Audit 11: The Bianchi identities — Same Maths, Different Names

Same maths, different names · Audit 11

The boundary of a boundary is nothing. Written as d² = 0, that one line is the identity dF = 0 behind half of Maxwell’s equations, the Bianchi identity that makes Einstein’s equations consistent, the rule that a dislocation or a vortex line cannot end inside a body, and the Maxwell relations of thermodynamics; the compatibility conditions of elasticity belong to a complex built from it. Each field has its own name for it. This audit records that they share one exterior-calculus structure, sometimes the identity itself and sometimes a complex derived from it; what each field has to add before the identity decides anything dynamical; and where much of the interesting physics lives, in the gap between “closed” and “exact”.

Version 0.4 · 12 September 2026 Series 11 of a first series of eleven (fifth in the order of work) Genre shared structure (an identity, not an equation of motion) Author Robert W. Harrison, with AI assistance (see method note)

In plain terms

Take a solid ball. Its boundary is a sphere. Now ask for the boundary of the sphere: there is none, because a sphere has no edge. Take a flat disc instead: its boundary is a circle, and the circle has no endpoints, so the boundary of the boundary is again nothing. This is true of every shape, in every number of dimensions, and it is not a fact about physics. It is a fact about what “boundary” means. Written in the calculus of differential forms it reads d² = 0, “the derivative of a derivative is zero”, and in the vector notation of a first physics course it is two familiar statements: the curl of a gradient is zero, and the divergence of a curl is zero.

What makes this worth a page is how much physics has been hung on it without acknowledging that it is the same peg. Two of Maxwell’s four equations, the ones with no charges or currents in them, say that the magnetic field has no sources and that a changing magnetic field circulates an electric one. Locally, both are the statement that the electromagnetic field can be written as the derivative of a potential, and the derivative of a derivative is zero; that is why magnetic field lines have no ends. In Einstein’s theory of gravity a set of four relations, named after Luigi Bianchi, holds for every curved space-time, and once Einstein’s equation is imposed they require whatever sources it to be conserved; they are the same statement about curvature, and they were known to geometers before Einstein needed them. An engineer checking whether a proposed pattern of strains in a steel beam can actually be produced by bending it uses Saint-Venant’s compatibility conditions, which come from a sequence of operators built out of the same statement, and a metallurgist knows that a dislocation, a line defect in a crystal, cannot simply stop in the middle of the crystal, which is the same statement once more. So is the rule that a vortex line in a fluid cannot end in the fluid, and so are the four Maxwell relations of thermodynamics, which let one measure a quantity that is hard to measure through one that is easy.

Two things make the audit worth doing rather than just the list. The first is that an identity constrains but does not decide. Once a measured quantity has been represented as the derivative of something, the identity says what patterns of that quantity are possible at all, and that is a physical commitment: an experiment cannot refute the identity, but it can refute the representation, which is what finding a magnetic charge would do. What the identity cannot do is supply the dynamics. The Bianchi identity plus Einstein’s equation gives the conservation of matter; the identity that the field derives from a potential plus Maxwell’s sourced equations gives the conservation of charge; the identity alone gives neither, and each row of the ledger has to say which equation it was joined to before a conservation law is claimed. The second is that the identity has a converse which is not an identity. “Every derivative has zero derivative” is always true; “everything with zero derivative is a derivative” is true in a region with no holes, and in a region with holes it is true of some things and not of others, depending on how the thing wraps the holes. The holes are where some of the best-known physics of the last century lives: the Aharonov–Bohm effect, Dirac’s magnetic monopole, the dislocations that Volterra found by cutting and re-gluing a ring. The page keeps the two statements apart throughout.

The rest of the page is a ledger. It lists the fields, what each calls the identity, when each first had it, what each joins it to, and a verdict: exact when the field’s statement is exactly d² = 0, or its covariant form, within the row’s setting, limited when it holds only after a named approximation or through a derived construction, analogy when only some features line up. Every claim carries a note on how far the source was actually read.

What is being audited

On any smooth manifold the exterior derivative d takes a k-form to a (k+1)-form, and applying it twice gives zero. Where there is a connection, the covariant exterior derivative D does not square to zero; its square is the curvature acting on the object it is applied to, and the curvature itself satisfies DF = 0. The Bianchi identity follows from that structure, not from any nilpotence of D. The identity, its covariant form, and the converse that is not an identity:

The Bianchi identities — d² = 0 (Poincaré, Cartan); DF = 0 for the curvature of a connection (Ricci–Padova 1889, Bianchi 1902, Cartan 1922); the contracted form used by Einstein and Hilbert in 1915–16 d(dα) = 0 for every form α ⇔ ∂²C = 0 for every oriented chain C (Stokes: ∫_∂Cα = ∫_Cdα)

vector calculus: ∇×(∇φ) = 0, ∇·(∇×A) = 0

connection: D²η = F∧η for η valued in an associated bundle (D²η = [F, η] in the adjoint); D is not nilpotent ⇒ DF = dF + [A, F] = 0 (the Bianchi identity of a gauge field; for the Levi-Civita connection, ∇_[aR_bc]d^e = 0)

with torsion T = Dθ: DT = R∧θ, DR = 0 (Cartan’s first and second identities)

contracted: ∇^aG_ab ≡ 0 for G_ab = R_ab − ½Rg_ab ⇒ with G_ab = 8πG T_ab, ∇^aT_ab = 0

the converse (not an identity): dα = 0 ⇒ α = dβ locally, for α of positive degree (Poincaré lemma); globally iff the class of α in the de Rham cohomology of the region vanishes

The first line is the starting structure, and d² = 0 follows from the definition of the exterior derivative: it is built so that Stokes’s theorem holds, Stokes’s theorem transfers the boundary operator to the forms, and the boundary of a boundary is empty. The rest of the page reaches that line in four ways, directly for a form, through the structure equation for a curvature, through a contraction or a specialisation of Noether’s theorem, and through a complex derived from it in elasticity, and the ledger says which for each row. The vector-calculus line is the case of three dimensions, where a 1-form is a vector field and a 2-form is another one. The connection line is the case that carries the name: a covariant derivative does not square to zero, its square is the curvature acting on whatever it is applied to, and the curvature in turn is “closed” with respect to the covariant derivative; that closure is a consequence of the connection structure and not of d² = 0 carried over unchanged. That is the Bianchi identity of a Yang–Mills field and, for the connection that comes with a metric, the second Bianchi identity of Riemannian geometry, ∇_[aR_bc]d^e = 0. Contracting it twice gives the four identities ∇^aG_ab = 0 on the Einstein tensor, which hold for every metric whatever, and which is why Einstein’s equations, once written down, force the conservation of whatever is put on their right-hand side.

Three conditions travel with the identity, and each row of the ledger is read against them.

First, an identity constrains and dynamics decides, and the two must not be confused in either direction. ∇^aG_ab = 0 is true of any metric, including metrics that solve no field equation at all, so by itself it supplies no field equation and establishes nothing about the conservation of a proposed matter source; joined to G_ab = 8πG T_ab it becomes ∇^aT_ab = 0, and that is a consequence of the field equation. In the same way dF = 0 is true of any F that is dA, and charge conservation is not in it; charge conservation comes from applying d to the sourced equation d⋆F = ⋆J, which gives d⋆J = 0. But the identity is not empty of physics. Once measured fields are represented as the curvature of a potential, dF = 0 constrains which fields can be measured; once a measured strain is represented as a symmetric gradient, compatibility constrains it. Those are commitments about the representation, and experiments test them. The verdict column judges whether the field’s statement is the identity; the “joined to” entry in the direction column says what has to be added before a conservation law follows.

Second, the geometric type of the object fixes what the identity says. Applied to a 0-form it says a gradient has no curl; applied to a 1-form, that a curl has no divergence; applied to the curvature 2-form of a connection, that curvature is covariantly closed. The names in the ledger attach to different types: Saint-Venant’s compatibility conditions are the composition identity of a complex derived from the de Rham complex, not d² applied to a form, the Riemannian Bianchi identity is the identity on a metric’s curvature, the dislocation-continuity rule is the identity on a torsion 2-form, and the Maxwell relations are the identity on a 1-form on the space of thermodynamic states. Getting the type right is what makes the dictionary column checkable.

Third, the identity has a converse which is not an identity. That a closed form of positive degree is locally exact is Poincaré’s lemma, a theorem about star-shaped regions; that a particular closed form is globally exact is the statement that its class in the de Rham cohomology of the region vanishes, which every closed form’s does when the cohomology group is zero, and which some closed forms’ still do when it is not. Most of the rows use both directions, and the physics that is usually described as “the potential having an effect where the field is zero” or “a line defect with a Burgers vector” is the converse failing on a region with a hole. The audit keeps the identity, which is always true, apart from its converse, which is true only under a topological hypothesis, and records which one each row is using.

Two columns in the ledger do the real work. The dictionary column says which form the identity is applied to and of what type, so that a verdict can be checked rather than believed. The direction column says which way the implication runs and what the identity is joined to: from an identity plus a field equation to a conservation law, or from a symmetry to an identity, which is Noether’s second theorem. The verdict judges the correspondence within the row’s stated setting; the adequacy of the field equation the identity is joined to is answered on the “model” line.

The ledger

exact the field’s statement is exactly d² = 0, or DF = 0, or the contracted identity, for the named object in the row’s setting limited the correspondence holds only after a named approximation, in a stated limit, or through a derived construction analogy selected features correspond; equivalence of the statements is not established

Each verdict applies to the correspondence named in that row’s dictionary column, not to the field as a whole. Whether the converse (closed implies exact) holds is stated in the row and does not enter the verdict, since it is not part of the identity.

