{"id":"41e26cc3-0326-4266-b02e-f72d82f20aef","arxiv_id":"2610.01976","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Magnetic Carrollian gravity is recast into dynamically equivalent Electric Carrollian, fracton-like tensor gauge, Maxwell electric, and dilaton theories by trading curvature for torsion and non-metricity.","lead":"This paper builds a Carrollian analog of the geometric trinity of gravity, showing that the curvature of Magnetic Carrollian gravity can be rewritten as kinetic terms of torsion or non-metricity. It maps the magnetic sector to Electric Carrollian gravity, fracton-like tensor gauge theories, Maxwell electric fields, and a dilaton scalar, forming a dictionary between Carrollian gravity and gauge and condensed matter theories.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"The equivalences rest on hand-picked gauge fixings with free parameters (α at (4.47), λ at (5.60)); Sec. 6 admits these need Lagrange-multiplier enforcement for genuine dynamical equivalence, never performed.","rationale":"I read the central claim as asserting that MCG curvature can be dynamically traded, through flat modified connections, for isolated kinetic sectors (ECG, spatial/temporal/clock torsion, non-metricity). The internal tensor algebra is consistent: the general metric-compatible connection (4.14), the torsion decomposition (4.27), and the curvature shifts (4.42, 4.60, 4.67, 4.81, 4.103, 5.58) follow through without apparent inconsistency, and the cancellations (for instance, the antisymmetric F-terms dropping from the scalar curvature in 4.59) are legitimate contractions. The load-bearing weakness is the status of the equivalences themselves. Each rests on a gauge fixing of the unconstrained ambiguity Xμν (or Υ) at a specific parameter value — X(μν) = αKμν with α set to −1 in theory I, Υ = λK with free λ in theory V — combined with an imposed global-flatness constraint. Because the geometry does not force these fixings, they are inputs, not derived choices; reproducing the known ECG action requires tuning α to −1, and the STECG action inherits the unspecified λ. The paper honestly acknowledges in Sec. 6 that these constraints and fixings must be enforced with Lagrange multipliers for genuine dynamical equivalence, and that this enforcement is not carried out. My concrete concern, which sharpens rather than replaces the reader's weakest_assumption, is that the decisive open question is whether that enforced variational system is multiplier-inert and admits non-trivial solutions with Kμν ≠ 0 and R̃ ≠ 0 on the two sides; if not, the equivalences reduce to on-shell identities at tuned constants, and the 'dynamical' language in (4.61) and (5.61) overstates the result. I agree with the reader's identification of the weakest assumption and with the CONDITIONAL verdict: the tensor algebra is defensible, and the paper flags its own gaps (the fracton and Maxwell analogies as lacking the required gauge symmetry, b311/b313/b326, and the multiplier gap in Sec. 6), so the residual risk is precisely whether the un-performed enforcement can close those gaps. My test is a bounded symbolic computation on the explicit system (4.45)–(4.61), which would settle whether the equivalence is dynamical or merely algebraic.","tokens_in":42414,"tokens_out":8676,"duration_ms":58372,"concrete_test":"Carry out the constrained variation for theory I that Sec. 6 calls for but does not perform. Write the full first-order action S = ∫e[ R̃ + Λ^{(μν)}(X(μν) − αKμν) + (Lagrange multipliers enforcing Riemann(ˇΓ)=0) ] with α = −1 in (4.47)–(4.61), vary all fields (h, τ, v, X, and the multipliers), and extract the complete Euler–Lagrange system. Then check two properties: (i) do the multiplier equations for the constraint X(μν) = αKμν reduce to algebraic identities (constraint dynamically inert), or do they impose genuine relations that force Kμν = 0 or R̃ = 0 on all solutions? (ii) do non-trivial solutions with Kμν ≠ 0 and R̃ ≠ 0 exist, reproducing the ECG kinetic term in the flat sector so that (4.61) is a genuine duality rather than a trivialization?","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that MCG curvature R̃ of a rigid, torsion-free Carrollian universe is