# Holographic Thermodynamics Requires a Chemical Potential for Color

Manus R. Visser  Department of Theoretical Physics, University of Geneva, 24
quai Ernest-Ansermet, 1211 Genève 4, Switzerland

###### Abstract

The thermodynamic Euler equation for high-energy states of large-$N$ gauge
theories is derived from the dependence of the extensive quantities on the
number of colors $N$. This Euler equation relates the energy of the state to
the temperature, entropy, number of degrees of freedom and its chemical
potential, but not to the volume or pressure. In the context of the
gauge/gravity duality we show that the Euler equation is dual to the
generalized Smarr formula for black holes in the presence of a negative
cosmological constant. We also match the fundamental variational equation of
thermodynamics to the first law of black hole mechanics, when extended to
include variations of the cosmological constant and Newton’s constant.

Introduction. The thermodynamics of black holes remains one of most important
theoretical advancements in gravitational physics of the past half-century. In
semiclassical general relativity the energy $E$, entropy $S$ and temperature
$T$ of a black hole can be identified with its mass, horizon area $A$ and
surface gravity $\kappa$, respectively, Bekenstein (1972); *Bekenstein:1973ur;
Hawking (1975); Bardeen  _et al._ (1973) (setting $k_{B}=\hbar=c=~{}1$)

$\displaystyle E=M\qquad\quad S=\frac{A}{4G}\qquad\quad
T=\frac{\kappa}{2\pi}.$ (1)

For static, asymptotically flat black holes these thermodynamic quantities are
simply related by the Smarr formula Smarr (1973)
$M=\frac{d-1}{d-2}\frac{\kappa A}{8\pi G}$ in $d+1$ spacetime dimensions. The
dimension dependent factors are a peculiar property of the Smarr formula,
which typically do not appear in the Euler equation of thermodynamics relating
all extensive and intensive quantities. Furthermore, for black holes in the
presence of a negative cosmological constant $\Lambda$ there is an additional
term in the generalized Smarr formula $M=\frac{d-1}{d-2}\frac{\kappa A}{8\pi
G}-\frac{1}{d-2}\frac{\Theta\Lambda}{4\pi G}$ Caldarelli  _et al._ (2000);
Kastor  _et al._ (2009). In an _extended_ version of black hole thermodynamics
the cosmological constant is interpreted as the pressure $P=-\Lambda/8\pi G$,
and is treated as a thermodynamic state variable in its own right Kastor  _et
al._ (2009); Dolan (2011a); *Dolan:2011xt; Cvetic  _et al._ (2011); Kubiznak
and Mann (2015). Its conjugate quantity $\Theta$ in the extended first law of
black holes is regarded as (minus) the thermodynamic volume. An important, but
slightly odd aspect of this interpretation is that the mass of the black hole
is identified with the enthalpy instead of the internal energy of the system
(see Kubiznak  _et al._ (2017) for a recent review).

From a holographic perspective the dimension dependent factors and the
$\Lambda$ term in the generalized Smarr formula remain somewhat elusive. In
holography, or gauge/gravity duality, the thermodynamics of black holes in the
‘bulk’ spacetime is equivalent to the thermodynamics of large-$N$ strongly-
coupled gauge theories living on the asymptotic boundary of the bulk spacetime
Maldacena (1999); Gubser  _et al._ (1998); Witten (1998a). In particular, the
thermodynamic variables (1) for black holes correspond with the energy $E$,
entropy $S$ and temperature $T$ of thermal states in the boundary theory
Hawking and Page (1983); Witten (1998b). The generalized Smarr formula
relating these variables in the gravity theory should be dual to the Euler
relation for the thermodynamic quantities in the field theory. But the
pressure interpretation of the $\Lambda$ term in the Smarr formula does not
directly carry over to the field theory, since the bulk pressure $P$ is not
dual to the boundary pressure $p$ and the bulk thermodynamic volume is not
related to spatial volume $V$ of the boundary Johnson (2014).

In several works Kastor  _et al._ (2009); Johnson (2014); Dolan (2014); Kastor
_et al._ (2014); Caceres  _et al._ (2015); Karch and Robinson (2015) it has
been suggested that varying $\Lambda$ is related to varying the number of
colors $N$, or the number of degrees of freedom $N^{2}$, in the boundary field
theory. For gauge theories arising from coincident D-branes, varying $N$
corresponds to varying the number of branes. Further, in conformal field
theories (CFTs) the number of degrees of freedom is given by the central
charge $C$, whose variation takes us from one CFT to another. In holographic
CFTs dual to Einstein gravity the central charge corresponds to $C\sim
L^{d-1}/G$ Henningson and Skenderis (1998); *Henningson:1998ey; Freedman  _et
al._ (1999); Myers and Sinha (2010); *Myers:2010tj, where $L$ is the curvature
radius of the bulk geometry, related to the cosmological constant via
$\Lambda=-d(d-1)/2L^{2}$, and $G$ is Newton’s constant in $d+1$ dimensions. So
varying $C$ in the boundary CFT is dual to varying $\Lambda$ and $G$ in the
bulk theory. In addition, it was argued in Karch and Robinson (2015) that
varying $\Lambda$ in the bulk does not only correspond to varying $C$ (or
$N$), but also to varying the volume $V$ of the spatial boundary geometry.
This is because the bulk curvature radius $L$ is equal to the boundary
curvature radius for a particular boundary metric. We show, however, that for
a different boundary metric varying $\Lambda$ only corresponds to varying $C$
(and $E$) in the boundary theory, and not to varying $V$. Overall, by building
on (and refining) the holographic dictionary in Karch and Robinson (2015), we
propose a precise boundary description of extended black hole thermodynamics.

In this paper we argue that the dual field theory description of black hole
thermodynamics requires a chemical potential $\mu$ for the central charge (see
also Dolan (2014); Kastor  _et al._ (2014)). From the large-$N$ scaling
properties of the field theory we derive the holographic Euler equation

$E=TS+\nu^{i}B_{i}+\mu C,$ (2)

and show that it is holographically dual to the generalized Smarr formula.
Here $\nu_{i}$ are additional chemical potentials for the conserved quantities
$B_{i}$ (such as charge and angular momentum). As expected, the dimension
dependent factors do not feature in (2), and the $\Lambda$ term is
incorporated in $\mu$. Moreover, $E$ is the standard energy of the field
theory and not the enthalpy. Strikingly though, there is no $pV$ term in the
large-$N$ Euler equation. We explain why this is consistent with the
fundamental equation of thermodynamics, $dE=TdS-pdV+\nu_{i}dB_{i}+\mu dC$, in
which both $V$ and $C$ are varied. Finally, we match this boundary variational
equation with the extended first law of black holes.

Thermodynamics of large-$N$ theories. We first derive the Euler equation from
the scaling properties of gauge theories at finite temperature in the
large-$N$ ’t Hooft limit ’t Hooft (1974), $N\to\infty$ for fixed coupling
$\lambda\equiv g^{2}N$. Large-$N$ $SU(N)$ gauge theories on compact spaces,
with fields in the adjoint representation, exhibit a separation between low-
energy states with energy of $\mathcal{O}(N^{0})$, and high-energy states for
which the energy scales as $E\sim N^{2}$ Gross and Witten (1980); Witten
(1998b); Sundborg (2000). This is because the low-energy excitations consist
of color singlets, whose energy is independent of $N$, and at high energies
all the $N^{2}$ adjoint degrees of freedom contribute on the same footing. The
low-energy states are in a confined phase and are characterized by a thermal
entropy that grows with energy, whereas the high-energy states are in a
deconfined phase and behave as a gas of free particles (at nonzero $\lambda$
there could exist an intermediate phase Aharony  _et al._ (2004)). Other gauge
theories at finite temperature display a similar (de)confinement phase
transition, but the energy in the deconfined phase may scale with a different
power of $N$, e.g. as $N^{3}$ for an exotic theory in $d=6$ with $(0,2)$
supersymmetry Witten (1995); *Strominger:1995ac.

In conformal theories the central charge $C$ counts the number of field
degrees of freedom. For $SU(N)$ gauge theories with conformal symmetry the
central charge scales as $C\sim N^{2}$ at large $N$, so high-energy states
satisfy $E\sim C$. Since holographic CFTs are the main examples of large-$N$
theories we have in mind, we denote the number of degrees of freedom simply as
$C$ for all large-$N$ theories.

High-energy states in large-$N$ theories obey interesting large-$N$ scaling
laws and are dual to black holes in holographic field theories. By definition
the internal energy of these equilibrium states depends on extensive
quantities, such as entropy $S$, volume $V$ and conserved quantities $B_{i}$,
and on the (intensive) central charge $C$, i.e. $E=E(S,V,B_{i},C).$ Formally,
we can vary the energy with respect to each of these quantities, while holding
the others fixed. This leads to Gibbs’ fundamental equation of thermodynamics

$\displaystyle dE$ $\displaystyle=TdS-pdV+\nu^{i}dB_{i}+\mu dC,$ (3)

where the temperature $T$, pressure $p$, chemical potentials $\nu^{i}$, and
the chemical potential $\mu$ conjugate to $C$ are defined as

$\displaystyle T$ $\displaystyle\equiv\left(\frac{\partial E}{\partial
S}\right)_{V,B_{i},C},\quad p\equiv-\left(\frac{\partial E}{\partial
V}\right)_{S,B_{i},C},$ (4) $\displaystyle\nu^{i}$
$\displaystyle\equiv\left(\frac{\partial E}{\partial
B_{i}}\right)_{S,V,C},\quad\mu\equiv\left(\frac{\partial E}{\partial
C}\right)_{S,V,B_{i}}.$

The variation of $C$ in (3) moves one away from the original field theory
content to a theory with a different number of degrees of freedom. On the
other hand, for variations which only compare different thermodynamic states
within the same theory, the variable $C$ is kept fixed. Hence, depending on
the ensemble, the central charge could be varied or fixed in the fundamental
equation of thermodynamics. However, we observe next that the central charge
necessarily has to appear in the large-$N$ Euler relation.

The entropy and conserved quantities scale with the central charge for high-
energy states, $S,B_{i}\sim C$, reflecting the contribution from all the
degrees of freedom. Thus, the energy function obeys the following scaling
relation

$E(\alpha S,V,\alpha B_{i},\alpha C)=\alpha E(S,V,B_{i},C),$ (5)

with $\alpha$ a dimensionless scaling parameter. This means that in the
deconfined phase of large-$N$ theories on compact spaces the energy is not an
extensive function. Differentiating with respect to $\alpha$ and putting
$\alpha=1$ leads to the Euler equation

$E=TS+\nu^{i}B_{i}+\mu C.$ (6)

Notice that pressure and volume do not appear in this Euler equation, since
the volume does not generically scale with $C$. It does scale with $C$ in the
infinite-volume limit of CFTs, i.e. $pV\\!=\\!-\mu C$ as $V\to\infty$ (see
Appendix C). By varying this relation and employing the fundamental
variational equation (3), we find a slightly unusual Gibbs-Duhem equation

$0=SdT+pdV+B_{i}d\nu^{i}+Cd\mu.$ (7)

The variation of volume (instead of pressure) features in this equation, since
the Euler relation does not involve a $pV$ term.

Furthermore, in the grand canonical ensemble the thermodynamic potential or
free energy is defined as

$W\equiv E-TS-\nu^{i}B_{i}=\mu C.$ (8)

It follows from the fundamental equation (3) that the grand canonical free
energy is stationary at fixed $(T,V,\nu^{i},C)$. The proportionality of free
energy with $C$ (or $N^{2}$) is a signature of deconfinement; in contrast, the
free energy of the confined phase is of order one Thorn (1981).

The Euler equation (6), or equivalently $W=\mu C$, only holds in a regime
where $1/C$ corrections can be neglected. For generic CFTs on compact spaces
this is the case in the high-temperature or large-volume regime $TR\gg 1$,
where $R$ is the curvature radius. On the other hand, for holographic and $2d$
sparse CFTs the free energy already scales with the central charge at low
temperatures $TR\sim\mathcal{O}(1)$ (i.e. if $ER\sim C$ with $C\gg 1$).
Further, the Euler equation is satisfied for any value of $\lambda$, at weak
and strong coupling, and for any large-$N$ field theory, including conformal
and confining theories, and theories with unusual scaling behavior like
Lifshitz theories. Each of these theories, though, satisfies a different
equation of state, which is not encoded in the large-$N$ Euler equation Karch
and Robinson (2015). For instance, the equation of state for conformal
theories is $E=(d-1)pV$, and for Lifshitz scale invariant theories with
dynamical scaling exponent $z$ it is given by $zE=(d-1)pV$ (see Appendix C).
The fact that the Euler relation applies to both conformal and Lifshitz
theories, means that it not only holds for relativistic, but also for non-
relativistic theories.

_Holographic black hole thermodynamics._ The large-$N$ Euler equation applies
in particular to strongly-coupled large-$N$ CFTs with a semiclassical,
gravitational dual description. We now investigate the gravity dual of the
Euler equation.

