{"id":"618a88c6-0541-499e-b21a-fbffd6728929","arxiv_id":"2506.14024","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"A globally rescaled Lagrangian is on-shell a total derivative of a unique pre-symplectic expression, and this identity yields generalized Komar charges for three gravity-matter theories.","lead":"This paper proves that any Lagrangian which rescales by a fixed weight under a global field rescaling can be written as a total derivative when evaluated on solutions of the equations of motion. The identity gives a systematic route to generalized Komar charges, the surface integrals used to compute black hole masses and thermodynamic potentials.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The reader's ACCEPT verdict is well founded. The main theorem (2.7) follows from the standard variational formula, and the on-shell exactness is immediate. Section 3 gives a compact but sufficient derivation of (3.17) from Noether identities and the Killing condition, so the alleged 'inherited exactness' is not an unexamined assumption. The examples are internally consistent and reproduce known results, providing independent support for the construction. Minor notational issues—the terse index convention in (2.10) and the apparent typo in (3.24b)—are cosmetic and do not change the verdict. Accordingly the verdict should remain ACCEPT/UNCHANGED.","tokens_in":12123,"tokens_out":36420,"duration_ms":349314,"concrete_test":"Independently verify Eq. (2.7) for a higher-derivative scalar toy model, L = (1/2)(□ϕ)^2 in flat space, by direct integration by parts: compute Θ(φ, δϕ) for δϕ = (1/2)ϕ and check that dΘ(φ, ϕ/2) equals L − (1/2)E_ϕ exactly off-shell, and that L = dJ_0 once E_ϕ = 0 is imposed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim—that a Lagrangian homogeneous under a global field rescaling is on-shell exact via Eq. (2.7)—is correctly derived from the variational identity (2.2). The derivation handles derivative dependence because the rescaling weights are constant. The reader's flagged exactness (3.17) is not an unproved extra assumption: Section 3 obtains it from Noether's second identity (3.3), the Killing condition δ_k φ ≑ 0, and local exactness of the closed form B(k, φ). Examples 4–6 carry through consistently. The only genuine weakness is presentational: Eq. (2.10) leaves implicit the index convention with α_0 = 1 and different meanings of ω^n_i for n = 0 versus n > 0, and Eq. (3.24b) appears to contain a typographical '=0'; neither affects the validity of the construction.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a field-theoretic analogue of Euler's homogeneous-function theorem: if a Lagrangian d-form L is homogeneous under a global rescaling of the fields with weights ω_i and transforms with weight ω_L, then L = (ω_i/ω_L) E_i∧φ_i + dΘ(...), so that on-shell L is an exact form with a solution-independent representative (Eq. 2.7). The paper then combines this identity with the generalized Komar charge algorithm of Refs. [12–16] to construct Komar charges in three examples: Einstein–Maxwell–Dilaton theory, the bosonic sector of minimal 5-dimensional supergravity, and a self-interacting scalar field coupled to gravity (using a Lagrange-multiplier extension for the non-linear potential). It also discusses the non-uniqueness of the total-derivative representative coming from additional scaling symmetries.","tokens_in":12310,"tokens_out":31467,"duration_ms":270998,"significance":"The central identity in Section 2 is clean, self-contained, and parameter-free; it follows directly from the variational identity and does not depend on the earlier Komar-charge algorithm. The application sections show that this identity supplies the needed on-shell total-derivative form in three non-trivial theories and reproduce known generalized Komar charges, which is a valuable consistency check. The Lagrange-multiplier trick in Section 6 extends the method to non-linear potentials. If correct, the paper provides a systematic and practical tool for constructing Smarr formulas and generalized Komar charges.","major_comments":[],"minor_comments":[{"comment":"The sign in Eq. (2.8) appears incorrect: from Eq. (2.6) with ω_L = 0 one obtains dJ_n = -ω^n_i E_i∧φ_i, as used in Eq. (4.11), not dJ_n = +ω^n_i E_i∧φ_i.","section":"Section 2, Eq. (2.8)"},{"comment":"The trailing '= 0' in Eq. (3.24b) is inconsistent with the definition of the generalized