Dependencies

For \(G:\mathcal W\times I\to [0,\infty ]\) and \(T\ge 0\),

In particular, the expected length is \(\int _{\mathcal W}|w|\, d\mu _T(w)=NT\).

The expected number of omitted letters is

With the same assumptions,

With the same assumptions, all integrands below are integrable and

For \(0\le p_0\le p_1\le 1\) and every integrable function \(L\) on \([0,1]\) with \(L(p)\le -\partial _p\log N(p)^2\) for \(0{\lt}p{\lt}1\), \(\int _{p_0}^{p_1}L\le \log N(p_0)^2-\log N(p_1)^2\).

For \(r\in \mathbb R_{\geq 0}\) and \(y,z\in Y\), define the real kernel

Thus \(K_r(y,z)\) is the number of anchor labels whose radius-\(r\) balls contain both sites.

For natural numbers \(d,K,k\) and \(c\in \mathbb {R}\), put

The estimates below concern positive \(k\).

For the chronological products \(D_m(w)\), define

Set \(G_i=(I-k_i)^{1/2}\). For a word \(w=(i_0,\ldots ,i_{m-1})\), set \(D_w=G_{i_{m-1}}\cdots G_{i_0}\), with \(D_{\varnothing }=I\). For \(v\in \mathcal H\), define

The copy permutations of a subsystem, the symmetric subspace, and the averaging projector \(\Pi _k=(k!)^{-1}\sum _{\pi \in S_k}U(\pi )\) onto it.

For every \(S\subseteq Y\) and every pair of real parameters \(a,\alpha \), define

There are no sign restrictions on \(a\) or \(\alpha \) in this definition.

Let \(G\) act on a finite set \(X\). The permutation operator \(U(g)\) on \(\mathbb {C}^X\) sends the basis vector \(e_x\) to \(e_{gx}\), and extends linearly to an algebra homomorphism \(\mathbb {C}[G]\to \mathrm{End}(\mathbb {C}^X)\). The label projector \(\pi ^\lambda \) is the image of \(e_\lambda \). For a real function \(f\) of the labels, the label observable is \(\sum _\lambda f(\lambda )\pi ^\lambda \); for \(f(\lambda )=\log d_\lambda \) it is the observable \(F=\sum _\lambda (\log d_\lambda )\pi ^\lambda \) of  [ Ope26b , Section 6.1 ] .

A weighted binary tree is either a leaf with a label \(j\) or an internal vertex with a parameter \(p\in [0,1]\) and two subtrees. Its root on inputs \((A_j)\) is \(A_j\) at a leaf and \(M_l\# _pM_r\) at an internal vertex with subtree roots \(M_l\), \(M_r\). The terminal weights are \(\delta _j\) at a leaf and \((1-p)w^{(l)}+p\, w^{(r)}\) at an internal vertex.

For \(\xi \in \mathcal H\otimes \mathcal K\) define the nonnegative observable

Put \(\nu _t=\bigotimes _{i\in I}\operatorname {Poisson}(t)\) and define

This samples independent counts and then a uniform compatible word. Every \(C_n\) is finite and nonempty, including at \(t=0\).

For words \(u\) and \(v\), their concatenation \(uv\) places all letters of \(v\) after all letters of \(u\). Thus \(|uv|=|u|+|v|\). Concatenation is associative, and the empty word is a two-sided identity.

For a word \(w\in I^{\{ 0,\ldots ,m-1\} }\), define \(C(w)=\sum _{j=0}^{m-1}e_{w(j)}\), where \(e_i\) is the vector equal to one at \(i\) and zero elsewhere. Thus \(C(w)_i\) is the number of occurrences of \(i\), and \(\sum _{i\in I}C(w)_i=m\) when \(I\) is finite. The empty word has count vector zero. For \(n\in \mathbb {N}^I\), put \(\mathcal W_n=\{ w\in \mathcal W:C(w)=n\} \). Every word in \(\mathcal W_n\) has length \(\sum _{i\in I}n_i\).

For a word \(w\) over \(I\sqcup J\), let \(L(w)\) retain its \(I\)-letters and let \(R(w)\) retain its \(J\)-letters, preserving their original order. Both filters send the empty word to the empty word, and \(|L(w)|+|R(w)|=|w|\). For prescribed words \(u\) over \(I\) and \(v\) over \(J\), put \(E(u,v)=\{ w:L(w)=u,\ R(w)=v\} \).

For \(a\in I\), write \([a]\) for the word with the single letter \(a\). For words \(u,v\in \mathcal W\), inserting \(a\) between \(u\) and \(v\) gives the word \(u[a]v\), of length \(|u|+|v|+1\).

A marked word is a pair \((w,j)\) with \(w\in \mathcal W\) and \(0\leq j{\lt}|w|\). The marked insertion of \((u,a,v)\) is \((u[a]v,|u|)\).

The bijection between marked words and \(\mathcal W\times I\times \mathcal W\) is

No finiteness assumption on \(I\) is needed. Repeated letters at distinct positions give distinct marked words; when \(I\) is empty, both sets in the bijection are empty.

Equip \(\mathcal W\) with its discrete \(\sigma \)-algebra and put

Let \(p\) be a predicate on an alphabet \(I\), and put \(S=\{ i\in I:p(i)\} \) and \(T=\{ i\in I:\neg p(i)\} \). Classify each letter by the bijection \(c:I\to S\sqcup T\), then define \(P_p(w)=(L(c^{\mathcal W}(w)),R(c^{\mathcal W}(w)))\). Write \(P_p(w)=(w_S,w_T)\). The empty word gives two empty words, and \(|w_S|+|w_T|=|w|\).

For a bijection \(e:I\to J\) of alphabets, let \(e^{\mathcal W}\) apply \(e\) to each letter of a finite word, in its original position. Thus \(e^{\mathcal W}(i_0,\ldots ,i_{m-1})=(e(i_0),\ldots ,e(i_{m-1}))\). This is a bijection of word spaces, preserves length, and sends the empty word to the empty word.

For a word \(w\) and \(k\geq 0\), write \(w_{{\lt}k}\) for its first \(\min (k,|w|)\) letters, and \(w_{\geq k}\) for the remaining \(\max (|w|-k,0)\) letters. In particular, \(|w_{{\lt}k}|=k\) whenever \(k\leq |w|\).

Let \(\Theta \) be the unit sphere of the one-copy space \(\mathcal V\), let \(e_a\) be a coordinate unit vector and let \(H\) be Haar probability measure on \(U(\mathcal V)\). For a matrix \(\rho \) on \(\mathcal V^{\otimes k}\), set \(q_\rho (U)=D_k\operatorname {Re}\operatorname {Tr}(\rho P_{Ue_a,k})\). Define

For \(\rho \ge 0\), the density is nonnegative and (1) is the coherent measure of  [ Ope26b , Proposition 7.4 ] .

