Local potentials and smooth circle-bundle connection data

Use the real ordered field of The real numbers and Complete ordered field (least-upper-bound property). For a nonempty set $S\subseteq \mathbb{R}$ bounded above, $\sup S$ denotes the unique real upper bound $u$ such that $u\le v$ for every upper bound $v$. Existence is the least-upper-bound property established for these reals by The reals are complete; if $u,u^{\mathrm{\prime }}$ both satisfy it, $u\le u^{\mathrm{\prime }}$ and $u^{\mathrm{\prime }}\le u$ imply equality. The notation is restricted to this domain.

For nonempty $S$ bounded below, use Greatest lower bound (infimum) and Every nonempty set bounded below has an infimum: $\inf S=-\sup (-S)$, where $-S=\{-s:s\in S\}$. Indeed a lower bound $l$ for $S$ gives the upper bound $-l$ for $-S$; negating its least upper bound gives a lower bound for $S$ greater than every other one.

Neither $\sup \mathrm{\varnothing }$ nor $\inf \mathrm{\varnothing }$ is defined in $\mathbb{R}$: every real is an upper and lower bound, with no least upper or greatest lower bound. If another construction stipulates the value zero for an empty path family, that stipulation is a separate total definition, not $\sup \mathrm{\varnothing }=0$. Infinite values require an explicitly specified extended ordered set; no extended-real field operations are introduced here.

Let $X$ be a set and use the ordered real field of The real numbers, Complete ordered field (least-upper-bound property) and Ordered field. In the convention of Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric, a real metric is a function $d:X\times X\to \mathbb{R}$ satisfying, for all $x,y,z\in X$, separation $d(x,y)=0⟺x=y$, symmetry $d(x,y)=d(y,x)$ and the triangle inequality $d(x,z)\le d(x,y)+d(y,z)$. A real pseudometric replaces separation by $d(x,x)=0$ and retains the other two axioms. All values are finite real numbers. Neither definition assumes nonnegativity as an additional axiom.

For either structure, put $B_d(x,r)=\{y\in X:d(x,y)<r\}$ for real $r>0$. The induced topology consists of the subsets $O\subseteq X$ for which every $x\in O$ admits $r>0$ with $B_d(x,r)\subseteq O$. After the nonnegativity deduction in the well-definedness lemma, these balls form a basis: $x\in B_d(x,r)$ by reflexivity, and if $z\in B_d(x,r)\cap B_d(y,s)$, the positive radius $t=\min \{r-d(x,z),s-d(y,z)\}$ has $B_d(z,t)$ contained in that intersection by the triangle inequality. This argument also works for pseudometrics and for empty $X$ (the empty topology). No infinite-distance, semimetric or quasimetric convention is consumed.

For any real metric or pseudometric $d$ on a set $X$ in Real metric and pseudometric convention, $d(x,y)\ge 0$ for every $x,y\in X$. Thus these three-axiom definitions already have values in $[0,\mathrm{\infty })$, including the vacuous assertion when $X=\mathrm{\varnothing }$.

For $a>0$, integer $m$ and integer $n\ge 1$, define the rational regime by

where $a^{1\mathrm{/}n}$ is the unique positive root of $s^n=a$ in Existence and uniqueness of $n$-th roots: a unique $a^{1\mathrm{/}n}\ge 0$ with $(a^{1\mathrm{/}n}{)}^n=a$, and integer powers include reciprocals for negative $m$. This is Rational powers $a^r$ of a positive base; Rational powers do not depend on the representative proves independence of the fraction representative and Laws of rational exponents gives the rational exponent laws. For the rational-exponent regime at base zero, retain the convention of Rational powers $a^r$ of a positive base and Integer powers $a^m$: $0^0=1$, $0^r=0$ for rational $r>0$, and $0^r$ is undefined for rational $r<0$. This records the integer zero-exponent convention together with the positive-rational supplementary clause; it does not extend the real-exponent convention.

