Proof. For one realization of the fresh exchanges and seam completions, the unkilled trajectories define a bijection between the endpoint hole sets. Deleting trajectories that encounter a killing set gives a partial permutation. Averaging proves the two substochastic bounds and the \(\ell^1\) and \(\ell^\infty\) contractions. Rowwise Cauchy–Schwarz gives the \(\ell^2\) contraction, as in [12]. Transposition gives the stated reverse realization and the same bounds.

On a smooth interval the generator is symmetric, with rate \(1/2\) on each available edge and Dirichlet killing rates. Summing by edges proves the norm-loss identity. Zero extension gives exactly (5): a surviving interior slot has no edge directly to the exterior of the cube. Instantaneous contractions can only add to the norm loss. Finally, for \(V=U\) or \(U^*\), the real quadratic forms agree and \[\lVert y\rVert_2^2-\lVert Vy\rVert_2^2 =2e_y-\lVert y-Vy\rVert_2^2.\] This identity, contraction, and the integrated norm loss prove (6). ◻

Proof. If \(T\) meets \(D\), the probability is zero. Otherwise set \(m=|T|\) and \(P'=D\cup T\). By (4), it suffices to prove \(r_T\le C\lambda\). We adapt the one-turn energy argument of [12], retaining all \(m\) sources.

Fix the exposure under \(\mathbb P_{P'}\) and use the time of \(T\) as the cut. Let \(U\) be the fresh one-period kernel killing only on the inner boundary, and put \(Q=(U+U^*)/2\) and \(\eta=\mathbf 1_T\). Let \(f=1\) on \(T\); off \(T\), let \(f\) be the probability of hitting \(T\) at a future cut before boundary killing. Finite-horizon approximation gives \(0\le f\le1\) and \(U^*f=f\) off \(T\), even if eventual killing is not almost sure. For \(g=f-\eta\), put \(e_g=\langle g,(I-Q)g\rangle\ge0\). Removing all pins of \(D\) except the boundary increases the return probability. Harmonicity off \(T\) and expansion of the quadratic form therefore give \[\begin{align*} m r_T &\le\mathbb E_{P'}\bigl[m-\langle f,(I-Q)f\rangle\bigr]\\ &=\mathbb E_{P'}\bigl[ \langle \eta,Q\eta\rangle+\langle (U+U^*)\eta,g\rangle-e_g\bigr]. \tag{19}\end{align*}\] This is also the multipoint hitting-energy identity in [13].

Here are the propagation estimates used to bound the two positive terms. For each \(j\in T\), let \(x_j\) be its site and choose radius \(R_0=\lceil a\beta\rceil\), with \(a\) a sufficiently large fixed constant. Let \(H_j\) be the event that the total nonregular time at this radius about \(x_j\) is at most \(\beta/100\). Equation (15) and Markov’s inequality give \[ \mathbb P_{P'}(H_j^c)+ \sup_t\mathbb P_{P'}(t\text{ is nonregular about }x_j) \le C\beta^{-27/11}=o(\lambda). \tag{20}\] The localization, diffusion, and spread-source estimates in [12], with one turn, have the following consequences. Replacing a propagation from \(j\) by the propagation killed on exit from \(B_{R_0}(x_j)\) loses expected mass at most \(Ce^{-c\beta}\). On \(H_j\), a localized propagation of duration at least \(\beta/3\) has \(\ell^2\) norm at most \(C\beta^{-3/4}\); localized full-period entries are at most \(C\lambda\). At a regular endpoint, a nonnegative localized mass \(\mu\) of mass at most one and norm at most \(C\beta^{-3/4}\) satisfies \[ |\langle \mu,Y\rangle|\le \bigl(C\beta^{-1/2}\mathcal D_t(Y)\bigr)^{1/2} \tag{21}\] for every finitely supported \(Y\). These estimates apply with pin law \(P'\): for the source \(j\), all other points of \(P'\) are allowed extra pins. They impose no separation on \(T\).

Since there are at most \(m^2\) return entries, localization, diffusion, and (20) imply \(\mathbb E_{P'}\langle \eta,Q\eta\rangle\le C\lambda\). For the cross term, consider one pairing \(\langle U_d\mathbf 1_{\{j\}},g\rangle\), where \(U_d=U\) or \(U^*\). Split it at elapsed time \(u\in I=[\beta/3,2\beta/3]\). Localize only the propagated starting mass, and let \(Y(u)\) be the adjoint suffix applied to \(g\). Both the mass and \(\lVert Y(u)\rVert_\infty\) are at most one. Discarding the localization loss, \(H_j^c\), and nonregular split slices therefore changes the expected pairing by \(o(\lambda)\). On retained slices use (21); then average over \(I\). Lemma 4 and Cauchy–Schwarz give, at each fixed exposure, the retained average bound \[\frac{C\beta^{-1/4}}{|I|} \int_I\sqrt{\mathcal D_{t(u)}(Y(u))}\,du \le C\beta^{-3/4}\sqrt{e_g}.\] In this application the energy uses boundary-only killing, as does \(U\), and its quadratic form equals that of \(Q\). There are \(2m\le12\) such pairings. Substituting in (19) and using \[C\beta^{-3/4}\sqrt{e_g}-e_g\le C'\beta^{-3/2}\] proves \(r_T\le C\lambda\). Equation (4) now proves (18). ◻

Proof. We first impose that the distinguished exposed cycle is colored down. Conditionally on a picture in \(\mathcal I\), this costs exactly a factor \(1/2\). We prove that the resulting joint path law is bounded by the base tagged-walk law.