For the terminal state \(\sigma _{j,u}\) of the transport tree, let \(d_j(u,U)=m_{1/4}(u)q_{\sigma _{j,u}}(U)\) and define

Neither a terminal weight nor a factor of two occurs in this definition. The parameter is clamped to \([0,1]\); a zero terminal weight gives the zero normalized derivative map, using the convention \(0^{-1}=0\).

For real \(C,c,\alpha \) and \(y,z\in Y\), define

Every radius \(n\) is regarded as a finite nonnegative real radius. The sum is the real infinite sum, with value zero if the series does not converge. Theorem 27.28.4 establishes convergence under the counting hypotheses used below.

For words \(u\) over \(A\) and \(v\) over \(B\), define the list \(I(u,v)\) recursively. If either word is empty, the list contains just the other word with its tag attached. If \(u=au'\) and \(v=bv'\) are nonempty, first list the words obtained by prepending the left-tagged \(a\) to each member of \(I(u',v)\), then list the words obtained by prepending the right-tagged \(b\) to each member of \(I(u,v')\).

For every \(r\in \mathbb R_{\geq 0}\) and \(y,z\in Y\),

The set on the right is a set of labels, including all multiplicities of the anchor positions.

For every \(r\in \mathbb R_{\geq 0}\) and \(y,z\in Y\), one has \(K_r(y,z)\geq 0\).

For \(r\in \mathbb R_{\geq 0}\) and \(y,z\in Y\), if \(\delta (y,z)=\infty \), then \(K_r(y,z)=0\).

Let \(x,y,z\in Y\) and \(r\in \mathbb R_{\geq 0}\). If \(\delta (y,x)\leq r\) and \(\delta (z,x)\leq r\), then

Let \(a,x,d,z\geq 0\) and \(0\leq \alpha \leq 1\). If \(x\leq d+z\), then

The endpoint \(\alpha =0\) is included.

For every \(a,\alpha \in \mathbb R\) and \(y\in Y\), \(r_{\varnothing ,a,\alpha }(y)=0\).

For all \(S\subseteq Y\) and \(a,\alpha \in \mathbb R\), if \(D_S(y)\ne \infty \), then

Let \(a\in \mathbb R\) and \(\alpha {\gt}0\). If \(y\in S\), then \(r_{S,a,\alpha }(y)=1\).

For all \(S\subseteq Y\), \(a,\alpha \in \mathbb R\), and \(y\in Y\), \(r_{S,a,\alpha }(y)\geq 0\).

Let \(S\subseteq Y\), \(a\geq 0\), and \(0{\lt}\alpha \leq 1\). For \(y,z\in Y\) with \(\delta (y,z)\ne \infty \),

No finiteness assumption is imposed on \(D_S(y)\) or \(D_S(z)\).

For all \(S\subseteq Y\), \(a,\alpha \in \mathbb R\), and \(y\in Y\),

If \(\mu (\{ x\} )=c\) for every \(x\in S\), where \(c\in [0,\infty ]\), then

These identities require neither \(S\ne \varnothing \) nor \(\mu (S){\gt}0\).

For every \(S\subseteq Y\), \(a\in \mathbb R\), \(\alpha {\gt}0\), and \(y\in Y\),

Here \(\mathbf1_S\) is the literal indicator of \(S\).

For a vector \(\theta \) on the sites and a partition \(P,x,Y_0,F\), the exponent of Lemma 23.11.3 is \(\eta _\theta =S_\theta (x|P)+S_\theta (x|Y_0)\), the marginals \(\rho _P,\rho _{Y_0}\) there are the states of \(\theta \) on \(P\) and \(Y_0\), and

For every \(n\in \mathbb {N}^I\),

At \(t=0\), every \(n\ne 0\) satisfies

The second equality uses the totalized conditional-measure convention in Lean: conditioning on a null event gives the zero measure. For \(n=0\), the fiber consists only of the empty word and has mass one, so the conditional law is its point mass.

For any alphabets \(I,J\), the set \(E(u,v)\) is finite. Every member has length \(|u|+|v|\), and \(|E(u,v)|=\binom {|u|+|v|}{|u|}\). This formula allows repeated letters in either prescribed word.

Suppose \(I\) and \(J\) are finite, with cardinalities \(a\) and \(b\). For \(m=|u|\) and \(n=|v|\), the joint event has mass

For every word \(w\) of length \(m\), the fiber

has exactly \(m+1\) elements. They are parametrized bijectively by \(k\in \{ 0,\ldots ,m\} \) through

If \(0\leq f(u)\leq C{\lt}\infty \), then \(f\in L^1(\mu _{I,t})\) for every \(t\geq 0\), and \(s\mapsto \int f\, d\mu _{I,\max (s,0)}\) is measurable on \(\mathbb R\).

Suppose \(I\) is finite and \(f:\mathcal W_I\to \mathbb R\) satisfies \(|f(w)|\leq C\) for every word \(w\), where \(C\in \mathbb R\). Then \(f\in L^1(\mu _{I,t})\) for every \(t\geq 0\), and the function \(s\mapsto \int f\, d\mu _{I,\max (s,0)}\) is measurable on \(\mathbb R\).

For every finite alphabet \(I\) and count vector \(n\in \mathbb {N}^I\), the set \(\mathcal W_n\) is finite and nonempty, with \(|\mathcal W_n|=(\sum _{i\in I}n_i)!/\prod _{i\in I}n_i!\). This includes \(I=\varnothing \) and the zero count vector; in both cases the cardinality is one.

For an observable \(f\) valued in a normed additive group, integrability is equivalent to summability of \(\sum _{w\in \mathcal W}e^{-Nt}t^{|w|}\| f(w)\| /|w|!\). If \(f\) is real and nonnegative, put \(A_m=\sum _{w\in I^{\{ 0,\ldots ,m-1\} }}f(w)\). Then integrability is equivalent to summability of \(\sum _{m\geq 0}e^{-Nt}t^m A_m/m!\), and in that case

Let \(f:\mathcal W_I\times Y\to \mathbb R\) and \(A:Y\times Y\to \mathbb R\). Suppose that, for every word \(u\) and every \(y\in Y\),

Then, for every \(m\geq 0\) and \(y\in Y\),

No sign assumptions on \(f\) or \(A\) are needed.

For every \(T\geq 0\) and \(m,n\in \mathbb {N}\),

The integrand is nonnegative and integrable. Equivalently, the same identity holds as a nonnegative Lebesgue integral on \((0,T]\) after viewing the weights in \([0,\infty ]\).