The separate real-exponent regime is Real powers for positive bases, with the zero-base positive-exponent convention: for $a>0$ and $x\in \mathbb{R}$, $a^x=\exp (x\log a)$, with the laws of The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents. For real exponents at base zero, set $0^x=0$ only when $x>0$; $0^0$ and $0^x$ for $x<0$ are undefined, exactly as in Real powers for positive bases, with the zero-base positive-exponent convention. The rational and real exponent operations are separate definitions; at zero base and exponent zero, state which regime is intended: the rational/integer convention gives $1$, while the real-power convention leaves the expression undefined. No arbitrary real power is assigned to a negative base. The real-power laws are supplied by that theorem, not inferred from the rational root formula. Use the real-number structure of The real numbers.

In particular $s(q)=q^{1\mathrm{/}2}$ for $q>0$ is the unique positive solution of $s(q{)}^2=q$. Its smoothness and derivative are a separate well-definedness result, not an assumption about fractional notation.

For $q>0$, let $s(q)$ be the unique positive root specified in Explicit rational and real power regimes. Then $s:(0,\mathrm{\infty })\to (0,\mathrm{\infty })$ is smooth and $s^{\mathrm{\prime }}(q)=1\mathrm{/}(2s(q))$.

Let $A\subseteq \mathbb{R}$, let $c\in \mathbb{R}$ be a limit point of $A$, let $f:A\to \mathbb{R}$ and $L\in \mathbb{R}$. Assume the indexed axiom of countable choice ${\mathrm{A}\mathrm{C}}_{\omega }$ of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). Then the epsilon-delta condition ${\lim }_{x\to c}f(x)=L$ of The $\epsilon$-$\delta$ limit ${\lim }_{x\to c}f(x)=L$ of $f:A\to \mathbb{R}$ at a limit point $c$ of $A$ is equivalent to the following: for every sequence $(x_n{)}_{n\in \mathbb{N}}$ with $x_n\in A\setminus \{c\}$ and $x_n\to c$, one has $f(x_n)\to L$. Sequence and sequence convergence mean Sequences of reals: bounded, eventually, frequently, tails, subsequences and Limits and Cauchy sequences of reals. The forward direction uses no choice; the reverse direction uses only the stated indexed ${\mathrm{A}\mathrm{C}}_{\omega }$, not arbitrary-index choice.

Let $M$ be a smooth Hausdorff second-countable manifold without boundary, in the sense of Smooth manifolds and their smooth charts. Let $g$ be a positive definite smooth Riemannian metric as in Riemannian metrics, symmetric cotangent-bundle connections, and covariant Hessians. Use the Levi-Civita connection of Levi civita connection, whose existence and uniqueness for the supplied smooth Riemannian metric are proved by Fundamental theorem of riemannian geometry. The coordinate formula is supplied by Christoffel formula for the levi civita connection: its coefficients are ${\mathrm{\Gamma }}_{ij}^k=\frac{1}{2}{g}^{kl}({\mathrm{\partial }}_i{g}_{jl}+{\mathrm{\partial }}_j{g}_{il}-{\mathrm{\partial }}_l{g}_{ij})$. A parametrized affine geodesic is a smooth curve $\gamma :J\to M$ on an interval with nonempty interior satisfying ${\mathrm{\nabla }}_{\dot{\gamma }}\dot{\gamma }=0$, equivalently ${\ddot{\gamma }}^k+{\mathrm{\Gamma }}_{ij}^k(\gamma ){\dot{\gamma }}^i{\dot{\gamma }}^j=0$, where covariant differentiation along the curve is defined by Covariant derivative along a curve; the coordinate equation is the expansion of that derivative in a coordinate basis. Its maximal domain is an interval on which it cannot be extended with the same affine parameter as a solution. Two-sided geodesic completeness means that every maximal affine geodesic is defined on all of $\mathbb{R}$.