Komar charge and with the non-trivial results in the subsequent examples; it should be removed, or replaced by a statement that dK[k] ≑ 0.","section":"Section 3, Eq. (3.24b)"},{"comment":"The placement of the parentheses in Eq. (4.22) gives the αQ_k term a factor of 1/2; the derivation via Eq. (4.21a) gives the term +αQ_k outside the factor 1/2, matching Eq. (4.23).","section":"Section 4, Eq. (4.22)"},{"comment":"The sign of τ(χ,φ) in Eq. (5.10) appears inconsistent with a direct explicit computation of δχJ0 from J0 = (1/3)⋆G∧V using Eq. (5.3b); please check and correct the sign.","section":"Section 5, Eqs. (5.9)–(5.10)"},{"comment":"The on-shell equivalence of the extended action (6.3) to the original theory is asserted but not proven; a short proof (integrating out C and g, or a brief citation plus explanation) would make the example self-contained.","section":"Section 6, Eq. (6.3)"},{"comment":"There are several small typos: 'generalized generalized Komar charges' before Eq. (4.23), 'a a matter of fact' after Eq. (2.10), 'Let as assume' at the start of Section 3, and 'equtions' in Section 3.1.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The paper is largely consistent with the authors' previous line of work and relies on Refs. [12–16] for the Komar-charge algorithm; this is appropriate given the scope, but a short self-contained justification of Eq. (3.17) would strengthen the manuscript. The numerous small sign and typographical errors in the equations should be corrected carefully before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know about 2506.14024: the engine is a two-line variational identity that is correct and genuinely useful, and the paper is worth referee time. The central result, Eq. (2.7), is a field-theoretic Euler theorem: for a Lagrangian that scales homogeneously under global field rescalings, the on-shell Lagrangian is the exterior derivative of the pre-symplectic potential evaluated at the scaling variation, L ≑ dJ0. No fitting, no hidden assumptions; it is a parameter-free consequence of the standard variational identity. The genuinely new part is not the identity itself — it is morally a scale-symmetry Noether current — but the systematic use of it inside the Ortín program for constructing generalized Komar charges, the non-uniqueness analysis (the α-family), and the coupling-promotion trick that extends the method to a non-linear scalar theory. The three worked examples all reproduce known charges from earlier work, which is the right kind of grounding: the method gives familiar answers before it gives new ones.\n\nThe soft spots are minor. The reader flagged Eq. (3.17) as an inherited assumption; on reading Section 3, that is too harsh. The exactness of ı_k L + B(Λ_k, φ) on-shell follows from the Noether second identity (3.3) together with the Killing condition δ_k φ ≑ 0, with the usual tacit Poincaré lemma for local exactness. What is genuinely under-discussed is the global side: the construction is local, and topological obstructions are never mentioned. The Lagrange-multiplier equivalence in Section 6 is asserted rather than checked in detail; it is a standard trick from their earlier work and it is plausible, but a referee should ask for the equations spelled out. Presentationally, Eq. (2.10) leaves the α_0 = 1 and ω^n_i conventions implicit, and Eq. (3.24b) carries a stray '= 0'. The prose has small typos ('a a matter of fact', 'the its definition'). None of this touches the main derivation.\n\nThe citation pattern is self-heavy, which is normal for a group mining its own program; the external anchors (Komar, Magnon, Pacilio) are real and the examples are independently reproducible.\n\nWho this is for: specialists in black hole thermodynamics and conserved-charge technology. It deserves a serious referee and, I think, acceptance after minor revisions. I would not desk-reject it.","headline":"A clean field-theoretic Euler identity that powers a worked Komar-charge construction; minor presentational gaps only, so it deserves a serious referee.","tokens_in":12808,"tokens_out":6802,"would_cite":true,"duration_ms":59798,"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":"A homogeneous Lagrangian is on-shell an exact form, and this identity builds generalized Komar charges.","keywords":["Komar charge","Noether–Wald charge","on-shell Lagrangian","Euler theorem for homogeneous functions","Smarr formula","black