G](p)=G. Two steps in the proof are not justified as written. First, Section 4 asserts T_1 Diff[G](p)=θ[G]_0(p) after proving only the forward inclusion. The reverse inclusion, that every germ η with η(0)=0 and Dη(y)∈g for all y arises from a curve in Diff[G](p), requires integrating η and showing Jφ_t∈G. This is true: Jφ_t solves d/dt Jφ_t = Dη(φ_t)Jφ_t, a linear ODE on G since Dη∈g; but the paper does not give this argument. Second, the proof of Theorem 7.1 (2)=>(3), deferred to 'the same method' as Theorem 4.6, contains a non-sequitur. For a curve φ_t(y)=A(t)y+h_t(y), the paper writes dφ_t/dt = η∘φ_{t0} and, because η has linear components, concludes dh_t/dt|_{t0}=0. This does not follow: η is linear in the target coordinate φ_{t0}(y), so dφ_t/dt|_{t0}(y)=B(t0)(A(t0)y+h_t0(y)); the higher-order part satisfies h'_t0 = B(t0)h_t0, not h'_t0=0. A correct proof uses the right logarithmic derivative v_t=dφ_t/dt∘φ_t^{-1}, shows Dv_t∈g, and then condition (2) forces v_t(y)=B(t)y; hence dφ_t/dt=B(t)φ_t and φ_t∈G. Without these repairs, Cor. 7.2 is not established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes A[G]-equivalence for smooth map germs (R^n,0)→(R^p,0), where G⊂GL(p,R) is a linear Lie group viewed as a G-structure on the target; A[GL(p,R)] recovers classical A-equivalence and A[{I_p}] recovers R-equivalence. It introduces the space θ[G](p) of vector-field germs whose Jacobian lies in the Lie algebra g, the algebra E_p[G] of functions compatible with θ[G]_0(p), and defines infinitesimal tangent spaces for A[G]- and R×G-equivalences. The paper computes these objects for SO(p), SL(p,R), Sp(2,R), block-diagonal and block-triangular groups, and relates them to existing classifications (isometric A-equivalence, unimodular geometry, bi-A-equivalence, Lagrangian equivalence). The main theorem (Thm. 7.1) asserts that, for connected linear G, three conditions are equivalent: (1) g≅θ[G]_0(p) as vector spaces, (2) every vector field in θ[G]_0(p) has linear components, and (3) Diff_0[G](p)=G; Corollary 7.2 concludes that under these conditions A0[G]-equivalence coincides with G-congruence and the relative infinitesimal moduli space vanishes.","tokens_in":25767,"tokens_out":10329,"duration_ms":91347,"significance":"If Theorem 7.1 is established, the paper gives a clean algebraic criterion for when the newly introduced A[G]-geometry collapses to classical G-geometry, and it explains why for SO(p) the two coincide while for SL(p,R) they differ. The examples connect the framework to substantial existing work (Domitrz–Rieger, Dufour, Ishikawa–Janeczko, Lagrangian singularity theory) and the paper explicitly flags when A[G] is or is not a geometric subgroup in Damon's sense. The manuscript contains a number of correct and useful computations (e.g., the finite-dimensionality of θ[SO(p)]_0 and the identification of θ[SL(p,R)] with exact (p−1)-forms). However, the proof of the central theorem has two real gaps — the unproved reverse inclusion in the identification T_1 Diff[G](p)=θ[G]_0(p) and a non-sequitur in the proof of (2)⇒(3) — so the central claim is not yet established as written.","major_comments":[{"comment":"The equality T_{1_{R^p}} Diff[G](p) = θ[G]_0(p) is asserted after proving only the inclusion 'tangent vectors lie in θ[G]_0(p)'. The reverse inclusion — that every germ η with η(0)=0 and Dη(y)∈g for all y is tangent to a curve in Diff[G](p) — is not shown. This matters because Theorem 7.1(3)⇒(1) and the identification of A[G]-infinitesimal data with g(f) in Corollary 7.2 both rely on this equality. The missing argument is standard: integrate the time-dependent vector field η to get a flow φ_t with φ_0=id; then D_y φ_t solves X'(t)=Dη(φ_t(y))X(t), X(0)=I, and since Dη(φ_t(y))∈g and G is a connected Lie subgroup of GL(p,R) with Lie algebra g, the solution X(t) lies in G for all t. Please add this proof.","section":"§4"},{"comment":"In the proof of Theorem 4.6, the step 'so that (dh_t/dt)|_{t=t0}(y)=0' is a non-sequitur. If dφ_t/dt|_{t=t0} = η∘φ_{t0} with η(y)=B(t0)y linear, and φ_t(y)=A(t)y+h_t(y), then comparing linear and higher-order parts gives A'(t0)=B(t0)A(t0) and