Independently, for a real metric $d$ in Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric, a sequence $(x_n)$ is Cauchy when for every $\epsilon >0$ there is $N$ with $d(x_m,x_n)<\epsilon$ whenever $m,n\ge N$. It converges to $x$ when for every $\epsilon >0$ eventually $d(x_n,x)<\epsilon$. Metric completeness means that every Cauchy sequence converges in the given metric space. In a Riemannian application the metric must be the distance induced by $g$ on a connected component: the infimum of lengths $\int \sqrt{g(\dot{\alpha },\dot{\alpha })}dt$ over piecewise-$C^1$ connecting curves. Its definition and finite metric property are supplied by Riemannian distance on a connected manifold and Riemannian distance is a metric. Specify that induced metric, rather than an arbitrary compatible metric. Distinct components are treated separately; no infinite distance is inserted into a real metric.

These are different completeness conditions. Their equivalence is a Hopf–Rinow theorem requiring its own hypotheses and proof; it is not part of this definition. In particular, a Lorentzian metric is not a positive definite distance, and this convention supplies no Lorentzian Hopf–Rinow implication.

The boundaryless restriction matters: the Euclidean interval $[0,1]$ is metrically complete, while the curve $\gamma (t)=1\mathrm{/}2+t$ stays inside it only for $-1\mathrm{/}2\le t\le 1\mathrm{/}2$ and cannot be continued there as a two-sided affine geodesic with velocity $1$. Completeness of this interval follows because a real Cauchy sequence converges in $\mathbb{R}$ by The reals are complete and its limit remains between $0$ and $1$. This boundary example explains the domain restriction; it is not an instance of the boundaryless equivalence.

Use the first-order language of The first-order language of set theory: $\in$, $=$, formulas with parameters, and class abbreviations: all quantified objects are sets, and a definable class $\{x:\phi (x,p)\}$ abbreviates a formula with set parameters $p$. It is not a new set or a domain for unrestricted quantification. Work with the usual set axioms, in particular The Axiom of Power Set: $\mathrm{\forall }x\mathrm{\exists }y\mathrm{\forall }z(\mathrm{\forall }t(t\in z\to t\in x)\to z\in y)$ and The Axiom Schema of Separation: for each formula $\phi$, $\mathrm{\forall }\bar{p}\mathrm{\forall }x\mathrm{\exists }y\mathrm{\forall }z(z\in y\leftrightarrow (z\in x\wedge \phi (z,\bar{p})))$, and the ordered pairs, products and functions of The Kuratowski ordered pair $(a,b):=\{\{a\},\{a,b\}\}$, The Cartesian product $A\times B:=\{z\in \mathcal{P}(\mathcal{P}(A\cup B)):\mathrm{\exists }a\in A\mathrm{\exists }b\in Bz=(a,b)\}$ and A function is a relation $f$ with $(a,b)\in f$ and $(a,c)\in f$ implying $b=c$; $f:A\to B$, the value $f(a)$, domain and codomain. The power-set axiom supplies a collecting set; Separation trims it to exactly $\mathcal{P}(X)=\{Y:Y\subseteq X\}$.

A small category has a set $O$ of objects, a set $A$ of arrows, source and target functions $s,t:A\to O$, an identity function $e:O\to A$, and a composition function on the set $\{(a,b)\in A\times A:s(a)=t(b)\}$, where $ab$ denotes $a\circ b$ and the laws are $s(e(x))=t(e(x))=x$, $s(ab)=s(b)$, $t(ab)=t(a)$, $e(t(a))a=a=ae(s(a))$, and $(ab)c=a(bc)$ whenever the displayed compositions are defined. It is locally small when each specified hom-collection is a set. A definable large category is a schema of formulas for its objects, arrows and operations, with those laws asserted as formulas; it is not a set of all objects or a category of all large categories. No Grothendieck universe is assumed.