The best-established example of holography, the Anti-de Sitter/Conformal Field
Theory (AdS/CFT) correspondence, states that the partition function of the CFT
and of the gravitational theory in asymptotically AdS spacetime are equal
$Z_{\text{CFT}}=Z_{\text{AdS}}$ Gubser  _et al._ (1998); Witten (1998a). For
field theories at finite temperature the thermal partition function is related
to the free energy via $W=-T\ln Z_{\text{CFT}}$. On the other hand, the
gravitational partition function is given by the Euclidean path integral,
which in the saddle-point approximation is computed by the on-shell Euclidean
action, $I_{\text{E}}=-\ln Z_{\text{AdS}}$ Gibbons and Hawking (1977). Since
thermal states in the CFT are dual to black holes in AdS, the on-shell action
should be evaluated on the classical black hole saddle. Here, we consider
rotating, charged black hole solutions Caldarelli  _et al._ (2000) to the
Einstein-Maxwell action with a negative cosmological constant, i.e.
$I_{\text{E}}=-\frac{1}{16\pi G}\int d^{d+1}x\sqrt{g}(R-2\Lambda-F^{2})$. In
the grand canonical ensemble (at fixed $T$ and $\Phi$) the free energy of the
holographic field theory corresponds to Chamblin  _et al._ (1999); Gibbons
_et al._ (2005a); Papadimitriou and Skenderis (2005)

$W=TI_{\text{E}}=M-\frac{\kappa A}{8\pi G}-\Omega J-\Phi Q.$ (9)

The final equality follows from evaluating the action – including the Gibbons-
Hawking boundary term Gibbons and Hawking (1977) and a background subtraction
term – on the black hole solution with angular momentum $J$ and electric
charge $Q$. The corresponding chemical potentials are the angular velocity
$\Omega$ and the electric potential $\Phi$ of the horizon. We note it is
straightforward to generalize this equation to black holes with multiple
electric charges and angular momenta Myers and Perry (1986); Hawking  _et al._
(1999); Gibbons  _et al._ (2004); *Gibbons:2004uw.

The thermodynamic Euler equation for these black holes follows from (9) by
inserting the holographic dictionary (1) for energy, entropy and temperature,
and the dictionary for the charge $\tilde{Q}=QL$ and potential
$\tilde{\Phi}=\Phi/L$ Chamblin  _et al._ (1999); Karch and Robinson (2015),
and using the relation (8) between free energy and chemical potential,

$E=TS+\Omega J+\tilde{\Phi}\tilde{Q}+\mu C.$ (10)

The thermodynamic variables in this equation are well-known black hole
parameters, except for the chemical potential and central charge. What is
their gravitational dual description? Essentially, $\mu C$ is the on-shell
Euclidean action (times $T$). In addition, a different expression for the
chemical potential is obtained from the generalized Smarr formula for AdS
black holes, which relates all the black hole parameters and is thus a
gravitational reorganization of the Euler equation Kastor  _et al._ (2009);
Caldarelli  _et al._ (2000); Dolan (2011a); *Dolan:2011xt; Cvetic  _et al._
(2011); Gibbons  _et al._ (2005a); Barnich and Compere (2005)

$M=\frac{d-1}{d-2}\left(\frac{\kappa A}{8\pi G}+\Omega J\right)+\Phi
Q-\frac{1}{d-2}\frac{\Theta\Lambda}{4\pi G}.$ (11)

The $\Lambda$ term is absent for asymptotically flat black holes, but is
necessary for the consistency of the Smarr formula of asymptotically AdS black
holes. The quantity $\Theta$ can be defined as
$\int_{\Sigma_{\text{bh}}}|\xi|dV-\int_{\Sigma_{\text{AdS}}}|\xi|dV$ Jacobson
and Visser (2019), where a subtraction with respect to the pure AdS background
is implemented to cancel the divergence at infinity. In this definition the
domain of integration $\Sigma_{\text{BH}}$ extends from the horizon to
infinity, while $\Sigma_{\text{AdS}}$ in the pure AdS integral extends across
the entire spacetime. Further, $\xi$ is the timelike Killing field
$\xi=\partial_{t}+\Omega\partial_{\phi}$, which (in the black hole geometry)
generates the event horizon, and $|\xi|=\sqrt{-\xi\cdot\xi}$ is its norm. In
the literature Dolan (2011a); *Dolan:2011xt; Cvetic  _et al._ (2011) (minus)
$\Theta$ is often called the “thermodynamic volume”, since it is the conjugate
quantity to $\Lambda$ (the bulk pressure) in the first law of black hole
mechanics, see equation (13). For our purposes, however, a geometric name is
probably more suitable, such as (background subtracted) “Killing volume”,
because we are interested in the field theory thermodynamics rather than the
bulk thermodynamics.

Comparing the Euler equation and the Smarr formula we see that the chemical
potential (times central charge) corresponds to three combinations of the
black hole paramaters

$\displaystyle\mu C$ $\displaystyle=M-\frac{\kappa A}{8\pi G}-\Omega J-\Phi
Q=\frac{1}{d-1}\\!\\!\left(M-\Phi Q-\frac{\Theta\Lambda}{4\pi G}\right)$
$\displaystyle=\frac{1}{d-2}\left(\frac{\kappa A}{8\pi G}+\Omega
J-\frac{\Theta\Lambda}{4\pi G}\right).$ (12)

Note that the dimension dependent factors in the Smarr formula are absorbed in
the chemical potential. The expression above for the chemical potential should
also follow from its definition in (4), $\mu\equiv\left(\frac{\partial
E}{\partial C}\right)_{S,V,J,\tilde{Q}}.$ We check this explicitly by
rewriting the extended first law of AdS black hole mechanics as a
thermodynamic variational identity, where $\mu$ plays the role of the
conjugate quantity to the central charge variation $dC$. For CFTs dual to
Einstein gravity the holographic dictionary for the central charge depends on
both the cosmological constant $\Lambda$ and Newton’s constant $G$. In order
to keep track of the central charge variation, we vary both coupling constants
as “bookkeeping devices” in the bulk first law Kastor  _et al._ (2014); Karch
and Robinson (2015).

The mass of rotating, charged AdS black holes can be regarded as the function
$M(A,J,Q,\Lambda,G)$. From a scaling argument Smarr (1973); Kastor  _et al._
(2009) and from the generalized Smarr formula (11) it follows that the
extended first law for these black holes is

$\displaystyle dM$ $\displaystyle=\frac{\kappa}{8\pi G}dA+\Omega dJ+\Phi
dQ+\frac{\Theta}{8\pi G}d\Lambda$ (13) $\displaystyle-\left(M-\Omega J-\Phi
Q\right)\frac{dG}{G}.$

Usually, in extended black hole thermodynamics only the variation of $\Lambda$
is taken into account in the first law, but the variation of Newton’s constant
can be easily included by noting that $M,J,Q\sim G^{-1}$ Kastor  _et al._
(2010); Sarkar and Visser (2020). Remarkably, the $\Lambda$ and $G$ variations
in (13) cannot be combined into one single term proportional to
$d(\Lambda/G)$, because there is a term remaining involving the variation of
$G$. This seems to imply that the standard interpretation of the extended
first law in terms of bulk pressure $P=-\Lambda/8\pi G$ is inconsistent, if
Newton’s constant is allowed to vary. On the other hand, we can find a
consistent boundary interpretation by expressing the right-hand side of the
first law in terms of variations of the entropy $S=A/4G$, electric charge
$\tilde{Q}=QL$, central charge $C\sim L^{d-1}/G$ and spatial volume $V\sim
L^{d-1}$ of the holographic field theory. To this end we rewrite the extended
first law above as

$\displaystyle dM$
$\displaystyle=\frac{\kappa}{2\pi}d\left(\frac{A}{4G}\right)+\Omega
dJ+\frac{\Phi}{L}d(QL)-\frac{M}{d-1}\frac{dL^{d-1}}{L^{d-1}}$
$\displaystyle+\left(M-\frac{\kappa A}{8\pi G}-\Omega
J-\frac{\Phi}{L}QL\right)\frac{d(L^{d-1}/G)}{L^{d-1}/G}.$ (14)

Here we used again the Smarr relation and $d\Lambda/\Lambda=-2dL/L$, and
observed that $(d-1)dL/L-dG/G$ is equal to the fraction in the final term. It
is crucial that the $L$ and $G$ variations appear in this combination,
otherwise the holographically dual first law would not involve a variation of
the central charge. Moreover, by allowing for variations of $G$ we can clearly
distinguish the variation of the spatial volume $V$ from that of the central
charge $C$ Karch and Robinson (2015). Consequently, from the holographic
dictionary we deduce that the extended first law for AdS black holes is dual
to the fundamental equation in thermodynamics

$dE=TdS+\Omega dJ+\tilde{\Phi}d\tilde{Q}-pdV+\mu dC.$ (15)

By comparing (Holographic Thermodynamics Requires a Chemical Potential for
Color) and (15) we see that the pressure $p$ satisfies the CFT equation of
state $E=(d-1)pV$, and the chemical potential $\mu$ fulfils the Euler equation
(10). This shows that our dictionary (Holographic Thermodynamics Requires a
Chemical Potential for Color) for the chemical potential is consistent, since
the same expression follows from equating bulk and boundary free energy and
from matching the “first laws”. Note that we only used the scaling properties
$V\sim L^{d-1}$ and $C\sim L^{d-1}/G$ to arrive at (15), but did not need
their proportionality constants, because the fractions $dV/V$ and $dC/C$
appear in the first law and hence the proportionality constants drop out.

As an aside, we mention that the precise match between the “first laws” can be
generalized to charged Lifshitz black holes Tarrio and Vandoren (2011) by
replacing the holographic dictionary with:
$E=ML^{1-z},\,T=(\kappa/2\pi)L^{1-z}$ and $\tilde{\Phi}=\Phi/L^{z}$ (see
Appendices C and D). The extended first law (13) with $J=0$ still holds for
Lifshitz black holes Brenna  _et al._ (2015), and is dual to the fundamental
equation (15), if $p$ satisfies the Lifshitz equation of state $zE=(d-1)pV$
and $\mu$ the Euler equation Visser (2021).

_Comparison with previous literature._ In references Dolan (2014); Zhang  _et
al._ (2015a); *Zhang:2015ova the chemical potential associated to the central
charge (or $N^{2}$) was defined as $\mu\equiv\left(\frac{\partial E}{\partial
C}\right)_{S}$ for AdS-Schwarzschild black holes. Compared to our (4) the
fixed volume requirement is lacking in this definition. From the extended
first law it follows that this definition of the chemical potential is
proportional to the Killing volume $\Theta$, which is inconsistent with the
Euler equation and our expressions for $\mu$ in (Holographic Thermodynamics
Requires a Chemical Potential for Color). This definition is especially
problematic since these references take the boundary curvature radius $R$ to
be equal to the bulk curvature radius $L$, which implies that the spatial
volume and central charge are both proportional to $L^{d-1}$. Thus, in Dolan
(2014); Zhang  _et al._ (2015a); *Zhang:2015ova they mix up the bulk duals to
spatial volume and central charge.

In Karch and Robinson (2015) this issue was resolved by allowing for
variations of $G$, in addition to $\Lambda$, so that the central charge
variation can be distinguished from the volume variation. Our matching above
between the “first laws” is based on this approach, but is still novel since
Karch and Robinson (2015) focussed on finding the boundary dual to the Smarr
relation and not to the first law. In Karch and Robinson (2015) the free
energy relation $W\sim N^{2}$ at large $N$, which is equivalent to our Euler
equation (8), was identified as the holographic origin of the Smarr formula.
In addition, the pressure and its equation of state played an important role
in their holographic derivation of the Smarr formula. However, their proof
only holds for a particular choice of CFT metric,
$ds^{2}=-dt^{2}+L^{2}d\Omega_{k,d-1}^{2}$, where $L$ is the AdS radius (see
Appendix E). The derivation can be extended to a more general CFT metric with
$R\neq L$ and, remarkably, it does not depend on the boundary pressure in this
case. Rescaling the CFT metric above with the Weyl factor $\lambda=R/L$
changes the CFT time into $Rt/L$ and the boundary curvature radius into $R$,
so that the spatial volume becomes $V\sim R^{d-1}$. The added benefit of this
more general boundary metric is that $V$ is clearly distinct from $C$ (and
$\tilde{Q}$). The (refined) holographic dictionary for this metric is Savonije
and Verlinde (2001)

$E=M\frac{L}{R},\quad
T=\frac{\kappa}{2\pi}\frac{L}{R},\quad\tilde{\Omega}=\Omega\frac{L}{R},\quad\tilde{\Phi}=\frac{\Phi}{R}.$
(16)

Importantly, with this dictionary the bulk and boundary variational equations
(13) and (15) still agree and the chemical potential again satisfies the Euler
relation. We can even keep $G$ fixed and only vary $\Lambda$ in the bulk,
since $dV\sim dR^{d-1}$ and $dC\sim dL^{d-1}\big{|}_{G}$ are obviously
distinguishable for $R\neq L$.

Now, we derive the bulk Smarr formula purely from the boundary Euler equation
and the holographic dictionary for $R\neq L$. The $\Lambda$ term in the Smarr
formula can be expressed as

$-\frac{\Theta\Lambda}{4\pi G}=L\left(\frac{\partial M}{\partial
L}\right)_{\\!\\!A,J,Q,G}\\!\\!=R\left(\frac{\partial E}{\partial
L}\right)_{\\!\\!A,J,Q,G}\\!\\!-E\frac{R}{L}.$ (17)

Note that the bulk quantities $A,J,Q$ and $G$ are kept fixed in the partial
derivative. The boundary energy depends implicitly on them as:
$E=E(S(A,G),J,\tilde{Q}(L,Q),V(R),C(L,G))$. This implies (see Appendix D for
more details)

$-\frac{\Theta\Lambda}{4\pi G}=\frac{R}{L}\left(\tilde{\Phi}\tilde{Q}+(d-1)\mu
C-E\right),$ (18)

where we used the definitions of $\tilde{\Phi}$ and $\mu$ from (4). Finally,
inserting the Euler equation and the holographic dictionary (16) precisely
yields the Smarr formula (11). Note that this derivation hinges on the
chemical potential and not on the pressure.

_Discussion._ In gauge/gravity duality, black holes in the bulk correspond to
thermal states in the boundary theory. We proposed a new dictionary between
the bulk and boundary thermodynamics, by introducing a chemical potential for
the number of colors in the gauge theory. The chemical potential is crucial
for the correspondence between the Euler equation for large-$N$ theories and
the Smarr formula relating the black hole parameters. Since the Euler relation
determines the energy as a function of other variables, it contains the
essential thermodynamic information about the field theory.

