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arxiv: 2505.02925 · v2 · submitted 2025-05-05 · 🌀 gr-qc · hep-th

Recognition: 5 theorem links

· Lean Theorem

Pre-geometric Einstein-Cartan Field Equations and Emergent Cosmology

Giuseppe Meluccio
Authors on Pith no claims yet

Pith reviewed 2026-05-06 18:26 UTC · model claude-opus-4-7

classification 🌀 gr-qc hep-th PACS 04.50.Kd04.20.Cv98.80.-k11.15.Ex
keywords pre-geometric gravityEinstein–Cartan theoryspontaneous symmetry breakinggauge theory of gravityMacDowell–MansouriWilczek actionemergent spacetimede Sitter cosmology
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The pith

Einstein and Cartan gravity emerge as two slices of one gauge-theoretic equation when a pre-geometric SO(1,4) symmetry breaks, and the resulting de Sitter solution extends smoothly through the Big Bang.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper takes seriously the idea that spacetime geometry is not fundamental but emerges from a Yang–Mills-like gauge theory on a bare four-manifold, and then asks what the equations of motion of such a theory actually look like — both before and after the symmetry breaking that produces a metric. The author shows that a single tensor field equation, written with mixed spacetime and internal indices, automatically packages Einstein's equations and Cartan's torsion equations together: they are simply different index components of one parent equation, distinguished only by which internal direction the Higgs-like field selects when it picks a vacuum. Matter is coupled to the pre-geometry through a single canonical source tensor that likewise splits into the energy–momentum tensor and the spin current after breaking. As an existence proof, an exact homogeneous and isotropic solution is constructed; in the broken phase it is ordinary de Sitter spacetime, and in the unbroken phase the gauge fields remain smooth, so the would-be Big Bang singularity is replaced by a phase transition between a non-metric and a metric era.

Core claim

The author derives the equations of motion for two pre-geometric gauge theories of gravity (Wilczek and MacDowell–Mansouri type), with and without matter, and shows that after spontaneous symmetry breaking of the SO(1,4) or SO(2,3) gauge symmetry down to the Lorentz group SO(1,3), a single tensor equation splits cleanly into the Einstein equations (for the index direction picked out by the Higgs-like field) and the Cartan torsion equations (for the remaining Lorentz directions). Matter is coupled to the pre-geometry through a canonical source tensor whose components reduce after the breaking to the energy–momentum tensor and the spin current. A first exact cosmological solution is given: und

What carries the argument

A pre-geometric Einstein–Cartan tensor G^μ_{AB}, built from the gauge field A and the Higgs-like field ϕ of an SO(1,4) (or SO(2,3)) Yang–Mills theory on a bare manifold, paired with a canonical source tensor T^μ_{AB} = (2m/|J|)·δ(|J|L_m)/δA^{AB}_μ where |J| is the Levi-Civita-contracted product of four covariant derivatives of ϕ. After ϕ acquires a vacuum expectation value along one internal direction, the (a,4) components of G^μ_{AB} = M_P^{-2} T^μ_{AB} become the Einstein equations and the (a,b) components become the Cartan (modified-torsion) equations.

If this is right

  • Einstein and Cartan equations share a single pre-geometric parent equation: energy–momentum and spin currents are two index-slices of one canonical source tensor.
  • The de Sitter solution extends smoothly through what would have been the Big Bang; the singularity is reinterpreted as the moment a metric structure crystallises out of a non-metric Yang–Mills phase.
  • The Higgs-like field that triggers the emergence of geometry leaves behind a supermassive scalar after symmetry breaking, turning low-energy gravity into an effective scalar–tensor theory with a third propagating mode.
  • Because the construction lives in a Yang–Mills framework on a manifold without a metric, the BGV past-incompleteness theorem does not apply to the pre-geometric phase — only to its broken descendant.
  • The same machinery accommodates either de Sitter or anti-de Sitter internal symmetry, with the cosmological constant emerging as a fixed multiple of the symmetry-breaking mass scale m.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The fact that energy–momentum and spin appear as orthogonal slices of one source tensor invites asking whether other matter properties (e.g. dilation currents) would emerge naturally in larger gauge groups such as the conformal or affine extensions.
  • The undetermined function E_s in the cosmological solution — the analog of lapse freedom — hints that time-reparametrisation invariance survives in a disguised form even without a metric, and might be promoted to a constraint algebra on the pre-geometric phase space.
  • If the supermassive ϕ fluctuation truly persists as a scalar–tensor mode after breaking, parameter-free tests in CMB B-modes or stochastic gravitational-wave backgrounds could discriminate Wilczek from MacDowell–Mansouri variants, since they differ in how ϕ enters the cosmological dynamics.
  • The Big Bang reinterpretation here is kinematic rather than dynamical: it does not yet specify what triggers the SO(1,4)→SO(1,3) phase transition, which remains the missing physical input.

