Proof of Discrete Fourier Analysis on the Lattice Torus: Counting, Orthogonality, Inversion, Parseval, and the Spin Sum

Each result cited is universally quantified over the data in its own statement.

Notation. $\phi_{a}$ ($a\in\mathbb{Z}$) is the one-dimensional system of The Trigonometric System on the Torus §one-dimensional, so that $\psi_{k}(x)=\prod_{i=1}^{n}\phi_{k_{i}}(x_{i})$ by The Trigonometric System on the Torus §system; $\sqrt{2}\sqrt{2}=2$ by the definition of $\sqrt{2}$ recorded there. Natural numbers are read in $\mathbb{R}$ through the canonical map $\iota$ of The Real Numbers: Standing Notation and Background §numbers; by Properties of the Canonical Map from the Natural Numbers to an Ordered Field §additive, Properties of the Canonical Map from the Natural Numbers to an Ordered Field §multiplicative and Properties of the Canonical Map from the Natural Numbers to an Ordered Field §base-step, the side $L=2M+1$ read in $\mathbb{R}$ is $2M+1$ formed in $\mathbb{R}$ with $M$ read in $\mathbb{R}$, so $L-1=2M$. Natural numbers lie in $\mathbb{Z}$ by Arithmetic, Order, Discreteness and Intervals of the Integers §sign, and $\mathbb{Z}$ is closed under $+$, $-$ and negation by Arithmetic, Order, Discreteness and Intervals of the Integers §ring. Put

$$E=\{a\in\mathbb{Z}:-M\le a\le M\},\qquad D=\{\nu\in\mathbb{Z}:0\le\nu\le 2M\},\qquad P=\{b\in\mathbb{Z}:1\le b\le M\},$$

nonempty finite sets by Arithmetic, Order, Discreteness and Intervals of the Integers §intervals. Points of $\mathbb{Z}^{n}\subseteq\mathbb{R}^{n}$ are maps $[n]\to\mathbb{R}$ by Euclidean Points as Tuples of Real Numbers; hence $\Gamma_{M}$ is the set $E^{n}$ of $n$-tuples in $E$ and $W_{L}$ is the set $D^{n}$ of $n$-tuples in $D$, in the sense of Tuples in a Set. For $y\in\mathbb{R}$ we write $y/L=yL^{-1}$.

Step 1 (Dirichlet values). For $r\in\mathbb{Z}$ with $-2M\le r\le 2M$ put $K(r)=1+2\sum_{a=1}^{M}\cos(2\pi a r/L)$. Then $K(0)=L$ and $K(r)=0$ for $r\neq0$. If $r=0$, each summand is $\cos0=1$ by Values at Zero, Parity, Addition and Double-Angle Identities for Sine and Cosine §zero; since $[M]$ has $M$ elements (claim 1 of Basic Properties of Finite Sets), Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §count and claim 1 of Properties of a Sum over a Finite Index Set give $\sum_{a=1}^{M}1=M$, so $K(0)=1+2M=L$. Let $r\neq0$. By The Dirichlet and Fejer Kernel Identities §dirichlet with $N=M$ and $t=r/L$,

$$K(r)\,\sin(\pi r/L)=\sin(L\pi r/L)=\sin(\pi r).$$

By Values at Zero, Parity, Addition and Double-Angle Identities for Sine and Cosine §double-angle and Quarter-Turn Identities and Periodicity of Sine and Cosine §integer (with $m=r$), $1-2(\sin(\pi r))^{2}=\cos(2\pi r)=1$, so $(\sin(\pi r))^{2}=0$ and $\sin(\pi r)=0$. On the other hand $\pi=2x_{0}>0$ by The Number Pi §pi and The Least Positive Zero of the Cosine §least-zero. If $0<r$, then $0<\pi r/L<\pi$ because $r\le 2M<L$, so $0<\sin(\pi r/L)$ by Quarter-Turn Identities and Periodicity of Sine and Cosine §positive. If $r<0$, the same applies to $-r$, and $\sin(\pi r/L)=-\sin(\pi(-r)/L)<0$ by Values at Zero, Parity, Addition and Double-Angle Identities for Sine and Cosine §parity. In both cases $\sin(\pi r/L)\neq0$, hence $K(r)=0$.