h'_{t0}(y)=B(t0)h_{t0}(y), not h'_{t0}=0. The conclusion φ_t∈SO(p) still follows because h_0=0 and the linear ODE h'=B h has the unique solution h≡0, but the proof must be corrected. The same flawed step is invoked in Theorem 7.1 for the implication (2)⇒(3) via 'the same method'. A cleaner proof uses the right logarithmic derivative v_t=dφ_t/dt∘φ_t^{-1}; then D v_t∈g, condition (2) forces v_t(y)=B(t)y, and hence dφ_t/dt=B(t)φ_t, so φ_t∈G.","section":"§4 and §7, Theorem 4.6 and Theorem 7.1"},{"comment":"The implication (3)⇒(1) is dismissed with the phrase 'If we consider the formal tangent space of Diff_0[G](p)=G, we can easily show...'. This is load-bearing and should be written out: from Diff_0[G](p)=G one obtains T_1 Diff_0[G](p)=g (embedded in θ(p) by linear vector fields X y·∂/∂y), and combining this with the equality T_1 Diff[G](p)=θ[G]_0(p) (whose proof is incomplete; see the first comment) gives θ[G]_0(p)≅g. Please provide the details, including the identification of the tangent space of Diff_0[G](p) at the identity.","section":"§7, proof of Theorem 7.1"}],"minor_comments":[{"comment":"The definition of A[G]-equivalence uses f∘φ=ψ∘g, while A-equivalence was defined by ψ∘f=g∘φ. Since φ and ψ range over all diffeomorphisms the two conventions are equivalent, but the reversal should be flagged for the reader.","section":"§2"},{"comment":"The notation θ[N]_0(p)=M_pθ(π_{p1}) is ambiguous; it should be clarified whether the module is over E_{p1} and what the subscript p denotes.","section":"Example 4.10(2)"},{"comment":"The entry '(x1x1x2+x4_2+Q)' appears to contain a typo; it should probably read '(x1,x1x2+x4_2+Q)'.","section":"Proposition 6.9"},{"comment":"The assertion that condition (2) of Theorem 7.1 holds for G=SO(p1,p2) is stated without proof; either give the proof or mark it as conjectural.","section":"After Corollary 7.2"},{"comment":"The phrase 'η_i(y) are linear function germs' for SO(p) relies on Example 4.4; for the benefit of the reader, the argument that θ[SO(p)]_0 is spanned by the angular vector fields should be restated briefly.","section":"Theorem 4.6"}],"recommendation":"major_revision","confidential_remarks":"The paper is essentially a survey with a new organizing framework; the computations in the examples are mostly sound and the connection to existing classifications is useful. The proof of the central Theorem 7.1 is incomplete in two places that are directly load-bearing for Corollary 7.2. Both gaps are repairable by standard ODE arguments, so major revision rather than rejection seems appropriate. The authors should also double-check the index convention in the embedding ι in §7 and the unproved claim about SO(p1,p2) after Corollary 7.2."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nYou asked for a read on arXiv:1908.08232. The paper is worth a serious look. It defines A[G]-equivalence for map germs into R^p with a linear Lie group G, develops the infinitesimal algebra (theta[G]_0, E_p[G]), and proves a clean structural result (Theorem 7.1) saying that under a linearity condition on theta[G]_0, the identity component Diff_0[G](p) is exactly G, so A0[G]-equivalence reduces to classical G-congruence. That gives one framework for Euclidean congruence (G=SO(p)), for separating unimodular from equi-affine geometry (G=SL(p)), and for block groups like bi-A and projection equivalences. Section 6 is largely a survey of prior classifications, but that is fine: it shows the framework's scope without overclaiming. The infinitesimal computations in the examples check out, and the paper is honest that A[G] is not always a geometric subgroup in Damon's sense.\n\nThe soft spot is in the proof of Theorem 7.1, and it is more than a typo. In Theorem 4.6 (the SO(p) case), the step concluding dh_t/dt = 0 from the linearity of eta is a non-sequitur: for phi_t = A(t)y + h_t(y), the equation dphi_t/dt = eta o phi_t0 gives h'_t0 = B(t0) h_t0, not zero. The same flaw is carried into Theorem 7.1 via the 'same method' argument. The theorem is true, but the proof as written does not establish it. A working proof exists: pass to the right logarithmic derivative v_t = (dphi_t/dt) o phi_t^{-1}, observe that the G-condition implies v_t in theta[G]_0, and condition (2) then forces v_t(y)=B(t)y, so phi_t solves a linear ODE and stays in G. Also, Section 4 asserts T_1 Diff[G](p)=theta[G]_0 with only one inclusion shown; the reverse direction needs the standard integration argument for the Jacobian ODE, which is missing.