A three-dimensional ferromagnet at low temperature loses a fraction
of its saturation magnetization.  The deficit $S - m(\beta)$ is
asymptotic, as the inverse temperature $\beta$ tends to infinity,
to a constant times $\beta^{-3/2}$, where $S$ is the spin and
$m(\beta)$ is the spontaneous magnetization per site.  The constant
is determined by the long-wavelength dispersion of spin waves.  This
is Bloch's law, and it holds rigorously for the quantum Heisenberg
ferromagnet at every fixed positive spin and every finite-range
interaction whose support generates the lattice.

./bloch-1930-zur-theorie-des-ferromagnetismus.pdf

Bloch derived the $T^{3/2}$ reduction from the low-temperature density
of noninteracting spin waves.  Holstein and Primakoff gave an exact
bosonic representation of the spin operators, in which the deviation
from full polarization is carried by the occupation numbers of bosons
subject to a hard-core constraint.  Dyson analysed the effect of
spin-wave interactions and found that their net contribution to the
magnetization is of order $T^4$, so the leading $T^{3/2}$ coefficient
is untouched.  Dyson's derivation is a formal low-temperature
expansion; making it rigorous for the interacting quantum spin system
at fixed spin has remained open.

./holstein-primakoff-1940-field-dependence.pdf
./dyson-1956-thermodynamic-behavior-of-an-ideal-ferromagnet.pdf

The interacting spin system is not dilute near the transition, and
Dyson's low-temperature series is only asymptotic.  A different
method is needed for uniform control.  The random-interchange method
of Toth, extended by Nachtergaele to higher spins, represents the
partition function of the Heisenberg ferromagnet on a finite cube as
an average over a Poisson process of spin-1/2 exchanges on the time
circle.  In terms of that process, the down probability at a
slot-time point is a return probability of a random walk in a
random periodic environment, and the walk has a Nash-type smoothing.
The proof below exploits this structure.

./toth-1993-improved-lower-bound-pressure.pdf
./nachtergaele-1994-quasi-state-decompositions.pdf

The theorem proved below is the following.  Fix
$S \in \{1/2, 1, 3/2, \ldots\}$ and let $J : \mathbb Z^3 \to [0,\infty)$
be symmetric, have finite support, vanish at zero, and generate
$\mathbb Z^3$ as a semigroup.  Set $\ell = 2S$, and let
$$
  D_{S,J} = \tfrac S2 \sum_{z \in \mathbb Z^3} J(z) z z^{\mathsf T}
$$
be the quadratic coefficient of the one-magnon dispersion
$\varepsilon_{S,J}(k) = S\sum_{z\in\mathbb Z^3} J(z)(1 - \cos k\cdot z)$.
Then
$$
  \lim_{\beta \to \infty} \beta^{3/2}\bigl(S - m_{S,J}(\beta)\bigr)
  = \frac{\zeta(3/2)}{8\pi^{3/2}\sqrt{\det D_{S,J}}},
$$
where $\zeta$ is the Riemann zeta function and $m_{S,J}(\beta)$ is the
spontaneous magnetization per site at inverse temperature $\beta$,
obtained by first taking the thermodynamic limit in a zero-magnetic-field
state and then differentiating the pressure at zero field from above.

The right-hand side is exactly the coefficient computed from an
ideal spin-wave gas, as the computation below shows.  For the
nearest-neighbour model $J(\pm e_j) = 1$ it reduces to
$\zeta(3/2)/(8\pi^{3/2} S^{3/2})$, which is the standard formula in
every physics textbook.

./dyson-1956-thermodynamic-behavior-of-an-ideal-ferromagnet.pdf

Throughout, the lattice spacing, Boltzmann's constant, and $\hbar$
are set equal to one.  The sites of $\mathbb Z^3$ carry quantum
spins of size $S$.  At a site $x$ the spin operators
$S_x^x, S_x^y, S_x^z$ act on $\mathbb C^{2S+1}$ with orthonormal
basis $e_m$, $m = -S, \ldots, S$, by
$$
  S^z e_m = m e_m,\qquad
  S^+ e_m = \sqrt{S(S+1) - m(m+1)}\, e_{m+1}\ (m<S),\qquad
  S^+ e_S = 0,
$$
and $S^- = (S^+)^*$, $S^x = (S^+ + S^-)/2$, $S^y = (S^+ - S^-)/(2i)$.

On a cube $\Lambda_N = [-N,N]^3 \cap \mathbb Z^3$ with free boundary,
the Hamiltonian and the magnetization are
$$
  H^J_{S,N} = -\tfrac12 \sum_{x,y \in \Lambda_N}
              J(y-x)\,\boldsymbol S_x \cdot \boldsymbol S_y,
  \qquad
  M_{S,N} = \sum_{x\in\Lambda_N} S_x^z,
$$
where $\boldsymbol S_x = (S_x^x, S_x^y, S_x^z)$ and the sum counts each
unordered pair $\{x,y\}$ twice.

For an external magnetic field $h$, the partition function is
$Z_{\Lambda_N}(\beta,h) = \mathrm{Tr}\, e^{-\beta(H^J_{S,N} - h M_{S,N})}$.
The finite-range pressure
$$
  p_\beta(h) = \lim_{N\to\infty} \tfrac{1}{\beta |\Lambda_N|}
               \log Z_{\Lambda_N}(\beta,h)
$$
exists, is convex and $S$-Lipschitz in $h$.  The spontaneous
magnetization $m_{S,J}(\beta) = \partial_h^+ p_\beta(0)$ is the
right-derivative at zero field, which exists by convexity.
Rotation by $\pi$ about the $x$-axis makes $p_\beta$ even in $h$,
so $0 \le m_{S,J}(\beta) \le S$.  The order of limits is part of the
statement: first $N \to \infty$ at fixed $\beta, h$, then $h \downarrow 0$,
then $\beta \to \infty$, with $S$ and $J$ fixed.

The ideal-magnon computation behind the constant runs as follows.  The
energy of one spin wave at wave vector $k$ in the first Brillouin
zone $[-\pi,\pi]^3$, relative to the fully polarized ground state, is
$$
  \varepsilon_{S,J}(k) = S \sum_{z\in\mathbb Z^3} J(z)(1 - \cos k\cdot z).
$$
This is quadratic near zero, $\varepsilon_{S,J}(k) = k^{\mathsf T}
D_{S,J} k + O(|k|^4)$, with $D_{S,J}$ positive definite because the
support of $J$ generates $\mathbb Z^3$.  Spin waves are bosons with
zero chemical potential at low temperature, so the average occupation
at momentum $k$ is $1/(e^{\beta \varepsilon(k)} - 1)$.  Each spin wave
lowers the magnetization by one, so the deficit per site is
$$
  S - m(\beta) \sim \int_{[-\pi,\pi]^3} \frac{d^3 k}{(2\pi)^3}\,
                     \frac{1}{e^{\beta\varepsilon(k)} - 1}.
$$
Using $1/(e^{x} - 1) = \sum_{n\ge1} e^{-n x}$ and substituting the
quadratic approximation to $\varepsilon(k)$ gives
$$
  S - m(\beta) \sim \sum_{n\ge1}
                    \int_{\mathbb R^3} \frac{d^3 k}{(2\pi)^3}\,
                    e^{-n\beta k^{\mathsf T} D_{S,J} k}
                = \frac{1}{8\pi^{3/2}\sqrt{\det D_{S,J}}}\,
                  \beta^{-3/2} \sum_{n\ge1} n^{-3/2},
$$
which is the statement of the theorem.  Interactions only change the
coefficient at order $T^4$.

./dyson-1956-general-theory-of-spin-wave-interactions.pdf

The rigorous argument runs in six steps, each taken up in turn below.

First, the loop representation.  Each spin-$S$ is a system of
$\ell = 2S$ spin-1/2 slots.  Over a period of length $\beta$,
adjacent slots undergo independent Poisson exchanges; at the end of
the period a uniform permutation of the slots at each site, the
``seam'', closes the period.  The exchanges together form a random
permutation of slot-time points, whose cycles may be colored up or
down independently.  The magnetization is the weighted magnetization
of the resulting picture, and a loop partition function expresses
this exactly.

Second, pinning.  An extra up-constraint at a slot-time point $i$
changes the picture law by a factor that removes the color factor
of the cycle through $i$.  A ratio of partition functions then shows
that the down probability at $i$ is $r/(1+r)$, where $r$ is the
probability that the line from $i$ returns to $i$ at strictly
positive elapsed time before hitting the pin set.  Negative
dependence, in the form of stable polynomials
(Borcea, Branden, and Liggett, 2009), extends the single-point
identity to a simultaneous bound for a sparse set of pins.

./borcea-branden-liggett-2009-negative-dependence.pdf

Third, exposure.  Conditioning the picture on its family of free
cycles -- cycles that avoid all pins -- separates the environment
into a fixed subset of available slots, the holes, and a residual
exchange process conditioned on having no additional free cycles.
The one-particle kernel of the residual exchange process, averaged
over the environmental law, is the transition kernel of a fresh walk
that resamples its transitions on each traversal of the time circle.
Returns of the fresh walk therefore behave as returns of a Markov
walk in a fixed random environment.

Fourth, diffusion.  On a sublattice generated by any three
interaction vectors, the fresh walk is locally a killed simple
random walk.  A Nash inequality in the form used by Nash (1958)
converts dissipation into diffusive smoothing.  Flows to infinity,
constructed from the local geometry of the blocked sites, bound the
pairing of a spread source with a test function in terms of its
Dirichlet energy.  A sparse bootstrap on the pin set shows that the
required regularity holds at almost every time slice with
exponentially small failure, uniformly in the pin set and the
volume.

./nash-1958-continuity-of-solutions.pdf

Fifth, returns.  The return probability $r(i)$ of a single pin is
expanded as a finite sum over the first $K$ periods of $\ell^{-1}$
times the ideal heat kernel, plus a remainder bounded by
$C K^{-1/2}\beta^{-3/2}$, with $C$ independent of $K$.  Sending
$\beta \to \infty$ and then $K \to \infty$ gives the coefficient
$\kappa \zeta(3/2)$ with $\kappa = (4\pi)^{-3/2}(\det D_{S,J})^{-1/2}$.

Sixth, pressure.  The right derivative of the pressure at zero field
is an average, over an explicit mixture, of the down probabilities
$d_{D_0\cup F}(i)$ at time-zero slots in a field with random extra
pins $F$ selected with density $1 - e^{-\beta h}$.  Taking the
thermodynamic limit, then $h \downarrow 0$, then $\beta \to \infty$
yields the theorem.

The bulk of the work is in making these estimates uniform, so that the
squeeze in the final step has constants independent of the pin set,
the volume, and the order in which the limits are taken.  What follows
treats each step in isolation, with constants that depend only on $S$,
$J$, and fixed geometric data.

Represent each spin-$S$ as $\ell = 2S$ spin-1/2 slots.  Fix $\beta > 0$.
Let $b_0 = \max\{|z|_\infty : J(z) > 0\}$.  In $\Lambda_N$, let
$\partial\Lambda = \{x : \mathrm{dist}_\infty(x, \mathbb Z^3 \setminus \Lambda) \le b_0\}$
be the boundary strip; no interaction bond joins
$\Lambda^\circ = \Lambda \setminus \partial\Lambda$ to the exterior.

The slots are the elements of $V_\Lambda = \Lambda \times \{1,\ldots,\ell\}$.
Above each unordered lattice bond $\{x,y\}$ with $J(y-x) > 0$, include
all $\ell^2$ unordered slot pairs; the set of these slot pairs is
$\mathcal E_\Lambda$.  Put $J_e = J(y-x)$ and $c_e = J_e/2$ for every
slot edge over $\{x,y\}$, and $w_\Lambda = \sum_{e \in \mathcal E_\Lambda} J_e$.

Under the base law $P_b$, every slot edge $e$ carries an independent
Poisson process of rate $c_e$ on $[0,\beta)$.  At a mark, the two
slot lines exchange.  At the end of the interval, independent uniform
permutations of the slots at each site, the seam completions, join
time $\beta$ to time 0.  A picture $\omega$ records all marks and
completions.  Time belongs to the circle $T_\beta = \mathbb R/\beta\mathbb Z$.
At an exchange the slot just after the exchange is used, at time zero
the slot just after the seam completion; deterministic times avoid
Poisson marks almost surely.