Step 2 (sums over $E$). For every map $g:E\to\mathbb{R}$,

$$\sum_{a\in E}g(a)=g(0)+\sum_{b=1}^{M}\bigl(g(b)+g(-b)\bigr).$$

By Arithmetic, Order, Discreteness and Intervals of the Integers §sign and Arithmetic, Order, Discreteness and Intervals of the Integers §discrete, $E$ is the union of the pairwise disjoint nonempty sets $\{0\}$, $P$ and $-P=\{-b:b\in P\}$. By Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §disjoint-union (twice) and claim 1 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set, $\sum_{a\in E}g(a)=g(0)+\sum_{b\in P}g(b)+\sum_{a\in -P}g(a)$. The map $b\mapsto-b$ is a bijection $P\to-P$, so claim 2 of Properties of a Sum over a Finite Index Set turns the last sum into $\sum_{b\in P}g(-b)$, and claim 3 of that lemma combines the two sums over $P$ into $\sum_{b\in P}(g(b)+g(-b))$. Finally the restriction of $\iota$ to $[M]$ is a bijection onto $P$: it is injective by Properties of the Canonical Map from the Natural Numbers to an Ordered Field §injective; it maps into $P$ by Properties of the Canonical Map from the Natural Numbers to an Ordered Field §lower-bound and Properties of the Canonical Map from the Natural Numbers to an Ordered Field §monotone; and every $b\in P$ equals $\iota(k)$ for some $k\in\mathbb{N}$ by Arithmetic, Order, Discreteness and Intervals of the Integers §sign, where $k\le M$ since $M<k$ would give $M<b$ by Properties of the Canonical Map from the Natural Numbers to an Ordered Field §monotone (and Properties of the Order on the Natural Numbers §trichotomy). So claims 2 and 1 of Properties of a Sum over a Finite Index Set give $\sum_{b\in P}(g(b)+g(-b))=\sum_{b=1}^{M}(g(b)+g(-b))$.

Step 3 (sample sums). For $m\in\mathbb{Z}$ with $-2M\le m\le 2M$, put $c_{m}=\sum_{\nu\in E}\cos(2\pi m\nu/L)$ and $s_{m}=\sum_{\nu\in E}\sin(2\pi m\nu/L)$. Then $c_{0}=L$, $c_{m}=0$ for $m\neq0$, and $s_{m}=0$. By Values at Zero, Parity, Addition and Double-Angle Identities for Sine and Cosine §parity the summand of $c_{m}$ takes equal values at $\nu$ and $-\nu$, and the summand of $s_{m}$ takes opposite values, vanishing at $\nu=0$ by Values at Zero, Parity, Addition and Double-Angle Identities for Sine and Cosine §zero. By Step 2 and claim 3 of Properties of Finite Sums, $c_{m}=\cos0+2\sum_{\nu=1}^{M}\cos(2\pi\nu m/L)=K(m)$, which is given by Step 1; and $s_{m}=0+\sum_{\nu=1}^{M}0=0$ by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §vanishing (with claim 1 of Properties of a Sum over a Finite Index Set).