Field	Name used · earliest source located	Dictionary	Direction of implication	Assumptions added or dropped · model · where it breaks	Verdict
Differential geometry	Bianchi identities (first and second); Cartan’s structure equations; d² = 0Padova 1889, from Ricci (statement without proof) · Bianchi 1902 (proof) · Cartan 1922–25 (with torsion) · Voss 1880 (contracted form, attributed; not verified)	d² = 0 on forms ↔ ∂² = 0 on oriented chains, by Stokes · the curvature 2-form Ω of any connection ↔ DΩ = 0 (second identity) · the torsion 2-form Θ ↔ DΘ = Ω∧θ (first identity); for a torsion-free connection, R_[abc]^d = 0 · the second identity for the Levi-Civita connection ↔ ∇_[aR_bc]d^e = 0 · its double contraction ↔ ∇^aG_ab = 0	Definitions → identities, with no hypothesis beyond smoothness: the identities are consequences of D²η = Ω∧η, the structure equation of the connection, which is where d² = 0 on the bundle enters. Nothing physical is assumed or implied.	adds a smooth manifold; a connection · model none; pure mathematicsbreaks nothing; but supplies no converse. Whether a closed form is exact is a separate theorem (Poincaré lemma, locally) and a topological fact (de Rham cohomology, globally). Distributional curvature, as at a conical singularity, needs the identities restated in the weak sense	exact
Electromagnetism	Homogeneous Maxwell equations; ∇·B = 0 and Faraday’s law; “no magnetic monopoles”; charge conservationMaxwell 1865 · Dirac 1931 (the monopole as the edge) · Wu & Yang 1975 (global formulation)	The field 2-form F = dA ↔ dF = 0 identically, which is ∇·B = 0 and ∇×E + ∂_tB = 0 · the sourced equations d⋆F = ⋆J ↔ the field equation, not an identity · d applied to it ↔ d⋆J = 0, the continuity equation ∂_tρ + ∇·j = 0, which is d² = 0 applied to ⋆F · a magnetic monopole ↔ dF = ⋆J_m ≠ 0 at the pole, with F closed but not exact on the region that excludes it	F = dA → dF = 0, exactly and with no dynamics. Field equation d⋆F = ⋆J → charge conservation, exactly, by the same identity. The converse, dF = 0 → F = dA, holds locally; globally it needs the class of F in the second cohomology of the region to vanish, which is automatic when that group is zero and can still hold when it is not.	adds that F is the curvature of a potential on each region where a potential exists; dF = 0 on a region does not by itself supply a global A there, and the exterior of a magnetic pole is the standard case where it does not · model Maxwell’s equations in vacuum, or in a medium with the sourced pair written either for the excitation form with a constitutive law or for F with J the total current, bound contributions included; the identity is unchanged by the mediumbreaks only by changing the object: with magnetic charge, dF ≠ 0 at the pole (Dirac 1931), and F is closed but not exact on the region outside it; the identity still holds for A on each patch where A exists (Wu and Yang 1975). Charge conservation breaks only if the sourced equation does. Nothing dynamical breaks dF = 0, and finding a magnetic charge would refute the representation of the field as the curvature of a global potential, not the identity	exact
Gauge theory	Bianchi identity of the Yang–Mills field; D_μF_νλ + cyclic = 0; integrability conditionYang & Mills 1954 (field; the identity is not stated in the paper) · Lubkin 1963 (geometric definition; Bianchi identities discussed, per the review) · Wu & Yang 1975 (global formulation) · Nakahara 2003, Sec. 10.3.5	The gauge potential A ↔ a connection on a principal bundle · the field strength F = dA + A∧A ↔ its curvature, with D²η = F∧η on charged fields and [F, η] on adjoint ones · DF = dF + [A, F] = 0 ↔ the Bianchi identity, exactly the second identity of the geometry row · the equation of motion D⋆F = ⋆J ↔ the Yang–Mills equation, not an identity · D applied to it ↔ D⋆J = 0, covariant conservation of the current	Structure equation → DF = 0, exactly. Yang–Mills equation → D⋆J = D²⋆F = [F, ⋆F] = 0, exactly, and the last step is a cancellation, not nilpotence: the spacetime pairing of the components is symmetric and the bracket antisymmetric. The covariant conservation law is not an ordinary one. DF = 0 is the curvature Bianchi identity; whether it supplies a gauge-invariant magnetic charge or a flux conservation law depends on how the current is defined, on the gauge group and on the global structure, and the non-nilpotence of D does not by itself decide it, since D² vanishes on particular arguments (D²F = [F, F] = 0, D²⋆F = [F, ⋆F] = 0) while not vanishing identically.	adds a Lie group and a representation; a bundle, possibly non-trivial · model classical Yang–Mills theory; the identity survives quantisation as an operator identity, which is outside this pagebreaks nothing; which conserved magnetic or topological quantities the theory has depends on the definition of the current, the gauge group, the matter fields and the global structure; the instanton number of the topology row is one such quantity, obtained from a characteristic class rather than from a current	exact
General relativity	Contracted Bianchi identity; “automatic conservation of the source”; the four constraint equations; “only six independent Einstein equations”Hilbert 1915 (Theorem I: four identities) · Einstein 1916, Sec. 18 (conservation from the field equations) · Arnowitt, Deser & Misner 1962, Sec. 3.4 (constraints preserved) · Misner, Thorne & Wheeler 1973, Ch. 15 and 17 · Carroll 1997, Ch. 4	The Levi-Civita curvature ↔ the connection row’s Ω · ∇^aG_ab ≡ 0 ↔ the double contraction of DΩ = 0, an identity for every metric · G_ab = 8πG T_ab ↔ the field equation · ∇^aT_ab = 0 ↔ identity joined to field equation · the four normal projections G_abn^b = 8πG T_abn^b onto a spacelike hypersurface ↔ constraints on initial data, preserved in time by the identity together with the evolution equations and a consistently evolved source · four coordinate freedoms ↔ the four identities among ten equations	Identity holds for every metric. Identity plus field equation → conservation of the matter tensor, exactly; for matter whose equations of motion are equivalent to that conservation (dust of non-vanishing density; a perfect fluid with its equation of state and particle conservation supplied) the equations of motion follow, and in general they do not (see the note). Identity, with the evolution equations → the constraints, if satisfied initially, remain satisfied (ADM, eq. 3.16). Diffeomorphism invariance of the action → the identity (Noether’s second theorem; next row).	adds the Einstein field equation, or the Einstein–Hilbert action; for the constraint reading, a foliation by spacelike surfaces · model general relativity, with any matter whose stress tensor is definedbreaks nothing within the theory. What the row does not give: ∇^aT_ab = 0 is a covariant conservation law and supplies no canonical globally conserved energy in a general spacetime; where a Killing vector ξ exists, T^abξ_b is a conserved current under the usual boundary conditions, and otherwise the pseudotensor and quasilocal literature is the response, not audited here. With the Einstein tensor kept on the left, a source that is not divergence-free is inconsistent, not “non-conserving”; a theory that changes the left-hand side has to be examined on its own equations	exact
Variational mechanics	Noether’s second theorem; gauge identities; Noether identities; “improper” conservation lawsHilbert 1915 (Theorem I) · Noether 1918, Theorem II, eq. 16 and Sec. 6 · Kosmann-Schwarzbach 2011 (history) · Rowe 2019	Invariance of an action under a group depending on ρ arbitrary functions ↔ ρ differential identities among the Euler–Lagrange expressions (Noether, Theorem II; eq. 16) · diffeomorphism invariance of the Einstein–Hilbert action ↔ the contracted Bianchi identity ∇^aG_ab = 0, as the identity so produced: this specialisation is exact · gauge invariance of the Yang–Mills action ↔ D(D⋆F) ≡ 0, the identity on the field equation, which is a different expression from the curvature identity DF = 0 · an “improper” divergence relation ↔ one whose current is built from the Lagrange expressions and their derivatives together with identically divergence-free terms (Noether, Sec. 6)	Symmetry → identity. Noether’s theorem produces identities among the field equations of any locally invariant action; for the Einstein–Hilbert action the identity produced is the contracted Bianchi identity, and for other actions it is an identity on that action’s Euler–Lagrange expressions, related to but not the same as the curvature identity. The geometric rows obtain the curvature identities from the connection structure with no action; both derivations are exact for what they derive.	adds a Lagrangian formulation; an invariance group parametrised by arbitrary functions of position · model any field theory with a local symmetrybreaks nothing within the theorem; for a theory without an action principle the row says nothing, though the geometric identity still holds for whatever curvature the theory contains. The row is limited because the correspondence with the Bianchi identity is exact for the gravitational specialisation and not for the theorem in general. Whether improper laws leave room for meaningful gravitational energy charges (with boundary terms, or with Killing vectors) is the question Hilbert, Klein and Noether argued over in 1917–18 and is not resolved here	limited
Einstein–Cartan gravity	Bianchi identities with torsion; conservation of energy–momentum and spinCartan 1922, 1923–25 · Hehl, von der Heyde, Kerlick & Nester 1976 · Trautman 2006	Torsion Θ = Dθ ↔ the first identity DΘ = Ω∧θ, no longer R_[abc]^d = 0 · the second identity DΩ = 0 ↔ unchanged · their contractions ↔ balance laws for the energy–momentum and the spin currents, with torsion and curvature source terms (Trautman’s “consequences of the Bianchi identities”) · the spin of matter ↔ the source of torsion, algebraically	Identities → balance laws, exactly, for the U₄ geometry; joined to the Einstein–Cartan field equations they become the conservation laws of matter with spin. The first identity is the new content: with torsion the algebraic symmetry of the curvature is lost and its replacement is a differential identity.	adds a metric-compatible connection with torsion; the Einstein–Cartan–Sciama–Kibble field equations, in which torsion is algebraically tied to spin and does not propagate · model Einstein–Cartan theory; Poincaré gauge theory more generallybreaks nothing; the identities hold for any connection. What is model-dependent is whether torsion is dynamical, and the row makes no claim about whether spacetime has torsion	exact
Cosmology	The continuity equation as a consequence of the Friedmann equations; “only two of the three are independent”Friedmann 1922 · Carroll 1997, eq. 8.20 (lecture notes)	The Friedmann equations ↔ the G₀₀ and G_ii components of the field equation on a homogeneous isotropic metric · ρ̇ + 3H(ρ + p) = 0 ↔ the time component of ∇^aT_ab = 0, hence of the contracted identity joined to the field equation · the redundancy among the three ↔ the identity, in the one place where it survives the symmetry	Contracted identity plus field equation → the continuity equation, exactly; equivalently, the Friedmann constraint and the acceleration equation together give continuity by differentiation and substitution. The other pairings need a condition: constraint plus continuity gives acceleration only where H ≠ 0, and acceleration plus continuity fixes the constraint only up to an integration constant set by initial data.	adds homogeneity and isotropy; a perfect-fluid source · model Friedmann–Lemaître–Robertson–Walker cosmologybreaks nothing; the row is the GR row on one metric. Where several fluids interact, the identity constrains only the total: individual components may exchange energy, and the “conservation” of each is an assumption about the model, not a consequence of the identity	exact
Elasticity	Saint-Venant’s compatibility conditions; the elasticity complex; Beltrami–Michell equations; Volterra distortionsSaint-Venant 1864 (in Navier’s Résumé, 3rd ed.) · Beltrami 1886 · Michell 1899 · Volterra 1907 (multiply connected bodies) · Gurtin 1972, Sec. 14 · Eastwood 2000 · Arnold, Falk & Winther 2007, 2010	The elasticity complex: displacement u → strain ε = sym ∇u → incompatibility inc ε, (inc ε)_ij = e_ikle_jmn∂_k∂_mε_ln → div ↔ a complex derived from a vector-valued de Rham complex by the Bernstein–Gelfand–Gelfand construction (Eastwood; Arnold, Falk and Winther), with a second-order operator in the middle: not d² = 0 with renamed variables · inc(sym ∇u) = 0 ↔ the first composition identity; its converse on a simply connected body, that inc ε = 0 implies ε = sym ∇u, is Saint-Venant’s compatibility theorem, the exactness of the complex · div(inc ε) = 0 ↔ the second composition identity, the linearised contracted Bianchi identity · Volterra’s distortions ↔ inc ε = 0 on a ring with no single-valued u: the exactness failing on a multiply connected body · the Beltrami–Michell equations ↔ compatibility rewritten in stress, which adds Hooke’s law and equilibrium and is not a kinematic synonym	Displacement → compatible strain, identically (first composition identity). Compatible strain → displacement, on a simply connected body (the converse, a theorem with a hypothesis). Symmetric strain is a symmetric 2-tensor, not a differential form, and the complex that carries these identities is derived from, not identical to, the de Rham complex; that is the limitation. Finite strain: vanishing Riemann curvature of the strained metric, exact but non-linear, and again a statement that curvature vanishes, not an identity that curvature satisfies.	adds small strains, for the linear complex; a simply connected body, for the converse; for Beltrami–Michell, a linear elastic constitutive law and equilibrium · model linear elasticitybreaks the converse, not the identities: on a multiply connected body a compatible strain field can have a multivalued displacement (Volterra 1907), which is the classical origin of the dislocation; and an incompatible strain (inc ε ≠ 0) is not a contradiction but the signature of a distributed defect, which is the next row. The first draft of this page called Saint-Venant compatibility “the Bianchi identity on the strain metric”; that conflated the vanishing of a curvature with an identity that curvature satisfies, and is withdrawn	limited
Defects in crystals	Conservation of the Burgers vector; “a dislocation line cannot end inside a crystal”; dislocation density as torsion; incompatibilityKondo 1952/1955 · Nye 1953 · Bilby, Bullough & Smith 1955 · Kröner 1958, 1981 · Lazar 2011 (read)	The plastic distortion β^p ↔ a (vector-valued) 1-form that is not a gradient · the dislocation density α = curl β^p (Nye’s tensor) ↔ its exterior derivative, a vector-valued 2-form; in the geometric picture, the torsion of the crystal connection (Kondo; Bilby, Bullough and Smith) · ∇·α = 0 ↔ d² = 0 on β^p, equivalently the first Bianchi identity dT = 0 of a connection with torsion and zero curvature (Lazar, eqs 28–34) · “dislocation lines do not end” ↔ the integral form: the net Burgers vector through any closed surface vanishes · the incompatibility tensor η = curl curl ε^p ↔ divergence-free identically, by the same identity	Distortion → divergence-free dislocation density, identically; the continuity of dislocation lines is the integral form of d² = 0, with no material assumption. The geometric reading (torsion) adds nothing to the identity but places it: it is the first Bianchi identity of the geometry row for a connection with curvature zero and torsion α.	adds a continuum description, with the dislocation density averaged over many lines; the identification of torsion with dislocation density, which is a modelling choice with a large literature · model the continuum theory of dislocations (Kröner)breaks nothing within the flat-connection model: a dislocation line ending in the interior would need a distortion whose curl has divergence, which no field has, and lines end on surfaces, at nodes where the Burgers vectors sum to zero, or on grain boundaries, all of which the integral form allows. With disclinations the connection acquires curvature and the first identity itself changes, to DT^a = R^a_b∧ϑ^b; dT = 0 is then no longer the statement, and the row’s exactness is for the translational model only	exact
Fluid kinematics	Solenoidal vorticity; constancy of vortex-tube strength along the tube; “vortex lines cannot end in the fluid”Helmholtz 1858 · Batchelor 1967, Sec. 2.6 · Saffman 1992, Ch. 1	Velocity v ↔ a 1-form · vorticity ω = ∇×v ↔ its exterior derivative, a 2-form · ∇·ω = 0 ↔ d² = 0 · constant strength along a regular vortex tube ↔ flux balance, by Gauss · no vortex line terminates at an interior point where ω ≠ 0 ↔ the local flow of a smooth, non-vanishing vector field, not Gauss; lines may be aperiodic, and may approach zeros of ω · the same for B ↔ the electromagnetism row	Definition of vorticity → solenoidal vorticity, identically, in any flow whatever, viscous or not, steady or not. This is the kinematic statement and is exact; it is not the dynamical theorem that vortex lines move with the fluid, which needs the ideal conditions and is Audit 10.	adds nothing beyond a velocity field smooth enough for the curl and, for the non-termination statement, for the local flow of ω to exist · model any continuum with a velocity fieldbreaks nothing; the identity is indifferent to viscosity, compressibility and body forces. Some textbooks list “vortex lines cannot end” among Helmholtz’s theorems and number it differently from one another; the page does not assign it a number, and states it as flux balance rather than as a claim that every line closes or reaches a boundary. A quantised vortex in a superfluid is the converse failing: v = (ħ/m)∇φ has zero curl where the fluid is, and the vorticity is concentrated on the line where it is not, so the vortex is a hole in the region on which ∇φ is defined	exact
Thermodynamics	Maxwell relations; “equality of mixed partial derivatives”; integrability of dUMaxwell 1871, Theory of Heat, Ch. IX (the four relations) · Carathéodory 1909 (Pfaffian forms) · Callen 1985, Ch. 7	The state space ↔ a manifold with coordinates (S, V, N, …) · the equilibrium fundamental relation U(S, V, N), with dU = T dS − p dV + μ dN identifying T, −p and μ as its partial derivatives ↔ a 0-form and its exterior derivative, an exact 1-form · d(dU) = 0, with the other variables held fixed ↔ (∂T/∂V)_S = −(∂p/∂S)_V, and one such relation per pair of variables: the Maxwell relations · the other potentials ↔ Legendre transforms, each supplying its own exact 1-form and its own relation · at fixed composition, a reversible cyclic process with non-zero net work ↔ the 1-form δW_rev = −p dV on the equilibrium state space, which is in general not closed; heat and work in irreversible processes ↔ not in general determined by the equilibrium state alone	Existence of U as a state function → the Maxwell relations, identically. The converse, that a 1-form with all the mixed partials equal is the differential of a state function, is Poincaré’s lemma on the state space and is what the first law asserts about δQ_rev + δW_rev at fixed composition, not about either separately.	adds equilibrium states forming a smooth manifold; a fundamental relation that is C² on the region considered; the first and second laws to supply the 1-form · model classical equilibrium thermodynamicsbreaks where the chosen potential ceases to be C², as at some phase transitions, so that the relation holds on each side or in the sense of distributions; away from equilibrium, where the equilibrium state space on which the relation is written is not the description in use. The identity never fails; the differentiability or the state space it needs is what is missing	exact
Topology of fields	Closed but not exact; de Rham cohomology; Aharonov–Bohm effect; Dirac quantisation; Chern–Weil theoryde Rham 1931 · Aharonov & Bohm 1959 · Wu & Yang 1975 · Chern & Simons 1974 · Kobayashi & Nomizu 1969, Ch. XII	d² = 0 ↔ the sequence of forms is a cochain complex; closed forms modulo exact forms ↔ de Rham cohomology, which measures how far the converse fails · the vector potential outside a solenoid ↔ a closed 1-form on a region with a hole, not exact there; its integral around the hole ↔ the flux, which the phase (q/ħ)∮A·dl of a charged particle measures modulo 2π · the monopole ↔ F closed on ℝ³ minus a point, not exact: an obstruction in H², on a region that is simply connected, where the solenoid’s is in H¹; A defined on two patches (Wu and Yang) · invariant polynomials in F, such as tr(F∧F) ↔ closed, by the Bianchi identity and the invariance; their cohomology classes ↔ independent of the connection (Chern–Weil); integrality ↔ a further fact, holding for suitably normalised polynomials integrated over integral cycles, of which the instanton number is the case used here; a polynomial rescaled by √2 is still closed and connection-independent and its non-zero integer periods are no longer integers	DF = 0 with invariance → closedness of the characteristic forms, exactly; Chern–Weil → the class is independent of the connection; an invariant polynomial representing an integral characteristic class, with its prescribed normalisation → integral periods. Three separate steps, of which the identity supplies the first. In the Aharonov–Bohm and monopole cases the direction is the converse: the identity holds, and the physics is that exactness fails.	adds the topology of the region or bundle, as data; for the physical rows, quantum mechanics to make the phase observable · model quantum mechanics of a charged particle; classical gauge theory for the characteristic classesbreaks nothing; the row is where the other rows’ “breaks” entries are explained. The Aharonov–Bohm phase is observable only modulo 2π, which is why the integral of a closed form around the hole, and not a local field, is what matters; the Dirac condition is the requirement that the two patches agree up to a single-valued gauge transformation	exact
Discrete geometry and lattices	“The boundary of a boundary is zero”; lattice Bianchi identity; Regge calculus; discrete exterior calculusRegge 1961 · Misner, Thorne & Wheeler 1973, Ch. 15 · Wilson 1974 · Batrouni 1982 · Miller 1986 · Regge & Williams 2000 · Hamber & Kagel 2004 · Desbrun, Hirani, Leok & Marsden 2005, Def. 5.7 and Remark 5.1	Three correspondences of different strength. (a) Discrete exterior calculus: a simplicial complex ↔ the manifold; ∂² = 0 ↔ a combinatorial identity (Desbrun et al., Def. 5.5); the discrete d ↔ the coboundary, the transpose of ∂, so that d² = 0 holds exactly (Remark 5.1). (b) Holonomy identities: group elements on links ↔ parallel transport; plaquette holonomies ↔ curvature; the ordered product of the plaquette holonomies around a cube, each transported to a common basepoint along specified links, ↔ the identity element, exactly (the abelian case needs no transports); in Regge calculus the ordered product of the rotation matrices of the hinges meeting on an edge ↔ the identity, for arbitrary deficit angles (Hamber and Kagel, eq. 2.10), with a contracted form (eq. 9.23). (c) Dependencies among the discrete field equations ↔ the analogue of the contracted identity as a relation among the Regge equations, exact in flat space and approximate in the nearly flat case (Regge and Williams, Sec. I.A)	(a) and (b) are exact: the identity is built in by construction, or holds as a geometric identity of holonomies without restriction on curvature. (c) is where the correspondence is limited: the continuum identity’s role as a dependency among the field equations survives discretisation only approximately away from flatness, and that is the statement the verdict attaches to.	adds a triangulation or lattice; for the holonomy identities, a choice of basepoint and connecting transports; for Regge calculus, piecewise-flat geometry with curvature concentrated on hinges · model numerical and lattice formulations of the continuum theories abovebreaks in (c), not in (a) or (b). In finite-difference schemes that do not build the discrete operators as coboundaries, div_hcurl_h = 0 can fail, so that a numerically evolved B acquires a discrete divergence, the “spurious magnetic charge” that constrained-transport and mimetic methods are designed to exclude; a discrete curl of a discrete gradient failing to vanish is a separate defect of the same kind	limited

