Continuity bounds the density on the compact unitary group. For \(\rho \ge 0\), \(\operatorname {Tr}(\rho P_{\theta ,k}) =\langle \theta ^{\otimes k},\rho \theta ^{\otimes k}\rangle \ge 0\). Change of measure gives (2). Taking \(f=1\) and using \(D_k\int P_{Ue_a,k}\, dH(U)=\Pi _k\) gives the mass statement.

Write \(\sigma _{j,u}=\Phi _j^*(|M^{-iu}v\rangle \langle M^{-iu}v|)\). The function \((W,U)\mapsto q_{\Phi _j^*(|Wv\rangle \langle Wv|)}(U)\) is continuous on \(U(\mathcal V^{\otimes k})\times U(\mathcal V)\), hence bounded. Since \(M^{-iu}\) is unitary and \(\int m_{1/4}=1/2\), this bounds the absolute joint density by an integrable function. The same argument includes a continuous spherical symbol. Positivity of the actual terminal state removes the positive part, and Fubini’s theorem gives (4). For nonzero terminal weight the trace of the state is one; at zero weight the normalized derivative map is zero. Apply the coherent mass formula and \(\int m_{1/4}=1/2\). Compactness at fixed \(p\) makes no uniform assertion in \(p\).

Write \(\log Y=\int _0^\infty [(1+s)^{-1}-(Y+s)^{-1}]\, ds\). For each \(s\) the positivity of \(g(X-g^{-1})P_\theta (X-g^{-1})\) with \(X=\Pi (Y+s)^{-1}\Pi \), averaged over \(\theta \), gives the resolvent form of the inequality; Fubini’s theorem exchanges the two integrals (06-transport.tex lines 526–546). The dimension of \(\mathcal S_k\) is at most \((k+1)^{\dim }\).

Integrating the pin gives \(D_k^{-1}\mathcal Q_k(e^\varphi )\le X\). Adding \(\varepsilon (I-\Pi )\) with \(\varepsilon I\le X\) keeps the inequality and makes the left side positive definite; operator monotonicity of the logarithm and the Jensen inequality conclude (06-transport.tex lines 520–575).

If \(P\) commutes with a Hermitian \(Z\) and \(PY=PZ\) for a Hermitian \(Y\), then \(PY^k=PZ^k\) for all \(k\); interpolating \(f\) by one polynomial on the two finite spectra gives \(Pf(Y)=Pf(Z)\). Applied with \(P=\Pi \) and \(P=I-\Pi \) this computes \(f(\Pi Y+(I-\Pi )Z)=\Pi f(Y)+(I-\Pi )f(Z)\), in particular the powers of \(\widetilde A\). The identity \(\widetilde A\widetilde B=\Pi AB+(I-\Pi )\) gives the commutation criterion, and positivity follows by splitting a vector into its components in \(\mathcal S_k\) and its complement.

Put \(X=\pi ^\lambda _{\{ a\} ,k}\). The one-site commutation theorem gives \(XW_{S,k}(t)=W_{S,k}(t)X\) for every region \(S\). Commutation is preserved by inversion: this follows by multiplying by the inverse on both sides when the matrix is invertible, and from the zero convention when it is singular. It is also preserved by products and powers, so \(XA=AX\) with the displayed factor order. Since \(\{ a\} \subseteq I\), the containing-subsystem symmetry theorem gives \(X\Pi =\Pi X\). Therefore

Each extended band metric is positive definite, and metrics at distinct bands commute. Their products over bands therefore give positive definite inputs. These inputs commute with copy permutations, so the symmetrizer commutes with every input and hence with the root \(M(p)\) of the interpolated tree. Since the symmetrizer fixes \(x\in \mathcal S_k\), it fixes the filtered vector \(M(p)^{-1/4}x/N(p)\). The symmetrizer therefore fixes the transport states by Lemma 7.9.5.3. The nonzero input vector and the positive leaf weights for \(0{\lt}p{\lt}1\) give trace one; positivity and the derivative formula follow from Lemma 7.9.5.1.

The extended band metrics of different bands commute, so the relative ratio factors over bands. Theorem 7.9.1.1 gives \(\log C_h=\sum _g\log C_{h,g}\). The relative pin and projection transfer give the conditional pin of each band (06-transport.tex lines 513–531), and the logarithmic passage turns it into the entropy lower bound (lines 495–583). For the energy, the support of each term commutes with every unsplit band, a single band remains at a split leaf (lines 732–735), the skew bound controls its symbol cost (lines 736–748), and Lemma 7.9.6.3 applies term by term with \(\bar Hx=E_0x\) (lines 588–766).