The measure \(\mu _t\) is a probability measure. The distribution of the length is Poisson with mean \(Nt\), and each individual word has the mass in (15). If \(t=0\) or \(I=\varnothing \), then \(\mu _t\) is the point mass at the empty word.

The length function belongs to \(L^1(\mu _{I,T})\) and

Let \(K\) be a finite alphabet, \(J\) an arbitrary alphabet, \(A\subseteq J\) finite, and \(F:\mathcal W_K\times J\to [0,C]\), where \(C{\lt}\infty \). Then

is integrable on \((0,T]\).

For every \(g:\mathcal W_I\to \mathbb R\) and \(m\geq 0\),

For \(s,t\ge 0\) and \(m\in \mathbb {N}\),

Let \(\mathcal Q_k(f)=D_k\int f(\theta )P_{\theta ,k}\, d\theta \) and let \(\Pi \) be the projection onto \(\mathcal S_k\). For continuous \(f{\gt}0\),

Moreover \(\log D_k=O(\log (k+1))\).

For fixed \(k\) and nonzero \(x\in \mathcal S_k\), the entropy-gain term of [ Ope26b , Proposition 7.4 ] is integrable in \(p\) on \([0,1]\).

Let \(X\) be positive definite and commute with copy permutations, and let \(\varphi \) be continuous with \(X\ge e^{\varphi (\theta )}P_{\theta ,k}\) for every unit vector \(\theta \). For every density matrix \(\rho \) on \(\mathcal S_k\), \(\int \varphi \, d\mu _\rho -\log D_k\le \operatorname {Tr}(\rho \log X)\).

Let \(\Pi \) be the projection onto \(\mathcal S_k\) and let \(A\) be a band metric, which commutes with \(\Pi \). The operator \(\widetilde A=\Pi A+(I-\Pi )\) is positive definite, agrees with \(A\) on \(\mathcal S_k\) and commutes with the copy permutations. Two such operators \(\widetilde A,\widetilde B\) commute exactly when \(A\) and \(B\) commute on \(\mathcal S_k\). Moreover \(\widetilde A^{-1/2}\widetilde B\widetilde A^{-1/2}=\Pi A^{-1/2}BA^{-1/2}+(I-\Pi )\), so every lower bound \(Q\le A^{-1/2}BA^{-1/2}\) with \(\Pi Q\Pi =Q\) holds for the extended operators, and for every \(X\) commuting with \(\Pi \) the skew squares of \(\widetilde A\) and of \(A\) have the same expectation in every density matrix on \(\mathcal S_k\).

Fix a finite set of sites with nonzero local Hilbert spaces, \(t\ge 0\) and a copy number \(k\). Suppose the transport data are admissible and the old and new band metrics from distinct bands commute on \(\mathcal S_k\). For a nonzero input vector \(x\in \mathcal S_k\) and \(0{\lt}p{\lt}1\), all transport states are density matrices supported on \(\mathcal S_k\). If moreover the set of histories is finite, the exact derivative formula of Lemma 7.9.5.1 holds for the replica inputs.

For every \(C\geq 0\), \(c,\alpha \in \mathbb R\), and \(y,z\in Y\), one has \(A_{C,c,\alpha }(y,z)\geq 0\).

For every \(C,c,\alpha \in \mathbb R\) and \(y,z\in Y\), if \(\delta (y,z)=\infty \), then \(A_{C,c,\alpha }(y,z)=0\).

The list \(I(u,v)\) has no repeated words and has \(\binom {|u|+|v|}{|u|}\) members, even if \(u\) or \(v\) repeats letters.

A word \(w\) occurs in \(I(u,v)\) if and only if \(\pi _A(w)=u\) and \(\pi _B(w)=v\).

Every word over \(A\sqcup B\) satisfies \(|\pi _A(w)|+|\pi _B(w)|=|w|\). Consequently, every member of \(I(u,v)\) has length \(|u|+|v|\).

Let a mean tree have positive weights \(\pi _j\), positive definite inputs \(A_j\) and root \(M\), let \(h\ge 0\) commute with \(A_j\) for every leaf \(j\) outside a finite set \(S\), and put \(W=\sum _{j\in S}\pi _j\). Let \(\mathsf D_j\) be the skew square of \(O_j=A_j^{-1/2}hA_j^{1/2}\). For every vector \(v\),

The transport states are density matrices. For \(0{\lt}p{\lt}1\),

The function \(N\) is continuous and positive on \([0,1]\). For \(0\le p_0\le p_1\le 1\) the right side of the exact derivative is integrable on \([p_0,p_1]\), and its integral equals \(\log N(p_0)^2-\log N(p_1)^2\).

If \(Z\) commutes with every input of a mean tree and fixes the vector \(v\), then \(Z\sigma _{j,u}=\sigma _{j,u}\) for every leaf \(j\) and every real \(u\).

For a finite alphabet \(I\), a nonnegative count vector \(n\in \mathbb {N}^I\), and \(m\in \mathbb {N}\) with \(\sum _{i\in I}n_i=m\), the number of length-\(m\) words with count vector \(n\) is

If \(\sum _{i\in I}n_i\ne m\), there are no such words.

Suppose also that \(Y\) is finite. Let \(r\in \mathbb R_{\geq 0}\), \(a,\alpha ,B,V\geq 0\), and fix \(y\in Y\). Assume the following bounds on the actual finite sets, with the second bound holding for every \(i\in I\):

Then

The radius factor in the exponent is exactly \(2r\). Both \(a=0\) and \(\alpha =0\) are allowed, with the nonnegative real-power convention \(s^0=1\), including \(s=0\).

Let \(d,k\) be positive integers, \(K\) a nonnegative integer, \(z{\gt}0\), \(c\ge 0\), and \(0\le 2s\le 1\). Let \(t,B,r\in \mathbb {R}\). Suppose that

Then \(r{\gt}0\) and

The remainder is independent of \(s,t,B,z\).

For every fixed \(d,K\in \mathbb {N}\) and \(c\in \mathbb {R}\),

Let \(A_0,\ldots ,A_{G-1}\) be positive definite operators on a finite-dimensional complex Hilbert space. Suppose that \(A_gA_h=A_hA_g\) whenever \(g\ne h\). Then

The empty product is the identity and the empty sum is zero; repeated operator values are allowed. This is the logarithmic identity used to separate commuting bands in [ Ope26b , Sections 7 and 8 ] .

