Proof. An empty word has \(n\) burger letters and \(n\) order letters. If \(f\) of its orders are flexible, its unconditioned probability is \[4^{-n}\left(\frac{1-b_{\mathrm F}}4\right)^{n-f} \left(\frac{b_{\mathrm F}}2\right)^f =c_{n,q}\left(\frac{2b_{\mathrm F}}{1-b_{\mathrm F}}\right)^f.\] The bijection has \(\ell=f+1\). Since \(2b_{\mathrm F}/(1-b_{\mathrm F})=\sqrt q\), the displayed weight is a factor independent of \((M,A)\) times \(q^{\ell/2}\). Rooted isomorphism classes correspond under the same bijection. Re-rooting does not change the weight. Equivalently, on an unrooted class one may choose a uniform dart; the automorphism group acts freely on darts, so this gives the asserted rooted law. A corner and its outgoing dart are equivalent root conventions. ◻

References and conventions. Part (i) is the bilateral inventory invariance principle of (Sheffield 2016b) and the finite-volume result (Gwynne and Sun 2015, Theorem 1.8). Parts (ii)–(iii) are the mating-of-trees results of (Duplantier et al. 2021) and (Miller and Sheffield 2019, Theorem 1.1); cone interval-surface determination in the beaded regime is stated explicitly in (Gwynne et al. 2026, Lemma 3.13). Independent random cuts follow by conditioning on their values, and countable intersections allow the stated later selections. This is intrinsic determination, modulo quantum-surface equivalence. The finite-sphere proof below instead transfers spatial observations using actual local fields; it does not require a sphere version of the isolated-interval reconstruction statement.

The finite-chain convention in (iii) is the strict-chord relation of (Duplantier et al. 2021, Lemma 8.9), with its generated classes identified in (Duplantier et al. 2021, Lemmas 8.14–8.15). A weak scalar chord splits into at most two strict chords: otherwise two distinct interior local minima would have the same height, excluded by 20. The same frontier-tree relation applies to the sphere’s whole-plane space-filling curve. Quantum-area reparametrization and its quantum boundary-length coordinates preserve the tree identifications; the initial and final times are identified at the exploration root.

For (iv), (Gwynne, Miller, et al. 2019, Lemma 2.4) reconstructs directed parameterized local traversals with a buffer; its quantum-area clock is described in the proof of (Gwynne, Miller, et al. 2019, Lemma 4.3). The frontier arcs are the labelled local flow-line boundaries in that construction. Their quantum lengths use the boundary Gaussian multiplicative chaos construction and its conformal coordinate rule (Sheffield 2016a, sec. 1.2 and 5.1). More explicitly, choose the specified side of a compact frontier subarc and a sided domain inside the buffer, then straighten that domain conformally and construct boundary length from the restriction of the actual normalized real field. The boundary coordinate rule identifies this measure with the same side’s global quantum boundary length and makes it independent of the chosen smaller domain. Exhaust the arc by compact subarcs to include its ends. The sided domains and prime-end labels come from the local imaginary field, so this construction uses only the indicated field restrictions. Contained Poisson-cell locality is also given in (Contreras Hip and Gwynne 2026, Lemma 3.7). We always shrink the testing region before using these locality statements.

The imaginary whole-plane field is taken modulo its angular period, with its stationary phase, rather than circle-average-pinned with a prescribed phase. Local decompositions consist of Dirichlet GFF parts and harmonic parts modulo that period. The real and imaginary fields are independent before the curve is clocked by real-field area. ◻

Proof. We use (Borga et al. 2026, Definition 2.2), whose three-marked construction agrees with the ordinary DMS sphere by (Aru et al. 2017, Theorem 1.1 and Remark 1.2). Write \(G_1\) for the whole-plane GFF pinned on the unit circle, and let \(d\) be the deterministic Liouville drift in that definition. The sphere field is obtained by a positive finite integrable weighting of \[N(G_1+d),\qquad N(H)=H-\gamma^{-1}\log\mu_H(\mathbb C).\] The drift has logarithmic singularities only at \(0,1,\infty\); on compact subsets avoiding \(0,1\) it has finite Dirichlet energy. In particular, the possible crease of \(\log(|z|\vee1)\) at the unit circle causes no difficulty for Cameron–Martin comparison.

Couple the pinned GFFs by \(G_1=G_R-(G_R)_1(0)\). Since \(N(H+c)=N(H)\), including for a field-dependent scalar \(c\), the normalized field is also \(N(G_R+d)\). The original weighting is still a positive finite integrable density on this Gaussian probability space, although its formula after changing the pin need not be the original one. It therefore suffices first to prove the domination for the unweighted field \(N(G_R+d)\).

Choose \(g\in C_c^\infty(B_R(0))\) with \(0\le g\le1\), equal to one on a neighborhood of \(\overline U\). Gaussian orthogonal decomposition in the Cameron–Martin direction \(g\) gives \(G_R=Xg+G^\perp\), where \(X\) is a nondegenerate real Gaussian independent of \(G^\perp\). Conditional on \(G^\perp\), set \(\nu=\mu_{G^\perp+d}\) and \[A(x)=\int_{\mathbb C}e^{\gamma xg(z)}\,\nu(dz), \qquad T(x)=x-\gamma^{-1}\log A(x).\] Almost surely \(\nu\) is finite, nonzero, and assigns positive mass to an open set outside \(\operatorname{supp}g\). These assertions follow from the corresponding properties of \(\mu_{G_R+d}\), since removing the bounded function \(Xg\) changes that measure by a bounded positive factor. The same comparison applies to every finite value of \(x\). Differentiation under the integral on compact \(x\)-intervals yields \[T'(x)= \frac{\displaystyle\int_{\mathbb C}(1-g(z))e^{\gamma xg(z)}\,\nu(dz)} {\displaystyle\int_{\mathbb C}e^{\gamma xg(z)}\,\nu(dz)}>0 .\] Thus \(T\) is a strictly increasing \(C^1\) diffeomorphism onto its image. Conditional on \(G^\perp\), the law of \(T(X)\) has a Lebesgue density and hence is absolutely continuous with respect to the nondegenerate Gaussian law of \(X\). On \(U\), \[N(G_R+d)|_U=G^\perp|_U+d|_U+T(X).\] Integrating the preceding conditional domination proves \[\operatorname{Law}(N(G_R+d)|_U) \ll \operatorname{Law}((G_R+d)|_U).\] A finite-energy cutoff equal to \(d\) near \(\overline U\), supported away from \(0,1\) and the pinning circle, is a Cameron–Martin shift of \(G_R\). Its restriction removes \(d|_U\) by mutual absolute continuity. Restoring the positive weighting proves the first assertion.

For the comparison with the cone, (Duplantier et al. 2021, Definition 4.10 and the following discussion) identifies its field inside the unit disk with \(G_1-\gamma\log|z|\). Under the deterministic coordinate map \(z\mapsto z/R\), its representation on \(B_R(0)\) is \[G_R-\gamma\log|z|+(\gamma-Q)\log R.\] On \(\overline U\), the added deterministic function has a finite-energy cutoff away from \(0\) and \(\partial B_R(0)\). Cameron–Martin gives mutual absolute continuity with \(G_R|_U\). In particular the direction needed for transferring a cone probability-one event is \[\operatorname{Law}(h|_U) \ll \operatorname{Law}(G_R|_U) \ll \operatorname{Law}(h^{\mathrm{cone},R}|_U).\] No comparison across the cone’s pinning circle is required.

Finally, the complete imaginary field is independent of the real field and has the same stationary-phase law in both constructions. Taking the product with this common law, and then taking measurable images to obtain its curve, preserves each asserted domination. ◻

References and use. Metric construction, locality, and Weyl scaling are supplied by (Gwynne and Miller 2021b); properness is (Dubédat et al. 2020, Lemma 3.8), and general conformal coordinate changes are (Gwynne and Miller 2021a, Theorem 1.3). Fixed-pair geodesic uniqueness is (Miller and Qian 2020, Theorem 1.2). The internal connection and dyadic-square estimates are (Dubédat et al. 2020, Lemma 3.20, Equations (3.66)–(3.67)). Strong confluence and its interior-endpoint form are (Bhatia and Kavvadias 2026, Theorem 2 and Proposition 6). Local statements transfer to fixed compact subsets of ordinary cone or sphere charts by local absolute continuity. For confluence, first take a shorter geodesic subarc in a buffered chart. Its sufficiently short neighboring connections cannot reach the chart boundary, whose internal distance from a compact inner neighborhood is positive. They are consequently determined by the same internal metric, so the whole-plane confluence event applies locally. This does not require absolute continuity of an entire surface’s global geodesics. Singular marked points will be treated separately.

For clarity, the open-square internal bound in (Dubédat et al. 2020, Proposition 3.10) does not by itself include boundary points. To access a specified flat-face or corner point of a rectangle of size \(\delta\), choose inward points at Euclidean distances comparable to \(2^{-j}\delta\). Consecutive inward points can be joined inside the rectangle by a bounded chain of balls or dyadic squares of that scale. The internal connection bound gives total cost at most \[C\sum_{j\ge0}(2^{-j}\delta)^\chi =\frac{C\delta^\chi}{1-2^{-\chi}}.\] The chain is Cauchy for the internal metric. Its Euclidean limit is the prescribed boundary point, and the same inward-chain construction between two approximations shows uniqueness in the internal completion. A bounded number of these connections also crosses any fixed-shape polygonal collar while staying inside that collar. No bound uniform over degenerating aspect ratios is claimed.

These properties are asserted only for \(D_h\). No analogous Weyl rule, metric property, or endpoint modulus is assumed for a subsequential discrete passage rule. ◻

Proof. Every burger produced inside the word is fresher than every burger in an initial past. Induction over symbols therefore shows that the available internal burgers and their internal matching are independent of that past. A typed order uses an internal burger of its type whenever one is available. An \(F\) uses the freshest internal burger whenever the internal stack is nonempty. A step which instead uses the past does not change the internal stack. In particular, each externally unmatched \(F\) occurs when the entire internal burger stack is empty. Replacing only these \(F\)’s by arbitrary typed orders therefore preserves every internal pair and every unpaired position.

In the resulting fully typed word let \(O_i(k)\) and \(B_i(k)\) count the unpaired orders and burgers of type \(i\). An order is unpaired exactly when the corresponding coordinate reaches a new negative running minimum. Thus \[O_i(k)=-\min_{0\le j\le k}Z_i^\sigma(j),\qquad B_i(k)=Z_i^\sigma(k)-\min_{0\le j\le k}Z_i^\sigma(j).\] Summing \(O_i+B_i\) proves Equation (12). Since \(Z^\sigma(0)=0\), its summands are bounded by twice the corresponding coordinate oscillations and by three times the maximum absolute coordinate height. This proves Equations (13) and (14). Also \(Z_i^\sigma=B_i-O_i\), so \(\|Z^\sigma(k)\|_1\le L_k(w)\); the triangle inequality proves Equation (15).

For Equation (16), process \(v\) on the retained burger stack of \(u\). All its internal pairs persist. Its orders which were internally unpaired either remain unpaired or consume a retained burger of \(u\). Such a consumption removes two letters from the two separate reductions; no new letter is created.

An order unpaired in a prefix stays unpaired in every longer prefix, since orders only consume earlier burgers. Hence \(F_k\) is nondecreasing. The increments for two exterior typings can differ only at these unpaired \(F\) positions, and each difference has \(\ell^1\) norm two. Summing the increment differences gives Equation (17). ◻

Proof. Adjoin an independent bilateral iid past and resolve the block using that past. Sheffield’s invariance principle (Sheffield 2016b, Theorem 2.5), equivalently (Gwynne, Mao, et al. 2019, Theorem 1.5), gives tightness of the family \[m^{-1/2}\max_{i,j\le m}|Z_i(j)|.\] Equation (14) gives tightness of the block-only random variables \(M_m/\sqrt m\). Choose \(A>0\) such that for every \(m\ge1\), with \(a_m=\lceil A\sqrt m\rceil\), \[\mathbb P[M_m\ge a_m]\le\tfrac12.\] Fix \(m\), write \(a=a_m\), and, in an infinite iid raw word, restart whenever the current block first has reduced length \(a\). More precisely, let \(S_0=0\) and \[S_{r+1}=\inf\{t>S_r:|R(X_{S_r+1}\cdots X_t)|=a\}.\] A single added symbol changes reduced length by one in absolute value, so these stopping times have no overshoot. They are finite almost surely: reduced length dominates absolute net burger count, which is a simple symmetric random walk. Strong Markov for the raw iid symbols makes \(S_r-S_{r-1}\) iid. Put \(N_m=\max\{r:S_r\le m\}\). By Equation (16), every prefix up to time \(m\) has reduced length at most \(a(N_m+1)\). Therefore, for integers \(r\ge0\), \[\mathbb P[M_m>ra]\le\mathbb P[N_m\ge r] \le\prod_{j=1}^r\mathbb P[S_j-S_{j-1}\le m] \le2^{-r}.\] Since \(a_m/\sqrt m\le A+1\), take \(r=\lfloor x\sqrt m/a_m\rfloor\) to obtain Equation (18) with \(C=2\) and \(c=\log(2)/(A+1)\). For \(0<\lambda<c\), integration of the tail gives \(K\le1+2\lambda/(c-\lambda)\). The estimate depends only on the raw block, proving its conditional version. Iterating this conditional moment estimate also bounds the exponential moment of a sum over successively exposed fresh blocks whose lengths are chosen before the respective blocks are inspected. ◻

Proof. The only additional public input is (Gwynne, Mao, et al. 2019, Lemma 3.7, Equation (3.15)). Put \[\mu'=\frac{\pi}{2\left(\pi+ \arctan\bigl(\sqrt{1-2b_F}/b_F\bigr)\right)}\in(1/3,1/2).\] That lemma says that for each \(\nu>\mu'\), \(\mathbb P[\exists i\ge n:F_i\ge i^\nu]\) decays faster than every negative power of \(n\). Choose \(\nu\in(\mu',1/2)\). It follows that \(F_m/\sqrt m\to0\) in probability. Its monotonicity, already proved in Lemma 12, simultaneously controls all prefixes.

Adjoin an independent bilateral past, and let \(\tau\) be its genuine typing. Theorem 2.5 of (Sheffield 2016b) gives \(m^{-1/2}Z^\tau(m\,\cdot)\Rightarrow\nu_{b_F}\). For each bounded \(1\)-Lipschitz test with absolute value at most one, Equation (17) bounds the difference between its expectations under \(\sigma\) and \(\tau\) by \[\mathbb E\left[\min\{2,2F_m/\sqrt m\}\right]\longrightarrow0.\] This bound is simultaneous over all \(\sigma\), proving Equation (20). Choose the auxiliary bilateral past independently of the previously observed data to obtain the conditional statement. For deterministic disjoint blocks choose independent auxiliary bilateral pasts for the respective blocks; their reference contours are independent, and the same deterministic error bound proves the finite-product assertion. ◻

Proof. Brownian scaling reduces the assertion to \(t=1\). Put \(D=W\cap B(0,1)\) and use polar coordinates \((r,\phi)\), with \(0<\phi<\theta\). The probability \(q(r,\phi)\) of leaving \(D\) through its circular arc before a side has the harmonic expansion \[q(r,\phi)=\frac4\pi \sum_{\substack{m\ge1\\m\text{ odd}}} \frac{r^{m\nu}}{m}\sin(m\nu\phi).\] For \(r<1\) this is the harmonic extension of the sine series of the constant function one on the circular arc, with zero side values. One can first take partial sums on smaller concentric sectors and then use uniqueness of bounded harmonic extensions. The two corner points have harmonic measure zero. Absolute summation gives \(q(r,\phi)\le C r^\nu\) for \(r\le1/2\); the bound for larger \(r\) follows from \(q\le1\).