Our field theory interpretation of the extended thermodynamics of black holes
stands in contrast to the common gravitational interpretation in terms of bulk
pressure and volume. One notable difference is that the black hole mass is
equivalent to the internal energy of the field theory, whereas in Kastor  _et
al._ (2009) it is identified with the enthalpy of the gravitational system.
Moreover, we found that the extended first law of black holes cannot be solely
written in terms of the variation of bulk pressure $P=-\Lambda/8\pi G$, if
both $\Lambda$ and $G$ are allowed to vary, but can be consistently
interpreted as a field theory first law. Thus, as the thermal field theory has
a natural thermodynamic description, the boundary interpretation seems
unavoidable.

As for future work, we expect that the dictionary for the chemical potential
can be generalized to a multitude of black holes in the presence of a
cosmological constant, such as black holes in higher-curvature gravity,
hyperscaling violating solutions, black rings and de Sitter black holes. On
the field theory side, an interesting problem is to extend the Euler equation
beyond the large-$N$ limit, by including $1/N$ corrections Karch and Robinson
(2015); Sinamuli and Mann (2017).

_Acknowledgments._ I would like to thank Jan de Boer, Pablo Bueno, Matthijs
Hogervorst, Arunabha Saha and Watse Sybesma for useful discussions, and
Andreas Karch, Juan Pedraza, Andrew Svesko and an anonymous referee for
detailed comments on a draft. This work was supported by the Republic and
canton of Geneva and the Swiss National Science Foundation, through Project
Grants No. 200020-182513 and No. 51NF40-141869 The Mathematics of Physics
(SwissMAP).

##  Appendix A Euler equation for two-dimensional CFTs

Examples of large-$N$ field theories are $2d$ modular invariant CFTs with
large central charge $c$. The microcanonical entropy for these theories is
given by the Cardy formula (setting $c_{L}=c_{R}=c$) Cardy (1986)

$S(E_{L},E_{R},c)=2\pi\sqrt{\frac{c}{6}E_{L}}+2\pi\sqrt{\frac{c}{6}E_{R}},$
(19)

with $E_{L,R}$ the left- and right-moving energies. On a circle of length
$V=2\pi R$, the total energy and angular momentum are, respectively,
$E=(E_{L}+E_{R})/R$ and $J=E_{L}-E_{R}$. The Cardy formula holds for CFTs with
a sparse light spectrum in the regime $C\to\infty$ with $ER\geq C$ Hartman
_et al._ (2014), where we normalized the central charge (conjugate to $\mu$)
as $C=c/12$. If we view the entropy (19) as the function $S=S(E,V,J,C)$, then
the fundamental variational equation of thermodynamics with
$\nu^{i}dB_{i}=\Omega dJ$, follows by taking partial derivatives of the
entropy function. Consequently, the products of thermodynamic quantities are

$\displaystyle TS$ $\displaystyle=\frac{4}{R}\sqrt{E_{L}E_{R}},\qquad\,\qquad
pV=E,$ (20) $\displaystyle\Omega J$
$\displaystyle=E-\frac{2}{R}\sqrt{E_{L}E_{R}},\qquad\mu
C=-\frac{2}{R}\sqrt{E_{L}E_{R}},$

where $\Omega$ is the angular potential. They satisfy the relation

$E=TS+\Omega J+\mu C.$ (21)

Hence, the large-$N$ Euler equation indeed holds for $2d$ CFTs. In fact, the
Euler relation splits up into two separate equations, $E=\Omega J-\mu C$ and
$TS=-2\mu C$.

In AdS3 gravity the Smarr formula for the outer horizon of a BTZ black hole is
given by $0=TS+\Omega J-\Theta\Lambda/4\pi G$ Frassino  _et al._ (2015).
Comparing this to the Euler equation (21) we find that the chemical potential
must correspond to $\mu C=E-\Theta\Lambda/4\pi G$. Using the holographic
dictionary for the central charge $c=3L/2G$ Brown and Henneaux (1986);
Strominger (1998), it can be shown that the chemical potential is dual to
$\mu\\!=\\!-(r_{+}^{2}-r_{-}^{2})/(L^{2}R),$ where $r_{\pm}$ are the outer and
inner horizon radii of the rotating BTZ black hole. Notably, $\mu$ vanishes
for extremal black holes, if $r_{+}=r_{-}$ or $ER=|J|$, which correspond to
CFT states with $E_{L}=0$ or $E_{R}=0$.

##  Appendix B The extended first law of entanglement

In this appendix we compare our chemical potential for AdS black holes to the
chemical potential in the extended first law for entanglement entropy of ball-
shaped regions in the CFT vacuum Blanco  _et al._ (2013); *Wong:2013gua;
Kastor  _et al._ (2014). This CFT first law takes the form

$d\bar{E}=\bar{T}dS_{\text{ent}}+\bar{\mu}dC,$ (22)

where $\bar{E}$ denotes the modular Hamiltonian expectation value,
$S_{\text{ent}}$ is the vacuum entanglement entropy of the ball-shaped region
and $C$ is the universal coefficient of the entanglement entropy (commonly
denoted as $a^{*}_{d}$) Myers and Sinha (2010); *Myers:2010tj; Caceres  _et
al._ (2017); Rosso and Svesko (2020). The CFT first law is dual to the first
law of static hyperbolic AdS black holes which are isometric to pure AdS space
Casini  _et al._ (2011); Faulkner  _et al._ (2014); Emparan (1999), a special
case of the black holes considered in the main text, with $J=Q=0$. The
boundary first law follows from reformulating our fundamental variational
equation in terms of dimensionless quantities $\bar{E}=ML,\bar{T}=\kappa
L/2\pi$, and $\bar{\mu}=\mu L$. The volume variation drops out of the first
law, since it is a dimensionful quantity. In the vacuum $\bar{E}=0$, hence the
chemical potential reduces to $\bar{\mu}=-\bar{T}S_{\text{ent}}/C$, which
agrees with the results in Kastor  _et al._ (2014) (where the temperature was
normalized as $\bar{T}=1$).

##  Appendix C Euler equation in flat space

In flat spacetime, static equilibrium states satisfy the standard
thermodynamic Euler equation,

$E=TS+\nu^{i}B_{i}-pV,$ (23)

which is often formulated instead in terms of densities since $V$ is infinite.
Note that the energy is purely extensive in this formula, since it satisfies
$E(\alpha S,\alpha B_{i},\alpha V)=\alpha E(S,B_{i},V)$. This Euler relation
applies in particular to conformal and Lifshitz theories on the plane (see
e.g. Natsuume (2015); Taylor (2016)). It is not immediately clear why this
equation is consistent with the large-$N$ Euler equation, therefore in this
appendix we explain the relation between the two for Lifshitz scale invariant
theories.

Anisotropic scaling symmetry
$\left\\{t,x^{i}\right\\}\to\left\\{\zeta^{z}t,\zeta x^{i}\right\\}$ with
dynamical scaling exponent $z$ implies that the product $TR^{z}$ is Lifshitz
scale invariant, where $R$ is the curvature radius of the compact space, such
as a sphere. Therefore, for Lifshitz theories with positive $z$ the infinite-
volume limit $R\to\infty$ is effectively the same as $T\to\infty$, so on the
plane these theories are essentially always in the high-temperature
deconfining phase. In this limit, the energy scales as $E\sim
T^{\frac{d-1+z}{z}}$ and entropy and conserved quantities as $S,B_{i}\sim
T^{\frac{d-1}{z}}$, so the scaling relation is
$E(\alpha^{\frac{d-1}{z}}S,V,\alpha^{\frac{d-1}{z}}B_{i},C)=\alpha^{\frac{d-1+z}{z}}E(S,V,B_{i},C)$.
This imposes the condition $(d-1+z)E=(d-1)(TS+\nu^{i}B_{i})$, which in
combination with the large-$N$ Euler equation yields $zE=-(d-1)\mu C$. We can
now compare this to the Lifshitz equation of state $zE=(d-1)pV$, which is a
consequence of the anisotropic scaling relation
$E(S,\alpha^{d-1}V,B_{i},C)=\alpha^{-z}E(S,V,B_{i},C)$. As a result, we find
$\mu C=-pV$ as $V\to\infty$, turning the large-$N$ Euler equation into the
standard one. The same argument works for conformal theories (by setting
$z=1$), hyperscaling violating theories and possibly other large-$N$ theories.
Notably, the standard Euler equation only applies in the infinite-volume limit
of large-$N$ theories. The large-$N$ Euler relation, on the other hand, also
holds at finite temperature on compact spaces for holographic field theories
and $2d$ sparse CFTs (but not for generic CFTs).

##  Appendix D Holographic derivation of the Smarr formula for Lifshitz black
holes

In this appendix we derive the Smarr formula for charged Lifshitz black holes
Tarrio and Vandoren (2011), with curvature radius $L$ and scaling exponent
$z$, from the holograhic Euler equation and the dictionary for the
thermodynamic quantities involved. We put the dual Lifshitz field theory on a
spatial geometry of curvature radius $R$. Our derivation generalizes section
2.3 of Karch and Robinson (2015) to $R\neq L$ and $z\neq 1$. Our aim is to
prove that even for Lifshitz black holes the boundary pressure and its
equation of state are not necessary input to deduce the Smarr formula
(although they are in the special case $R=L$ considered in Karch and Robinson
(2015)).

The holographic dictionary for Lifshitz black holes reads

$\displaystyle E=M\frac{L}{R^{z}},\quad
T=\frac{\kappa}{2\pi}\frac{L}{R^{z}},\quad\tilde{\Phi}=\frac{\Phi}{R^{z}},\quad\tilde{Q}=QL.$
(24)

Note that the factors of $R$ and $L$ are chosen such that the products
$ER^{z}$, $TR^{z}$ and $\tilde{\Phi}R^{z}$ are Lifshitz scale invariant (see
Appendix C). First, we express the $\Lambda$ term in the Smarr formula in
terms of the boundary energy $E$

$-\frac{\Theta\Lambda}{4\pi G}=L\left(\frac{\partial M}{\partial
L}\right)_{\\!\\!A,Q,G}\\!\\!=R^{z}\left(\frac{\partial E}{\partial
L}\right)_{\\!\\!A,Q,G}-E\frac{R^{z}}{L}.$ (25)

The strategy is to show that the right-hand side satisfies the Smarr formula.
Note that the bulk quantities $A,Q$ and $G$ are fixed in the partial
derivative with respect to $L$. The boundary energy depends on these bulk
quantities as follows

$E=E(S(A,G),\tilde{Q}(L,Q),V(R),C(L,G)).$ (26)

Note that $J=0$. The partial derivative is hence given by

$\displaystyle\left(\frac{\partial E}{\partial L}\right)_{\\!\\!A,Q,G}\\!\\!$
$\displaystyle=\left(\frac{\partial
E}{\partial\tilde{Q}}\right)_{\\!\\!S,V,C}\\!\\!\left(\frac{\partial\tilde{Q}}{\partial
L}\right)_{\\!\\!Q}\\!\\!+\left(\frac{\partial E}{\partial
C}\right)_{\\!\\!S,V,\tilde{Q}}\left(\frac{\partial C}{\partial
L}\right)_{\\!\\!G}$
$\displaystyle=\frac{1}{L}\\!\left(\tilde{\Phi}\tilde{Q}+(d-1)\mu C\right).$
(27)

In the second line we used $\tilde{Q}=QL$ and $C\sim L^{d-1}/G$, and we
recognized the definitions of the electric potential $\tilde{\Phi}$ and
chemical potential $\mu$ (see (4) in the main text). Thus, we find

$\displaystyle-\frac{\Theta\Lambda}{4\pi G}$
$\displaystyle=\frac{R^{z}}{L}\\!\left(\tilde{\Phi}\tilde{Q}+(d-1)\mu
C-E\right)$ (28)
$\displaystyle=\frac{R^{z}}{L}\\!\left((d-2)E-(d-1)TS-(d-2)\tilde{\Phi}\tilde{Q}\right).$

Finally, by inserting the holographic dictionary (24) we recover the Smarr
formula. Note that the Smarr formula for Lifshitz black holes does not involve
$z$ and is hence the same as for black holes in Einstein gravity Brenna  _et
al._ (2015). Crucially, the holographic Euler equation was employed in the
second line of (28) and is therefore dual to the Smarr formula, as pointed out
in Karch and Robinson (2015). We emphasize that the boundary pressure does not
play a role in this derivation, whereas the chemical potential $\mu$ does. For
$R=L$ the pressure does feature in the derivation, since in that case the
boundary volume depends on the bulk radius, i.e. $V(L)$, which yields an extra
term $-(d-1)pV/L$ in (D). But ultimately the result in (28) remains the same,
since this pressure term cancels, due to the Lifshitz equation of state,
against a new term $zEL^{z-1}$ on the right side of (25).

##  Appendix E The renormalized holographic Euler equation

In the main text the energy was defined with respect to the ground state, so
the vacuum energy was effectively set to zero. However, CFTs on a curved
background exhibit the Casimir effect, which implies that the ground state
could have non-vanishing energy. In AdS/CFT the ground-state energy can be
computed with the method of holographic renormalization, by regularizing the
gravitational action with local counterterms at the boundary Henningson and
Skenderis (1998); Balasubramanian and Kraus (1999). In this appendix we derive
the renormalized holographic Euler equation for static vacuum AdS black holes,
and find that the ground-state energy contributes a constant term to the
chemical potential.