Fix a path \(X\) starting at \(v\), with successive slots joined by slot edges and jump times \(s<\tau_1<\cdots<\tau_j<t\). Let \(A_X\) consist of the edge–time pairs \((e,u)\) with \(s<u<t\), \(u\) not a jump time, and \(e\) incident to \(X(u)\); let \(\mu\) be the base Poisson intensity measure. The path requires its \(j\) jump marks and the absence of all other marks in \(A_X\). Poisson disintegration gives the density \[q(X)=2^{-j}\exp\{-\mu(A_X)\}\] with respect to counting measure on such slot lists and Lebesgue measure on ordered jump times. The complementary clocks and seams retain their independent base laws. Denote those variables by \(\zeta\), and let \(N_X(\zeta)\) count the admissible color histories for which \(\mathcal I\) holds and the distinguished cycle is down.

Recolor only the segment \(X\) up. Every other slot at its site or a neighboring site is up, since isolation makes those slots unexposed. At a prescribed jump the colors change from \((\downarrow,\uparrow)\) to \((\uparrow,\downarrow)\); after recoloring, both slots are up on both sides, so the mark can be deleted. Fill \(A_X\) with arbitrary additional marks. Each new mark joins two up slots, so the recolored history remains valid. Its only color discontinuities are the prescribed \(\downarrow\!\to\!\uparrow\) flip at \((v,s)\) and \(\uparrow\!\to\!\downarrow\) flip at \((X(t),t)\). All pins remain up, because the original segment belongs to an exposed cycle. Figure 1 depicts the deleted jumps, the arbitrary up–up fillings, and the oriented endpoint flips.

For fixed \(X\), \(\zeta\), and filling, the map on color histories is injective: recoloring the specified segment down recovers the original history. In the filled picture, a cycle containing a prescribed oriented flip has at most one compatible coloring. Every other cycle has at most its usual one choice if pinned and two choices if unpinned. Writing \(w_P=2^{n_P}\) and \(\omega'\) for the filled picture, we therefore have \(N_X(\zeta)\le w_P(\omega')\) for every filling.

For fixed \(X\), the set \(A_X\) is deterministic. Independent Poisson filling on \(A_X\), together with the complementary clocks and seams, has exactly the unconditioned base law. Averaging yields \[\mathbb E_\zeta N_X(\zeta)\le\mathbb E_{\mathrm b}w_P=Z_P.\] The down-strand path density is \(q(X)\mathbb E_\zeta N_X(\zeta)/Z_P\), hence at most \(q(X)\). Summing over jump lists and integrating their ordered times proves the assertion for every measurable \(\mathcal A\); finite volume gives finitely many marks almost surely. Removing the down-color condition supplies the factor two. At seam endpoints the flips are inserted on the specified sides of the unchanged seam mapping. Reversing time gives the reverse assertion, with the flip directions reversed. ◻

Proof. First control the number of strand steps. A chronological connected list of \(j\) inter-site marks in an interval of length \(u\), anchored at a prescribed site, has probability at most \[ \frac{(Cu)^j}{j!}. \tag{25}\] Indeed, there are at most \(C^j\) successive site and slot-edge choices, and the ordered time simplex has volume \(u^j/j!\); apply the factorial bound in Proposition 3. The same count works in reverse order. It bounds lists anywhere in the interval, so it also covers a segment beginning at an arbitrary intermediate visit; no union over real starting times is necessary. Taking \(j=c_0L\) on blocks, with \(c_0\) large, gives \(e^{-cL}\) tails. Taking \(j=C\log\beta\) on bins gives rapidly small tails by the factorial in (25). A polynomial union over sites, blocks, bins, and both directions proves the first property with rapidly small error. For this count and the next one enlarge the region to \(B_{20R_*}(x)\).

At a fixed block cut and center, six obstacles in a ball of radius \(c_1L\) supply six distinct slot points there. There are at most \(CL^{18}\) such six-tuples. Proposition 8, followed by a union over centers and at most \(C\beta\) cuts, bounds this failure probability by \[ C\beta R_*^3L^{18}\lambda^6 \le C\beta^{-3+18\delta} =C\beta^{-2.91}=o(\beta^{-5/2}). \tag{26}\] Seam permutations preserve the site counts, including multiplicities, so this estimate holds on either seam side.

If six blocked sites were present in one of the required fixed-radius balls during a block, one obstacle at each site could be traced back to the block cut. The first property would place their six distinct starting slots in a ball of radius \(c_1L\), once \(c_1\) is large enough. This is impossible. Thus local regularity holds at all times. For the remaining condition, Proposition 7 bounds the expected measure of times when distant-component regularity fails in any block \(I\) by \(C|I|e^{-cR_*^\xi}\). Markov’s inequality and a union over blocks show, with rapidly small error, that it fails on less than half of every block. This proves the second property.

The step bound also ensures that an isolated strand cannot meet another obstacle during its block. Any potential encounter can be traced back in both strands to their distinct block-start slots, at total distance at most a fixed multiple of \(c_0L\). Choose \(c_2\) larger than that multiple. This argument also covers strands entering the region, by tracing backward from the encounter.