For the exit time of \(D\), set \[f(\phi)=\frac12\left(\frac{\cos(2\phi-\theta)}{\cos\theta}-1\right).\] Since \(\cos\theta<0\) and \(\cos(2\phi-\theta)\ge\cos\theta\), one has \(f\le0\). The function vanishes at the two sides and \(f''+4f=-2\). Choose \(A<\infty\) such that \(-f(\phi)\le A\sin(\nu\phi)\); the ratio has finite limits at the two endpoints because both numerator and denominator vanish linearly. Then \[v(r,\phi)=A r^\nu\sin(\nu\phi)+r^2 f(\phi)\] is nonnegative on \(D\), satisfies \(-\tfrac12\Delta v=1\), and obeys \(v\le A r^\nu\). Nonnegativity follows from \(\nu<2\) and \(r^2 f\ge r^\nu f\). Applying the stopped Itô formula on compact subdomains, then increasing the subdomains to \(D\), gives \(\mathbb E_z\tau_D\le v(z)\le A r^\nu\). Therefore \[\mathbb P_z(\tau_W>1) \le q(z)+\mathbb P_z(\tau_D>1) \le C r^\nu.\] The one-dimensional assertion follows from the reflection principle. ◻

Proof. Whitening and Lemma 16 prove Equation (21). Let \(p_t^D(x,y)\) be the killed transition density in the half-line or whitened wedge. Its symmetry, the semigroup identity, and the bounded free Gaussian transition density imply \[p_1^D(x,y) =\int_Dp_{1/2}^D(x,z)p_{1/2}^D(z,y)\,dz \le C\int_Dp_{1/2}^D(z,y)\,dz =C\mathbb P_y(\tau_D>1/2).\] Integrating over the whitened landing set, which has volume \(O(h^k)\) and lies at distance \(O(h)\) from the corner, proves Equation (22), uniformly in \(x\). Scaling handles the duration range. Conditioning on past-measurable starting data leaves the same Brownian increment law, proving the last assertion. ◻

Proof. For fixed \(R\), Proposition 14 gives uniform weak convergence of the block increment paths. Enlarge all path and landing tolerances by a fixed small multiple of \(h\); the corresponding closed Brownian events bound the desired limsup. Their probabilities are bounded by Lemma 17, with a changed constant. Uniformity over starting points in a fixed compact set follows by a finite net of starting heights and one more tolerance enlargement. For the landing test, starting points outside a sufficiently large compact set require a correspondingly large block fluctuation; Proposition 13 makes that probability smaller than any prescribed fixed power of \(R\) by choosing the compact set after fixing \(R\). Starting below a wall has probability zero unless an already allowed tolerance covers the discrepancy. Finally choose the block length large enough that the uniform weak-convergence error is less than the desired small power of \(R\). This proves the stated bounds, including conditioning on a fresh raw past. ◻

Proof. We first record the required survival bounds. For fixed \(v>0\) and \(0<r\le1\), the reflection principle gives \[\mathbb P\left[\inf_{0\le u\le v}L(u)\ge-r\right]\le C_vr.\] Whitening the covariance sends the two lower walls to a wedge of opening \(\theta\). Lemma 16 and Brownian scaling give \[ \mathbb P\left[\inf_{0\le u\le v}L(u)\ge-r, \inf_{0\le u\le v}R(u)\ge-r\right] \le C_v r^{2d_2}. \tag{23}\] Only this elementary Brownian survival estimate is used here; no lattice or passage-kernel estimate is involved.

Partition \(K\) into intervals \(I=[s,s+h]\) of length \(h=2^{-n}\), allowing two extra intervals at its ends. Fix \(\varepsilon\in(0,1/2)\) and put \(r=h^{1/2-\varepsilon}\). Almost surely, for all sufficiently large \(n\), both coordinate oscillations on every such interval are at most \(r\). For example this follows from Brownian Hölder continuity with exponent \(1/2-\varepsilon/2\). Suppose such an interval contains a record time \(t\). If a required arm points forward, then its constraint implies \[\inf_{0\le u\le\delta/2} \{Z_i(s+h+u)-Z_i(s+h)\}\ge-r;\] if it points backward, the analogous constraint is \[\inf_{0\le u\le\delta/2} \{Z_i(s-u)-Z_i(s)\}\ge-r.\] These implications hold when \(h\le\delta/2\). They use deterministic walls \(-r\) relative to the respective cell endpoints, so their probability bounds do not condition on a selected record time.

For two arms on the same side, Equation (23) bounds this necessary exterior event by \(Cr^{2d_2}\). For arms on opposite sides the two events use disjoint time increments, and the reflection principle bounds their product by \(Cr^2\). In the last assertion of the Lemma, the necessary exterior event uses two coordinate arms on one side and one coordinate arm on the other; its probability is at most \(Cr^{2d_2+1}\).

Here is the resulting covering calculation, including the dimension needed below. If the survival power is \(\alpha\), the number \(N_h\) of cells satisfying the necessary exterior event obeys \[ \mathbb E N_h\le C h^{-1}r^\alpha =C h^{-1+(1/2-\varepsilon)\alpha}. \tag{24}\] For \(\alpha\le2\) and every \(\zeta>0\), Markov’s inequality and Borel–Cantelli imply, along the dyadic sequence, \[N_h\le h^{-1+(1/2-\varepsilon)\alpha-\zeta} \quad\text{eventually almost surely}.\] Each occupied cell contributes a set of diameter at most \(\sqrt2r\) to the \(Z\)-image and a set of diameter at most \(r\) to a coordinate image. Consequently their upper box dimensions are at most \[\frac{1-(1/2-\varepsilon)\alpha+\zeta} {1/2-\varepsilon}.\] Letting \(\varepsilon,\zeta\) decrease to zero through countable sequences gives \(2-\alpha\). Taking \(\alpha=2d_2\) or \(\alpha=2\) proves the two image assertions. If \(\alpha=2d_2+1>2\), choose \(\varepsilon\) so small that \((1/2-\varepsilon)\alpha>1\). The right side of Equation (24) is summable over dyadic \(h\), so eventually there are no candidate cells. This proves the nonexistence assertion. ◻

Proof. We shall use the following precise source of the height densities. Take deterministic intervals \(I=[p,q]\) and \(J=[r,v]\) with \(q<r\). The two paths \[(Z(t)-Z(q))_{t\in I},\qquad (Z(t)-Z(r))_{t\in J}\] and the increment \(G=Z(r)-Z(q)\) are independent. The last has a nonsingular Gaussian density on \(\mathbb R^2\). Its coordinate projections have Gaussian densities on \(\mathbb R\). Thus, conditional on the two displayed paths, a prescribed Lebesgue-null set of relative heights is missed almost surely. Conditioning is only on increments inside the two separated windows; it is never on a putative contact or chord.

For completeness this also proves the scalar minimum facts. The minimum values on two disjoint compact rational intervals have probability zero to agree: express the left minimum relative to its right endpoint and the right minimum relative to its left endpoint, and condition on both window paths. Equality would prescribe one value for the independent scalar Gaussian increment in the gap. Taking a countable union shows that no such pair of rational intervals has equal minima. Two distinct minimizers on a compact rational interval could be enclosed in disjoint compact rational subintervals of that interval with the same minimum, so each rational compact interval has a unique minimizer. Every local minimum is the minimizer on one of these intervals. This makes it strict, makes the set countable, and also shows that distinct local minima have distinct values. If a scalar horizontal class contained four ordered visits, its middle two visits would be distinct local minima at the same height. This proves (i).

Suppose that \(s<t\) were a chord in both coordinates. Choose disjoint rational windows \(I,J\) containing \(s,t\) in their interiors, and a positive rational \(\delta\) small enough that the forward arms from \(s\) and the backward arms from \(t\) of length \(\delta\) lie in these windows. Such choices are countable and cover every proposed nonzero chord. Define compact sets of relative heights using all these local joint record times in the respective windows: \[A=\{Z(u)-Z(q):u\in I\text{ has both forward arms of length }\delta\},\] \[C=\{Z(w)-Z(r):w\in J\text{ has both backward arms of length }\delta\}.\] Here the record times are restricted to closed subintervals leaving the indicated arm margins; countably many choices of these subintervals suffice. By Lemma 19, both sets have upper box dimension at most \(2(1-d_2)<1\). Hence their difference set \(A-C\) has two-dimensional Lebesgue measure zero. Explicitly, for any \(a\in(2(1-d_2),1)\) each set has an \(\epsilon\)-cover with \(O(\epsilon^{-a})\) balls; pairwise differences cover \(A-C\) with \(O(\epsilon^{-2a})\) balls, whose total area tends to zero. The endpoint equality \(Z(s)=Z(t)\) would imply \(G\in A-C\). The conditional Gaussian density therefore excludes the proposed common chord, proving (ii).

Now let \(a<b<c\) be an \(L\)-branch triple, and suppose an \(R\)-chord has an endpoint \(t\in\{a,b,c\}\) and another endpoint \(u\ne t\). If \(t=b\), the \(L\)-coordinate has lower-wall arms on both sides of \(b\), and the \(R\)-chord supplies a lower-wall arm on the side pointing toward \(u\). Truncate all three arms to a common positive rational length. The final assertion of Lemma 19 excludes this case, since \(d_2+1/2>1\).

If \(t=a\) or \(t=c\), retain a small window around \(t\) disjoint from a small window around \(b\). At \(t\) the \(L\)-arm points toward \(b\), and the \(R\)-arm points toward \(u\). Truncate both to a common positive rational length so that they lie in the outer-visit window. They can point in the same direction or in opposite directions; both possibilities are included. In relative \(L\)-heights, let \(A\) be the image of all local joint record times of this specified orientation in the outer-visit window. Lemma 19 makes \(A\) a one-dimensional Lebesgue-null set: its upper box dimension is at most \(2(1-d_2)<1\) in the first case and zero in the second.

In the middle-visit window let \(C\) be the countable set of minimum values on its compact rational subintervals, measured relative to the window’s reference endpoint. Since \(b\) is a local minimum, its relative \(L\)-height belongs to \(C\). Use the right endpoint as reference for the earlier window and the left endpoint for the later window. The equality \(L(t)=L(b)\) would force the scalar Gaussian increment between the windows to belong to either \(A-C\) or \(C-A\). Both are Lebesgue-null, being countable unions of translates of a null set. Conditional on the two window paths the gap increment has a density, so the equality has probability zero. Countably many rational windows, arm lengths, and orientations cover every outer contact. This proves (iii); interchanging the two coordinates gives the other color. The argument allows \(u\) to equal another member of the branch triple, as well as to lie between or outside its visits.

Every exclusion just proved is a null event specified in one window or in finitely many separated windows together with their relative heights. It therefore transfers under the stated joint local absolute continuity. On strict interior windows a Brownian quadrant excursion has this absolute continuity by its killed transition densities and the Markov bridge construction, first restricting to compact positive endpoint ranges and then exhausting them. The excursion’s common initial/final endpoint is outside the assertion. ◻

Proof. We first state the elementary Brownian estimates used in the proof. For a Brownian block of fixed duration \(s\) and \(k\in\{1,2\}\) constant lower walls, let \(\tau_k\) be the first violation, in the constrained coordinates. For \(0<h\le1\), \[\begin{align*} \sup_{z\in[0,h]^k}\mathbb P_z(\tau_k>s) &\le C_s h^{2d_k},\tag{27}\\ \sup_{z\in[0,\infty)^k} \mathbb P_z(\tau_k>s,\ Z(s)\in[0,h]^k) &\le C_s h^{k+2d_k}. \tag{28}\end{align*}\] Fixed multiples of \(h\) change only \(C_s\). Both bounds follow from Lemma 17, after translating the constant walls to zero and fixing the positive duration. The estimates remain valid conditionally when the walls and starting point are known before the block is exposed. An initial point below a tested wall has zero probability of satisfying the gate.

Choose \(\vartheta\in(0,1/2)\) with \(2\beta e\vartheta<\varepsilon\), and put \(\eta=\delta^{1/2-\vartheta}\). Cover each \(K_i\) by intervals of a deterministic mesh of width \(\delta\), retaining the finitely many mesh cells which meet \(K_i\). For sufficiently small \(\delta\) these cells, and an \(s\)-interval on either side of each cell, lie in \(J_i\). Let \(\mathcal H_\delta\) be the event that both coordinate oscillations on every such cell are at most \(\eta\). The reflection principle and a union bound give \[ \mathbb P(\mathcal H_\delta^c) \le C\delta^{-1}\exp\{-c\eta^2/\delta\} =C\delta^{-1}\exp\{-c\delta^{-2\vartheta}\}. \tag{29}\] This bound is uniform in all starting vectors.

Fix one cell \([a_i,b_i]\) for each \(i\). Suppose the graph is realized with \(t_i\in[a_i,b_i]\), and suppose \(\mathcal H_\delta\) holds. For one coordinate, the scalar minimum-chord relation is transitive. Indeed all endpoints in a connected component have the same height, and the union of the spans along a path between two vertices covers the interval between their times; every such span stays above that height. Thus, in each component of \(G_c\), the chronologically consecutive selected times themselves form minimum chords. Use these sorted chains, with one comparison per unit of monochrome rank. Let \(k_i^-\) and \(k_i^+\) be the numbers of colors with a predecessor and successor, respectively, at vertex \(i\). Then \[ k_i^-,k_i^+\in\{0,1,2\},\qquad \sum_i k_i^- =\sum_i k_i^+=e. \tag{30}\] Write \(h_{i,c}=X_{i,c}(a_i)\).

Use the left slot \([a_i-s,a_i]\) and the right slot \([b_i,b_i+s]\). All these slots are disjoint. If color \(c\) has predecessor \(j\) at \(i\), let \(H_c\) be their common actual chord height. Then \(|h_{j,c}-H_c|\le\eta\) and \(|h_{i,c}-H_c|\le\eta\). The entire left slot lies between the two actual chord endpoints, so, with the wall \(w_{i,c}^-=h_{j,c}-\eta\), \[ \inf_{[a_i-s,a_i]}X_{i,c}\ge w_{i,c}^-,\qquad 0\le X_{i,c}(a_i)-w_{i,c}^-\le3\eta. \tag{31}\] If color \(c\) has a successor at \(i\), the entire right slot likewise lies in that chord span. With \(w_{i,c}^+=h_{i,c}-\eta\), cell oscillation and the chord minimum give \[ \inf_{[b_i,b_i+s]}X_{i,c}\ge w_{i,c}^+,\qquad 0\le X_{i,c}(b_i)-w_{i,c}^+\le2\eta. \tag{32}\] These are necessary gates involving only fixed-slot path restrictions, not a conditioning on the actual selected times or on \(\mathcal H_\delta\).

Condition on all starting vectors and expose the independent Brownian pieces in the order \(J_1,\ldots,J_m\), each in increasing time. Every left wall in Equation (31) uses a representative in an earlier window. Every right wall in Equation (32) uses the already exposed left endpoint of its cell. Their heights are therefore known before the respective slots. Applying Equations (27) and (28) and the tower property bounds the probability of all gates for this cell tuple by \[C_m\eta^{\sum_i(k_i^-+2d_{k_i^-}+2d_{k_i^+})}.\] The bound is uniform in the starting vectors, so it also holds before conditioning on them. Since \(d_k\ge k d_2/2\) for \(k\in\{0,1,2\}\), Equation (30) shows that the exponent is at least \[e+2d_2e=2\beta e.\] There are \(O(\delta^{-m})\) tuples of cells. Let \(\mathcal F_{G,\delta}\) be the union of the gate events for these tuples, together with \(\mathcal H_\delta^c\). All its constituents are measurable, and the preceding deterministic implications give \(\mathcal E_G\subset\mathcal F_{G,\delta}\). Consequently \[\mathbb P(\mathcal F_{G,\delta}) \le C\delta^{\beta e-m-2\beta e\vartheta} +C\delta^{-1}e^{-c\delta^{-2\vartheta}}.\] Equation (26) follows by the choice of \(\vartheta\); the case \(e=0\) also follows directly from the trivial bound one. If \(\beta e>m\), choose \(\varepsilon\in(0,\beta e-m)\) and send \(\delta\) to zero.

For completeness the realization event can also be taken measurable without selecting a completion. Replace every missing gap by affine interpolation in each coordinate. If any completion satisfies the chords, every gap traversed by a chord has both its endpoint values above that chord height, so affine interpolation also satisfies the chords. Thus a completion exists exactly when this fixed interpolation has a realizing tuple. Minimum values on compact intervals vary continuously with the path and with the interval endpoints. Taking the minimum of the total nonnegative chord discrepancies over the compact set \(\prod_i K_i\) makes realization a measurable event. ◻

Proof. Absolute continuity transfers each null event. The quantitative truncated assertion is simply integration of the density over \(\mathcal F_{G,\delta}\). There is no assertion of a uniform density bound without the displayed truncation.

For a connected union \(G_1\cup G_2\) on \(m\) vertices, take a spanning tree of the union. Its edges of either color form an independent set in that monochrome graph, so \(e\ge m-1\). If \(m>\beta/(\beta-1)\), then \(\beta e\ge\beta(m-1)>m\), and the diagram is polar. There are finitely many two-colored graphs on a fixed finite vertex set, and finitely many window orders. Countable buffered localization therefore excludes all such connected diagrams simultaneously. If a generated class had at least \(m_0=\lfloor\beta/(\beta-1)\rfloor+1\) eligible vertices, finite chains connecting them would give a finite connected graph on at least \(m_0\) vertices. A spanning tree can be reduced by removing leaves to a connected subgraph on exactly \(m_0\) vertices. Its edges are still actual chords, contradicting the preceding exclusion. This proves Equation (33). ◻

Proof. The hyperspace of closed subsets of a compact metric space is compact, with the empty set adjoined as an isolated point. The countable product is compact and metrizable. Joint tightness with the contours and a Skorokhod realization therefore give convergence of all the closed observation sets.