Step 4 (shifting the samples). Let $h:\mathbb{R}\to\mathbb{R}$ satisfy $h(x+1)=h(x)$ for every $x\in\mathbb{R}$. Then $\sum_{\nu\in D}h(\nu/L)=\sum_{\nu\in E}h(\nu/L)$. Define $\tau:E\to D$ by $\tau(\nu)=\nu$ if $0\le\nu$ and $\tau(\nu)=\nu+L$ if $\nu<0$, and $\rho:D\to E$ by $\rho(s)=s$ if $s\le M$ and $\rho(s)=s-L$ if $M<s$; all these values are integers by Arithmetic, Order, Discreteness and Intervals of the Integers §ring. Below we add a number to both sides of an inequality and chain inequalities by the order rules of Elementary Order Arithmetic in an Ordered Field (in force by The Real Numbers: Standing Notation and Background §background), and we use $L=2M+1$ (Notation paragraph) and $0<M$, which holds by Arithmetic, Order, Discreteness and Intervals of the Integers §sign since $M$ lies in the image of $\mathbb{N}$; in particular $-M\le0\le M\le 2M$. Since the order of $\mathbb{R}$ is total, every $\nu\in E$ falls under exactly one of the cases (a), (b), and every $s\in D$ under exactly one of (c), (d). (a) $\nu\in E$, $0\le\nu$: then $\tau(\nu)=\nu$ satisfies $0\le\nu\le M\le 2M$, so it lies in $D$, and since $\tau(\nu)\le M$ we get $\rho(\tau(\nu))=\nu$. (b) $\nu\in E$, $\nu<0$: then $\nu+1\le0$ by Arithmetic, Order, Discreteness and Intervals of the Integers §discrete, so $-M\le\nu\le-1$, and adding $L=2M+1$ gives $M+1\le\nu+L\le 2M$. So $\tau(\nu)=\nu+L$ lies in $D$ (as $0<M+1$), and since $M<M+1\le\tau(\nu)$ we get $\rho(\tau(\nu))=(\nu+L)-L=\nu$. (c) $s\in D$, $s\le M$: then $-M\le0\le s\le M$, so $\rho(s)=s$ lies in $E$, and since $0\le\rho(s)$ we get $\tau(\rho(s))=s$. (d) $s\in D$, $M<s$: then $M+1\le s$ by Arithmetic, Order, Discreteness and Intervals of the Integers §discrete, and $s\le 2M$; subtracting $L=2M+1$ gives $-M\le s-L\le-1$. So $\rho(s)=s-L$ lies in $E$, and since $s-L\le-1<0$ we get $\tau(\rho(s))=(s-L)+L=s$. By (a) and (b), $\tau$ maps $E$ into $D$ and $\rho\circ\tau$ is the identity of $E$; by (c) and (d), $\rho$ maps $D$ into $E$ and $\tau\circ\rho$ is the identity of $D$. So $\tau$ is a bijection by claim 3 of Inverse of a Bijection. Since $h(\tau(\nu)/L)$ is $h(\nu/L)$ or $h((\nu+L)/L)=h(\nu/L+1)=h(\nu/L)$, claim 2 of Properties of a Sum over a Finite Index Set gives $\sum_{\nu\in D}h(\nu/L)=\sum_{\nu\in E}h(\tau(\nu)/L)=\sum_{\nu\in E}h(\nu/L)$.

Every $\phi_{a}$ satisfies $\phi_{a}(x+1)=\phi_{a}(x)$: this is clear for $a=0$, and for $a\neq0$ it is Quarter-Turn Identities and Periodicity of Sine and Cosine §integer applied with $m=a$ or $m=-a$, because $2\pi a(x+1)=2\pi ax+2\pi a$. Hence so does every product $\phi_{a}\phi_{b}$.