For a move of \(x\subseteq Y\) to \(P\), the parts \(P,x,Y\setminus x,F\) partition the sites, and Theorem 23.11.5 bounds the relative metric with \(\eta _\theta =S_\theta (x|P)+S_\theta (x|Y\setminus x)=I_\theta (x:F|P)\), the entropy of the move (06-transport.tex lines 315–328). A move to \(F\) is the same after exchanging the outer parts, and a choice making no move has relative metric \(I\), which dominates every coherent projector. A quadratic-form bound \(c\, |\langle \theta ^{\otimes k},w\rangle |^2\le \langle w,Mw\rangle \) for Hermitian \(M\) is the operator bound \(cP_{\theta ,k}\le M\). The polynomial factor \(p(k)\) gives \(-\log p(k)^{-1}=O(\log (k+1))\), and \(\log (e\dim x)\) is the logarithm of \(e\) times the product of the site dimensions.

Integrate the pointwise bound and apply Lemma 7.9.5.2 (06-transport.tex lines 427–429 and 769–779).

Each summand is \(G(p)=\int m_{1/4}(u)\operatorname {Tr}(\sigma _{h,u}Q)\, du\) with \(Q\) Hermitian. Replacing the old input \(A_h\) by \(A_h^{1/2}e^{\delta Q}A_h^{1/2}\) changes \(-\log N(p)^2\) at \(\delta =0\) at rate \((1-p)w_hG(p)\), by the norm-derivative formula and the leaf identity. For fixed \(\delta \) the perturbed norm is continuous in \(p\), so \((1-p)w_hG(p)\) is a pointwise limit of continuous functions on \((0,1)\) and hence measurable. The states are density matrices, so \(|G(p)|\le \| Q\| /2\). The source asserts the integration (06-transport.tex lines 427–429) and bounds the integrand (lines 769–779); measurability is the remaining point.

For a split towards \(P\) put \(P_0=D\cap P\) and \(x=D\cap Y\). Then \(D=P_0\cup x\), so \(h\) is the lift of an operator \(0\le K\le 1\) on \(P_0x\) and \(\dim D=(\dim P_0)(\dim x)\). The skew operator of  [ Ope26b , Lemma 6.4 ] takes its absolute values on \(\mathcal S_k\); for \(Z\) commuting with the projection \(\Pi \) onto \(\mathcal S_k\), \(\Pi |\Pi Z|=\Pi |Z|\), so the two skew operators have the same expectation in every density matrix on \(\mathcal S_k\). The entropy \(S_\theta (x|P)+S_\theta (x|Y\setminus x)\) of the skew bound is \(I_\theta (x:F|P)\). A split towards \(F\) is the same with the outer parts exchanged. The skew bound holds with an additive \(\varepsilon \) for all large \(k\); for fixed \(k\) the excess over symmetric density matrices is bounded by the operator norm of the skew square, and its supremum is the remainder \(r_k\).

Combine Theorem 27.3.5 with Theorem 27.3.8. For the integrated form, apply Corollary 27.3.6 with the entropy-gain lower bound, which is integrable by Lemma 27.3.7.

For each round \(j\), choose the sequences \(\beta _{j,k}\) and \(r_{j,k}\) from Theorem 27.3.9, before fixing a replica count or input vector. Set \(\beta _k=\sum _j|\beta _{j,k}|\) and \(r_k=\sum _j|r_{j,k}|\). Both sequences are nonnegative and dominate the corresponding signed error in every round. Finite sums preserve convergence to zero and the logarithmic asymptotic bound. Since \(\log (k+2)\ge \log 2{\gt}0\), one can enlarge its coefficient to absorb the finitely many initial indices, even when \(\beta _0\ne 0\). The derivative identity is unchanged. Subtracting the larger entropy error gives the common pointwise bound; adding the larger energy remainder gives the common energy bound. Finally, the entropy gain is integrable by Lemma 27.3.7, so Corollary 27.3.6 applies with the common error, including both endpoints.

The empty products give \(I_IZ=ZI_J\). If \(LZ=ZM\), then \((A_gL)Z=A_g(ZM)=(A_gZ)M=Z(B_gM)\). Induction over the ordered factors proves the assertion.

Each \(A_{j,g}\) is positive definite and commutes with \(\Pi \). The equations for \(Z\) give \(\Pi Z=Z\), hence \(A_{j,g}Z=ZB_{j,g}\) and \(\widetilde A_{j,g}Z=ZB_{j,g}\). Thus the compressed matrices are positive definite and metrics from distinct bands commute after compression. The roots of their common tree consequently commute, so their ordered product is positive definite.

The extended metrics commute across distinct bands. Factor the tree evaluation of their products in (5) into the ordered product of the band roots. Intertwining each band root with its compressed root and multiplying these identities gives \(\widehat M_pZ=ZM_p\).

Choose the exponent from Theorem 23.45.1 for the fixed system and \(t\). Its identity-extension inequality is

The band metric commutes with \(\Pi _k\), and \(\Pi _k^2=\Pi _k\). Hence \(\Pi _kA_k(\pi )\Pi _k=\Pi _kA_k(\pi )\), proving the assertion.