For a point of a limiting closed set, Hausdorff convergence supplies points in the approximating closed sets. A second approximation replaces those points by actual records. Any open restriction containing the limiting record is eventually satisfied. This proves (i). Compactness proves (ii). Recording the marked subpaths as one tuple proves (iii); addition is continuous where the finite total cost bounds all entries, and the inequalities pass to the limit there.

For (iv), consecutive path vertices have asymptotically equal projected locations, by the contour chord rules and continuity of \(\eta\); the same holds for different representatives of one vertex. Join their projected locations inside balls of vanishing radius. The resulting connected compact sets have Hausdorff limits equal to the projected recorded ranges. Hausdorff limits of nonempty connected compact sets are connected. This argument is purely topological and puts no bound on the length of the small joining pieces. ◻

Proof. A burger produced inside the word is newer than every input burger. Whenever a compatible locally produced burger is available, a rigid order uses the newest such burger of its type, and an \(F\) uses the newest of all locally produced burgers. These choices do not depend on the input. Induction through the word proves both assertions. In particular, removing an input burger at an externally unresolved order does not change either local stack. ◻

Proof. The sets \(K_\epsilon\) are compact and decrease to \(K_0\) as \(\epsilon\downarrow0\). Property (i) implies \(s_1\ne s_2\) on \(K_0\), so \(|s_1-s_2|\) has a positive minimum there if \(K_0\) is nonempty. Let \(T_+\) and \(T_-\) be the projections to \(t\) of the two closed sign parts of \(K_0\). They are compact. If a time belonged to both, the supplier in at least one coordinate would have two distinct choices. Those choices together with \(t\) would make a scalar triple, while the other coordinate has a horizontal pair ending at \(t\), contrary to (ii). Thus \(T_+\) and \(T_-\) are disjoint compact subsets of the time interval. Choose disjoint finite unions of open intervals with rational endpoints around them. Decreasing \(\epsilon_0\) if necessary makes every triple in \(K_{\epsilon_0}\) belong to the corresponding sign neighborhood; otherwise a sequence of contrary triples has a limit in \(K_0\). The resulting finite partition gives the assertion. Uniform perturbation of the path only increases the two errors in the definition of \(K_\epsilon\) by a fixed multiple of that perturbation. ◻

Proof. If a first incorrect decision exists, all preceding resolutions are correct. Its actual candidates can therefore be read from the already correct past contour. By Lemma 27, both candidates precede the extension. They form approximate horizontal pairs with height error at most one lattice unit, hence belong to \(K_{\epsilon_0}\) after rescaling. Lemma 28 gives exactly the freshness comparison prescribed by the instruction, a contradiction. Apply this argument successively to the finitely many portions. Supplier truncation is robust: a left cut whose two contour heights are strictly below all tested heights forces every required backward first hit to occur after the cut. Strict positivity supplies the no-empty-stack condition in a finite interior excursion chart. ◻

Proof. Use the labelled frontier strands in 8(iv). In the imaginary-geometry construction, strands of the same labelled family merge on meeting. Their shared arcs give scalar contour gluing with the corresponding side convention; see (Gwynne, Miller, et al. 2019, sec. 2.1.3 and Lemma 2.4). The block boundaries are the first and last frontier portions of the clocked local traversal, so these labels do not require an absolute exterior contour height.

For each pair of ports, take the closure of their correspondence on nontrivial common arcs of the same labelled family, with the incoming and outgoing sides specified. Initially this is a relation on their length-coordinate intervals. For an ordered block pair \(i<j\), pull it back to endpoint times by the right-port map \(s\mapsto X(b_i)-X(s)\) and the left-port map \(t\mapsto X(a_j)-X(t)\); restrict to times passing the retained suffix and prefix minimum tests, which ensure membership in these ports. Denote this pulled-back closed relation by \(E_{ij}\). Every such common arc gives a scalar contact. The scalar equal-minimum relation is closed, so taking the closure adds no false scalar contacts. Conversely, a scalar contact between interior times of two retained blocks is approached by ordinary scalar chords with endpoints in those same blocks. For an ordinary excursion pair use nearby higher levels. At a branch triple use higher levels for the adjacent pairs and lower levels for the outer pair. Choose levels avoiding the countable set of scalar local-minimum heights. Strict Brownian local minima and their distinct heights justify these approximations. Thus every protected scalar contact belongs to \(E_{ij}\). This uses 20 for the Brownian sewing event and the scalar approximation argument detailed in 33.

The relation is not defined by arbitrary equality of spatial points: that equality can be caused through the other colour. Keeping the labelled common arcs is what distinguishes the two relations. Two tree lineages intersect in a lineage, so the common length coordinates of one pair of ports form an interval. There are finitely many ports. Consequently their relations have finitely many such pieces, even if the associated surfaces have infinitely many beads or the strands have infinitely many individual points of contact.

The real field gives the quantum-length coordinates on the retained arcs. Evaluating the two coordinates at a common arc point determines a single additive difference of block-start heights. A spanning forest of the nontrivial-overlap graph stores all differences in each connected component. A protected isolated endpoint contact also joins the same component, by the ordinary-chord approximation just proved. No difference between distinct components is recovered or needed.

These are measurable local data. The strands, their side labels, local clock, and length coordinates are measurable by the cited construction. Their common-arc correspondence is a measurable closed relation: it can be tested on compact parameter subintervals, with positive common length exhausted by rational lower bounds. Standard measurable selection on these compact relations chooses a common point; every choice gives the same additive difference. Under the countable family of reference laws a common completed-measurable version has a Borel version, by using their probability-weighted sum. Choosing a spanning forest by the fixed finite order is then measurable. ◻

Proof. Interval minima are continuous in their endpoints, so \(C_{ij}\) and \(H_{ij}\) are compact. Suppose a candidate has height \(m_{ij}\), and let \(\tau\in G_{ij}\) minimize the untested contour. All heights between the candidate endpoints are then at least \(m_{ij}\). Hence \(s,t,\tau\) belong to one scalar class and \(\eta(\tau)=\eta(s)\in K_*\). An omitted gap or cut cannot contain such a visit, by full-preimage retention. If \(\tau\) belongs to another retained block, its scalar contact with \(s\), approximated by common arcs as in 30, puts that block in the same overlap component. This too contradicts its inclusion in \(G_{ij}\). Thus \(m_{ij}\notin H_{ij}\), proving positive separation.

The untested minimum is used to prove separation, but is not an input to the local selection. Instead intersect \(C_{ij}\) with the local relation \(E_{ij}\) from 30. Call the resulting candidates good, and the other candidates bad. By the preceding separation, their respective height sets satisfy \[H_{\rm good}\subset(-\infty,m_{ij}-\delta],\qquad H_{\rm bad}\subset[m_{ij}+\delta,\infty)\] for some \(\delta>0\). Both sets are compact, including the bad set. If both are nonempty, choose the first rational \(\alpha\) in a fixed enumeration with \(\sup H_{\rm good}<\alpha<\inf H_{\rm bad}\), and allow heights below \(\alpha\). If only bad candidates exist, allow none; if only good candidates exist, choose \(\alpha>\sup H_{\rm good}\). An empty candidate set requires no test. All these choices use only local data and the finite order. Extrema and the first-admissible-rational choice are measurable; equivalently exhaust complements of closed relations by positive rational distance and take compact-set extrema.

For stability, suppose uniformly convergent contours, convergent cuts and a vanishing height mesh gave incorrectly classified protected lattice candidates. After taking endpoint subsequences, all exact retained tests pass to the limiting weak tests, so the limit lies in \(C_{ij}\). The combined untested minima converge to \(m_{ij}\). At every such limiting candidate both its relation to \(m_{ij}\) and its classification by the chosen band have strict and agreeing signs. This contradicts incorrect classification. The same argument covers the empty-good or empty-bad case. It proves correctness throughout one sufficiently small contour tube, rather than only along the originally sampled sequence. ◻

Proof. Choose nested neighborhoods \[K\subset\operatorname{int}K_*\Subset U_0\Subset U_1\Subset U.\] Cover the full preimage of \(K_*\) by used intervals whose entire ranges have closure in \(U_0\), and give them left extensions whose entire ranges have closure in \(U_1\). Transience supplies a bounded enclosing window. For a bilateral discrete application enlarge the window first to coordinate descent cuts strictly below every tested height, so all needed partners are in the window. This removes the possibility of an unrecorded supplier escaping to infinity. Record every directed \(U_1\)-traversal reaching \(\overline U_0\). There are finitely many: each crosses a fixed positive distance, and uniform continuity on the bounded time window bounds its duration below. The dense local-traversal construction in the proof of (Gwynne, Miller, et al. 2019, Lemma 2.4) includes them all. The real field needed to clock them is restricted to \(U_1\subset U\); a slightly larger imaginary buffer suffices to reconstruct them.

Apply 30 separately in the two coordinates, storing each component’s offsets in a forest. Apply 31 to each pair of blocks in a component. Test intervening blocks only within that component; all other blocks belong to the combined untested set in that Lemma. No protected incidence can join different components for infinitely many scales, since its limiting scalar contact would join them by 30. There are finitely many block pairs, so one contour tube works for all their band tests.

Keep the exact discrete sewing dictionary at the retained pieces. Primal contour-vertex identifications use the appropriate equal-height weak-minimum tests. An edge supplied by the other coordinate uses a genuinely matched up-step and down-step, with the strict one-step-above-base condition in its excursion interior and the prescribed endpoint corner offsets. These tests are functions of the actual raw steps and exact lattice forest shifts. In particular a weak horizontal pair in the other coordinate is not by itself declared to be a primal edge. For example, the scalar word with heights \(0,1,0,1,0\) has a weak pair between its first and last visits but two separate matched excursions. Every retained suffix, prefix and intervening minimum, including a critical branch minimum, is evaluated exactly in the discrete word. Only an untested obstruction is replaced by a rational band. 31 permits the bounded height-mesh errors from the step convention and proves that this replacement preserves each relevant incidence.

Exterior flexible resolutions require another step. A flexible order unresolved within its extended raw block, but lying in its used portion, has both candidate suppliers before the extension and hence at a positive limiting time separation from the used portion. Each positive-span supplier chord has the same spatial image as its decision. Since the whole used image lies in \(U_0\), every supplier’s local traversal belongs to the finite \(U_1\)-traversal family just retained. The no-common-chord and mixed-branch exclusions of 20 separate the two possible supplier orders into compact disjoint sets of decision times. The finite rational instructions and first-error proof in [loc:finite-instructions,loc:first-error] therefore make the resolved used steps exact on a sufficiently small tube. No recursively enlarged collection of raw supplier blocks is needed: their continuum traversal labels and finite order supply the comparison. Externally unmatched flexible orders in the unused extension need not be genuinely typed to compute the internal matches or used increments. Their surrogate paths enter only the macroscopic reference input, using 14; the exact density identities of [sec:local-kernels] keep genuine and surrogate extension increments distinct. Anchoring exact heights at used starts prevents these unused types from changing any retained incidence.

We have reconstructed every primal incidence used by a path or an ordered subpath record confined to \(K\), with a margin inside \(K_*\). Rational intrinsic time covers with image in \(K_*\) give the corresponding confined chart graphs for the universal endpoint tests. Their spatial interpretation uses the open loss specified above. Poisson refinements, rational bands, finite instruction lists and finite traversal orders give countably many descriptions; their durations, paths and forest shifts remain continuous or lattice input variables. Refinements accommodate any finite collection of observations. Disjoint spatial buffers have disjoint time preimages, and the unused extensions can be trimmed inside the respective buffers. Thus their retained raw blocks are disjoint, proving all four assertions. ◻

Proof. Write \(a=X(s)=X(t)\). First suppose \(X>a\) on \((s,t)\). For a level \(a+\delta\) approaching \(a\) from above, chosen to avoid the countable set of local-minimum heights, take the component of \(\{X>a+\delta\}\) containing a fixed interior compact subinterval. Its endpoints tend to \(s,t\) as \(\delta\downarrow0\). Indeed, \(X-a\) has a strictly positive minimum on each compact subinterval of \((s,t)\), whereas its endpoint values are zero. At each boundary of the chosen component there are strictly lower values arbitrarily close on the exterior side: otherwise that boundary would be a local minimum at the excluded level. Thus, for fixed sufficiently small \(\delta\), uniform contour convergence supplies a discrete excursion above a nearby lattice level whose entrance and exit are as close to these endpoints as desired. The entrance up-step and exit down-step are a matched pair. Let \(\delta\) tend to zero sufficiently slowly with the discrete index. The sewing dictionary gives the claimed identification or primal diagonal, with endpoint errors tending to zero.

There is at most one time \(r\in(s,t)\) with \(X(r)=a\): two such times would be local minima of the same height. If one exists, there are values below \(a\) arbitrarily close to \(s\) on its exterior side and to \(t\) on its exterior side. Otherwise \(s\) or \(t\) would itself be a local minimum of height \(a\), distinct from \(r\). For \(\delta>0\), again avoiding local-minimum heights, take instead the component of \(\{X>a-\delta\}\) containing \([s,t]\). The just noted exterior values show that its endpoints tend to \(s,t\) as \(\delta\downarrow0\). Apply the same discrete matched-step argument at a nearby lattice level below \(a\). This proves the assertion for the outer pair at a scalar branch triple. Each consecutive pair at that triple has strictly positive excursion interior and is covered by the first argument. The excursion end convention is covered by the first argument as well. A constant generator needs no edge.

For finitely many prescribed generators, take one diagonal choice of the level errors and the discrete index thresholds. The number of matched-step links per prescribed scalar generator is one and does not depend on the discrete index. No rate of convergence is needed. ◻

Proof. First take primal vertices. If the assertion fails, there are an \(\epsilon>0\), a subsequence, and vertices \(x_j=c_j(s_j)\), \(y_j=c_j(t_j)\) whose projected distance tends to zero but which cannot be joined as asserted. Pass to limits \(s_j\to s\), \(t_j\to t\). By (T1), \(\eta(s)=\eta(t)=z\). Choose one finite chain from (T3). For each of its generators use its discrete approximating path. Join an approximating endpoint to the endpoint used by the next generator by the corresponding segment of \(c_j\). Those chronological intervals shrink to the shared limiting time, so their projected images converge uniformly to \(z\), by (T1) and continuity of \(\eta\). Use the same chronological connectors at \(s_j,t_j\). There are only finitely many pieces. Their concatenation eventually projects into \(B(z,\epsilon/2)\), a contradiction.

For arbitrary surface points, first use (T2), then the vertex version with a smaller error, and concatenate. Finally, (T1) and surjectivity of \(\eta\) give asymptotic density of \(\pi_j(\Sigma_j)\). Notice that chronological connectors in this proof can contain unboundedly many primal edges. ◻

Proof. Choose \(r<r_1<r_2<R\) and a joining error much smaller than both \(r_1-r\) and \(R-r_2\). For large \(j\), choose one anchor lift whose image lies outside \(B(z,R)\). Such a lift exists by asymptotic density. The exterior of \(\overline{B(z,r_2)}\) is path connected. Join the image of any prescribed lift of a point outside \(B(z,R)\) to the anchor image by a target path in that exterior. Subdivide the path into sufficiently small steps. Asymptotic density supplies lifts whose images are sufficiently close to the subdivision points; retain the prescribed lifts at the ends. Apply the uniform joining property to successive lifts. The resulting upstairs path has projection outside \(\overline{B(z,r_1)}\).

Take the union of the anchor and these paths for every lift in \(\pi_j^{-1}(\mathbb S^2\setminus B(z,R))\), and then take its closure in \(\Sigma_j\). This is a compact connected set \(K_j\) containing that entire preimage. Its projection remains outside \(B(z,r_1)\), so it is disjoint from the preimage of \(\overline{B(z,r)}\). Every loop in the latter preimage lies in a component of \(\Sigma_j\setminus K_j\). That component is simply connected and is contained in \(\pi_j^{-1}(B(z,R))\), proving the filling claim. The joining and density bounds used here are independent of \(z\). ◻

Proof of the approximate right inverse and degree. Fix a sufficiently small error \(\epsilon\). First choose a loop-image radius much smaller than \(\epsilon\), small enough that Lemma 35 fills loops of that radius inside an \(\epsilon/3\) ball. Next choose a joining error much smaller than that loop-image radius, and then a target triangulation so fine that the endpoints of each of its edges can be joined using Lemma 34. For sufficiently large \(j\), choose for each target vertex a lift with projection arbitrarily close to that vertex. Connect the chosen lifts along each target edge by the joining property. This defines \(g_j\) on the one-skeleton.