It remains to bound the isolated strand’s displacement. The site coordinate of the base walk in Lemma 9 jumps at rate \(S\) to each nearest neighbor. Exponential martingales in each coordinate give, for \(u\le2L\), \[\mathbb P_{\mathrm b}\left(\max_{0\le v\le u} |X(v)-X(0)|_\infty>CL^{3/4}\right) \le C e^{-c\sqrt L}.\] For example, use an exponential parameter of order \(L^{-1/4}\); the martingale compensator is at most a constant times \(uL^{-1/2}\). On the step event already established, strands beginning in the region stay interior. Lemma 9 therefore bounds the joint event of isolation and excessive displacement. A polynomial union over slots, blocks, and directions is rapidly small. Seams make no spatial displacement; for a seam block, enumerate also the slots on its other seam side. Combining these exceptional sets proves (24) and the third property. ◻

Proof. We use Nash’s energy-dissipation argument [11]. For a nonnegative input of mass at most one, let \(l=\lVert v\rVert_2^2\). The all-time local regularity in Proposition 10 permits (16) at every smooth time, so \(l'\le-cl^{5/3}\). Instantaneous operations are contractions by Lemma 4. Integrating gives \(\lVert v\rVert_2\le C(1+u)^{-3/4}\) for a unit point input. Split the kernel at its midpoint and express an entry as the inner product of a forward mass and an adjoint mass, each propagated for \(u/2\). Their two norm bounds give (27). ◻

Proof. Consider first a piece of one block, of length at most \(2L\), starting at any slice and over a site \(z\) in \(B_{9R}(x)\). Stop on exiting \(B_L(z)\), including its exit step in the displacement estimate. Every obstacle strand visiting this ball or its immediate neighbors traces back at most \(c_0L\) steps to a distinct slot at the deterministic block cut. The obstacle count in Proposition 10 thus shows that at most five strands can be relevant before stopping. Their instantaneous neighborhoods have uniformly bounded size.

The rate jumps of each site coordinate have bounded rates and unit increments. Their drift vanishes away from the neighboring obstacles and is uniformly bounded elsewhere. Lemma 11, with the stopping-ball killing, bounds the expected occupation time near the relevant strands by \[C\int_0^{2L}(1+t)^{-3/2}\,dt\le C.\] The expectation is uniform in the starting slot and slice, including either side of any prescribed operation.

A forced transport occurs only when the fresh walk is at its prescribed departure hole. In any bin there are at most \(C\log\beta\) relevant transports, by the bin step bound for the five strands. Apply (27) to the probability of presence just before each such operation. Summing over bins, using their lengths between one and three, bounds the expected number of forced steps before stopping by \[C\log\beta\sum_{n\ge0}(1+n)^{-3/2}\le C\log\beta.\] The same estimate holds from an arbitrary intermediate slice: a unit-time interval intersects only a bounded number of the original bins. These estimates use deterministic transport times at fixed exposure, with no conditional Poisson assertion.

Let \(A\) be the accumulated occupation time near obstacles plus the number of forced steps. From every stopping slice and surviving state, the conditional expected remaining accumulation is at most \(C\log\beta\). The Markov property and Markov’s inequality show that each further increment of size \(2C\log\beta+1\) has conditional probability at most \(1/2\). Occupation time is continuous and the transport-count overshoot is at most one. Restarting at successive thresholds therefore gives, for each fixed \(a>0\), \[\Pr(A>aL^{3/4})\le C_a \exp\{-c_aL^{3/4}/\log\beta\}.\] This controls both the integrated drift and the forced displacement.

Subtract the integrated drift from the ordinary rate jumps. For each coordinate, bounded rates and unit jumps imply that its compensated process \(M_t\) has \(\exp\{\theta M_t-C\theta^2t\}\) as a supermartingale for \(|\theta|\le1\), with a fixed \(C\). The maximal inequality with \(\theta\) of order \(L^{-1/4}\) gives \[\Pr\bigl(\max_{t\le2L}|M_t|>aL^{3/4}\bigr) \le C_a e^{-c_a\sqrt L}.\] The three displacement contributions prove the claimed bound on one block piece. Since \(CL^{3/4}<L\) for large \(\beta\), they also bound the probability of reaching the stopping sphere. Seam operations change no site coordinate.

An interval of length \(u\) meets at most \(C(1+u/L)\) block pieces. Apply the preceding uniform bound successively and add their displacements. There are polynomially many pieces, so the total failure probability remains rapidly small. Throughout an interval of length at most \(2\beta\), the resulting displacement is \(O(\beta L^{-1/4})=o(R)\). The walk therefore stays in the local region used above, and in \(B_{R_*}(x)\). This proves the lemma. ◻

Proof. Where both edge distances from the source sites are at most \(R_*\), use the two inverse-square envelopes. Their convolution in dimension three is bounded by \(C/(1+b)\) at source separation \(b\): balls of radius \(b/2\) about either source contribute \(C/(1+b)\), and dyadic shells outside them have the same total bound.

If one source distance exceeds \(R_*\), both are at least a fixed multiple of \(R_*\), since \(a,a'\) lie over \(B_{6R}(x)\) and \(R=o(R_*)\). The global envelopes then bound the remaining sum by \[C R_*^{3-2(199/100)} =C R_*^{-98/100} \le C R^{-1}.\] The last inequality follows from \((5/3)(98/100)=49/30>3/2+\delta/4\). As \(b\le12R\), this is bounded by the right-hand side of (29). ◻

Proof. We first record the heat-kernel estimates before applying the cutoff. For every fixed integer \(m\ge0\) and \(j=0,1,2\), Fourier inversion gives \[ |\nabla^j p_t(z)| \le C_{m,j}(1+t)^{-(3+j)/2} \left(1+\frac{|z|_\infty}{\sqrt{1+t}}\right)^{-m}. \tag{41}\] Here \(j\) nearest-neighbor differences may be taken in arbitrary coordinate directions. One time derivative satisfies the same bound with \(j=2\). To verify the estimates, write \(\omega(k)=2S\sum_j(1-\cos k_j)\) and use \(\omega(k)\ge c_S|k|^2\) on \([-\pi,\pi]^3\). A \(j\)-fold difference inserts a multiplier vanishing to order \(j\) at zero; a time derivative inserts \(-\omega\), which vanishes to order two. The integral without frequency differentiation is thus \(O((1+t)^{-(3+j)/2})\). For the spatial weight, integrate by parts \(m\) times in a coordinate for which \(|z_j|=|z|_\infty\). The integral of the absolute value of the differentiated multiplier is at most \(C_{m,j}(1+t)^{(m-3-j)/2}\): near zero this follows by rescaling \(k\) by \((1+t)^{-1/2}\), using \(\nabla\omega=O(|k|)\) and bounded higher derivatives; away from zero the exponential dominates every power of \(t\). The periodic integrands have no boundary terms. Combining this bound with the bound without integration by parts proves (41). The case \(0\le t\le1\) follows from uniform bounds on the same derivatives.