Choose \(C\) from Theorem 27.5.1 and put \(c=(k+2)^{-C}{\gt}0\). Every terminal input is bounded below by the positive definite matrix \(cI\). Monotonicity of the mean tree and its value on a constant input give

Choose \(C\) from Theorem 27.5.1 and set \(b=(k+2)^{-C}{\gt}0\). The commuting-product bound gives \(b^KI\le A_{j,0}\cdots A_{j,K-1}\) for each \(j\). Since \(b^K=(k+2)^{-CK}\), monotonicity and the constant-input identity for the mean tree give

Since \(M\) is positive definite, spectral functional calculus gives \(N^2=\operatorname {Re}(\xi ^*M^{-1/2}\xi )\). Theorem 7.16.3 gives \(M^{-1/2}\le b^{-1/2}I\). The quadratic-form order then gives the stated bound, using \(\operatorname {Re}(\xi ^*\xi )=\| \xi \| ^2\).

If \(\xi =0\), then \(N=0\), and the right-hand side is nonnegative. Otherwise \(N{\gt}0\), since \(M^{-1/4}\) is invertible. The squared-norm bound gives \(N^2\le (r^{-C})^{-1/2}\| \xi \| ^2\le r^{C/2}\). Taking logarithms and using \(\log (r^{C/2})=(C/2)\log r\) gives the result.

The validity of the moves ensures that every terminal triple is a partition. The commutation assumption, applied with the same terminal label on both sides, gives pairwise commutation within each product \(A_j\). Apply Theorem 27.6.1 to these partitions and the given tree \(\mathcal T_{\bar q}\).

Choose \(C_0\) from Theorem 27.8.1. Every terminal input is positive definite, and so is its mean-tree root \(M(q)\). Since \(C_0K\ge 0\) and \(k+2\ge 1\), the scalar floor \((k+2)^{-C_0K}I\le M(q)\) satisfies the hypotheses of Theorem 27.7.2, which gives the bound.

There are \(N^m\) words of length \(m\), so their total mass is \(e^{-Nt}(Nt)^m/m!\). Summing these nonnegative masses gives \(e^{-Nt}\sum _{m\geq 0}(Nt)^m/m!=1\). The same length-wise calculation gives the Poisson distribution. At zero time all positive lengths have zero mass; for an empty alphabet there are no words of positive length.

Integrate against the sum of weighted point masses. Absolute summability is precisely the integrability criterion. For nonnegative real observables, regrouping the series by length gives (16).

The exponential series gives \(\sum _{m\geq 0}e^{-Nt}t^m a^m C/m!=e^{(a-N)t}C\). The nonnegative terms in (16) are bounded by this summable series. The comparison proves both integrability and (17).

Splitting a word at position \(k\) and concatenating the two parts returns the original word. Conversely, if \(uv=w\), then \(|u|\leq m\), and splitting \(w\) at \(|u|\) returns \(u\) and \(v\). The prefix of the split at \(k\leq m\) has length \(k\), so the maps in (19) are inverse. This also proves finiteness, including the unique split of the empty word.

For \(0\le k\le m\), the identity \(\binom {m}{k}k!(m-k)!=m!\) expresses each summand as \(e^{-N(s+t)}s^kt^{m-k}\binom {m}{k}/m!\). Factor out \(e^{-N(s+t)}/m!\) and apply the binomial theorem. Only positive factorials are canceled, so zero times are included.

Fix a word \(w\) of length \(m\). Its concatenation fiber consists of precisely the \(m+1\) pairs formed by its prefix of length \(k\) and suffix of length \(m-k\), with \(0\le k\le m\). The product measure assigns to each pair the product of the corresponding singleton masses. Summing over the fiber and using (20) gives \(e^{-N(s+t)}(s+t)^m/m!\), the singleton mass of \(w\) under \(\mu _{s+t}\). Since the word space is countable, agreement on every singleton proves equality of the measures.

The prefix of \(u[a]v\) before position \(|u|\) is \(u\), the letter at that position is \(a\), and the suffix after it is \(v\). Conversely, splitting at \(j\) gives \(w=w_{{\lt}j}[w_j]w_{\geq j+1}\) and \(|w_{{\lt}j}|=j\). Thus reconstruction recovers both the word and its marked position. The cases \(j=0\) and \(j=|w|-1\) have empty prefix and suffix, respectively.

At \(T=0\), both sides vanish. For \(T{\gt}0\), the beta integral and its relation to the gamma function at positive integers give

On the integration interval, the two exponential factors multiply to \(e^{-NT}\). Factoring out \(e^{-NT}/(m!n!)\) proves the identity. Continuity proves integrability; nonnegativity identifies the real and nonnegative integrals. Writing \(s_+=\max (s,0)\) extends the two factors to continuous functions \(s\mapsto w_{s_+}(m)\) and \(s\mapsto w_{(T-s)_+}(n)\) on the real line. These extensions agree with the original weights on \([0,T]\).