For a unit basis vector \(e\), \(\int P_{Ue,k}\, dU=D_k^{-1}\Pi _k\), where \(P_{\theta ,k}=|\theta ^{\otimes k}\rangle \langle \theta ^{\otimes k}|\), \(\Pi _k\) is the projector onto the symmetric subspace and \(D_k=\operatorname{Tr}\Pi _k\).

Let \((A_0,\ldots ,A_{m-1})\) be any finite list of complex matrices indexed by \(I\), and let \(b\ge 0\). Suppose that \(bI\le A_j\) for every position \(j\) and that the matrices at distinct positions commute. Then \(b^m I\le A_0\cdots A_{m-1}\). Repeated matrices are allowed. For the empty list, the product is \(I\) and \(b^0=1\), including when \(b=0\).

Let \(k_i\le I\), let \(\Omega \in \mathcal H\) satisfy \(k_i\Omega =0\), and suppose \(\sum _i k_i\ge g(I-|\Omega \rangle \langle \Omega |)\) for \(g\in \mathbb {R}\). If \(g\le N\) and \(\langle \Omega ,\xi _r\rangle =0\) for every spectator coordinate, then

Suppose \(k_i\le I\) and \(\sum _i k_i\ge g(I-|\Omega \rangle \langle \Omega |)\) with \(g{\gt}N\). Every joint vector whose slices are orthogonal to \(\Omega \) is zero. If \(\| \Omega \| =1\), then \(((I-|\Omega \rangle \langle \Omega |)\otimes I_{\mathcal K})\zeta =0\) for every \(\zeta \). These conclusions require no condition on \(k_i\Omega \).

If \(\| \Omega \| =1\), set \(Q=I-|\Omega \rangle \langle \Omega |\). Then \(\langle \Omega ,Qv\rangle =0\) for every \(v\in \mathcal H\). For every \(\zeta \in \mathcal H\otimes \mathcal K\), all slices of \(\xi =(Q\otimes I_{\mathcal K})\zeta \) are orthogonal to \(\Omega \). If \(k_i\le I\), \(k_i\Omega =0\), \(\sum _i k_i\ge gQ\), and \(g\le N\), this gives

For every operator \(A\) on \(\mathcal H\) and every joint vector \(\xi \),

Tensoring with \(I_{\mathcal K}\) preserves the chronological recurrence:

For the chronological square-root products of Definition 27.30.1, concatenation satisfies

The identity depends only on the ordered matrix product, without a commutation, order, ground-state, or gap hypothesis on the matrices \(k_i\). It holds for tuples of fixed lengths and for arbitrary finite words, including either empty word.

Every \(D_w\) fixes \(\Omega \) and preserves \(\Omega ^\perp \). For every \(v\perp \Omega \) and every \(m\ge 0\),

If \(g\le N\), then \(A_m(v)\le (N-g)^m\| v\| ^2\). If \(g{\gt}N\), then \(v=0\). The one-step estimate below and this last conclusion require only \(k_i\le I\) and the gap bound, without the condition \(k_i\Omega =0\). These assertions allow an empty label set and require no commutation between distinct \(k_i\).

Suppose \(Y\) is finite. Let \(S\subseteq Y\), \(a\geq 0\), \(0{\lt}\alpha \leq 1\), and \(v\in \mathbb R\). Let \(A:Y\times Y\to \mathbb R\) be a nonnegative kernel satisfying, for every \(y,z\in Y\),

Then, for every \(y\in Y\),

There is no separate hypothesis \(v\geq 0\). The zero-entry condition at infinite pair distance is part of the hypotheses, even though \(\infty _{\mathbb R}=0\) in the displayed row weight.

Suppose additionally that \(\mu \) is finite and \(\mu (S)\ne 0\). If all singletons in \(S\) have the same mass, then

Let \(R\) be a semiring, let \(I,J\) be finite index sets, and fix \(Z\in R^{I\times J}\). For \(0\leq g{\lt}K\), let \(A_g\in R^{I\times I}\) and \(B_g\in R^{J\times J}\) satisfy \(A_gZ=ZB_g\). Then

where both products are taken in increasing order. The assertion includes \(K=0\) and empty index sets.

Let \(A_1,B_1{\gt}0\) on \(\mathcal H_1\) and \(A_2,B_2{\gt}0\) on \(\mathcal H_2\) satisfy \(A_1J=JA_2\) and \(B_1J=JB_2\). For every real \(p\),

Let \(T\) be a fixed weighted binary tree whose vertex weights lie in \([0,1]\). If its positive definite inputs \(A_j,B_j\) satisfy \(A_jJ=JB_j\) for every label \(j\), and its roots are \(M_A=T(A)\) and \(M_B=T(B)\), then \(M_AJ=JM_B\). Repeated labels and zero vertex weights are permitted.

Let a weighted binary tree have terminal weights \(w_j\) and positive definite inputs \(A_j\), and let \(M\) be its root. Then \(M\) is positive definite, commutes with invertible congruence, is monotone in the inputs, and satisfies \(\mathcal T(c_jA_j:j)=\bigl(\prod _jc_j^{w_j}\bigr)M\) for \(c_j{\gt}0\). A tree with constant positive definite input \(X\) has root \(X\). For every vector \(z\),

If \(P\) is positive semidefinite (in particular, an orthogonal projection) and \(A_j\ge c_jP\) with \(c_j{\gt}0\), then \(M\ge \bigl(\prod _jc_j^{w_j}\bigr)P\). Finally, if \(A_j=\prod _{g=1}^KA_{j,g}\) with positive definite factors and every factor from band \(g\) commutes with every factor from band \(g'\ne g\), then \(M=\prod _{g=1}^KM_g\), where \(M_g\) is the root of the same tree with inputs \(A_{j,g}\), and the factors \(M_g\) commute.

Let \(A,B\subseteq F\) satisfy \(A\subseteq B\). For every irreducible label \(\lambda \) of \(S_k\),

This includes empty coordinate sets and \(k=0\), and follows from [ Ope26b , Lemma 6.1(2) ] .

If \(\mu _t(C_n)\ne 0\), then the actual conditional word law is

For \(t{\gt}0\), every count vector \(n\) satisfies \(\mu _t(C_n){\gt}0\), so this conditional identity holds for every \(n\in \mathbb {N}^I\).

Let \(\Omega \in \mathcal H\), set \(P_\Omega =|\Omega \rangle \langle \Omega |\), and assume \(k_i\le I\), \(k_i\Omega =0\), and \(\sum _i k_i\ge g(I-P_\Omega )\) for \(g\in \mathbb {R}\). If \(\xi \in \Omega ^\perp \otimes \mathcal K\), then \(F_\xi \) is integrable and, for every \(t\ge 0\),

If \(\| \Omega \| =1\), then \(P_\Omega \) is an orthogonal projection, and in particular this holds for \(\xi =((I-P_\Omega )\otimes I_{\mathcal K})\zeta \) and arbitrary \(\zeta \in \mathcal H\otimes \mathcal K\).