dynamically traded for isolated torsion/non-metricity kinetic sectors. The mechanism in each case is identical: relax rigidity or metric-compatibility, compute the shifted Ricci scalar (e.g., 4.60: ˇR = R̃ + α(KμνKμν − K²)), impose global flatness (ˇR = 0), and read off an isolated kinetic term. The step that makes each identification precise is a hand-picked gauge fixing of the unconstrained clock-form ambiguity: X(μν) = αKμν with α tuned to −1 in theory I (4.47, 4.61), and Υ = λK with λ completely free in theory V (5.60, 5.61). These fixings are not forced by the geometry: Xμν is arbitrary subject only to Xμνv^ν = 0, and Υμν retains all traceless freedom. Consequently each equivalence matches a known action only at a special parameter value. The paper's own Sec. 6 concedes that for these to be true dynamical equivalences, the flatness constraints and connection-ambiguity fixings must be enforced in the action with Lagrange multipliers, and that this is not done. The internal tensor algebra (4.14, 4.27, 4.42, 4.56–4.60, 5.15, 5.55–5.58) appears consistent, and the cancellations (e.g., F-terms dropping from 4.59) are legitimate. The unresolved and load-bearing question is the status of the constrained theory: if the multipliers enforcing X(μν) = αKμν and flatness are dynamically active, the equivalence breaks off-shell and the claimed indistinguishability of a curved rigid spacetime from a flat shearing one — the headline instance (4.61) — reduces to an on-shell identity at a tuned constant rather than a dynamical duality. This is not a new finding against the authors; it is the gap they themselves flag in Sec. 6, and the work of making it precise is exactly what is missing.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"This paper extends the 'Geometrical Trinity of Gravity' paradigm to Carrollian geometry. From the most general metric-compatible Carroll connection (§4.1, Eq. (4.14)), the authors relax metric-compatibility or zero-torsion constraints, compute shifted Ricci scalars for five modified connections, and impose global flatness on each to trade the Magnetic Carrollian gravity (MCG) curvature R̃ for isolated kinetic sectors: Electric Carrollian gravity (Theory I, Eqs. (4.60)–(4.61)), a spatial-torsion tensor gauge theory (Theory II, Eq. (4.68)), a temporal-torsion dilaton (Theory III, Eq. (4.82)), a Maxwell-like electric sector (Theory IV, Eq. (4.105)), and a symmetric-teleparallel isotropic electric theory (Theory V, Eq. (5.61)). The central thesis is that a curved, rigid Carrollian universe (K_μν = 0, R̃ ≠ 0) is dynamically indistinguishable from globally flat configurations carrying isolated kinetic sectors, with some targets being non-gravitational (fracton, Maxwell, dilaton). The shifted-curvature identities and torsion decomposition (4.27) are internally consistent, but the claimed equivalences rest on hand-picked gauge fixings with free parameters ((4.47), (5.60)) and on constraint enforcement that the authors themselves concede in §6 is not carried out.","tokens_in":42933,"tokens_out":9411,"duration_ms":64667,"significance":"If the equivalences held as genuine dynamical ones, this would be a substantial contribution: a Carrollian analog of the GR trinity with five target theories, two non-gravitational, and an explicit geometric dictionary connecting magnetic curvature to gauge and scalar sectors. The algebra is a real asset: the shifted Ricci identities (4.56–4.60, 4.67, 4.81, 4.103, 5.55–5.58) and the torsion decomposition (4.27) are internally consistent and derivable, and the paper is transparent about where gauge symmetries are absent (the fracton and Kalb-Ramond identifications, §4.2.3) and where dynamical enforcement is missing (§6). The paper would be a useful toolkit for Carrollian holography and condensed-matter analogies even at the level of on-shell algebraic relations. However, the central claim as advertised — dynamical equivalence — is not yet established: the target actions arise at tuned values of free gauge constants and under a stationarity restriction (L_v τ = 0), and the constrained variations that would make the equivalences dynamical are sketched but absent. This gap is acknowledged and, in