Step 5 (one-dimensional orthogonality). For $a,b\in E$, $\sum_{\nu\in D}\phi_{a}(\nu/L)\phi_{b}(\nu/L)$ equals $L$ if $a=b$ and $0$ if $a\neq b$. By Step 4 the sum equals $\sum_{\nu\in E}\phi_{a}(x_{\nu})\phi_{b}(x_{\nu})$ with $x_{\nu}=\nu/L$. The expression is symmetric in $a$ and $b$, so it suffices to treat the following cases, in which we use Step 3 together with claims 3 and 4 of Properties of a Sum over a Finite Index Set; all frequencies below lie between $-2M$ and $2M$. (i) $a=b=0$: the summand is $1=\cos(2\pi\cdot0\cdot \nu/L)$, so the sum is $c_{0}=L$. (ii) $a=0<b$: the sum is $\sqrt{2}\,c_{b}=0$. (iii) $a=0>b$: the sum is $\sqrt{2}\,s_{-b}=0$. (iv) $a,b>0$: with $u=2\pi ax_{\nu}$, $v=2\pi bx_{\nu}$, Product-to-Sum Formulas for Sine and Cosine §cosine-cosine gives $\phi_{a}\phi_{b}=2\cos u\cos v=\cos(2\pi(a-b)x_{\nu})+\cos(2\pi(a+b)x_{\nu})$, so the sum is $c_{a-b}+c_{a+b}$; here $2\le a+b\le 2M$, so $c_{a+b}=0$, and $c_{a-b}$ is $L$ if $a=b$ and $0$ otherwise. (v) $a,b<0$: with $a'=-a$, $b'=-b$, Product-to-Sum Formulas for Sine and Cosine §sine-sine gives $\phi_{a}\phi_{b}=2\sin(2\pi a'x_{\nu})\sin(2\pi b'x_{\nu})=\cos(2\pi(a'-b')x_{\nu})-\cos(2\pi(a'+b')x_{\nu})$, so the sum is $c_{a'-b'}-c_{a'+b'}$, which is $L$ if $a=b$ and $0$ otherwise, as in (iv). (vi) $a>0>b$: with $b'=-b$, Product-to-Sum Formulas for Sine and Cosine §sine-cosine (with $u=2\pi b'x_{\nu}$, $v=2\pi ax_{\nu}$) gives $\phi_{a}\phi_{b}=2\sin(2\pi b'x_{\nu})\cos(2\pi ax_{\nu})=\sin(2\pi(b'+a)x_{\nu})+\sin(2\pi(b'-a)x_{\nu})$, so the sum is $s_{b'+a}+s_{b'-a}=0$, and $a\neq b$.

Step 6 (one-dimensional dual orthogonality). For $\nu,\nu'\in D$, $\sum_{a\in E}\phi_{a}(\nu/L)\phi_{a}(\nu'/L)$ equals $L$ if $\nu=\nu'$ and $0$ if $\nu\neq\nu'$. Put $x=\nu/L$, $y=\nu'/L$. For $b\in P$, The Trigonometric System on the Torus §one-dimensional gives $\phi_{b}(x)\phi_{b}(y)=2\cos(2\pi bx)\cos(2\pi by)$ and $\phi_{-b}(x)\phi_{-b}(y)=2\sin(2\pi bx)\sin(2\pi by)$; adding Product-to-Sum Formulas for Sine and Cosine §cosine-cosine and Product-to-Sum Formulas for Sine and Cosine §sine-sine with $u=2\pi bx$, $v=2\pi by$, their sum is $2\cos(2\pi b(x-y))=2\cos(2\pi br/L)$ with $r=\nu-\nu'\in\mathbb{Z}$, $-2M\le r\le 2M$. Since $\phi_{0}(x)\phi_{0}(y)=1$, Step 2 and claim 3 of Properties of Finite Sums give

$$\sum_{a\in E}\phi_{a}(x)\phi_{a}(y)=1+2\sum_{b=1}^{M}\cos(2\pi br/L)=K(r),$$

which by Step 1 is $L$ if $r=0$, that is $\nu=\nu'$, and $0$ otherwise.