Expanding the atomic word measure expresses the left side as a sum over marked words. The bijection of Definition 27.10.1.3 rewrites this sum as

On the right side of (25), expand both atomic measures. Each summand is a measurable nonnegative function of time, so Tonelli’s theorem moves the sums outside the time integral. Formula (23) then gives (26). This argument includes zero singleton masses multiplied by an infinite value of \(F\), using \(0\cdot \infty =0\).

Apply (25) to \(F(u,i,v)=G(u,i)\). The suffix measure has total mass one, giving (27). Taking \(G=1\) makes the inner sum on the left equal to the length and the time integrand equal to \(N\).

Expand \((\sum _{i\in I}X_i)^m\) by choosing a variable at each of the \(m\) positions. The monomial contributed by \(w\) is \(\prod _{j=0}^{m-1}X_{w(j)}=X^{C(w)}\). Hence the coefficient of \(X^n\) counts exactly the words with \(C(w)=n\). The multinomial theorem gives the same coefficient as the right side of (28).

Every element of \(\mathcal W_n\) has length \(\sum _i n_i\), so retaining or forgetting this fixed length gives a bijection with the finite set counted in Lemma 27.12.1.2. Its cardinality is the multinomial coefficient, which is positive.

Every word in \(C_n\) has length \(m\). Sum its common singleton mass over the finite set \(C_n\). The multinomial identity \(|C_n|\prod _i n_i!=m!\) cancels the factorial in the word weight. The remaining factors satisfy \(\prod _i e^{-t}=e^{-Nt}\) and \(\prod _i t^{n_i}=t^m\). All factorials are positive, so this argument also applies at \(t=0\).

The set \(\mathbb {N}^I\) is countable. Formula (29) equates the singleton masses of the count law and the product of the Poisson laws, so the two measures agree. Coordinate projection gives each marginal law. Equality of the joint law with the product of its marginals proves mutual independence.

A finite set is the disjoint union of its singletons. Summing their masses proves the first identity. For every measurable \(A\), apply the same identity to \(A\cap S\) to obtain \(\mu (A\cap S)=c|A\cap S|\). Equality on measurable sets proves the restriction identity.

Positive event mass implies \(S\ne \varnothing \) and \(c{\gt}0\). Finiteness of \(\mu \) implies \(c{\lt}\infty \), and finiteness of \(S\) implies \(|S|{\lt}\infty \). Substituting (30) in the conditional measure therefore permits cancellation of \(c\). The assumption \(\mu (S)\ne 0\) is essential for this cancellation.

The event \(C_n\) is finite. Every word in it has length \(m=\sum _i n_i\) and the same singleton mass \(e^{-Nt}t^m/m!\), where \(N=|I|\). Theorem 27.14.2 applies to this finite probability measure whenever the event mass is nonzero. For \(t{\gt}0\), that common mass is strictly positive, and \(C_n\) is nonempty. Consequently \(\mu _t(C_n)=|C_n|e^{-Nt}t^m/m!{\gt}0\).

At zero time \(\mu _0\) is the point mass at the empty word, whose count vector is zero. Thus \(C_n\) has zero mass when \(n\ne 0\); the definition of conditioning then gives the zero measure. A word with zero total count has length zero.

Fix a word \(w\). In (34), only \(n=C(w)\) can contribute to its singleton mass: all other uniform measures are supported on disjoint fibers. The count law gives \(\nu _t(\{ n\} )=\mu _t(C_n)\). If this mass is positive, (32) shows that its contribution is \(\mu _t(C_n)\mu _t(\{ w\} \mid C_n)=\mu _t(\{ w\} )\). If the fiber has zero mass, both the contribution and \(\mu _t(\{ w\} )\) vanish. Thus the two measures agree on every singleton of the countable word space, proving equality and normalization.

Induct on the two prescribed words. When one is empty, every letter must carry the other tag, so the remaining filtered word uniquely determines \(w\). When both are nonempty, the first tag of \(w\) determines which recursive case applies. Removing that first letter reduces the assertion to the corresponding shorter pair of words.

Induct on the two prescribed words. Prepending a fixed letter preserves distinctness, and the two nonempty recursive cases have different first tags. Thus each recursive step preserves the absence of repeated words. If either word is empty, there is one interleaving. Otherwise the number of interleavings is the sum of the two shorter counts, which equals \(\binom {|u|+|v|}{|u|}\) by Pascal’s identity.

Each letter contributes one to exactly one of the two filtered lengths. Induction on \(w\) gives the first identity. Substitute the two prescribed filtered words to obtain the consequence.

The words in \(E(u,v)\) are precisely the tagged interleavings of \(u\) and \(v\). Their duplicate-free enumeration has \(\binom {|u|+|v|}{|u|}\) members. The two filters partition the positions of each member.