For each triangular face, the image under \(\pi_j\circ g_j\) of its boundary lies in a ball of the chosen loop-image radius about one of its vertices. Lemma 35 extends \(g_j\) continuously over this face, with projected image inside the \(\epsilon/3\) ball about that vertex. The extensions agree on shared edges. Fineness of the target mesh gives the asserted uniform error on every face. Short round geodesics give a homotopy from \(\pi_j\circ g_j\) to the identity. Thus \[\deg(\pi_j)\deg(g_j)=1,\] and both integer degrees are \(+1\) or both are \(-1\). In particular \(\pi_j\) is onto. No embedding or injectivity of \(g_j\) is claimed. ◻

Proof. All lifts of \(a\) can be joined to any one prescribed lift by paths projecting into \(B(a,\rho/3)\), using the uniform joining property. After closure, their union is compact, connected, contains the entire fiber of \(a\), and is disjoint from \(\lambda\). The same holds for \(b\). If an upstairs path connected the prescribed lifts while avoiding \(\lambda\), its union with these two compact connected sets would be a compact connected set \(K\) containing both entire fibers and disjoint from \(\lambda\). The elementary complement fact above would contract \(\lambda\) in \(\Sigma_j\setminus K\). Projection of this contraction would contract \(\pi_j\circ\lambda\) in \(\mathbb S^2\setminus\{a,b\}\), contradicting its nonzero homology class. This proves separation even if exact fibers are disconnected: their pieces were first joined away from the loop. ◻

Proof. Call the horizontal path \(\alpha\) and the vertical one \(\beta\). Close \(\alpha\) upstairs by a path whose projection follows an arc from its right endpoint to its left endpoint going down outside \([-a,a]\times[-2,2]\), around the bottom of this rectangle, and up its other side. Such an upstairs path is obtained by a finite chain of the small-image joins, starting and ending at the prescribed lifts. Choose its projected error smaller than the fixed clearance from the vertical tube. It is consequently disjoint from \(\beta\).

Let \(b_-\) and \(b_+\) be the actual projected endpoints of \(\beta\). The projected closed loop has winding difference one, up to sign, about \(b_-,b_+\). To check this without a simplicity assumption, homotope the horizontal piece inside its horizontal strip to a straight left-to-right segment, and homotope the added piece in its thin exterior tube to the chosen closing arc. These homotopies avoid both endpoint neighborhoods. The resulting rectangle loop contains \(b_-\) and excludes \(b_+\), giving the stated winding difference. Lemma 37 forces \(\beta\) to meet the closed loop. It cannot meet the added piece and therefore meets \(\alpha\). Two paths in an embedded primal graph can intersect only on a common vertex or a common edge; a common edge also gives a common vertex. Thus the intersection is an actual primal graph intersection, at which their walks may be stitched. ◻

Proof. Choose nested disk collars strictly between \(E\) and \(F\). We connect the lifted core, take a regular neighborhood, and fill its holes except for the component containing the prescribed exterior.

Connect the core. For large \(j\), the projection is onto by Proposition 36. Connect every lift over \(E\) to one anchor by finite chains of small-image joins along target paths in \(E\). Choose the joining error smaller than the first collar. The closure of the union is a compact connected set \(H_j\) containing \(\pi_j^{-1}(E)\) and projecting inside that collar. The triangles meeting \(H_j\) form a connected finite subcomplex, whose full cell star lies in the next collar by Equation 34.

Take a regular neighborhood. Barycentrically subdivide the triangulation. Give a new vertex value zero if its original simplex belongs to the connected subcomplex, and value one otherwise. Extend linearly and let \(N_j\) be the sublevel set at \(1/2\). The boundary avoids new vertices and cuts each crossed triangle in a segment, with exactly two segments meeting across each shared edge. It is therefore a disjoint union of polygonal circles, and \(N_j\) is a compact subsurface. Normalizing the barycentric weights of the zero-valued vertices retracts each piece to its zero face. These retractions agree on common faces, so \(N_j\) is connected and contains the original subcomplex in its interior. It lies in the full original cell star. Subdivide along the level segments and triangulate the resulting polygons to make \(N_j\) cellular. This resolves pinches at original vertices without modifying the primal graph.

Fill the interior holes. The exterior-connection construction of Lemma 35, using the intermediate disk collars in place of round disks, gives a compact connected set \(K_j\) containing \(\pi_j^{-1}(\mathbb S^2\setminus\operatorname{int}F)\) and disjoint from \(N_j\). The proof is unchanged: the exterior of a topological disk is path connected, and each of the finitely many collars has positive separation from the next. All of \(K_j\) lies in one component \(O_j\) of \(\Sigma_j\setminus N_j\).

A connected proper compact subsurface of a sphere has complementary components which are open disks, one for each boundary component. This follows directly by applying the Jordan–Schoenflies Theorem to its disjoint boundary circles and using connectedness of the subsurface. Fill every complementary component of \(N_j\) except \(O_j\), and set \(\Delta_j=\Sigma_j\setminus O_j\). This is a closed disk containing \(H_j\) in its interior. It avoids \(K_j\), hence projects inside \(\operatorname{int}F\). Filling the complementary components preserves cellularity after the same subdivision. ◻

Proof. We first construct the annulus, then retain its incidence layers and extract a separating primal circuit. Apply Lemma 40 to \((E_0,E_1)\) and \((E_2,E_3)\). Since \(E_1\subset\operatorname{int}E_2\), the resulting disks are nested as claimed. Their difference is an annulus; the displayed inclusions give (i).

For (ii), Equation 34 and the margin between \(E_3,E_4\) place all the star triangles, and their chronological images in Equation 35, inside \(E_4\) for large \(j\). The compact set \(\eta^{-1}(E_4)\) is contained in the open set \(J\). Vanishing chronological mesh therefore places each whole triangle interval in \(J\), not just one representative time.

Every point of \(A_j\) projects outside \(E_0\) and inside \(E_3\). The margins to \(E_{-1}\) and \(E_4\) show that every triangle in its fixed star buffer has chronological images in \(E_4\setminus\operatorname{int}E_{-1}\) for large \(j\). The inverse image of this compact set is a compact subset of \(J\) disjoint from \(S\). A slightly larger compact time neighborhood still in \(J\setminus S\) contains the full triangle intervals. A finite interval cover, with slightly larger buffers still off \(S\), proves (iii). Any additional finite number of incidence guards is covered by increasing \(r\) before taking \(j\) large.

For (iv), use Corollary 38 for transverse crossings. To obtain the surrounding circuit, decompose a closed primal walk into finitely many simple circuits. Projected winding is additive, so at least one has nonzero winding difference. By Lemma 37, this circuit separates a lift of \(z\) from a lift of an exterior point. It lies in \(A_j\), whereas \(\operatorname{int}\Delta_j^-\) is connected and disjoint from it; therefore all of \(\pi_j^{-1}(E_0)\) lies on its inner side. The complement of \(\Delta_j^+\) is connected, lies on the exterior side, and contains the exterior lift. The chosen circuit interior is thus contained in \(\Delta_j^+\). Its fixed star buffers are retained by (ii). The circuit uses only edges of the original walk. ◻

Proof. The sphere map. The first-coordinate up-steps and their stack matches give a rooted plane tree. Thicken it on the sphere. Its complement is a disk following the tree contour, with corners marked by the intervening second-coordinate steps. Draw an arc in this disk for each matched second-coordinate pair. Stack matching makes the arcs noncrossing; their endpoints are the specified primal corners, with loops and repeated endpoints allowed. Cutting along the arcs leaves disks, giving a cellular sphere embedding. The arcs are the primal diagonals of the sewn quadrangles, and their duals form the complementary dual tree. If there are \(n_1,n_2\) up-steps of the two colors, respectively, the map has \(n_1+1\) vertices, \(n_1+n_2\) edges, and \(n_2+1\) faces, consistently with this construction. The case of a one-vertex initial tree is included. Notice that the word with steps \((1,0),(-1,0)\) still gives a sphere map; its identically zero second contour would collapse the literal time-circle quotient to a point.

Discrete fillers. First move each coordinate monotonically from \(a\) to \(b\). This takes \(\lVert b-a\rVert_1\) steps and stays nonnegative. Any even excess is supplied by cancelling positive-then-negative pairs in one coordinate. This proves sufficiency and necessity.

Approximation of continuous fillers. Fix a polygonal time partition on which the continuous filler oscillates by less than \(\epsilon\). Round the positions at its partition points to the required parity lattice. On each block the available number of steps is of order \(n\), whereas the endpoint displacement is of order \(\sqrt n\). The preceding inequality and parity conditions therefore hold for large \(n\). Use monotone coordinate moves between the rounded endpoints and cancelling pairs for the excess. The path stays in their coordinate rectangle, enlarged by one height-mesh unit. This is within a fixed multiple of \(\epsilon\) of the continuous path throughout the block. Interior positivity follows from its positive minimum on compact interior time intervals. At an excursion endpoint the first two steps, or the last two steps, may be arranged to enter, or leave, the strictly positive quadrant; these modifications have vanishing time and height size. Let \(\epsilon\downarrow0\) by a diagonal choice of partitions. Even cuts and parity-preserving translations give the required compatibility at retained block boundaries. ◻

Proof. Protect the discrete incidences. Start with a bounded window containing the desired triangles. In each coordinate find a visit to the left and a visit to the right strictly below every corner height and adjacent step level used in that window. Retain a window beyond all four visits. The two colors need not attain their low values at the same times. A scalar chord or matched step from the original window to outside the enlarged one would cross its coordinate’s lower visit, which is impossible. Consequently every scalar partner needed for one full incidence layer is now in the enlarged window. Repeat this finite enlargement for the prescribed number of layers, and once more to protect their incidences. In a contour coupling, take strictly positive macroscopic height margins at these lower visits; uniform convergence preserves the inequalities, including adjacent lattice levels.

Keep this entire final window unchanged. Translate each coordinate by a sufficiently large lattice constant so that the window lies in the positive quadrant, respecting the parity lattice. Attach nonnegative axial paths from zero to its initial point and from its final point back to zero. Lemma 42 gives an actual sphere map. All relations inside the retained window use the same equalities and intervening minima, and the retained lower visits exclude any partner in a cap. This proves the exact incidence claim. Independent choices of lower visits in the two coordinates suffice throughout.

Preserve the continuum sphere quotient. Let \(W\) be an increment block of duration \(T\), and consider a Brownian quadrant excursion of duration \(T+2\). The law of its middle increment block, from time \(1\) to time \(T+1\), has a strictly positive density with respect to Brownian increments of duration \(T\). Indeed, the Markov bridge decomposition writes that density, up to normalization, as \[g(w)=\int_{\mathbb R_{>0}^2} e_1(a)e_1(a+w(T)) \boldsymbol 1_{\{a+w(t)\in\mathbb R_{>0}^2\ \text{for all }0\le t\le T\}} \,da,\] where \(e_1\) is the positive entrance density, equal to the exit density by reversal, with the fixed Brownian covariance understood. These densities have Gaussian decay times a fixed harmonic power. The integral is finite and is positive for every bounded continuous \(w\), since all sufficiently large coordinate translations place its range strictly inside the quadrant. This is also the usual killed Brownian bridge decomposition: the indicator supplies the killing during the middle block.

Sample the start height and the two caps from this excursion law conditionally on its middle increments being \(W\). Since the given law of \(W\) is absolutely continuous with respect to Brownian increments, and \(g>0\), the resulting entire excursion law is absolutely continuous with respect to the reference excursion law. Its Radon–Nikodym derivative is the density of the prescribed middle increments divided by the normalized middle-increment density of the reference excursion. The middle block is reproduced exactly up to translation. Its caps are continuous and positive in their interiors. Theorem 8(ii)–(iii) gives the sphere quotient, and Lemma 42 approximates these caps by admissible axial fillers while retaining the actual middle steps. A countable choice among deterministic bounded windows does not change the absolute-continuity assertion, after conditioning on each chosen window of positive probability. ◻

Proof. The iid-block contour invariance principle, applied backwards, and continuity of Brownian first passage at a positive level imply \[\left(m^{-2}\sum_{j\le m}T_j, m^{-1}\sum_{j\le m}Y_j\right) \ \Longrightarrow\ (\tau_1,V_1),\] with density \(g_1\). The same argument gives the stopped contour path. Brownian first passage is continuous under uniform convergence on compact sets at almost every Brownian path: before each strictly earlier time the level has not been reached, and after the hitting time it is crossed arbitrarily soon. Truncation to a large time window removes no limiting mass.

Let \(\phi(s,t)=\mathbb E e^{i(sT_1+tY_1)}\). The characteristic-function powers converge uniformly on compact sets under the anisotropic scaling \((s,t)\mapsto(m^{-2}s,m^{-1}t)\). The limiting modulus is strictly below one away from the origin. To obtain a uniform bound near zero, set \(r=\sqrt{|s|}+|t|\) and choose \(m\) comparable to \(r^{-1}\). The rescaled frequency stays in a compact annulus. Uniform convergence of the powers there yields \[ 1-|\phi(s,t)|\ge c\bigl(\sqrt{|s|}+|t|\bigr). \tag{38}\] There are no further periodic obstructions. Indeed the ladder-block support contains \[(1,0),\qquad (2,1),\qquad (2,-1),\qquad (3,0),\] realized respectively by \(h\), \(hc\), \(hC\), and \(hcC\). The differences generate the lattice \(\{(k,\ell):k+\ell\text{ is even}\}\), and every ladder block has \(T+Y\) odd. Hence the only points of modulus one on the frequency torus are \((0,0)\) and \((\pi,\pi)\). Away from neighborhoods of these points, the powers decay exponentially. Fourier inversion, using Equation (38), now gives an integrable majorant \(\exp\{-c(\sqrt{|s|}+|t|)\}\) in the rescaled frequencies. Dominated convergence proves Equation (37); the second dual point gives the factor two. The same domination proves the stated uniformities.

For conditional paths, fix \(\varepsilon\in(0,1/2)\) and reserve the last \(\lfloor\varepsilon m_N\rfloor\) ladder blocks. Conditional on the unreserved stopped path, the probability of the required remaining endpoint is a ladder local-limit mass. Dividing by the total endpoint mass, which is bounded below on the specified compact sets after multiplication by \(N^{3/2}\), gives a bounded continuous limiting tilt for the unreserved path. For fixed \(\varepsilon\), the density \(g_{\varepsilon h}\) extends continuously by zero to nonpositive times and is bounded. Thus the unreserved path converges to the corresponding initial portion of the Brownian bridge.

The omitted portion is negligible as \(\varepsilon\downarrow0\). Its conditional probability of a specified duration or oscillation event is bounded by a constant times its unconditional probability: use the uniform upper local-limit bound for the unreserved endpoint and the positive lower bound for the total endpoint. The constant is uniform for \(\varepsilon<1/2\) on the compact terminal parameter set. Unconditional first-passage convergence and Brownian scaling make those probabilities tend to zero. Holding the unreserved path constant after its stopping time then changes the full stopped path by at most the omitted oscillation. This proves the conditional convergence and its sequential uniformity. The Brownian Markov transition densities in the half-plane are strictly positive; inserting finitely many intermediate open balls and then refining them proves full support of each compatible path tube. The same density formulas and the preceding truncation prove continuity of the bridge kernel. ◻

Proof. Multiply Equation (37) by \(|[a\sqrt N,2a\sqrt N]\cap\mathbb Z|^{-1}\). To obtain the duration marginal, apply the one-dimensional Fourier argument directly to \(\phi(s,0)\). Its support has span one since \(T=1\) and \(T=2\) both have positive probability. It gives \[\mathbb P\left[\sum_{j\le m}T_j=k\right] =N^{-1}\left(f_{m/\sqrt N}(k/N)+o(1)\right),\qquad f_h(t)=\frac{h}{\sigma_q\sqrt{2\pi}\,t^{3/2}} e^{-h^2/(2\sigma_q^2t)},\] uniformly on the required compact sets. Averaging the \(O(\sqrt N)\) values of \(m\) with their normalized uniform weights yields the positive duration density \(a^{-1}\int_a^{2a}f_h(t)\,dh\). Thus no absolute two-dimensional LLT error is summed over an unbounded displacement range. Dividing Equation (39) by this duration mass proves the last assertion. The path assertions are unchanged by conditioning on \(H_N\). ◻

Proof. Choose integer bases for the lattice, its kernel, and its image, or use Smith normal form. Each nonempty lattice fiber is a translate of the same kernel lattice. In these coordinates the assertion is the uniform Riemann-sum approximation of a smooth function on parallel affine fibers. On compact supports uniform continuity controls every cell error. The determinants of the three fixed bases give the covolume factors. ◻

Proof. For a bounded continuous test \(f\), the density error bounds the error in its weighted numerator by \(\|f\|_\infty\) times the displayed \(L^1\) error. Weak convergence and almost-everywhere continuity then give \[\int f(c,y,v)W^e(c,v)\,\eta(dc,dv)K_c(dy).\] The same calculation with \(f=1\) gives the strictly positive limiting denominator. Division and disintegration prove the kernel statement. The pushforward assertion follows because the density depends only on the macroscopic variables. For random frozen data, couple their convergent sequence almost surely and use the asserted sequential hypothesis, then integrate. Bounded test functions permit dominated convergence. Compact truncations are removed only after controlling their omitted mass under the actual experiment. ◻

Proof. Removing an original burger of rank at most \(m_i\) during a replacement would lower that stack size below \(m_i\), which is excluded. During the suffix, equal prescribed types give equal heights. Removing above rank \(m_i\) does not change a protected rank; removing at such a rank removes the same identity in both runs. A frozen suffix addition at such a rank also has the same identity in both runs. Induction proves agreement of all occupied protected ranks, even after the suffix dips below its original floor. At the specified \(F\) step the two available top burgers, or their absence, therefore agree exactly. Their creation-time comparison agrees as well. ◻

Proof. Choose each \(c_i\) below half the minimum of the limiting coordinate on the mutable interval. Its positive minimum exists by strict interior excursion positivity. The excursion coordinates tend to zero at the terminal endpoint, so choose a collar with both coordinates below \(c_i/2\). Uniform convergence, with sufficiently small tube radius, gives the asserted integer inequalities. The provisional suffix has the same heights in both runs because its entrance heights and its resolved increments are fixed. Apply Lemma 48 at a first allegedly incorrect terminal decision. Nonnegative actual heights exclude unfilled orders, and zero final heights leave no burgers. ◻

Proof. Expand the conditional probability as a finite sum over raw words and auxiliary depth choices. The iid letter weight factors over the disjoint pieces. Being a guard or a noise is a property of its own word: its successive retained type-1 burgers are detected by backwards inspection, and every \(F\) there resolves internally. Thus summing over noise words with a specified duration and displacement gives exactly \(q_{j,N}\), and conditioning on those parameters gives \(\mathcal B_{j,N}\). The guard and exterior-prefix probabilities are constant in the ratio. All interior ordinary pieces retain their independent raw letter weights. Lemma 29 makes the imposed ordinary resolutions genuine for every word in the stable cell. Changing variables by Equation (41) and dividing by the reference mass \(r_N\) proves the iid identity.