The same identity appears in several places not given rows: the closure of the symplectic form dω = 0 in Hamiltonian mechanics (Audit 03 and Audit 10 touch it), the Jacobi identity as d² = 0 on a Lie algebra, and the “conservation of the source” in any theory whose field equation is D⋆F = ⋆J. These are noted under extensions and not separately audited.

Historical relationships

The identity is older than any of its names. Its vector-calculus forms, that a gradient has no curl and a curl no divergence, were in use before the geometric identity was named, and the thermodynamic version was published by Maxwell in his Theory of Heat, whose ninth chapter gives the four relations between the physical properties of a substance (the chapter and its table-of-contents entry were checked in the 1872 third edition; the 1871 first edition was not seen). Saint-Venant’s compatibility conditions for the strain in an elastic body are of the same decade, in his 1864 edition of Navier’s lectures; Todhunter dates the result to 1860, and I have not resolved the difference. On the geometric side the priority is more tangled than the name suggests. Levi-Civita’s 1926 textbook, quoted at second hand, records that the identities on the curvature tensor were stated without proof by Padova in 1889, on a verbal communication from Ricci, and that Bianchi rediscovered them and published a direct proof in 1902; the widely repeated claim that Ricci found them in 1880 appears to conflate this with Voss’s 1880 paper, which is said to contain the contracted form, and I have verified neither the Voss attribution nor the Ricci date. The statement “Ricci 1880” should not be repeated on this page’s authority.