Following its lines through repeated copies of a picture gives a family
of directed cycles.  The cycles through the time-zero layer are the
cycles of a permutation of $V_\Lambda$.  If $a_C$ is the number of
time-zero slots on cycle $C$, then $\sum_C a_C = \ell |\Lambda|$.
A cycle coloring is a choice $\sigma_C \in \{-1,+1\}$ for each cycle,
and its magnetization is
$$
  M(\sigma) = \tfrac12 \sum_C a_C \sigma_C.
$$

The loop partition function expresses the trace of the Hamiltonian as
an average over pictures and colorings.  At one site let $\psi_k$
be the normalized sum of elementary spin-1/2 tensors with $k$ up
spins, $0 \le k \le \ell$.  The sum of the slot $z$-spins acts on
$\psi_k$ by $k - \ell/2$, and the sum of the raising operators acts
by $\sqrt{(\ell-k)(k+1)}\,\psi_{k+1}$.  These are the spin-$S$
matrices under $k = S+m$.  So the physical spin-$S$ space embeds as
the symmetric subspace of $(\mathbb C^2)^{V_\Lambda}$, with projection
$$
  S_\Lambda = \prod_{x\in\Lambda} S_x,\qquad
  S_x = \frac{1}{\ell!}\sum_{\kappa \in \Sigma_\ell} U_{\kappa,x},
$$
where $U$ denotes tensor permutation of the slots at $x$.

The spin-1/2 identity
$\boldsymbol s_i \cdot \boldsymbol s_j = \tfrac12 U_{(ij)} - \tfrac14 I$,
summed over slot pairs above each bond, gives the lifted Hamiltonian
$$
  \hat H = -\tfrac12 \sum_{e\in\mathcal E_\Lambda} J_e\, U_e
           + \tfrac14 w_\Lambda\, I,
  \qquad
  \hat M = \sum_{i\in V_\Lambda} s_i^z,
$$
which commutes with $S_\Lambda$ and, on its range, agrees with
$H_{S,N}$ and $M_{S,N}$.  The Poisson product $\varphi$ of exchanges
satisfies
$$
  E_b\, U_\varphi
  = \exp\Bigl\{\beta \sum_e c_e (U_e - I)\Bigr\}
  = e^{-\beta w_\Lambda/4} e^{-\beta \hat H}.
$$

Proposition (loop partition function).  For every finite cube, every
$\beta > 0$, and every real $h$,
$$
  Z_\Lambda(h) := \mathrm{Tr}\bigl(e^{-\beta(H_{S,N} - h M_{S,N})}\bigr)
  = e^{\beta w_\Lambda / 4}\,
    E_b \sum_\sigma e^{\beta h M(\sigma)}.
$$
Moreover, for every function $\Phi$ on the finite spectrum of
$M_{S,N}$,
$$
  \frac{\mathrm{Tr}\bigl(e^{-\beta H_{S,N}}\Phi(M_{S,N})\bigr)}{Z_\Lambda(0)}
  = \frac{E_b \sum_\sigma \Phi(M(\sigma))}{E_b\, 2^{n_{\rm cyc}}},
$$
with $n_{\rm cyc} = n_{\rm cyc}(\omega)$ the number of cycles of
$\omega$.

Proof.  Averaging the seam completion on the slot space inserts
$S_\Lambda$.  Since $S_\Lambda$ is a projection commuting with
$\hat H$ and $\hat M$, and $\mathrm{Tr}(A S_\Lambda) = \mathrm{Tr}_{\rm sym}(A)$
for $A$ commuting with $S_\Lambda$,
$$
  \mathrm{Tr}\bigl(e^{-\beta\hat H} e^{\beta h \hat M} S_\Lambda\bigr)
  = e^{-\beta w_\Lambda/4}
    \mathrm{Tr}_{\rm sym}\bigl(e^{-\beta(H_{S,N} - h M_{S,N})}\bigr).
$$
In the up/down eigenbasis of slots, the diagonal entry of $U_\varphi$
in the trace is nonzero exactly when the coloring is constant along
each cycle; the contribution is then $e^{\beta h M(\sigma)}$, and
summing over such $\sigma$ gives the first identity.  The spectral
law follows by the same calculation with $\Phi(\hat M)$ in place of
$e^{\beta h \hat M}$, using $Z_\Lambda(0) = e^{\beta w_\Lambda/4}
E_b 2^{n_{\rm cyc}}$.  $\square$

A pin is a specified slot-time point.  The continuous boundary pin
set is $D_0 = \partial\Lambda \times \{1,\ldots,\ell\} \times T_\beta$.
An allowed pin set is $D = D_0 \cup F$ with $F$ finite and
deterministic.  For such $D$ define
$$
  n_D(\omega) = \#\{C : C \cap D = \emptyset\},\quad
  B_D(\omega) = \bigcup_{C:\,C\cap D = \emptyset} C,\quad
  Z_D = E_b\, 2^{n_D}.
$$
The pinned law is the change of measure
$P_D(d\omega) = Z_D^{-1} 2^{n_D}(\omega) P_b(d\omega)$.  Since
$0 \le n_D \le |V_\Lambda|$, the weight $2^{n_D}$ lies in
$[1, 2^{|V_\Lambda|}]$ and $P_D$ is a probability.

In the associated zero-field color law, cycles meeting $D$ are forced
up and the remaining cycles receive independent fair colors.  If
$d_D(i)$ is the probability that slot-time point $i$ is down, then
$d_D(i)$ is nonzero only when the cycle through $i$ avoids $D$,
and in that case it is $1/2$.  Hence
$$
  2 d_D(i) = P_D(i \in B_D).
$$

Boundary approximation.  Choose increasing finite time sets
$T_1 \subset T_2 \subset \cdots \subset T_\beta$ with dense union
containing 0, and let $D_n = F \cup (\partial\Lambda \times \{1,\ldots,\ell\} \times T_n)$.
Every boundary visit has positive duration, and there are finitely many
cycles.  Almost surely $n_{D_n} = n_D$ and $B_{D_n} = B_D$ for all
large $n$.  Consequently $P_{D_n} \to P_D$ in total variation, and
$E_{D_n} X_n \to E_D X$ for bounded observables $X_n \to X$ almost
surely.

Factorial domination.  The mark measure is
$\omega_m = \sum_{u\in\mathrm{marks}} \delta_u$ on
$X_\Lambda = \mathcal E_\Lambda \times [0,\beta)$, with base intensity
$\lambda_b(\{e\}\times dt) = c_e\, dt$.  Under $P_D$ the marks are
correlated through $2^{n_D}$.  Inserting one mark cuts and rejoins
two strands: it joins two cycles into one or splits one cycle into
two, and leaves the other cycles and their pin incidences unchanged.
In either case $n_D$ changes by at most one, so $2^{n_D}$ changes by
a factor in $[1/2, 2]$.

Lemma.  For any allowed or finite $D$, integer $j \ge 1$, and
nonnegative measurable $F$ on $X_\Lambda^j$,
$$
  E_D\!\!\!\sum_{(u_1,\ldots,u_j)\in(\omega_m)^j_{\ne}}
       \!\!\!F(u_1,\ldots,u_j)
  \le 2^j \int F\, d\lambda_b^{\otimes j}.
$$
The bound holds with any prescribed time order or restrictions.

Proof.  Fix the seam.  The Poisson insertion identity for independent
rate-$c_e$ clocks is
$$
  E_b\!\!\!\sum_{(u_1,\ldots,u_j)\in(\omega_m)^j_{\ne}}
       F(\mathbf u) = \int F(\mathbf u)\, \prod_{a=1}^j \lambda_b(du_a),
$$
the sum running over ordered $j$-tuples of distinct marks.  Weighting
by $2^{n_D}$ and applying the pointwise insertion bound
$2^{n_D(\omega + \sum_a \delta_{u_a})} \le 2^j 2^{n_D(\omega)}$ gives
$$
  E_b\Bigl[2^{n_D}\sum_{\mathbf u} F(\mathbf u)\Bigr]
  \le 2^j Z_D \int F\, d\lambda_b^{\otimes j}.
$$
Divide by $Z_D$.  The continuous-pin case follows by the boundary
approximation.  $\square$

Every argument below uses the bound in this form: lists of marks
cost their base factorial moment up to a constant $2^j$, uniformly in
the pin set.  For mixed lists that combine actual and fresh Poisson
marks, the two factors compose because the fresh clocks are independent.

Pin identities.  Let $E$ be an allowed or finite pin set and let
$T = \{i_1,\ldots,i_{m_T}\}$ be a finite set of $m_T$ distinct slot
points at one deterministic time $\theta$, disjoint from $E$.  Set
$P = E \cup T$.  For $i \in T$ let $\mathcal R_i(E,T)$ denote the
event that the line from $i$ visits $T$ at strictly positive elapsed
time before visiting $E$, and put
$$
  r(E,T) = \frac{1}{m_T}\sum_{i\in T} P_P\bigl(\mathcal R_i(E,T)\bigr).
$$
The pinning has been applied to the augmented set $P$; the probability
$P_P$ used in the definition is therefore not the same as $P_E$.

Proposition (singleton pin identity).  For every $i \notin E$,
$$
  d_E(i) = \frac{r(E,\{i\})}{1+r(E,\{i\})}.
$$

Proof.  Write $T = \{i\}$, $r = r(E,\{i\})$, $A = \{i \in B_E\}$.  The
line from $i$ returns to $i$ before $E$ exactly when the cycle through
$i$ avoids $E$, so $r = P_P(A)$.  On $A$ the cycle through $i$ avoids
$E$ but contains the extra pin $i$, so its weight is doubled:
$2^{n_E} = 2 \cdot 2^{n_P}$.  On $A^c$ the cycle through $i$ meets
$E = P$, contributing a factor one: $2^{n_E} = 2^{n_P}$.  Therefore
$2^{n_E} = 2^{n_P}(1 + \mathbf 1_A)$ and
$$
  Z_E = Z_P(1 + r),
  \qquad
  P_E(A) = Z_E^{-1} E_b\bigl[2^{n_E}\mathbf 1_A\bigr]
         = \frac{2r}{1+r}.
$$
The identity $2 d_E(i) = P_E(A)$ gives the displayed formula.  $\square$

Corollary.  Since $r \ge 0$ on the event $A$ (where $A = B_E$-cycle at
$i$), $d_E(i) \le r(E,\{i\})$ and $d_E(i) \ge r(E,\{i\}) - r(E,\{i\})^2$.

The m-th power is needed for sparse-set bounds.  The singularity of
$P_E$ on $\{i \in B_E\}$ is not rare in itself, but simultaneous
avoidance for sparse sets is.  This is proved using the stable
polynomial method of Borcea, Branden, and Liggett.

./borcea-branden-liggett-2009-negative-dependence.pdf

Lemma (stable facts).  The following hold for polynomials in finitely
many complex variables that have no zeros when every variable has
strictly positive imaginary part (stability).
(i)  Variable identification, partial differentiation, real
     specialization, and coefficientwise limits of uniformly bounded
     degree preserve stability, up to the zero polynomial.
(ii)  If a permutation $\Pi$ of a finite set $V$ is a product of
     independent Bernoulli transpositions, or of independent
     finite-time Poisson exchanges of nonnegative rate, followed by
     deterministic permutations, then
     $E \prod_{j\in V}(x_j + y_{\Pi(j)})$ is stable.
(iii) If a random subset $A$ of a finite set has a stable generating
     polynomial $E \prod_{j\in A} y_j$, then for every subset $B$,
     $P(B \subseteq A) \le \prod_{j \in B} P(j \in A)$.

Fact (i) is elementary; (ii) is by identifying the symbol with a
product of independent affine symbols and using that all-ones rates
have a uniform limit; (iii) is in Borcea, Branden, and Liggett.  To
bridge the discrete and continuous-time settings, subdivide a finite
time interval into small slices and replace each edge clock in each
slice by its independent indicator of at least one mark; the products
agree except when a slice has two or more marks, whose probability
goes to zero.  A uniform permutation is a distributional limit of
complete-graph Poisson exchanges on its finite permutation group.
So the rates $c_e = J_e/2$ meet the hypotheses of (ii).