For the conditioned experiment, initially retain both indicators: the concatenated word reduces to the empty word, and its suffix has the prescribed resolved types. The empty-word normalizing constant and the raw exterior weights cancel on the frozen atom. The exact entrance height of the suffix is determined by its prescribed net increment and the terminal value zero. It is the \(h_*\) in Equation (41). Positivity and the stable instructions make the two retained indicators equal to one before the terminal collar; Corollary 49 makes them equal to one there as well. The remaining ratio is exactly Equation (42). ◻

Proof. First keep all noise durations fixed and use full height parameters. Each noise mass is \(N^{-2}\) times a positive continuous function, up to a uniform \(o(N^{-2})\), by Corollary 45. The reference mass for \(r\) two-dimensional heights is \(N^{-r}\) times its positive continuous density and its fixed lattice cell volume. With a terminal noise the quotient in Equation (42) therefore has common factor \(N^{-r-2}\). Without that noise the corresponding factor is \(N^{-r}\). These factors cancel in the normalized conditional law. The remaining quotient converges uniformly on the compact parameter set to the product of the translated functions in Equation (36) divided by the reference height density.

The translations in Equation (41) are exact functions of the raw words, guard context and global instructions. Their normalized limits depend only on the recorded increment paths. In particular the true extension increments in \(A_i\) need not equal the locally recorded surrogate increments: their difference is bounded by twice the corresponding raw unidentified-\(F\) counts. Unresolved flexible counts in independent raw blocks are \(o_{\mathbb P}(\sqrt N)\), so the macroscopic block limits are the ordinary Brownian increment laws, for every one of the finitely many instructions. The extra global extension instructions do not enter the local observation: its exact heights are anchored at \(u_i\) and it uses only the local used-portion instruction. Changing those extra instructions changes neither \(Y_N\) nor the reference limiting input, and hence does not change the extracted joint law or its kernel. Every interior ordinary gap is likewise a free raw nuisance block. Its actual resolution under the global guard context may depend on neighboring retained words, but its difference from any fixed local surrogate is bounded by twice its own unidentified-\(F\) count. The same unconditioned estimate therefore gives its macroscopic increment path. Only the exterior prefix and the finite suffix with prescribed types enter through fixed resolved paths; guards resolve internally. No conditional unidentified-letter estimate for frozen interior words is used.

Conditional noise contours converge by Lemma 44, uniformly on the compact endpoint set. Their exact endpoints may depend on additional raw extension data and global instructions beyond \(C_N\). Those normalized endpoints differ by \(o_{\mathbb P}(1)\) from the endpoints computed from \(C_N\) and the macroscopic ordinary nuisance paths. Uniform bridge-kernel convergence and continuity therefore replace the former by the latter in \(L^1\) for bounded continuous conditional tests. This proves the limiting factorization with \(K_c\); finite-\(N\) conditional independence given \(C_N\) alone is not required. They can therefore be adjoined as continuous conditional nuisance kernels in Lemma 47. Concatenation of finitely many continuous paths with positive durations and the prescribed endpoint translations is continuous. For each of the countably many tube centers and decomposition patterns, mix the limiting reference resampling law over the actual limiting frozen-data law. A real-valued tube-distance variable has only countably many atoms under this mixture. Choose a countable dense family of radii avoiding the union of these atom sets. Fubini’s theorem then gives a full-measure set of frozen values on which all the chosen boundaries have zero conditional reference mass. Multiplication by the bounded limiting density preserves that assertion. For a sequence of frozen data converging to a value in this set, conditional bridge convergence and continuity of concatenation imply that the cell indicators converge in \(L^1\) in an almost sure coupling. Together with the uniform density approximation this is the required \(L^1\) macroscopic tilt approximation. This argument asserts a common full-measure set for the actual mixture; it does not assert conditional boundary continuity at every possible frozen contour.

If only forest parameters are retained, complete them by auxiliary full heights with a positive smooth density and integrate along fibers. Lemma 46 proves the same assertion with the correct projected lattice factor. If durations are integrated, sum the local-limit masses against the original smooth cut density. Its Riemann-sum factors appear identically in numerator and denominator. One must not freeze a randomized first-depth count in this calculation: that would remove one of the free displacement coordinates.

For positivity, take a compatible parameter/path tuple in the interior of the cell. Shrink to a product of path tubes and a small parameter neighborhood still in the cell. The endpoint densities and reference height density have positive lower bounds there. Free Brownian blocks have positive probability to follow each required tube, and the conditional noise bridges do too by Lemma 44. Thus the limiting weighted integral over this smaller set is positive. This argument is stable in a neighborhood of the frozen macroscopic data. Its lower bound may depend on that data and the chosen cell; no uniform lower bound as cells shrink is asserted. ◻

Proof. It suffices initially to test finitely many bounded continuous functions of the observations and of the contour on a bounded time window. Choose deterministic enclosing slots inside the positive gaps between the extended blocks. In each slot use a noise and guard as above. A countable choice of enclosing slots suffices because the gap conditions are strict. If the block cuts are Poisson cuts, retain their smooth joint density inside these enclosing slots rather than conditioning on each of their microscopic durations. Noise and guard endpoints may be conditioned on separately; they are determined in the reserved slots.

The noise and guard pieces fit with probability tending to one as \(a\downarrow0\), after \(N\to\infty\). Indeed their total backwards depth is at most \(3a\sqrt N+O(1)\). The contour invariance principle identifies their duration with a backwards Brownian first passage to a level tending to zero at an ordinary interior time. That duration tends to zero in probability. This applies under the finite excursion law by its interior absolute continuity. With \(a\) fixed, guards and noises have positive limiting durations. These durations, supplier windows, depths, and displacements can be truncated to compact interior sets with arbitrarily high actual probability.

On this truncated event, Brownian genericity and Lemma 28 give finitely many stable signs on a sufficiently small contour tube. Before the first reserved slot use the actual unchanged past. Between slots apply Lemma 29 successively. In the finite experiment, freeze the raw exterior prefix and suffix, the suffix’s resolved types, guard words and endpoint cuts. The height at the end of the mutable region is then the negative of the prescribed suffix increment. Keep the randomized noise depths free. The protected-rank Corollary 49 handles the final collar; the separated comparison instructions handle the remaining suffix.

Disintegrate after restricting to the actual stable cell. Every frozen atom under consideration then has positive probability in that cell and supplies an original correctly resolved word satisfying its protected floors. The exact conditional distribution on each such atom is Lemma 51. Along convergent frozen data, Proposition 52 gives a macroscopic density with positive limiting denominator. The actual limiting paths are in the required support: the free blocks are Brownian paths under interior excursion absolute continuity, and the selected noises are first-passage portions with positive duration and interior randomized depth. An open tube about such a tuple contains a smaller compatible product tube of the type used in the positivity proof.

Apply Lemma 47. The full limiting contour on the window is reconstructed by concatenating the retained block paths, noise paths, integrated interior ordinary paths, and the fixed resolved exterior paths, with the displayed height translations and exterior absolute time origins. The retained cut and mark data locate each chart in that contour, so the pair determines its intrinsic input \(C\). Conditioning on this pair therefore leaves the reference observation kernel unchanged. For deterministic cuts the extra cut data are redundant; for random cuts they are part of the conditioning sigma-field. In the bilateral experiment there is no terminal empty-word factor. Any further exterior contour window may be included among the nuisance pieces; unidentified flexible effects there are negligible in rescaled increments. In the finite experiment the prescribed exterior increments are exact after the terminal consistency argument.

After extracting the actual limiting frozen-data law, choose continuity cells as in Proposition 52. Intersect its full-measure admissible set with the generic, strict-margin set. The preceding assertions hold along every sequence of frozen data converging to a value in this intersection. Couple the actual frozen data, apply this sequential assertion almost surely, and integrate as in Lemma 47. Exhaust the compact parameter cutoffs, the supplier windows and the countable stable cells. Their omitted actual probabilities tend to zero. A monotone-class argument for continuous cylinder tests and increasing time windows gives conditioning on the complete contour.

For disjoint buffers, Lemma 32 uses disjoint raw block families. Sample their shift parameters independently in the product reference experiment. The union of their coordinate forests is again an admissible forest; the calculation above permits the actual macroscopic height constraints to couple those parameters but not their conditional observation kernels. Hence the reference product kernels remain products after conditioning on the complete contour. All arguments applied to arbitrary \(E\)-valued measurable functions of the finite chart, so they also apply to the universal endpoint variables. ◻

Proof. Use a metric \(d_{\rm w}\) for weak convergence on the countable product of the observation spaces and their Polish contour and auxiliary spaces. Let \(\mathcal L_j\) be admissible laws converging to \(\mathcal L\). By admissibility, for row \(j\) choose a discrete approximant whose law is within \(1/j\) of \(\mathcal L_j\) in \(d_{\rm w}\). Choose it far enough along its defining sequence that each of the first \(j\) retained time units exceeds \(j\) lattice steps and all the prescribed finite coordinate approximations hold. The triangle inequality gives a single diagonal array converging to \(\mathcal L\).

If a continuum coordinate has no discrete definition, retain its actual joint coupling as follows. Couple a row’s discrete raw arrays to their limit almost surely, adjoin the row’s canonical continuum tuple with its prescribed law conditional on the complete limiting contour and cut data, independently of the entire coupled actual and reference observation array, and attach the same tuple along the row. This gives joint convergence and preserves every discrete raw-word marginal. The resampling identity is applied to those raw coordinates; the canonical contour-to-surface and field-to-metric relations are used only in the limiting continuum readout.

Each diagonal row is still an actual raw-word experiment with its independent reference experiments. Equation (42) is valid there before any limit. Apply its local-limit and macroscopic transfer proof to this diagonal sequence, choosing continuity cells for its actual limiting frozen-data mixture. The retained canonical joint marginal supplies genericity, positive durations and gaps, and the required finite-volume endpoint margins in the limiting chart. Thus the free raw letters, positive-duration compact truncations and cost compactification are the same as in Theorem 53. The time unit is allowed to vary with the row, and the proof imposes no restriction on the deterministic cost denominator. Thus the theorem produces the new conditional kernels. Products of independent reference laws remain products under joint weak convergence; their disintegrations give the asserted factorization. Countable selection and the same local field-density argument then give the spatial conclusions.

For a change of base units, first apply that fixed change in row \(j\), including it in every retained actual and reference coordinate, and only then choose its sufficiently fine discrete approximant. This ensures the required divergence of every fixed rescaled lattice time unit even when the base change varies with \(j\). Include the countable union of the row-wise relabeled coordinates, and sample each target reference density directly in its normalized units. Projection onto the retained continuum coordinates proves the last assertion. In particular no continuity of the map from a field to its internal metric is needed: the entire contour/mark/field/metric coupling is retained. For a random auxiliary continuum coordinate \(A\) beyond the complete contour and cut data \(\mathcal C\), the extension of conditioning follows from its fixed joint law with \(\mathcal C\). Given a bounded continuous test \(\phi(A)\), approximate \(\mathbb E[\phi(A)\mid\mathcal C]\) in \(L^1\) of the fixed \(\mathcal C\) marginal by bounded continuous functions of \(\mathcal C\). The row-wise conditional identity, tested against bounded continuous functions of \((\mathcal C,Y)\), then passes to the limit with an error uniform in the row. Letting the \(L^1\) error vanish preserves conditional independence of \(A\) and the observations given \(\mathcal C\). Deterministically reconstructed continuum coordinates are the immediate special case. This proof uses discrete approximation and the exact density again; it does not assert weak closure of abstract conditional-independence relations or pointwise convergence of regular conditional kernels. ◻

Proof. Multiply the conditional identity for \(Y_j\) by \(\mathbf 1_{\{J=j\}}\) and sum over \(j\). This proves the first identity. The assumed measurability of \(Q\) and the tower property give the second. Products of test functions prove the finite joint assertion, and a monotone-class argument using cylinder functions proves the exhaustion assertion. ◻

Proof. Choose nested neighborhoods \[K\subset\operatorname{int}K_*\Subset U_0\Subset U_1\Subset U.\] The imaginary field on \(V\) reconstructs all directed \(U_1\)-traversals reaching \(\overline U_0\). There are finitely many such traversals, as in Lemma 32. The real field on \(U\) gives their quantum-area clocks and the quantum lengths of their labelled frontier arcs. Retain their genuine relative order \(O\). This order compares visits in distinct traversals; the directed clocks compare visits in the same traversal. No absolute origin on the global tour is required for these comparisons.

We describe the local tests used to select a protected chart. At a fixed Poisson refinement, choose finitely many used blocks inside these clocked traversals, with their closures in \(U_0\), and left extensions with closures in \(U_1\). Require their used interiors to cover every visit to \(K_*\). This is a test on the reconstructed finite traversal family: every visit to \(K_*\) lies in one of its members. The marks in \(U\) locate the cuts along each directed clock. They therefore give the block durations and extension offsets, even though they do not give the absolute global Poisson indices of those cuts. The local frontier lengths give the two increment paths, pinned at each used start.

For these blocks, Lemma 30 gives the two labelled port relations and the forest differences within each overlap component. Evaluate the retained suffix, prefix and intervening minimum tests using these paths and differences. Among the protected candidates passing those tests, the port relation distinguishes the genuine contacts from the other candidates. Select the rational band which strictly separates their height sets, with the empty-set conventions in Lemma 31. Its existence follows from that lemma. Its selection uses the two local height sets, not the value of a minimum in an omitted gap.

The flexible-order instruction is selected from the same data. Consider the possible pairs of scalar suppliers for a decision in a used block, with each supplier at or before that block’s left-extension start. Their scalar contacts are read from the labelled frontier relations. Each supplier has the same spatial image as the decision, so its visit occurs in the retained family of \(U_1\)-traversals. The closed scalar contact relations on the finitely many closed traversal records make the candidate set compact. Positive extension duration separates its suppliers from the used decision times. The order \(O\) and the directed clocks decide which member of the pair is later. The two possible orders therefore give compact disjoint sets of decision times, by the argument of Lemma 28. Choose a finite rational partition of the used clock whose boundaries avoid these sets, and prescribe the corresponding type on each part. This is the local test for an admissible instruction. A bounded global supplier window and a small tube for the complete contour are used in Lemma 29 to prove that the selected instruction is eventually correct in the discrete words. They are not inputs to its local selection. The same distinction applies to the omitted minima used to prove band clearance.

Thus the data in Equation (43) determine a finite intrinsic description: the retained block pattern, side labels, forests, rational bands and flexible instructions. They also determine its continuous input: intrinsic durations and extension offsets, used-start-pinned increment paths, and the forest differences. All choices are taken from the countable atlas and may be made by its fixed enumeration. Extension-only external \(F\)’s keep their deterministic surrogate types. Their normalized typing error vanishes under the independent reference law by Proposition 14; their exact types do not enter the local observation because its heights are anchored at used starts. The extra global instructions used to prove the resampling identity likewise contribute no additional input to this observation.