All indexed families in the local bundle construction have a supplied set index $I$. If $U_i\subseteq M$ and $G$ is a set, their tagged disjoint union is the subset $D=\{(i,x,z)\in I\times M\times G:x\in U_i\}$, formed by Separation. Given a set equivalence relation $E\subseteq D\times D$, each class $[u{]}_E=\{v\in D:(u,v)\in E\}$ is a subset of $D$, and the quotient is the set

Thus the quotient is obtained by Separation in $\mathcal{P}(D)$, without choosing representatives. Transition-map graphs and connection data are subsets of specified finite products and power sets, not arbitrary proper-class objects.

For the Cauchy-development coding application, fix a set manifold $\mathrm{\Sigma }$ and the tensor-bundle total spaces over $\mathrm{\Sigma }\times \mathbb{R}$. A code consists of an open set $D\subseteq \mathrm{\Sigma }\times \mathbb{R}$, graphs of a metric and a two-form on $D$ inside those fixed total spaces, a time orientation, and a fixed initial embedding. All of its coordinates belong to fixed products and power sets. Separation selects the tuples satisfying the specified smoothness, Lorentzian, field-equation and initial-data formulas. Hence these codes form a set. This size assertion does not prove field-equation existence, maximality or uniqueness, and makes no proper-class representative choice.

Let $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$ be the ordered set of The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined. Define a new total operation $\star :\bar{\mathbb{R}}\times \bar{\mathbb{R}}\to \bar{\mathbb{R}}$ as follows. If $a=0$ or $b=0$, put $a\star b=0$. If both factors are finite, put $a\star b=ab$ using real multiplication. In the remaining cases both factors are nonzero and at least one is infinite; put

$a\star b:=\{\begin{array}{ll}+\mathrm{\infty },& \mathrm{sgn}(a)\mathrm{sgn}(b)=1,\\ -\mathrm{\infty },& \mathrm{sgn}(a)\mathrm{sgn}(b)=-1,\end{array}$

where $\mathrm{sgn}(a)=1$ for $a>0$ and $\mathrm{sgn}(a)=-1$ for $a<0$ in the extended order. The clauses exhaust the domain and agree when a zero factor is finite, so they specify a unique value for every pair. Thus $0\star (\pm \mathrm{\infty })=0$, $(+\mathrm{\infty })\star (-2)=-\mathrm{\infty }$, and $(-\mathrm{\infty })\star (-\mathrm{\infty })=+\mathrm{\infty }$.

For any set $X$ and maps $f,g:X\to \bar{\mathbb{R}}$, their extended pointwise product with zero at infinite factors is the map $f\star g:X\to \bar{\mathbb{R}}$ defined by $(f\star g)(x)=f(x)\star g(x)$.

This explicitly chosen operation does not redefine the partial extended-real arithmetic of The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined, where $0\cdot \mathrm{\infty }$ is left undefined. It is not a limit rule or a field multiplication. No measurability theorem is part of this definition: such a theorem must specify the measurable structures and check the finite, zero, and positive or negative infinite factor sets separately.

Let $(X,\mathrm{\Sigma },\mu )$ be a measure space in Measure spaces. Give $\mathbb{R}$ the order topology ${\mathcal{T}}_{<}$ of The order topology of a linearly ordered set, with the open rays as a subbasis; order-convex sets, order-density, the least upper bound property, and linear continua, which that supplier identifies with the usual real topology, and let $\mathcal{B}(\mathbb{R}):={\sigma }_{\mathbb{R}}({\mathcal{T}}_{<})$ be its Borel sigma-algebra as in The Borel sigma-algebra of a topological space. A measurable real function is a map $f:X\to \mathbb{R}$ measurable from $(X,\mathrm{\Sigma })$ to $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ in the sense of A measurable function between measurable spaces. For $1\le p<\mathrm{\infty }$, let the representative space be ${\mathcal{L}}^p(\mu )=\{f:X\to \mathbb{R}:f\text{ is measurable and }{\int }_X\mathrm{\mid }f{\mathrm{\mid }}^{p}d\mu <\mathrm{\infty }\}$, exactly the supplier definition The function space ${\mathcal{L}}^p(\mu )$ for $0<p<\mathrm{\infty }$. For $p=\mathrm{\infty }$, the representatives are the measurable real functions for which $\mu (\{x:\mathrm{\mid }f(x)\mathrm{\mid }>C\})=0$ for some finite $C\ge 0$. The level sets used here are measurable by Closure properties of measurable functions used by the integral. This finite-bound condition is exactly $\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}<\mathrm{\infty }$ in The essential supremum of a measurable function with respect to a measure and The space $L^{\mathrm{\infty }}(\mu )$ of essentially bounded measurable functions: if such a $C$ exists then the essential supremum is at most $C$; conversely, if the essential supremum is finite, The essential supremum is attained as the least essential bound proves that this finite value itself is an essential bound.