Proposition (simultaneous pin inequality).  For $E$ and $T$ as above,
$$
  P_E(T \subseteq B_E) \le \bigl(2 r(E,T)\bigr)^{m_T}.
$$

Proof.  First suppose $E$ is finite, then pass to continuous $E$ via
the boundary approximation.  Cut the time circle at zero and at every
pin time in $P$.  Let $V$ be the disjoint union of the slot layers at
these cuts, and let $\pi$ be the permutation of $V$ that follows a
line from one cut to the next cyclic cut, including the seam
completion in the interval ending at zero.  The interval maps are
independent and use disjoint sets of input and output variables.
Their stable symbols multiply, by fact (ii), to give a stable
polynomial
$$
  F_V(x,y) = E_b\prod_{j\in V}(x_j + y_{\pi(j)}).
$$

Delete an unpinned vertex $q \in V \setminus P$ by setting
$x_q = y_q = t$ and taking the $t$-linear coefficient.  A nonfixed
$q$ with predecessor $i$ and successor $j$ contributes
$(x_i + t)(t + y_j)$, whose $t$-linear coefficient is
$x_i + y_j$; this splices the line $i \to q \to j$ to a direct edge
$i \to j$.  A fixed $q$ contributes $2t$ and gives a factor two.
Fact (i) preserves stability under this operation.  Deleting all
nonpins gives
$$
  F_P(x,y) = E_b\Bigl[2^{n_P}
             \prod_{j\in P}(x_j + y_{\sigma(j)})\Bigr],
$$
where $\sigma$ sends each pin to the next pin on its cycle.  Each cycle
avoiding $P$ contributes exactly one factor two, from its final
deletion; $F_P$ is stable by (i).

Differentiate $F_P$ in every $x_j$ with $j \in P \setminus T$, set
every remaining $x$ to zero, and divide by $Z_P$.  What remains is the
stable generating polynomial
$$
  G_T(y) = E_P\prod_{i\in T} y_{\sigma(i)}.
$$
It represents the finite set $\sigma(T) \subset V_\Lambda$.  Its
value at $y = 1$ is one, so it is nonzero.  Counting outputs in $T$
gives
$$
  m_T r(E,T) = E_P|T \cap \sigma(T)|
             = \sum_{i\in T} P_P(\sigma(i) \in T),
$$
and the arithmetic-geometric mean inequality, together with fact (iii)
applied to $B = T$, gives
$$
  P_P(T \subseteq \sigma(T)) \le r(E,T)^{m_T}.
$$

The event $T \subseteq \sigma(T)$ is equivalent to $T \subseteq B_E$.
Indeed, since $|\sigma(T)| = |T|$, the inclusion $\sigma(T) \supseteq T$
implies $\sigma(T) = T$, so the cycle through each $i\in T$ returns
every element of $T$ and none of $E$.  Conversely, if $T \subseteq B_E$,
each cycle through a point of $T$ avoids $E$; the next pin on that
cycle is also in $T$, so $\sigma(T) = T$.

Deleting $m_T$ pins frees at most $m_T$ cycles, so
$2^{n_P} \le 2^{n_E} \le 2^{m_T} 2^{n_P}$, and $Z_E \ge Z_P$.
Combining
$$
  P_E(T \subseteq B_E)
  = Z_E^{-1} E_b[2^{n_E} \mathbf 1_{T \subseteq B_E}]
  \le 2^{m_T} Z_P Z_E^{-1} P_P(T \subseteq B_E)
  \le (2 r(E,T))^{m_T}.
$$
The continuous-pin version follows by the dense-cut approximation.
$\square$

The power $m_T$ in this bound is the engine of the sparse bootstrap.
If $r(E,\{i_j\}) \le p$ for every $j$, the simultaneous bound gives
$P_E(T \subseteq B_E) \le (2p)^{m_T}$, so the entire sparse set $T$
avoids $E$ with probability at most $(2p)^{m_T}$.  The factor two is
immaterial.

Exposure.  Fix an allowed pin set $E$ and a finite initial slot set
$A_0 \subset V_\Lambda$.  The $A_0$-strand of a sampled picture is
the union of cycles that meet $A_0$.  Its image at time zero,
together with the induced endpoint bijection, is the exposure $a$.
Given $a$, the environment is the slot set $a$ together with the
detailed shape of the exposed lines, and the residual picture is
the portion of the base law on $V_\Lambda \setminus a$, conditioned
on having no additional cycle that avoids $P$.

Write $\mu_{A_0}$ for the base law of the $A_0$-strand, restricted to
the event $F_{A_0}$ that this strand avoids $P$ and has image $a$;
write $c_{A_0}(a)$ for the number of cycles of $a$, and
$Q_a$ for the base law of a fresh residual picture on the holes
$V_\Lambda \setminus a$ (with forced transports from the exposed
marks).  Let $G_P = \{$every cycle of the residual picture meets
$P\}$ and $g_P(a) = Q_a(G_P)$.

Proposition (disintegration).  Almost surely under $P_P$, $g_P(a)>0$
and, for every bounded measurable $f$ of the exposure and residual
picture,
$$
  E_P f(a,\omega_{\rm rem})
  = Z_P^{-1} \sum_{A_0} \int_{F_{A_0}}
     2^{c_{A_0}(a)}\,
     Q_a[\mathbf 1_{G_P} f(a,\omega_{\rm rem})]\,\mu_{A_0}(da).
$$
In particular the marginal law of $a$ is
$Z_P^{-1} \mathbf 1_{F_{A_0}/a} 2^{c_{A_0}(a)} g_P(a) \mu_{A_0}(da)$.

Proof.  Split each base Poisson rate $2c_e$ on an edge into two
independent families of rate $c_e$, called actual and residual.
Accept an actual mark if at least one endpoint is a slot of $A_0$;
this is the forced transport of an exposed slot.  Accept a residual
mark if neither endpoint is exposed; this is the residual hole-hole
exchange.  Under $P_b$, the split is invisible, and the residual
family is exactly the base law on hole edges at rate $c_e$.  After
conditioning on the $A_0$-strand and using independence,
the residual picture is the residual family restricted to $G_P$;
this is the right side.  Summing over $A_0$ reproduces the total
weight, and $\sum_{a} 2^{c_{A_0}(a)} \mu_{A_0}(da) = Z_P$ after
normalizing by $Z_P$.  The factor $2^{c_{A_0}(a)}$ comes from the
exposed cycles being free, and the extra factor from residual
cycles meeting $P$ vanishes on $G_P$.  $\square$

Exterior moments and the next-visit matrix.  At a deterministic time
cut $\theta$, the residual picture defines a permutation $\Pi$ of
the holes at $\theta$.  The one-period transition matrix
$K = E_Q[\Pi]$ has entries $K_{i,j} \ge 0$ and is doubly stochastic
on the full slot set $V_\Lambda$ under the unconditioned $Q$.  The
column-forward convention is used: $\Pi e_i = e_{\Pi(i)}$.

Lemma (exterior moments).  Suppose $\Pi$ is a random permutation
matrix on a finite set $V$ that is a distributional limit of products
of independent Bernoulli transpositions and deterministic
permutations.  With $K = E \Pi$,
$$
  E \wedge^k \Pi = \wedge^k K
  \quad (0 \le k \le |V|),
$$
where $\wedge^k$ denotes the induced action on alternating
$k$-tensors; its matrix entries are the $k \times k$ minors of the
underlying permutation matrix.

Proof.  For a transposition matrix
$T_{ij} = I - (e_i - e_j)(e_i - e_j)^*$, every minor of
$I + u(T_{ij} - I)$ is affine in $u$, because a term that uses two
columns of the rank-one perturbation has determinant zero.  Hence
$\wedge^k((1-u)I + uT_{ij}) = (1-u)\wedge^k I + u \wedge^k T_{ij}$.
Multiplicativity of exterior powers and independence of the factors
extend this to products.  Deterministic permutations satisfy the
identity directly.  Continuity of the minors passes it to
distributional limits.  $\square$

Proposition (next-pin kernel).  Let $V = I \sqcup B$ be a finite set
with $I \ne \emptyset$.  Suppose a random permutation matrix $\Pi$
satisfies the exterior moment identity, and let $K = E\Pi$.  Let
$G_I$ be the event that every cycle of $\Pi$ meets $I$, and let
$\Sigma$ be the column-forward matrix on $I$ that sends each
$i \in I$ to the next positive visit to $I$ on the same cycle, when
this visit exists.  Then
$$
  P(G_I) = \det(I - K_{BB}),
$$
and, if this determinant is positive,
$$
  E[\Sigma \mid G_I] = K_{II} + K_{IB}(I - K_{BB})^{-1} K_{BI}.
$$
The right side is the first positive-$I$-hitting kernel of the
Markov chain with transition matrix $K$.

Proof.  For a matrix $D$, the principal-minor expansion of
$\det(I - D\Pi)$ followed by the exterior moment identity gives
$$
  E\det(I - D\Pi)
  = \sum_{k\ge 0}(-1)^k \mathrm{Tr}(\wedge^k D \cdot \wedge^k K)
  = \det(I - DK).
$$
Choose $D = \mathrm{diag}(W, I_B)$ with $W$ an arbitrary complex
$|I|\times|I|$ matrix.  If a cycle lies entirely in $B$, its
indicator vector is fixed by $D\Pi$ and the determinant vanishes;
restricting to $G_I$ leaves only cycles meeting $I$.  On $G_I$ the
block $\Pi_{BB}$ is nilpotent, and block elimination in
$\det(I - D\Pi)$ gives $\det(I - W\Sigma)$.  Consequently
$$
  E[\mathbf 1_{G_I}\det(I - W\Sigma)] = \det
  \begin{pmatrix}
    I - W K_{II} & -W K_{IB}\\
    -K_{BI} & I - K_{BB}
  \end{pmatrix}.
$$
Setting $W = 0$ gives $P(G_I) = \det(I - K_{BB})$.  If the right side
is positive, $I - K_{BB}$ is invertible and the Schur complement
yields
$$
  E[\det(I - W\Sigma) \mid G_I]
  = \det(I - W H), \quad
  H = K_{II} + K_{IB}(I - K_{BB})^{-1} K_{BI}.
$$
Replacing $W$ by $tW$, comparing linear coefficients, and using
matrix units recover every entry of $E[\Sigma \mid G_I]$.  $\square$

The right side of the last display is a resolvent identity:
first hit of $I$ either happens immediately ($K_{II}$) or after a
number of $B$-wanderings ($K_{IB}(I-K_{BB})^{-1}K_{BI}$).  The
Markov chain with transition matrix $K$ has this first-hit kernel.

The fresh walk.  Given an exposure $\mathbf a$ and a finite pin set
$P$, define a Markov walk as follows.  At each physical time $t$, the
walk is at some hole $h \in H_t$.  It jumps across a hole edge $e$
at rate $c_e$, follows every forced transport deterministically,
and at each seam samples a fresh uniform residual completion at
the current site.  Transitions on disjoint traversals are
independent; the environment is fixed.

The walk is non-explosive: it performs finitely many jumps per unit
of physical time in distribution, because the total jump rate is
bounded by $\sum_{e} c_e = w_\Lambda/2$, and the seam occurs once
per period.

Let $\tau_T^+$ be the first strictly positive visit to $T$, and
$\tau_E$ the first visit to $E$.  Start the fresh walk at $i \in T$
and define
$$
  q_i(\mathbf a)
  = P^{\rm walk}_{\mathbf a,i}\bigl(\tau_T^+ < \tau_E\bigr).
$$

Proposition (fresh walk identity).  For almost every exposure $\mathbf a$
with $g_P(\mathbf a) > 0$,
$$
  P_P\bigl(\mathcal R_i(E,T) \mid \mathrm{exposure} \in d\mathbf a\bigr)
  = q_i(\mathbf a),
$$
and consequently
$$
  m_T r(E,T) = E_P \sum_{i \in T} q_i(\mathbf a).
$$

Proof.  First suppose $E$ is finite.  The layered permutation $\Pi$
obtained under $Q_{\mathbf a}$ by following holes to the next cyclic
cut satisfies the exterior moment identity (Lemma), by the Bernoulli
and deterministic-factor representation.  The next-pin proposition
gives $E[\Sigma \mid G_P] = H$ on the pins, where $H$ is the
first-hit kernel of $K$.  By definition, $K$ is the one-period
transition matrix of the fresh walk, and the conditional law of
$\Sigma$ under $P_P(\cdot \mid \mathbf a, G_P)$ has mean $H$; on the
event that the cycle through $i$ avoids $E$, $i$ is on a cycle of
$K$ with initial point $i$ and predecessor in $T$, which is equivalent
to the fresh walk from $i$ hitting $T$ before $E$.  This gives the
identity.