To apply countable selection, distinguish this intrinsic description from the global labels locating its blocks in the complete contour. Those labels are measurable from the complete contour and cut data. They need not be measurable from the local fields. The reference experiment, however, samples only the intrinsic raw blocks, durations and shifts; its observation uses no absolute tour start or omitted-gap duration. Two realizations of the same intrinsic description therefore use the same reference experiment and observation after relabelling their blocks. Smooth changes of the reference duration or shift densities leave its conditional kernel unchanged by Lemma 47. For the countable atlas, choose these common kernel versions simultaneously. Consequently the reference kernel evaluated at the selected chart depends only on the local intrinsic description and input just constructed.

Theorem 53 supplies the conditional identity for every globally located member of the atlas. Lemma 55 now applies because its selected evaluated kernel is local. Exhaust the finite observations and rational open losses to obtain the asserted spatial kernel. This selection is performed on the limiting continuum data, after the fixed-chart identities have been proved. Because the chart observations were extracted jointly, restriction from a larger protected chart to a smaller one is a measurable projection of that same family, so the kernels inherit restriction compatibility. Versions may be fixed for the countable atlas and the countable deterministic field changes used later, since absolute continuity preserves their common null sets.

For disjoint \(U\) the protected retained block families are disjoint. The reference experiment and Theorem 53 therefore give product kernels conditional on the complete continuum data. Selection of their indices by that data preserves the product identity. Overlap of the imaginary buffers is harmless here, because the entire imaginary field is conditioned on. To use independent Dirichlet field parts in a later resampling experiment one separately chooses disjoint imaginary testing domains.

The complete cone or sphere contour supplies its ambient decorated surface by Theorem 8(ii). Apply the local atlas directly to that surface’s fields and traversals. Write \(\ell\) for the data in Equation (43), \(s(\ell)\) for the selected intrinsic description, and \(c(\ell)\) for its intrinsic input. The same data give a map \(\Phi_{s(\ell)}^\ell\) from its intrinsic time records to spatial records along the local traversals. The spatial conditional kernel is consequently \[Q_\ell =\bigl(\Phi_{s(\ell)}^\ell\bigr)_* K^{s(\ell)}_{c(\ell)}.\] All three constructions use the same local data in the cone and sphere settings, and Theorem 53 supplies the common reference kernel \(K^s_c\) in both. The clocks, quantum frontier lengths, used-start height differences and side labels are intrinsic under conformal coordinate changes. Transporting the local fields and traversals therefore transports the spatial record while preserving its intrinsic conditional law. This proves covariance and the common cone/sphere interpretation with the actual field normalization retained. ◻

Proof. If a joint law has disintegration \(\mu(dc)K_c(dy)\) and the field law is tilted by a density \(R(c)\), its joint law is \(R(c)\mu(dc)K_c(dy)\). Thus the conditional kernel is unchanged as a function of the realized macroscopic input. Bounded density approximations and truncation give the assertion for an arbitrary absolutely continuous tilt. The fields and any embedding gauge may be included in the macroscopic data: the gauge is adjoined conditionally independently of the microscopic observations given the contours.

For \(f\) in the specified Cameron–Martin space, the Cameron–Martin theorem gives the field density with the actual normalization retained. A shift with nonzero average on a fixed pinning circle would move the field to a disjoint affine support, so that case is not covered. Apply Theorem 56 to the shifted interior and unchanged exterior patches. Resample the area marks inside the changed region with their correct conditional intensity \(\mu_{h+f}\) and retain the marks outside, where the intensity is unchanged. All chart refinements can be coupled this way. Conditional product kernels then give the stated joint law on the two separated regions; extend to the unobserved region by a regular conditional distribution. Only afterward restrict to a selected input event and bound the Cameron–Martin likelihood. In particular, a larger imaginary buffer in Theorem 56 does not enlarge the support required for this real-field modification. ◻

Proof. Work throughout with unnormalized iid cylinder weights. At the fixed right anchor in a safety band, read backwards a guard and then a noise, so in forward order the noise precedes the guard. Expand the event as a sum over the guard stopping word \(g\), noise duration \(t\), ordinary nuisance words \(u\), and retained word \(b\). Write \(w(g),w(u),w(b)\) for their raw iid letter weights; keep any anchor-mark probability as a separate factor. The randomized depth probability is included in \(q_N\). On the compatible-instruction subevent, a prescribed retained starting-height difference is an exact affine constraint on the corresponding noise displacement coordinate. The translation is determined by the incoming history, \(g,u,b\) and the instruction. Thus the summand, after summing over noise words, is bounded by \[w(g)w(u)w(b)\,q_N(t,d), \qquad q_N(t,d)\le C N^{-2},\] with a sum over any unretained displacement coordinates. The local-limit bound is uniform over the truncated displacement range, so its translation may depend on \(g,u,b\). All additional compatibility indicators can be dropped for this upper bound.

There are \(O(N)\) admissible noise durations in the slot. Summing over them therefore leaves at most \(C N^{-1}\) for the two-dimensional free displacement. Retaining only one coordinate sums over \(O(\sqrt N)\) values of the unused coordinate and gives at most \(C N^{-1/2}\). Retaining none gives a bounded factor. These are precisely the orders of the corresponding smooth reference lattice masses, whose densities are bounded below on the chosen compact target ranges. The projected parity constraints change only fixed covolume factors. For each fixed \(g,t\), the weights of all ordinary words of the resulting lengths sum to one. The backwards guard is a stopping word read from a fixed anchor, so the sum of its stopped-cylinder weights over all allowed lengths and words is at most one. Summing the independent mark weights also contributes at most one. These statements concern raw weights: no division by the probability of a guard, fit event, or successful instruction is made. There is consequently no extra sum over possible search anchors. The bound is uniform in the incoming stacks because the fresh raw letters and the backwards ladder test inside the safety band do not depend on those stacks.

Apply this calculation in chronological order. Previously retained parameters are fixed at each step; the forest condition ensures that there is at most one new target for each smoothed coordinate. Thus no second exact constraint is imposed on the same free coordinate. Multiply the finitely many bounds and integrate the previous parameters. Sum over the finite instruction list if necessary. Pushing the dominated law through a measurable observation preserves the inequality. ◻

Proof. For \(B_N=A_N\cap G_N\), domination gives \(P_N^0(B_N)\le C'Q_N(A_N)\to0\). The binary relative-entropy inequality gives, when this probability lies in \((0,1)\), \[P_N(B_N)\log\frac1{P_N^0(B_N)} \le \operatorname{Ent}(P_N\mid P_N^0)+\log2.\] The conclusion follows. Conditional entropies integrate to joint entropy when the history marginal is unchanged, giving the last assertion. A tight, rather than deterministically bounded, conditional entropy can first be truncated. ◻

Proof. Let \(\mu(dc)\) be the retained reference input law, \(K_c\) its conditional observation kernel, and \(\lambda_c(dv)\) the conditional law of the ordinary and bridge filling pieces. On a compact stable cell, write \(a(c,v)\) for the positive translated endpoint-density multiplier in Proposition 52. Thus it includes the product of noise endpoint densities divided by the reference height density, together with any duration or ordinary-input density factors. The ordinary sewing induces, up to a common positive normalization, the retained subprobability law \[f(c)\,\mu(dc)K_c(dy),\qquad f(c)=\int a(c,v)\mathbf1_{\mathcal T}(c,v)\,\lambda_c(dv).\] This follows from Theorem 53 and the exact macroscopic density: the multiplier depends on the retained input and filling, so leaves \(K_c\) unchanged. Open fillability and positivity of \(a\) on a smaller compatible tube give \(f(c)>0\). In particular the weight is the density-weighted filling integral, rather than only the unweighted probability of finding a compatible path. If \(B\) is a local failure event, measurable in the retained input and observation, which is null in the filled ordinary sewing, then \[0=\int f(c)K_c(B_c)\,\mu(dc).\] Hence \(K_c(B_c)=0\) for \(\mu\)-almost every openly fillable input. One may restrict first to \(f(c)\ge\varepsilon\) and then let \(\varepsilon\downarrow0\); no uniform lower bound on filling probabilities is needed. Countable protected charts and rational open losses give a common reference-null set. Tight absolute continuity, or Proposition 59 followed by Lemma 60, transfers that null set to the selected law. For a sequence of scale-dependent reference laws the same argument applies in each jointly extracted limit, so it also applies to mixtures over the viewing level. All retained slots avoid the distinguished times, and every filling restriction at their omitted neighborhoods has strict clearance; the proof therefore makes no claim about the law at those times. ◻

Proof. Consider certificates from \(x\) to \(z\) and from \(z\) to \(y\), with \(x,y\ne z\). Choose a circuit surrounding \(z\) that avoids \(x\) and \(y\) and has cost less than \(\epsilon\). In sufficiently accurate realizations both paths must cross this circuit: their endpoints near \(z\) lie on its inner side and their other endpoints on its outer side. Robust separation makes the crossings actual common primal vertices. Cut at such vertices and join along the circuit. The resulting cost is at most the sum of the original costs plus \(\epsilon\). Simultaneous marked extraction retains that inequality. Letting the circuit shrink proves concatenation; the cases with a constant certificate are dealt with by the cost-zero diagonal.

For a path \(P\) compactly confined to \(U\), apply the upper comparison with endpoint neighborhoods shrinking to its endpoints. Extract the resulting certificates in a fixed compact confinement, first enlarging the range of \(P\) slightly within \(U\). Their endpoints become exact and their costs are at most \(C\operatorname{len}_D(P)\). Taking the infimum over such paths proves Equation (44). The triangle inequality just proved and symmetry give \[|F_U(x,y)-F_U(x',y')| \le C\{D_U(x,x')+D_U(y,y')\}\] whenever these quantities are finite. On a compact subset of \(U\), \(D_U\) and \(D\) agree for sufficiently close pairs: the compact set has positive \(D\)-distance from the complement of \(U\), and a sufficiently short near-minimizing path cannot exit \(U\). Continuity follows. ◻

Proof. At the current point of \(P\), choose a disk whose hole contains it and stop at its first exit. Repeat, stopping before an incomplete last traversal. Completed traversal intervals are disjoint. They are finite in number when the total charge is finite, since each costs at least \(a\min_{B\in\mathcal B_n}w_B>0\). Their weights therefore sum to at most the total charge divided by \(a\).

Consecutive selected domains intersect. Delete domains contained in others, and replace each domain in the original chain by a maximal selected domain containing it. The resulting intersection graph connects a domain containing the initial point to a domain containing the final completed exit. Choose a simple graph path between them. Two consecutive maximal Jordan domains have intersecting boundaries: by the Jordan separation theorem, otherwise one of the two intersecting domains would contain the other. Follow the boundary connections between intersection points. No retained boundary is used twice, so the cost is at most \(A\) times the sum of the selected weights. Containment deletion moves an endpoint by at most \(2\epsilon_n\), and an incomplete last disk adds at most \(2\epsilon_n\). If no traversal is completed, both endpoints are in one disk and only this endpoint error remains. ◻

Proof. The first assertion follows by applying Lemma 66 to a near-minimizing \(D_U\)-path and then sending \(n\) to infinity. For the second, perform the construction on discrete realizing paths at each fixed finite disk family. Move stopping circles within their open buffers to preserve strict crossings. If the number of completed traversals were unbounded, extract any prescribed finite number of marked traversals. The additive cost inequality would give a cost greater than the total budget once their number exceeds that budget divided by \(a\min_B w_B\). Thus finitely many marked pieces suffice. The \(D\)-connection supplied by the lemma has cost at most \(A/a\) times the budget, and its endpoint errors tend to zero in \(D_U\). This proves the lower assertion. Only \(D\)-continuity removes endpoint errors in this direction. ◻

Proof. Mark approximating paths at rational fractions of their total costs. An individual overshoot is at most one edge length and tends to zero. After simultaneous extraction, marks \(x_t\), \(t\in\mathbb Q\cap[0,1]\), satisfy \(D(x_s,x_t)\le a|s-t|/c\). Their Lipschitz extension is a path of \(D\)-length at most \(a/c\) in the retained compact subset of \(U\), with the required endpoints. The definition of \(D_U\) gives the result. ◻

Proof. Given the local fields, write \(p_i^o\) for the failure probability under \(\mathcal K_i^o\). The genuine-order failure probability is \(\mathbb E\sum_o w_op_i^o\). Adjoin conditionally independent candidate draws for all feasible orders. On the event that every positive weight is at least \(\tau\), the union bound gives \[\sum_o p_i^o\leq \tau^{-1}\sum_o w_op_i^o.\] This proves Equation (48). The minimum positive weight is strictly positive almost surely because there are finitely many orders. Its distribution is independent of \(i\).

For the coupling claim, first condition on the whole fields and retain the actual draw at its actual order. Sample the other components independently from their specified local kernels. The conditional law of the full vector is the product of all the feasible-order kernels, regardless of which component was retained. This product depends only on the local fields. The assumed product law for actual observations in disjoint domains therefore gives the claimed factorization after integrating the independent interior fields. An order of conditional weight zero is excluded; the actual order has positive conditional weight almost surely. ◻

Proof. The Cameron–Martin density is \(Z_u=\exp((h,u)_\nabla-\|u\|_\nabla^2/2)\), with the sum of the two Dirichlet inner products understood. Hence \(\mathbb E Z_u^2=\exp(\|u\|_\nabla^2)\). Adjoining the same kernel on both sides leaves the density equal to \(Z_u\). Cauchy–Schwarz proves the bound. Conditioning fixes the index and shifts. Multiply a harmonic residual by a fixed smooth cutoff supported in \(A'\) and equal to one on \(S\); its Dirichlet norm is bounded by a fixed multiple of its \(H^2(A')\) norm. For the imaginary field, first choose the residual constant in one fixed interval of period length. ◻

Proof. Take \(T\) large and \(c_*\) small so that the closed annuli are disjoint and each finer outer disk lies in the preceding inner hole. On the fixed compact subset, the covariance kernel is \(G_V(u,v)=-\log|u-v|+g_V(u,v)\), where \(g_V\) is smooth and separately harmonic. Averaging nested circles gives \[\operatorname{Cov}(H_j,H_k) =\min(t_j,t_k)-\log 8+g_V(z_j,z_k).\] The logarithmic term has independent disjoint increments and increment variance \(T\). The mixed differences of the smooth term are bounded by \(C|z_{j+1}-z_j||z_{k+1}-z_k|\leq Cr_jr_k\). Therefore the covariance operator of the increment vector is uniformly bounded and its trace is \(O(m)\).

For the profiles, represent \(\Phi_j(z)\) as pairing \(h\) with the signed measure \(\omega_j^z-\sigma_j\), where \(\omega_j^z\) is harmonic measure on the two annulus boundaries and \(\sigma_j\) is uniform measure on the outer circle. These measures have total mass zero and bounded total variation. The same properties hold for normalized spatial derivatives through order two: on \(A'\) the differentiated annular Poisson series is bounded by a convergent geometric series. At distinct scales \(j<k\), the logarithmic potential of the scale-\(j\) signed measure has gradient bounded by \(C/r_j\) on the finer hole. Cancellation against the scale-\(k\) measure bounds the covariance by \(Cr_k/r_j\). Cancellation in both variables bounds the smooth contribution by \(Cr_jr_k\). These estimates hold also for the indicated derivatives. On one scale, logarithmic singularities are integrable against the smooth boundary densities, so the diagonal variances and their integrals over \(A'\) are bounded.

Consequently the centered Gaussian vector \((\Phi_j)\) in \(\bigoplus_{j=1}^m H^2(A')\) has covariance operator bounded by a constant and covariance trace at most \(Cm\). The operator bound follows from the summable estimate \(Ce^{-T|j-k|}\) and Cauchy–Schwarz. If the covariance eigenvalues satisfy \(\lambda_n\leq C\) and \(\sum_n\lambda_n\leq Cm\), then for \(\theta<1/(4C)\), \[\mathbb E e^{\theta\|X\|^2} =\prod_n(1-2\theta\lambda_n)^{-1/2} \leq\exp\!\left(2\theta\sum_n\lambda_n\right).\] Apply this to the profiles and increments. For the whole-plane imaginary field the additive normalization terms disappear under the zero-mass pairings, giving the same estimates. Its phase modulo the fixed period does not enter the oscillatory profiles. ◻

Proof. Write \(G_V(u,v)=-\log|u-v|+g_V(u,v)\) as above. Subtracting the zero-boundary disk Green function and then the harmonic value at the center gives \[\begin{aligned} \operatorname{Cov}(\Phi_{z,r}(x),\Phi_{z,r}(y)) ={}&g_V(z+rx,z+ry)-g_V(z+rx,z)\\ &-g_V(z,z+ry)+g_V(z,z)-\log|1-x\overline y|. \end{aligned}\] In particular the divergent coarse variance \(-\log r\) has canceled. The displayed kernel and the derivatives through order two in each variable are bounded on the fixed smaller disk, uniformly in \(z,r\). The covariance operator of each \(H^2\) profile is therefore bounded and has bounded trace. The Gaussian eigenvalue calculation in Lemma 71 proves the claim. This is a marginal bound for each disk; no independence of their harmonic profiles is asserted or needed. The imaginary-field normalization disappears in the same centered differences. ◻

Proof. Choose \(0<\epsilon<Q-2\) and then \(a,B\) with strict margins \[ \gamma(Q+2+\epsilon)a<\tfrac12d, \qquad \gamma(Q-2-\epsilon)B>3d. \tag{55}\] Use the separation \(T\) of Lemma 71, with \(t_j=jT\) from the first \(j\) satisfying \(t_j\geq aN\) to the last satisfying \(t_j\leq BN\). There are at most \(C_2N\) scales. At every scale take a deterministic square mesh of spacing \(cr_j\) in a fixed neighborhood of \(K\). Here \(c>0\) is small enough for Lemma 71 and the hole geometry. For each point \(y\) of the finest mesh, follow its nearest center \(z_j(y)\) on each coarser mesh. These centers are deterministic and satisfy \(|z_j(y)-y|\leq cr_j\) after absorbing a fixed mesh factor into \(c\).