principle, closable.","major_comments":[{"comment":"The Theory I and Theory V equivalences are established only at tuned values of free gauge constants. The sole geometric constraint on the clock-form ambiguity is X_μν v^ν = 0 (§4.1); the symmetric part X_(μν) is otherwise free. The ansatz X_(μν) = αK_μν (4.47) is a postulate, and the match to the standard ECG Lagrangian ∫e(K_μνK^μν − K²) in (4.60)–(4.61) fixes α = −1 by hand. Similarly, Υ = λK (5.60) is introduced as a postulate, and the resulting STECG action (5.61) inherits the completely free parameter λ, so any λ is allowed. As written, these are algebraic identities at special parameter values rather than dynamical equivalences forced by the geometry. The paper should either derive these fixings from a geometric criterion (e.g., the embedding argument of §5.1) or explicitly rescope the claims as equivalences that hold only at tuned parameter values.","section":"§4.2.2 and §5.2.1, Eqs. (4.47), (4.60)–(4.61), (5.60)–(5.61)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The authors concede that the constraints defining each equivalence — global flatness, the torsion/non-metricity restrictions, and the fixings (4.47) and (5.60) — 'must be dynamically enforced' via Lagrange multipliers and that this is not done in the paper. This concession is the load-bearing gap: the flatness condition is precisely the mechanism that trades curvature for the kinetic sectors, so without the constrained variations one cannot determine whether the constrained theories share the solution space, symmetries, and degrees of freedom of the baseline ∫eR̃. A necessary addition is the explicit Lagrange-multiplier action for at least one representative (e.g., Theory I) with the multipliers included, followed by a comparison of the equations of motion, in particular a counting of propagating degrees of freedom before and after enforcement. Without this, 'dynamical equivalence' is an announcement rather than a demonstrated property.","section":"§6, closing paragraph","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"Combining the ansatz X_μν = αK_μν − ½F_μν with the transversality condition X_μν v^ν = 0 forces L_v τ_μ = 0, i.e., a stationary clock form with vanishing Carrollian gravitoelectric field. This is a substantive restriction on the headline duality that a curved rigid universe is indistinguishable from a globally flat expanding/shearing one: the equivalence holds only on configurations with no gravitoelectric field. The text acknowledges this under 'Stationary time', but the condition is part of the hypotheses of the result and should be stated as such in the statement of the equivalence, along with the analogous L_v φ = 0 condition in (4.96) for the dilatonic sector.","section":"§4.2.2, Eqs. (4.47)–(4.50); §4.2.4, Eq. (4.96)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The identifications with Kalb-Ramond and scalar-charge fracton theories are made at the level of kinetic terms only, and the author's own text states the required gauge symmetries are absent (δB = dΛ for the axial sector; the full fracton gauge freedom for V⊕q). The degree-of-freedom argument (9 − 1 = 8) used to match the traceless fracton magnetic field rests on discarding the axial sector by ansatz, not by a geometric constraint. This is an acknowledged limitation, but the 'dictionary' claim would be strengthened if the paper stated explicitly which geometric conditions select the V⊕q sector, or alternatively if it presented the identifications as structural analogies rather than equivalences.","section":"§4.2.3, Eqs. (4.68)–(4.72)","authors_acknowledge":"However,A still lacks the gauge symmetry required for a complete realization of the scalar-charge Fracton theory. 