We say that a ternary groupoid $(X,[\ ])$ is a ternary semigroup if the operation $[\ ]$ is associative, that is:

$$[[abc]de]=[a[bcd]e]=[ab[cde]].$$

Let $z_{000},\dots,z_{111}\in\operatorname{\mathbb{C}}$ be $8$ numbers indexed by the vertices of a cube, as shown in Figure 1 . We say that these $8$ numbers satisfy the Kashaev equation if

(1.1)

$\displaystyle 2(a^{2}+b^{2}+c^{2}+d^{2})-(a+b+c+d)^{2}+4(s+t)=0,$

where $a,b,c,d,s,t$ are the monomials defined in Figure 1 . Notice that the equation ( 1.1 ) is invariant under the symmetries of the cube. Thus, reindexing the $8$ values using an isomorphic labeling of the cube does not change the Kashaev equation.

A $JB^{\ast}-$ triple is a Banach space $Z$ with a triple product $\{\ ,\ ,\ \}:Z^{3}\longrightarrow Z$ that is linear and symmetric in the first and third variables (symmetric in the sense that $\{x,y,z\}=\{z,y,x\}$ for all $x,z$ ) and antilinear in the second variable and which satisfies,
$(i)$ the mapping $x\Box x$ , given by $x\Box x(z)=\{x,x,z\}$ is Hermitian, $\sigma(x\Box x)\geq 0$ and $\|x\Box x\|=\|x\|^{2}$ ,
$(ii)$ for every $a,b,x,y,z\in X$ , the Jordan triple identity

$$\{a,b,\{x,y,z\}\}=\{\{a,b,x\},y,z\}-\{x,\{b,a,y\},z\}+\{x,y,\{a,b,z\}\}$$

holds.

A naïve mechanism consists of an action set $\mathcal{A}$ and an associated mapping from any action to a possible outcome, i.e., $\langle\boldsymbol{x},p\rangle:\mathcal{A}\mapsto\mathcal{X}\times\mathbb{R}_{+}$ .

In particular, there exists a special action $\bot$ meaning “exiting the mechanism” such that

$\displaystyle\boldsymbol{x}(\bot)=\mathbf{0},p(\bot)=0.$

( Exit )

A sesquilinear form on $V$ is a map $V\times V\to\mathbb{F}$ , $(u,v)\mapsto\langle u,v\rangle$ such that for all $\alpha,\beta\in\mathbb{F}$ , $u,v,w\in V$ :

1. (i)

   $\left\langle u,\alpha v+\beta w\right\rangle=\alpha\left\langle u,v\right\rangle+\beta\left\langle u,w\right\rangle$ (linear in the second ^{(1) (1)} (1) Note that this convention varies in the literature; the one here is used in almost every physics text. argument),

2. (ii)

   $\left\langle\alpha u+\beta v,w\right\rangle=\bar{\alpha}\left\langle u,w\right\rangle+\bar{\beta}\left\langle v,w\right\rangle$ (conjugate linear in the first argument).

A hermitian form on $V$ is a sesquilinear form $\left\langle\cdot,\cdot\right\rangle$ satisfying, in addition:

1. (iii)

   $\left\langle u,v\right\rangle=\overline{\left\langle v,u\right\rangle}$ (symmetry).

An inner product or scalar product on $V$ is a hermitian form $\left\langle\cdot,\cdot\right\rangle$ satisfying, in addition:

1. (iv)

   $\left\langle v,v\right\rangle>0$ for $v\neq 0$ (positive definite).

A (sesqui-) quadratic form on $V$ is a map $q\colon V\to\mathbb{F}$ such that for $\alpha\in\mathbb{F}$ , $u,v\in V$ :

1. (i)

   $q(\alpha v)=\bar{\alpha}\alpha q(v)$ (scaling quadratically),

2. (ii)

   $\left\langle u,v\right\rangle_{q}:=\frac{1}{4}\bigl{(}q(u+v)-q(u-v)+iq(u-iv)-iq(u+iv)\bigr{)}$ is sesquilinear in $u,v$ .

A norm on $V$ is a map $V\to\mathbb{R}_{\mathchoice{\vbox{\hbox{$\scriptstyle+$}}}{\vbox{\hbox{$\scriptstyle+$}}}{\vbox{\hbox{$\scriptscriptstyle+$}}}{\vbox{\hbox{$\scriptscriptstyle+$}}}}:=[0,\infty)$ , $v\mapsto\|v\|$ , such that for all $\alpha\in\mathbb{F}$ , $u,v\in V$ :

1. (i)

   $\left\|\alpha v\right\|=|\alpha|\left\|v\right\|$ (scaling linearly),

2. (ii)

   $\left\|u+v\right\|\leq\left\|u\right\|+\left\|v\right\|$ (triangle inequality),

3. (iii)

   $\left\|v\right\|=0$ if and only if $v=0$ (positive definite).