hole thermodynamics","momentum maps","quasi-homogeneous Lagrangians"],"falsifier":"Take the proposed generalized Komar charge for Einstein–Maxwell–Dilaton theory (Eq. (4.23)) and evaluate its exterior derivative on a known non-extremal black-hole solution; if $dK[k]$ does not vanish on-shell, or if the horizon integral differs from the integral at infinity, the exactness assumption is false. A simpler algebraic check is to substitute arbitrary field configurations into Eq. (2.7) and verify the identity term by term; any counterexample would falsify the Euler-theorem claim.","tokens_in":11937,"feed_emoji":"🕳️","tokens_out":9370,"duration_ms":82125,"temperature":0.7,"pith_summary":"The paper proves a functional version of Euler's theorem for homogeneous functions: if the fields rescale with weights $\\omega_i$ and the Lagrangian rescales with weight $\\omega_L$, then $L=\\frac{\\omega_i}{\\omega_L}E_i\\wedge\\phi^i+d\\Theta(\\phi,\\frac{\\omega}{\\omega_L}\\phi)$ holds identically, with $E_i$ the Euler–Lagrange forms and $\\Theta$ the pre-symplectic potential, the boundary term generated by varying the action. On solutions, $E_i=0$, so the Lagrangian is the total derivative of a universal expression that depends only on the fields and the fixed weights, not on the particular solution. Because the generalized Komar construction needs the on-shell pullback of the Lagrangian by a Killing vector to be exact, this identity supplies exactly the boundary term needed to close the charge. The authors demonstrate the method on Einstein–Maxwell–Dilaton gravity, minimal five-dimensional supergravity, and a self-interacting scalar, where nonlinearity is handled by promoting the coupling constant to a constrained field. The result is a systematic route to conserved surface charges of the form needed for Smarr formulas in black-hole thermodynamics.","feed_headline":"Euler theorem turns on-shell Lagrangians into boundary terms","feed_subtitle":"Universal boundary terms from the identity produce generalized Komar charges and Smarr formulas.","key_machinery":"The load-bearing identity is the functional Euler theorem (Eq. (2.7)): $L=\\frac{\\omega_i}{\\omega_L}E_i\\wedge\\phi^i+d\\Theta(\\phi,\\frac{\\omega}{\\omega_L}\\phi)$, where $E_i$ are the Euler–Lagrange forms and $\\Theta$ is the pre-symplectic potential, the boundary term generated by an arbitrary field variation. On solutions it becomes $L\\doteq dJ_0$ with $J_0=\\Theta(\\phi,\\frac{\\omega}{\\omega_L}\\phi)$. This identity is the engine of the paper: it turns the on-shell Lagrangian into a total derivative without solving the field equations, and when combined with the generalized-Komar requirement $i_k L+B(\\Lambda_k,\\phi)\\doteq d\\omega(k,\\phi)$, it yields the exact $(d-2)$-form $\\omega(k,\\phi)$ and hence the closed charge $K[k]$.","core_discovery":"The central claim is that for any Lagrangian $d$-form that transforms homogeneously under a global transformation of the fields, the Lagrangian can be written identically as a sum of the equations of motion contracted with the fields plus an exact piece: $L=\\frac{\\omega_i}{\\omega_L}E_i\\wedge\\phi^i+d\\Theta(\\phi,\\frac{\\omega}{\\omega_L}\\phi)$. On-shell this reduces to $L\\doteq dJ_0$, where $J_0=\\Theta(\\phi,\\frac{\\omega}{\\omega_L}\\phi)$ is solution-independent. The paper shows that $J_0$ is unique only up to conserved currents $J_n$ associated with scaling symmetries of the theory, and that the same Euler-theorem logic works for any global transformation with nonzero weight, not just rescalings. It then plugs this identity into the generalized Komar algorithm: given a Killing vector $k$, the on-shell interior product $i_k L$ becomes an exact $(d-2)$-form, and the resulting closed charge $K[k]$ reproduces and organises the known charges of Einstein–Maxwell–Dilaton theory and minimal $d=5$ supergravity, and gives a potential-times-charge form for the self-interacting scalar.","pith_inferences":["The paper does not pursue it, but Eq. (2.7) also gives a shortcut for Euclidean on-shell actions: the boundary term is read off from the pre-symplectic potential before solving equations, which could simplify free-energy computations for stationary black holes.","Because the identity is algebraic in the fields, it should extend to higher-curvature or Chern–Simons theories; testing it on Lovelock gravity or generalized Proca models would be a direct next step.","Applying