4.2.4 Equivalent Theory III: Temporal Torsion and Dilatonic Scaling To explore another sector of equivalent theories, in the following we will isolate the torsion terms originating from the clock formFµν and the tensorXµν while eliminating all others.","authors_acknowledge_p":0.84}],"minor_comments":[{"comment":"'stracturally similar' is misspelled in the abstract and in §1; also 'we prove that the curvature of a Magnetic Carrollian universe can be dynamically traded' overstates what §6 demonstrates, so the wording 'show' or 'argue' would better match the body of the paper.","section":"Abstract and §1","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The claim that ̃∇_μs^μ + s_μs^μ reproduces the transformation of the Ricci scalar under a conformal Weyl rescaling is asserted without giving the explicit transformation identity; adding it would let the reader verify the dilaton mapping in (4.94).","section":"§4.2.4, Eq. (4.87) and following text","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The factor-2 discrepancy between the first-order gauged Ricci scalar (A.38) and the second-order scalar (4.42) used throughout the main text is flagged but not reconciled. Since both are presented as the baseline MCG action, the paper should state which one is the operative baseline or justify the relative weight of the boost-curvature sector.","section":"Appendix A, Eq. (A.38)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The coefficients c1, c2, c3 in (4.69) are left as 'specific dimensional constants'; they are fixed numerical coefficients of the SO(d) orthogonal decomposition of A_σμν and should be computed explicitly, since the fracton and Kalb-Ramond identifications depend on their relative values.","section":"§4.2.3, Eq. (4.69) and Table 1","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The suppression of the shear term K_μνK^μν in Theory V follows directly from the trace-gauge Υ = λK, since only the trace Υ couples at (5.57); this should be flagged as a consequence of the gauge choice rather than a general property of minimal non-metricity.","section":"§5.2.1, Eqs. (5.57)–(5.61)","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"The abstract and introduction announce results stronger than what the body demonstrates: the equivalences are evaluated at tuned values of free parameters ((4.47), (5.60)) and the constraint enforcement the authors themselves identify as necessary in §6 is not performed. The two realistic paths for revision are (i) carrying out the Lagrange-multiplier analysis for at least one representative theory, demonstrating that the constrained action shares the baseline solution space, or (ii) rescoping the announced 'dynamical equivalences' to on-shell algebraic relations at tuned parameters, with the dictionary presented as structural. The paper cites Vigneron et al. [28] for the structural existence of the Carrollian trinity; the claimed novelty is the explicit dynamical Lagrangians, which is precisely where the gaps lie, so the novelty claim and the central results stand or fall together. On a fairness note, the internal tensor algebra is consistent and the authors are honest about the gauge-symmetry and constraint limitations; the assessment is a correctness-risk concern with a concrete test (write the constrained actions, compare solution spaces), not a circularity objection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"Read this one over coffee. The genuinely new thing is that the paper works intrinsically in Carrollian geometry — no c→0 limit of a Lorentzian parent — and produces explicit equivalent Lagrangians for Magnetic Carrollian gravity, which is what Refs. [27] and [28] left open. The general metric-compatible connection (4.14), the torsion decomposition (4.27), and the curvature shifts (4.56–4.60, 4.67, 4.81, 4.103, 5.55–5.58) are correct formal algebra as far as I can check; the F-term cancellations at (4.59), for instance, are legitimate. And the observation that a Carrollian trinity can exit the gravitational sector entirely — into Maxwell, fracton, and dilaton-type theories, where the Lorentzian trinity stays inside gravity — is a genuine extension of the program.