For every \(t\ge 0\),

In particular, the count-and-order sampler is a probability measure.

Under \(\mu _t\), the count vector \(w\mapsto (n_i(w))_{i\in I}\) has law \(\bigotimes _{i\in I}\operatorname {Poisson}(t)\). Thus the label counts are mutually independent, and each has Poisson law of rate \(t\).

For \(w=(i_0,\ldots ,i_{m-1})\), regard the letters of \(w_S,w_T\) as letters of \(I\) through their inclusions. Then

where both lists retain increasing order of the original positions.

Suppose \(f\geq 0\), \(a,C\geq 0\), and \(A_m\leq a^m C\) for every \(m\geq 0\). Then \(f\) is integrable and

The comparison series has sum \(e^{(a-N)t}C\) for arbitrary real \(a,C\).

Use the notation of Theorem 27.19.4, with a finite alphabet \(I\), retained subset \(S\), arbitrary \(x\in M\), and \(T\geq 0\). For \(u\in \mathcal W_S\), let \(R(u)=A(u)x\), regarding its letters as elements of \(I\). Then

All integrals are finite. The assertion includes an empty alphabet and \(T=0\).

For \(T\ge 0\) and any function \(F:\mathcal W\times I\times \mathcal W\to [0,\infty ]\),

Both sides may be infinite. No integrability assumption on \(F\) is required.

Let \(I\) be finite, \(T\geq 0\), and \(f:\mathcal W_I\to \mathbb R\). Suppose \(|f(w)|\leq C\) for every \(w\), where \(C\in \mathbb R\). Then

Let \(I\) be finite, \(T\geq 0\), and \(f:\mathcal W_I\to \mathbb R\) with \(|f(w)|\leq C\) for all \(w\), where \(C\in \mathbb R\). Suppose \(G:\mathcal W_I\times I\to \mathbb R\) and \(D\in \mathbb R\) satisfy \(0\leq G(u,i)\leq D\) and \(f(u[i])-f(u)\leq G(u,i)\) for every \(u,i\). Then

Under the hypotheses and notation of Theorem 27.19.3, give the additive group of \(M\) a seminorm and suppose that \(\| E_i y\| \leq \| y\| \) for every \(i\in I\) and \(y\in M\). Then

Let \(K\) be a semiring and \(M\) a \(K\)-module with an abelian additive group. Let \(E_i:M\to M\) be linear for each letter \(i\in I\). For a word \(u=(i_0,\ldots ,i_{k-1})\), put \(A(u)=E_{i_{k-1}}\cdots E_{i_0}\), with \(A(\varnothing )=1\). Fix \(x\in M\) and a word \(w=(i_0,\ldots ,i_{m-1})\), and write \(R_j=A((w_{{\lt}j})_S)x\). Then

The sum is over positions, so repeated letters contribute separately.

Suppose \(I\) is finite and \(t\geq 0\). For a single sample \(w\) with law \(\mu _{I,t}\), its two ordered subwords have joint law

In particular, \(w_S\) and \(w_T\) are independent with laws \(\mu _{S,t}\) and \(\mu _{T,t}\). This holds when either subset is empty and at \(t=0\).

For \(T\geq 0\) and any \(F:\mathcal W_S\times I\to [0,\infty ]\),

These are nonnegative integrals; both sides may be infinite.

For any alphabet \(I\), any \(f:\mathcal W_I\to \mathbb R\), and any word \(w\), the next prefix is \(w_{{\lt}j+1}=w_{{\lt}j}[w_j]\) for \(0\leq j{\lt}|w|\), and

No finiteness assumption on \(I\) or bound on \(f\) is needed.

Let \(S=\{ i\in I:p(i)\} \) and let \(R_p\) retain precisely the letters in \(S\). For \(F:\mathcal W_S\times I\to [0,C]\) with \(C{\lt}\infty \),

The marked sum, inner kernels, and time integrand are integrable.

For \(F:\mathcal W_I\times I\to [0,C]\) with \(C{\lt}\infty \), the actual marked-prefix sum is integrable and satisfies

Every inner word expectation and the outer time integrand are integrable.

For finite alphabets \(I,J\), a bijection \(e:I\to J\), and every \(t\geq 0\), the map \(e^{\mathcal W}\) preserves the Poissonized word measures:

This includes empty alphabets and zero time.

For \(s,t\geq 0\), the pushforward of the product measure under concatenation satisfies

Equivalently, independent word samples with durations \(s\) and \(t\) concatenate to a word with law \(\mu _{s+t}\).

Let \(t\geq 0\), \(C,B,V,M\geq 0\), \(c{\gt}0\), and \(0{\lt}\alpha \leq 1\). Let \(S\subseteq Y\) be arbitrary and let \(f:\mathcal W_I\times Y\to \mathbb R\) be nonnegative at every argument. Suppose that, for every \(n\in \mathbb N\), \(y\in Y\), and \(i\in I\), the literal label and site counts satisfy

Suppose also that, for every word \(u\) and every \(y\in Y\),

Define the spatial and temporal rates by

Then \(a{\gt}0\) and \(v\geq 0\). For every \(y\in Y\), the function \(u\mapsto f(u,y)\) is integrable with respect to \(\mu _{I,t}\) and

If \(D_S(y)=\infty \), then \(f(u,y)=0\) for every \(u\in \mathcal W_I\). At finite distance from \(S\), the right side of (104) is \(M e^{vt-a(D_S(y)_{\mathbb R})^\alpha }\). The amplitude \(C\) belongs to the shell kernel, while \(M\) bounds the initial observable.

Let \(T\geq 0\) and \(F:\mathcal W_I\times I\to \mathbb R\), where \(I\) is finite. Suppose \(|F(u,i)|\leq C\) for every \(u,i\), with \(C\in \mathbb R\). Then the marked-prefix sum \(w\mapsto \sum _{j{\lt}|w|}F(w_{{\lt}j},w_j)\) belongs to \(L^1(\mu _{I,T})\), and

is integrable on \((0,T]\). Each kernel \(F(\cdot ,i)\) belongs to \(L^1(\mu _{I,s})\) for every \(s\geq 0\).