At either endpoint scale \(t\), there are \(O(e^{2t})\) distinct endpoint centers. Equation (51) and a Gaussian union bound show that the probability of any violation of \(|H_j|\leq(2+\epsilon)t_j\) there is at most \[O(e^{2t})\exp\!\left( -\frac{(2+\epsilon)^2t^2}{2(t+O(1))}\right) =O\!\left(e^{-(2\epsilon+\epsilon^2/2)t+O(1)}\right).\] This is exponentially small in \(N\). The endpoint meshes are counted at their own scales, not at the finest scale. Put \(\Lambda_j=\gamma(Qt_j-H_j)\). For all sufficiently large \(N\), Equation (55) gives \(\Lambda_{\mathrm{first}}<Nd\) and \(\Lambda_{\mathrm{last}}>2Nd\) simultaneously for all ancestries.

Fix one ancestry. Assign each target \(id\), \(i\in I_N\), to its first upward crossing, the first index \(j\) with \(\Lambda_j\geq id\). No monotonicity of \((\Lambda_j)\) is required. Fix an exponent \(b>2B+4\). Equation (53), with the fixed drift \(\gamma QT\) absorbed, gives constants \(\theta_1,C_3>0\) such that \[\mathbb E\exp\!\left(\theta_1\sum_j |\Lambda_j-\Lambda_{j-1}|^2\right)\leq e^{C_3N}.\] For \(L\geq2d\), an upward jump \(s>L\) receives at most \(1+s/d\leq2s/d\) targets. Therefore the number of labels assigned to jumps larger than \(L\) is bounded by \[ \frac{2}{Ld}\sum_j|\Lambda_j-\Lambda_{j-1}|^2. \tag{56}\] Choose \(L\) large enough that this number exceeds \(\alpha N/8\) with probability at most \(e^{-bN}\). This is possible by exponential Markov applied to the preceding quadratic moment. At every remaining jump, alignment holds with error \(L\), and at most \(J=\lceil L/d\rceil+2\) targets are assigned to one shell.

Condition on the exterior of the union of the disjoint open annuli of this ancestry, for both fields. For the imaginary field retain the lifted exterior differences, with its remaining common constant taken modulo its period. Each annulus then contains independent Dirichlet parts. The outer circle averages and harmonic extensions are exterior measurable; boundary averages can equivalently be defined as limits from outside the open annuli. Thus the first-crossing indices and their large-jump exclusions are exterior measurable as well.

Let \(\widehat\Phi_j\) be the imaginary oscillatory harmonic profile. After \(J\) has been fixed, Equation (52) allows a cutoff \(R\) so large that \[\mathbb P\!\left[ \#\{j:\|\Phi_j\|_{H^2(A')}+ \|\widehat\Phi_j\|_{H^2(A')}>R\} >\frac{\alpha N}{8J}\right]\leq e^{-bN}.\] Indeed, the event forces the quadratic sum of the two profile norms to exceed a constant times \(R^2\alpha N/J\), and exponential Markov applies. Discarding such profiles loses at most another \(\alpha N/8\) labels. At least \(3\alpha N/4\) favorable labels remain. Keep one label by a deterministic rule at each remaining shell. For \(\nu=\alpha/(2J)\) there are at least \(\nu N\) distinct retained trials.

At a retained trial with index \(i\), the normalized real field equals its fresh Dirichlet part plus \[\Phi_j+H_j-Qt_j+\gamma^{-1}id.\] Alignment bounds the displayed constant by \(L/\gamma\). The imaginary residual constant is chosen in one period. The profile cutoff therefore gives a deterministic bound \(R_1=R_1(R,L)\) on the pair of Cameron–Martin extensions used in Lemma 70. The common intrinsic kernel identity identifies the conditional physical test with the same index-\(i\) reference kernel under this shifted field law.

Use Lemma 69 and Equation (50) to require that the all-order reference event has failure probability at most \(\delta_*\). Then Lemma 70 bounds every retained conditional failure probability by \(e^{R_1^2/2}\sqrt{\delta_*}\). Choose \(\eta>0\) so small that \[2^{C_2N}\eta^{\nu N/2}\leq e^{-bN},\] then choose \(\delta_*\) so that this conditional failure bound is at most \(\eta\), and finally choose the genuine-order threshold \(\varepsilon_*\) small enough to imply that value of \(\delta_*\). Conditional independence of the fields and candidate kernel vectors implies that fewer than \(\nu N/2\) successes has probability at most \(2^{C_2N}\eta^{\nu N/2}\). This calculation conditions on the selected indices and profiles; neither is required to be independent of the exterior. Every successful all-order event ensures success of the actual observation, whatever its global order.

This auxiliary coupling may be constructed separately for each fixed ancestry: it bounds that ancestry’s actual failure event. No compatible coupling of counterfactual candidates in overlapping ancestries is needed. There are \(O(e^{2BN})\) finest-mesh points. A union bound over them for large-jump, profile, and actual-trial failures is exponentially small because \(b>2B+4\). Add the already bounded endpoint failure. Every \(x\in K\) is within \(cr_{\mathrm{last}}\) of a finest-mesh point. Reducing \(c\) once puts it in the central hole of every selected annulus in that point’s ancestry. Set \(\beta=\nu/2\).

Replacing \(v_i\) by \(M\lambda^{-i}\) replaces each target by \(id-\log M\). The fixed endpoint margins absorb this shift for all sufficiently large \(N\); all increment, profile, and likelihood estimates are unchanged. ◻

Proof. The joint density change is a function of the fields alone: adjoining their same conditional passage kernel does not alter the Radon–Nikodym derivative. Thus a measurable local comparison event of probability one keeps probability one. Use a countable collection of rational ports, confinements, and cost tolerances, and then continuum joining, to obtain the simultaneous assertion. Inverse absolute continuity proves that a strict improvement under either law would improve the other law, so the optimal constants coincide. For a surface application, this last assertion uses the actual chart-covering and quantum-unit identities stipulated above. It does not infer a comparison outside the tested charts from a single local absolute-continuity statement.

For the lower constant, the local event is the intrinsic comparison. An ambient lower bound \(c>0\) implies it by Lemma 68; for \(c=0\) it is automatic. Conversely, divide a compact certificate into finitely many marked pieces within a rational chart cover. The local intrinsic bounds, additive cost inequality, and ambient triangle inequality give the same ambient lower bound for that certificate. Hence the ambient optimal definition and these measurable local tests are equivalent.

A fixed additive real-field change on the observation patch is locally absolutely continuous under the stated Gaussian reference: extend it smoothly across its buffer with support disjoint from the fixed pinning circle. The extension is a Cameron–Martin direction of that actual Gaussian law. In Equation (45), multiply the comparison under \(g+c_i\) by \(v_i^{-p}\). The reference Weyl identity gives \(v_i^{-p}D_{g+c_i}=D_g\). The numerical comparison constant therefore does not change. Countably many fixed indices can be handled simultaneously. This argument transports an almost-sure assertion; it does not identify individual passage samples before and after the field change. ◻

Proof. Fix the annulus threshold for favorable fraction \(1/4\). On a reference regularity event of sufficiently high probability, all separated crossings in the fixed shell cost at least \(a_0>0\) in \(D\), and surrounding reference circuits have cost at most \(A_0<\infty\). These choices do not involve \(\delta\). Choose \(b\) smaller than one quarter of the allowable failure probability.

If (i) fails, at least one quarter of the labels in arbitrarily late windows have near-saturation probability less than \(b\). On the complementary good event, every indicated geodesic traversal can be followed at a saving of at least \(\delta a_0w_i\) relative to the old upper comparison. Annulus transfer makes these events available simultaneously at every point. Stop a compact reference geodesic successively when it exits good surrounding separators. In each completed interval retain the subarc from its last visit to the inner testing circle until its first subsequent visit to the outer testing circle. This is minimizing in \(W\), and has length at least \(a_0w_i\). The retained intervals are disjoint.

The reference separator chain gives \[\operatorname{len}_D(P)\le A_0\sum_iw_i+o(1).\] Replacing the retained subarcs and following the gaps at upper factor \(C\) therefore gives total cost at most \[C\operatorname{len}_D(P)-\delta a_0\sum_iw_i+o(1).\] At each fixed finite family of separators choose the total realization and splicing error arbitrarily small. Then shrink the separators. This improves the upper factor to any number strictly above \(C-\delta a_0/A_0\), contradicting optimality.

For (ii), first choose a fixed small square size in the middle of the shell. On every hole-to-outer crossing, mark a point in this middle region, follow the path until it first exits the square centered there, and choose the exit face. Start the marked piece at its last visit to the parallel midline. It lies in the resulting half-square and crosses opposite faces; its width and its collar are fixed positive fractions of the shell radius. A high-probability reference event bounds the \(D\)-distance of every pair of such opposite-face endpoints below by \(a_0w_i\): these pairs have uniformly positive Euclidean separation on a compact set.

If shortcut probability were small on at least one quarter of the labels, annulus transfer would make every marked rectangular piece cost at least \(c_1D_{R^+}(x,y)\). For \(c>0\), Lemma 68 gives the old intrinsic lower bound on the same piece and on every gap; for \(c=0\) this assertion is automatic. Thus its extra cost is at least \((c_1-c)a_0w_i\). The positive extra charge bounds the number of marked pieces at a fixed finite annulus family even when \(c=0\), by simultaneous marked extraction and the additive cost inequality. Reference separator chaining now improves the lower comparison to any number strictly below \(c+(c_1-c)a_0/A_0\), a contradiction.

The marked half-square itself is retained as the closed envelope; thus the shortcut event throughout this contradiction is the stronger event in (ii). Use an intermediate threshold between \(c\) and \(c_1\) before selecting rational outward collar widths and gridded parameter bins. Keep the actual envelope and its opposite-face endpoints inside each bin, rather than replacing their range by the enlarged collar. Strict slack preserves the inequality. Finally, compactness of the family of reference rectangles, continuity of \(D\), and internal box connection bounds give a reference regularity event of probability greater than \(1-b_1/2\) with the additional bounds in (ii). Intersecting it with each witness event retains probability at least \(b_1/2\). ◻

Proof. Choose \(c<c_1<c_2<C\) and \(\delta_k\downarrow0\). The two sets of labels in Proposition 75 intersect in more than half of every sufficiently late window. Choose a common label tending to infinity. Retain the two witnesses separately, using a dummy witness off each event. Their simultaneous occurrence is unnecessary. The extraction property in Definition 63(iv) realizes the further law by a diagonal discrete array with these witnesses, all fixed offsets, and the common reference experiments retained. Its kernels are the newly extracted reference kernels; they need not be limits of kernel values in the preceding laws.

The near-saturation paths have their entire ranges in a fixed closed band \(K\) and minimize in its open collar \(W\). Retain \(D_W|_{K\times K}\) as well as the paths. On compact reference regularity sets, their lengths lie in a fixed interval \([a,A]\subset(0,\infty)\) and their parametrizations have a common Euclidean modulus. Under a joint coupling, a subsequence converges uniformly to a nonconstant \(D_W\)-geodesic \(P\), since the identities \(D_W(P_k(s),P_k(t))=|s-t|\operatorname{len}_D(P_k)\) pass to the limit. Exhausting these regularity sets loses arbitrarily little probability.

If a limiting competitor in \(W\) were strictly cheaper than \(C\operatorname{len}_D(P)\), its compact range would lie in a member of a fixed countable exhaustion of \(W\). Hausdorff attainability of the full certificate records realizes it in the approximating laws, with nearby endpoints and almost the same cost. The common upper comparison and the continuum joining property correct those endpoints at vanishing cost, uniformly on the retained compact reference regularity set. This contradicts the near-saturation inequalities. Thus the limiting path is saturated. Any cheaper competitor for an interior subsegment could be joined to the unchanged remaining pieces at factor \(C\), contradicting saturation of the full path; every interior subsegment is saturated as well.

For the low witness retain its closed rectangular envelope \(R_k\), its endpoints, its certificate range, and its two outer collars. The envelope centers and side lengths lie in a compact parameter set with positive lower width; the possible orientations form a finite set. Uniform reference moduli and the slack \(c_1<c_2\) allow a finite gridded library for the denominator alone: choose a fixed open rectangle \(\Omega\) inside the original denominator collar, still containing \(R_k\) with uniform positive clearance, and retain the internal metric on \(\Omega\) and a further fixed collar. Shrinking the denominator can only increase its intrinsic distance. Extract the actual envelope parameters as well as the marked certificate and endpoints. Along \(R_k\to R\), closed range inclusion and the opposite-face endpoint conditions pass to the limit. The limiting range therefore remains inside \(R\) itself, regardless of the open collar used to realize it. Its ratio relative to \(D_\Omega\) is at most \(c_2=C-\delta_0\). Retaining this envelope will allow the forcing construction to preserve the certificate on a thin real-field patch still contained in the denominator domain. If its lower comparison were \(C>0\), Lemma 68 would contradict this witness.

Transfer to a whole-plane field in a buffered patch using the same local kernels. Choose its actual Gaussian reference to be the whole-plane GFF \(G_R\) pinned on \(\partial B_R(0)\), with this fixed circle strictly outside the buffered patch. All later candidate supports are compactly contained in that patch, after discarding finitely many coarse scales. Thus every catalogue shift has zero average on the pinning circle and belongs to the Cameron–Martin space of this one reference law. The annular Markov trials use Dirichlet interiors conditioned on their exterior harmonic parts; no change of density across the pinning circle is required. Choose an interior segment shorter than its \(D\)-distance to the patch boundary; it is an ambient geodesic because any exiting competitor is longer. Strong confluence places a nonconstant part of it on a geodesic between two points of a fixed countable dense set. Fixed-pair uniqueness holds simultaneously for these pairs. Choose rational fractional times inside the common segment and an open finite union of rational disks with closure inside the saturation neighborhood. This is a countable union of events covering a positive-probability event; one choice has positive probability and gives Equation (58). ◻

Proof. Use the target regular conditional law of the full passage object given the two retained restrictions and the output fields. Before imposing \(A\), the locality and product hypotheses give exactly the claimed output law. Its density relative to \(\mu\) is the Gaussian shift density \[L_f(h')=\exp\{W_f(h')-\tfrac12\|f\|_{\mathcal H}^2\}.\] At \(h'=h+f\) this is \(\exp\{W_f(h)+\tfrac12\|f\|_{\mathcal H}^2\}\). Restriction to \(A\) decreases the unselected output measure and confines it to the set where this density is at most \(\Lambda\). This proves the assertion, regardless of whether \(A\) is measurable from the output. ◻

Proof. Domination, support, and Tonelli’s theorem give \[\alpha N \le \sum_\beta w_\beta\nu_\beta(H_N\cap B_\beta) \le \Lambda\int\mathbf 1_{H_N} \sum_\beta w_\beta\mathbf 1_{B_\beta}\,d\mu \le \Lambda K N\mu(H_N).\] For the last assertion, \(\mathbf 1_{H_N}\to0\) pointwise, including on \(\{R=C\}\), so bounded convergence yields a contradiction. ◻

Proof. Write \(P'\) for a new geodesic. It must enter \(K\), since otherwise its length is an old path length, contradicting (i) and \(B<g\). Its portion between the first and last visits to \(\partial K\) has length at most \(B\): its exterior prefix and suffix have costs at least \(\ell_0\) and \(\ell_1\).

Gate contacts below may lie in the closures of the open overlaps. The outer \(D\)-distance bounds extend to these contacts by continuity. For terminal contacts, \(\overline N\subset\Omega\), and the internal metric \(D_b(\cdot,\cdot;\Omega)\) is locally continuous: sufficiently close points are joined by near-minimizing paths contained in a fixed small ball of \(\Omega\). Thus its distance bound to \(q_j\) also extends from \(G_j\) to \(\overline G_j\). Realizing these infimal bounds may add an arbitrarily small error, which the two strict displayed budget inequalities absorb.