We consider static, vacuum asymptotically AdS black holes with hyperbolic,
planar and spherical horizons Birmingham (1999)

$ds^{2}=-f_{k}(r)dt^{2}+\frac{dr^{2}}{f_{k}(r)}+r^{2}d\Omega_{k,d-1}^{2},$
(29)

where

$f_{k}(r)=k+\frac{r^{2}}{L^{2}}-\frac{16\pi GM}{(d-1)\Omega_{k,d-1}r^{d-1}}.$
(30)

For $k=1$ the unit metric $d\Omega_{k,d-1}^{2}\\!$ is the metric on a unit
$S^{d-1}\\!$ sphere, for $k=0$ it is the dimensionless metric
$\frac{1}{L^{2}}\\!\sum_{i=1}^{d-1}dx_{i}^{2}$ on the plane
$\mathbb{R}^{d-1}$, and for $k=-1$ the unit metric on hyperbolic space
$H^{d-1}$ is $du^{2}+\sinh^{2}ud\Omega_{k=1,d-2}^{2}$. The mass parameter $M$
is related to the horizon radius $r_{+}$ via

$M=\frac{(d-1)\Omega_{k,d-1}r_{+}^{d-2}}{16\pi
G}\left(\frac{r_{+}^{2}}{L^{2}}+k\right).$ (31)

According to the GKPW prescription in AdS/CFT Gubser  _et al._ (1998); Witten
(1998a) the CFT metric is identified with the boundary metric of the dual
asymptotically AdS spacetime up to a Weyl rescaling, i.e.
$g_{\text{CFT}}=\lim_{r\to\infty}\lambda^{2}(x)g_{\text{AdS}}$ where
$\lambda(x)$ is a Weyl scale factor. As $r\to\infty$ the boundary metric
approaches

$ds^{2}=\frac{r^{2}}{L^{2}}dt^{2}+\frac{L^{2}}{r^{2}}dr^{2}+r^{2}d\Omega_{k,d-1}^{2}.$
(32)

A common choice of Weyl factor is $\lambda=L/r$, so that the CFT metric
becomes $-dt^{2}+L^{2}d\Omega_{k,d-1}^{2}$. The boundary curvature radius is
then equal to the AdS radius and the volume is
$V=\Omega_{k,d-1}L^{d-1}/(d-1)$. Moreover, the CFT time is the same as the
global AdS time $t$, which implies that the CFT energy $E$ can be identified
(up to a constant) with the ADM mass $M$, the conserved charge associated to
time $t$ translations.

The temperature, entropy and energy of the black holes are

$\displaystyle T$ $\displaystyle=\frac{d\,r_{+}^{2}+k(d-2)L^{2}}{4\pi
L^{2}r_{+}},\qquad S=\frac{\Omega_{k,d-1}r_{+}^{d-1}}{4G},$ (33)
$\displaystyle E_{\text{ren}}$
$\displaystyle=\frac{(d-1)\Omega_{k,d-1}L^{d-2}}{16\pi
G}\left(\frac{r_{+}^{d}}{L^{d}}+k\frac{r_{+}^{d-2}}{L^{d-2}}+\frac{2\epsilon_{k}^{0}}{d-1}\right).$

The energy was derived from the renormalized boundary stress-energy tensor in
Balasubramanian and Kraus (1999) and from the on-shell Euclidean gravitational
action with counterterms in Emparan  _et al._ (1999). The resulting energy,
$E_{\text{ren}}=M+E_{k}^{0}$, differs from the mass paramater by a constant
term, the Casimir energy of the dual field theory

$E_{k}^{0}=\frac{\Omega_{k,d-1}L^{d-2}}{8\pi G}\epsilon_{k}^{0},$ (34)

with $\epsilon^{0}_{k}=0$ for odd $d$ and equal to Emparan  _et al._ (1999)

$\epsilon_{k}^{0}=(-k)^{d/2}\frac{(d-1)!!^{2}}{d!}\qquad\text{for even $d$}.$
(35)

For instance, $\epsilon_{k}^{0}=-k/2$ for $d=2$ and
$\epsilon_{k}^{0}=3k^{2}/8$ for $d=4.$ The renormalized version of the Smarr
formula reads

$E_{\text{ren}}=\frac{d-1}{d-2}TS-\frac{1}{d-2}\frac{\Theta_{\text{ren}}\Lambda}{4\pi
G},$ (36)

with a new (counterterm subtracted) Killing volume

$\Theta_{\text{ren}}=-\frac{\Omega_{k,d-1}}{d}\left(r_{+}^{d}-\frac{d-2}{d-1}L^{d}\epsilon_{k}^{0}\right).$
(37)

The holographic Euler equation still takes the form

$E_{\text{ren}}=TS+\mu_{\text{ren}}C,$ (38)

since the Casimir energy is also proportional to the central charge, which we
normalize here as $C=\Omega_{k,d-1}L^{d-1}/16\pi G$. But the chemical
potential is not given by equation (Holographic Thermodynamics Requires a
Chemical Potential for Color) in the main text anymore, since it receives a
constant contribution from the vacuum energy

$\mu_{\text{ren}}=-\frac{r_{+}^{d-2}}{L^{d-1}}\left(\frac{r_{+}^{2}}{L^{2}}-k\right)+\frac{2}{L}\epsilon_{k}^{0}.$
(39)

For $d=2$ we find the chemical potential $\mu_{\text{ren}}=-r_{+}^{2}/L^{3}$,
which agrees with the expression found in Appendix A (for $r_{-}=0$ and
$R=L$). For planar black holes ($k=0$) or very large hyperbolic or spherical
black holes (with $r_{+}\gg L$), the Casimir energy is effectively zero and
hence there is no distinction between the renormalized energy and the vacuum-
subtracted energy. As can be seen from (33) and (39), there are additional
thermodynamic relations for these black holes

$E=-(d-1)\mu C\qquad\text{and}\qquad TS=-d\,\mu C,$ (40)

consistent with the infinite-volume limit of Appendix C.

## References

  * Bekenstein (1972) Jacob D. Bekenstein, “Black holes and the second law,” Lett. Nuovo Cim. 4, 737–740 (1972).
  * Bekenstein (1973) Jacob D. Bekenstein, “Black holes and entropy,” Phys. Rev. D 7, 2333–2346 (1973).
  * Hawking (1975) Stephen W. Hawking, “Particle Creation by Black Holes,” Commun. Math. Phys. 43, 199–220 (1975), [Erratum: Commun.Math.Phys. 46, 206 (1976)].
  * Bardeen  _et al._ (1973) James M. Bardeen, B. Carter, and S.W. Hawking, “The Four laws of black hole mechanics,” Commun. Math. Phys. 31, 161–170 (1973).
  * Smarr (1973) Larry Smarr, “Mass formula for Kerr black holes,” Phys. Rev. Lett. 30, 71–73 (1973), [Erratum: Phys.Rev.Lett. 30, 521–521 (1973)].
  * Caldarelli  _et al._ (2000) Marco M. Caldarelli, Guido Cognola, and Dietmar Klemm, “Thermodynamics of Kerr-Newman-AdS black holes and conformal field theories,” Class. Quant. Grav. 17, 399–420 (2000), arXiv:hep-th/9908022 .
  * Kastor  _et al._ (2009) David Kastor, Sourya Ray, and Jennie Traschen, “Enthalpy and the Mechanics of AdS Black Holes,” Class. Quant. Grav. 26, 195011 (2009), arXiv:0904.2765 [hep-th] .
  * Dolan (2011a) Brian P. Dolan, “The cosmological constant and the black hole equation of state,” Class. Quant. Grav. 28, 125020 (2011a), arXiv:1008.5023 [gr-qc] .
  * Dolan (2011b) Brian P. Dolan, “Pressure and volume in the first law of black hole thermodynamics,” Class. Quant. Grav. 28, 235017 (2011b), arXiv:1106.6260 [gr-qc] .
  * Cvetic  _et al._ (2011) M. Cvetic, G.W. Gibbons, D. Kubiznak, and C.N. Pope, “Black Hole Enthalpy and an Entropy Inequality for the Thermodynamic Volume,” Phys. Rev. D 84, 024037 (2011), arXiv:1012.2888 [hep-th] .
  * Kubiznak and Mann (2015) David Kubiznak and Robert B. Mann, “Black hole chemistry,” Can. J. Phys. 93, 999–1002 (2015), arXiv:1404.2126 [gr-qc] .
  * Kubiznak  _et al._ (2017) David Kubiznak, Robert B. Mann, and Mae Teo, “Black hole chemistry: thermodynamics with Lambda,” Class. Quant. Grav. 34, 063001 (2017), arXiv:1608.06147 [hep-th] .
  * Maldacena (1999) Juan M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Int. J. Theor. Phys. 38, 1113–1133 (1999), arXiv:hep-th/9711200 .
  * Gubser  _et al._ (1998) S.S. Gubser, Igor R. Klebanov, and Alexander M. Polyakov, “Gauge theory correlators from noncritical string theory,” Phys. Lett. B 428, 105–114 (1998), arXiv:hep-th/9802109 .
  * Witten (1998a) Edward Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. 2, 253–291 (1998a), arXiv:hep-th/9802150 .
  * Hawking and Page (1983) S.W. Hawking and Don N. Page, “Thermodynamics of Black Holes in anti-De Sitter Space,” Commun. Math. Phys. 87, 577 (1983).
  * Witten (1998b) Edward Witten, “Anti-de Sitter space, thermal phase transition, and confinement in gauge theories,” Adv. Theor. Math. Phys. 2, 505–532 (1998b), arXiv:hep-th/9803131 .
  * Johnson (2014) Clifford V. Johnson, “Holographic Heat Engines,” Class. Quant. Grav. 31, 205002 (2014), arXiv:1404.5982 [hep-th] .
  * Dolan (2014) Brian P. Dolan, “Bose condensation and branes,” JHEP 10, 179 (2014), arXiv:1406.7267 [hep-th] .
  * Kastor  _et al._ (2014) David Kastor, Sourya Ray, and Jennie Traschen, “Chemical Potential in the First Law for Holographic Entanglement Entropy,” JHEP 11, 120 (2014), arXiv:1409.3521 [hep-th] .
  * Caceres  _et al._ (2015) Elena Caceres, Phuc H. Nguyen, and Juan F. Pedraza, “Holographic entanglement entropy and the extended phase structure of STU black holes,” JHEP 09, 184 (2015), arXiv:1507.06069 [hep-th] .
  * Karch and Robinson (2015) Andreas Karch and Brandon Robinson, “Holographic Black Hole Chemistry,” JHEP 12, 073 (2015), arXiv:1510.02472 [hep-th] .
  * Henningson and Skenderis (1998) M. Henningson and K. Skenderis, “The Holographic Weyl anomaly,” JHEP 07, 023 (1998), arXiv:hep-th/9806087 .
  * Henningson and Skenderis (2000) Mans Henningson and Kostas Skenderis, “Holography and the Weyl anomaly,” Fortsch. Phys. 48, 125–128 (2000), arXiv:hep-th/9812032 .
  * Freedman  _et al._ (1999) D.Z. Freedman, S.S. Gubser, K. Pilch, and N.P. Warner, “Renormalization group flows from holography supersymmetry and a c theorem,” Adv. Theor. Math. Phys. 3, 363–417 (1999), arXiv:hep-th/9904017 .
  * Myers and Sinha (2010) Robert C. Myers and Aninda Sinha, “Seeing a c-theorem with holography,” Phys. Rev. D 82, 046006 (2010), arXiv:1006.1263 [hep-th] .
  * Myers and Sinha (2011) Robert C. Myers and Aninda Sinha, “Holographic c-theorems in arbitrary dimensions,” JHEP 01, 125 (2011), arXiv:1011.5819 [hep-th] .
  * ’t Hooft (1974) Gerard ’t Hooft, “A Planar Diagram Theory for Strong Interactions,” Nucl. Phys. B 72, 461 (1974).
  * Gross and Witten (1980) D.J. Gross and Edward Witten, “Possible Third Order Phase Transition in the Large N Lattice Gauge Theory,” Phys. Rev. D 21, 446–453 (1980).
  * Sundborg (2000) Bo Sundborg, “The Hagedorn transition, deconfinement and $N=4$ SYM theory,” Nucl. Phys. B 573, 349–363 (2000), arXiv:hep-th/9908001 .
  * Aharony  _et al._ (2004) Ofer Aharony, Joseph Marsano, Shiraz Minwalla, Kyriakos Papadodimas, and Mark Van Raamsdonk, “The Hagedorn - deconfinement phase transition in weakly coupled large N gauge theories,” Adv. Theor. Math. Phys. 8, 603–696 (2004), arXiv:hep-th/0310285 .
  * Witten (1995) Edward Witten, “Some comments on string dynamics,” in  _STRINGS 95: Future Perspectives in String Theory_ (1995) pp. 501–523, arXiv:hep-th/9507121 .
  * Strominger (1996) Andrew Strominger, “Open p-branes,” Phys. Lett. B 383, 44–47 (1996), arXiv:hep-th/9512059 .
  * Thorn (1981) Charles B. Thorn, “Infinite $N_{c}$ QCD at finite temperature: is there an ultimate temperature?” Phys. Lett. B 99, 458–462 (1981).
  * Gibbons and Hawking (1977) G.W. Gibbons and S.W. Hawking, “Action Integrals and Partition Functions in Quantum Gravity,” Phys. Rev. D 15, 2752–2756 (1977).
  * Chamblin  _et al._ (1999) Andrew Chamblin, Roberto Emparan, Clifford V. Johnson, and Robert C. Myers, “Charged AdS black holes and catastrophic holography,” Phys. Rev. D 60, 064018 (1999), arXiv:hep-th/9902170 .
  * Gibbons  _et al._ (2005a) G.W. Gibbons, M.J. Perry, and C.N. Pope, “The First law of thermodynamics for Kerr-anti-de Sitter black holes,” Class. Quant. Grav. 22, 1503–1526 (2005a), arXiv:hep-th/0408217 .
  * Papadimitriou and Skenderis (2005) Ioannis Papadimitriou and Kostas Skenderis, “Thermodynamics of asymptotically locally AdS spacetimes,” JHEP 08, 004 (2005), arXiv:hep-th/0505190 .
  * Myers and Perry (1986) Robert C. Myers and M.J. Perry, “Black Holes in Higher Dimensional Space-Times,” Annals Phys. 172, 304 (1986).
  * Hawking  _et al._ (1999) S.W. Hawking, C.J. Hunter, and Marika Taylor, “Rotation and the AdS / CFT correspondence,” Phys. Rev. D 59, 064005 (1999), arXiv:hep-th/9811056 .
  * Gibbons  _et al._ (2004) G.W. Gibbons, H. Lu, Don N. Page, and C.N. Pope, “Rotating black holes in higher dimensions with a cosmological constant,” Phys. Rev. Lett. 93, 171102 (2004), arXiv:hep-th/0409155 .
  * Gibbons  _et al._ (2005b) G.W. Gibbons, H. Lu, Don N. Page, and C.N. Pope, “The General Kerr-de Sitter metrics in all dimensions,” J. Geom. Phys. 53, 49–73 (2005b), arXiv:hep-th/0404008 .
  * Barnich and Compere (2005) G. Barnich and G. Compere, “Generalized Smarr relation for Kerr AdS black holes from improved surface integrals,” Phys. Rev. D 71, 044016 (2005), [Erratum: Phys.Rev.D 73, 029904 (2006)], arXiv:gr-qc/0412029 .
  * Jacobson and Visser (2019) Ted Jacobson and Manus Visser, “Gravitational Thermodynamics of Causal Diamonds in (A)dS,” SciPost Phys. 7, 079 (2019), arXiv:1812.01596 [hep-th] .
  * Kastor  _et al._ (2010) David Kastor, Sourya Ray, and Jennie Traschen, “Smarr Formula and an Extended First Law for Lovelock Gravity,” Class. Quant. Grav. 27, 235014 (2010), arXiv:1005.5053 [hep-th] .
  * Sarkar and Visser (2020) Debajyoti Sarkar and Manus Visser, “The first law of differential entropy and holographic complexity,” JHEP 11, 004 (2020), arXiv:2008.12673 [hep-th] .
  * Tarrio and Vandoren (2011) Javier Tarrio and Stefan Vandoren, “Black holes and black branes in Lifshitz spacetimes,” JHEP 09, 017 (2011), arXiv:1105.6335 [hep-th] .
  * Brenna  _et al._ (2015) Wilson G. Brenna, Robert B. Mann, and Miok Park, “Mass and Thermodynamic Volume in Lifshitz Spacetimes,” Phys. Rev. D 92, 044015 (2015), arXiv:1505.06331 [hep-th] .
  * Visser (2021) Manus R. Visser, “The Holographic Euler Equation for Lifshitz and Hyperscaling Violating Black Holes,” In preparation (2021).
  * Zhang  _et al._ (2015a) Jia-Lin Zhang, Rong-Gen Cai, and Hongwei Yu, “Phase transition and thermodynamical geometry for Schwarzschild AdS black hole in AdS5 $\times$ S5 spacetime,” JHEP 02, 143 (2015a), arXiv:1409.5305 [hep-th] .
  * Zhang  _et al._ (2015b) Jia-Lin Zhang, Rong-Gen Cai, and Hongwei Yu, “Phase transition and thermodynamical geometry of Reissner-Nordström-AdS black holes in extended phase space,” Phys. Rev. D 91, 044028 (2015b), arXiv:1502.01428 [hep-th] .
  * Savonije and Verlinde (2001) Ivo Savonije and Erik P. Verlinde, “CFT and entropy on the brane,” Phys. Lett. B 507, 305–311 (2001), arXiv:hep-th/0102042 .
  * Sinamuli and Mann (2017) Musema Sinamuli and Robert B. Mann, “Higher Order Corrections to Holographic Black Hole Chemistry,” Phys. Rev. D 96, 086008 (2017), arXiv:1706.04259 [hep-th] .
  * Cardy (1986) John L. Cardy, “Operator Content of Two-Dimensional Conformally Invariant Theories,” Nucl. Phys. B 270, 186–204 (1986).
  * Hartman  _et al._ (2014) Thomas Hartman, Christoph A. Keller, and Bogdan Stoica, “Universal Spectrum of 2d Conformal Field Theory in the Large c Limit,” JHEP 09, 118 (2014), arXiv:1405.5137 [hep-th] .
  * Frassino  _et al._ (2015) Antonia M. Frassino, Robert B. Mann, and Jonas R. Mureika, “Lower-Dimensional Black Hole Chemistry,” Phys. Rev. D 92, 124069 (2015), arXiv:1509.05481 [gr-qc] .
  * Brown and Henneaux (1986) J. David Brown and M. Henneaux, “Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity,” Commun. Math. Phys. 104, 207–226 (1986).
  * Strominger (1998) Andrew Strominger, “Black hole entropy from near horizon microstates,” JHEP 02, 009 (1998), arXiv:hep-th/9712251 .
  * Blanco  _et al._ (2013) David D. Blanco, Horacio Casini, Ling-Yan Hung, and Robert C. Myers, “Relative Entropy and Holography,” JHEP 08, 060 (2013), arXiv:1305.3182 [hep-th] .
  * Wong  _et al._ (2013) Gabriel Wong, Israel Klich, Leopoldo A. Pando Zayas, and Diana Vaman, “Entanglement Temperature and Entanglement Entropy of Excited States,” JHEP 12, 020 (2013), arXiv:1305.3291 [hep-th] .
  * Caceres  _et al._ (2017) Elena Caceres, Phuc H. Nguyen, and Juan F. Pedraza, “Holographic entanglement chemistry,” Phys. Rev. D 95, 106015 (2017), arXiv:1605.00595 [hep-th] .
  * Rosso and Svesko (2020) Felipe Rosso and Andrew Svesko, “Novel aspects of the extended first law of entanglement,” JHEP 08, 008 (2020), arXiv:2003.10462 [hep-th] .
  * Casini  _et al._ (2011) Horacio Casini, Marina Huerta, and Robert C. Myers, “Towards a derivation of holographic entanglement entropy,” JHEP 05, 036 (2011), arXiv:1102.0440 [hep-th] .
  * Faulkner  _et al._ (2014) Thomas Faulkner, Monica Guica, Thomas Hartman, Robert C. Myers, and Mark Van Raamsdonk, “Gravitation from Entanglement in Holographic CFTs,” JHEP 03, 051 (2014), arXiv:1312.7856 [hep-th] .
  * Emparan (1999) Roberto Emparan, “AdS / CFT duals of topological black holes and the entropy of zero energy states,” JHEP 06, 036 (1999), arXiv:hep-th/9906040 .
  * Natsuume (2015) Makoto Natsuume,  _AdS/CFT Duality User Guide_, Vol. 903 (Springer Japan, 2015) arXiv:1409.3575 [hep-th] .
  * Taylor (2016) Marika Taylor, “Lifshitz holography,” Class. Quant. Grav. 33, 033001 (2016), arXiv:1512.03554 [hep-th] .
  * Balasubramanian and Kraus (1999) Vijay Balasubramanian and Per Kraus, “A Stress tensor for Anti-de Sitter gravity,” Commun. Math. Phys. 208, 413–428 (1999), arXiv:hep-th/9902121 .
  * Birmingham (1999) Danny Birmingham, “Topological black holes in Anti-de Sitter space,” Class. Quant. Grav. 16, 1197–1205 (1999), arXiv:hep-th/9808032 .
  * Emparan  _et al._ (1999) Roberto Emparan, Clifford V. Johnson, and Robert C. Myers, “Surface terms as counterterms in the AdS / CFT correspondence,” Phys. Rev. D 60, 104001 (1999), arXiv:hep-th/9903238 .