\n\nThese are repairable gaps, not a dead end. The central claim holds up, and the definition of A[G]-equivalence plus Theorem 7.1 is a real contribution. The reliance on an unpublished Mather note [29] for a standard fact is mildly annoying but not wrong. No fitting or invented entities; the self-citations (e.g., reference [20], in preparation) are not load-bearing.\n\nWho reads this? Singularity theorists and differential geometers working on G-structures and normal forms. It deserves peer review, with a referee who can fill the ODE argument; this is not a desk reject. I would cite it as the standard reference for A[G]-equivalence.\n\nBest.","headline":"A useful unifying framework for A[G]-equivalence with a true but under-proved main theorem; the gap in Theorem 7.1 is standard and fixable.","tokens_in":26363,"tokens_out":5524,"would_cite":true,"duration_ms":52342,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["58K40","53C10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For connected linear Lie groups whose infinitesimal vector fields are linear, A0[G]-equivalence of map germs is exactly G-congruence, making singularity A0[G]-geometry coincide with classical G-geometry.","keywords":["G-structure","A-equivalence","singularities of map germs","A[G]-equivalence","G-congruence","infinitesimal tangent spaces","Lie group actions"],"falsifier":"For a candidate group such as G=SO(p1,p2) with p1,p2>0, compute the space θ[G]_0(p) explicitly. If it contains a vector field with a non-linear component whose Jacobian matrix still lies in so(p1,p2) at every point, then condition (2) of Theorem 7.1 fails and the theorem predicts Diff_0[G](p)≠G; finding such a vector field, or proving none exists, would settle whether the linearity criterion is the correct dividing line. A more direct refutation would be to produce a single η∈θ[G]_0(p) whose integral flow leaves Diff_0[G](p) even though condition (1) holds.","tokens_in":25190,"feed_emoji":"📐","tokens_out":11007,"duration_ms":89354,"temperature":0.7,"pith_summary":"The paper introduces A[G]-equivalence, a geometric variant of the classical right-left equivalence of smooth map germs: two germs f,g:(R^n,0)→(R^p,0) are equivalent when g can be obtained from f by a right change of coordinates in the source and a left change of coordinates in the target whose Jacobian lies in a prescribed linear Lie group G⊂GL(p,R) at every point. The central result, Theorem 7.1, is a criterion for when this new equivalence is not new at all. For a connected linear Lie group G, the following are equivalent: the Lie algebra g is isomorphic to the space θ[G]_0(p) of infinitesimal G-preserving vector fields; every vector field in θ[G]_0(p) has linear component functions; and the identity-component group Diff_0[G](p) of G-preserving diffeomorphism germs equals G itself. When any of these holds, A0[G]-equivalence coincides with G-congruence (the classical R×G-equivalence), so singularity-theoretic A0[G]-geometry is the classical G-geometry. The criterion is genuinely discriminating: it holds for the orthogonal group SO(p), making isometric A-equivalence the usual Euclidean congruence of curves and surfaces, and it fails for SL(p,R), where the equivalence is larger and carries infinite-dimensional moduli.","feed_headline":"Map-germ equivalence reduces to classical congruence","feed_subtitle":"For connected Lie groups with linear infinitesimal vector fields, A0[G]-geometry and classical G-geometry coincide.","key_machinery":"The load-bearing object is θ[G]_0(p), the space of vector field germs on (R^p,0) that vanish at the origin and whose Jacobian matrix lies in the Lie algebra g of G at every point; it is exactly the formal tangent space T_1 Diff[G](p) of the group of G-preserving diffeomorphism germs. The