Load-bearing premise

That matter couples to the pre-geometric gauge field in exactly the way the correspondence principle requires for the Einstein and Cartan equations to fall out cleanly after symmetry breaking — a coupling postulated, with a specifically chosen normalisation factor, rather than derived from a deeper principle.

What would settle it

Identify the predicted supermassive scalar (the surviving quantum fluctuation of the Higgs-like field ϕ around its vacuum expectation value) in cosmological observables — inflationary perturbations, gravitational-wave spectra, or torsion-induced four-fermion signatures — at amplitudes consistent with a Planck-scale symmetry breaking. Absence of any such scalar-tensor deviation, or detection of a pre-Big-Bang epoch incompatible with a smooth gauge-theoretic phase, would falsify the picture.

read the original abstract

The field equations of pre-geometric theories of gravity are derived and analysed, both without and with matter. After the spontaneous symmetry breaking that reduces the gauge symmetry of these theories \`a la Yang-Mills, a metric structure for spacetime emerges and the field equations recover both the Einstein and the Cartan field equations for gravity. A first exact solution of the pre-geometric field equations is also presented. This solution can be considered as a pre-geometric de Sitter universe and provides a possible resolution for the problem of the Big Bang singularity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

4 major / 7 minor

Summary. The paper derives the equations of motion for two pre-geometric gauge theories of gravity (Wilczek and MacDowell–Mansouri), without and with matter, in a Yang–Mills-like formulation on a four-manifold without metric structure. It shows that, after spontaneous symmetry breaking (SSB) along a fixed internal direction, the gauge equations of motion split into components that reproduce the vacuum Einstein and Cartan equations (Eqs. (39), (40)), and more generally, with matter, into the Einstein and Cartan field equations sourced by the canonical energy–momentum and spin tensors (Eqs. (69), (75)). A symmetry-reduced cosmological ansatz (Eq. (41)) is then solved, yielding a one-parameter family of pre-geometric solutions (Eqs. (43), (45)) that, after SSB and a time-reparametrization gauge fixing, reproduce the de Sitter scale factor (Eq. (54) → Eq. (51)). The author argues this provides a non-singular "pre-geometric de Sitter" cosmology and a resolution of the Big Bang singularity.

Significance. The construction is a useful and pedagogically clear addition to the literature on gauge-theoretic and pre-geometric formulations of gravity. The main technical contribution — explicit derivation and SSB-reduction of the field equations of the Wilczek and MacDowell–Mansouri theories, and a unified pre-geometric Einstein–Cartan tensor whose components carry both the Einstein and Cartan equations after SSB (Eq. (70)) — is a natural and non-trivial complement to the Lagrangian and Hamiltonian analyses already published by the author and collaborators (Refs. [52], [53]). The matter-coupling proposal via the |J|-normalised action (Eq. (58)) is concrete and falsifiable in the sense that it ties matter sourcing to a specific scalar density built from ∇ϕ. The cosmological application is preliminary but suggestive: it shows that the formalism is at least compatible with a regular de Sitter–like history. The Big Bang–resolution claim is more aspirational than established (see major comments) but the framework offers a definite computational target for future work.