Step 7 (factorisation). Let $G$ be a nonempty finite set and $j\in\mathbb{N}$. For all maps $g_{1},\dots,g_{j}:G\to\mathbb{R}$,

$$\sum_{w\in G^{j}}\prod_{i=1}^{j}g_{i}(w_{i})=\prod_{i=1}^{j}\Bigl(\sum_{e\in G}g_{i}(e)\Bigr).$$

Here $G^{j}$ is nonempty and finite by claim 3 of Finiteness of Cartesian Products, Tuple Sets, and Permutation Sets. Let $J$ be the set of $j\in\mathbb{N}$ for which this holds for every family $g_{1},\dots,g_{j}$; we apply Principle of Induction for the Natural Numbers. For $j=1$: $[1]=\{1\}$ by claim 2 of Basic Properties of Initial Segments of the Natural Numbers. By Tuples in a Set an element $w\in G^{1}$ is a map $[1]=\{1\}\to G$, so it is determined by its value $w_{1}$ at $1$, and every $e\in G$ is such a value (of the map sending $1$ to $e$); hence $w\mapsto w_{1}$ is a bijection $G^{1}\to G$; both products reduce to their only factor by claim 1 of Properties of Finite Products, and claim 2 of Properties of a Sum over a Finite Index Set gives the identity. Let $j\in J$ and let $g_{1},\dots,g_{S(j)}$ be given. Let $q:G^{j}\times G\to G^{S(j)}$ be the bijection of claim 2 of Finiteness of Cartesian Products, Tuple Sets, and Permutation Sets. For $(v,e)\in G^{j}\times G$, the tuple $q((v,e))$ has $i$th component $v_{i}$ for every $i\in[j]$ and $S(j)$th component $e$, by the component property of $q$ in claim 2 of Finiteness of Cartesian Products, Tuple Sets, and Permutation Sets; so claim 1 of Properties of Finite Products gives $\prod_{i=1}^{S(j)}g_{i}(q((v,e))_{i})=\bigl(\prod_{i=1}^{j}g_{i}(v_{i})\bigr)g_{S(j)}(e)$. Hence, by claim 2 of Properties of a Sum over a Finite Index Set, then The Product of Two Sums over Finite Index Sets is a Sum over the Cartesian Product, then the hypothesis $j\in J$ for the family $g_{1},\dots,g_{j}$, then claim 1 of Properties of Finite Products again,

$$\sum_{u\in G^{S(j)}}\prod_{i=1}^{S(j)}g_{i}(u_{i})=\sum_{(v,e)\in G^{j}\times G}\Bigl(\prod_{i=1}^{j}g_{i}(v_{i})\Bigr)g_{S(j)}(e)=\Bigl(\sum_{v\in G^{j}}\prod_{i=1}^{j}g_{i}(v_{i})\Bigr)\Bigl(\sum_{e\in G}g_{S(j)}(e)\Bigr)=\prod_{i=1}^{S(j)}\Bigl(\sum_{e\in G}g_{i}(e)\Bigr).$$

So $S(j)\in J$, and $J=\mathbb{N}$.

Step 8 (reduction to one dimension). The map $\beta:W_{L}\to\mathbb{L}_{M}$, $\beta(w)=L^{-1}w$, is surjective by The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field §lattice, and injective because $(L^{-1}w)_{i}=L^{-1}w_{i}$ by Scalar Multiple of a Point of $\mathbb{R}^n$ and $L^{-1}\neq0$, together with claim 1 of Euclidean Points as Tuples of Real Numbers; so it is a bijection. By The Trigonometric System on the Torus §system and claim 2 of Properties of Finite Products, for $k,l\in\Gamma_{M}$ and $w,w'\in W_{L}$,

$$\psi_{k}(\beta(w))\,\psi_{l}(\beta(w'))=\prod_{i=1}^{n}\phi_{k_{i}}(w_{i}/L)\,\phi_{l_{i}}(w'_{i}/L).$$

Hence claim 2 of Properties of a Sum over a Finite Index Set and Step 7 (with $G=D$, $W_{L}=D^{n}$, respectively $G=E$, $\Gamma_{M}=E^{n}$) give

$$\sum_{z\in\mathbb{L}_{M}}\psi_{k}(z)\psi_{l}(z)=\prod_{i=1}^{n}\Bigl(\sum_{\nu\in D}\phi_{k_{i}}(\nu/L)\phi_{l_{i}}(\nu/L)\Bigr),\qquad \sum_{k\in\Gamma_{M}}\psi_{k}(\beta(w))\psi_{k}(\beta(w'))=\prod_{i=1}^{n}\Bigl(\sum_{a\in E}\phi_{a}(w_{i}/L)\phi_{a}(w'_{i}/L)\Bigr).$$