The identity entered physics as an identity, with its name, through general relativity, and the history is instructive because the people who needed it did not at first know it. Hilbert’s 1915 paper proves as its first theorem that a generally invariant action yields four identities among the field equations, and Einstein’s 1916 exposition derives the conservation of matter’s energy and momentum from the field equations in its eighteenth section, by way of his own variational identities rather than by the name Bianchi; Janssen and Renn’s study of Einstein’s route to the field equations shows the conservation-compatibility conditions of his 1914–15 formalism becoming the contracted Bianchi identities only once the Lagrangian is the curvature scalar, and Rowe’s account records that none of Hilbert, Lorentz, Einstein or Weyl was at that time versed in the full identities, citing Pais. Noether’s 1918 paper is where the two readings meet: her second theorem states that invariance under a group depending on arbitrary functions yields differential identities among the Lagrange expressions, and its sixth section addresses the “improper” energy law of general relativity that Hilbert had asserted. Cartan’s papers of 1922 to 1925 supplied the form in which the identities are now written, as structure equations for a connection with torsion. The gauge-field version has its own history: Yang and Mills’s 1954 paper does not state the identity; Lubkin’s 1963 paper on the geometric definition of gauge invariance discusses Bianchi identities for gauge fields, according to the review of this page (the paper’s record is verified and its content was not inspected); and Wu and Yang’s 1975 paper gave the global formulation in terms of bundles and patches that the page uses.

The material rows are older than their geometric reading and were unified late. Volterra’s 1907 memoir on multiply connected elastic bodies constructed the distortions now called dislocations by cutting a ring, displacing the faces and re-gluing, which is the converse failing in the elasticity row, and the recognition that the continuum dislocation density is the torsion of a connection came from Kondo in 1952 and from Bilby, Bullough and Smith in 1955, with Nye’s 1953 tensor as the object and Kröner’s 1958 monograph as the systematic theory. The “boundary of a boundary” phrase is Wheeler’s, and the fifteenth chapter of Misner, Thorne and Wheeler makes it the organising principle for the Bianchi identities; the discrete versions, in Regge’s 1961 calculus and in the lattice gauge theory that followed Wilson’s 1974 paper, inherit the identity from the combinatorics of the lattice. The pattern for the series, as far as the located sources show it: one identity, in use as vector calculus and in Maxwell’s thermodynamics before it had a geometric name, named in geometry in 1889–1902, needed in physics in 1915 by people who had to be told it, and given its general form by Cartan and by Noether, from different directions, within a few years of each other.

Notes by row

Differential geometry exact

The identity is a consequence of definitions and the row is exact for that reason, but the mechanism differs between the two lines and the page states it precisely. On forms, d is defined so that Stokes’s theorem holds, and ∂²C = 0 for oriented chains, the boundaries cancelling in pairs, gives d² = 0 by transposition. On a bundle with connection, the covariant exterior derivative D is not nilpotent: D²η = Ω∧η for a form η valued in an associated bundle, and D²η = [Ω, η] for an adjoint-valued one, which is the structure equation. Applying D once more to a section s gives D³s two ways, as D(Ω∧s) = (DΩ)∧s + Ω∧Ds and as Ω∧Ds, so (DΩ)∧s = 0 for every s, which is the Bianchi identity DΩ = 0. The first identity concerns torsion: DΘ = Ω∧θ, which for a torsion-free connection is the algebraic cyclic symmetry R_[abc]^d = 0. The textbook forms are in Kobayashi and Nomizu’s first volume (Chapter III, cited at record level; the theorem numbers were not verified) and in Nakahara (Section 7.8 for the structure equations and 10.3.5 for the gauge-field form, both from the table of contents). Priority is discussed above: Padova 1889 from Ricci, Bianchi 1902 with proof, the contracted identity attributed to Voss 1880 without verification. The row supplies no converse, and the page’s repeated distinction between the identity and the Poincaré lemma belongs here: a closed form of positive degree is exact on a star-shaped region, and on a general region a particular closed form is exact exactly when its class in the de Rham cohomology vanishes, which is the topology row.

Electromagnetism exact

Half of Maxwell’s equations are the identity and half are the field equation, and the difference is invisible in the vector form. With F = dA, the equations ∇·B = 0 and ∇×E + ∂_tB = 0 are the components of dF = 0 and hold for any A whatever; they carry no information about sources, media or dynamics. They are not thereby empty: once the measured fields are represented as the curvature of a potential, dF = 0 constrains which field configurations can be measured, and a magnetic charge, if found, would refute that representation, which is a physical outcome even though no experiment can refute an identity. The sourced pair, d⋆F = ⋆J, is the field equation; in a medium it is written either for the excitation form with a constitutive law relating it to F, or for F itself with J the total current, bound contributions included, and the identity is untouched either way. Charge conservation is the identity applied to the sourced pair: d(d⋆F) = 0 gives d⋆J = 0, which is ∂_tρ + ∇·j = 0. So conservation of charge is a consequence of Maxwell’s sourced equations plus d² = 0, and not of dF = 0, a point that accounts of “the two Maxwell equations that are identities” sometimes blur. The first draft of this page said that the row assumes “a global potential, or equivalently no magnetic charges”, and that was wrong on both counts: dF = 0 on a region does not by itself give a global A on that region, and the exterior of a magnetic pole is the standard example, with dF = 0 everywhere on it and no global potential because the flux through an enclosing sphere is not zero. Maxwell’s 1865 paper is cited at record level; the covariant reading is standard and no textbook pinpoint was verified for it, which the references disclose. Dirac’s 1931 paper, which was read, does not use the language of forms: it finds that a wave function in the field of a magnetic pole must have a nodal line running from the pole, that the potential is singular along a line that may be chosen arbitrarily, and that the pole strength is quantised (his equation 9, p. 68, in units where his h is ħ). Wu and Yang’s 1975 paper, whose abstract was read and whose two-patch construction is taken from Heras’s review, is the global formulation: A is defined on two overlapping regions with a gauge transformation on the overlap, F is closed on ℝ³ minus the pole and not exact, and Dirac’s condition is the requirement that the transition function be single-valued. That reading is the topology row. The verdict is exact because dF = 0 is d² = 0 and nothing else.

Gauge theory exact

The Yang–Mills field strength is the curvature of a connection, and its Bianchi identity DF = 0 is the geometry row’s second identity, with nothing added. Yang and Mills’s 1954 paper, which was read, defines the field strength (their equation 4) and writes the field equation with its current (equations 12 and 13); it does not state the identity D_μF_νλ + cyclic = 0. The geometric reading came in between: Lubkin’s 1963 paper is reported in the review of this page to discuss Bianchi identities for gauge fields, and its record is verified though its content was not inspected here; Wu and Yang’s 1975 paper is credited for the global formulation the page uses, not for the beginning of the geometric interpretation. Two things distinguish the row from the abelian one, and the first needs a step the first draft omitted. Applying D to the Yang–Mills equation gives D⋆J = D²⋆F = [F, ⋆F], and this vanishes not because D² is zero, which it is not, but because the spacetime pairing of the components of F with those of ⋆F is symmetric while the Lie bracket is antisymmetric, so the terms cancel in pairs. The resulting covariant conservation law is not an ordinary one, and the conserved isotopic-spin quantity of Yang and Mills’s equation 14 includes a contribution from the gauge field itself. Second, the magnetic side. In the abelian case dF = 0 can be read as the vanishing of a magnetic source current and, separately, as the statement that F is a closed 2-form whose periods are conserved. In the non-abelian case DF = 0 is the curvature Bianchi identity, and whether it supplies a gauge-invariant magnetic charge or a flux conservation law depends on how the current is defined, on the gauge group and on the global structure. The first draft of this page argued that D² ≠ 0 settles the matter, and that inference was invalid: an operator that is not identically zero can vanish on particular arguments, and it does here, D²F = [F, F] = 0 because the graded bracket of an even form with itself vanishes, and D²⋆F = [F, ⋆F] = 0 as just shown. The instanton number of the topology row, obtained from a characteristic class, is one conserved topological quantity among those the structure may or may not admit. Even in Maxwell’s theory two things are called a magnetic current: the vanishing source current on the right of dF = 0, and the closed 2-form F itself, whose flux through closed surfaces is conserved for that reason; they are different objects. Nakahara’s Section 10.3.5 is the textbook location, from the table of contents only.

General relativity exact

Every metric satisfies ∇^aG_ab = 0, so the four contracted identities are not a property of solutions; they are a property of the left-hand side of Einstein’s equation, and they force the right-hand side to be divergence-free. Three readings follow and all three are exact, with a qualification on the first that the first draft of this page did not make. The first is Misner, Thorne and Wheeler’s “automatic conservation of the source” (Chapter 17, cited at record level): joined to the field equation the identity gives ∇^aT_ab = 0. For matter whose equations of motion are equivalent to that statement, dust of non-vanishing density, or a perfect fluid once its equation of state p = p(ρ) and its particle conservation are supplied, the equations of motion follow from the field equations, which is what made the theory consistent in 1915. In general they do not. For a minimally coupled scalar field, ∇^aT_ab = (□φ − V′(φ))∇_bφ, and a constant φ₀ with V′(φ₀) ≠ 0 has a conserved stress tensor −V(φ₀)g_ab, which can source an Einstein metric with an effective cosmological constant, while the scalar’s own equation fails; the conservation law is satisfied and the matter equation is not. “The equations of motion are contained in the field equations” is therefore true for some matter and not as a universal, and the page withdraws the universal. The second reading is the constraint structure: the four normal projections G_abn^b = 8πG T_abn^b onto a spacelike hypersurface contain no second time derivatives and constrain the initial data, and the identity, together with the evolution equations and a consistently evolved source, guarantees that the constraints once satisfied remain satisfied; Arnowitt, Deser and Misner state this at their equation 3.16 (“the maintenance in time of these constraints is guaranteed by the Bianchi identities”), which was read in the 2008 reprint; the first draft wrote the constraints as the covariant components G_0μ, which is the normal-projection statement only in coordinates adapted to the foliation with zero shift, and the projection form is kept throughout. The third is the counting: four identities among ten equations leave six, matching the four coordinate functions that the metric’s ten components carry without physical content; Carroll’s lecture notes make the count in Chapter 4, and the same passage records that the four constrained components cannot be used to evolve the data. Hilbert’s 1915 first theorem is the earliest located statement of the four identities in this context, and Einstein’s 1916 exposition derives conservation from the field equations in Section 18, by his own route (see the history). What the row does not supply is a canonical globally conserved energy in a general spacetime: a Killing vector ξ supplies a conserved current T^abξ_b under the usual boundary conditions (a sufficient condition, not a necessary one: a conformal Killing vector does the same for a traceless conserved stress tensor), and where no such vector exists the pseudotensor and quasilocal-energy problem takes over and is left aside. Finally, with the Einstein tensor kept on the left, a source that is not divergence-free makes the equation inconsistent rather than “non-conserving”; a theory that alters the left-hand side has to be examined on its own complete equations.