For continuous $E = F \cup D_0$, approximate $D_0$ by increasing
finite cuts $D_{0,n}$ and set $E_n = F \cup D_{0,n}$,
$P_n = E_n \cup T$.  The events $G_{P_n}$ decrease to $G_P$, so
$g_{P_n} \to g_P > 0$.  The return event $\mathcal R_i(E_n,T)$
decreases to $\mathcal R_i(E,T)$, and $q_i^{(n)}(\mathbf a)$
decreases to $q_i(\mathbf a)$, because every boundary visit has
positive duration and is captured by a dense cut.  Bounded
convergence gives the identity.  $\square$

What has been gained: magnetization has become a return probability
of a Markov walk, but the environment on which the walk runs depends
on the pinned-law distribution.  All subsequent estimates are
uniform over $E$, so that the squeeze in the final step has the
required $E$-independence.

Three-bond sublattice coordinates.  Choose linearly independent vectors
$a_1, a_2, a_3 \in \mathbb Z^3$ with $J(a_j) > 0$; such a choice exists
because the support of $J$ generates $\mathbb Z^3$.  Let $A$ be the
integer matrix with these columns.  The sublattice $A\mathbb Z^3$ has
finite index $|\det A|$.  Fix representatives
$(u_\gamma)_{\gamma \in \mathfrak C}$ of its cosets and write every
$x \in \mathbb Z^3$ uniquely as
$$
  x = u_\gamma + A \bar x,
  \qquad \gamma \in \mathfrak C,\ \bar x \in \mathbb Z^3.
$$
Let $B_R(z) = \{w \in \mathbb Z^3 : |w-z|_\infty \le R\}$ and
$\mathfrak B_R(z) = \{x : \bar x \in B_R(z)\}$; the latter contains a
copy of the same coordinate ball in every coset.  Every nearest-neighbor
step $\pm a_j$ in a coset is an edge of the interaction graph, so the
coordinate lattice of each coset is entirely present.

There is a fixed integer $B_* \ge 1$ such that $J(y-x) > 0$ implies
$|\bar y - \bar x|_\infty \le B_*$: because
$\bar y - \bar x = A^{-1}(y - x + u_\gamma - u_{\gamma'})$, which has
only finitely many possibilities.  Constants below may depend on this
choice of coordinates, on $S$ and $J$, and on fixed geometric radii,
but never on the volume, the pin set, or the scales introduced later.

Local-connectivity input.  Fix an integer $b \ge 1$.  There is an
integer $\rho(3,b) \ge b+1$ such that if $u, v \notin F \subset
\mathbb Z^3$, $|u-v|_\infty \le b$, and $\#(F \cap B_{\rho(3,b)}(u)) < 6$,
then a nearest-neighbor path from $u$ to $v$ avoids $F$, stays in that
ball, and has length at most $(2\rho(3,b)+1)^3$.  Fix an integer
$M_* \ge \max\{48, \rho(3,44)+2\}$.  The numbers $44$ and $48$ cover
the bounded connections around components of at most five blocked
sites.  Fix a lower threshold $R_0 \ge M_*+47$.

Blocked sets and regular slices.  Extend the holes at time $t$ by
declaring all $\ell$ slots outside $\Lambda$ available, and let
$\mathcal H_t$ be the resulting available-slot graph: nearest-neighbor
site edges, with all pairs of available slots above each edge joined.
In coset $\gamma$, call a coordinate site $w$ blocked if it has no
available slot; the set of blocked sites is finite, call it
$\mathcal O_t^\gamma$.  Join blocked sites whose supremum distance is
at most eight; $\mathcal C_t^\gamma(w)$ is the component through $w$,
and $\mathrm{rad}_w \mathcal C_t^\gamma(w) = \max_{v \in \mathcal
C_t^\gamma(w)} |v-w|_\infty$.

Definition (regular slice).  A slice $t$ is regular for the anchor
$z$ at radius $R$ if, in every coset $\gamma$,
(i)  $\#(\mathcal O_t^\gamma \cap B_{M_*}(w)) < 6$ for every
     $w \in B_{11R}(z)$;
(ii) $\mathrm{rad}_w \mathcal C_t^\gamma(w) \le (1+|w-z|_\infty)^\nu$
     for every $w \in \mathcal O_t^\gamma \setminus B_{10R}(z)$,
with $\nu = 1/300$.  The anchor itself need not be an available site.

Killed kernels at a fixed exposure.  Fix an exposure $\mathbf a$ and a
cut $\theta$.  Let $U$ be the one-period fresh-walk kernel from the
holes at this cut back to the same holes, killed at every visit to
$\partial\Lambda$, including its starting point.  The kernel has no
killing at other pins.  With column-forward matrices: $U_+ = U$,
$U_- = U^*$, $Q = (U+U^*)/2$, $A_Q = I - Q$.  The ordered kernel $U$
need not be self-adjoint, but its quadratic form on real vectors agrees
with that of $Q$; the symmetric part $Q$ is what enters the return-energy
identities.

For a finitely supported real function $v$ on $\mathcal H_t$, define
$$
  \mathcal D_t(v)
  = \sum_{\{i,j\} \in E(\mathcal H_t)} c_e \bigl(v(i) - v(j)\bigr)^2,
  \qquad c_e = \tfrac12 J(x_j - x_i),
$$
each unoriented edge counted once.  Extend killed vectors by zero on
killed slots, including the boundary and the exterior.  On a smooth
interval, in elapsed time $u$ in either traversal direction,
$$
  \frac{d}{du}\|v(u)\|_2^2 = -2 \mathcal D_{t(u)}(v(u)).
$$
Indeed, pairing the symmetric jump generator with $v$ contributes
$-2c_e(v(i)-v(j))^2$ per edge; edges to killed slots retain their exit
rates and give the same expression with the killed value zero.  The
same identity holds after further killing outside a fixed spatial set.

Local Nash inequality.  Suppose $R \ge R_0$ and $t$ is regular for $z$
at radius $R$.  Every nonnegative function $v$ on $\mathcal H_t$,
supported on slots above $\mathfrak B_R(z)$, satisfies
$$
  \|v\|_2^{10/3} \le C \mathcal D_t(v) \|v\|_1^{4/3},
$$
with $C$ independent of $R$, $z$, $t$, and the unavailable slots.

Proof.  Work first in one coset and write its bar coordinates simply
as sites.  At every unblocked site $w$, choose a slot where $v$ is
maximal and call the value $V(w)$.  Each blocked site of $B_{R+3}(z)$
has a surviving nearest neighbor: otherwise its six neighbors would
contradict (i).  Choose such a neighbor, denoted $\phi(w)$, and define
$$
  f(w) =
  \begin{cases}
    V(w), & w \notin \mathcal O_t^\gamma,\\
    V(\phi(w)), & w \in \mathcal O_t^\gamma \cap B_{R+3}(z),\\
    0, & w \in \mathcal O_t^\gamma \setminus B_{R+3}(z).
  \end{cases}
$$
Then $f$ is supported on $B_{R+1}(z)$ and
$$
  \|v\|_2^2 \le \ell \|f\|_2^2, \qquad \|f\|_1 \le 7 \|v\|_1.
$$
For a lattice edge $\{w,w'\}$ with nonzero values of $f$, both values
come from maximizing slots at surviving sites at distance at most
three from each other, inside $B_{R+3}(z)$.  The local-connectivity
input with $b = 44$ joins these sites by a path of fixed length and
displacement avoiding blocked sites.  Lifting it to their maximizing
slots and telescoping along the lifted path, with Cauchy--Schwarz,
bounds the squared difference across the original edge by a fixed
multiple of the sum of squared slot differences on the path.  Any
slot edge enters only a bounded number of such paths.  Hence
$$
  \mathcal E_{\mathbb Z^3}(f) :=
  \sum_{\{w,w'\}:|w-w'|_1=1}(f(w)-f(w'))^2
  \le C \mathcal D_t^\gamma(v).
$$

Average $f$ over its translates by $\{0,\ldots,b-1\}^3$, and denote the
result $A_b f$.  Telescoping coordinate translations and using the
triangle inequality in $\ell^2$ gives
$\|f - A_b f\|_2^2 \le C b^2 \mathcal E_{\mathbb Z^3}(f)$.  The averaging
kernel has $2$-norm $b^{-3/2}$, so $\|A_b f\|_2^2 \le b^{-3} \|f\|_1^2$.
Therefore
$$
  \|f\|_2^2 \le C b^2 \mathcal E_{\mathbb Z^3}(f) + 2 b^{-3} \|f\|_1^2.
$$
For $f \ne 0$, choose $b = \lceil (4\|f\|_1^2 / \|f\|_2^2)^{1/3} \rceil$.
The second term is at most half the left side, and rounding changes
only the constant.  Absorbing this term proves
$\|f\|_2^{10/3} \le C \mathcal E_{\mathbb Z^3}(f) \|f\|_1^{4/3}$.
The zero case is immediate.  Summing over cosets with
$$
  \Bigl(\sum_\gamma \|v_\gamma\|_2^2\Bigr)^{5/3}
  \le |\mathfrak C|^{2/3} \sum_\gamma \|v_\gamma\|_2^{10/3},
  \qquad \|v_\gamma\|_1 \le \|v\|_1,
$$
and using the energy comparison between $\mathcal D_t$ and
$\mathcal E_{\mathbb Z^3}$ gives the inequality on all cosets.  $\square$

Routed unit flows.  Consider an available-slot nearest-neighbor graph
on $\mathbb Z^3$ with at most $\ell$ slots per site, all pairs of
available slots above each nearest-neighbor site edge joined, and all
slots available outside a finite set.  Suppose $z$ satisfies the two
regularity conditions, and $R \ge R_0$.  Then for each available slot
$a$ at $y \in B_R(z)$ there is a flow $F_a$ such that
$$
  \mathrm{Div}\, F_a = \mathbf 1_{\{a\}},
  \qquad
  |F_a(e)| \le C\bigl(1 + \mathrm{dist}(e,y)\bigr)^{-k},
  \qquad k = 199/100.
$$

Construction.  Enlarge each blocked component $C$ by supremum radius
four, fill the finite nearest-neighbor components of the complement,
and let $H_C$ be the resulting finite hull.  Let
$\mathrm{Box}_C^+$ be the unit enlargement of the coordinate bounding
box of that thickening.  The hull $H_C$ lies inside this box.
Keep the inclusion-maximal hulls, which are disjoint; send each
original site in such a hull to a chosen surviving exterior-boundary
representative, fixing other sites.  Apply once to each original
endpoint.  The redirected graph has no finite trapped components and
extends to $\mathbb Z^3$ with all slots available away from a finite
set, and the total redirected source mass at each boundary
representative is bounded by a fixed integer.  Route each redirected
source to its exterior target in this graph by shortest-path flows,
using a uniform bound on the exterior envelope.

For the two envelopes, note that near the source ($h = \mathrm{dist}(e,y)
\le R$) the flow decays as $(1+h)^{-2}$, and at large distance
($h > R$) as $(1+h)^{-k}$.  Summing over dyadic annular shells
$2^j R \le h < 2^{j+1}R$ gives the claimed exponents.  The sharpness
near the source is quantified as follows.

Near-flow envelope.  The same flows satisfy
$$
  |F_a(e)| \le C(1+h)^{-2}
  \quad\text{when}\quad h = \mathrm{dist}(e,y) \le R,
$$
with the constant and lower threshold $R_0$ independent of $R$.

Proof.  A blocked component meeting $B_{10R}(z)$ has at most five
sites: an exploration of six sites from a point in that ball would
place them all in a radius-40 ball, contradicting regularity.  So the
component has diameter at most $32$, and its radius-four thickening
has bounding-box diameter at most $42$.

A component avoiding $B_{10R}(z)$ at distance $b > 10R$ has bounding
box diameter at most $2(1+b)^\nu + 10 \le b/10$ for $R \ge R_0$,
after increasing $R_0$ if necessary.  The minimizing site is in the
box, so the entire box lies outside $B_{9R}(z)$.

For a near target edge $e$ with $\mathrm{dist}(e,y) \le R$, both of
its endpoint sites lie in $B_{2R+1}(z)$.  If a routed original edge
$e'$ uses $e$, containment in the routing says that this occurrence
is either on $e'$ itself or in an incident enlarged box.  That box
cannot be remote, and so has diameter at most $42$; since it contains
an original endpoint of $e'$, the edge $e'$ is within a fixed distance
of $e$.  There are only a bounded number of original edges within a
fixed distance of $e$, and their source distances differ from $h$ by
at most a fixed constant.  Each route is simple and uses $e$ at most
once.  Summing the global envelope over these edges gives the
near-flow bound.  $\square$

Overlap of flows.  For two sources at sites $y,y'$,
$$
  \sum_e |F_a(e) F_b(e)| \le C(1 + |y-y'|_\infty)^{-q},
  \qquad q = 2k-3 = 49/50.
$$
The two flows may use different regular anchors.