Step 9 (products of Kronecker factors). Let $d_{1},\dots,d_{n}\in\{0,1\}$. If $d_{i}=1$ for every $i$, then $\prod_{i=1}^{n}(L\,d_{i})=L^{n}$ by Natural Number Power of an Element of a Field; otherwise some factor vanishes and $\prod_{i=1}^{n}(L\,d_{i})=0$ by claim 4 of Properties of Finite Products. By claim 1 of Euclidean Points as Tuples of Real Numbers, $k=l$ in $\Gamma_{M}$ exactly when $k_{i}=l_{i}$ for every $i$, and $\beta(w)=\beta(w')$ exactly when $w_{i}=w'_{i}$ for every $i$ ($\beta$ being injective). Combining this with Steps 5, 6 and 8:

$$\sum_{z\in\mathbb{L}_{M}}\psi_{k}(z)\psi_{l}(z)=L^{n}\delta_{kl}\quad(k,l\in\Gamma_{M}),\qquad \sum_{k\in\Gamma_{M}}\psi_{k}(y)\psi_{k}(z)=\begin{cases}L^{n}&y=z,\\0&y\neq z\end{cases}\quad(y,z\in\mathbb{L}_{M}).\tag{$\ast$}$$

Claim 2 (Counting the modes). By The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points §count with $N=M$, $\sum_{k\in\Gamma_{M}}1=(2M+1)^{n}=L^{n}$, using the Notation paragraph.

Claim 1 (Counting the sites). By Euclidean Points as Tuples of Real Numbers (claim 2) define $\gamma:\Gamma_{M}\to W_{L}$ by $\gamma(k)_{i}=k_{i}+M$ and $\gamma':W_{L}\to\Gamma_{M}$ by $\gamma'(w)_{i}=w_{i}-M$ ($i\in[n]$); they land in the stated sets because $-M\le k_{i}\le M$ if and only if $0\le k_{i}+M\le 2M=L-1$, and they are mutually inverse by claim 1 of that lemma, so $\gamma$ is a bijection by claim 3 of Inverse of a Bijection. Reindexing along $\beta$ and $\gamma$ by claim 2 of Properties of a Sum over a Finite Index Set, and using claim 2 of this lemma (proved above),

$$\sum_{z\in\mathbb{L}_{M}}1=\sum_{w\in W_{L}}1=\sum_{k\in\Gamma_{M}}1=L^{n}.$$

Claim 3 (Orthogonality). Multiply the first identity of $(\ast)$ by $L^{-n}$.

Claim 4 (Inversion formula). Let $f$ be a lattice field and $y\in\mathbb{L}_{M}$. By The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field §coefficients, claim 4 of Properties of a Sum over a Finite Index Set, and claim 5 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set (interchange over $\Gamma_{M}\times\mathbb{L}_{M}$),

$$\sum_{k\in\Gamma_{M}}\hat{f}(k)\psi_{k}(y)=\sum_{k\in\Gamma_{M}}\Bigl(\sum_{z\in\mathbb{L}_{M}}L^{-n}f(z)\psi_{k}(z)\psi_{k}(y)\Bigr)=\sum_{z\in\mathbb{L}_{M}}L^{-n}f(z)\Bigl(\sum_{k\in\Gamma_{M}}\psi_{k}(z)\psi_{k}(y)\Bigr).$$

By the second identity of $(\ast)$ the summand vanishes for $z\neq y$ and equals $L^{-n}f(y)L^{n}=f(y)$ for $z=y$, so Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §vanishing (with the nonempty subset $\{y\}$) and claim 1 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set show that the sum is $f(y)$.