Variational mechanics limited

Noether’s second theorem, in the Tavel translation that was read, states that if an action is invariant under a group whose transformations depend on ρ arbitrary functions and their derivatives, then ρ identities hold among the Lagrange expressions and their derivatives (Theorem II, with the identities at her equation 16 in Section 2). Applied to the Einstein–Hilbert action, whose invariance group is the diffeomorphisms, the identity so obtained is ∇^aG_ab = 0, and for that specialisation the correspondence with the contracted Bianchi identity is exact. The theorem is broader than the identity, though, and the first draft of this page ran the two together. Applied to the Yang–Mills action the identity produced is D(D⋆F) ≡ 0, an identity on the field equation, which is related to the curvature identity DF = 0 through the structure equation but is a different expression; the general theorem yields identities among the Euler–Lagrange expressions of whatever action is invariant, and those are d² = 0 or DF = 0 only in particular cases. That is why the row is limited. Noether’s Section 6 is also where “improper” is defined, and more narrowly than the first draft had it: a divergence relation is improper when its current can be composed from the Lagrange expressions and their derivatives together with terms whose divergence vanishes identically, and she shows that the energy relations of a theory invariant under the displacement group become improper exactly when the theory is invariant under an infinite group containing it. Three things are then to be kept apart: the identity among the equations, the conservation statement that holds on solutions, and the boundary contribution that the identically divergence-free terms carry; improperness constrains the second and does not by itself exclude meaningful gravitational energy charges defined through the third, or through Killing vectors, which is the question Hilbert, Klein and Noether argued over in 1917–18 and Rowe’s 2019 account, read, describes. The geometric rows obtain the curvature identities from the connection structure with no action, and this row obtains identities on field equations from a symmetry with no geometry; both derive what they derive, and neither is the unique derivation. Kosmann-Schwarzbach’s history is cited through Olver’s review.

Einstein–Cartan gravity exact

With torsion the algebraic identity R_[abc]^d = 0 is lost and replaced by the differential identity DΘ = Ω∧θ, while the second identity DΩ = 0 is untouched; Cartan wrote both in his 1922 note and his 1923–25 memoir (records), and the modern statement is Hehl and Obukhov’s equation 5, read. Trautman’s encyclopaedia article, which was read, states the two identities and then derives, under the heading “consequences of the Bianchi identities: conservation laws”, the balance equations for the energy–momentum current and the spin current, with curvature and torsion appearing as source terms on the right-hand sides; joined to the Einstein–Cartan field equations, in which torsion is algebraically tied to the spin density, these become the conservation laws of matter with spin. Hehl, von der Heyde, Kerlick and Nester’s 1976 review is the standard account and is cited at abstract level, since the full text was not accessible. The row is exact because the identities hold for any metric-compatible connection; what is model-dependent is whether spacetime torsion exists and whether it propagates, and the page takes no position on either.

Cosmology exact

The row is the general-relativity row evaluated on one metric and is included because the redundancy it produces is often presented as a coincidence. On a homogeneous isotropic metric the field equations reduce to two: the Friedmann constraint for H² and the acceleration equation for ä/a. The continuity equation ρ̇ + 3H(ρ + p) = 0 is the time component of ∇^aT_ab = 0, which Carroll’s notes derive directly (their equation 8.20, read), and it also follows from the constraint and the acceleration equation by differentiating the first and substituting the second; the two derivations agree because the contracted identity makes them agree. Only two of the three equations are independent, but the first draft’s remark that the choice of which two to keep is a convenience was too quick. Constraint plus acceleration gives continuity without condition. Constraint plus continuity gives the acceleration equation only after dividing by H, so on an interval where a and ρ are constant the two can hold while an unsuitable pressure violates the third. Acceleration plus continuity determine the constraint only up to an integration constant, which initial data must fix. The page records one further caution: the identity constrains only the total stress tensor, so where several components (matter, radiation, a scalar field) are present, the conservation of each separately is a modelling assumption about their interactions and is not delivered by the identity. Friedmann’s 1922 paper is cited at record level; the publisher’s record spells the name “Friedman”.

Elasticity limited

The first draft of this page described Saint-Venant’s compatibility conditions as “the Bianchi identity on the strain metric”, and that was wrong in a way worth spelling out, because the correct statement is better. For a symmetric strain ε define the incompatibility operator (inc ε)_ij = e_ikle_jmn∂_k∂_mε_ln. Two different identities then hold: inc(sym ∇u) = 0 for every displacement u, and div(inc ε) = 0 for every symmetric ε. The first says that the strain of a displacement is compatible, and its converse on a simply connected body, that a strain with inc ε = 0 is the strain of some displacement, is Saint-Venant’s compatibility theorem; the second is the linearised form of the contracted Bianchi identity. The vanishing of inc ε is the vanishing of the linearised curvature of the metric δ + 2ε, and a curvature vanishing is not the same thing as an identity that every curvature satisfies: curved metrics satisfy the Bianchi identities too. Nor is ε a differential form. It is a symmetric 2-tensor, and sym ∇ is not d applied to u. What carries these identities is the elasticity complex, displacement → strain → incompatibility → divergence, with a second-order operator in the middle, and Eastwood’s 2000 paper, read, constructs it from a vector-valued de Rham complex by the Bernstein–Gelfand–Gelfand procedure, remarking that “anything which is true of the de Rham complex should have a counterpart for linear elasticity” (p. 26, under “Consequences”); Arnold, Falk and Winther’s 2007 paper, read in its arXiv version, gives the same derivation in Section 4 with the two composition identities J∘ε = 0 and div∘J = 0 at their equation 3.2, and their 2010 survey calls the elasticity complex “quite different from the de Rham complex” (introduction; its Section 7 on the complex was not reached). So the correspondence is exact through a derived construction and not by renaming, which is what the verdict “limited” records. The original is in Saint-Venant’s 1864 edition of Navier’s lectures (record; the appendix was not located; Todhunter dates the result to 1860), Beltrami’s 1886 paper is credited with the rigorous proof, and Michell’s 1899 paper gives the stress form; the Beltrami–Michell equations add Hooke’s law and equilibrium to compatibility and are not a kinematic synonym for it. Gurtin’s 1972 treatise, Section 14, is the modern reference, cited from the table of contents. The converse is where the holes enter: Volterra’s 1907 memoir on multiply connected bodies found that on a ring a compatible strain can correspond to a displacement that is multivalued, changing by a rigid motion on going once round, and the six distortions he classified are the dislocations and disclinations of the defect row, produced by cutting the ring, displacing or rotating the faces and re-gluing. That is the exactness of the complex failing on a region with a hole, as in the Aharonov–Bohm and monopole cases of the topology row. The finite-strain statement is the vanishing of the full Riemann tensor of the strained metric, exact but non-linear, and again a curvature vanishing rather than an identity on curvature.

Defects in crystals exact

A distributed dislocation density is a plastic distortion whose curl does not vanish, and the statement that dislocation lines cannot end inside the crystal is d² = 0 applied to that distortion: α = curl β^p has zero divergence identically, so the net Burgers vector through any closed surface is zero and every line that enters a volume leaves it, or ends on another line at a node where the Burgers vectors sum to zero, or on a boundary. The geometric reading, that α is the torsion of a connection on the crystal and the continuity rule is Cartan’s first Bianchi identity for a connection with zero curvature, goes back to Kondo (1952, with the 1955 RAAG memoir as the located reference) and to Bilby, Bullough and Smith (1955), with Nye’s 1953 tensor as the object being described; Kröner’s 1958 monograph and his 1981 Les Houches lectures are the systematic treatments, both cited at record level. Lazar’s 2011 paper, which was read, states the identification and the identity in the form used here, for the translational, teleparallel model: torsion equals the exterior derivative of the translational gauge potential, T^a = dϑ^a (his equations 28 and 29), and dT^a = 0 is “the well-known conservation law of dislocations”, with the consequence that a dislocation line cannot end inside the body (equation 34). The incompatibility tensor of the elasticity row, inc of the plastic strain, is divergence-free by the second composition identity of that row, though no pinpoint for the statement was read and it is recorded as standard. The verdict is exact for the model stated: within a connection with torsion and zero curvature the continuity of dislocations is an identity, not a material law. It is not an unrestricted statement about crystals. When disclinations are present the connection acquires curvature and the first Bianchi identity itself changes, to DT^a = R^a_b∧ϑ^b (the geometric identity in Trautman’s form), so dT = 0 is no longer the statement, and disclinations are not audited.

Fluid kinematics exact

This row is the identity, and Audit 10 was the theorem; the page keeps them apart because textbooks often do not. Vorticity is the curl of velocity, so it is divergence-free in any flow at all, viscous, compressible, unsteady, and the integral form says that a regular vortex tube has the same strength along its length and that the vorticity flux through any closed surface is zero. The first draft went on to say that vortex lines therefore close on themselves or end on boundaries, and that is more than the identity gives: a vortex line cannot terminate at an interior point where the vorticity is non-zero, which follows from the local existence of the flow of a smooth non-vanishing vector field rather than from Gauss, but it can wind aperiodically without closing, and it can approach a point where the vorticity vanishes. Batchelor’s Section 2.6, on the vorticity distribution, is the textbook location (from the table of contents; the sentence itself was not seen), and Saffman’s first chapter covers the same ground (record). Helmholtz’s 1858 paper states the constancy of tube strength, and some presentations count “vortex lines cannot end” among his theorems while numbering it differently from one another; the page assigns it no number. The dynamical theorem, that vortex lines move with the fluid, is what needs the ideal conditions and is what the previous audit was about. The kinematic identity needs nothing. One consequence is worth stating because it is the converse again: in a superfluid the velocity is (ħ/m)∇φ and its curl vanishes wherever the phase is defined, so a quantised vortex is not a place where the identity fails but a line along which the phase is undefined, a hole in the region, and the circulation around it is the integral of a closed 1-form that is not exact there. That is the subject of Audit 13. What the identity gives is that closedness makes the circulation the same around any two loops that are homologous in the region; the quantisation of its values is supplied by the phase relation, not by the identity.

Thermodynamics exact

The Maxwell relations are d² = 0 on the space of equilibrium states, starting from the fundamental relation. Given the equilibrium relation U(S, V, N) with dU = T dS − p dV + μ dN, which is what identifies T, −p and μ as the partial derivatives of U, the statement d(dU) = 0, that is, the equality of the mixed second derivatives with the other variables held fixed, reads (∂T/∂V)_S = −(∂p/∂S)_V; each Legendre transform of U supplies another exact 1-form and another relation, and the four for a simple system are the four Maxwell published in the ninth chapter of the Theory of Heat (the 1872 edition’s chapter and its table-of-contents entry were checked). Callen’s textbook derives them in the same way, from the equality of mixed second partial derivatives, in its seventh chapter (from the table of contents). Carathéodory’s 1909 paper is the source of the Pfaffian-form language, and it is cited for that and not for the relations. The existence of U alone does not give the relations; the identification of its derivatives with the measurable T, p and μ is the fundamental relation, and that is where the physics enters. The row says where else it enters: at fixed composition, for reversible changes on the equilibrium state space, the heat 1-form δQ_rev = T dS and the work 1-form δW_rev = −p dV are in general not closed, and the first law is the statement that their sum is exact, which is the converse direction and needs U to be a state function; the second law is that δQ_rev has an integrating factor, which is Carathéodory’s subject. “In general” is needed: a fundamental relation of the form U = f(S) + g(V) makes both T dS and −p dV exact on their own, so non-closure is the typical case and not a theorem. Heat and work in irreversible processes are not in general determined by the equilibrium state alone, and nothing on this row is said about them. The mixed-partial argument needs a fundamental relation that is C² on the region considered, and the relations fail where that hypothesis does: where the chosen potential ceases to be C², as at some phase transitions, the relation holds on each side or in the sense of distributions; away from equilibrium the equilibrium state space on which the relation is written is not the description in use, whatever one takes the internal energy to be there. The identity never fails; the differentiability, or the state space, it needs is what is missing.