∥

# The Third Trick

Hannes Diener  School of Mathematics and Statistics, University of Canterbury,
Christchurch, New Zealand<EMAIL_ADDRESS>and  Matthew
Hendtlass  School of Mathematics and Statistics, University of Canterbury,
Christchurch, New Zealand<EMAIL_ADDRESS>
###### Abstract.

We prove a result, similar to the ones known as Ishihara’s First and Second
Trick, for sequences of functions.

##  1\. Introduction

In Bishop’s constructive mathematics (BISH)111Informal mathematics using
intuitionistic logic and an appropriate set-theoretic or type-theoretic
foundation such as [1]. See [2, 3] for details. We do assume
dependent/countable choice, but will explicitly label any use thereof., one
often has to navigate around the reality that one cannot make use of the law
of excluded middle. Even though constructivists assume that we can make
decision about _finite_ objects, case distinctions of the kind of

(1) $\forall{x\in\mathbb{R}}:{x<0\lor x=0\lor x>0}$

are unavailable. Most of these general disjunctions are actually false in
constructive varieties such as Brouwer’s intuitionism (INT) or Russian
recursive mathematics (RUSS), while in BISH they are merely not acceptable
since the latter is consistent with not just constructive varieties, but also
classical mathematics (CLASS). One easily reaches the conclusion that there
are no such disjunctions available in BISH.

This is what makes the results nowadays known as Ishihara’s First and Second
Trick [11] so deliciously surprising: they allow us to make an interesting and
non-trivial decision about ideal objects. Both Tricks assume a strongly
extensional222defined below mapping $f$ of a complete metric space $(X,\rho)$
into a metric space $(Y,\rho)$333We will use $\rho$ as denoting the metric on
any metric space we consider for simplicity. There is no point at which this
leads to ambiguity., and a sequence $(x_{n})_{n\geqslant 1}$ in $X$ converging
to a limit $x$.

######  Proposition 1 (Ishihara’s First Trick).

For all positive reals $\alpha<\beta$,

$\exists{n\in\mathbb{N}}:{\rho(f(x_{n}),f(x))>\alpha}\ \lor\
\forall{n\in\mathbb{N}}:{\rho(f(x_{n}),f(x))<\beta}\ .$

######  Proposition 2 (Ishihara’s Second Trick).

For all positive reals $\alpha<\beta$, either we have

  * •

$\rho(f(x_{n}),f(x))<\beta$ eventually,

  * •

or $\rho(f(x_{n}),f(x))>\alpha$ infinitely often.

Even though it is wrong (see, however, Proposition 4), it is _helpful_ to
think of Ishihara’s Second Trick as saying that a strongly extensional
function is either sequentially continuous or it is discontinuous.

In this note, we will show that we can prove a similar result for a sequence
of functions converging point-wise to another function. Vaguely speaking we
can show that one can decide whether convergence happens somewhat uniform, or
in a discontinuous fashion.

The motivating example for our results is the following. Consider the sequence
of functions $(f_{n})_{n\geqslant 1}$ defined by

$f_{n}(x)=\begin{cases}2nx&\text{if }x\in[0,\frac{1}{2n}]\\\ 1-2nx&\text{if
}x\in[\frac{1}{2n},\frac{2}{2n}]\\\ 0&\text{if }x\in[\frac{2}{2n},1]\
.\end{cases}$

$0$$\frac{1}{2n}$$\frac{1}{n}$$1$$1$$f_{n}$

The sequence $(f_{n})_{n\geqslant 1}$ is one of the standard examples in
classical analysis (see, for example, [8, VII.2 Problem 2]) of a sequence of
functions on the unit interval that converges point-wise but not uniformly.
However, constructively this example breaks, since the assumption that we have
point-wise convergence implies the limited principle of omniscience (LPO),
which states that for every binary sequence $(a_{n})_{n\geqslant 1}$ we have

$\forall{n\in\mathbb{N}}:{a_{n}=0}\lor\exists{n\in\mathbb{N}}:{a_{n}=1}\ ,$

which is, under the assumption of countable choice, equivalent to Equation 1.

######  Proposition 3.

LPO is equivalent to the statement that $(f_{n})_{n\geqslant 1}$ defined as
above converges point-wise to $0$.

###### Proof.

Let $(a_{n})_{n\geqslant 1}$ be a binary sequence. Without loss of generality
we may assume that $(a_{n})_{n\geqslant 1}$ is increasing. Now consider the
sequence $(x_{n})_{n\geqslant 1}$ in $[0,1]$ defined by

$x_{n}=\begin{cases}\frac{1}{2m}&\text{if }a_{n}=1\text{ and
}a_{m}=1-a_{m+1}\\\ 0&\text{if }a_{n}=0\ .\end{cases}$

Using the notation of [6, 7] this is simply the sequence
$\left((a_{n})\circledast(\frac{1}{2n})\right)$. It is easy to see (or
formally proven in [6, Lemma 2.1]), that $(x_{n})_{n\geqslant 1}$ is a Cauchy
sequence. It therefore converges to a limit $z\in[0,1]$. Since $f_{n}(z)\to
0$, there exists $N\in\mathbb{N}$ such that

(2) $\forall{n\geqslant N}:{f_{n}(z)<\frac{1}{2}}\ .$

Now either $a_{N+1}=1$ or $a_{N+1}=0$. In the first case we are done. In the
second case there cannot be $m>N+1$ such that $a_{m}=1$. For assume there is
such an $m$, then we can find $N<m^{\prime}<m$ such that $a_{m^{\prime}}=0$
and $a_{m^{\prime}+1}=1$. In that case $x_{n}$ is eventually constant on
$\frac{1}{2m^{\prime}}$, which means that $z=\frac{1}{2m^{\prime}}$. However
$f_{m}(z)=1$ and $m\geqslant N$, which would be a contradiction to Equation 2.
Thus $a_{n}=0$ for all $n\in\mathbb{N}$. Hence LPO holds.