\n\nThe soft spot is structural and it is the whole game. Every equivalence runs the same play: relax rigidity or metric compatibility, shift the scalar curvature, impose flatness, and read off an isolated kinetic term. Each identification leans on a hand-picked fixing of the clock-form ambiguity. Theory I locks X_{(μν)} = αK_{μν} with α tuned to match the known ECG action (4.47, 4.61); Theory V postulates Υ = λK with λ entirely free, which writes the answer ∫−λK² into the action (5.60, 5.61). Neither fixing is forced by geometry — X_{μν} is arbitrary subject to X_{μν}v^ν = 0. So the headline claim, that a curved rigid universe is dynamically indistinguishable from a flat shearing one, reduces to an on-shell identity at a tuned constant. The fracton and Maxwell mappings are kinetic analogies the authors themselves flag as lacking the required gauge symmetry (b311, b313, b326). And in Sec. 6 they concede that the flatness constraint and these gauge fixings must be enforced with Lagrange multipliers for genuine dynamical equivalence — work they do not actually do. The concession is honest; the missing enforcement is exactly the load-bearing part.\n\nBottom line: a construction worth refereeing, with claims not yet established off-shell. A referee should push either for the multiplier enforcement or for an explicit downgrade to on-shell equivalence at special parameter values. I'd bring it to reading group to argue about what \"dynamical equivalence\" should mean here, but I wouldn't cite the equivalences as established in my own work.","headline":"A genuinely intrinsic Carrollian trinity with correct curvature algebra, but the equivalences are on-shell identities at hand-picked constants, and Sec. 6 concedes the missing Lagrange-multiplier enforcement.","tokens_in":43587,"tokens_out":4354,"would_cite":false,"duration_ms":30291,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek/deepseek-v4-flash","headline":"Magnetic Carrollian gravity is shown to be dynamically equivalent to flat-space gauge and scalar theories by trading curvature for torsion or non-metricity.","keywords":["Carrollian gravity","geometric trinity","teleparallel gravity","non-metricity","torsion","electric Carroll gravity","fracton gauge theory","Maxwell electric sector"],"falsifier":"Compute the off-shell action with the flatness condition and the gauge fixing $X_{(\\mu\\nu)}=\\alpha K_{\\mu\\nu}$ enforced by Lagrange multipliers. If the resulting field equations do not reduce to those of Electric Carroll gravity for generic initial data, or if the free constant $\\alpha$ is not fixed by consistency, the claimed dynamical equivalence would fail. Equivalently, an independent Hamiltonian analysis of the constraint algebra would settle whether the free parameter $\\alpha$ is a physical coupling or a removable gauge artifact.","tokens_in":45678,"feed_emoji":"🌀","tokens_out":4096,"duration_ms":36671,"temperature":0.7,"pith_summary":"This paper tries to establish that the geometric trinity of gravity, well known for general relativity, has a Carrollian analogue with a richer structure. The central claim is that the curvature of a magnetic, rigid Carrollian universe can be dynamically traded, under a global flatness condition, for the kinetic energy densities of torsion (in several isolated sectors) or of minimal non-metricity. Each trade yields a distinct equivalent theory: electric Carroll gravity, a fracton-like tensor gauge theory, a Maxwell electric sector, a dilaton scalar, and an isotropic electric Carroll gravity. The equivalence is stated as an algebraic substitution of the scalar curvature $\\tilde{R}$ by these kinetic densities, valid once the connection ambiguities are fixed by chosen gauge conditions with free constants.","feed_headline":"Carrollian gravity's magnetic curvature trades for flat-space gauge theories","feed_subtitle":"A dictionary maps the rigid curved sector to electric Carroll, fracton, Maxwell, and dilaton dynamics under flat connections.","key_machinery":"The central mechanism is the decomposition of the most general metric-compatible Carrollian connection (Equation 4.14) into geometric pieces: the clock-form ambiguity $X_{\\mu\\nu}$, spatial contorsion $A^\\lambda{}_{\\mu\\nu}$, extrinsic curvature $K_{\\mu\\nu}$, and the clock-form field strength $F_{\\mu\\nu}$. By selectively setting some of these to zero, different pure torsion sectors are isolated, and by imposing global flatness (Riemann$=0$), the Magnetic Ricci scalar is algebraically equated to the quadratic kinetic density of the surviving sector. The same logic applies to the non-metric connection where minimal non-metricity is fixed by $Q_{\\rho\\mu\\nu}=-\\tau_\\mu K_{\\rho\\nu}-\\tau_\\nu K_{\\rho\\mu}$.","core_discovery":"The