Topology of fields exact

This row is the converse made into a subject. Because d² = 0, the closed forms contain the exact ones, and the quotient, de Rham’s cohomology (his 1931 thesis, record), measures how far “closed” falls short of “exact” on a given region; on a region without holes the quotient is trivial and the two coincide for forms of positive degree, which is Poincaré’s lemma. The condition for a particular closed form to be exact is that its own class vanish, which is weaker than the vanishing of the group: a region with non-trivial cohomology admits non-exact closed forms, and still has exact ones. The physical cases on the page are of one kind, but not of one degree. Outside a solenoid the field vanishes and A is a closed 1-form on a region with a hole; it is not exact there, its integral around the hole is the enclosed flux, and Aharonov and Bohm’s 1959 paper, which was read, shows that the phase of a charged particle measures that integral even though no force acts anywhere on the particle’s path (their Section 2); the obstruction is in the first cohomology of the region. Around a magnetic pole F is a closed 2-form on ℝ³ minus a point and is not exact, so no single A serves; the region is simply connected, so this is an obstruction in the second cohomology and not the first, and Wu and Yang’s two patches with a gauge transformation on the overlap are the standard construction, with the Dirac condition the statement that the transformation is single-valued. In non-abelian theory the same identity does positive work, in three separate steps that the first draft compressed into one. Because DF = 0 and the polynomials are invariant, forms such as tr(F∧F) are closed; Chern–Weil theory (Kobayashi and Nomizu’s second volume, Chapter XII, cited at record level; Chern and Simons 1974, record) shows their cohomology classes to be independent of the connection; and for suitably normalised polynomials integrated over integral cycles the classes are integral, which is what makes the instanton number an integer. The third step is not a consequence of the first two: a polynomial rescaled by √2 is still closed and still connection-independent, and its non-zero integer periods are no longer integers. The forms in question live on the base manifold, and their classes lie in its cohomology while encoding the topology of the bundle over it; pulled back to the bundle’s total space they become exact, which is the transgression Chern and Simons begin from (record). The verdict is exact because all of this is the identity plus topology; the physics that makes the phases observable is quantum mechanics and is not audited.

Discrete geometry and lattices limited

Wheeler’s phrase, “the boundary of a boundary is zero”, is the identity in its combinatorial form, and the fifteenth chapter of Misner, Thorne and Wheeler, whose section titles were checked, builds the Bianchi identities on it. Three things then have to be kept apart, and the first draft of this page attached its verdict to the wrong one. First, discrete exterior calculus takes the phrase literally: Desbrun, Hirani, Leok and Marsden, whose paper was read, define the boundary of a simplicial chain (Definition 5.5, with ∂∂ = 0), define the discrete exterior derivative as its transpose, the coboundary (Definition 5.7), and observe that d² = 0 then follows immediately (Remark 5.1), so that the identity is exact on the mesh by construction. Second, holonomy identities. With group elements on links and the curvature on plaquettes, the ordered product of the plaquette holonomies around an elementary cube is the identity element, exactly; in the abelian case this is because each link appears twice with opposite orientation, and in the non-abelian case the plaquettes must first be parallel-transported to a common basepoint along specified links and multiplied in a specified order, since holonomies at different points cannot be compared otherwise. Batrouni’s 1982 paper treats the lattice identity (abstract) and Halpern’s dual formulation relies on its continuum form. Regge calculus has the same kind of identity, and it is exact: Hamber and Kagel’s 2004 paper, read, states that the ordered product of the rotation matrices of the hinges meeting on an edge is the identity (their equation 2.10), gives the fully contracted form for arbitrary deficit angles (equation 9.23), and says in its abstract that the identity “is valid for arbitrarily curved manifolds without a restriction to the weak field small curvature limit, but is in general not linear in the curvatures”. Third, and separately, the continuum identity’s role as a dependency among the field equations: Regge and Williams record that the Regge analogues of the Bianchi identities provide exact relations between sets of the Regge equations in flat space and approximate relations in the nearly flat case (Section I.A, read), so that the equations may not determine all the edge lengths. It is this third correspondence, and not the holonomy identity, that is limited, and the verdict now attaches to it; Miller’s 1986 paper derives the identity from ∂∂ = 0 on the skeleton and gives one contracted identity per vertex (abstract). The practical consequence is the numerical schemes. A finite-difference scheme that does not build its discrete divergence and curl as coboundaries can fail to satisfy div_hcurl_h = 0, so that a magnetic field evolved by a discrete curl acquires a discrete divergence, which is the “spurious magnetic charge” that constrained-transport and mimetic methods are built to exclude; a discrete curl of a discrete gradient that fails to vanish is a separate defect of the same origin, and not itself a magnetic charge.

The dictionary

Terms that name the same object across rows, within each row’s setting, except where an entry says otherwise.

The identity: d² = 0 · ∂² = 0 on oriented chains (Wheeler’s “boundary of a boundary”) · D²η = Ω∧η, the structure equation, from which DΩ = 0 follows · curl grad = 0 and div curl = 0 (vector calculus) · DΩ = 0 and DΘ = Ω∧θ (connections; Bianchi, Cartan) · ∇_[aR_bc]d^e = 0 (Riemannian geometry) · ∇^aG_ab = 0 (the contracted form) · dF = 0 (electromagnetism) · DF = 0 (gauge theory) · inc(sym ∇u) = 0 and div(inc ε) = 0 (the elasticity complex; Saint-Venant is the converse of the first) · ∇·α = 0 (dislocations) · ∇·ω = 0 (vorticity) · the Maxwell relations (thermodynamics) · the Noether identities (variational mechanics), of which ∇^aG_ab = 0 is the gravitational case
The object it acts on: a 0-form: potential φ, internal energy U · a 1-form: A, v, dU, the plastic distortion · a connection’s curvature 2-form: Ω, F, R_ab · a torsion 2-form: Θ, the dislocation density α · the elasticity complex, derived from a vector-valued de Rham complex: displacement, symmetric strain, incompatibility · chains: regions, plaquettes, simplices, and holonomies transported to a common basepoint
What it is joined to: additional dynamical or constitutive input, where a conservation law or an equation of motion is claimed · d⋆F = ⋆J (Maxwell) · D⋆F = ⋆J (Yang–Mills) · G_ab = 8πG T_ab (Einstein) · the Einstein–Cartan equations (torsion tied to spin) · the two Friedmann equations · the first law, U a state function · the constitutive law of the crystal, for what the dislocation density does, as opposed to what it must satisfy
What it then gives: charge conservation d⋆J = 0 · covariant current conservation D⋆J = 0, not an ordinary conservation law · ∇^aT_ab = 0, and the preservation of the four constraints · balance laws for energy–momentum and spin (Trautman) · the continuity equation of cosmology · the four Maxwell relations · the continuity of dislocation and vortex lines · closedness of the characteristic forms (Chern–Weil)
The converse: closed ⇒ exact: Poincaré’s lemma, locally, for positive degree; globally, when the form’s own cohomology class vanishes, which the vanishing of the group guarantees and non-vanishing does not exclude · not an identity, and not part of the verdict · its failure, by degree: the Aharonov–Bohm phase (H¹ of the region outside a solenoid), the Dirac monopole (H² of ℝ³ minus a point, which is simply connected), Volterra’s distortions and the multivalued displacement on a ring, the quantised vortex; and characteristic classes, closed forms on the base whose classes encode the topology of a bundle over it rather than an excluded point or line
The direction: definition → identity (geometry) · identity + field equation → conservation law (electromagnetism, gravity, cosmology) · symmetry → identity on the field equations (Noether’s second theorem) · construction → identity (discrete exterior calculus, holonomies) · identity + a representation of measured quantities → a constraint on what can be measured (electromagnetism, elasticity) · in no row does the identity alone → dynamics, or the conservation of a proposed source
Improper conservation law: Noether’s term (Sec. 6) for a divergence relation whose current can be composed from the Lagrange expressions and their derivatives together with identically divergence-free terms; the energy relations of general relativity are improper in this sense. The identity among the equations, the conservation statement on solutions and the boundary contribution of the divergence-free terms are three different things, and improperness does not by itself exclude conserved gravitational charges defined through the third or through Killing vectors

Verdict

Three claims are usually run together. The ledger supports the first once the operators and their domains are named in each row, supports the second only once the joined equation is named, and supports the third only as a statement about topology rather than about the identity.

First: the homogeneous Maxwell equations, the Bianchi identities of Riemannian geometry and of gauge theory, the contracted identity that makes Einstein’s equations consistent, the compatibility and incompatibility identities of elasticity, the continuity of dislocation and vortex lines, the Maxwell relations of thermodynamics and Wheeler’s boundary of a boundary share one exterior-calculus organisation: differential forms, curvatures and compatibility operators satisfy exact identities once their objects and operators are specified. Some rows use the identity directly, as d² = 0 on a form or DΩ = 0 on a curvature; others reach it through a contraction, through a specialisation of Noether’s theorem to one action, or through a complex derived from the de Rham complex rather than equal to it, as in elasticity. Ten of the thirteen rows are exact and three are limited: two because of the scope of the correspondence claimed or the construction that intervenes, in Noether’s theorem and in elasticity, whose composition identities are themselves exact; the third because the dependencies among the Regge field equations hold only approximately away from flatness. The structure is an identity wherever it applies: there is no ideal condition to buy it with and no regime in which it fails, and where a row’s “breaks” entry is not empty, what breaks is the object the identity is applied to, the model in which that object was defined, or the converse, never the identity.

Second: these identities constrain possible fields and the consistency of their equations, and they do not determine the dynamics. Both halves matter. Once measured quantities are represented as derivatives or curvatures, the identities say which configurations are possible at all, and that is physical content: dF = 0 constrains the fields of a theory that represents them as the curvature of a potential, and compatibility constrains the strains of a theory that represents them as symmetric gradients, so that an experiment which cannot refute an identity can still refute a representation. What the identities cannot do is supply the equation of motion or establish that a proposed source is conserved. Charge conservation is d² = 0 applied to Maxwell’s sourced equation, not to the homogeneous one; the conservation of energy and momentum in general relativity is the contracted identity applied to Einstein’s field equation, and without the field equation the identity establishes no conservation law for the proposed source; the Maxwell relations follow from the fundamental relation of equilibrium thermodynamics, and it is that relation that carries the physics. The recurring error the audit found in accounts of the subject is to let the identity carry the content of the equation it is joined to: “the Bianchi identity implies energy conservation” is true only with the field equation understood, and even then it gives the matter equations of motion only for matter whose equations are that conservation law. Noether’s second theorem runs the other way, from a local symmetry to identities on the field equations, of which the contracted Bianchi identity is the gravitational case; it and the geometric derivation each derive what they derive, and the page does not rank them.