The pair $(V,\left\|\cdot\right\|)$ is called a normed linear space , the pair $(V,q)$ a (sesqui-) quadratic space , and the pair $(V,\left\langle\cdot,\cdot\right\rangle)$ an inner product space or a pre-Hilbert space .

A Poisson algebra $\mathcal{A}$ is a vector space over $\mathbb{F}$ ( $\mathbb{R}$ or $\mathbb{C}$ ) equipped with an $\mathbb{F}$ -bilinear and associative product

$$\mathcal{A}\times\mathcal{A}\to\mathcal{A},\quad(f,g)\mapsto fg,$$

and an additional product (typically non-associative, called a Poisson bracket )

$$\mathcal{A}\times\mathcal{A}\to\mathcal{A},\quad(f,g)\mapsto\{f,g\},$$

satisfying, for all $\alpha,\beta\in\mathbb{F}$ , and $f,g,h\in\mathcal{A}$ :

1. (i)

   linearity: $\{f,\alpha g+\beta h\}=\alpha\{f,g\}+\beta\{f,h\}$ ,

2. (ii)

   antisymmetry: $\{f,g\}=-\{g,f\}$ ,

3. (iii)

   Jacobi identity: $\{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0$ ,

4. (iv)

   Leibniz rule: $\{f,gh\}=\{f,g\}h+g\{f,h\}$ .

A simple upper test function is a function of the form

$$\psi=\sigma\circ\varphi\circ a,$$

where $\sigma\in C^{\infty}(\mathbb{R})$ , $\sigma^{\prime}\geq 0$ , $\sigma^{\prime\prime}\geq 0$ , $a\in C^{\infty}(\mathbb{R}^{d},\mathbb{R}^{d})$ , $Da$ constant, and $\det Da=1$ .

A simple lower test function is a function of the form

$$\psi=\sigma\circ\varphi\circ a,$$

where $\sigma\in C^{\infty}(\mathbb{R})$ , $\sigma^{\prime}\geq 0$ , $\sigma^{\prime\prime}\leq 0$ , $a\in C^{\infty}(\mathbb{R}^{d},\mathbb{R}^{d})$ , $Da$ constant, and $\det Da=1$ .

Given a complex ramified cover $\varphi:C\to\mathbf{CP}^{1}$ , a real structure on $\varphi$ is a antiholomorphic involution $\iota:C\to C$ such that

$$\varphi\circ\iota=\text{conj}\circ\varphi.$$

The tuple $(\varphi,\iota)$ is a real ramified cover . An isomorphism of real ramified covers $(\varphi:C\to\mathbf{CP}^{1},\iota)$ and $(\psi:D\to\mathbf{CP}^{1},\kappa)$ is an isomorphism of complex ramified covers $\alpha:C\to D$ such that

$$\alpha\circ\iota=\kappa\circ\alpha.$$

A complete theory ${T}$ is said to be irreducible if

$${{\rm acl}(\emptyset)={\rm dcl}(\emptyset)}$$

(in ${T^{{\rm eq}}}$ ). We refer to the expansion of a theory ${T}$ by constants for ${{\rm acl}(\emptyset)}$ as its irreducible component .

For $p\geq-1$ an integer, we set

$$[p]=\{0,\ldots,p\}.$$

Consider a unital subalgebra $\mathcal{A}\subset\mathcal{C}^{\infty}(M)$ of smooth functions on a Banach manifold $M$ , i.e. $\mathcal{A}$ is vector subspace of $\mathcal{C}^{\infty}(M)$ containing the constants and stable under pointwise multiplication. A $\mathbb{R}$ -bilinear operation $\{\cdot,\cdot\}~{}:\mathcal{A}\times\mathcal{A}\rightarrow\mathcal{A}$ is called a Poisson bracket on $M$ if it satisfies :

1. (i)

   anti-symmetry  : $\{f,g\}=-\{g,f\}~{};$

2. (ii)

   Jacobi identity : $\{\{f,g\},h\}+\{\{g,h\},f\}+\{\{h,f\},g\}=0$ ;

3. (iii)

   Leibniz formula : $\{f,gh\}=\{f,g\}h+g\{f,h\}$ ;

Let $Loop_{3}(X/\!\!/G)$ denote the groupoid whose objects are $(\sigma,\gamma)$ with $\sigma\in G$ and $\gamma:\mathbb{R}\longrightarrow X$ a continuous map such that $\gamma(s+1)=\gamma(s)\cdot\sigma$ , for any $s\in\mathbb{R}$ . A morphism $\alpha:(\sigma,\gamma)\longrightarrow(\sigma^{\prime},\gamma^{\prime})$ is a continuous map $\alpha:\mathbb{R}\longrightarrow G$ satisfying $\gamma^{\prime}(s)=\gamma(s)\alpha(s)$ . Note that $\alpha(s)\sigma^{\prime}=\sigma\alpha(s+1)$ , for any $s\in\mathbb{R}$ .