Claim 5 (Bijection). Write $\mathcal{A}(f)=\hat{f}$. The vector-space operations of $\mathrm{Map}(\mathbb{L}_{M},\mathbb{R})$ and $\mathrm{Map}(\Gamma_{M},\mathbb{R})$ are pointwise by The Real Vector Space of Real-Valued Functions on a Set §vector-space. For lattice fields $f,g$, $\lambda\in\mathbb{R}$ and $k\in\Gamma_{M}$, claims 3 and 4 of Properties of a Sum over a Finite Index Set give $\widehat{f+g}(k)=\hat{f}(k)+\hat{g}(k)$ and $\widehat{\lambda f}(k)=\lambda\hat{f}(k)$, so $\mathcal{A}$ is linear in the sense of Linear Map. For $b\in\mathrm{Map}(\Gamma_{M},\mathbb{R})$ let $\mathcal{S}(b)$ be the lattice field $z\mapsto\sum_{k\in\Gamma_{M}}b(k)\psi_{k}(z)$. By claim 4, $\mathcal{S}(\mathcal{A}(f))(z)=f(z)$ for every $z$, so $\mathcal{S}\circ\mathcal{A}$ is the identity. For $b$ and $l\in\Gamma_{M}$, the same three sum rules as in claim 4 and claim 3 give

$$\mathcal{A}(\mathcal{S}(b))(l)=\sum_{z\in\mathbb{L}_{M}}L^{-n}\Bigl(\sum_{k\in\Gamma_{M}}b(k)\psi_{k}(z)\Bigr)\psi_{l}(z)=\sum_{k\in\Gamma_{M}}b(k)\Bigl(L^{-n}\sum_{z\in\mathbb{L}_{M}}\psi_{k}(z)\psi_{l}(z)\Bigr)=\sum_{k\in\Gamma_{M}}b(k)\delta_{kl}=b(l),$$

the last step by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §vanishing and claim 1 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set. So $\mathcal{A}\circ\mathcal{S}$ is the identity, and claim 3 of Inverse of a Bijection shows that $\mathcal{A}$ is a bijection with inverse $\mathcal{S}$.

Claim 6 (Parseval). Let $f,g$ be lattice fields. Expanding $g(z)$ by claim 4 and using claim 4 of Properties of a Sum over a Finite Index Set and claim 5 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set,

$$L^{-n}\sum_{z\in\mathbb{L}_{M}}f(z)g(z)=\sum_{z\in\mathbb{L}_{M}}\Bigl(\sum_{k\in\Gamma_{M}}L^{-n}f(z)\hat{g}(k)\psi_{k}(z)\Bigr)=\sum_{k\in\Gamma_{M}}\hat{g}(k)\Bigl(L^{-n}\sum_{z\in\mathbb{L}_{M}}f(z)\psi_{k}(z)\Bigr)=\sum_{k\in\Gamma_{M}}\hat{f}(k)\hat{g}(k).$$

Claim 7 (Spin sum). For $s\in\mathbb{R}$ put $\ell(s)=\log(\exp(s)+\exp(-s))$, defined as recorded in the statement; then $\exp(\ell(s))=\exp(s)+\exp(-s)$ by The Natural Logarithm. The set $\{-1,1\}=\{-1\}\cup\{1\}$ is nonempty and finite by claim 2 of Basic Properties of Finite Sets and claim 1 of Peeling an Element off a Finite Set, and Unions of Finite Sets, so $\mathrm{Map}(F,\{-1,1\})$ is nonempty and finite for every nonempty finite set $F$ by The Maps from a Finite Set to a Nonempty Finite Set Form a Nonempty Finite Set §finite. Let $Z$ be the set of those $p\in\mathbb{N}$ such that for every set $F$ with $p$ elements and every map $t:F\to\mathbb{R}$

$$\sum_{\sigma\in\mathrm{Map}(F,\{-1,1\})}\exp\Bigl(\sum_{z\in F}\sigma(z)t(z)\Bigr)=\exp\Bigl(\sum_{z\in F}\ell(t(z))\Bigr).\tag{$\ast\ast$}$$

We show $Z=\mathbb{N}$ by Principle of Induction for the Natural Numbers; the claim is then $(\ast\ast)$ for $F=\mathbb{L}_{M}$, which is nonempty and finite by The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field §lattice, and so has $p$ elements for some $p$ by Finite Set.