Third: that the identity, being always true, is uninteresting. It is uninteresting in exactly the way that makes its converse interesting. That every exact form is closed is d² = 0; that a closed form of positive degree is locally exact is Poincaré’s lemma; whether it is globally exact depends on its own class in the cohomology of the region, and the failures are a large part of the physics on this page: the Aharonov–Bohm phase, which is (q/ħ)∮A·dl modulo 2π, the integral of a closed but not exact 1-form around a solenoid, an obstruction in H¹; the Dirac monopole, which is a closed but not exact 2-form around a point, an obstruction in H² on a simply connected region; Volterra’s distortions, which are compatible strains without single-valued displacements on a ring; the quantised vortex, which is a curl-free velocity with circulation around a line where the phase is undefined; and the characteristic classes of a gauge field, closed forms on the base manifold whose classes encode the topology of the bundle over it: the identity makes them closed, Chern–Weil makes them connection-independent, and an integral class requires the appropriate invariant polynomial and normalisation. None of these is a failure of the identity, and quantisation in none of them comes from the identity alone: it needs a single-valued phase, or an integral class with the right normalisation, added. Each is the identity holding on a region whose topology, or whose bundle’s topology, prevents the converse, and a page that reported d² = 0 without the converse would have reported the less interesting half.

What this means for a hydrodynamic theory of quantum mechanics or gravity. Any theory that describes gravity by an effective metric inherits the contracted Bianchi identity for free, because every metric satisfies it; Audit 01’s acoustic metric does, with no field equation behind it. So the identity is not evidence for such a theory and its satisfaction is not a prediction. What a theory must supply is the equation the identity is joined to: if the theory has an Einstein-like equation G_ab = κT_ab, then the identity forces its source to be divergence-free and the theory must say what that source is and why it is conserved, by a matter-theory argument the identity does not provide; if the theory has no such equation, the identity still constrains its geometry, but the contracted identity alone establishes no conservation law for the theory’s proposed matter source. The same holds for the converse. A theory in which vortices carry quantised circulation, or in which fields have monopole-like sources, is asserting that the relevant closed form is not exact on the region the theory uses, which is a claim about topology, and it must say where the obstruction lives: a line on which a phase is undefined, a point excluded from the domain, or a non-trivial bundle over the region, detected by a class on the region itself with no excluded point at all, as the characteristic classes show. It must also say what supplies the quantisation, since the identity does not: a single-valued phase, or an integral class with the right normalisation. The identity itself will hold in any such theory, on whatever objects it defines, and provides no discrimination among theories whatever; what does discriminate is the dynamics, the identification of the observable quantities with the geometric objects, and the global structure. The obligation the identity imposes is to name all three; the page’s ledger is a list of fields that did.

What this page does not claim

It does not claim that the identity determines any dynamics or conserves any proposed source on its own; nor does it claim the identity is empty of physics, since joined to a representation of measured quantities it constrains them. It does not claim that Saint-Venant compatibility is the Bianchi identity on a strain metric, or that the elasticity complex is the de Rham complex renamed. It does not claim that Noether’s second theorem is the Bianchi identity, only that its gravitational specialisation is. It does not claim that Einstein’s equations contain the equations of motion of every kind of matter. It does not claim that the converse holds in any row without a topological hypothesis, that the vanishing of a cohomology group is needed for a particular form to be exact, or that closedness and connection-independence of a characteristic form make its integrals integers; and it does not use “closed” and “exact” interchangeably. It does not claim that Ricci found the Bianchi identities in 1880, or that Voss found the contracted form; the first is unsupported by the located sources and the second is unverified. It does not claim that Einstein used the Bianchi identity by that name in 1915 or 1916, or that Yang and Mills stated the identity for their field; the located sources say otherwise. It does not claim that the covariant conservation law ∇^aT_ab = 0 defines a conserved energy, or resolve the argument over improper conservation laws. It does not claim that the Regge holonomy identities are approximate, since Hamber and Kagel’s is exact for arbitrary deficit angles; the limitation is in the relations among the Regge equations. It does not claim that spacetime has torsion, that dislocation density is torsion in any sense beyond the modelling identification the defect row records, or that dislocation continuity survives unchanged when disclinations are present. It does not claim that every vortex line closes or reaches a boundary. It does not claim priority or independence for any field’s version of the identity beyond what the sources located show, and the reading depth of each source is stated.

Extensions of the structure, and what is known about each

Every entry below was searched before being written.

Hamiltonian mechanics. A symplectic form is by definition a closed non-degenerate 2-form; closure is a condition imposed, not a consequence of d² = 0 for an arbitrary 2-form, and the Poisson bracket’s Jacobi identity is equivalent to it; the Jacobi identity itself can be read as d² = 0 on the Chevalley–Eilenberg complex of the Lie algebra. Both are standard and neither was separately sourced; they are the identity on objects this series meets in Audit 03 and Audit 10. Not audited here.

Anomalies. In quantum field theory the axial current is not conserved and, for massless fermions in four dimensions, the anomalous contribution to its divergence is proportional to tr(F∧F) (Adler 1969; Bell and Jackiw 1969, abstracts). The anomaly density tr(F∧F) is locally exact by the topology row, so the anomalous contribution can be expressed using a current that is not gauge invariant. Massive fermions also contribute an explicit mass term to the axial-current divergence. The identity is intact; the classical conservation law is what fails, and the failure is measured by a form the identity makes closed. Not audited here.

Constrained-transport and mimetic numerics. Numerical schemes that preserve ∇·B = 0 to round-off do so by building the discrete curl as a coboundary, so that the discrete identity holds exactly; Audit 10 recorded Evans and Hawley’s scheme for this reason. The discrete row records the general mechanism. Not audited.

Disclinations and the second identity. Rotational defects carry curvature rather than torsion and their continuity rule is the second Bianchi identity of the crystal connection; Kröner’s 1981 lectures treat them (record). Not audited.

Quantised circulation. The superfluid vortex in the fluid row is the converse failing on a line; the quantisation of the circulation needs the single-valuedness of a wavefunction and is the subject of Audit 13.

Verified since the first draft. Eastwood 2000 (read; pp. 23–26). Arnold, Falk and Winther 2007 (arXiv version read; Sections 1, 3 and 4). Hamber and Kagel 2004 (read; abstract, eq. 2.10, eq. 9.23). Regge and Williams 2000 (read; Section I.A). Noether 1918, Section 6 (read).

Still unverified on this page. Lubkin 1963 (record only; the statement that it discusses Bianchi identities is the reviewer’s). Arnold, Falk and Winther 2010 beyond the introduction. The Voss 1880 attribution for the contracted identity, and the “Ricci 1880” date (both from Wikipedia-level accounts; Levi-Civita’s 1926 text is quoted at second hand). Bianchi 1902 (record). Saint-Venant 1864, including which appendix (record; Todhunter’s 1860 dating not reconciled). Beltrami 1886, Michell 1899, Volterra 1907 (records). Gurtin 1972, Section 14, and Love and Sokolnikoff (tables of contents or records). Kondo’s 1952 paper (the 1955 memoir is the located reference). Nye 1953, Bilby, Bullough and Smith 1955 (records); Kröner 1958 and 1981 (records; every internal claim taken from Lazar 2011). Batchelor’s Section 2.6 sentence and Saffman’s chapter (tables of contents). Maxwell’s 1871 first-edition pagination (the 1872 third edition’s chapter was checked). Callen Chapter 7 and Frankel (tables of contents). Kobayashi and Nomizu theorem numbers; Nakahara section contents beyond the table of contents. Textbook pinpoints for the covariant form of Maxwell’s equations and for the Yang–Mills Bianchi identity (none read). Hilbert 1915 (record, with the theorem from Rowe and Byers); Klein 1918 (pages not located). Einstein 1916 (Section 18 read in the Bose translation, not the Einstein Papers Project’s). Wu and Yang’s own section numbers (the two-patch construction from Heras 2018). Chern and Simons 1974 (record). Hehl et al. 1976 beyond the abstract; Wald and Misner, Thorne and Wheeler pinpoints (records and chapter titles only). Regge 1961 (record); Hamber 2009 (record). Wilson 1974 (abstract). Dirac’s equation 9, whose “= 2” was read from a noisy scan and should be checked against a clean copy.

Next in the series

Audit 13 is quantised circulation: the statement that the circulation of a superfluid, and the fluxoid of a superconducting ring, come in integer multiples of a fixed quantum, the magnetic flux itself being quantised only where the current contribution to the fluxoid vanishes, as Audit 10 recorded. It is the first place in the series where the failure of this page’s converse does physical work, though not on its own: the quantisation is the integral of a closed 1-form around a hole, and it is the phase relation and the single-valuedness of the phase, added to the topology, that fix its values. The audit will ask which fields have the phase and which have only the hole.

References

Bibliographic details for every entry (authors, title, year, journal, volume, page) were verified against the publisher’s record or an equivalent index, with the exceptions marked at the entry: the Saint-Venant appendix; the journal pagination of Arnold, Falk and Winther 2007, taken from the review; the Beltrami and Padova page ranges; Kondo 1952; the Einstein 1916 record, resolved by DOI only; and the journal paginations of Hehl and Obukhov, Heras, Lazar, and Regge and Williams, which were read in their arXiv versions. The flag beside each says how far the content itself was consulted. Where a claim in the text rests on a specific passage, the section, equation or page is given in the text.

full text full text or the relevant section consulted abstract abstract consulted secondary known through a secondary account only record bibliographic record only; content not inspected

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Arnold, D. N., Falk, R. S. & Winther, R. (2010). Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Amer. Math. Soc. 47, 281–354. abstract (introduction read; Sec. 7, “The elasticity complex”, not reached)

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Desbrun, M., Hirani, A. N., Leok, M. & Marsden, J. E. (2005). Discrete exterior calculus. arXiv:math/0508341. full text (Defs 5.5–5.7, Remark 5.1)

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Gurtin, M. E. (1972). The linear theory of elasticity. In C. Truesdell (ed.), Handbuch der Physik VIa/2, Springer, 1–295. record (Sec. 14, “Compatibility”, from the table of contents)

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Hehl, F. W., von der Heyde, P., Kerlick, G. D. & Nester, J. M. (1976). General relativity with spin and torsion: foundations and prospects. Rev. Mod. Phys. 48, 393–416. abstract

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Kröner, E. (1981). Continuum theory of defects. In R. Balian, M. Kléman & J.-P. Poirier (eds), Physics of Defects (Les Houches 1980), North-Holland, 215–315. record

Lazar, M. (2011). On the fundamentals of the three-dimensional translation gauge theory of dislocations. arXiv:1003.3549 (published in Math. Mech. Solids, 2011; pagination not verified). full text (arXiv version; eqs 28, 29 and 34)

Levi-Civita, T. (1926). The Absolute Differential Calculus. Blackie. secondary (p. 182, on Padova, Ricci and Bianchi, as quoted in encyclopaedia accounts)