The slot-uniform part of the smooth evolution is a sum of site kernels at times \(n\beta+s\), \(n\ge1\). For \(j=0,1,2\), choose \(m>j+1\) in (41). Splitting the time sum at \(n\beta+s\) of order \(r_z^2\), or comparing it with its integral on dyadic intervals, gives \[ \sum_{n\ge1}(n\beta+s)^{-(3+j)/2} \left(1+\frac{r_z}{\sqrt{n\beta+s}}\right)^{-m} \le \frac{C}{\beta}(D_*+r_z)^{-1-j}. \tag{42}\] For completeness, when \(r_z\le D_*\) the sum without its spatial factor is \(O(\beta^{-(3+j)/2})\). When \(r_z>D_*\), the terms with \(n\beta+s\ge r_z^2\) sum to \(C\beta^{-1}r_z^{-1-j}\). For the remaining terms, use the bound \((1+r_z/\sqrt t)^{-m}\le(\sqrt t/r_z)^m\) and sum \(t^{(m-3-j)/2}\) up to \(r_z^2\); the same bound results. This proves (42) and the asserted pointwise convergence of \(v\). It gives the value and difference estimates in (39); for time derivatives use \[\beta^{-1}(D_*+r_z)^{-3}\le W(z)/D_*.\]

For the point part, (41) gives (40) with \(m=12\). Formula (31) adds only a site-supported slot deviation decaying as \(e^{-3\ell s}\). This deviation, its spatial differences, and its time derivative satisfy the same bounds, with constants depending on \(S\). In the reverse direction the initial seam removes that deviation; at a seam its change is bounded by the incoming absolute value of the slot deviation and hence by \(C W_d(z)\). The smooth part is unchanged because it is slot-uniform.

It remains to justify multiplication by \(\chi\). Value and time derivative estimates are preserved. In a spatial difference the extra term contains a difference of \(\chi\), supported where \(r_z\) is of order \(R\). There \(G(z)/R\le C W(z)\), and \(P_d(z)/R\le C W_d(z)\) since \(d\le\sqrt{1+\beta}\ll R\). Thus the difference estimates are preserved as well. Finally, if \(s\ge a\beta\), then \(d\) is comparable with \(D_*\); direct comparison of the envelopes gives \(P_d\le C_aG\), \(W_d\le C_aW\), and \(W_d/d\le C_aW/D_*\). This proves the last assertion. ◻

Proof. At every smooth time, all neighboring slots of \(Z_s\) are holes, by isolation. Summing (48) over these slots therefore gives \[\frac12\sum_{j\sim Z_s} (\varphi_s(j)-\varphi_s(Z_s)) = (\mathscr L\varphi_s)(Z_s) = (\partial_s\varphi_s)(Z_s)+\gamma_s(Z_s).\] The transport contribution (49) is exactly the profile increment along each jump of \(Z_s\).

It remains to check the seam, where the two hole sets can differ. At the strand’s site let \(H_-,H_+\) be the incoming and outgoing holes, and let \(Z_-,Z_+\) be the corresponding obstacle slots. The actual hole map preserves the sum over holes. The full-profile seam preserves the sum over all slots. Since the obstacle is the only one at this site, the total seam source is exactly \[\begin{align*} \sum_{H_+}(K\varphi_- -\varphi_+) &=\sum_{H_-}\varphi_- -\sum_{H_+}\varphi_+\\ &=\varphi_+(Z_+)-\varphi_-(Z_-). \end{align*}\] Thus it also supplies the increment along the obstacle strand. Integrating the time derivatives and summing all increments telescopes to (50). Reverse traversal has the same conservation of counting sums and uses the inverse transport, so this proof applies with reverse elapsed time as well. ◻

Proof. The step bounds in Proposition 10 keep the strand and its adjacent source sites within \(CL\) of its starting site. The weights \(W\) are comparable throughout this region, because \(L\ll D_*\). The stated hypotheses give the same comparability for \(W_d\). Each strand has bounded degree, contributes for at most \(2L\) units of continuous time, and makes at most \(CL\) jumps. The difference bounds of Lemma 14, together with (48) and (49), give (51). At a seam, fixed slot multiplicity bounds the number of terms, and the difference of the two averages is bounded by the slot-difference bound. This adds at most \(C\mathcal W(z)\).

For an isolated strand use Lemma 15. Its endpoint displacement is at most \(CL^{3/4}\). The spatial endpoint difference therefore costs at most \(CL^{3/4}W(z)\) in the smooth case. Time variation costs \(CLW(z)/D_*\), and a seam increment costs at most \(CW(z)\). On the cutoff annulus, \(b_*\le CW(z)/D_*\), so the integral in (50) has the same time bound. Since \(1+L/D_*\le CL^{3/4}\), this proves (52) for the smooth part.