paper claims that the Ricci scalar $\\tilde{R}$ of Magnetic Carrollian gravity, defined on a torsion-free metric-compatible connection with rigid spatial geometry ($K_{\\mu\\nu}=0$), can be exactly replaced by specific kinetic terms for torsion or non-metricity sectors of alternative connections, provided those connections are globally flat (Riemann$=0$). In the teleparallel sector, isolating intrinsic torsion maps $\\tilde{R}$ to $\\alpha(K_{\\mu\\nu}K^{\\mu\\nu}-K^2)$ (Equation 4.61), giving Electric Carroll gravity; spatial torsion maps it to $A_{\\alpha\\beta\\gamma}A^{\\gamma\\alpha\\beta}$ (4.68), resembling tensor gauge theories; temporal vector torsion gives $-s_\\mu s^\\mu$ (4.82), a dilatonic scalar; and clock-form torsion yields $-\\tfrac{1}{4}E_\\mu E^\\mu$ (4.105), a Maxwell electric sector. In the symmetric teleparallel sector, minimal non-metricity coupled with gauge fixing $\\Upsilon=\\lambda K$ gives $-\\lambda K^2$ (5.61), an isotropic electric Carroll gravity. The cleanest instance claimed is that a curved rigid spacetime is dynamically indistinguishable from a globally flat, expanding and shearing spacetime once the clock-form ambiguity is fixed as $X_{(\\mu\\nu)}=\\alpha K_{\\mu\\nu}$.","pith_inferences":["A natural testable extension is to enforce the gauge fixings (e.g., $X_{(\\mu\\nu)}=\\alpha K_{\\mu\\nu}$ or $\\Upsilon=\\lambda K$) through explicit Lagrange multipliers in the action; the paper acknowledges this is required but does not carry it out, leaving open whether the equivalences survive as dynamical constraints rather than on-shell algebraic identities.","The constant parameters $\\alpha$ and $\\lambda$ in the gauge fixings appear free; if they are truly undetermined, the equivalences hold only at special tuned values, which would weaken the claim of general dynamical equivalence. A concrete calculation of the reduced phase space could settle whether the free constants are pure gauge or physical.","The mapping of spatial torsion to fracton theories stops short of reproducing the required gauge symmetries (e.g., $\\delta B=d\\Lambda$ for the Kalb-Ramond sector). An editorial extension would be to investigate whether the connection ambiguities can be restricted so that these symmetries emerge dynamically, connecting Carrollian geometry directly to fracton phases.","The paper's framework suggests a computational rule: any kinetic term quadratic in a purely spatial or temporal Carrollian tensor can be realized as an equivalent formulation of magnetic Carroll gravity by an appropriate flat connection, which could guide future searches for new dualities in condensed matter and holography."],"forward_implications":["If correct, the magnetic and electric Carrollian sectors are not independent regimes; a single flat connection can describe phenomena that appear as either a curved rigid spacetime or an expanding shearing spacetime.","Carrollian gravity can be translated into non-gravitational languages: the fracton magnetic Hamiltonian, Maxwell electrodynamics, and dilaton (Newtonian-like) scalar theory become geometric sectors of a single underlying connection.","The symmetric teleparallel construction shows that an isotropic, shear-free electric Carroll model, with action $\\int -\\lambda K^2$, is a valid equivalent form of magnetic Carroll gravity under minimal non-metricity.","The dictionary suggests a broader pattern: any isolated torsion or non-metricity sector of a globally flat Carrollian connection can be viewed as an equivalent formulation of the curvature-based magnetic theory, potentially extending to further matter couplings."],"supporting_citations":[{"why":"Establishes the baseline Magnetic Carrollian gravity action $S=\\int e\\tilde{R}$ and the electric sector $K_{\\mu\\nu}K^{\\mu\\nu}-K^2$, which the equivalences are built upon.","marker":"[10]"},{"why":"Defines the original geometric trinity of gravity, the conceptual template the paper extends to Carrollian geometry.","marker":"[21]"},{"why":"Provides the foundations of Carrollian geometry, including the non-metricity conditions and the decomposition of the clock form, which the paper uses to construct the minimal non-metric connection.","marker":"[12]"},{"why":"Establishes