Sum the singleton masses over \(E(u,v)\). The factorial identity \(\binom {m+n}{m}m!n!=(m+n)!\), exponential addition, and \(t^{m+n}=t^mt^n\) give (36). Only positive factorials are canceled, so the calculation also applies at \(t=0\).

The set of pairs of finite words is countable. Equality of its singleton masses in (36) proves (37). The coordinate projections give the two marginals, and the product joint law gives independence.

The inverse image of a singleton \(\{ v\} \) is the singleton \(\{ (e^{\mathcal W})^{-1}(v)\} \). Its word has length \(|v|\), and \(|I|=|J|\), so the two singleton masses are both \(e^{-|J|t}t^{|v|}/|v|!\). Equality on singletons proves equality of the measures on the countable word space. Every map between these discrete spaces is measurable.

For each letter \(i\), the left filter retains \(c(i)\) exactly when \(p(i)\) holds, and the right filter retains it exactly when \(\neg p(i)\) holds. In either case the inclusion into \(I\) returns \(i\). Applying this identity at every position proves both equalities, preserving the original order.

Put \(H_j=A(w_{\geq j})R_j\). The endpoint identities are \(H_0=A(w)x\) and \(H_m=A(w_S)x\). If \(p(i_j)\) holds, then \(R_{j+1}=E_{i_j}R_j\) and \(H_j-H_{j+1}=0\). Otherwise, \(R_{j+1}=R_j\), and linearity gives

Summing these consecutive differences proves the identity.

Every chronological composition of the maps is contractive: the inequality \(\| y\| \leq \| x\| \) is preserved at each step starting from \(x\). Apply the triangle inequality to the telescoping sum, and then use contractivity of each full suffix on its corresponding defect.

Relabeling gives \((c^{\mathcal W})_*\mu _{I,t}=\mu _{S\sqcup T,t}\). Applying the joint filtered-word law to this measure gives

Both factors have mass one, so coordinate projections give the stated marginals. Their product is the joint law, which proves independence.

Apply the prefix insertion identity to \(G(u,i)=F(R_p(u),i)\). For each insertion time \(s\), the retained marginal of a word with law \(\mu _{I,s}\) is \(\mu _{S,s}\) by subset thinning. Transporting the inner integral along \(R_p\) gives the asserted formula. No independence assumption about the different random prefixes is needed.

Replace \(F(u,i)\) by zero for \(i\in S\) and leave it unchanged otherwise in (47).

Set \(F=1\) in (48). Each retained-word measure has mass one, leaving \(|I\setminus S|\) integrated over the time interval.

The previously established nonnegative first moment is \(|I|T{\lt}\infty \). The length is measurable because the word space is discrete. The finite nonnegative integral gives integrability, and conversion to a real integral gives the displayed value.

Dominate \(f\) by the integrable constant \(C\) under the probability measure. For measurability, expand its nonnegative expectation into

Each atomic time weight is continuous, so this countable nonnegative sum is measurable. Its finite value converts to the real expectation.

Each expectation is measurable and lies in \([0,C]\). The sum is bounded by \(|A|C\) on an interval of finite volume.

Pointwise, the marked-prefix sum lies between zero and \(C|w|\). Lemma 27.21.1 proves its integrability. Apply the nonnegative prefix identity to \(\operatorname {ofReal}\circ F\). Finite-sum compatibility and the integrability of both sides justify conversion to the asserted real equality.

The composed kernel \(F(R_p(u),i)\) obeys the same uniform bound, so its marked sum is integrable by length domination. Convert the existing nonnegative retained-prefix identity using the integrability just proved. That identity already derives the retained law from actual subset thinning.

Set the kernel to zero on retained labels and apply the preceding theorem. For time integrability use the finite subset \(A=I\setminus S\) in Lemma 27.21.3.

Contractivity bounds both the discrepancy and each one-map defect by \(2\| x\| \). The discrepancy is therefore integrable under the probability law. The omitted-position sum is dominated by \(2\| x\| |w|\), and hence is integrable by the finite first moment of word length. Integrate the pointwise norm bound and apply the omitted-prefix occupation formula to \(F(u,i)=\| E_iR(u)-R(u)\| \).

Taking the first \(j+1\) letters amounts to taking the first \(j\) and appending the letter at position \(j\). Substituting this equality into the sum gives successive differences \(f(w_{{\lt}j+1})-f(w_{{\lt}j})\). They telescope between \(w_{{\lt}0}=\varnothing \) and \(w_{{\lt}|w|}=w\).

Every function on the discrete word space is measurable. Domination by the integrable constant \(C\) gives integrability. Write \(f=f^+-f^-\), where \(f^+=\max (f,0)\) and \(f^-=\max (-f,0)\). Both parts take values in \([0,C]\) and have measurable expectations by Lemma 27.21.2. Their difference is the expectation of \(f\) and is therefore measurable.