the coupling-promotion trick to a cosmological constant rather than a scalar coupling should yield a Komar charge whose boundary integral is the pressure–volume term of extended black-hole thermodynamics."],"forward_implications":["In any theory with a homogeneous rescaling of the fields, the action evaluated on solutions is a pure boundary term, $S_{\\mathrm{on-shell}}=\\int dJ_0$.","For every Killing vector $k$, the on-shell interior product $i_k L$ is exact, so the generalized Komar charge $K[k]$ is closed on-shell and satisfies a Gauss law.","The ambiguity in $J_0$ by conserved currents is physical: in Einstein–Maxwell–Dilaton theory, adding $\\alpha J_1$ shifts the Komar charge by a term proportional to the conserved scalar charge $Q_k$, while leaving Smarr formulas unchanged when the scalar-charge Gauss law is used.","Promoting a dimensionful coupling constant to a constrained scalar field extends the Euler-theorem identity to nonlinear Lagrangians, as demonstrated for a self-interacting scalar; the authors state that the same trick applies to nonlinear electrodynamics coupled to gravity."],"supporting_citations":[{"why":"Supplies the algorithm that produces a non-trivial $\\omega(k,\\phi)$ and the commutator formula $dK[k]=[i_k,\\mathcal{O}_s]L$ used throughout the paper.","marker":"[16]"},{"why":"Establishes the generalized Komar framework for higher-order curvature theories and the role of dimensionful coupling constants in black-hole chemistry.","marker":"[12]"},{"why":"Motivates promoting coupling constants to constrained scalar fields so that they contribute to Komar charges and Smarr formulas.","marker":"[14]"},{"why":"Provides the magnetic momentum-map technique used to compute the magnetic potential and charge terms in the examples.","marker":"[15]"},{"why":"Supplies the treatment of compensating gauge transformations and momentum maps for the first law in Einstein–Maxwell theory.","marker":"[24]"},{"why":"Gives the earlier four-dimensional self-interacting scalar example whose result is reproduced and rewritten in potential-times-charge form.","marker":"[17]"},{"why":"Gives the conserved 2-form charge $Q_k$ for the dilaton that appears in the Einstein–Maxwell–Dilaton Komar charge and Smarr relations.","marker":"[28]"},{"why":"Underlies the discussion of electric-magnetic duality and the choice $\\alpha=0$ in the Einstein–Maxwell–Dilaton example.","marker":"[13]"}],"fun_headline_variants":["Euler's identity makes Lagrangians exact on-shell","Solution-independent boundary terms from Euler's theorem","Generalized Komar charges from Euler's theorem identity","Universal boundary terms from homogeneous Lagrangians"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that a symmetry that leaves all fields fixed also makes the on-shell Lagrangian pullback exact: there must always exist some $(d-2)$-form $\\omega(k,\\phi)$ with $i_k L+B(\\Lambda_k,\\phi)\\doteq d\\omega(k,\\phi)$. The paper takes this exactness from the earlier generalized-Komar framework rather than proving it; if it fails, the constructed charge is not conserved on-shell and the Gauss law behind Smarr formulas is lost.","fun_headline_variants_meta":{"raw":{"variants":["Euler's identity makes Lagrangians exact on-shell","Solution-independent boundary terms from Euler's theorem","Generalized Komar charges from Euler's theorem identity","Universal boundary terms from homogeneous Lagrangians"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000741,"raw_usage":{"total_tokens":3258,"prompt_tokens":846,"completion_tokens":2412,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":462,"completion_tokens_details":{"reasoning_tokens":2352}},"tokens_in":462,"tokens_out":2412,"duration_ms":18299,"temperature":1.0,"reasoning_tokens":2352,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:56:24.944228+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the proposed generalized Komar charge for Einstein–Maxwell–Dilaton theory (Eq. (4.23)) and evaluate its exterior derivative on a known non-extremal black-hole solution; if $dK[k]$ does not vanish on-shell, or if the horizon integral differs from the integral at infinity, the exactness assumption is false. A simpler algebraic check is to substitute arbitrary field configurations into Eq. (2.7) and verify the identity term by term; any counterexample would falsify the Euler-theorem claim.","supporting_citations":[],"review_version":2}