Conversely, it is easy to see that LPO in the form of Equation 1 is enough to
show that $(f_{n})_{n\geqslant 1}$ converges point-wise. ∎

That means that in varieties of BISH in which LPO is false the above sequence
of function actually does not provably converge point-wise; which means the
above example cannot serve to show that uniform convergence is not implied by
point-wise convergence. As it turns out there are (constructive) scenarios
where the two notions coincide.

Looking at this from a different angle one could say that the assumption of
point-wise convergence is constructively a stronger assumption than
classically.

Of course, working in BISH we only know that LPO is not provable, but not
whether $\neg\textrm{LPO}$ is provable. That is if, in BISH, we encounter a
sequence of functions converging point-wise there are, intuitively speaking,
two options: the convergence is actually uniform or it is not, in which case
LPO holds.

This is not just an external disjunction: surprisingly we can make this
decision within BISH and without knowing whether LPO holds or fails.

##  2\. The Third Trick

We will first consider a restricted version of our results applying only to
functions of a specific type signature. In our opinion, this has the advantage
of making the actual underlying ideas and structure of our proofs cleaner and
clearer.

As Escardó has shown [9, 10], the natural setting for Ishihara’s tricks is
$\mathbb{N}_{\infty}$—the space of all increasing binary sequences with the
metric induced by the usual one on Cantor space. The space
$\mathbb{N}_{\infty}$ contains the sequences $\underline{n}=0^{n}1\dots$ for
any $n\in\mathbb{N}$, and the sequence $\omega=000\dots$. Using classical
logic we have that

$\mathbb{N}_{\infty}=\mbox{$\left\\{\,\underline{n}\,\middle|\,n\in\mathbb{N}\,\right\\}$}\cup\left\\{\omega\right\\}\
,$

however this cannot be proven with intuitionistic logic alone, since that
statement is actually equivalent to LPO.

We also assume that the set $\mathbb{N}_{\infty}$ is equipped with the reverse
lexicographic order $\leqslant$; so, for example,
$\underline{n}\leqslant\underline{n+1}<\omega$.444Of course, this order is not
decidable, constructively.

For functions $\mathbb{N}_{\infty}\to\mathbb{N}$ Ishihara’s Second Trick
becomes

######  Proposition 4 (Ishihara’s Second Trick for
$\mathbb{N}_{\infty}\to\mathbb{N}$).

If $f:\mathbb{N}_{\infty}\to\mathbb{N}$ is strongly extensional, then

$\exists{N\in\mathbb{N}}:{\forall{n\geqslant N}:{f(\underline{n})=f(\omega)}}\
\lor\
\exists{\alpha\in\mathbb{N}^{\mathbb{N}}_{\textrm{inc}}}:{\forall{n\in\mathbb{N}}:{f(\underline{\alpha(n)})\neq
f(\omega)}}\ ,$

where $\mathbb{N}^{\mathbb{N}}_{\textrm{inc}}$ is the space of all strictly
increasing functions $\mathbb{N}^{\mathbb{N}}$.

Notice that for a function $f$ of type $\mathbb{N}_{\infty}\to\mathbb{N}$ this
really states that we can decide whether $f$ is continuous or find a witness
of discontinuity.

Similarly to Ishihara’s Second Trick being an iteration of the first one, our
Third Trick will also rely on iterating a simpler result, which is Part 2 of
the following.

######  Lemma 5.

Assume that $(f_{n})_{n\geqslant 1}:\mathbb{N}_{\infty}\to\mathbb{N}$ is such
that $f_{n}(x)\to 0$ for all $x\in\mathbb{N}_{\infty}$.

  1. (1)

For any $\gamma\in\mathbb{N}_{\infty}$

$\exists{i\in\mathbb{N}}:{f_{i}(\gamma)\neq
0}\>\>\lor\>\>\forall{i\in\mathbb{N}}:{f_{i}(\gamma)=0}\ .$

  2. (2)

Either

     * •

there exists $i\in\mathbb{N}$ and $\alpha\in\mathbb{N}_{\infty}$ such that
$f_{i}(\alpha)\neq 0$

     * •

or, for all $i\in\mathbb{N}$ and $\alpha\in\mathbb{N}_{\infty}$, we have that
$f_{i}(\alpha)=0$

###### Proof.

  1. (1)

For any $\gamma\in\mathbb{N}_{\infty}$, since $f_{n}(\gamma)\to 0$, there
exists $N_{\gamma}$ such that $f_{i}(\gamma)=0$ for all $i\geqslant
N_{\gamma}$. Because we only need to check finitely many terms, either there
exists $i<N_{\gamma}$ such that $f_{i}(\gamma)\neq 0$ or not.

  2. (2)

Using the first part of this lemma we can, first exclude the case that there
exists $i\in\mathbb{N}$ such that $f_{i}(\omega)\neq 0$, since that means we
are done. So, for the rest of this proof we can assume that $f_{i}(\omega)=0$
for all $i\in\mathbb{N}$.

Also, using the first part of this lemma, we can build (using unique choice) a
binary sequence $\lambda_{k}$ such that

$\displaystyle\lambda_{k}=0$
$\displaystyle\implies\forall{i}:{f_{i}(\underline{k})=0}\ ,$
$\displaystyle\lambda_{k}=1$
$\displaystyle\implies\exists{i}:{f_{i}(\underline{k})\neq 0}\ .$

Now define a sequence $\beta_{k}\in\mathbb{N}_{\infty}$ by

$\displaystyle\beta_{k}=\underline{n}$
$\displaystyle\iff\forall{i<n}:{\lambda_{n}=0}\ ,$
$\displaystyle\beta_{k}=\beta_{k-1}$
$\displaystyle\iff\exists{i<n}:{\lambda_{i}=1}\ .$

In words, going through the sequence $\lambda$, as long as $\lambda_{k}$ is
$0$ we just set $\beta_{k}=\underline{k}$, but as soon as we hit a $k$ with
$\lambda_{k}=1$ we stay constant at that first $\underline{k}$. This is a
Cauchy sequence, which therefore converges to a limit
$\beta\in\mathbb{N}_{\infty}$. Now, by Lemma 5.1 either there exists
$j\in\mathbb{N}$ such that $f_{j}(\beta)\neq 0$ or not. In the first case we
are done, so let us focus, for the rest of the proof, on the second case.

In this second case we claim that we must have $\lambda_{k}=0$ for all $k$.
For assume there is $k$ such that $\lambda_{k}=1$. We may assume that $k$ is
the minimal such number. In that case we have $\beta=\underline{k}$ and there
exists $i$ such that $f_{i}(\beta)=f(\underline{k})\neq 0$. But that is a
contradiction to the case we are in, and therefore $\lambda_{k}=0$ for all
$k\in\mathbb{N}$. Now there cannot be a $\alpha\in\mathbb{N}_{\infty}$ and
$i\in\mathbb{N}$ such that $f_{i}(\alpha)\neq 0$. For assume there is such
$\alpha$ and $i$. Then $\alpha=\omega$, since, if there exists $n$ such that
$\alpha(n)=1$, we would have $\lambda_{n}=1$. But that means that $0\neq
f_{i}(\alpha)=f_{i}(\omega)$, which we excluded at the beginning of the proof.
∎

######  Theorem 6 (The Third Trick for $\mathbb{N}_{\infty}\to\mathbb{N}$).

If $(f_{n})_{n\geqslant 1}:\mathbb{N}_{\infty}\to\mathbb{N}$ such that
$f_{n}(x)\to 0$ for all $x\in\mathbb{N}_{\infty}$, then either

  * •

there exists $N\in\mathbb{N}$ such that for all $i\geqslant N$ and
$\alpha\geqslant\underline{N}$ we have $f_{i}(\alpha)=0$, that is the
convergence is uniform,

  * •

or there exists a sequence $\alpha_{n}$ in $\mathbb{N}_{\infty}$ and sequence
$k_{n}$ such that $k_{n}\geqslant n$, $\alpha_{n}\geqslant\underline{n}$, and
$f_{k_{n}}(\alpha_{n})\neq 0$, that is there is a witness showing that the
convergence is not uniform.

###### Proof.

Since $f_{n}(\omega)\to 0$ we may assume, without loss of generality, that
$f_{n}(\omega)=0$ for all $n\in\mathbb{N}$. Using the second part of the
previous lemma, fix a binary sequence $\lambda_{n}$ (using countable choice to
collect the $\alpha_{n}$) such that

$\displaystyle\lambda_{n}=0$ $\displaystyle\implies\exists{i\geqslant
n,\alpha_{n}\geqslant\underline{n}}:{f_{i}(\alpha_{n})\neq 0}\ ,$
$\displaystyle\lambda_{n}=1$ $\displaystyle\implies\forall{i\geqslant
n,\alpha\geqslant\underline{n}}:{f_{i}(\alpha)=0}\ .$

The sequence $\lambda_{n}$ is increasing, and we may assume that
$\lambda_{1}=0$, since we are otherwise done.

Now define a sequence $(\beta_{n})_{n\geqslant 1}$ in $\mathbb{N}_{\infty}$ by

$\displaystyle\lambda_{n}=0$ $\displaystyle\implies\beta_{n}=\alpha_{n}\ ,$
$\displaystyle\lambda_{n}=1$ $\displaystyle\implies\beta_{n}=\alpha_{m},\text{
where }\lambda_{m}=0\land\lambda_{m+1}=1\ .$

In words, as long as $\lambda_{n}=0$ we set $\beta_{n}=\alpha_{n}$, and as
soon as we find the first term such that $\lambda_{n}=1$, we stay constant on
the previous term $\alpha_{n-1}$. This ensures that all $\beta_{n}$ are such
that there exists $i$ with $f_{i}(\beta_{n})\neq 0$. Again,
$(\beta_{n})_{n\geqslant 1}$ is easily seen to be a Cauchy sequence, which
therefore converges to a limit $\gamma$ in $\mathbb{N}_{\infty}$.

Since $f_{n}(\gamma)\to 0$ there is $N_{\gamma}$ such that $f_{n}(\gamma)=0$
for all $n\geqslant N_{\gamma}$, which means that by checking all
$i<N_{\gamma}$ we can decide whether either $f_{i}(\gamma)=0$ for all
$i\in\mathbb{N}$ or whether there is $k$ such that $f_{k}(\gamma)\neq 0$ but
$f_{j}(\gamma)=0$ for $j>k$.

In the first case we must have $\lambda_{n}=0$ for all $n\in\mathbb{N}$: for
assume there exists $m$ such that $\lambda_{m}=1$. That means we can find
$m^{\prime}<m$ such that $\lambda_{m^{\prime}}=0$ and
$\lambda_{m^{\prime}+1}=1$. But that implies that
$\beta_{n}=\alpha_{m^{\prime}}$ for all $n>m^{\prime}$, and therefore
$\gamma=\alpha_{m^{\prime}}$. Thus there exists $i>m^{\prime}$ such that
$f_{i}(\gamma)\neq 0$; a contradiction to the case we are in.

In the second case we must have $\lambda_{k^{\prime}}=1$, since otherwise we
can reach the following, two-step contradiction. If there is $k^{\prime}>k$
such that $\lambda_{k^{\prime}}=0$ and $\lambda_{k^{\prime}+1}=1$, then
$\gamma=\alpha_{k^{\prime}}$ and
$\exists{i\geqslant{k^{\prime}}}:{f_{i}(\gamma)\neq 0}$, which contradicts our
choice of $k$. Thus $\lambda_{n}=0$ for all $n\in\mathbb{N}$, which means
$\gamma=\omega$, but that is also a contradiction, since we assumed that
$f_{n}(\omega)=0$ for all $n\in\mathbb{N}$.

Together we can decide whether $\forall{n\in\mathbb{N}}:{\lambda_{n}=0}$ or
whether $\exists{n\in\mathbb{N}}:{\lambda_{n}=1}$ which means we are done, by
the definition of $(\lambda_{n})_{n\geqslant 1}$. ∎

######  Proposition 7.

In the second case of the previous proposition LPO holds.

###### Proof.

Let $(a_{n})_{n\geqslant 1}$ be a binary sequence. Without loss of generality
we may assume that $(a_{n})_{n\geqslant 1}$ is increasing. Now consider the
sequence $(x_{n})_{n\geqslant 1}$ in $\mathbb{N}_{\infty}$ defined by

$x_{n}=\begin{cases}\alpha_{m}&\text{if }a_{n}=1\text{ and }a_{m}=1-a_{m+1}\\\
0&\text{if }a_{n}=0\ .\end{cases}$

Using the notation of [6, 7] this is simply the sequence
$(a_{n})\circledast(\alpha_{n})$. It is easy to see (or follows from Lemma 2.1
of [6]), that $(x_{n})_{n\geqslant 1}$ is a Cauchy sequence. It therefore
converges to a limit $z$. Since $f_{n}(z)\to 0$ we can find $N_{z}$ such that

(3) $\forall{n\geqslant N_{z}}:{f_{n}(z)=0}\ .$

Now either $a_{N_{z}}=1$ and we are done, or $a_{N_{z}}=0$. In this second
case there cannot be $n>N_{z}$ such that $a_{n}=1$: for assume there is such
$n$. Then we can find $N_{z}\leqslant m<n$ such that $a_{m}=1-a_{m+1}$, which
means that $z=\alpha_{m}$. By assumption there exists $j\geqslant m$ such that
$f_{j}(\alpha_{m})\neq 0$, but that is a contradiction to Equation 7. Thus, if
$a_{N_{z}}=0$, then $\forall{n\in\mathbb{N}}:{a_{n}=0}$. Altogether LPO holds.
∎

######  Corollary 8.

Under the assumption of $\neg\textrm{LPO}$, if $(f_{n})_{n\geqslant
1}:\mathbb{N}_{\infty}\to\\{0,1\\}$ is such that $f_{n}(x)\to 0$ point-wise
then $f_{n}\to 0$ uniformly.