major comments (4)
  1. [§IV, Eqs. (42)–(45) and (54)] The 'first exact solution' is in fact a one-function family in the Wilczek case (E_s(t) arbitrary, Φ constant) and a two-function family in the MacDowell–Mansouri case (E_s(t) and Φ(t) both arbitrary). The author acknowledges this after Eq. (43) but defends it by analogy with time-reparametrization gauge freedom. This analogy needs a more careful justification: in metric FLRW, fixing the lapse uses one function of t; here the residual freedom corresponds to a single scalar function E_s, so the analogy is in fact tight in vacuum — but the paper should state explicitly that the de Sitter form a(t)=exp(√(Λ/3) t) emerges only after imposing the gauge E_t=m on Eq. (54), and that without such a gauge choice the solution is not unique. Currently the manuscript presents the de Sitter behaviour as a prediction, when it is more accurately a consistency check after gauge fixing.
  2. [§IV, last paragraph; §VII] The claim that the pre-geometric phase resolves the Big Bang singularity rests on the pre-geometric fields being regular at all times. But within the same one-parameter family of solutions there exist E_s(t) that vanish or diverge at finite t, which through Eq. (43) make E_t = (√2/2) Ė_s/|E_s| singular. The non-singularity of the pre-geometric configuration is therefore a property of the chosen representative, not a dynamical consequence of the field equations. The paper should either (i) identify a dynamical or symmetry criterion that selects regular E_s within the family, or (ii) soften the singularity-resolution claim to a statement of consistency rather than prediction. As written, §VII overstates what §IV establishes.
  3. [§V, footnote 14, Eqs. (61)–(67)] The 'prescription of dropping the explicit antisymmetrisation of the internal indices' when computing the SSB of G along the 4th direction is presented as a fix for a factor-of-2 mismatch (Eqs. (65) vs (66)). This is a load-bearing step in showing that SSB of the pre-geometric Einstein–Cartan tensor reproduces the geometric Einstein and Cartan tensors with the correct coefficients (Eqs. (67)–(68)). The justification given is essentially that the result must come out right; this is circular. A proper derivation should treat the symmetric and antisymmetric components of A^{AB} on the same footing throughout — perhaps by varying the action subject to the antisymmetry constraint via a Lagrange multiplier — and recover the SSB limits without an ad hoc prescription. Please supply this or explain why the prescription is forced by an unambiguous geometric argument.
  4. [§V, Eq. (58)] The matter action is normalised by |J|/(24 v^5 m^4), constructed from ∇ϕ so that |J| → e after SSB (Eq. (59)). This is acceptable as an existence proof for a coupling that reduces to ∫ e L_m d^4x. However, the construction is not unique: any internal-index scalar built from ∇ϕ, ϕ that limits to e after SSB would do. The paper should comment on this ambiguity, as different choices will generally give physically distinct field equations away from the SSB scale (e.g. during the transition itself). Without such a comment, the matter sector of the unbroken phase is underspecified.
minor comments (7)
  1. [§I, double-sign convention] The convention 'first sign for SO(1,4), second for SO(2,3)' interacts with the separate ± from square-root branches in Eqs. (43), (45), and the author flags this in footnote 10. Consider using a different symbol (e.g. ε = ±1 for the gauge-group choice) to avoid reader confusion.
  2. [§III, Eqs. (35)–(36)] The introduction of the modified torsion tensor T̄ via the Gauss–Bonnet contribution should mention that for a torsion-free background T̄ reduces to T̃, so the difference is genuinely beyond Einstein–Cartan. A short comment on the propagating-torsion content of the MacDowell–Mansouri theory (Refs. [66, 67]) would help.
  3. [§IV, Eq. (47) and following] The choice of Cartesian rather than spherical coordinates for the flat FLRW metric is unusual; the diagonal tetrads in (48) work but readers may wonder about the off-diagonal structure expected in spherical coordinates. A line stating that this is purely for notational simplicity would help.
  4. [§IV, Eq. (54) and footnote 11] The discussion of time reparametrization in pre-geometric vs. metric theories is good but could be condensed and put in the main text rather than a long footnote, since it is essential to the interpretation of the solution.
  5. [§VI, Eqs. (78)–(79)] These equations are written in a sgn(·)/|·| form that obscures their structure. Consider either choosing definite signs at the outset (consistent with §IV) or rewriting in a manifestly covariant form. As printed, Eqs. (78a)–(79c) are unwieldy and hard to check.
  6. [Bibliography] Several self-references appear ([52], [53], [65]) that are essential to the argument; brief one-line summaries of what each contributes (Lagrangian SSB, Hamiltonian DOF count, alternative SSB mechanisms) would help readers who do not have those papers at hand.
  7. [§VII] The paragraph on the BGV theorem could be sharpened. As stated, it is correct that BGV is a kinematic past-incompleteness statement; but the claim that pre-geometry 'resolves' it should be qualified — BGV's hypothesis (average expansion) is itself a metric statement, and what pre-geometry does is move outside its scope rather than refute it.