For measurable $f,g:X\to \mathbb{R}$, the difference $f-g$ is measurable by the addition and scalar-multiplication clauses of Closure properties of measurable functions used by the integral. Therefore

$\{x:f(x)\mathrm{\ne }g(x)\}=(f-g{)}^{-1}(\mathbb{R}\setminus \{0\})\in \mathrm{\Sigma },$

because $\mathbb{R}\setminus \{0\}$ is open in the stated topology and hence Borel. On either representative space put

$f\sim g⟺\mu (\{x\in X:f(x)\mathrm{\ne }g(x)\})=0.$

By The null subspace of measurable functions that vanish almost everywhere and Measure-null sets and almost-everywhere statements relative to a measure, this is equivalent to $f-g\in \mathcal{N}(\mu )$, so it is exactly the relation in The space $L^p(\mu )$ as the quotient by null functions. The element $[f]\in L^p(\mu )$ is the equivalence class of $f$; an explicit representative is a specified measurable function belonging to that class. The quotient norms are

$\mathrm{\parallel }[f]{\mathrm{\parallel }}_p={\left({\int }_X\mathrm{\mid }f{\mathrm{\mid }}^{p}d\mu \right)}^{1\mathrm{/}p}(1\le p<\mathrm{\infty }),\mathrm{\parallel }[f]{\mathrm{\parallel }}_{\mathrm{\infty }}={\mathrm{ess\hspace{0.17em}sup}}_X\mathrm{\mid }f\mathrm{\mid },$

with independence of representative and the norm axioms supplied by The $L^p$ norm descends to the quotient and makes $L^p$ a normed space for $1\le p\le \mathrm{\infty }$.

A pointwise statement such as $f(x)=0$ concerns the selected representative. A property invariant under almost-everywhere equality, such as $f=0$ almost everywhere, concerns the class. No value of $[f]$ at a particular point is defined by this quotient.

This definition uses real-valued representatives. When a coupled Sobolev construction uses complex functions, it specifies their real and imaginary representatives separately; this item does not assert a complex-valued $L^p$ quotient or choose representatives simultaneously for an arbitrary family of classes. The definition imposes no additional choice principle.

Let $(M,g,N,o)$ be the smooth, Hausdorff, second-countable, boundaryless oriented and time-oriented Lorentzian four-manifold of Oriented Lorentzian fields, observers and coordinate conventions. Its smooth real $k$-forms are sections of ${\bigwedge }^k{T}^{\ast }M$, written ${\mathrm{\Omega }}^k(M)$, and $d$ is the exterior derivative of Lorentzian exterior calculus and Hodge star. A smooth two-form $F$ is closed when $dF=0$ and exact when $F=dA$ for some smooth real one-form $A$ on the whole base.

By De rham cohomology, $H_{\mathrm{d}\mathrm{R}}^2(M)=Z^2(M)\mathrm{/}{B}^2(M)$, where $Z^2(M)$ is the vector space of closed smooth two-forms and $B^2(M)$ is the vector space of exact smooth two-forms. Thus, for closed $F$, the statement $[F]=0$ means exactly that $F=dA$ for a global smooth one-form $A$. This equivalence is the quotient definition; it does not assert that every closed form is exact.