The triangle inequality gives \(|\sum _{j{\lt}|w|}F(w_{{\lt}j},w_j)|\leq C|w|\). The length is integrable by Lemma 27.21.1, so the marked-prefix sum is integrable. For each \(i\), the function \(F(\cdot ,i)\) is integrable and has a measurable expectation by Lemma 27.22.2. Since \(\mu _{I,s}\) has mass one, its expectation has absolute value at most \(C\). It is therefore integrable on an interval of finite length; summing over the finite alphabet proves time integrability.

Set \(F^+=\max (F,0)\) and \(F^-=\max (-F,0)\), so that \(F=F^+-F^-\) and \(0\leq F^\pm \leq C\). The bounded nonnegative word kernels, marked-prefix sums, and time integrands are integrable. Thus subtraction commutes with each finite sum and integral in (57). Apply Theorem 27.21.4 to \(F^+\) and \(F^-\) and subtract the two equalities.

The increment kernel \(F(u,i)=f(u[i])-f(u)\) satisfies \(|F(u,i)|\leq |f(u[i])|+|f(u)|\leq 2C\). Apply (57) to \(F\). By (55), its marked-prefix sum equals \(f(w)-f(\varnothing )\) for every word. The integrability of \(f\) and the unit mass of \(\mu _{I,T}\) turn this expectation into \(\int f\, d\mu _{I,T}-f(\varnothing )\).

Sum the increment bounds along the actual prefixes of a fixed word. The telescope (55) gives \(f(w)-f(\varnothing )\leq \sum _{j{\lt}|w|}G(w_{{\lt}j},w_j)\). Both sides are integrable: \(f\) is bounded, and the right side is at most \(D|w|\). Integrate this inequality, apply the real marked-prefix identity (51) to \(G\), and add \(f(\varnothing )\).

Every word of length \(m+1\) has a unique prefix of length \(m\) and a unique last letter. Thus \((u,i)\mapsto u[i]\) is a bijection from \(\mathcal W_{I,m}\times I\) to \(\mathcal W_{I,m+1}\). Reindexing the finite sum by this bijection gives (60).

Rearranging (61) gives \(\sum _{i\in I}f(u[i],y)\leq Nf(u,y)+\sum _{z\in Y}A(y,z)f(u,z)\). Sum over \(u\in \mathcal W_{I,m}\) and apply (60). Interchanging the two finite sums on the right gives (62).

Induct on \(m\), proving the bound for all \(y\) at once. For \(m=0\), it is the initial bound in (63). For the induction step, set \(q_m=(N+v)^m C\geq 0\). By (62), the induction hypothesis, and the nonnegativity of \(N\) and \(A\),

The last inequality uses (63) and \(q_m\geq 0\). No division by a profile coordinate occurs.

Theorem 27.23.3 gives \(S_m(y)\leq (N+v)^m C r(y)\). Apply Theorem 27.9.4 to \(f(\cdot ,y)\) with \(a=N+v\geq 0\) and the nonnegative constant \(C r(y)\). Its comparison series, with the actual Poisson word weights, is

This summable majorant proves integrability as well as (66).

Fix a word \(u\) and put \(m=|u|\). All terms in \(S_m(y)\) are nonnegative, and \(f(u,y)\) is one of them. By Theorem 27.23.3,

Thus vanishing holds at every word, independently of any choice of time.

Monotonicity and subadditivity of the nonnegative real power give \(x^\alpha \leq (d+z)^\alpha \leq d^\alpha +z^\alpha \). Multiplication by \(a\geq 0\) and rearrangement give \(-az^\alpha \leq ad^\alpha -ax^\alpha \). Apply the increasing exponential function and use \(e^{p+q}=e^pe^q\).

The infinite-distance branch is zero; the other branch is a positive real exponential.

The infinite-distance branch is zero, and a real exponential never vanishes.

Use the finite-distance branch of the definition.

Membership gives \(D_S(y)=0\). Since \(\alpha {\gt}0\), one has \(0^\alpha =0\), so the profile is \(e^0=1\).

For \(y\in S\), both sides equal one by Lemma 27.24.6. For \(y\notin S\), the left side is zero and the right side is nonnegative by Lemma 27.24.3.

The extended distance to the empty set is infinite, so the infinite-distance branch applies.

If \(D_S(z)=\infty \), the left side is zero by Lemma 27.24.4, and the right side is nonnegative by Lemma 27.24.3. Otherwise the extended triangle inequality gives

All three quantities now have their ordinary finite real values. Apply Lemma 27.24.1 to the three scalar arguments \(D_S(y)_{\mathbb R}\), \(\delta (y,z)_{\mathbb R}\), and \(D_S(z)_{\mathbb R}\), and identify the two profiles using Lemma 27.24.5.

Write \(r=r_{S,a,\alpha }\). For a pair at infinite distance, the corresponding kernel entry is zero. At finite pair distance, multiply Lemma 27.24.9 by \(A(y,z)\geq 0\). Summing both cases and factoring out \(r(y)\) gives

The final inequality uses the full-row bound and \(r(y)\geq 0\) from Lemma 27.24.3.