Simulated Author's Rebuttal

4 responses · 2 unresolved

We thank the referee for a careful and constructive report and for the recommendation of minor revision. The four major comments identify genuine weaknesses in the presentation rather than in the underlying construction, and we agree with all of them in substance. We will revise the manuscript as follows: (i) §IV and the corresponding wording in §VII will be reformulated so that the de Sitter scale factor is presented as a gauge-fixed consistency check on a one-parameter (Wilczek) or two-function (MacDowell–Mansouri) family of solutions, not as a prediction; (ii) the singularity-resolution claim will be softened from a dynamical prediction to a consistency statement, since regularity is a property of the chosen representative within the family, and the existence of singular representatives will be acknowledged explicitly; (iii) footnote 14 and the surrounding derivation in §V will be rewritten so that the antisymmetry of A^{AB} is enforced uniformly throughout the variation, removing the appearance of an ad hoc prescription while leaving Eqs. (67)–(68) and (69)–(76) unchanged; and (iv) §V will include an explicit discussion of the non-uniqueness of the |J|-normalisation of the matter action, identifying our choice as the minimal one consistent with the correspondence principle and general covariance, and flagging the classification of alternatives as an open problem. Two genuinely open issues — a selection rule for regular E_s and a first-principles fixing of the matter norma

read point-by-point responses

  1. Referee: The 'first exact solution' is a one- (Wilczek) or two-function (MacDowell–Mansouri) family; the de Sitter form a(t)=exp(√(Λ/3) t) only emerges after fixing E_t=m on Eq. (54). The paper currently presents this as a prediction rather than a consistency check after gauge fixing.

    Authors: We agree with the referee that the wording in §IV is sharper than warranted, and we will adjust it. The intended logical content is precisely the one the referee describes: in vacuum, the symmetry-reduced pre-geometric equations leave one function (E_s) undetermined, in tight analogy with the lapse freedom of metric FLRW; the exponential scale factor is recovered only after imposing the gauge E_t(t)=m on Eq. (54), as we already note in the paragraph following it. We will (i) state explicitly at the beginning of the discussion of Eqs. (43) and (45) that the solution constitutes a one-parameter (resp. two-function) family; (ii) reformulate the conclusion as a consistency check — namely, that the family contains the de Sitter spacetime as the gauge-fixed representative — rather than as a prediction; and (iii) revise the analogous wording in §VII accordingly. We thank the referee for pointing out that the analogy with metric time-reparametrisation is in fact stronger than our presentation made it sound, and we will state this more clearly. revision: yes

  2. Referee: The Big Bang singularity-resolution claim is overstated: within the same family there exist E_s(t) that vanish/diverge at finite t, making E_t = (√2/2) Ė_s/|E_s| singular. Non-singularity is a property of the chosen representative, not a consequence of the field equations.

    Authors: This is a fair criticism and we will soften the claim accordingly. The referee is correct that, since E_s(t) is undetermined by the symmetry-reduced equations in vacuum, the regularity of the pre-geometric configuration depends on the choice of representative within the family. What the field equations do guarantee is that there exists a regular pre-geometric continuation through (and past) what would be the Big Bang singularity in the broken phase — i.e. the singularity resolution is consistent with, but not forced by, the dynamics at this level of analysis. We will (i) rephrase the last paragraph of §IV and the corresponding passage in §VII to state this as a statement of consistency rather than a prediction; (ii) explicitly acknowledge that singular representatives also exist in the family; and (iii) flag the identification of a dynamical or symmetry criterion selecting regular E_s — possibly via the inclusion of kinetic/potential terms for A and ϕ omitted in the present vacuum analysis (footnote 9) — as an open question for future work. We are not in a position to provide such a criterion in the present manuscript. revision: yes

  3. Referee: The footnote-14 prescription (dropping the explicit antisymmetrisation of internal indices when computing the SSB of G along the 4th direction) is load-bearing for Eqs. (67)–(68) but is justified circularly. A proper derivation should treat A^{AB} symmetric/antisymmetric parts on the same footing, e.g. via a Lagrange multiplier, and recover the SSB limit without ad hoc steps.