Identify $\mathbb{C}$ with ${\mathbb{R}}^2$ by $z=x+iy$ and set $U(1)=S^1:=\{(x,y)\in {\mathbb{R}}^2:x^2+y^2=1\}$. Its topology is the subspace topology. Specify its smooth structure by the four charts

$V_{x,ϵ}=\{(x,y)\in S^1:ϵx>0\},(x,y)\mapsto y,y\mapsto (ϵ\sqrt{1-y^2},y),$

and

$V_{y,ϵ}=\{(x,y)\in S^1:ϵy>0\},(x,y)\mapsto x,x\mapsto (x,ϵ\sqrt{1-x^2}),$

for $ϵ\in \{-1,1\}$, with each coordinate range $(-1,1)$. They cover $S^1$; their transition maps are restrictions of the positive square-root branch and coordinate projections, hence are smooth by The positive square-root branch is smooth and $C^k$ Euclidean maps are closed under componentwise algebra and composition. In these coordinates multiplication is the restriction of $(x,y)(u,v)=(xu-yv,xv+yu)$ and inversion is $(x,y)\mapsto (x,-y)$, so both are smooth; they are also continuous for the stated topology. Thus this is a smooth Lie group and, in particular, a topological group as required by Principal g bundle and associated fiber bundle.

Let $E:\mathbb{R}\to U(1)$ be $E(t)=e^{it}=(\cos t,\sin t)$; the Cartesian formula is $\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$. The trigonometric derivative identities of The derivatives of sine and cosine are cosine and minus sine, iterated, show that $E$ is smooth in the $C^{\mathrm{\infty }}$ sense of $C^k$ Euclidean maps and diffeomorphisms: every derivative of either coordinate is $\pm \sin$ or $\pm \cos$, and these functions are continuous by A function differentiable at $c$ is continuous at $c$. Surjectivity is $t\mapsto (\cos t,\sin t)$ is a bijection from $[0,2\pi )$ onto the real unit circle. If $E(t)\in V_{x,ϵ}$, the chart coordinate $\sin t$ has derivative $\cos t=x\mathrm{\ne }0$; if $E(t)\in V_{y,ϵ}$, the coordinate $\cos t$ has derivative $-\sin t=-y\mathrm{\ne }0$. The smooth inverse-function theorem Choice-free smooth inverse function theorem in Euclidean space therefore makes $E$ a diffeomorphism from some open interval about $t$ onto an open circle arc. Every smooth map from an open subset of $M$ to $U(1)$ consequently has, near each point, a smooth real phase lift obtained by composing with such a local inverse.

A normalized circle-connection datum on $M$ consists of a supplied countable open cover $(U_i{)}_{i\in I}$, smooth transition maps $g_{ij}:U_i\cap U_j\to U(1)$, smooth real one-forms $a_i\in {\mathrm{\Omega }}^1(U_i)$, and a fixed constant ${\kappa }_0>0$. Require $g_{ii}=1$ and $g_{ij}{g}_{jk}=g_{ik}$ on every triple intersection. Around each point of $U_i\cap U_j$, take any smooth real phase lift $f_{ij}$ with $g_{ij}=e^{i{f}_{ij}}$; require

on every such phase-lift neighborhood. If $f$ and $\tilde{f}$ are two lifts on a connected neighborhood, $\ker (\exp )=2\pi i\mathbb{Z}$, and $\exp z=\exp w$ exactly when $z-w\in 2\pi i\mathbb{Z}$ gives $f-\tilde{f}\in 2\pi \mathbb{Z}$ pointwise; continuity makes this difference constant, so $df=d\tilde{f}$. Thus the compatibility condition is independent of the chosen lift. The globally defined form $\lambda (g):=-i{g}^{-1}dg$ is real, and The complex exponential is entire and its complex derivative is itself together with The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$ gives $\lambda (g)=df$ on every phase-lift neighborhood. Hence the equivalent lift-free condition is $a_j=a_i+\lambda (g_{ij})$.