Lubkin, E. (1963). Geometric definition of gauge invariance. Ann. Phys. (N.Y.) 23, 233–283. record (content not inspected; cited on the reviewer’s report that it discusses Bianchi identities)

Maxwell, J. C. (1865). A dynamical theory of the electromagnetic field. Phil. Trans. R. Soc. Lond. 155, 459–512. record

Maxwell, J. C. (1871). Theory of Heat. Longmans. secondary (Ch. IX, “Relations between the physical properties of a substance”, checked in the 1872 third edition’s table of contents, pp. 163–169 there)

Michell, J. H. (1899). On the direct determination of stress in an elastic solid, with application to the theory of plates. Proc. Lond. Math. Soc. 31, 100–124. record

Miller, W. A. (1986). The geometrodynamic content of the Regge equations as illuminated by the boundary of a boundary principle. Found. Phys. 16, 143–169. abstract

Misner, C. W., Thorne, K. S. & Wheeler, J. A. (1973). Gravitation. W. H. Freeman. record (Ch. 15, “Bianchi identities and the boundary of a boundary”, section titles from the publisher’s contents; Ch. 17 at record level)

Nakahara, M. (2003). Geometry, Topology and Physics, 2nd ed. Institute of Physics Publishing. record (Secs 7.8 and 10.3.5 from the table of contents)

Navier, C. L. M. H. (1864). Résumé des leçons données à l’École des Ponts et Chaussées sur l’application de la mécanique, 3rd ed., with notes and appendices by A. J. C. Barré de Saint-Venant. Dunod. record (the appendix containing the compatibility conditions not located; Todhunter dates the result to 1860)

Noether, E. (1918). Invariante Variationsprobleme. Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1918, 235–257. English translation by M. A. Tavel, Transport Theory Stat. Phys. 1, 186–207 (1971); arXiv:physics/0503066. full text (translation; Theorem II, eq. 16 in Sec. 2, and the definition of “improper” in Sec. 6)

Nye, J. F. (1953). Some geometrical relations in dislocated crystals. Acta Metall. 1, 153–162. abstract

Padova, E. (1889). Sulle deformazioni infinitesime. Rend. Accad. Naz. Lincei, ser. 4, 5, 174–178. secondary (via Levi-Civita 1926; the page range beyond p. 176 not verified)

Regge, T. (1961). General relativity without coordinates. Nuovo Cimento 19, 558–571. record

Regge, T. & Williams, R. M. (2000). Discrete structures in gravity. arXiv:gr-qc/0012035 (published in J. Math. Phys. 41, 2000; pagination not verified). full text (arXiv version, Sec. I.A: the relations concern the Regge equations)

Rowe, D. E. (2019). Emmy Noether on energy conservation in general relativity. arXiv:1912.03269. full text (Secs 4, 7 and 9)

Saffman, P. G. (1992). Vortex Dynamics. Cambridge University Press. record (Ch. 1)

Trautman, A. (2006). Einstein–Cartan theory. In J.-P. Françoise, G. L. Naber & T. S. Tsou (eds), Encyclopedia of Mathematical Physics, vol. 2, Elsevier, 189–195; arXiv:gr-qc/0606062. full text (arXiv version; the sections on the Bianchi identities and their consequences)

Volterra, V. (1907). Sur l’équilibre des corps élastiques multiplement connexes. Ann. Sci. École Norm. Sup. 24, 401–517. record

Voss, A. (1880). Zur Theorie der Transformation quadratischer Differentialausdrücke und der Krümmung höherer Mannigfaltigkeiten. Math. Ann. 16, 129–179. record (the attribution of the contracted identity to this paper is not verified)

Wilson, K. G. (1974). Confinement of quarks. Phys. Rev. D 10, 2445–2459. abstract

Wu, T. T. & Yang, C. N. (1975). Concept of nonintegrable phase factors and global formulation of gauge fields. Phys. Rev. D 12, 3845–3857. abstract (the two-patch construction from Heras 2018)

Yang, C. N. & Mills, R. L. (1954). Conservation of isotopic spin and isotopic gauge invariance. Phys. Rev. 96, 191–195. full text (eqs 4, 12–14; the Bianchi identity is not stated)

Change log

0.4 · 12 Sep 2026 Final consistency edits after a third external review: the fluid ledger separates tube strength (flux balance, Gauss) from non-termination (local flow of a smooth non-vanishing field), with the smoothness stated on the assumptions line; the topology ledger’s direction column states integrality as requiring an invariant polynomial representing an integral class with its prescribed normalisation; “the whole structure” becomes “the starting structure”; the plain-terms Maxwell sentence qualified as local; the anomaly extension separates the mass term from the exactness statement. Reviewer’s assessment: ready to publish with the existing reading-depth disclosures. Published.

0.3 · 12 Sep 2026 Consistency pass after a second external review, which found that several corrections made in 0.2 had not reached the opening, the dictionary and the closing note, and that one argument remained invalid. Opening: gravity sentence conditioned on Einstein’s equation; elasticity described as a complex built from the identity; the plain-language converse no longer says “only in a region with no holes”; “definition rather than a theorem” replaced; the four routes to the identity named after the equation box. Dictionary: “without which the identity says nothing physical” removed. HQG note and Audit 13 teaser narrowed as the review specified. Yang–Mills: the inference from D² ≠ 0 to the absence of a magnetic-current reading withdrawn as invalid (D²F = [F, F] = 0 and D²⋆F = 0 vanish though D² does not), replaced by the dependence on current definition, gauge group and global structure. Thermodynamics: fixed composition; heat and work “in general” not closed, with the U = f(S) + g(V) counterexample; C² fundamental relation; “where the chosen potential ceases to be C²”; irreversible heat and work “not in general determined by the equilibrium state alone”. Chern–Weil: classes on the base manifold encoding bundle topology, not cohomology “of the bundle”; integrality requires the appropriate polynomial and normalisation; the √2 example stated for non-zero periods. Pinpoints: Lazar eq. 35 removed from the curvature–torsion sentence (it is the dislocation continuity relation); Eastwood quotation at p. 26; Arnold, Falk and Winther 2007 pagination added from the review, and the 2010 survey’s Section 7 named. The explanation of the three limited verdicts corrected to match the Regge row. Scope: perfect-fluid closure named; Killing vectors sufficient not necessary; zero-shift qualification; vortex-line non-termination from the flow, not Gauss; circulation invariance under homologous loops, quantisation from the phase; anomaly scoped to massless four-dimensional fermions; Aharonov–Bohm phase written with its factor. Still draft.

0.2 · 11 Sep 2026 Revised after external review; the identification was overstated in the first draft and three rows are corrected in substance. Central statement: D is not nilpotent, D²η = F∧η, and the Bianchi identity follows from that structure; ∂² = 0 written for oriented chains. Elasticity: Saint-Venant compatibility is not the Bianchi identity on a strain metric; the row is rebuilt on the elasticity complex, derived from a vector-valued de Rham complex by the Bernstein–Gelfand–Gelfand construction (Eastwood; Arnold, Falk and Winther), with its two composition identities inc∘sym∇ = 0 and div∘inc = 0 distinguished; Beltrami–Michell named as constitutive plus equilibrium; verdict limited. Noether: the second theorem is broader than the identity, exact only in its gravitational specialisation, the Yang–Mills Noether identity being D(D⋆F) ≡ 0; “improper” redefined from Section 6; “the one place where the identity is derived” removed; verdict limited. Regge: the holonomy identity of Hamber and Kagel is exact for arbitrary deficit angles, and the limitation is moved to the relations among the Regge equations; non-abelian plaquette comparison at a common basepoint; spurious magnetic charge as failure of div_hcurl_h = 0. “An identity says nothing physical until joined to a field equation” withdrawn as too strong: identities constrain represented fields and the consistency of equations, and do not determine dynamics or the conservation of a proposed source; the falsifiability sentence and “in no row does the identity alone → physics” removed. Electromagnetism: the false assumption “a global potential exists, or equivalently no magnetic charges” corrected; exactness conditioned on the form’s own cohomology class; H¹ and H² obstructions distinguished; medium formulation stated. Chern–Weil: closedness, connection-independence and integrality separated. Yang–Mills: the cancellation [F, ⋆F] = 0 supplied; the magnetic-current discussion rewritten. General relativity: “matter equations contained in the field equations” withdrawn as a universal (scalar-field counterexample); constraints as normal projections; Killing-vector currents; modified equations scoped. Cosmology: the pairings of the three equations stated with their conditions. Defects: exactness scoped to the flat translational model, with DT = R∧ϑ when curvature is present. Fluids: endpoint claim reduced to flux balance. Thermodynamics: fundamental relation made the starting point; reversible heat and work scoped; “the object stops existing” replaced. History: “a century” withdrawn; Lubkin 1963 added at record level. Hamiltonian extension and Audit 13 teaser corrected. Still draft.

0.1 · 11 Sep 2026 First draft on the series template. Sources verified before drafting; the verification changed the text in four places. The “Ricci 1880” priority claim was not supported and is replaced by Levi-Civita’s account (Padova 1889 from Ricci; Bianchi 1902), with the Voss attribution left unverified. Yang and Mills’s 1954 paper was read and found not to state the Bianchi identity, so the row credits the geometric reading to Wu and Yang. Einstein’s 1916 derivation of conservation was read and found to proceed by his own identities rather than by the name Bianchi, and the history follows Janssen and Renn and Rowe on this. Carroll’s notes derive the continuity equation from ∇^aT_ab = 0 directly rather than from the Friedmann pair, and the cosmology row gives both routes. Still draft.

Erratum · 12 Sep 2026 The dates on the entries above were first printed as 14–15 September 2026 in error; they now give the days on which each revision was made and the page published, Sydney time.

Method note. This page was drafted with AI assistance (Anthropic’s Claude) working from my brief and revised with me. Bibliographic details were verified against publisher records for every reference before the text was written, except for the gaps explicitly identified in the references; the flag beside each says how far the content itself was consulted, and where a claim rests on a particular passage the section, equation or page is given in the text. Claims that a topic was unexplored were searched before being kept. The verdicts are mine and I expect some of them to be argued with; corrections are welcome and will be logged. Content on this site is licensed CC BY-SA 4.0.

11 · The Bianchi identities

In plain terms

What is being audited

The ledger

Historical relationships

Notes by row

Differential geometry exact

Electromagnetism exact

Gauge theory exact

General relativity exact

Variational mechanics limited

Einstein–Cartan gravity exact

Cosmology exact

Elasticity limited

Defects in crystals exact

Fluid kinematics exact

Thermodynamics exact

Topology of fields exact

Discrete geometry and lattices limited

The dictionary

Verdict

What this page does not claim

Extensions of the structure, and what is known about each

Next in the series

References

Change log

Like this:

In plain terms

What is being audited

The ledger

Historical relationships

Notes by row

Differential geometry exact

Electromagnetism exact

Gauge theory exact

General relativity exact

Variational mechanics limited

Einstein–Cartan gravity exact

Cosmology exact

Elasticity limited

Defects in crystals exact

Fluid kinematics exact

Thermodynamics exact

Topology of fields exact

Discrete geometry and lattices limited

The dictionary

Verdict

What this page does not claim

Extensions of the structure, and what is known about each

Next in the series

References

Change log

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