For the point part the time derivative is bounded by \(CW_d/d\). On the cutoff annulus its commutator is bounded by \[C\bigl(W_d/R+P_d/R^2\bigr)\le CW_d/d,\] because \(d\le\sqrt{1+\beta}\ll R\). The same endpoint argument gives \(C[L^{3/4}+1+L/d]W_d(z)\), which proves the stated bound. Finally Lemma 14 bounds the point part by the smooth envelopes when \(s\ge\beta/2\). Addition proves the last assertion; the division at \(\beta/2\) is a block endpoint. ◻

Proof. Proposition 10 bounds both displacement and the number of jumps in a bin by \(C\log\beta\). Continuous contributions have bounded time length. Within a bin, \(1+s\) changes by a bounded factor, while a spatial shift of \(C\log\beta\) changes any of the polynomial envelopes in Lemma 14 by at most a fixed power of \(1+\log\beta\). The formulas for removed edges, transports, and seam averaging now give the result. ◻

Proof. Here is the exact telescoping calculation for instantaneous factors; it also fixes the endpoint convention in the integrals. Let \(K_j:V_{j-1}\to V_j\), \(1\le j\le n\), be the chronological factors between counting spaces, and write \(a_j,b_j\) for comparison values at those slices, with \(a_0=b_n=h\). In this calculation \(b_j\) is indexed in forward order. Put \[p_j=K_j a_{j-1}-a_j,\qquad q_j=K_j^*b_j-b_{j-1},\qquad P_{r,l}=K_r\cdots K_{l+1}.\] The forward error is \(\sum_jP_{n,j}p_j\). Reverse telescoping gives \[P_{n,j}^*h=b_j+\sum_{k>j}P_{k-1,j}^*q_k.\] Their pairing is consequently \[\sum_j\langle p_j,b_j\rangle +\sum_{j<k}\langle P_{k-1,j}p_j,q_k\rangle.\] The indices \(j<k\) express that \(p_j\) is inserted after its factor, whereas \(q_j\) is inserted before that factor. On each intervening smooth interval the variation-of-constants formula gives the same identity with time integrals. Composing these formulas with the finite telescoping calculation proves (55). Two continuous source measures have no diagonal mass; atomic terms retain the strict slice order above. In particular the forward terminal seam source pairs directly with \(h\) in the first sum. The reverse seam source lies before that operation and pairs only with earlier forward sources, through kernels ending before the seam. No kernel in this pairing crosses that seam twice. ◻

Proof. Expand \(\sum_Q A_Q^2\). Terms using the same slot are bounded by \(C\lambda\|w\|_2^2\) by Proposition 8. Terms using two distinct slots, including two slots at the same site, are bounded by \(C\lambda^2\sum_Q(\sum_{z\in Q}w(z))^2\). Taking a square root proves (62). At a seam the same bound holds on either side: each \(A_Q\) depends only on the number of obstacle slots at each site, and these numbers are preserved by the seam.

Distance at most \(CF\) restricts the two anchors to cubes whose indices differ by one of a bounded number of offsets. For each offset, Cauchy–Schwarz in the product of the picture probability space and the cube index set gives \[\mathbb E_P\sum_Q A_Q A'_{Q+v} \le\left(\mathbb E_P\sum_Q A_Q^2\right)^{1/2} \left(\mathbb E_P\sum_Q (A'_Q)^2\right)^{1/2}.\] Sum the bounded number of offsets. Both factors use their own snapshot estimate, so no joint law at the two times is required. ◻

Proof. Every obstacle source in the block is associated with a strand starting within \(B_{4R}(x)\). Sources of that strand are within \(CL\) of its start, and have total variation at most \(CLW(z_\alpha)\). Lemma 12 permits their propagation to the next block to be killed outside a larger ball of radius \(CL\) about \(z_\alpha\), at a rapidly small relative loss of total variation. Apply this restriction separately to each strand. Apply the same restriction about each injection site to the cutoff sources. The elapsed propagation time is at most \(4L\), also for the final block continued into the copied period.

These balls lie in the unpinned interior. Up to the first exit from its chosen ball, the propagation conserves counting mass: the local heat evolution, hole transports, and seam averages all do so, and there is no pin killing there. The localized propagation loses mass only at that exit. Its signed net mass therefore differs from the signed sum of the injected sources by at most their total variation times the rapidly small exit probability. All source variations and the number of injection sites are polynomially bounded in \(\beta\) on \(\mathcal G\). Their aggregate loss is consequently rapidly small, uniformly in the volume and in the averaging slice.

Fix a fully regular averaging slice. For each \(\alpha\), choose one available representative slot \(a_\alpha\) within a bounded distance of \(z_\alpha\). Proposition 7 provides this slot and paths from every slot carrying the localized mass to \(a_\alpha\), of length and displacement at most \(CL\). Erase loops from these paths. Routing a signed mass along them produces a flow whose divergence is that mass minus its net mass at \(a_\alpha\). Its norm is at most the total variation times the largest path norm, hence its squared norm is bounded by \[CL\bigl(LW(z_\alpha)\bigr)^2=CL^3W(z_\alpha)^2.\] At a given edge, only strands starting within \(CL\) can contribute. There are at most \(CL^3\) such starting slots, by deterministic counting and fixed slot multiplicity. Cauchy–Schwarz on each edge therefore bounds the squared norm of the sum of these local flows by \[ CL^6Q_j. \tag{78}\]

Let \(q_\alpha\) be the remaining net mass at \(a_\alpha\). The cancellation in Lemma 16, and the preceding conservation observation, give \[|q_\alpha|\le CL\theta_\alpha W(z_\alpha).\] Indeed the error from localization is rapidly small relative to \(LW(z_\alpha)\), so for an isolated strand it is absorbed into \(CL^{3/4}W(z_\alpha)\). Send \(q_\alpha\) to infinity along the unit flow from \(a_\alpha\). Bounded displacement of these representatives preserves the inverse-distance overlap estimate in Lemma 13, up to a constant. The squared norm of the resulting sum of flows is thus at most \[ C\sum_{\alpha,\alpha'} \frac{|q_\alpha q_{\alpha'}|} {1+|z_\alpha-z_{\alpha'}|_\infty} \le CL^2C_j. \tag{79}\] This is the only part of the obstacle flow that carries mass over arbitrarily large distances, and it is the part on which isolated-strand cancellation saves a factor \(L^{-1/2}\) after squaring.