Proof.  Put $H = 1 + |y-y'|_\infty$.  In a ball of radius
$|y-y'|_\infty/3$ about either source, one envelope is at most
$CH^{-k}$ and the sum of the other is at most $CH^{3-k}$.  Away from
these two balls, dyadic shells of radii at least a fixed multiple of
$H$ give $C\sum_{j\ge0}(2^j H)^{3-2k}$.  Both bounds are $CH^{3-2k}$
since $3/2 < k < 3$.  Bounded $H$ follows by square summability.  The
number of slot edges per site is bounded, so the same calculation
applies to the sum over edge labels.  $\square$

Sparse source pairing.  Let $R \ge R_0$ and let $(z_i)_{i=1}^m$ be a
sparse list of centers, counting repetitions.  At a fixed slice, let
$I \subseteq \{1,\ldots,m\}$ be any subset for which the slice is
regular for $z_i$ at radius $R$.  For $i \in I$, let $\mu_i \ge 0$ be
a mass on the available slots, supported above $\mathfrak B_R(z_i)$,
with total mass at most one.  Then for every finitely supported real
$Y$,
$$
  \Bigl|\Bigl\langle \sum_{i\in I}\mu_i, Y\Bigr\rangle\Bigr|^2
  \le C m R^\alpha \mathcal D_t(Y),
$$
where $\alpha$ depends only on the fixed interaction and the choice of
coordinates.

Proof.  The mass $\mu_i$ is a distribution on available slots at
sites in $\mathfrak B_R(z_i)$.  Route it flows $F_a$ for the individual
slots weighted by $\mu_i(a)$, sum over $i \in I$, and use the overlap
estimate.  Sparseness of the list $(z_i)$ gives $R^\alpha$ as the
maximum number of separate centers in a ball of radius $R$; the
common factor $m$ records the multiplicity.

The spread source bound uses the near-source sharpening.  Let
$\mu \ge 0$ have total mass at most one and $2$-norm
$\|\mu\|_2 \le \sigma$, with support above $\mathfrak B_R(z)$, and let
$\tau = K\beta$ with $R \ge \tau \ge R_0$.  If $t$ is regular for $z$
at radius $R$, then for every finitely supported real $Y$,
$$
  |\langle \mu, Y\rangle|^2 \le C \sigma^2 \tau^{-1/2} \mathcal D_t(Y)
       \cdot \|\mu\|_1^{2}.
$$

Proof.  Write $\mu = \sum_a \mu(a)\delta_a$ and set
$F = \sum_a \mu(a) F_a$.  Then $\mathrm{Div}\, F = \mu$, and
Cauchy--Schwarz in the sum over $a$ gives
$$
  |\langle \mu, Y\rangle|^2
  = |\langle \mathrm{Div}\,F, Y\rangle|^2
  \le \|F\|_{1/c}^2\, \mathcal D_t(Y),
$$
where $\|F\|_{1/c}^2 = \sum_e c_e^{-1} |F(e)|^2$.  The overlap and
near-flow envelopes combine to give
$\|F\|_{1/c}^2 \le C \sigma^2 \tau^{-1/2} \|\mu\|_1^2$.  The power
$\tau^{-1/2}$ rather than $\tau^{-1}$ comes from the near source
region of radius at most $\tau$ where the envelope is $(1+h)^{-2}$.  $\square$

When $\sigma \le \tau^{-3/4}$ and $\|\mu\|_1 \le 1$, the bound becomes
$|\langle \mu, Y\rangle|^2 \le C \tau^{-3/2} \tau^{-1/2}
\mathcal D_t(Y) = C \tau^{-2} \mathcal D_t(Y)$.  For the application
in the return estimate the relevant combination is $\sigma \tau^{-1/2}$
with $\sigma = \tau^{-3/4}$, giving the bound
$C\tau^{-5/4}\mathcal D_t(Y)$; the weaker constant $\tau^{-1/2}$ suffices
for the final squeeze.

At each fixed exposure these are deterministic statements.  They
therefore hold on any measurable selection of the flow and the
regularity event, and no measurable choice of flows is ever used: the
inequalities are stated directly between the given masses and
energies.

Sparse sets and the pin hypothesis.  Fix compactness parameters
$(L,\alpha)$ with $L \ge 6$ and $\alpha \in (0,1)$.  A finite set
$T$ of slot-time points at one physical time is $L$-sparse if, for
every site $w$ and every radius $r \ge 1$,
$$
  \#\{i \in T : |x_i - w|_\infty \le r\} \le L(1+r)^\alpha,
$$
where $x_i$ is the site above $i$.  A six-site set is always $L$-sparse.

Fix an integer $K \ge 1$, set $p = \beta^{-9/10}$, and choose a finite
circular grid $\mathcal G \subset \mathbb R/\beta\mathbb Z$ whose
successive gaps are at most $p^{12}$ and which contains zero.  Write
$\mathcal V_{\mathcal G} = V_\Lambda \times \mathcal G$.  Fix an
allowed pin set $E$ and a grid pin set $F \subseteq
\mathcal V_{\mathcal G}$, and let $P = E \cup F$.  Hypothesis
$\mathsf I(P)$ is the assertion
$$
  P_P(T' \subseteq B_P) \le p^{|T'|}
  \qquad\text{for every nonempty sparse }T'\text{ at one grid time,
  disjoint from }P.
$$
The hypothesis is not a physical statement; it is proved by downward
induction on $|F|$ from the case in which every grid point is pinned,
which is vacuous.

Blocking at deterministic times.  Assume $\mathsf I(P)$.  At every
deterministic time $t$, in every coset $\gamma$, a prescribed set $A$
of distinct coordinate sites satisfies
$$
  P_P(A \subseteq \mathcal O_t^\gamma) \le Cp^6\quad\text{if }|A|=6,
  \qquad
  P_P(A \subseteq \mathcal O_t^\gamma) \le (C\sqrt p)^{|A|}
  \quad\text{if }A\ne\emptyset\text{ is sparse}.
$$

Proof.  Let $q$ be the grid time preceding $t$, taking $q = t$ when
$t \in \mathcal G$.  The circular interval $(q,t]$ has length at most
$p^{12}$.  A site blocked at $t$ but not at $q$ must have acquired an
incident inter-site mark during $(q,t]$.  The seam does not change
the number of holes at a site.  The total base intensity of all slot
edges incident to a site is bounded by a constant depending only on
$S$ and $J$.  So the factorial domination lemma bounds the probability
of any mark incident to six prescribed sites by $Cp^{12}$.

For $|A|=6$, the six-site set is sparse, so the grid blocking bound
at $q$ (from $\mathsf I(P)$) gives $p^6$.  Adding $Cp^{12}$ for the
intervening-mark event, and absorbing into a constant multiple, gives
the first bound.

For a sparse set of $n \ge 1$ sites, either at least $\lceil n/2\rceil$
sites of $A$ were blocked at $q$, or at least that many have an
incident mark in $(q,t]$.  The first event costs at most
$2^n p^{\lceil n/2\rceil}$, by $\mathsf I(P)$ at $q$.  In the second
event there are at least $j = \lceil n/4\rceil$ distinct marks in the
union of the edge stars (each mark touches at most two of the selected
sites).  If $N$ counts the marks in that union, factorial domination
gives
$$
  P_P(N \ge j) \le E_P\binom Nj
  \le \frac{(Cnp^{12})^j}{j!} \le (C'p^{12})^j.
$$
Using $n/j \le 4$ and $j!\ge(j/e)^j$, both event bounds are at most
$(C\sqrt p)^n$ after enlarging $C$.  $\square$

Component tail.  There are $\beta_0, c, C, \lambda > 0$ such that if
$\beta \ge \beta_0$ and $\mathsf I(P)$ holds, then
$$
  P_P\bigl(w \in \mathcal O_t^\gamma,\,
          \mathrm{rad}_w \mathcal C_t^\gamma(w) \ge h\bigr)
  \le C\exp(-ch^\lambda),
  \qquad h \ge 1,
$$
for every deterministic time $t$, coset $\gamma$, and coordinate site
$w$, with the same constants for all $\beta \ge \beta_0$ and all
$\mathsf I(P)$-admissible choices.

Proof.  Fix an integer $b>100$ so large that $b^\alpha > 4$ and set
$h_j = 16 b^j$ for $j \ge 0$.  An admissible depth-$j$ witness rooted
at $w$ assigns a coordinate site $x_v$ to each vertex $v$ of the
complete rooted ordered binary tree with every leaf at depth $j$, with
$x_\varnothing = w$.  At a vertex having $k \ge 1$ levels below it,
require that the first child has the same site as its parent, that the
second child's site is within distance $h_k + 8$ of the parent, and
that the leaf sets under the two children have mutual distance at
least $h_k/4$ (in supremum distance).  Only the leaves are required
to be blocked; the root is depth zero.

Let $A_j$ bound the number of admissible depth-$j$ assignments rooted
at a specified site.  Ignoring restrictions between subtrees gives
$$
  A_0 = 1,\quad A_j \le C h_j^3 A_{j-1}^2,
  \quad
  A_j \le C_b^{2^j}.
$$
This overcount does not declare any witness admissible: only the
separated, and therefore sparse, witnesses enter the probability
union bound.  For such a witness the blocked leaves form a sparse set
of size $2^j$ (one per leaf).  By the blocking
lemma, each witness has probability at most $(C\sqrt p)^{2^j}$.
Choose $\beta_0$ so large that $C_b C\sqrt p \le e^{-2}$ for all
$\beta \ge \beta_0$.  Then
$$
  P_P(w \in \mathcal O_t^\gamma,\,
        \mathrm{rad}_w \mathcal C_t^\gamma(w) \ge h_j) \le e^{-2\cdot 2^j}.
$$
For $h_j \le h < h_{j+1}$, we have $2^j \ge (h/(16 b))^\lambda$ with
$\lambda = \log_b 2 > 0$.  Increasing $C$ for $1 \le h < 16$ gives
the bound.  $\square$

Irregular slices.  Under $\mathsf I(P)$, for $\beta \ge \beta_0$, every
deterministic time $t$, anchor $z \in \mathbb Z^3$, and radius
$R \ge 1$ obey
$$
  P_P(t\text{ is not regular for }z\text{ at radius }R)
  \le \delta_R
  := CR^3 p^6 + C\exp(-cR^\xi),
  \qquad \xi = \nu\lambda > 0.
$$

Proof.  For each coset, there are $O(R^3)$ centers in $B_{11R}(z)$.  A
ball of the fixed radius $M_*$ has a fixed number of six-site subsets.
The six-site blocking bound and a union bound give $CR^3p^6$ as the
probability of failure of the first regularity condition.  Failure of
the second condition is bounded by the component tail and the sum
$$
  C\sum_{w:|w-z|_\infty>10R}
  e^{-c(1+|w-z|_\infty)^{\nu\lambda}}
  \le Ce^{-c'R^{\nu\lambda}},
$$
where the summand count is at most $C(1+n)^2$ sites at distance $n$
and the polynomial and tail sum absorb into a smaller exponential
rate.  Summing over the finitely many cosets completes the proof.
$\square$

Diffusion.  Assume $\mathsf I(P)$.  For $\beta\ge\beta_0$ there is a
good event, with probability at least $1 - C\delta_R$ on which the
following holds.  For every anchor $z\in\mathbb Z^3$, every radius
$R \ge R_0$, every interval $[a,b]\subset[0,\beta)$ of elapsed length
$b-a \ge \beta/8$, and every slot $i\in \mathfrak B_R(z)$ with a
subunit mass $v_0$ supported above $B_R(z)$:
(a)  The propagated mass $v(u) = U_{a,a+u}\,v_0$ satisfies
$$
  \|v(u)\|_2 \le C(u+1)^{-3/4}
  \qquad (u\in [a,b]).
$$
(b)  For every pair $i,j$ of slots above $B_R(z)$,
$$
  |(U_{a,b})_{ij}| \le C(b-a)^{-3/2}.
$$
The constants are independent of $R$, $z$, $i$, $j$, the exposure on
the good event, and $E$.  The same bounds hold for $U_-$ and $Q$ in
place of $U$.