argument compares two formal tangent spaces at a map germ f: the A[G]-tangent space TA[G](f)=tf(M_nθ(n))+ω_f(θ[G]_0(p)) and the R×G-tangent space T(R×G)(f)=tf(M_nθ(n))+g(f), where g(f)={X∘f : X∈g}. The embedding ι(e_ij)=y_i∂/∂y_j identifies g with the linear vector fields inside θ[G]_0(p), and Theorem 7.1 uses this identification to show that linearity of θ[G]_0(p) is equivalent to Diff_0[G](p)=G. Corollary 7.2 then turns that group equality into equality of tangent spaces and of orbits, so every A0[G]-equivalence is a G-congruence.","core_discovery":"The core discovery is Theorem 7.1 together with Corollary 7.2. For a connected linear Lie group G⊂GL(p,R), let θ[G]_0(p) be the R-vector space of vector field germs η=Ση_i ∂/∂y_i on (R^p,0) whose Jacobian matrix (∂η_i/∂y_j)(y) lies in the Lie algebra g for every y and which vanish at the origin. The following are equivalent: (1) g≅θ[G]_0(p) as R-vector spaces; (2) the component functions η_i of every element of θ[G]_0(p) are linear functions; (3) Diff_0[G](p)=G, where Diff_0[G](p) is the group of diffeomorphism germs isotopic to the identity through diffeomorphisms whose Jacobians stay in G. Under any of these conditions, the formal tangent space of the A0[G]-orbit equals the tangent space of the R×G-orbit for every map germ f, and two map germs are A0[G]-equivalent exactly when they are G-congruent. Thus, precisely in this situation, A0[G]-geometry and classical G-geometry are identical.","pith_inferences":["The same infinitesimal linearity test could be applied to the indefinite orthogonal groups SO(p1,p2), which the paper mentions but does not fully analyze; deciding whether θ[G]_0(p) is linear for those groups would map out exactly where A[G]-geometry collapses into classical pseudo-Riemannian congruence.","A natural extension is to source-side G'-structures—the R[G'], A[G';G], and K[G';G] equivalences the introduction mentions but leaves aside; one would expect an analogue of Theorem 7.1 to characterize when those equivalences reduce to the corresponding classical actions on the source.","The relative moduli space M(A[G];R×G)(f) could serve as a quantitative measure of how many independent geometric invariants a singular germ carries; computing it for the known normal forms of cuspidal edges and swallowtails would connect the formalism to existing curvature calculations.","If the criterion holds broadly, singularity classification and geometric classification become the same task for the affected groups; a practical consequence is that existing lists of G-congruence normal forms can be read as complete A0[G]-classifications, and vice versa."],"forward_implications":["For G=SO(p), Euclidean curve and surface geometry—curvatures, Frenet-type invariants, Monge normal forms—is exactly the corresponding A0[SO(p)]-singularity theory; the paper makes the classical identification precise.","Whenever the theorem's conditions hold, the relative infinitesimal moduli space M(A[G];R×G)(f) vanishes for every germ f, so no functional moduli separate the two geometries.","For G=SL(p,R), the conditions fail; θ[SL(p,R)] is infinite-dimensional, A[SL] is not a geometric subgroup in the sense used in the paper, and the relative moduli space is infinite-dimensional, so volume-preserving (unimodular) classification is genuinely broader than equi-affine congruence.","For H<G with G satisfying the theorem, the quotient M(A[G];A[H])(f) equals M(R×G;R×H)(f) and its dimension is bounded by dim G−dim H; the gap between two such geometries is finite-dimensional and computable from the Lie algebras.","The paper's proposed semi-finite determinacy problem follows: when the conditions fail, finite Taylor jets can still encode all geometric invariants of interest, but the classical finite-determinacy theorems no longer apply, and the tangent-space structure described here is offered as the guide."],"supporting_citations":[{"why":"Supplies the definitions of A-equivalence and the tangent-space machinery (θ(n), θ(p), t_f, ω_f) that Section 3 adapts.","marker":"[27]"},{"why":"Defines geometric subgroups of A and K and provides the framework the paper uses