For the cutoff sources, each injection site contributes total variation at most \(CLb_*\) in the block. Send their localized descendants directly to infinity using Lemma 13. If two injection sites are within \(CL\), use the constant overlap bound. There are at most \(CR^3L^3\) such ordered pairs. For sites farther apart, the descendants are still separated by a constant multiple of their starting distance, after increasing the threshold \(CL\). The inverse-distance sum over a ball of radius \(CR\) is \(CR^5\). Their total squared flow norm is at most \[ CL^2b_*^2(R^3L^3+R^5)\le CL^2b_*^2R^5. \tag{80}\] Adding the three flows and using the squared triangle inequality proves (75). Every source of these flows is within \(B_{6R}(x)\), since \(L=o(R)\).

We next average the bound over the exposure. Radial shells give \[ \sum_{|z-x|_\infty\le4R}W(z)^2 \le C\beta^{-2}D_*^{-1}, \qquad \sum_{|z-x|_\infty,|z'-x|_\infty\le4R} \frac{W(z)W(z')}{1+|z-z'|_\infty} \le C\beta^{-2}R. \tag{81}\] For the second estimate one may enlarge \(W(z)\) to \(\beta^{-1}(1+|z-x|_\infty)^{-2}\). Two comparable dyadic shells of radius \(b\) contribute at most \(C\beta^{-2}b\): for each site in one shell, the inverse-distance sum over the other is at most \(Cb^2\). Shells of radii \(c\le b/2\) contribute at most \(C\beta^{-2}c\). Summing first over \(c\) and then over the dyadic values of \(b\le CR\) proves the estimate without a logarithmic factor.

Proposition 8 bounds the expectation of \(Q_j\) by \(C\lambda\sum W^2\). To estimate \(C_j\), first take pairs whose starting sites are at distance at most \(C_1L\), where \(C_1\) is a sufficiently large fixed constant compared with the isolation radius. Charge only one obstacle, with probability \(C\lambda\), and allow \(CL^3\) possible second slots. The envelope \(W\) is comparable at these distances because \(L=o(D_*)\). This portion of \(L^2C_j\) is bounded in expectation by \[CL^5\lambda\sum W^2,\] which is absorbed into the \(L^6\) term in (76).

For a more distant pair, the two starting slots are distinct. If both are isolated, their occupation probability is at most \(C\lambda^2\), and \(\theta_\alpha\theta_{\alpha'}=L^{-1/2}\). If at least one is not isolated, a third distinct obstacle slot must lie within distance \(c_2L\) of that member. The separation ensures that this third slot is different from the other member of the pair. A union bound over its \(CL^3\) possible positions and the three-point snapshot bound give probability at most \(CL^3\lambda^3\). Applying (81) proves (76). Only one-time snapshot probabilities have been used; no independence between strands or between blocks is needed.

Finally, at the terminal cut, the localized mass from one strand has \(\ell^2\) norm at most its variation \(CLW(z_\alpha)\). The mass from one cutoff injection site has norm at most \(CLb_*\). Their supports have overlap at most \(CL^3\). Cauchy–Schwarz at each slot proves (77). The displayed bounds use measurable finite snapshot sums; choosing the flows measurably is unnecessary. ◻

Proof. For a bin beginning at elapsed time \(s\), put \(d_s=\sqrt{1+s}\). Lemma 17 bounds its source variation on \(\mathcal G\) by \(H_\beta\sum_\alpha W_{d_s}(z_\alpha)\), where the sum is over the obstacle slots above \(B_{4R}(x)\) at that bin’s starting slice. The single-box case of Lemma 20 gives \[\begin{align*} \left\|\sum_\alpha W_{d_s}(z_\alpha)\right\|_{L^2(\mathbb P_P)} &\le C\left(\sqrt\lambda\,\lVert W_{d_s}\rVert_2 +\lambda\lVert W_{d_s}\rVert_1\right)\\ &\le C\left(\beta^{-3/4}d_s^{-5/2} +\beta^{-3/2}d_s^{-1}\right)=J(d_s). \end{align*}\] Minkowski’s inequality in the probability space now gives \[\|\mathbf 1_{\mathcal G}M\|_{L^2(\mathbb P_P)} \le H_\beta\sum_{\mathrm{bins}}J(d_s) \le H_\beta\beta^{-3/4}.\] For the last inequality, bins have lengths between \(1\) and \(3\), so the sum of \(d_s^{-5/2}=(1+s)^{-5/4}\) is bounded and the sum of \(d_s^{-1}=(1+s)^{-1/2}\) is at most \(C\sqrt\beta\). Absorbing the square of a logarithmic factor into \(H_\beta\) proves (88). This use of Minkowski requires no independence of the bin snapshots.