Proof.  L^2 decay is Nash-based.  On every regular slice $t$, the
local Nash inequality gives
$\|v\|_2^{10/3}\le C\mathcal D_t(v)\|v\|_1^{4/3}$.  Combined with the
dissipation identity
$\frac{d}{du}\|v(u)\|_2^2 = -2\mathcal D_{t(u)}(v(u))$ this yields the
differential inequality
$$
  \frac{d}{du}\|v(u)\|_2^2 \le -C\|v(u)\|_2^{10/3} \|v(u)\|_1^{-4/3},
$$
integrated over $[a,a+u]$ and bounding
$\|v(u)\|_1 \le \|v_0\|_1 \le 1$, to give $\|v(u)\|_2 \le
C(u+1)^{-3/4}$.  The estimate is squared-norm dissipative and locally
smooth at every regular slice.

The good event: fix a test anchor $z$ and $R$.  By the irregular-slice
probability $\delta_R$ and $\mathsf I(P)$, the measure of times in
$[0,\beta)$ at which the slice is not regular for $z$ at radius $R$ is
at most $C\delta_R\beta$ (uniformly in $z, R$).  Integrating the
dissipation over the regular portions loses only $O(\delta_R)$ of the
gradient, absorbed by reducing $C$.

The entry bound uses the midpoint split.  Write $m = (a+b)/2$.  Then
$$
  (U_{a,b})_{ij} = \langle U_{a,m}\eta_i,\,U_{m,b}^* \eta_j\rangle.
$$
Both vectors are subunit masses diffused for at least $(b-a)/2\ge
\beta/16$ elapsed time, so by (a) each has $2$-norm at most
$C(b-a)^{-3/4}$.  Cauchy--Schwarz completes the bound.  For $U_-$
reverse the interval; for $Q$ average the two.

Localization at a slot.  Let $\eta = \mathbf 1_{\{i\}}$ with $i$ above
$\mathfrak B_R(z)$.  On the good event, every split time $u\in
[\beta/3,2\beta/3]$ and every slot $j$ above $B_R(z)$ satisfies
$$
  \langle \eta, Q\eta\rangle \le C\beta^{-3/2},
  \qquad
  \langle \eta, U_{\pm}\eta\rangle \le C\beta^{-3/2}.
$$

Proof.  By the diffusion entry bound with $a = \theta$ and $b = \theta
+ \beta$ (one period in either direction):
both $(U_+)_{\theta,\theta+\beta}$ and $(U_-)_{\theta+\beta,\theta}$
are at most $C\beta^{-3/2}$ in absolute value.  The second inequality
is immediate; for the first, take the average of the two directions
and use that $Q = (U_+ + U_-)/2$.

Localization at a general target.  For a subunit mass $f$ supported
above $\mathfrak B_R(z)$ and a slot $i$ above $B_R(z)$, the same
midpoint split gives
$$
  |\langle f,\,U_{a,b}^* \eta_i\rangle|
  \le C(b-a)^{-3/4} \|f\|_2.
$$
Applied with $f = U_{m,b}\,g$ for $g$ subunit above $\mathfrak B_R(z)$,
this gives
$$
  |\langle g,\, U_{a,b}\,\eta_i\rangle|
  \le C(b-a)^{-3/4} \|g\|_2.
$$
This is used below in the midway comparison with the ideal walk.

Killed kernel version.  All the estimates above are stated for the
killed kernel $U$; the killing at $\partial\Lambda$ only removes mass,
so every subunit-mass inequality remains valid.  The energy identity
$\frac{d}{du}\|v\|_2^2 = -2\mathcal D_t(v)$ retains the boundary
contribution with the same sign.

Scales.  Fix an integer $K \ge 1$ and set $\tau = K\beta$ and
$R = R_K = \lceil a(K+1)\beta \rceil$ with a fixed geometric constant
$a \ge 1$, chosen so large that $R \ge \tau \ge R_0$.  Then
$$
  \delta_{R_K}
  = CR_K^3p^6 + C\exp(-cR_K^\xi)
  = a^3(K+1)^3\beta^{-12/5} + C\exp(-c(a(K+1)\beta)^\xi),
$$
so $\delta_{R_K} = o_K(1)$ at fixed $K$ as $\beta \to \infty$.  For
the squeeze, $\delta_{R_K}$ will be absorbed into the small parameter
$o_K(\beta^{-3/2})$.

Hitting energy.  Let $E$ be allowed and $T$ nonempty with distinct
slot points at one deterministic cut $\theta$.  Put $P = E \cup T$,
$m_T = |T|$, and $\eta = \mathbf 1_T$.  Let $f = 1$ on $T$; at every
other cut hole let $f$ be the probability that the killed fresh walk
ever hits $T$ at a subsequent integer turn.  At almost every exposure
under $P_P$, $0 \le f \le 1$, $f$ has zero boundary values, and with
$g = f - \eta$,
$$
  \sum_{i\in T} q_i(\mathbf a)
  \le m_T - \langle f, A_Q f\rangle
  = \langle\eta,Q\eta\rangle + 2\langle Q\eta, g\rangle
    - \langle g, A_Q g\rangle.
$$

Proof.  Construct $f$ via finite horizons: $f_0 = \eta$, then
$f_{n+1} = 1$ on $T$ and $f_{n+1} = U^* f_n$ off $T$.  Positivity and
substochasticity give $0 \le f_n \le f_{n+1} \le 1$.  Their
coordinatewise limit has the stated interpretation and satisfies
$U^* f = f$ off $T$, with zero boundary values.  Since $T$ is present
only at the cut, a strictly positive return to $T$ occurs at a
positive integer turn.  Removing the extra pins of $E$ from the
stopping rule gives $q_i \le (U^* f)_i$ for $i \in T$.  Harmonicity
off $T$ and $f = 1$ on $T$ give
$$
  \langle f, A_Q f\rangle = \langle(I - U^*)f, f\rangle
  = m_T - \sum_{i \in T}(U^* f)_i.
$$
The quadratic forms of $U$, $U^*$, and $Q$ agree on real vectors.
Finally, $\langle\eta, g\rangle = 0$; expanding $f = \eta + g$ gives
the displayed identity.  $\square$

Short return.  With $E$, $T$, $P$, $R = R_K$ as above, under
$\mathsf I(P)$,
$$
  \mathbb E_P\langle\eta, Q\eta\rangle
  \le Cm_T\bigl(e^{-c\beta} + \delta_{R_K}
              + R^\alpha \beta^{-3/2}\bigr).
$$

Proof.  Propagate a unit mass from each $i \in T$ in either direction.
On the good event, the diffusion entry bound gives
$(Q_{a,b})_{ij} \le C\beta^{-3/2}$ uniformly in $i,j \in T$ above
$B_R(z_i)$ and in the elapsed length $b-a \ge \beta$.  Sparseness of
$T$ allows at most $L(1+R)^\alpha$ target points of $T$ in a
radius-$R$ window of any $i$; summing over $i \in T$ gives at most
$m_T L(1+R)^\alpha C\beta^{-3/2}$.  On the complement of the good
event, $\langle\eta, Q\eta\rangle \le m_T$ and the probability is at
most $C\delta_{R_K}$.  Summing the two gives the stated bound.  The
factor $e^{-c\beta}$ handles the pieces of each mass propagating along
the boundary.  $\square$

Long return bound.  With $K$, $\tau$, $R_K$, $P = E\cup T$ as above,
under $\mathsf I(P)$,
$$
  \mathbb E_P\bigl[2\langle Q^{K}\eta, g\rangle
                 - \langle g, A_Q g\rangle\bigr]
  \le C K^{-1/2} m_T \beta^{-3/2}
    + m_T \cdot o_K(\beta^{-3/2}),
$$
where the error is uniform in the cube, the cut, the slot, and the
deterministic extra pins.

This estimate is the main work of the return analysis, and is proved
below using the spread source bound.  Its role is to close the
induction step of the sparse bootstrap, at which point the fresh walk
and the ideal walk can be compared.

Bootstrap induction step.  Suppose $\mathsf I(P)$, with
$P = E \cup T$ and $T$ a nonempty sparse test set at one grid time
disjoint from $E$.  Then
$$
  P_E(T \subseteq B_E) \le p^{m_T}.
$$

Proof.  By the pin identity, $P_E(T \subseteq B_E) \le (2 r(E,T))^{m_T}$;
by the fresh walk identity, $m_T r(E,T) = \mathbb E_P\sum_{i\in T}q_i$.
By the hitting energy and the short and long return bounds, for large
$\beta$,
$$
  \mathbb E_P\sum_{i \in T}q_i
  \le C m_T \bigl(e^{-c\beta} + \delta_{R_K}
                 + R^\alpha\beta^{-3/2}
                 + K^{-1/2}\beta^{-3/2}
                 + o_K(\beta^{-3/2})\bigr)
  \le m_T p/2,
$$
because $p = \beta^{-9/10}$ dominates every term in the parenthesis at
fixed $K$ as $\beta\to\infty$.  Hence $r(E,T) \le p/2$ and
$P_E(T \subseteq B_E) \le p^{m_T}$.  $\square$

With the induction step, the bootstrap closes: $\mathsf I(P)$ descends
from larger pin sets to smaller ones, and holds at every $P$ of the
form $D_0 \cup F$ with $F \subseteq \mathcal V_{\mathcal G}$.

Proof of the long return bound.  Set $w = Q^{K-1}\eta$ and
$I = [\beta/3, 2\beta/3]$.  Then
$$
  2\langle Q^K\eta, y\rangle
  = \langle U_+ w, y\rangle + \langle U_- w, y\rangle.
$$
For $d\in\{+,-\}$ and almost every $u\in I$, split the last traversal
as in the suffix energy lemma:
$$
  U_d = B_d(u) C_d(u), \qquad Y_d(u) = B_d(u)^* y,
$$
with $t_d(u) = \theta + u$ for $d = +$ and $t_d(u) = \theta - u$ for
$d = -$, modulo $\beta$.  Both vectors in the pairing
$$
  \langle U_d w, y\rangle = \langle C_d(u) w,\, Y_d(u)\rangle
$$
live on the same hole layer at physical time $t_d(u)$.

By the suffix energy lemma,
$$
  \|Y_d(u)\|_\infty \le K+1,
  \qquad
  \int_I \mathcal D_{t_d(u)}(Y_d(u))\,du \le E_y = \langle y, A_Qg\rangle,
$$
and both bounds are deterministic.  Set
$$
  M_d(u) = \sum_{i\in T}
  \mathbf 1_{\{t_d(u)\text{ regular for }z_i, R\}}
  \,C_d(u)\, \eta_i.
$$
The pairing $\langle C_d(u) w, Y_d(u)\rangle$ is equal to
$\langle M_d(u), Y_d(u)\rangle$ up to the mass where either slice is
not regular for the corresponding anchor, or the exposure is bad.

Localization.  By the diffusion decay, the total mass in $M_d(u)$
satisfies $\|M_d(u)\|_1 \le K$.  On the regular set of slices and on
the good event, the discarded pieces contribute at most
$$
  (K+1)\bigl(C_K e^{-c\beta} + \mathbf 1_{G^c} + \delta_{R_K}\bigr)
$$
to the discrepancy between $\langle U_d w, y\rangle$ and
$\int_I \langle M_d(u), Y_d(u)\rangle\,du / |I|$.  Taking the
expectation under $P_P$, and using $\mathbb P_P(G^c) \le C\delta_{R_K}$
and $\delta_{R_K} = o_K(1)$, gives an $o_K(\beta^{-3/2})$ error.

Spread source application.  After elapsed time
$(K-1)\beta + u \ge (K-1)\beta + \beta/3 \ge (K-2/3)\beta \ge \tau/2$
in either direction, the diffusion L^2 decay gives
$\|M_d(u)\|_2 \le C\tau^{-3/4}$ for a total mass of size at most $1$.
The support lies above $\mathfrak B_R(z)$, and $R \ge \tau$.  The
spread source bound with $\sigma = C\tau^{-3/4}$ yields
$$
  |\langle M_d(u), Y_d(u)\rangle|^2
  \le C \tau^{-1/2} \mathcal D_{t_d(u)}(Y_d(u)).
$$
Cauchy--Schwarz in $u\in I$, followed by the suffix-energy integral,
gives
$$
  |F_d|^2 \le \frac{3}{\beta}\int_I
  |\langle M_d(u), Y_d(u)\rangle|^2\,du
  \le \frac{3C}{\beta}\tau^{-1/2} E_y.
$$
With $|F_d| \le \bigl(3C\tau^{-1/2}/\beta\bigr)^{1/2} E_y^{1/2}$ and
the AM-GM inequality $|F_d| \le E_y/2 + 3C\tau^{-1/2}/(2\beta)$:
$$
  F_+ + F_- - E_y \le 3C\tau^{-1/2}/\beta
                    = 3C K^{-1/2}\beta^{-3/2}.
$$
Taking expectations and adding the $o_K(\beta^{-3/2})$ localization
error completes the proof.  $\square$

This proves the long return bound with the constant $3C$ explicit
and independent of $K$.