For a finite alphabet \(I\), \(T\geq 0\), and \(F:\mathcal W_I\times I\to \mathbb R\) satisfying \(|F(u,i)|\leq C\) for a real constant \(C\) and every \(u,i\),

Let \(t\geq 0\), \(a\geq 0\), \(0{\lt}\alpha \leq 1\), and \(v,C\geq 0\). Let \(f:\mathcal W_I\times Y\to \mathbb R\) and \(A:Y\times Y\to \mathbb R\) be nonnegative at every argument, and let \(S\subseteq Y\) be arbitrary. Assume that, for every word \(u\) and all \(y,z\in Y\),

Then, for every \(y\in Y\), the function \(u\mapsto f(u,y)\) is integrable with respect to \(\mu _{I,t}\), and

At finite distance from \(S\), the right side is \(C e^{vt-a(D_S(y)_{\mathbb R})^\alpha }\); at infinite distance it is zero. The initial bound uses the actual indicator of \(S\), and the increment hypothesis sums all right-appended letters. No uniform bound on \(f\) is required.

Let \(a\geq 0\), \(0{\lt}\alpha \leq 1\), \(v,C\geq 0\), and let \(f:\mathcal W_I\times Y\to \mathbb R\) and \(A:Y\times Y\to \mathbb R\) be nonnegative at every argument. For an arbitrary \(S\subseteq Y\), assume all four conditions in (80). If \(D_S(y)=\infty \), then, for every \(u\in \mathcal W_I\),

This assertion is independent of time and holds at every word. In particular, when \(S=\varnothing \), \(f\) vanishes identically.

For finite alphabets \(I,J\) and every \(t\geq 0\), the joint distribution of \(L(w)\) and \(R(w)\) under \(\mu _{I\sqcup J,t}\) is

Thus the two ordered subwords are independent, with respective laws \(\mu _{I,t}\) and \(\mu _{J,t}\). Either alphabet may be empty.

Let \(t\geq 0\), \(f:\mathcal W_I\times Y\to \mathbb R\), \(A:Y\times Y\to \mathbb R\), \(r:Y\to \mathbb R\), and \(v,C\geq 0\). Suppose \(f(u,y)\geq 0\), \(A(y,z)\geq 0\), and \(r(y)\geq 0\) for every \(u\in \mathcal W_I\) and \(y,z\in Y\). Assume (61) for every \(u,y\) and (63) for every \(y\). Then, for each \(y\in Y\), the function \(u\mapsto f(u,y)\) is integrable with respect to \(\mu _{I,t}\), and

Neither a uniform bound on \(f\) nor strict positivity of \(r\) is required.

Let \(f:\mathcal W_I\times Y\to \mathbb R\), \(A:Y\times Y\to \mathbb R\), and \(r:Y\to \mathbb R\). Suppose \(A(y,z)\geq 0\) for every \(y,z\in Y\), and let \(v,C\geq 0\). Assume (61) for every \(u,y\), and suppose that, for every \(y\in Y\),

Then, for every \(m\geq 0\) and \(y\in Y\),

Both \(f\) and \(r\) may take negative values.

Let \(f:\mathcal W_I\times Y\to \mathbb R\), \(A:Y\times Y\to \mathbb R\), \(r:Y\to \mathbb R\), and \(v,C\geq 0\). Suppose \(f(u,z)\geq 0\) and \(A(z,z')\geq 0\) for every word \(u\) and every \(z,z'\in Y\). Assume (61) for every word and coordinate, and (63) for every coordinate. For each \(y\in Y\) with \(r(y)=0\), one has \(f(u,y)=0\) for every \(u\in \mathcal W_I\). No sign assumption on the other coordinates of \(r\) is needed.

Let \(M\) be positive definite, let \(C\ge 0\) and \(r\ge 1\), and suppose that \(r^{-C}I\le M\). For any vector \(\xi \) with \(\| \xi \| \le 1\), set \(N^2=\| M^{-1/4}\xi \| ^2\). Then \(\log N^2\le (C/2)\log r\). Here the real logarithm is assigned the value zero at zero.

Let \(A,B\) be complex matrices indexed by the same finite set. If \(A\leq B\), then every complex vector \(v\) on that set satisfies \(\operatorname {Re}(v^*Av)\leq \operatorname {Re}(v^*Bv)\).

There are universal constants \(C,c{\gt}0\) such that, for \(0{\lt}t\le 1\), \(a=2t\) and \(a\log (e\dim x)\le c\), there is a polynomial \(p(k)\), uniform in \(\theta \), with

where \(\eta _\theta =S_\theta (x|P)+S_\theta (x|Y_0)=I_\theta (x:F|P)=I_\theta (x:F|Y_0)\). The same statement holds for a move to \(F\) after exchanging the outer parts: \(A_k^{-1/2}A_k(P,Y_0,Fx)A_k^{-1/2}\) satisfies the bound with \(\eta _\theta =S_\theta (x|F)+S_\theta (x|Y_0)\).

The measure \(\mu _\rho \) is finite for every matrix \(\rho \). For \(\rho \ge 0\) and every continuous real function \(f\) on \(\mathcal V\),

If in addition \(\operatorname {Tr}\rho =1\) and \(\Pi _k\rho =\rho \), then \(\mu _\rho (\Theta )=1\). Also \(\mu _0=0\).

For fixed real \(p\), replica number \(k\) and supplied vector, \(d_j\) is jointly measurable and absolutely integrable, and \(\nu _j\) is finite. For admissible data, nonnegative regulator parameter and cross-band commutation on \(\mathcal S_k\), every continuous real \(f\) on \(\mathcal V\) satisfies

Under the same structural hypotheses, a nonzero symmetric supplied vector gives \(\nu _j(\Theta )=1/2\) if the terminal weight is nonzero, and \(\nu _j=0\) if that weight is zero. Thus every interior leaf has mass \(1/2\), the new leaves vanish at \(p=0\), and the old leaves vanish at \(p=1\). A zero supplied vector gives zero measure. These assertions include \(k=0\).

Let \(t\ge 0\) and assume every factor is nonzero. The metric \(W_{Q,k}=\sum _\lambda w_k(\lambda )\pi ^\lambda _{Q,k}\) is positive definite, commutes with the copy permutations of \(Q\) and with the metrics of nested and disjoint subsystems, and \(W_{Q,0}=I\). On the symmetric subspace, \(W_{Q,k}=W_{Q^c,k}\).

Let \(0\le h\le 1\) have designated support \(D\) meeting \(Y\) and exactly one of the outer parts of a partition, let \(0{\lt}t{\lt}1/4\), \(a=2t\) and \(a\log (e\dim D)\le 1/8\). Then, uniformly over density matrices \(\rho \) on \(\mathcal S_k\), \(\operatorname {Tr}(\rho \mathsf D_k)\le 2C a^2\log ^4(e\dim D)\int \eta ^{1/8}\, d\mu _\rho +r_k\) with \(r_k\to 0\), where \(\mathsf D_k\) is the skew square of the band metric and \(\eta \) is the entropy of the split.