Propagate all these sources to a slice in a block contained in the last quarter of the traversal. Every propagation has elapsed time at least \(\beta/4\). Localize to \(B_{6R}(x)\); by Lemma 12, the discarded total variation is rapidly small on \(\mathcal G\). Lemma 11 and substochasticity show that a localized unit point mass has total mass at most one and maximum entry at most \(C\beta^{-3/2}\). The additional spatial restriction only decreases the kernel. Consequently the localized signed source mass \(\nu_t\) is dominated in absolute value by a nonnegative mass \(\sigma_t\) such that \[\lVert \sigma_t\rVert_1\le M, \qquad \lVert \sigma_t\rVert_\infty\le CM\beta^{-3/2}.\]

At a fully regular such slice, use the unit flows of Lemma 13. For every slot \(a\), splitting distances at \(D_*\) gives \[\begin{align*} \sum_{a'}\frac{\sigma_t(a')}{1+\mathop{\mathrm{dist}}(a,a')} &\le CM\beta^{-3/2}D_*^2+CM D_*^{-1}\\ &\le CM\beta^{-1/2}. \end{align*}\] Superposing the unit flows with coefficients \(\nu_t(a)\) therefore gives a flow with divergence \(\nu_t\) and squared norm at most \[C\sum_{a,a'}\frac{\sigma_t(a)\sigma_t(a')} {1+\mathop{\mathrm{dist}}(a,a')} \le CM^2\beta^{-1/2}.\] There are blocks of total length at least \(c\beta\) contained in the last quarter, and at least half of each is fully regular. Average (74) over this regular set. Cauchy–Schwarz in time and (73) give (89). ◻

Proof of Proposition 21. The two collections exhaust the sources in the chosen direction. Combine (85) and (89), and choose a measurable coefficient with \[X_d^2\le C\left(\sum_{j=1}^nF_j/L+K_n+M^2\beta^{-3/2}\right).\] All expressions on the right are measurable functions of the exposure; none depends on the test vector. Equations (86), (87), and (88) bound its expectation on \(\mathcal G\). With \(L=\beta^\delta\) and \(R=\beta^{3/2+\delta/4}\), their sum is at most \[ C\left[ \beta^{-3+4\delta} +\beta^{-5/2-\delta/4} +\beta^{-4+13\delta/4} +\beta^{-4+5\delta} +\beta^{-13/2+17\delta/4} +H_\beta\beta^{-3}\right] =o(\beta^{-5/2}), \tag{90}\] since \(\delta=1/200\). The term \(\beta^{-5/2-\delta/4}\) covers both the isolated net-mass flows and the cutoff flows. All localization errors were estimated in \(\ell^1\) against test vectors of supremum norm at most a constant times \(C_0\), so they give the uniform rapidly small remainder in (71). The argument used elapsed time and the ordered kernels throughout; it applies to \(U\) and to \(U^*\) with the same bounds. Adding the two expected squares proves (72). ◻

Proof of Proposition 5. Fix the cube, pins, and starting slot in the proposition, and expose the cycles under \(\mathbb P_P\), \(P=E\cup\{i\}\). Use the trial vector \(h\) and constant \(c\) from (34) and (32). On the good event \(\mathcal G\) of Proposition 10, Proposition 21, applied with a fixed supremum bound \(2\), gives a measurable \(X\) such that \[\lvert \langle \mu_d,z\rangle\rvert\le X\sqrt{e_z}+\varepsilon_\beta, \qquad d=+,-,\] for vectors vanishing on killed cut slots with \(\lVert z\rVert_\infty\le2\), where \(\varepsilon_\beta\) is rapidly small and \[\mathbb E_P[\mathbf 1_{\mathcal G}X^2]=o(\beta^{-5/2}).\] If the two directions initially give different coefficients, take their maximum; this preserves the displayed expectation bound. The source identities (54) identify these residuals with those in (8). Since \(0\le h\le1\), the averaged capacity criterion (14) applies. Proposition 19 and the probability estimate in Proposition 10 give \[ r:=\mathbb E_Pq_i=1-c+o(\beta^{-5/2}). \tag{91}\] Every error here is uniform over the cube and the allowed pin sets. In particular, the good-event estimate is used only through its probability on the complement; there \(q_i\in[0,1]\) suffices.

By Lemma 14, \(v(x)=O(\beta^{-3/2})\), so \[1-c=\frac{v(x)}{1+v(x)}=v(x)+O(\beta^{-3}).\] The exact pin identity (3) now yields \[ d_E(i)=\frac{r}{1+r} =v(x)+o(\beta^{-5/2}) =\frac1\ell\sum_{n\ge1}p_{n\beta}(0) +o(\beta^{-5/2}). \tag{92}\] Indeed \(r=O(\beta^{-3/2})\), so the second rational transformation also costs only \(O(\beta^{-3})\). By (43), the last sum is \(n_S^{\rm sw}(\beta)\). The uniform little-oh in (92) is equivalent to the asserted function \(\varepsilon_S(\beta)\to0\) after choosing a sufficiently large \(\beta_0(S)\). Equations (91) and (43), together with \(1-c=v(x)+O(\beta^{-3})\), also give \(r=\ell^{-1}n_S^{\rm sw}(\beta)+o(\beta^{-5/2})\), as asserted. ◻

Proof. After extracting the all-up factor, the field-tilted picture weight is \[W_t(\omega)=\prod_C(1+e^{-\beta t a_C}).\] Let \(\mathbb E_t\) denote expectation under this tilted picture law. Given the picture, each cycle is up independently with probability \((1+e^{-\beta t a_C})^{-1}\ge1/2\). If \(b_0\) counts cycles meeting \(D_0\), Jensen’s inequality therefore gives \[1\ge\frac{Z_N^+(t)}{Z_N(t)} \ge\mathbb E_t2^{-b_0} \ge\exp[-(\log2)\mathbb E_t b_0].\] Let \(J_\partial\) count marks on slot edges incident to a boundary site, and let \(q_\partial\) be the number of those edges. Every boundary-meeting cycle either has a boundary slot at time zero or passes through one of these marks. A mark belongs to at most two cycles, so \[b_0\le\ell|\partial\Lambda_N|+2J_\partial, \qquad q_\partial\le6\ell^2|\partial\Lambda_N|.\] Insertion of one mark joins two cycles or splits one. Each factor of \(W_t\) lies in \([1,2]\), and at most two new factors appear, whence \(W_t(\omega\cup\{u\})\le4W_t(\omega)\) for almost every inserted mark \(u\). The Poisson insertion formula at rate \(1/2\) implies \[\mathbb E_t J_\partial\le2\beta q_\partial \le12\beta\ell^2|\partial\Lambda_N|.\] The preceding inequalities, divided by \(\beta V_N\), prove (94). The pressure limit follows because the boundary-to-volume ratio tends to zero. ◻