##  3\. The General Case

The situation is a bit more intricate when we move to more general metric
spaces. Notice that, for example, in the case of
$\mathbb{N}_{\infty}\to\mathbb{N}$ being sequentially continuous is equivalent
to being point-wise continuous, and even equivalent to being uniformly
continuous.

The following definitions mirror these different levels of continuity.

######  Definition 9.

Let $(f_{n})_{n\geqslant 1}$ be a sequence of functions. We say that
$(f_{n})_{n\geqslant 1}$ converges

  1. (1)

_sequentially semi-uniform at $x$_, if for all $x_{n}\to x$ and all
$\varepsilon>0$ there exists $N\in\mathbb{N}$ such that

$\forall{n,i\geqslant N}:{\rho\left(f(x_{n}),f_{i}(x_{n})\right)<\varepsilon}\
;$

  2. (2)

_semi-uniform at $x$_, if for all $\varepsilon>0$ there exists
$N\in\mathbb{N}$ and $\delta>0$ such that for all $y\in B_{x}(\delta)$

$\forall{i\geqslant N}:{\rho\left(f(y),f_{i}(y)\right)<\varepsilon}\ .$

It is easy to see that we can also combine $N$ and $\delta$ in the second
definition by using $\delta=\nicefrac{{1}}{{N}}$.

Trivially we have

$\text{ uniform }\implies\text{ semi-uniform }\implies\text{ seq.\ semi-
uniform }\implies\text{ point-wise }\ .$

In this section we will show that all three implications can be reversed,
assuming certain principles and working on certain spaces.

######  Theorem 10 (General metric space version of the Third Trick).

Consider a sequence of functions $(f_{n})_{n\geqslant 1}:X\to Y$ defined on a
complete555We would like to mention that here, and in the following one can
replace completeness with the much weaker notion of complete enough [6, 7].
metric space $X$ into an arbitrary metric space $Y$ converging point-wise to
$f:X\to Y$; let $(x_{n})_{n\geqslant 1}$ be a sequence in $X$ converging to
$x\in X$, and consider $\varepsilon>0$. Either

  * •

there exists $N\in\mathbb{N}$ such that

$\forall{n,i\geqslant N}:{\rho\left(f(x_{n}),f_{i}(x_{n})\right)<\varepsilon}\
;$

  * •

or there exists a sequence $z_{n}\to x$ and $k_{n}$ such that $k_{n}\geqslant
n$, and

$\forall{n\in\mathbb{N}}:{\rho\left(f_{k_{n}}(z_{n}),f(z_{n})\right)>\frac{\varepsilon}{4}}\
.$

Furthermore, in case the second alternative holds, LPO holds.

The proof very much follows the lines of the proof of Lemma 5, Theorem 6, and
Proposition 7 and has been moved to an appendix.

######  Corollary 11.

Assuming $\neg\textrm{LPO}$, if $f_{n}\to f$ point-wise then $f_{n}\to f$
sequentially semi-uniform.

To get from sequential semi-uniform convergence to semi-uniform convergence we
can use Ishihara’s boundedness principle BD-N, which states that every
countable pseudo-bounded subset of $\mathbb{N}$ is bounded. Here, a subset $S$
of $\mathbb{N}$ is _pseudo-bounded_ if $\lim_{n\to\infty}s_{n}/n=0$ for each
sequence $(s_{n})_{n\geqslant 1}$ in $S$. In [13, Lemma 3], it was shown that
a set $S$ of natural numbers is pseudo-bounded if and only if for each
sequence $(s_{n})_{n\geqslant 1}$ in $S$, $s_{n}<n$ for all sufficiently large
$n$. Every bounded subset of $\mathbb{N}$ is trivially pseudo-bounded and,
conversely, every inhabited, _decidable_ , and pseudo-bounded subset of
$\mathbb{N}$ is easily seen to be bounded. However, in the absence of
decidability, this is not guaranteed anymore.

Overall BD-N is a very weak principle that is true in CLASS but which also
holds in INT and RUSS. Indeed, there are very few known models in which BD-N
fails. The first such model was a realizability model described in [14] and
the second one was a topological model [15].

######  Proposition 12.

BD-N implies that for $(f_{n})_{n\geqslant 1}$ and $f$ defined on a countable,
complete space $X$ such that $f_{n}\to f$ sequentially semi-uniform, also
converges semi-uniformly.

###### Proof.

Assume $f_{n}\to f$ sequentially semi-uniform. Let $X=\left\\{r_{i}\right\\}$,
and consider $\varepsilon>0$, and $x\in X$. Consider the set

$S=\mbox{$\left\\{\,n\in\mathbb{N}\,\middle|\,\exists{i,j\geqslant
n}:{r_{i}\in B_{x}(\frac{1}{n})\ \land\
\rho(f(r_{i}),f_{j}(r_{i}))>\varepsilon}\,\right\\}$}\cup\left\\{0\right\\}\
.$

This set is easily seen to be countable. We will show that $S$ is pseudo-
bounded. To this end let $(s_{n})_{n\geqslant 1}$ be a sequence in $S$. Define
a sequence $x_{n}$ by

$\displaystyle s_{n}\leqslant n$ $\displaystyle\implies x_{n}=x$
$\displaystyle s_{n}>n$ $\displaystyle\implies x_{n}=r_{i},\text{ where
}i\text{ is as in the definition of }S\ .$

Since, for all $n$, we have $\rho(x,x_{n})<\frac{1}{n}$, this sequence
converges to $x$. Since $(f_{n})_{n\geqslant 1}$, by assumption, converges
sequentially semi-uniform there exists $N$ such that $\forall{i,n\geqslant
N}:{\rho(f(x_{n}),f_{i}(x_{n}))<\varepsilon}$. That means, that for all
$n\geqslant N$ we must have $s_{n}\leqslant n$, since otherwise $x_{n}$ would
be such that there is $j\geqslant N$ with
$\rho(f(x_{n}),f_{j}(x_{n})>\varepsilon$, by the definition of $S$. Thus, by
BD-N, the set $S$ is bounded. That means that for all $\varepsilon>0$ there
exists $N\in\mathbb{N}$ such that for all $y\in
B_{x}(\frac{1}{N})\cap\mbox{$\left\\{\,r_{i}\,\middle|\,i\in\mathbb{N}\,\right\\}$}$

$\forall{i\geqslant N}:{\rho\left(f(y),f_{i}(y)\right)<\varepsilon}\ ,$

and since $\varepsilon$ and $x$ were arbitrary we are done. ∎

######  Corollary 13.

BD-N implies that for every sequence of non-discontinuous functions
$(f_{n})_{n\geqslant 1}$ and $f$ defined on a complete and separable space $X$
such that $f_{n}\to f$ sequentially semi-uniform, also converges semi-
uniformly.

For the next corollary we remind the reader that the existence of a
discontinuous function is equivalent to the weak limited principle of
omniscience WLPO, and hence the negation of the latter implies that all
functions are non-discontinuous [12, Theorem 1].

######  Corollary 14.

Assuming $\neg\textrm{WLPO}$ and BD-N (both hold in RUSS and INT). For a
sequence of functions $f_{n},f:X\to Y$ defined on a separable $X$ and into an
arbitrary metric space $Y$, point-wise and semi-uniform convergence are
equivalent.

Finally, in order to also make the step to uniform convergence, we need to
assume a form of Brouwer’s fan theorem: FAN${}_{\Pi^{0}_{1}}$. All versions of
Brouwer’s fan theorem them enable one to conclude that a bar is uniform. Here,
a bar $B$ is a subset of the space of all finite binary sequences $2^{\ast}$
such that for all infinite binary sequences $\alpha$ there is $n$ such that
$\overline{\alpha}n$—the initial segment of $\alpha$ of length $n$—is in $B$.
A bar is uniform if this happens uniformly for all $\alpha\in 2^{\mathbb{N}}$,
that is if

$\exists{N}:{\forall{\alpha\in 2^{\mathbb{N}}}:{\exists{n\leqslant
N}:{\overline{\alpha}n\in B}}}\ .$

A bar $B$ is called ${\Pi}_{1}^{0}$-bar, if there exist a set $S\subset
2^{\ast}\times\mathbb{N}$ such that

$u\in B\iff\forall{n\in\mathbb{N}}:{(u,n)\in S}\ ,$

and

$(u,n)\in S\implies(u\ast 0,n)\in S\land(u\ast 1,n)\in S\ .$

FAN${}_{\Pi^{0}_{1}}$ holds in INT as well as in CLASS and is slightly
stronger than the uniform continuity theorem (UCT), which is the statement
that all point-wise continuous functions $2^{\mathbb{N}}\to\mathbb{R}$ are
uniformly continuous.

######  Proposition 15.

FAN${}_{\Pi^{0}_{1}}$ implies that every sequence of non-discontinuous
functions $f_{n},f:2^{\mathbb{N}}\to\mathbb{R}$ such that $f_{n}\to f$ semi-
uniformly, also converges uniformly.

Conversely, the latter statement implies UCT.

###### Proof.

Let $f_{n}:2^{\mathbb{N}}\to\mathbb{R}$ be such that $f_{n}\to f$ semi-
uniform. Fix $\lambda_{u,n}$ such that

$\displaystyle\lambda_{u,n}=0$ $\displaystyle\implies\rho\left(f_{n}(u\ast
000\dots),f(u\ast 000\dots)\right)<\varepsilon\ ,$
$\displaystyle\lambda_{u,n}=1$ $\displaystyle\implies\rho\left(f_{n}(u\ast
000\dots),f(u\ast 000\dots)\right)>\nicefrac{{\varepsilon}}{{2}}\ .$

Define a decidable set $S\subset 2^{\ast}\times\mathbb{N}$ by

(4) $(u,n)\in S\iff\forall{\left\lvert u\right\rvert\leqslant
i\leqslant\left\lvert u\right\rvert+n}:{\forall{w\in 2^{\ast}}:{\left\lvert
w\right\rvert\leqslant n-\left\lvert u\right\rvert\implies\lambda_{u\ast
w,i}=0}}\ .$

To obtain a $\Pi_{1}^{0}$-set define $B$ by $u\in
B\iff\forall{n\in\mathbb{N}}:{(u,n)\in S}$. We claim that $B$ is a
$\Pi_{1}^{0}$-bar. Firstly, notice that the condition “$\left\lvert
w\right\rvert\leqslant n-\left\lvert u\right\rvert$” easily ensures that if
$(u,n)\in S$ then also $(u\ast i,n)\in S$ for $i=0,1$, as required. To see
that $B$ is a bar let $\alpha\in 2^{\mathbb{N}}$ be arbitrary. By the
assumption of semi-uniformity there exists $N$ such that for all $\beta\in
2^{\mathbb{N}}$ and all $n\geqslant N$ we have

$\rho\left(f_{n}(\overline{\alpha}N\ast\beta),f(\overline{\alpha}N\ast\beta)\right)<\nicefrac{{\varepsilon}}{{2}}\
.$

In particular we have that

(5) $\forall{w\in 2^{\ast}}:{\forall{i\geqslant
N}:{\lambda_{\overline{\alpha}N\ast w,i}=0\ .}}$

That ensures that $\overline{\alpha}N\in B$, since it actually over-fullfills
4.

Applying FAN${}_{\Pi^{0}_{1}}$ we get a uniform bound for $B$; that is, there
is $M$ such that $\overline{\alpha}M\in B$ for all $\alpha\in 2^{\mathbb{N}}$.
Now let $v\in 2^{\ast}$ and $n\in\mathbb{N}$ such that both $\left\lvert
v\right\rvert,n\geqslant M$. Let $w$ be the suffix of $v$:
$v=\overline{v}M\ast w$.

Let $k=\max\\{n-M,|w|+M\\}$. Then $n\leqslant M+k$, and $\left\lvert
w\right\rvert\leqslant k-M$. That means that, since $(\overline{v}M,k)\in S$,
we have $\lambda_{v,n}=0$, and hence, by the definition of $\lambda$ we have
$\rho\left(f_{n}(v\ast 000\dots),f(v\ast 000\dots)\right)<\varepsilon$ for
$\left\lvert v\right\rvert\geqslant M$ and $n\geqslant M$. Since we can pad
out any $v\in 2^{\ast}$ if it is shorter than $M$ by $0$s that actually means
that for _any_ $v\in 2^{\ast}$ and all $n\geqslant M$ we have
$\rho\left(f_{n}(v\ast 000\dots),f(v\ast
000\dots)\right)\leqslant\varepsilon$. Finally, by the non-discontinuity this
means that for any $\alpha\in 2^{\mathbb{N}}$ and $n\geqslant M$ we have
$\rho\left(f_{n}(\alpha),f(\alpha)\right)<\varepsilon$. Hence the convergence
is uniform.

Conversely to see that UCT holds, it suffices to show that every point-wise
continuous $f:2^{\mathbb{N}}\to\mathbb{R}$ is bounded [4, Theorem 10]. We may
assume that $f\geqslant 0$. Let $f_{n}=\min\\{f,n\\}$. Then $f_{n}\to f$ semi-
uniformly, since a point-wise continuous function is locally bounded. Now if
$f_{n}\to f$ uniformly then there exists $M\in\mathbb{N}$ such that
$\left\lvert f_{M}(\alpha)-f(\alpha)\right\rvert<1$ for all $\alpha\in CS$.
That means that there cannot be $\alpha$ such that $f(\alpha)>M+1$, since in
that case $f(\alpha)>M+1=f_{M}(\alpha)+1$, which means that $\left\lvert
f_{M}(\alpha)-f(\alpha)\right\rvert>1$. So $f(\alpha)\leqslant M+1$ for all
$\alpha\in 2^{\mathbb{N}}$ and we are done. ∎

######  Corollary 16.

Assuming FAN${}_{\Pi^{0}_{1}}$. For a sequence of non-discontinuous functions
$f_{n},f:X\to Y$ defined on a compact666As common in constructive analysis we
define compact as complete and totally bounded. $X$ and into an arbitrary
metric space $Y$, semi-uniform and uniform convergence are equivalent.