Fix a finite tensor product \(\mathcal V=\bigotimes _{v\in V}\mathbb {C}^{n_v}\), with \(n_v{\gt}0\), and let \(t{\gt}0\). The replica metrics use one common label weight padded to the original full dimension \(\dim \mathcal V\). There exists \(C\ge 0\), chosen before the copy number and partition, such that every \(k\ge 0\) and every partition \(\pi =(P,Y,F)\) satisfy

where \(\widetilde A_k(\pi )=\Pi _kA_k(\pi )+(I-\Pi _k)\) is the existing identity extension of the symmetric band metric and \(A_k(\pi )=(W_{P,k}^{-1}W_{F,k}^{-1}W_{Y,k})^2\). Empty parts, \(k=0\), and an empty set of sites are included.

Let \(V\) be a finite set of sites, and let \(\iota _v\) be a finite nonempty coordinate set for every \(v\in V\). Put \(\mathcal V=\bigotimes _{v\in V}\mathbb {C}^{\iota _v}\) and \(d=\dim \mathcal V=\prod _{v\in V}|\iota _v|\). Fix \(t{\gt}0\). For every subsystem \(Q\subseteq V\), use the common replica metric \(W_{Q,k}=\sum _\lambda w_k(\lambda )\pi ^\lambda _{Q,k}\) from Theorem 23.5.2, with every label padded to the same full dimension \(d\).

There exists \(C\ge 0\), depending on the fixed system and \(t\), such that the following holds for every \(k\ge 0\) and every pairwise disjoint cover \(P\cup Y\cup F=V\). Let \(\Pi _k\) be the orthogonal projection onto \(\operatorname {Sym}^k\mathcal V\), and define the full-space operators and the scalar

Then \(0{\lt}c_k\le 1\), and, in the positive-semidefinite order,

The exponent \(C\) is independent of \(k\) and of \(P,Y,F\). Empty parts, \(k=0\), and the empty set of sites are included, with the empty tensor product equal to \(\mathbb {C}\).

Fix the same finite tensor product and \(t{\gt}0\). There exists \(C\ge 0\) such that, for every \(k\ge 0\), every label set \(J\), not necessarily finite, every family of partitions \((\pi _j)_{j\in J}\), and every nonempty finite rooted weighted binary tree \(\mathcal T\) with terminal labels in \(J\), its root satisfies

The exponent is chosen before the copy number, label set, partitions, tree and interpolation parameters. Repeated terminal labels, zero terminal weights, and interpolation parameters at either endpoint are allowed.

Fix a finite tensor product \(\mathcal V=\bigotimes _{v\in V}\mathbb {C}^{n_v}\), with \(n_v{\gt}0\), and let \(t{\gt}0\). There exists \(C\ge 0\) such that the following holds for every \(K,k\ge 0\), every label set \(J\), and every family of partitions \(\pi _{j,g}\) indexed by \(j\in J\) and \(0\le g{\lt}K\). Put \(A_{j,g}=\widetilde A_k(\pi _{j,g})\), using the identity-extended band metric of Theorem 27.5.1. Suppose that, for each \(j\), the matrices \(A_{j,g}\) commute at distinct positions \(g\). Then every nonempty finite rooted weighted binary tree \(\mathcal T\) with terminal labels in \(J\) satisfies

The exponent \(C\) is chosen before \(K,k,J\), the partitions, the tree, and its interpolation parameters. The label set need not be finite. Empty site sets, \(K=0\), \(k=0\), repeated labels, and interpolation parameters at either endpoint are included; the empty product is \(I\).

The conclusions of  [ Ope26b , Proposition 7.4 ] hold under the hypotheses of the source: admissible finite history and choice data, cross-band commutation on \(\mathcal S_k\), positive contractions with designated supports compatible with the terminal partitions, and \(\ell \ge 1\) bounding the logarithmic dimensions with \(a\ell \) below a universal threshold. Namely, for every \(k\) and nonzero \(x\in \mathcal S_k\): at every \(0{\lt}p{\lt}1\) the exact derivative, the entropy-gain lower bound and the energy bound hold, and the entropy-gain lower bound integrates over every \([p_0,p_1]\subseteq [0,1]\) to \(\log N(p_0)^2-\log N(p_1)^2\).

Assume the conclusion of the relative coherent pin  [ Ope26b , Lemma 6.3 ] and of the skew bound  [ Ope26b , Lemma 6.4 ] at a split leaf, and that metrics from different bands commute on \(\mathcal S_k\), for every \(k\), as in the source. The mean trees are evaluated on the band metrics extended by the identity off \(\mathcal S_k\) (Lemma 27.3.1), which agree with the source’s operators on \(\mathcal S_k\); an earlier formalization assumed commutation on the full tensor power instead  [ con26a ] . Then the conclusions of  [ Ope26b , Proposition 7.4 ] hold: with \(a=2t\), \(\ell \ge 1\) bounding \(\log (e\dim x)\) for every transferred subsystem and \(\log (e\dim D_i)\) for every split term, and \(a\ell \) below a universal threshold, for every \(k\), nonzero \(x\in \mathcal S_k\) and \(0{\lt}p{\lt}1\),

with \(\beta _k=O(\log (k+1))\) independent of \(p\) and \(x\); and if \(\bar Hx=E_0x\), then \(\langle v,\bar Hv\rangle \le 2E_0+Ca^2\ell ^C\sum _iW_i(p)\sum _{j}\pi _j\int m_{1/4}\int \eta _{i,j}^{1/8}\, d\mu _{\sigma _{j,u}}\, du+o_k(1)\), uniformly in \(p\) and \(x\).

Choose the universal constants of Theorem 27.3.9 once. Fix a finite family of admissible transports with the same one-copy dimensions, band count, supported energy terms, positive parameter \(a\), and dimension bound \(\ell \ge 1\), satisfying all hypotheses of that theorem in each round. The history and conditional choice sets may vary between rounds. There exist nonnegative sequences \(\beta _k\) and \(r_k\), and a constant \(C_\beta \ge 0\), such that \(\beta _k\le C_\beta \log (k+2)\) for every \(k\ge 0\) and \(r_k\longrightarrow 0\). These same sequences give the entropy and energy error bounds for every round, every replica count, every nonzero symmetric input vector, and every interior interpolation parameter. The energy estimate retains its replica-energy eigenvector hypothesis with the stated eigenvalue. The exact derivative identity and the entropy inequality integrated over every closed subinterval of \([0,1]\) also hold. The family may be empty. Only \(C_\beta \) and the two sequences may depend on the fixed family; the universal constants are unchanged.