Proof of Theorem 1. Fix \(S\) and a sufficiently large \(\beta\). The finite-volume magnetization is bounded, so differentiation under the sum in (93) is justified. Equations (96) and (95) give \[ S-(p_N^+)'(t) =\frac1{V_N}\widehat\mathbb E_{N,t}N_\downarrow =\frac1{V_N}\sum_{j\text{ at time }0} \widehat\mathbb E_{N,t}d_{D_0\cup A}(j). \tag{98}\] Here \(d_E(j)\) means the down probability also for pinned slots, where it is zero. Let \(\rho_N\) be the fraction of sites \(x\) for which \(B_{50R_*}(x)\) is interior to \(\Lambda_N\). For such a site let \(\mathcal A_x\) be the event that no selected pin lies above that ball. A union bound using (97) gives \[\widehat\mathbb P_{N,t}(\mathcal A_x^c) \le C_{S,\beta}\,b_\beta(t),\] where \(C_{S,\beta}\) is finite and independent of \(N\). On \(\mathcal A_x\), Proposition 5 applies to each of the \(\ell\) slots at \(x\), for every conditional pin set \(A\). On its complement, and at sites outside the interior region, use the boundedness of down probabilities. Summing the slot estimates in (98) gives \[ \lvert S-(p_N^+)'(t)-n_S^{\rm sw}(\beta)\rvert \le \ell\varepsilon_S(\beta)\beta^{-5/2} +C_S(1-\rho_N)+C_{S,\beta}b_\beta(t). \tag{99}\] Here \(n_S^{\rm sw}(\beta)\) is bounded for the large values of \(\beta\) under consideration, so its contribution on exceptional sites is included in \(C_S\).

Integrate (99) over \(t\in[0,h]\), for \(h>0\), and divide by \(h\). Now take \(N\to\infty\), keeping \(\beta\) and \(h\) fixed. The radius is fixed in this limit, so \(\rho_N\to1\), and Lemma 24 gives convergence of \(p_N^+\) at both field endpoints. We obtain \[\lvert S-\frac{p_{S,\beta}(h)-p_{S,\beta}(0)}h -n_S^{\rm sw}(\beta)\rvert \le \ell\varepsilon_S(\beta)\beta^{-5/2} +\frac{C_{S,\beta}}h\int_0^h b_\beta(t)\,dt.\] For this fixed \(\beta\), the last term tends to zero as \(h\downarrow0\). Convexity and the uniform magnetization bound give the right derivative of the limiting pressure. Therefore \[\lvert S-m_S(\beta)-n_S^{\rm sw}(\beta)\rvert \le\ell\varepsilon_S(\beta)\beta^{-5/2}.\] Finally let \(\beta\to\infty\). Since \(S\) is fixed, multiplication by \((\beta S)^{5/2}\) proves the theorem. ◻

Proof of Corollary 2. Put \(t=\beta S\). Independence of the three coordinates gives \[p_\beta(0)=I(t)^3,\qquad I(t)=\frac1{2\pi}\int_{-\pi}^{\pi} e^{-2t(1-\cos k)}\,dk.\] We claim, uniformly for \(t\ge1\), that \[ I(t)=\frac1{\sqrt{4\pi t}} \left(1+\frac1{16t}+O(t^{-2})\right). \tag{100}\] To see this, set \(u=\sqrt t\,k\) and first restrict to \(|u|\le t^{1/10}\). Taylor’s formula gives \[2t(1-\cos(u/\sqrt t)) =u^2-\frac{u^4}{12t}+O(u^6/t^2),\] and hence, on this interval, \[e^{-2t(1-\cos(u/\sqrt t))} =e^{-u^2}\left(1+\frac{u^4}{12t} +O\!\left(\frac{u^6+u^8}{t^2}\right)\right).\] The bound \(2(1-\cos k)\ge c k^2\) controls the omitted interval by \(O(e^{-c t^{1/5}})\) after the same rescaling. Extending the Gaussian integrals to \(\mathbb R\) has an error of this size as well. Since \[\int_\mathbb Re^{-u^2}\,du=\sqrt\pi, \qquad \int_\mathbb Ru^4e^{-u^2}\,du=\frac34\sqrt\pi,\] the formula (100) follows, with a constant that can be enlarged to cover all \(t\ge1\). Cubing gives \[p_\beta(0) =(4\pi t)^{-3/2}\left(1+\frac3{16t}+O(t^{-2})\right).\] Apply this uniform estimate at \(nt\), \(n\ge1\), and sum. The remainder is bounded by \(Ct^{-7/2}\sum_{n\ge1}n^{-7/2}\), so \[ n_S^{\rm sw}(\beta) =\frac{\zeta(3/2)}{8\pi^{3/2}}(\beta S)^{-3/2} +\frac{3\zeta(5/2)}{128\pi^{3/2}}(\beta S)^{-5/2} +O((\beta S)^{-7/2}). \tag{101}\] Theorem 1 now gives the asserted expansion of \(S-m_S(\beta)\). ◻