It is isomorphic to $Loop_{2}(X/\!\!/G)$ . We can construct the isomorphism $\phi:Loop_{2}(X/\!\!/G)\longrightarrow Loop_{3}(X/\!\!/G)$ in this way: an object $(\sigma,\gamma)$ is sent to

$$\phi(\sigma,\gamma)=\gamma\ast\gamma\sigma\ast\gamma\sigma^{2}\ast\cdots$$

where $\ast$ is the composition of paths. For any real number $r$ between positive integers $n$ and $n+1$ , $\phi(\sigma,\gamma)(r)=\gamma(r-n)\sigma^{n}$ . A morphism $\alpha:(\sigma,\gamma)\longrightarrow(\sigma^{\prime},\gamma^{\prime})$ is mapped to the morphism

$$\phi(\alpha)=\alpha\ast\sigma^{-1}\alpha\sigma\ast\sigma^{-2}\alpha\sigma^{2}\ast\cdots.$$

For any $r$ between positive integers $n$ and $n+1$ , $\phi(\alpha)(r)=\sigma^{-n}\alpha(r-n)\sigma^{n}$ .

Let $\mathbb{G}$ be a groupoid. The Inertia groupoid $I(\mathbb{G})$ of $\mathbb{G}$ is defined as follows.

An object $a$ is an arrow in $\mathbb{G}$ such that its source and target are equal. A morphism $v$ joining two objects $a$ and $b$ is an arrow $v$ in $\mathbb{G}$ such that

$$v\circ a=b\circ v.$$

In other words, $b$ is the conjugate of $a$ by $v$ , $b=v\circ a\circ v^{-1}$ .

The torsion Inertia groupoid $I^{tors}(\mathbb{G})$ of $\mathbb{G}$ is a full subgroupoid of of $I(\mathbb{G})$ with only objects of finite order.

[ 3 ] The Fourier transform on the Hilbert space $L^{2}\big{(}\mathbb{R}^{2d}\big{)}$ is defined by:

$$\hat{g}(\epsilon)=(2\,\pi)^{-d}\int e^{i\,\left\langle\epsilon,u\right\rangle}g(u)\,du.$$

(6.1)

Let $L$ be a partially ordered set. An $L$ -colored graph is an ordered triple $(V,\chi^{\prime},\chi^{\prime\prime})$ such that $V$ is a nonempty set, $\chi^{\prime}:V\to L$ is an arbitrary function and $\chi^{\prime\prime}:V^{2}\to L$ is a function satisfying:

1. 1.

   $\chi^{\prime\prime}(x,x)=0$ , and

2. 2.

   $\chi^{\prime\prime}(x,y)=\chi^{\prime\prime}(y,x)$ .

A multicolored graph is an $L$ -colored graph, where $L$ is the powerset of some set with set inclusion as the ordering relation.

Let $\bar{x}>0$ be given. We say that a couple $(\bar{u},\bar{v})\in[0,1[\times[0,+\infty[$ is a constant strategy for $\bar{x}$ if

$\displaystyle\begin{cases}\left[\left(\dfrac{\lambda+r}{\bar{p}}-\lambda-\mu-\bar{v}\right)\bar{x}-\dfrac{\bar{u}}{\bar{p}}\right]~{}=~{}0,\\ \\ \bar{p}~{}=~{}\dfrac{r+\lambda}{r+\lambda+\bar{v}},\end{cases}$

where the second relation comes from taking $T_{b}=+\infty$ in ( 4.3 ).

Let $\psi\equiv\square\varphi$ be a past-dependent formula and $\mathbf{x}=\mathcal{M}(\mathbf{u})$ be a finite length signal until the time $t$ . We define the reward $\operatorname{\mathsf{reward}}(\psi,\mathbf{x})$ as

$$\operatorname{\mathsf{reward}}(\psi,\mathbf{x})=\exp(-\operatorname{\rho}(\varphi,\mathbf{x},t))-1$$

(4)

Let $a_{0},a_{1}:I\to A$ be $A$ -paths with the same end points. A genus homotopy between $a_{0}$ and $a_{1}$ is a Lie algebroid map $h:T\Sigma\to A$ , where $\Sigma$ is a compact surface with connected boundary $\partial\Sigma$ , together with a diffeomorphism $\phi:U\to V$ between collar neighborhoods of $\partial(I\times I)$ and $\partial\Sigma$ , such that

$$h\circ\mathrm{d}\phi=a(t,\varepsilon)\mathrm{d}t+b(t,\varepsilon)\mathrm{d}\varepsilon$$

satisfy the boundary conditions ( 3 ).

where

$$r=(x^{2}+y^{2})^{1/2},\hskip 28.452756pt\theta=\tan^{-1}{\frac{y}{x}}$$

(1.2)

We write $x=Re(z)$ (read “ $x$ is the real part of $z$ ”), $y=Im(z)$ (read “ $y$ is the imaginary part of $z$ ”). Furthermore, $r=|z|$ is the absolute value of $z$ , and $\theta=arg(z)$ is the argument of $z$ , which is determined only up to multiples of $2\pi$ .