    Authors: We acknowledge that the presentation in footnote 14 is unsatisfactory and we will rework it. The factor-of-2 mismatch between Eqs. (65) and (66) is genuinely an artefact of differentiating an antisymmetric tensor (A^{AB}) versus a tensor with a single internal index (e^a) — it is not a free choice — but the manner in which we handled it in the text reads, as the referee correctly notes, as if the prescription were chosen to make the answer come out right. In the revised version we will give the derivation in a manifestly unambiguous form by enforcing the antisymmetry of A^{AB} from the outset (using either the symmetric projector δ^C_{[A}δ^D_{B]} consistently throughout the variation, or, equivalently, a Lagrange-multiplier implementation of the antisymmetry constraint), and then taking the SSB limit. The factor of 2 then arises from the index structure of the projector rather than from a separate prescription, and footnote 14 can be replaced by a short identity-level remark. We expect Eqs. (67)–(68) and the matter-coupled Eqs. (69)–(70), (75)–(76) to be unchanged. revision: yes

  4. Referee: The matter action normalised by |J|/(24 v^5 m^4) is not unique: any internal-index scalar built from ∇ϕ, ϕ that limits to e after SSB would do. Different choices give physically distinct equations away from the SSB scale, so the unbroken-phase matter sector is underspecified.

    Authors: The referee is right, and we will add an explicit comment to §V. The role of |J| in Eq. (58) is to provide the simplest scalar density built covariantly from ϕ and ∇ϕ that reduces to e under SSB, and the correspondence principle as stated only fixes its broken-phase limit; any other Lorentz-invariant density with the same SSB limit would be admissible at the level of the correspondence principle alone. Different such choices coincide on shell after SSB (and hence are indistinguishable in the low-energy Einstein–Cartan regime) but in general differ during the transition and in the unbroken phase, where they would source physically distinct pre-geometric Einstein–Cartan equations. In the revision we will (i) state this non-uniqueness explicitly after Eq. (59); (ii) note that |J| is the lowest-derivative, lowest-order-in-ϕ choice consistent with the SSB limit and with general covariance, which is our motivation for selecting it; and (iii) flag the systematic classification of admissible pre-geometric matter normalisations, and possible criteria (e.g. derivative order, behaviour during the phase transition, observational signatures) to discriminate among them, as an open problem for future work. revision: yes

standing simulated objections not resolved
  • A dynamical or symmetry criterion that selects regular E_s(t) within the symmetry-reduced family — and thereby would upgrade the Big Bang singularity-resolution from a consistency statement to a prediction — is not provided in the present work and is left as an open problem.
  • A unique, first-principles selection rule for the matter-action normalisation in the unbroken phase (beyond the requirement of reducing to e under SSB) is not established here; the choice of |J| is motivated by minimality but is not forced.

Circularity Check

3 steps flagged

The "pre-geometric de Sitter solution" is built from an ansatz selected a posteriori to reproduce de Sitter under SSB, and leaves the scale-factor function E_s entirely free; the singularity-free claim picks the smooth element of that family by hand.

specific steps
  1. ansatz smuggled in via citation [Sec. IV, Eq. (41) and surrounding text]
    "The choice to motivate the ansatz (41) a posteriori, rather than a priori, aims at stressing the fact that any solution of the equations of motion for a pre-geometric theory is in principle unrelated to gravity – it is just a solution of a gauge theory of fields under certain symmetry assumptions."

    The author admits the three-function ansatz A^{04}_0=E_t, A^{i4}_i=E_s, A^{0i}_i=Ω was chosen so that after SSB it matches the de Sitter tetrad/spin-connection structure (48). The 'derivation' of pre-geometric de Sitter is then a consistency check of an ansatz selected to map to de Sitter, not an independent derivation from pre-geometric symmetry principles.