Set \(r=r_{S,a,\alpha }\). Lemma 27.24.3 gives \(r\geq 0\). Since \(C\geq 0\), Lemma 27.24.7 and the initial bound imply

Theorem 27.24.10 supplies \(\sum _{z\in Y}A(y,z)r(z)\leq v r(y)\), including coordinates at infinite distance from \(S\). These are precisely the profile hypotheses of Theorem 27.23.4. Apply it with the unchanged summed append-increment bound in (80) to obtain integrability and (81).

Use \(r=r_{S,a,\alpha }\). As in (82), indicator domination and \(C\geq 0\) give \(f(\varnothing ,z)\leq Cr(z)\) at every coordinate \(z\). Theorem 27.24.10 gives the kernel supersolution bound, while Lemma 27.24.4 gives \(r(y)=0\). Apply Theorem 27.23.5 with the original summed append-increment hypothesis. The empty-set conclusion follows because \(D_{\varnothing }(y)=\infty \) for every \(y\).

The exponential \(e^{-ct}\) decays faster than the power \(t^{(-p-2)/\alpha }\) as \(t\to \infty \). Substitute \(t=n^\alpha \); since \(\alpha {\gt}0\), this tends to infinity and gives \(e^{-cn^\alpha }\leq n^{-p-2}\) for all sufficiently large \(n\). For these \(n\), also require \(n\geq 1\). Then

The summands are nonnegative, and \(\sum _{n\geq 1}n^{-2}\) converges. Comparison outside a finite initial segment proves the claim.

The series is summable by Theorem 27.26.1 and has nonnegative summands. Its \(n=0\) term is exactly one, because \(0^\alpha =0\) for \(\alpha {\gt}0\).

Both the full-rate and half-rate series are summable by Theorem 27.26.1. For every natural number \(n\geq r\), monotonicity of nonnegative real powers gives \(r^\alpha \leq n^\alpha \). Splitting the exponential rate yields

Sum over the natural numbers with \(r\leq n\), enlarge to all natural numbers using nonnegativity, and take the constant \(e^{-(c/2)r^\alpha }\) outside the sum.

Every summand in (88) is zero or one.

The extended triangle inequality gives \(\delta (y,z)\leq \delta (y,x)+\delta (z,x)\leq r+r{\lt}\infty \). This establishes finite distance before passing to real values. The resulting real inequality is \(\delta (y,z)_{\mathbb R}\leq 2r\).

By Lemma 27.27.3, no finite-radius anchor ball can contain both \(y\) and \(z\). Every indicator in (88) is therefore zero.

For each \(i\), the indicator that both sites belong to \(N_r(x_i)\) is at most the indicator that \(y\) belongs to \(N_r(x_i)\). Sum over \(I\); the latter sum is the displayed cardinality.

Put \(E=e^{a(2r)^\alpha }{\gt}0\). For \(y,z\in N_r(x_i)\), Lemma 27.27.3 first gives finite pair distance and then \(\delta (y,z)_{\mathbb R}\leq 2r\). Since \(a,\alpha \geq 0\), it follows that \(e^{a(\delta (y,z)_{\mathbb R})^\alpha }\leq E\). Exchange the two finite sums and count sites inside each ball:

The last two inequalities use (91) and \(V,E\geq 0\).

Each summand is nonnegative by Lemma 27.27.2 and positivity of the exponential. Its infinite sum is nonnegative as well.

Lemma 27.27.4 gives \(K_n(y,z)=0\) at every natural radius \(n\). The series is identically zero.

The summands are nonnegative by Lemma 27.27.2. Lemma 27.27.5 and (95) give, for every \(n\),

The right side is summable by Theorem 27.26.1 with \(p=2\), followed by multiplication by \(CB\). Comparison of norms proves convergence. In particular, this is absolute convergence of the actual shell series.

Since \(a2^\alpha \geq 0\), the hypothesis \(\eta {\gt}0\) implies \(c{\gt}0\). Theorem 27.28.4 proves convergence for each entry. Multiplying each entry’s summands by the fixed nonnegative weight \(e^{a(\delta (y,z)_{\mathbb R})^\alpha }\) preserves convergence. Since \(Y\) is finite, the resulting finite sum of convergent series is convergent, and its finite site sum may be exchanged with the infinite radius sum.

At radius \(n\), apply Theorem 27.27.6 with the label and site bounds \(B(1+n)^2\) and \(V(1+n)^2\) from (97). Multiplication by \(C e^{-cn^\alpha }\geq 0\) gives

The equality uses \((2n)^\alpha =2^\alpha n^\alpha \) for nonnegative bases, including \(n=0\). The majorant is summable by Theorem 27.26.1 with \(p=4\) and rate \(\eta {\gt}0\). Sum (99) over \(n\) using the established convergence, then factor out \(CBV\). This gives (98).