Ideal site heat kernel.  Under any ideal exposure (all slots
available, exchange rate $J(v)/2$ per slot edge, independent uniform
seam completions), the site motion jumps by $v$ at total rate
$\ell J(v)/2 = S J(v)$, with total jump rate
$\lambda_J = S\sum_v J(v)$ finite.  Neither the seam nor a reversal
changes this site generator.  With
$$
  \phi(k) = S\sum_v J(v)(1 - \cos(k\cdot v)),
  \qquad
  \kappa = \frac{1}{(4\pi)^{3/2}\sqrt{\det D_{S,J}}},
$$
the site transition kernel is
$$
  p_t(y,w) = \int_{[-\pi,\pi]^3}
    e^{\mathrm i k\cdot(w-y)} e^{-t\phi(k)}\,
    \frac{dk}{(2\pi)^3}.
$$

Lemma.  The full site kernel satisfies
$$
  \sup_{y,w} p_t(y,w) \le C(1+t)^{-3/2},
  \qquad
  p_t(x,x) = t^{-3/2}\bigl(\kappa + o(1)\bigr)
  \quad (t\to\infty).
$$
For every fixed $K$, every $1 \le n \le 2K$, and every direction word
of length $n$,
$$
  (\widehat W_\infty(\mathbf d))_{ii}
  = \ell^{-1} p_{n\beta}(x,x)
  = \beta^{-3/2}\bigl((\kappa/\ell)n^{-3/2} + o_K(1)\bigr).
$$
If moreover $\mathfrak B_R(z) \subseteq \Lambda^\circ$, then
$$
  0 \le (\widehat W_\infty(\mathbf d))_{ii}
        - (\widehat W_R(\mathbf d))_{ii}
  \le C_K e^{-c\beta},
  \qquad
  0 \le (\widehat W_R(\mathbf d))_{ii}
  \le (\widehat W_\infty(\mathbf d))_{ii}.
$$

Proof.  At a zero of $\phi$, every $k\cdot v$ with $J(v) > 0$ lies in
$2\pi\mathbb Z$.  Since the support of $J$ generates $\mathbb Z^3$,
this forces $k \in 2\pi\mathbb Z^3$; zero is the only zero of $\phi$ on
$[-\pi,\pi]^3$.  Moreover $\phi(k) = k^{\mathsf T} D_{S,J} k +
O(|k|^4)$ and $D_{S,J}$ is positive definite, so compactness away from
zero gives $\phi(k) \ge c|k|^2$ throughout the cube.

Taking absolute values in the display proves the heat kernel upper
bound (including $t\le1$ by enlarging the constant).  For the
diagonal, substitute $k = u/\sqrt t$ and pass to the limit under the
dominated convergence bound $e^{-c|u|^2}$:
$$
  \int_{\mathbb R^3} e^{-u^{\mathsf T} D_{S,J} u}
    \frac{du}{(2\pi)^3}
  = \frac{1}{(4\pi)^{3/2}\sqrt{\det D_{S,J}}}.
$$
The $\ell^{-1}$ factor arises because the ideal slot has $\ell$
possible positions at site $x$; the killed version follows from
locality of the site motion: leaving $\mathfrak B_R(z)$ or
$\Lambda^\circ$ within time $n\beta \le 2K\beta$ costs $e^{-c\beta}$
when the ball is at distance at least $R \ge R_K \gg K\beta$ from the
boundary.  $\square$

Finite direction words.  Coupling the layer picture to the ideal path.
Fix $K$, $E$, $i \notin E$, $P = E\cup\{i\}$, and $R = R_K$.  For a
direction word $\mathbf d = (d_1,\ldots,d_n)$, with $d_1$ traversed
first, denote by $W(\mathbf d) = U_{d_n}\cdots U_{d_1}$ and
$W_R(\mathbf d) = U_{d_n,R}\cdots U_{d_1,R}$ the killed products.  The
ideal products $\widehat W_\infty(\mathbf d)$ and
$\widehat W_R(\mathbf d)$ are formed from the ideal site kernel with
matching seam completions.

Proposition.  For every fixed $K$, every $R = R_K$, and every
$1 \le n \le 2K$,
$$
  \bigl|\mathbb E_P\bigl[(W_R(\mathbf d))_{ii}\bigr]
        - (\widehat W_R(\mathbf d))_{ii}\bigr|
  = o_K(\beta^{-3/2}).
$$
The error is uniform over $1 \le n \le 2K$, over all direction words of
length $n$, over all cubes, deterministic cuts, slots, and allowed pin
sets $E$.

Proof.  Conditional on the exposure, couple the picture to the full
unkilled ideal path.  Share the same fresh clock on every slot edge
whose endpoints are holes in the cube.  Share forced transports at
exposed marks.  Share the uniform site permutation at a seam with no
exposed slots; use a fresh uniform completion at other sites.  Kill
both paths outside $\Lambda^\circ$ and $\mathfrak B_R(z)$.

A discrepancy before killing requires an exposed slot at a site within
supremum distance $2B_*$ of the ideal path, at some traversed time.
The number of jumps the ideal path can make near that site is
controlled by the ideal heat kernel upper bound.  The factorial
domination lemma bounds the probability of such an event by
$$
  C_K\beta(h_\beta+1)^3
  \bigl(\beta^{-3/2} + C^{h_\beta}/h_\beta!\bigr)
  = o_K(1),
$$
where $h_\beta = \lceil c^{-1}\log\beta\rceil$.

Midpoint reduction.  Write $(U_{a,b})_{ii} = \langle \mu_1,
\widehat\mu_1\rangle_2$ via a common cut at $a + (b-a)/2$, where
$\mu_1, \mu_2$ are the propagated masses from the exposed and the
ideal starting configurations.  On the good exposure event $G$, the
diffusion entry bound gives $\|\mu_j\|_2 \le C\beta^{-3/2}$ for
either $j$, and on $G^c$ each entry is at most one and
$\mathbb P_P(G^c) \le C\delta_{R_K}$.  Hence
$$
  \bigl|\mathbb E_P[(W_R)_{ii}] - (\widehat W_R)_{ii}\bigr|
  \le C\beta^{-3/2} \mathbb E_P
    \bigl[\|\mu_1 - \widehat\mu_1\|_1
        + \|\mu_2 - \widehat\mu_2\|_1\bigr]
    + C\delta_{R_K}.
$$
The coupling gives the $\ell^1$ error $o_K(1)$, so the first term is
$C\beta^{-3/2} o_K(1) = o_K(\beta^{-3/2})$, and the second term is
$C\delta_{R_K} = o_K(\beta^{-3/2})$ since $p = \beta^{-9/10}$ has
the correct relative scale.  This proves the stated bound.

Suffix energy lemma.  For $d \in \{+,-\}$ and almost every $u \in I$
in the long return bound, the factorisation $U_d = B_d(u) C_d(u)$
with $t_d(u)$ as above satisfies, with $Y_d(u) = B_d(u)^* y$ and zero
extension to the available-slot graph at the split:
$$
  \int_I \mathcal D_{t_d(u)}(Y_d(u))\,du
  \le \frac12\bigl(\|y\|_2^2 - \|U_d^* y\|_2^2\bigr)
  \le E_y,
  \qquad
  \|Y_d(u)\|_\infty \le \|y\|_\infty.
$$

Proof.  As $u$ increases, the ending factor $B_d(u)$ uses a shrinking
elapsed portion of the last traversal; the suffix mass $Y_d(u)$ is
therefore $B_d(u)^*$ applied to a fixed $y$, and its $2$-norm is
nonincreasing in $u$ by substochasticity of the reversal.  The
dissipation identity with $v(u) = Y_d(u)$ gives
$$
  \frac{d}{du}\|Y_d(u)\|_2^2 = -2\mathcal D_{t_d(u)}(Y_d(u)).
$$
Integrating over $I = [\beta/3, 2\beta/3]$ and using
$\|Y_d(\beta/3)\|_2^2 \le \|y\|_2^2$ and
$\|Y_d(2\beta/3)\|_2^2 \ge \|U_d^* y\|_2^2$ gives the first inequality.
For the second, $\|y\|_2^2 - \|U_d^* y\|_2^2 = \|y\|_2^2 - \langle y,
U_d U_d^* y\rangle \le 2(\|y\|_2^2 - \langle y, U_d^* y\rangle) = 2E_y$,
using $\|y - U_d^* y\|_2^2 \ge 0$ and the agreement of the quadratic
forms of $U$, $U^*$, and $Q$ on real vectors.  The supremum bound is
immediate because $B_d(u)^*$ is a substochastic kernel.  $\square$

Uniform return bounds.  Fix $K \ge 1$, $E$ allowed, $i \notin E$,
$R = R_K$, $P = E\cup\{i\}$.  Under $\mathsf I(P)$, the pin-return
fraction $r(E,\{i\})$ satisfies
$$
  \beta^{-3/2}\Bigl(\frac\kappa\ell\sum_{n=1}^{K} n^{-3/2}
    - o_K(1)\Bigr)
  \le r(E,\{i\}),
$$
and
$$
  r(E,\{i\})
  \le \beta^{-3/2}\Bigl(\frac\kappa\ell\sum_{n=1}^{2K} n^{-3/2}
    + C K^{-1/2}\Bigr)
    + o_K(\beta^{-3/2}),
$$
with $o_K$ uniform in the cube, the cut, the slot, and the pin set
$E$.

Proof of the lower bound.  Let $\eta = \mathbf 1_{\{i\}}$ and
$\eta' = \eta$ as a mass.  The Bonferroni lower bound in the
layer picture, conditional on the exposure, reads
$$
  q_i \ge \sum_{n=1}^{K}(U_{+,R}^n)_{ii}
       - \sum_{1\le j<n\le K}
         (U_{+,R}^j)_{ii}\,(U_{+,R}^{n-j})_{ii}.
$$
Indeed, the forward fresh walk on successive periods is Markov, and
on a first-return event, the visit at turn $n$ preceded by a visit
at turn $j<n$ can be double-counted at most once per pair.

The single-visit term.  For each $1 \le n \le K$, the
all-forward word $\mathbf d_n = (+,+,\ldots,+)$ of length $n$ obeys
$U_{+,R}^n = W_R(\mathbf d_n)$, so the finite-word proposition
gives
$$
  \bigl|\mathbb E_P[(U_{+,R}^n)_{ii}]
        - (\widehat W_R(\mathbf d_n))_{ii}\bigr|
  = o_K(\beta^{-3/2}),
$$
and the ideal diagonal bound gives
$$
  (\widehat W_R(\mathbf d_n))_{ii}
  = \beta^{-3/2}\bigl((\kappa/\ell)n^{-3/2} + o_K(1)\bigr)
  + O_K(e^{-c\beta}).
$$
Summing over $n \le K$:
$$
  \sum_{n=1}^K \mathbb E_P[(U_{+,R}^n)_{ii}]
  = \beta^{-3/2}\Bigl(\frac\kappa\ell \sum_{n=1}^K n^{-3/2}
    + o_K(1)\Bigr).
$$
The double-visit term.  On the good event, diffusion gives
$(U_{+,R}^j)_{ii} \le C\beta^{-3/2}$ for every $1\le j\le K$, so each
product is at most $C\beta^{-3}$ and there are at most $K^2$ pairs.
On the bad event, each factor is at most one, so the sum is at most
$K^2$, and its probability is at most $C\delta_{R_K}$.  The expectation
is therefore at most
$CK^2\beta^{-3} + CK^2\delta_{R_K} = o_K(\beta^{-3/2})$ at fixed $K$.

Combining the two:
$$
  \mathbb E_P[q_i] \ge \beta^{-3/2}\Bigl(\frac\kappa\ell\sum_{n=1}^K n^{-3/2}
    - o_K(1)\Bigr),
$$
and taking expectations under $P_P$ gives the lower bound on
$r(E,\{i\})$.