the existence of affine connections permitting both torsion and non-metricity on Carrollian structures, the structural input the paper turns into explicit dynamical Lagrangians.","marker":"[28]"},{"why":"Supplies the gauged Carroll algebra and the first-order formulation, used in the appendix to derive the Magnetic Carrollian action and factors of 2.","marker":"[34]"}],"fun_headline_variants":["Carrollian gravity's curvature becomes flat torsion and non-metricity","Magnetic Carroll gravity trades curvature for flat gauge sectors","A geometric dictionary maps Carrollian gravity to gauge theories","Flat connections turn Carrollian curvature into torsion and Maxwell","Carrollian gravity's magnetic sector finds flat dynamical twins"],"cache_read_input_tokens":0,"weakest_assumption_plain":"Each equivalence relies on a manually chosen gauge fixing with a free constant (e.g., $X_{(\\mu\\nu)}=\\alpha K_{\\mu\\nu}$ or $\\Upsilon=\\lambda K$), and the global flatness of the modified connection is imposed without proving these constraints are dynamically forced rather than merely convenient algebraic choices.","fun_headline_variants_meta":{"raw":{"variants":["Carrollian gravity's curvature becomes flat torsion and non-metricity","Magnetic Carroll gravity trades curvature for flat gauge sectors","A geometric dictionary maps Carrollian gravity to gauge theories","Flat connections turn Carrollian curvature into torsion and Maxwell","Carrollian gravity's magnetic sector finds flat dynamical twins"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000157,"raw_usage":{"cost":0.000156828,"is_byok":false,"cost_details":{"upstream_inference_cost":0.000156828,"upstream_inference_prompt_cost":0.000123788,"upstream_inference_completions_cost":3.304e-05},"total_tokens":1207,"prompt_tokens":1089,"completion_tokens":118,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":256,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":36}},"tokens_in":1089,"tokens_out":118,"cost_source":"openrouter_usage","duration_ms":2260,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":36,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-02T07:23:57.406115+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Compute the off-shell action with the flatness condition and the gauge fixing $X_{(\\mu\\nu)}=\\alpha K_{\\mu\\nu}$ enforced by Lagrange multipliers. If the resulting field equations do not reduce to those of Electric Carroll gravity for generic initial data, or if the free constant $\\alpha$ is not fixed by consistency, the claimed dynamical equivalence would fail. Equivalently, an independent Hamiltonian analysis of the constraint algebra would settle whether the free parameter $\\alpha$ is a physical coupling or a removable gauge artifact.","supporting_citations":[{"cited_title":"Carroll Expansion of General Relativity","cited_arxiv_id":"2112.12684","evidence_quote":"Establishes the baseline Magnetic Carrollian gravity action $S=\\int e\\tilde{R}$ and the electric sector $K_{\\mu\\nu}K^{\\mu\\nu}-K^2$, which the equivalences are built upon."},{"cited_title":"The Geometrical Trinity of Gravity","cited_arxiv_id":"1903.06830","evidence_quote":"Defines the original geometric trinity of gravity, the conceptual template the paper extends to Carrollian geometry."},{"cited_title":"Foundations of Carrollian Geometry","cited_arxiv_id":"2510.21651","evidence_quote":"Provides the foundations of Carrollian geometry, including the non-metricity conditions and the decomposition of the clock form, which the paper uses to construct the minimal non-metric connection."},{"cited_title":"Affine connections for Galilean and Carrollian structures: a unified perspective","cited_arxiv_id":"2506.03936","evidence_quote":"Establishes the existence of affine connections permitting both torsion and non-metricity on Carrollian structures, the structural input the paper turns into explicit dynamical Lagrangians."},{"cited_title":"Magnetic Carrollian gravity from the Carroll algebra","cited_arxiv_id":"2207.14167","evidence_quote":"Supplies the gauged Carroll algebra and the first-order formulation, used in the appendix to derive the Magnetic Carrollian action and factors of 2."}],"review_version":1}