Since \(2^\alpha {\gt}0\) and \(c{\gt}0\), the chosen \(a\) is positive. Direct cancellation gives \(c-\bigl(c/(2\cdot 2^\alpha )\bigr)2^\alpha =c/2{\gt}0\). Apply Theorem 27.28.5 with this value of \(a\).

Apply Theorem 27.28.6 to (101). It gives \(a{\gt}0\) and, for every \(y\in Y\),

Every term of the series defining \(v\) is nonnegative, and \(C,B,V\geq 0\), so \(v\geq 0\). Lemma 27.28.2 gives \(A_{C,c,\alpha }(y,z)\geq 0\), and Lemma 27.28.3 gives \(A_{C,c,\alpha }(y,z)=0\) whenever \(\delta (y,z)=\infty \). Together with (102) and (105), these facts satisfy the hypotheses of Theorem 27.25.1, with initial amplitude \(M\). That theorem gives integrability and (104) at every site. The same hypotheses give vanishing at every word when \(D_S(y)=\infty \) by Theorem 27.25.2.

Functional calculus gives \(G_i\Omega =\Omega \) and \(G_i^2=I-k_i\). Since \(G_i\) is self-adjoint, it preserves \(\Omega ^\perp \). For \(v\perp \Omega \),

The operator order makes each \(k_i\) self-adjoint, so the quadratic forms are real. Summing gives

Appending the last letter gives \(A_{m+1}(v)=\sum _{w\in \mathcal I^m}\sum _i\| G_iD_wv\| ^2\). Apply (109) to each excited vector \(D_wv\) to obtain (107). For \(g\le N\), induction from \(A_0(v)=\| v\| ^2\) proves the power bound. For \(g{\gt}N\), nonnegativity of the left side of (109) forces \(\| v\| ^2=0\).

Induct on the length of \(v\), appending its last letter \(i\). The empty word gives \(D_{u\varnothing }=D_u=ID_u\). For the induction step, \(D_{u(vi)}=G_iD_{uv}=G_i(D_vD_u)=(G_iD_v)D_u=D_{vi}D_u\). Thus the factors of the later word multiply on the left, as in [ Ope26b ] , 09-amplification.tex, lines 49–54.

The matrix entries are \((A\otimes I)_{(i,r),(j,s)}=A_{ij}\delta _{rs}\). Summing over \(s\) gives the first equality. Interchanging the two finite coordinate sums gives the norm identities.

Use \((AB)\otimes I=(A\otimes I)(B\otimes I)\) for the recurrence. Apply (111) to each word and interchange the word and spectator sums. At length zero the word product is \(I\), and every product sends zero to zero.

The finite-word estimate applies to each \(\xi _r\), so \(A_m^{\mathcal K}(\xi )=\sum _r A_m(\xi _r) \le (N-g)^m\sum _r\| \xi _r\| ^2=(N-g)^m\| \xi \| ^2\).

The rank-one action gives \(\langle \Omega ,Qv\rangle =\langle \Omega ,v\rangle (1-\| \Omega \| ^2)=0\). Each slice of \((Q\otimes I)\zeta \) equals \(Q\zeta _r\). Apply (114).

For each excited slice \(v=\xi _r\), the one-step inequality gives \(0\le \sum _i\| \sqrt{I-k_i}v\| ^2\le (N-g)\| v\| ^2\). Since \(g{\gt}N\), this forces \(v=0\); this argument uses no condition on \(k_i\Omega \). Thus every joint coordinate is zero. For the projected statement, normalization gives the required slice orthogonality.

Suppose first that \(g\le N\). The finite-word bound gives \(\sum _{|w|=m}F_\xi (w)\le (N-g)^m\| \xi \| ^2\). The weighted summation theorem proves integrability and gives

If \(g{\gt}N\), the one-step gap inequality forces \(\xi =0\), so both sides vanish. The complementary ground projection puts every physical coordinate slice in \(\Omega ^\perp \), proving the last assertion.

The left side of the assumed inequality and the coefficient of \(r\) are positive. Thus \(r{\gt}0\). Taking logarithms, expanding the products and powers, and dividing by \(k\) gives the stated leading terms with remainder

Use \(\log (k+1)\le \log (k+2)\) and \(2s\le 1\) to bound this remainder by \(R_{d,c,K}(k)\). This is the scalar step in [ Ope26b , 07-comparators.tex, lines 153–163 and 332–381 ] . In the physical comparison, \(r\) is the squared inverse-power norm of the original auxiliary-label prevector; the assumed overlap inequality must be obtained for that same vector.

Both \(1/k\) and \(\log (k+2)/k\) tend to zero. The coefficients \(\log 4\) and \(d^2+cK\) are fixed. This gives the vanishing error used in the replica limit in [ Ope26b , 07-comparators.tex, lines 659–667 ] .