  2. fitted input called prediction [Sec. IV, after Eqs. (43) and (45)]
    "As a result, the function Es remains undetermined. ... leaving Φ undetermined as the Eq. (44c) is automatically satisfied by the solutions (45). These solutions automatically satisfy the component (E)=(4) of the equations of motion (28) too, implying that also Es remains undetermined."

    The pre-geometric equations only fix two algebraic relations among {Ω, E_t, E_s}; the function E_s(t) is free. The exponential de Sitter form (50) is recovered only by additionally imposing the gauge-fix E_t(t)=m and identifying e^i_i ← E_s/m, then using the broken-phase torsion-free relation (46). The singularity-free 'pre-geometric de Sitter' is therefore the smooth element of a function family the dynamics does not fix; pathological E_s (e.g. vanishing at finite t) belongs to the same solution family.

  3. self definitional [Sec. IV, Eqs. (49)–(54)]
    "Going back to the solutions (43) and (45), after the SSB their form becomes ... ω^{0i}_i=√(Λ/3) e^i_i, e^0_0=√(3/Λ) ė^i_i/|e^i_i|. ... if the gauge-fixing condition Et(t)=m, which gives e^0_0 → 1, is applied to the solution (54), then one can solve for e^i_i ← Es/m and the exponential form of Eq. (50) for the scale factor is correctly recovered in both pre-geometric theories."

    The exponential scale factor a(t)=exp(√(Λ/3) t) emerges only after (a) imposing torsion-free vacuum so ω is determined by ∂e (geometric input), and (b) gauge-fixing E_t=m. Within the pre-geometric phase alone, the same equations are satisfied by any E_s. The 'recovery' of (50) is thus a relabeling of the de Sitter Hubble relation H=√(Λ/3) under SSB identifications, not a pre-geometric prediction of singularity-free cosmology.

full rationale

The paper's mathematical chain in Secs. III–IV is largely self-contained: variational principle → Euler–Lagrange equations (20),(21),(27),(28); under SSB substitutions (22), the latter reduce to the standard Einstein and Cartan field equations (39),(40). That reduction is real and is not circular — it is a direct algebraic consequence of the substitution rules, and the relations (24),(31) are derived rather than fitted. The circularity is concentrated in Sec. IV, where the headline novel result lives. Two issues compound: (i) The symmetry ansatz (41) is admitted to be motivated post hoc by what reproduces de Sitter after SSB. The author writes explicitly: "The choice to motivate the ansatz (41) a posteriori, rather than a priori, aims at stressing the fact that any solution of the equations of motion for a pre-geometric theory is in principle unrelated to gravity." This is honest but means the ansatz was selected by working backward from the desired SSB image (48). (ii) Within that ansatz, the field equations (42)/(44) impose only algebraic relations among Ω, E_t, E_s (and force Φ = const in the Wilczek case). The author notes plainly that "the function E_s remains undetermined" (after Eq. 43) and similarly for the MM case after Eq. (45). The de Sitter scale factor (50) is then "recovered" only after imposing the gauge fixing E_t(t)=m AND identifying e^i_i ← E_s/m — at which point ω^{0i}_i = √(Λ/3) e^i_i combined with the broken-phase torsion-free spin connection (46) gives ė ∝ e, hence exponential. The exponential form is therefore not predicted by the pre-geometric equations alone; it is forced by combining a free function with the geometric torsion-free relation imported from the SSB phase. The Sec. VII statement that the pre-geometric framework "resolves the Big Bang singularity" inherits this freedom: the same family contains E_s with zeros at finite t, which after SSB would yield singular tetrads. The non-singular interpretation selects the smooth representative. This is moderate circularity — not a fitted-parameter-renamed-as-prediction, and not a self-citation chain, but a chosen ansatz plus an undetermined function that together import the answer being "predicted." Score 5.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Model omitted the axiom ledger; defaulted for pipeline continuity.

pith-pipeline@v0.9.0 · 9373 in / 7690 out tokens · 113669 ms · 2026-05-06T18:26:51.182619+00:00 · methodology

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