to distinguish A[G] from R×G.","marker":"[8]"},{"why":"Provides the classification of volume-preserving A[SL]-simple germs that the paper interprets as the unimodular instance of A[G].","marker":"[9]"},{"why":"Provides the Frenet-type formulae and uniqueness theorem for frontals used for the A0[SO(2)] curve example.","marker":"[14]"},{"why":"Supplies the equi-affine curvature that serves as the complete invariant for R×SL(2,R)-equivalence.","marker":"[30]"},{"why":"Provides the theory of Lagrangian singularities and generating families underlying the A[L(2n)] example.","marker":"[2]"},{"why":"Supports Proposition 6.11, relating A-equivalence of the second component to A[T*_r]-equivalence of the pair.","marker":"[16]"}],"fun_headline_variants":["Map-germ geometries coincide under linear infinitesimal condition","When Lie algebra matches, map germs are G-congruent","A0[G]-geometry equals G-geometry under linear fields","Theorem: A0[G]-equiv iff G-congruent for linear Lie groups","Linear Jacobians collapse A0[G]-equivalence to G-congruence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that every vector field whose derivative matrix lies in the Lie algebra g integrates to a flow of diffeomorphism germs whose Jacobians remain in G; without that integration step, the identification T_1 Diff[G](p)=θ[G]_0(p) cannot force Diff_0[G](p)=G, and the paper proves only one direction of it.","fun_headline_variants_meta":{"raw":{"variants":["Map-germ geometries coincide under linear infinitesimal condition","When Lie algebra matches, map germs are G-congruent","A0[G]-geometry equals G-geometry under linear fields","Theorem: A0[G]-equiv iff G-congruent for linear Lie groups","Linear Jacobians collapse A0[G]-equivalence to G-congruence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000817,"raw_usage":{"total_tokens":3536,"prompt_tokens":857,"completion_tokens":2679,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":473,"completion_tokens_details":{"reasoning_tokens":2585}},"tokens_in":473,"tokens_out":2679,"duration_ms":18023,"temperature":1.0,"reasoning_tokens":2585,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:46:46.190923+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a candidate group such as G=SO(p1,p2) with p1,p2>0, compute the space θ[G]_0(p) explicitly. If it contains a vector field with a non-linear component whose Jacobian matrix still lies in so(p1,p2) at every point, then condition (2) of Theorem 7.1 fails and the theorem predicts Diff_0[G](p)≠G; finding such a vector field, or proving none exists, would settle whether the linearity criterion is the correct dividing line. A more direct refutation would be to produce a single η∈θ[G]_0(p) whose integral flow leaves Diff_0[G](p) even though condition (1) holds.","supporting_citations":[{"cited_title":"Mather, Stability of C ∞-Mappings III","cited_arxiv_id":null,"evidence_quote":"Supplies the definitions of A-equivalence and the tangent-space machinery (θ(n), θ(p), t_f, ω_f) that Section 3 adapts."},{"cited_title":"Damon, The unfolding and determinacy theorems for sub groups ofA andK","cited_arxiv_id":null,"evidence_quote":"Defines geometric subgroups of A and K and provides the framework the paper uses to distinguish A[G] from R×G."},{"cited_title":"Domitrz and J","cited_arxiv_id":null,"evidence_quote":"Provides the classification of volume-preserving A[SL]-simple germs that the paper interprets as the unimodular instance of A[G]."},{"cited_title":"Fukunaga and M","cited_arxiv_id":null,"evidence_quote":"Provides the Frenet-type formulae and uniqueness theorem for frontals used for the A0[SO(2)] curve example."},{"cited_title":"Nomizu and T","cited_arxiv_id":null,"evidence_quote":"Supplies the equi-affine curvature that serves as the complete invariant for R×SL(2,R)-equivalence."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the theory of Lagrangian singularities and generating families underlying the A[L(2n)] example."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supports Proposition 6.11, relating A-equivalence of the second component to A[T*_r]-equivalence of the pair."}],"review_version":1}