###### Proof.

By [16, Proposition 7.4.3] there exists a uniformly continuous and surjective
function $F:2^{\mathbb{N}}\to X$. Now let $(g_{n})_{n\geqslant
1}:2^{\mathbb{N}}\to\mathbb{R}$ be defined by

$g_{n}(\alpha)=\rho(f_{n}(F(\alpha)),f(F(\alpha)))\ .$

It is easy to see that $g_{n}$ is non-discontinuous and that $g_{n}\to 0$
semi-uniform. So by Proposition 15 $g_{n}\to 0$ uniformly, which means for an
arbitrary $\varepsilon>0$ there exists $N\in\mathbb{N}$ such that $\left\lvert
g_{n}\right\rvert<\varepsilon$ for all $n\geqslant N$. Since $F$ is
surjective, for all $x\in X$ and $n\geqslant N$ we have
$g_{n}(\alpha)=\rho(f_{n}(x),f(x))$. Hence $f_{n}\to f$ uniformly. ∎

Since every compact space is, by definition, totally bounded, which in turns
implies separability, we get the following, final corollary.

######  Corollary 17.

Assuming $\neg\textrm{WLPO}$, BD-N, and FAN${}_{\Pi^{0}_{1}}$ (all these hold
in INT). For a sequence of functions $f_{n},f:X\to Y$ defined on a compact $X$
and into an arbitrary metric space $Y$, point-wise and uniform convergence are
equivalent.

This generalises the result of [5, §1 and §5], which there is proven with the
help of continuous choice and the full fan theorem.

##  4\. Conclusion—from the Second to the Third Trick?

We should concede that, while our results are related to Ishihara’s First and
Second Trick, naming it the “Third Trick” might be slightly misleading. As
mentioned above, the Second Trick is an iteration of the First Trick, so one
might expect that the Third Trick is an iteration—or at least follows—from the
Second one. This is not the case. One is tempted, in the situation that
$f_{n}\to f$, where $f_{n},f:\mathbb{N}^{\mathbb{N}}\to\mathbb{N}$, to
consider

$F(\alpha)=\sum_{n\geqslant 0}\left\lvert f_{n}(\alpha)-f(\alpha)\right\rvert\
.$

This is a well-defined function, since the sum is finite, because of the
point-wise convergence. If we could apply Ishihara’s Second Trick to $F$ we
would immediately get (our main) Theorem 6. However, Ishihara’s Second Trick
requires strong extensionality, and that is not guaranteed with $F$. In [6]
the first author has given a version of Ishihara’s second trick, which does
not rely on strong extensionality, but which in turn has weaker consequences,
which are not strong enough to deduce Theorem 6.

## References

  * [1] P. Aczel and M. Rathjen.  Notes on constructive set theory.  Technical Report 40, Institut Mittag-Leffler, The Royal Swedish Academy of Sciences, 2001.
  * [2] E. Bishop.  Foundations of Constructive Analysis.  McGraw-Hill, 1967.
  * [3] E. Bishop and D. Bridges.  Constructive Analysis.  Springer-Verlag, 1985.
  * [4] D. Bridges and H. Diener.  The pseudocompactness of $[0,1]$ is equivalent to the uniform continuity theorem.  Journal of Symbolic Logic, 72(4):1379–1384, 2007.
  * [5] H. de Swart.  Elements of intuitionistic analysis. rolle’s theorem and complete, totally bounded, metric spaces.  Mathematical Logic Quarterly, 22(1):289–298, 1976.
  * [6] H. Diener.  Variations on a theme by Ishihara.  Mathematical Structures in Computer Science, 25:1569–1577, 10 2015\.
  * [7] H. Diener and M. Hendtlass.  Complete: When enough is enough.  submitted for publication, Mar 2017.
  * [8] J. Dieudonné.  Foundations of Modern Analysis.  Academic Press, 1960.
  * [9] M. Escardó.  Infinite sets that satisfy the principle of omniscience in any variety of constructive mathematics.  J. Symbolic Logic, 78(3):764–784, 09 2013.
  * [10] M. Escardó.  Constructive decidability of classical continuity.  Mathematical Structures in Computer Science, 25(07):1578–1589, 2015\.
  * [11] H. Ishihara.  Continuity and nondiscontinuity in constructive mathematics.  Journal of Symbolic Logic, 56(4):1349–1354, 1991.
  * [12] H. Ishihara.  Continuity properties in constructive mathematics.  Journal of Symbolic Logic, 57(2):557–565, 1992.
  * [13] S. Ishihara, Hajime; Yoshida.  A constructive look at the completeness of the space $\mathcal{D}(\textbf{R})$.  Journal of Symbolic Logic, 67(4):1511—1519, 2002.
  * [14] P. Lietz.  From Constructive Mathematics to Computable Analysis via the Realizability Interpretation.  PhD thesis, TU Darmstadt, 2004.
  * [15] R. S. Lubarsky.  On the failure of BD-N and BD, and an application to the anti-specker property.  Journal of Symbolic Logic, 78(1):39–56, 2013.
  * [16] A. S. Troelstra and D. van Dalen.  Constructivism in mathematics. Vol. II, volume 123 of Studies in Logic and the Foundations of Mathematics.  North-Holland Publishing Co., Amsterdam, 1988.

## Appendix: Proof of Theorem 10

###### Proof.

The proof is analogous to Lemma 5, Theorem 6, and Proposition 7, which
correspond to Part (b), (c), and (d) respectively. The only real added effort
is in Part (b), where the loss of decidability is compensated by using
approximate decisions for $\frac{\varepsilon}{4},\frac{\varepsilon}{2}$, and
$\varepsilon$.

  1. (a)

Let $x_{n}\to x$, and $\varepsilon>0$. First, we may assume, without loss of
generality, that
$\forall{n\in\mathbb{N}}:{\rho(f_{n}(x),f(x))<\frac{\varepsilon}{2}}$, since
we can otherwise consider an appropriate tail of the sequence
$(f_{n})_{n\geqslant 1}$. Secondly, we fix a modulus of convergence
$\mu\in\mathbb{N}^{\mathbb{N}}$ such that $\rho(x_{n},x)<\frac{1}{k}$ for all
$n\geqslant\mu(k)$.

  2. (b)

Our first claim is that for any $n\in\mathbb{N}$ we can decide whether there
is $z_{n}$ and $i\in\mathbb{N}$ such that $\rho(z_{n},x)\leqslant\frac{1}{n}$
and

$\rho(f_{i}(z_{n}),f(z_{n}))>\frac{\varepsilon}{4}$

or whether $\rho(f_{i}(x_{m}),f(x_{m}))<\varepsilon$ for all $i\geqslant n$
and $m\geqslant\mu(n)$.

To this end, first notice that for fixed $m$ there exists $N_{m}$ such that
$\rho(f_{i}(x_{m}),f(x_{m}))<\varepsilon$ for all $i\geqslant N_{m}$. Thus we
can decide whether there exists $i\geqslant n$ such that
$\rho(f_{i}(x_{m}),f(x_{m}))>\frac{\varepsilon}{2}$ or whether
$\rho(f_{i}(x_{m}),f(x_{m}))<\varepsilon$ for all $i\geqslant n$. Choose a
binary sequence $\gamma_{m}$ that flags these possibilities by $1$ and $0$
respectively.

Now define a sequence $(w_{m})_{m\geqslant\mu(n)}$ such that $w_{m}=x$, if
$\gamma_{m}=0$, and if $\gamma_{m}=1$ set $w_{m}=x_{m^{\prime}}$ where
$m^{\prime}$ is the smallest index such that $\gamma_{m}=1$. In words, as long
as $\gamma_{m}=0$ we stay constant on $x$ and if we ever hit a term such that
$\gamma_{m}=1$, we switch to $x_{m}$ and stay constant from then on. The
sequence $w_{m}$ is easily seen to be a Cauchy sequence, and therefore
converges to a limit $w$. Now, choose $N_{w}$ such that
$\rho(f_{i}(w),f(w))<\frac{\varepsilon}{2}$ for all $i\geqslant N_{w}$. Since
we only need to check for the indices between $n$ and $N_{w}$ (if any), we can
decide whether either $\rho(f_{i}(w),f(w))<\frac{\varepsilon}{2}$ for all
$i\geqslant n$ or whether there is $j\geqslant n$ such that
$\rho(f_{i}(w),f(w))>\frac{\varepsilon}{4}$. In the latter case we are done,
since $\rho(w,x)\leqslant\frac{1}{n}$, so we can choose $z_{n}=w$. In the
first case we must have $\gamma_{m}=0$ for all $m\geqslant\mu(n)$: for assume
there is $\gamma_{m}=1$. Then we can find the first such index $m^{\prime}$,
which means that $w=x_{m^{\prime}}$ and there exists $i\geqslant n$ such that
$\rho(f_{i}(x_{m^{\prime}}),f(x_{m^{\prime}}))>\frac{\varepsilon}{2}$. This is
a contradiction to $\rho(f_{i}(w),f(w))<\frac{\varepsilon}{2}$ for all
$i\geqslant n$.

  3. (c)

Using the previous part, fix a binary sequence $(\lambda_{n})_{n\geqslant 1}$
such that

$\displaystyle\lambda_{n}=0$ $\displaystyle\implies\exists{i\geqslant
n,z_{n}\in
X}:{\rho(x,z_{n})\leqslant\frac{1}{n}\land\rho(f_{i}(z_{n}),f(z_{n}))>\frac{\varepsilon}{4}}$
$\displaystyle\lambda_{n}=1$ $\displaystyle\implies\forall{i\geqslant
n,m\geqslant\mu(n)}:{\rho(f_{i}(x_{m}),f(x_{m}))<\varepsilon}\ .$

We may assume that the sequence $\lambda_{n}$ is increasing, and that
$\lambda_{1}=0$, since we are otherwise done.

Now define a sequence $(y_{n})_{n\geqslant 1}$ in $X$ by

$\displaystyle\lambda_{n}=0$ $\displaystyle\implies y_{n}=z_{n}\ ,$
$\displaystyle\lambda_{n}=1$ $\displaystyle\implies y_{n}=z_{m},\text{ where
}\lambda_{m}=0\land\lambda_{m+1}=1\ .$

In words, as long as $\lambda_{n}=0$ we set $y_{n}=z_{n}$, and as soon as we
find the first term such that $\lambda_{n}=1$, we stay constant on the
previous term $z_{n-1}$. Again, $(y_{n})_{n\geqslant 1}$ is easily seen to be
a Cauchy sequence, which therefore converges to a limit $y$ in $X$.

Since $f_{n}(y)\to f(y)$ there is $N_{y}$ such that

(6) $\rho(f_{n}(y),f(y))<\frac{\varepsilon}{4}\text{ for all }n\geqslant
N_{y}\ .$

Now either $\lambda_{N_{y}}=1$ or $\lambda_{N_{y}}=0$.

In the second case there cannot be $n>N_{y}$ with $\lambda_{n}=1$: In that
case we could find $N_{y}\leqslant n^{\prime}<n$ such that
$\lambda_{n^{\prime}}=0$ and $\lambda_{n^{\prime}+1}=1$. But then
$y=z_{n^{\prime}}$ and there is $i\geqslant n^{\prime}$ such that
$\rho(f_{i}(y),f(y))>\frac{\varepsilon}{4}$, which is a contradiction to
Equation 6, since $i\geqslant n^{\prime}\geqslant N_{y}$. Hence
$\lambda_{n}=0$ for all $n>N_{y}$, which means that $\lambda_{n}=0$ for all
$n$, since $(\lambda_{n})_{n\geqslant 1}$ is increasing.

Together we can decide whether $\forall{n\in\mathbb{N}}:{\lambda_{n}=0}$ or
whether $\exists{n\in\mathbb{N}}:{\lambda_{n}=1}$ which means—by the
definition of $(\lambda_{n})_{n\geqslant 1}$—we have proven the main claim of
the theorem.

  4. (d)

To prove the “LPO claim” of this theorem let $(a_{n})_{n\geqslant 1}$ be a
binary sequence. Without loss of generality we may assume that
$(a_{n})_{n\geqslant 1}$ is increasing. Now consider the sequence
$(v_{n})_{n\geqslant 1}$ in $X$ defined by

$v_{n}=\begin{cases}z_{m}&\text{if }a_{n}=1\text{ and }a_{m}=1-a_{m+1}\\\
x&\text{if }a_{n}=0\ .\end{cases}$

Again, this is a Cauchy sequence, which therefore converges to a limit $v$.
Since $f_{n}(v)\to 0$ we can find $N_{v}$ such that

(7) $\forall{n\geqslant N_{v}}:{\rho(f_{n}(v),f(v))<\frac{\varepsilon}{4}}\ .$

Now either $a_{N_{v}}=1$ and we are done, or $a_{N_{v}}=0$. In this second
case there cannot be $n>N_{v}$ such that $a_{n}=1$: for assume there is such
$n$. Then we can find $N_{v}\leqslant m<n$ such that $a_{m}=1-a_{m+1}$, which
means that $v=z_{m}$. Also, by assumption there exists $j\geqslant m$ such
that $\rho(f_{j}(z_{m}),f(z_{m}))>\frac{\varepsilon}{4}$, but that is a
contradiction to Equation 7. Thus, if $a_{N_{v}}=0$, then
$\forall{n\in\mathbb{N}}:{a_{n}=0}$. Altogether LPO holds. ∎