$1\in Z$. A set with $1$ element is $\{x\}$ for some $x$, as $[1]=\{1\}$ by claim 2 of Basic Properties of Initial Segments of the Natural Numbers. $\mathrm{Map}(\{x\},\{-1,1\})$ is the union of the singletons $\{\sigma_{+}\}$ and $\{\sigma_{-}\}$, where $\sigma_{\pm}(x)=\pm1$, which are disjoint since $-1\neq1$. By Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §disjoint-union and claim 1 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set, the left side of $(\ast\ast)$ is $\exp(t(x))+\exp(-t(x))=\exp(\ell(t(x)))$, which is the right side by claim 1 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set.

If $p\in Z$ then $S(p)\in Z$. Let $F$ have $S(p)$ elements and $t:F\to\mathbb{R}$. By claim 2 of Peeling an Element off a Finite Set, and Unions of Finite Sets, $F=F'\cup\{x\}$ with $F'$ having $p$ elements (so $F'\neq\emptyset$) and $x\notin F'$. Write $\Sigma'=\mathrm{Map}(F',\{-1,1\})$ and, for $\varepsilon\in\{-1,1\}$, $A_{\varepsilon}=\{\sigma\in\mathrm{Map}(F,\{-1,1\}):\sigma(x)=\varepsilon\}$; then $A_{1}\cap A_{-1}=\emptyset$ and $A_{1}\cup A_{-1}=\mathrm{Map}(F,\{-1,1\})$. Let $\chi_{\varepsilon}:\Sigma'\to A_{\varepsilon}$ send $\tau$ to its extension by $x\mapsto\varepsilon$; restriction to $F'$ is a two-sided inverse, so $\chi_{\varepsilon}$ is a bijection by claim 3 of Inverse of a Bijection, and $A_{\varepsilon}$ is nonempty and finite by claim 4 of Basic Properties of Finite Sets. For $\tau\in\Sigma'$, claim 2 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set and claim 1 of Basic Properties of the Exponential Function give

$$\exp\Bigl(\sum_{z\in F}\chi_{\varepsilon}(\tau)(z)t(z)\Bigr)=\exp\Bigl(\sum_{z\in F'}\tau(z)t(z)+\varepsilon t(x)\Bigr)=\exp(\varepsilon t(x))\exp\Bigl(\sum_{z\in F'}\tau(z)t(z)\Bigr).$$

By claims 2 and 4 of Properties of a Sum over a Finite Index Set and the hypothesis $p\in Z$ applied to $F'$ and the restriction of $t$, the sum of the left side over $\sigma\in A_{\varepsilon}$ is $\exp(\varepsilon t(x))\exp\bigl(\sum_{z\in F'}\ell(t(z))\bigr)$. Adding the cases $\varepsilon=1$ and $\varepsilon=-1$ by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §disjoint-union, and using claim 1 of Basic Properties of the Exponential Function and claim 2 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set,

$$\sum_{\sigma\in\mathrm{Map}(F,\{-1,1\})}\exp\Bigl(\sum_{z\in F}\sigma(z)t(z)\Bigr)=\exp(\ell(t(x)))\exp\Bigl(\sum_{z\in F'}\ell(t(z))\Bigr)=\exp\Bigl(\sum_{z\in F}\ell(t(z))\Bigr).$$

So $S(p)\in Z$, hence $Z=\mathbb{N}$, which proves claim 7.