Let the site dimensions be positive and let \(t\geq 0\). Fix admissible transport data: a weighted history tree and its conditional choice trees, with positive history and conditional weights, old partitions in \(K\) bands, and moves that transfer subsets of the middle regions. For a terminal label \(j\) and band \(g\), let \(A_{j,g}\) be the raw band metric of its actual partition. Assume that metrics from distinct bands commute on the copy-symmetric subspace \(\mathcal S_k\).

Write \(\Pi \) for the orthogonal projection onto \(\mathcal S_k\) and let \(Z:\mathbb C^r\longrightarrow \mathcal H^{\otimes k}\) satisfy \(Z^*Z=I\) and \(ZZ^*=\Pi \). For \(p\in \mathbb R\), put \(\bar p=\max (0,\min (1,p))\) and let \(T_{\bar p}\) be the actual interpolated transport tree. Define

Then \(M_p\) is positive definite and \(\widehat M_p Z=ZM_p\). This is the symmetric-coordinate form of the band factorization in [ Ope26b , Sections 7 and 8 ] ; it uses the same tree in every band, including at the endpoints of the interpolation.

Fix \(\mathcal V=\bigotimes _{v\in V}\mathbb {C}^{n_v}\) with \(n_v{\gt}0\) and \(t{\gt}0\). There exists \(C_0\ge 0\) such that the following holds for every admissible transport round with \(K\ge 0\) bands and every copy count \(k\ge 0\). The old parts form partitions, each move transfers a subsystem of its middle part, and the history and conditional choice weights are positive. Suppose that the identity-extended band metrics at distinct band positions commute for every pair of terminal labels. Write \(\pi _{j,g}\) for the partition at terminal label \(j\) and band \(g\), and set \(A_j=\prod _{g=0}^{K-1}\widetilde A_k(\pi _{j,g})\). For \(q\in \mathbb {R}\), let \(\bar q=\max (0,\min (1,q))\) and let \(\mathcal T_{\bar q}\) be the interpolated terminal tree of the given round. Its root \(M(q)=\mathcal T_{\bar q}(A_j:j)\) satisfies

The constant is chosen before the number of bands, history and choice sets, transport round, copy count and parameter. The history and choice sets are not assumed finite.

Fix the physical dimensions and \(t{\gt}0\) as in Theorem 27.8.1. There exists \(C_0\ge 0\) such that every admissible transport round satisfying the same band commutation assumption obeys the following bound. For its root \(M(q)\), with any \(K,k\ge 0\), \(q\in \mathbb {R}\), and \(\xi \) with \(\| \xi \| \le 1\),

The constant is chosen before the history and choice sets, transport round, numbers of bands and copies, interpolation parameter and vector. The real logarithm is assigned the value zero at zero.

The relative coherent pin holds in the form assumed in Theorem 27.3.4. Hence the conclusions of  [ Ope26b , Proposition 7.4 ] hold assuming only the skew bound at a split leaf.

Let \(M\) be a positive definite complex matrix on a finite-dimensional space, which may have dimension zero. Suppose that \(b{\gt}0\) and \(bI\le M\). For any vector \(\xi \), set \(N^2=\| M^{-1/4}\xi \| ^2\). Then

Let \(M\) be a positive definite complex matrix on a finite-dimensional space, which may have dimension zero. Suppose that \(b{\gt}0\), \(s\le 0\), and \(bI\le M\). Then \(M^s\le b^sI\), with powers defined by spectral functional calculus.

Let \(I\) be a finite set of sites with local dimensions \(n_i\ge 0\), let \(k\ge 0\), and let \(t\in \mathbb {R}\). Write \(W_{S,k}(t)\) for the regional replica metric, using the label weights of the full one-copy space, and let \(\Pi \) project onto simultaneous copy symmetry. For arbitrary subsets \(P,Y,F\subseteq I\), set

Here the inverse of a singular matrix is interpreted as zero. For every site \(a\in I\) and irreducible label \(\lambda \) of \(S_k\),

Empty local coordinate sets and zero replicas are included. The regional commutation is a consequence of [ Ope26b , Lemma 6.1(2) ] , applied to the band metric of [ Ope26b , Proposition 7.4 ] .

Let \(a\in F\), let \(S\subseteq F\), and let \(f\) be a real-valued function on the irreducible labels of \(S_k\). For every label \(\lambda \),

Empty local coordinate sets and \(k=0\) are allowed. This is the one-site consequence of  [ Ope26b , Lemma 6.1(2) ] .

Suppose \(Y\) is finite. Let \(C,B,V\geq 0\), \(c{\gt}0\), and \(\alpha {\gt}0\), and assume (97) for all \(n\in \mathbb N\), \(y\in Y\), and \(i\in I\). Set \(a=c/(2\cdot 2^\alpha )\). Then \(a{\gt}0\), and for every \(y\in Y\),

Let \(C\geq 0\), \(c{\gt}0\), \(\alpha {\gt}0\), \(B\in \mathbb R\), and fix \(y,z\in Y\). Suppose that, for every \(n\in \mathbb N\),

Then the series \(\sum _{n=0}^{\infty }C e^{-cn^\alpha }K_n(y,z)\) is summable. No finiteness of \(Y\), site-count bound, or separate sign assumption on \(B\) is required; the \(n=0\) label bound already implies \(B\geq 0\).

Suppose \(Y\) is finite. Let \(C,a,B,V\geq 0\) and \(\alpha {\gt}0\), and let \(c\in \mathbb R\) satisfy \(\eta =c-a2^\alpha {\gt}0\). Assume the following bounds for every \(n\in \mathbb N\), \(y\in Y\), and \(i\in I\):

Then, for every \(y\in Y\),

The remaining decay rate is exactly \(c-a2^\alpha \). The fourth moment is finite, and no assumption \(\alpha \leq 1\) is needed.

For every \(p\in \mathbb N\), \(c{\gt}0\), and \(\alpha {\gt}0\), one has \(1\leq M_p(c,\alpha )\).

For every \(p\in \mathbb N\), \(c{\gt}0\), and \(\alpha {\gt}0\), the series in (84) is summable.

Let \(p\in \mathbb N\), \(c{\gt}0\), \(\alpha {\gt}0\), and \(r\in \mathbb R\) with \(r\geq 0\). Then

The constant multiplying the decay at \(r\) is the full moment \(M_p(c/2,\alpha )\), which is finite and at least one by Theorems 27.26.1 and 27.26.2. The cutoff need not be an integer, and the boundary \(n=r\) is included whenever \(r\) is a natural number.