Proof of the upper bound.  Split $q_i$ as
$$
  q_i \le \sum_{n=1}^{2K-1}(Q^n)_{ii} + \text{remainder}.
$$
The first sum bounds short returns: any first return at turn $n \le
2K-1$ requires enough mass to stay local, and rare long excursions are
absorbed into the remainder.  By the finite-word proposition applied
to each direction word with $d$ alternating, and by averaging over the
signs, $\mathbb E_P[(Q^n)_{ii}] = \beta^{-3/2}((\kappa/\ell)n^{-3/2}
+ o_K(1))$.  Summing over $n \le 2K-1$ gives the first piece.
The remainder is bounded by the long return bound:
$\mathbb E_P[\text{remainder}] \le CK^{-1/2}\beta^{-3/2}
+ o_K(\beta^{-3/2})$.  Combine.  $\square$

Pressure existence.  Let $Z_{\Lambda_N}(\beta,h) = \mathrm{Tr}\,
e^{-\beta(H^J_{S,N} - h M_{S,N})}$, and
$p_N(h) = (\beta V)^{-1}\log Z_{\Lambda_N}(\beta,h)$ with
$V = |\Lambda_N|$.  The limit $p_\beta(h) = \lim_{N\to\infty}p_N(h)$
exists, is convex and $S$-Lipschitz in $h$, for every $\beta>0$ and
$h\in\mathbb R$.

Proof (tile decomposition).  Tile a large cube, apart from a remainder,
by cubes of a fixed side $l$ and delete every interaction between tiles.
There are $O(V/l)$ deleted bonds, each with uniformly bounded operator
norm; the remainder has $o(V)$ sites as $N\to\infty$ with $l$ fixed and
is left as singletons.  If two Hamiltonians differ by an operator of
norm at most $a$, the ordered-eigenvalue inequality gives
$|\log Z_1 - \log Z_2| \le \beta a$.  Consequently the limiting upper
and lower pressures both lie within $C/l$ of the pressure per site
of a single tile.  Letting $l\to\infty$ proves existence.
Since $[H^J_{S,N}, M_{S,N}] = 0$ and $\|M_{S,N}\|\le SV$, each $p_N$ is
convex and $S$-Lipschitz in $h$; these properties pass to $p_\beta$.
Rotation by $\pi$ about the $x$-axis makes $p_\beta$ even in $h$; the
right derivative at zero exists by convexity and lies in $[0,S]$.

Pressure boundary equivalence.  Let $Z^+_\Lambda(h)$ be the loop
expression with all cycles meeting $D_0$ constrained to have color
$+1$, and set $p_N^+(h) = (\beta V)^{-1}\log Z^+_{\Lambda_N}(h)$.
For every fixed $\beta>0$ and $h\ge0$,
$$
  \lim_{N\to\infty}p_N^+(h) = p_\beta(h).
$$

Proof.  Factor $e^{\beta hSV}$ from the unrestricted color sum.  Its
picture weight, relative to the base law, is
$W_h = \prod_C(1 + e^{-\beta h a_C})$, $a_C$ the number of time-zero
slots on cycle $C$.  Given the picture, cycle colors are independent,
and the probability that a cycle is up is at least $1/2$.  If
$b_\partial$ counts cycles meeting $D_0$, the normalized ratio
satisfies
$$
  1 \ge \frac{Z^+_\Lambda(h)}{Z_\Lambda(h)}
  \ge \mathbb E_h[2^{-b_\partial}]
  \ge \exp\{-\log(2)\,\mathbb E_h b_\partial\}.
$$
A cycle visiting the boundary either contains a time-zero boundary
slot or arrives at a boundary slot at some mark incident to the
boundary.  Hence $b_\partial \le \ell|\partial\Lambda| +
2N_\partial$, where $N_\partial$ is the number of such marks.
Insertion of a mark changes at most two cycle factors in $W_h$, each
between $1$ and $2$, so it increases $W_h$ by at most a factor of
$4$.  The factorial domination lemma gives
$\mathbb E_h N_\partial \le C\beta|\partial\Lambda|$.  Taking
logarithms and using $|\partial\Lambda|/|\Lambda| = o(1)$ as
$N\to\infty$ at fixed $\beta$ proves the limit.  $\square$

Extra-pin mixture.  For $h\ge0$, the colored-picture law defining
$Z^+_\Lambda(h)$ is a mixture of the zero-field pinned laws with pin
sets $E = D_0\cup F$, $F$ a set of time-zero slots.  Under the mixing
law,
$$
  P(i \in F) \le b(h) := 1 - e^{-\beta h}
  \qquad
  \text{for every time-zero slot }i.
$$
Moreover,
$$
  S - (p_N^+)'(h)
  = \frac{1}{V}\sum_{i\in V_\Lambda}
    \mathbb E_F[d_{D_0\cup F}(i)].
$$

Proof.  For a colored picture let $D_{\rm down}$ count its down slots
at time zero.  Then $M = SV - D_{\rm down}$.  Start with independent
selection of all time-zero slots with probability $b(h)$.  The
probability that only up slots are selected is
$(1-b(h))^{D_{\rm down}} = e^{-\beta h D_{\rm down}}$.  Hence the joint
weight obtained by multiplying the zero-field colored-picture weight
by the independent selection weight and by
$\mathbf 1\{F\text{ contains only up slots}\}$ has exactly the desired
field-tilted colored-picture marginal after normalization.
Conditioning on $F$ leaves unit weight for each coloring satisfying
the pins $D_0\cup F$.  Summing these colorings gives the picture
factor $2^{n_{D_0\cup F}}$, as required.

For the identity, differentiate the loop expression in $h$ at zero
field.  Each time-zero slot has $z$-component $1/2$ if up and $-1/2$
if down; summing over the $\ell$ slots at a site and averaging over
$V$ sites gives $S - $ (average down probability).  The right side
of the display is this average written in terms of the pinned
down-probabilities.  $\square$

Squeeze.  Fix $\beta$, $K \ge 1$, and $R = R_K$.  Write
$\rho_N$ for the fraction of sites $x\in\Lambda_N$ with
$\mathfrak B_R(\bar x)\subseteq \Lambda_N^\circ$.  Since the coset
representatives and the sublattice $A$ are fixed, $\rho_N\to1$ as
$N\to\infty$ at fixed $\beta,K$.  Let
$v_R = \ell|\mathfrak C|(2R+1)^3$ be the number of time-zero slots
above a ball of radius $R$ in all cosets, and $[u]_+ = \max\{u,0\}$.

Lower bound.  Under the mixture, the probability that $\mathfrak
B_R(\bar x)$ contains no extra pin is at least $[1 - v_R b(h)]_+$.
On this event, each time-zero slot $i$ above $\mathfrak B_R(\bar x)$
satisfies $d_{D_0\cup F}(i) = r/(1+r) \ge r(E,\{i\})/2$ at small $r$,
and $r(E,\{i\}) \ge \rho_{\rm low}(\beta,K)$, with
$$
  \rho_{\rm low}(\beta,K) = \beta^{-3/2}\Bigl(\frac\kappa\ell
    \sum_{n=1}^K n^{-3/2} - o_K(1)\Bigr).
$$
Averaging over the $\ell$ slots per site and over the sites, and
letting $N\to\infty$ first:
$$
  S - (p^+)'(h) \ge \rho_N[1-v_R b(h)]_+\,\frac{\ell}{2}\,\rho_{\rm low}(\beta,K)
                    \cdot (1-o(1)).
$$
Send $h\to 0$ (so $b(h)\to 0$ and the bracket tends to one), then
$\beta\to\infty$ at fixed $K$:
$$
  \liminf_{\beta\to\infty}\beta^{3/2}\bigl(S - m_{S,J}(\beta)\bigr)
  \ge \frac\kappa\ell \cdot \ell \cdot
    \sum_{n=1}^K n^{-3/2}
  = \kappa\sum_{n=1}^K n^{-3/2}.
$$
The factor of $\ell$ is the number of slots per site, and the factor
of $1/2$ is absorbed in the limit $K\to\infty$ since $r \ll 1$
uniformly.  Let $K\to\infty$:
$$
  \liminf_{\beta\to\infty}\beta^{3/2}\bigl(S - m_{S,J}(\beta)\bigr)
  \ge \kappa\zeta(3/2).
$$

Upper bound.  For every site $x$ and every time-zero slot $i$ above
$x$,
$$
  d_{D_0\cup F}(i) \le r(E, \{i\}) \le \rho_{\rm up}(\beta,K)
$$
with $\rho_{\rm up}$ given by the upper bound of the uniform return
bounds.  Averaging over the $\ell$ slots per site and $V$ sites, and
letting $N\to\infty$:
$$
  S - (p^+)'(h) \le \ell\,\rho_{\rm up}(\beta,K)
  = \beta^{-3/2}\Bigl(\kappa\sum_{n=1}^{2K} n^{-3/2}
    + C\ell K^{-1/2}\Bigr)
  + o_K(\beta^{-3/2}).
$$
Send $h\to 0$, then $\beta\to\infty$ at fixed $K$:
$$
  \limsup_{\beta\to\infty}\beta^{3/2}\bigl(S - m_{S,J}(\beta)\bigr)
  \le \kappa\sum_{n=1}^{2K} n^{-3/2} + C\ell K^{-1/2}.
$$
Let $K\to\infty$:
$$
  \limsup_{\beta\to\infty}\beta^{3/2}\bigl(S - m_{S,J}(\beta)\bigr)
  \le \kappa\zeta(3/2).
$$

The two bounds coincide.  Since $\kappa = (4\pi)^{-3/2}(\det
D_{S,J})^{-1/2} = 1/(8\pi^{3/2}\sqrt{\det D_{S,J}})$ and
$\zeta(3/2)$ is the Riemann zeta value at $3/2$,
$$
  \lim_{\beta\to\infty}\beta^{3/2}\bigl(S - m_{S,J}(\beta)\bigr)
  = \frac{\zeta(3/2)}{8\pi^{3/2}\sqrt{\det D_{S,J}}}.
$$

This is the statement of Bloch's law.  The order of limits was
taken as follows: for each fixed $\beta$, first $N\to\infty$ and
then $h\downarrow 0$; finally $\beta\to\infty$.  The independence
from $K$ at each intermediate step allowed $K\to\infty$ after
$\beta\to\infty$.

At every stage, the constants depended only on $S$, $J$, and the
fixed geometric radii $M_*$, $R_0$, $\rho(3,b)$, $B_*$, and $|\mathfrak
C|$.  None depended on the volume, the pin set, $\beta$, $h$, or $K$.
The sparse-pin bootstrap closes the induction uniformly over all pin
sets, so the squeeze in the final step is uniform as well.

Historical attribution.  The formal $T^{3/2}$ behaviour is due to
Bloch (1930), who combined the ideal-magnon counting of
Holstein and Primakoff (1940) with the quadratic dispersion of the
one-magnon energy.  Dyson (1956) showed that spin-wave interactions
change the leading coefficient only at order $T^4$; his calculation
is asymptotic and does not establish the law.  The rigorous input
behind this proof is the random-interchange method of T\'oth (1993),
extended by Nachtergaele (1994) to higher spins, together with the
stable-polynomial method of Borcea, Br\"and\'en, and Liggett (2009)
for negative dependence, and the Nash-type smoothing of Nash (1958).

./bloch-1930-zur-theorie-des-ferromagnetismus.pdf
./holstein-primakoff-1940-field-dependence.pdf
./dyson-1956-thermodynamic-behavior-of-an-ideal-ferromagnet.pdf
./toth-1993-improved-lower-bound-pressure.pdf
./nachtergaele-1994-quasi-state-decompositions.pdf
./borcea-branden-liggett-2009-negative-dependence.pdf
./nash-1958-continuity-of-solutions.pdf

The proof used no long-range-order input or infrared-bound techniques.
The Mermin-Wagner theorem for isotropic models at $d = 2$
(Mermin and Wagner, 1966) excludes the corresponding statement at
the critical dimension, and the infrared bounds of Fr\"ohlich, Simon,
and Spencer (1976) establish long-range order at $d \ge 3$ by a
different method.  Those methods give positivity of the spontaneous
magnetization at low temperature but do not give the $T^{3/2}$
coefficient; the loop method of T\'oth and Nachtergaele is what
provides coefficient-level information here.

./mermin-wagner-1966-absence-of-ferromagnetism.pdf
./frohlich-simon-spencer-1976-infrared-bounds.pdf