The right principal $U(1)$-bundle convention is as defined in Principal g bundle and associated fiber bundle: it has a continuous right action and locally trivial equivariant product charts, with free and transitive action on each fibre. The compatible transition functions are the gluing data for such a bundle, in the convention that a point with fibre coordinate $z_i$ in chart $i$ has coordinate $z_j=g_{ij}^{-1}{z}_i$ in chart $j$. No global real-valued angle on $U(1)$ is assumed.

The constant ${\kappa }_0$ is a chosen normalization with units of webers; it is not Planck's constant and is not a unit of electric charge. On a chart whose coordinates have length units, the normalized connection components $a_i$ have units ${\mathrm{m}}^{-1}$; the corresponding local potential $A_i:={\kappa }_0{a}_i$ has components $\mathrm{W}\mathrm{b}{\mathrm{m}}^{-1}$, and its curvature $F_i:=d{A}_i={\kappa }_{0}d{a}_i$ has components $\mathrm{W}\mathrm{b}{\mathrm{m}}^{-2}$. This definition imposes no flux or charge quantization.

The typed conventions formerly reserved on EMG-MG01 are authored earlier on this page: Finite-real supremum and infimum domain, Real metric and pseudometric convention, Metric nonnegativity from the three axioms, Explicit rational and real power regimes, Sequence-limit criterion with explicit choice use, Boundaryless domain for geometric completeness, and Set-sized objects and definable-class schemas. They fix the finite-real, topology, power, choice, geometric-domain and set-size interfaces. The finite chart formulas here make no countable selection; the explicit countable-choice hypothesis of the sequence criterion is spent only in that lemma's reverse implication. Authored carriers and local proofs do not themselves assert independent acceptance or repair an unrelated original supplier.

Let $D\subseteq {\mathbb{R}}^4$ be an open set star-shaped about $p\in D$, meaning $p+t(x-p)\in D$ for every $x\in D$ and $0\le t\le 1$. Let $F=\frac{1}{2}{F}_{ab}d{x}^a\wedge d{x}^b$ be a smooth real closed two-form on $D$. Use Einstein summation over coordinate indices $a,b,c\in \{0,1,2,3\}$. Define

Then $A=A_{b}d{x}^b$ is a smooth real one-form on $D$ and $dA=F$. More generally, if $\alpha ={\alpha }_{a}d{x}^a$ is a smooth closed one-form on $D$, then

is smooth and $d\chi =\alpha$. The first formula gives a local potential for every smooth closed two-form in a star-shaped coordinate chart; it does not assert global exactness.

The radial calculation uses only the stated smoothness and star-shaped-domain hypotheses and the local exterior-derivative formula. The additional typed dependencies preserve the approved foundational conventions; their statements do not introduce a choice step into this explicit finite-interval calculation.

Let $M$ be a smooth, Hausdorff, second-countable oriented four-manifold, and let $(U_i,g_{ij},a_i;{\kappa }_0)$ be a normalized circle-connection datum as in Local potentials and smooth circle-bundle connection data. Use the set-sized-object and quotient convention of Set-sized objects and definable-class schemas. Then the chart data construct a smooth Hausdorff second-countable right principal $U(1)$-bundle $\pi :P\to M$ and a global smooth real connection one-form $\omega$ on $P$. Its curvature is basic: there is a smooth closed two-form $f\in {\mathrm{\Omega }}^2(M)$ with $d\omega ={\pi }^{\ast }f$ and $f{\mathrm{\mid }}_{U_i}=d{a}_i$. The local potentials $A_i={\kappa }_0{a}_i$ satisfy $d{A}_i=F$, where $F={\kappa }_{0}f$ is global. For two connections on the same fixed bundle with the same transition functions, their local-form differences are restrictions of one global base one-form; adding a global one-form to all local representatives stays in that bundle sector.