The exterior gap also gives \(D(x_0,x_1)\ge g\), by concatenating the two minimizing exterior approaches with a connection between \(x_0\) and \(x_1\). The first corridor visited by \(P'\) is \(T_0\). Indeed, if it were \(T_1\), the preceding part lies in \(O\), where its old cost is at most its new cost. Its cost is at most \(\ell_0+B\), since the final exterior approach costs at least \(\ell_1\). Appending the gate correction and the old exterior approach from \(x_1\) would give an old path of length at most \(\ell_0+\ell_1+B+e\), a contradiction. The last corridor is similarly \(T_1\). A path remaining in \(O\) is excluded in the same way. No excursion in \(O\) can connect the two outer gates: such an excursion, of cost at most \(B\), would give \(D(x_0,x_1)\le B+2e<g\).

The finite incidence graph consequently contains an inner \(T_0-N-T_1\) passage along \(P'\). For precision, a finite subdivision of the compact curve subordinate to the open region cover gives a finite region itinerary; repetitions can be retained or erased. There is no need to sum corrections over repeated gate visits. The first full approach to \(N\) uses \(T_0\), and the last full exit from \(N\) uses \(T_1\). Condition (iv) supplies points \(z_0,z_1\) on these passages, in this order, with \(D_f(z_j,q_j;W)\le e\).

Between these two points there is a central passage from the last \(T_0\) contact to the first following \(T_1\) contact. This passage lies in \(N\), avoids the effective corridors in its interior, and has multiplier at least \(e^{\xi b}\). Correcting its terminal points inside \(\Omega\) by (iii) shows that its new length is at least \(d-2e\). In particular \[D_f(z_0,z_1)=\operatorname{len}_{D_f}(P'[z_0,z_1])\ge d-2e.\] Upper comparison on the two short approach joins and the preserved certificate now give \[F'_W(z_0,z_1) \le (C-\delta)d+2Ce <C(d-2e) \le C D_f(z_0,z_1).\] The strict middle inequality leaves room for the arbitrarily small concatenation losses. ◻

Proof. We first make all Euclidean choices; no metric-dependent curve is used to choose the shape of a shift.

Proof. Choose constants \(0<m\le1\le M<\infty\) with \(mD\le D_f\le MD\). Properness puts all the relevant geodesics in one compact \(D\)-ball. Let \(\omega_n\) be the supremum of the old \(D\)-diameters of suitably enlarged possible support balls. The Euclidean radii tend uniformly to zero, so continuity gives \(\omega_n\to0\).

Replace the portion of an old geodesic between its first and last support visits by an old minimizing connection. Its new length is at most \(M\omega_n\). For the reverse comparison, cut a new geodesic at its first and last support visits and insert an old minimizing connection of cost at most \(\omega_n\). Outside the support the lengths coincide. Thus \[|D_f(x,y)-D(x,y)|\le(M+1)\omega_n\] for the endpoints under consideration. Moreover, the new geodesic segment spanning all its support visits has new length at most \(M\omega_n\), and old length at most \(m^{-1}M\omega_n\). Hence the old length of the entire new geodesic tends to \(D(a_0,a_1)\).

The new geodesics are uniformly Lipschitz in \(D\) under their fractional parametrizations. Every subsequential limit is an old minimizing path; equivalently, the uniform distance comparison shows that each of its subsegments has the appropriate fractional distance. Uniqueness identifies the limit with \(P\) with its specified parametrization. A contradiction-subsequence argument proves uniformity over all support locations and bounded shifts. Write the uniform endpoint discrepancy as \(\eta_n\to0\).

Put \(q_0=P(t_1)\), \(q_1=P(t_2)\) and \(q'_0=P'(t_1)\), \(q'_1=P'(t_2)\). Consider an arbitrary output certificate from \(q'_0\) to \(q'_1\) compactly in \(U\). If it enters the modification neighborhood, cut its approximants at their first and last visits to a ball strictly larger than the ball removed from the preserved exterior patch. Marked subpath extraction gives retained prefix and suffix certificates in that open exterior patch, with total cost no larger than the original cost. Transfer them to the old sample. Join their cut points at old passage cost at most \(C\omega_n+o(1)\), and correct the two endpoints at total cost at most \(2C\eta_n+o(1)\). All corrections lie in \(U\), by its clearance and the fact that short old geodesics remain in small neighborhoods. For a certificate avoiding the larger ball, transfer it directly and make only the endpoint corrections. The retained pieces use their own positive clearance from \(\partial U\); a common clearance for all competing certificates is unnecessary.

Taking infima yields, with \(\ell=(t_2-t_1)D(a_0,a_1)\), \[C\ell\le F'_U(q'_0,q'_1)+C(\omega_n+2\eta_n).\] The new denominator \(\ell'\) tends to \(\ell>0\), proving the ratio bound. Finally, the old middle subarc is separated from \(P([0,t_1]\cup[t_2,1])\). Uniform convergence therefore places every visit of \(P'\) to the shrinking modification neighborhood strictly between \(t_1\) and \(t_2\). Concatenating the strict shortcut with upper-comparison paths along the remaining geodesic portions gives the strict upper inequality. ◻

Proof. Transience confines the visits to \(\overline U\) to a compact time interval. Uniform continuity gives a positive lower duration for each \(U\)-traversal reaching \(K\), because it must cross \(\operatorname{dist}(K,\partial U)>0\). There are therefore only finitely many such traversals. Property (i) ensures that each contains a rational-point visit. In the locally reconstructed collection select the first rational representative in a fixed enumeration for each traversal; this selection does not use their chronological order.

Apply (ii) again to the disk \(B_R\). For every selected rational representative, reconstruct its full directed \(B_R\)-traversal, with time zero at entry. Let \(D_R\) be the event that all these records are identical. It is measurable from the buffered imaginary-field restriction. Distinct traversal intervals cannot have identical records: they have positive time length and hence positive area, whereas ranges of disjoint time intervals overlap in zero area. On \(D_R\), the common directed parametrization and (iii) specify the order of the representatives exactly, and therefore the order of their distinct small-domain traversals.

Almost surely the finitely many representative visit times have a finite minimum and maximum. By continuity, the image of the interval between them is compact and lies in \(B_R\) for all sufficiently large \(R\). They then lie in one common \(B_R\)-traversal, so \(D_R\) holds eventually. This proves the exhaustion. ◻

Proof. First truncate the normalized small-box and face modulus at a loss smaller than \(\beta/10\). Choose a small mesh width \(t\) using this common internal box modulus and the prescribed terminal error budget. Grid both the closed-envelope parameters and the positions of the two endpoints. For each resulting finite bin, choose an outward rectangular preservation collar \(V\) containing every envelope in the bin with positive margin. Each indicated face of \(V\) is within \(O(t)\) of the corresponding envelope face. Choose terminal caps of diameter \(O(t)\) straddling these two faces, containing the endpoint bins and fixed stop points outside \(\overline V\). The internal box modulus on their collars gives the desired arbitrarily small terminal connection errors. The entire certificate range is in \(V\) because it was retained inside its closed envelope. Choose an intermediate denominator domain containing \(\overline V\) inside the original one. Its intrinsic denominator is at least the old denominator, so the strict low-ratio bound remains valid.

The possible endpoints lie in deterministic compact subsets of finitely many newly thin domains. Their internal reference connection diameters are finite almost surely. Fix an upper bound \(A'\) with a further loss smaller than \(\beta/10\). These cap sizes, thin domains, and \(A'\) remain fixed during all later tube truncations.

For a singleton order input, take its determination event to be the whole sample space. Otherwise choose a compact \(K\Subset U\) covering the prepared local traversals. By Lemma 85, a fixed imaginary buffer makes the sum of order-determination failure probabilities over the finite geometry bins smaller than \(\beta/10\). Write \(E_{v,a}\) for the prepared success event in bin \(a\) at favorable index \(v\), intersected with its determination event, and put \(E_v=\bigcup_a E_{v,a}\). The genuine order on each of these events is a function of the buffered imaginary field. Thus the success union \(E_v\) is local and retains a common positive probability, say \(\beta_1\). No individual bin is required to have positive probability.

Choose a larger fixed conditioning disk around the complete dependency region. Local absolute continuity transfers the reference fields to a pair of Dirichlet GFFs on this disk, restricted to its interior. The field density is independent of the index. Truncate this single density to conclude that the success union \(E_v\) has probability at least \(\beta_2>0\) under this fixed Dirichlet reference law, uniformly in the index.

For an affine trial copy, condition on the fields outside its conditioning disk. The fields inside are Dirichlet fields plus exterior-measurable harmonic functions. On an acceptable profile event, subtract the real coarse constant and reduce the imaginary constant modulo its period. Smooth cutoffs extend the remaining profiles on the dependency region to Cameron–Martin functions of uniformly bounded energy. The same remains true after adding any real residual constant in \([-1,1]\).

Let \(P\) be the fixed Dirichlet reference law, including an independent auxiliary kernel seed, and \(Q\) one such conditional law. The passage kernel is the same function of the field inputs under both laws. A uniform inverse likelihood second moment gives, for the success union \(E_v\), \[\beta_2^2 \le Q(E_v)\,\mathbb E_Q\left[ \left(\frac{dP}{dQ}\right)^2\right], \qquad Q(E_v)\ge\beta_*:=\beta_2^2/K_*>0.\] The profile cutoff and \(\beta_*\) are independent of the number of trial copies: the normalized domains and buffer ratios are fixed, and (iv) controls their harmonic oscillations uniformly.

Choose that cutoff so the expected fraction of unacceptable profiles is below \(\theta/100\). Markov’s inequality implies that at least half the trials are acceptable except with probability at most \(\theta/50\). Choose \(J\) with \[(1-\beta_*)^{J/2}<\theta/10,\] and pack \(J\) deterministic disjoint copies of the entire conditioning disk into the prescribed shell subregion. All spatial ratios are now fixed. Before choosing \(M\), allocate a fixed portion of the remaining failure probability to the outer reference shell: fix a positive normalized crossing lower bound \(a\), a finite boundary-connection upper bound, and a reference connection modulus there. The normalized reference field law and shell geometry are fixed, so these choices do not depend on \(M\). Write \(\varepsilon\) for the already fixed normalized terminal error budget. Choose the fixed quantum separation \(M\) required by the forcing budget, using this \(a\), the already fixed \(A'\) and cap errors; in particular impose \(A'+4\varepsilon<aM^p\) in the units of Lemma 90. Assumption (iii) absorbs its logarithmic displacement into the desired compensation, so it does not change \(\beta_*\).

The \(J\) desired compensations are exterior-measurable. Their normalized vector is uniformly tight: the favorable index cancels in (iii), leaving only the fixed affine charts, their coarse field values, and the now fixed displacement from \(M\). Truncate this vector to a compact interval with loss at most \(\theta/10\), and cover that interval by a finite grid \(\mathcal B\) of mesh at most two. Select the nearest grid point using only exterior data. Its residual lies in \([-1,1]\), precisely the range in the preceding likelihood bound.

For every exact \(b\in\mathcal B\), draw a fresh auxiliary observation array for the prepared patches from the common kernel at the exactly shifted real field, and restrict it to each \(V\). Use independent seeds across disks and grid values, conditional on the fields. For the selected grid point declare success on the union of its events \(E_{v,a}\); select the first successful bin in the fixed finite ordering. On that bin’s determination event the genuine order equals the locally computed order. Hence the success indicator is a function of that disk’s field restriction, its independent seed, and the exterior-measurable grid choice. The indicators are conditionally independent given the exterior of the disjoint disks. Each acceptable disk has success probability at least \(\beta_*\), and the chance of no success is at most \((1-\beta_*)^{J/2}\).

The successful certificate is kept at the exact grid value at which it was drawn. Rounding has been handled by changing the field law in a uniform absolute-continuity estimate; no claim about persistence of an individual certificate under rounding has been made.

Finally impose the remaining regularity and finite internal bounds for all the gridded routing, core, gap, and access domains, with total additional failure smaller than the unused portion of \(\theta\). These later bounds affect only the wall and core heights and the smoothing widths. They do not change the cap modulus or \(A'\). After the resulting finite smooth recipe list is fixed, truncate its finitely many Gaussian linear pairings. For a smooth compactly supported recipe, this linear functional is a distributional pairing against its Laplacian, so it depends only on the real field on the recipe support. Clipping is therefore part of the same local shell event. The uniformly bounded energies give a fixed clipping threshold with arbitrarily small failure. Allocating the remaining losses proves the claimed probability. ◻

Proof. If \(c<C\), take the equality tangent and the positive-probability event in Equation (58). Restrict its interval length below by a deterministic positive number and its total anchor length above by a deterministic finite number; some such restriction still has positive probability. Apply Lemma 73 to the favorable finer labels given by Proposition 75, with volume multiplier \(M\) and with the full good event (a)–(c).

On the retained middle subarc, every point lies in at least \(\beta N\) good holes. A good ball accounts for at most \(K_0w\) of geodesic length: the time span between its first and last visits is bounded by a reference boundary connection, since a geodesic minimizes between any two of its points. Integration over the subarc gives \[\sum_{\text{good hit balls}}w\ge K_0^{-1}\beta N \operatorname{len}_D(\text{middle subarc}).\] Choose a recipe by a fixed ordering among the successful members of the finite catalogue. This partitions source events and adds no probability conditioning factor. Intersect with the deterministic uniform-error events of Remark 84; their lost probability tends to zero. The expected source weight is still at least \(\alpha_0N\) for some \(\alpha_0>0\).

For each deterministic candidate index, perform its fixed shift and complete the passage law before restricting to the selected source event. Lemma 77 gives an output submeasure \(\nu_\beta\le\Lambda\mu\) with the same fixed common law \(\mu\). Gate forcing gives a strict shortcut on the output geodesic. Shrinking-modification stability puts the gain in the same fixed fractional interval and shows that the output ratio from Equation (59) belongs to \[H_N=\{C-\epsilon_N\le R<C\},\qquad \epsilon_N\downarrow0.\] Here \(R\) is the same function of the output fields and passage data, and \(\mu\) is the common dominating law.

The output traversal tests in (d), bounded overlap at each spatial radius, \(O(N)\) radii, and boundedly many aligned labels and recipes give total output weight at most \(K_1N\) times its anchor length. The source length cutoff and the common amplitude bound give a deterministic output length cutoff. Thus the hypotheses of Lemma 78 hold and force \(\mu(H_N)\ge\alpha_0/(\Lambda K_2)>0\). Bounded convergence gives \(\mu(H_N)\to0\), a contradiction.

We have \(c=C>0\). The upper bound is Equation (44), and the lower bound follows from Lemma 68. Therefore \(F_U=CD_U\). ◻

Proof. The low event is Proposition 75(ii), with its closed rectangular envelope and actual internal denominator metric. The kernel restriction and product properties give condition (i) of Proposition 87; in the one-order case its local order event is the whole sample space. Otherwise Definition 63(vi) and Lemma 85 provide that event using a larger imaginary buffer. The exact volume and affine identities give condition (iii), and Lemma 72 gives condition (iv). The reference internal box and face bounds give condition (ii). Thus the complete local catalogue, including fresh observations at each exact shift and likelihood clipping, has probability as close to one as required by Lemma 73.

We verify explicitly how a hit of the old reference geodesic supplies the deterministic forcing data. Write \(u=\lambda^{-ip}\) and \(w=M^p u\). Include in the good event a lower reference crossing bound \(aw\) between the central hole and the outer circle, a surrounding boundary connection bound \(Aw\), and fixed reference connection moduli. On a deep hit, the reference geodesic pays at least \(aw\) between first entry and last exit from the outer circle. Its exterior pieces pay at least the two exterior infima \(\ell_0,\ell_1\). Hence \[D(a_0,a_1)\ge\ell_0+\ell_1+aw.\] Its exterior minimizing landing points have reference distance at least \(aw\), by concatenating their exterior approaches. The truncated reference modulus separates them by a fixed positive Euclidean amount, so they fit a finite outer-cap grid with the required small errors.

Let the prepared low denominator satisfy \(d\ge a'u\). Choose \(e=\varepsilon u\) with \(4C\varepsilon<\delta_0a'\). The thin preservation patches, terminal caps, and their connection bound \(A_1u\) have already been fixed in the catalogue-availability proof. Choose the fixed \(M\) so large that \[A_1u+4e<aw, \qquad B=A_1u+2e.\] All later core, gap and access bounds concern a finite family of new domains and affect only wall heights, core heights and smoothing widths. Lemma 81 therefore applies without changing \(A_1\) or the terminal errors. Its budget and gate conditions give \(B+2e<aw\) and \(4Ce<\delta_0d\), precisely the strict inequalities of Lemma 80. The resulting strict shortcut has compact confinement in the image of \(B_3(0)\) around the hit ball. These neighborhoods shrink uniformly to the old compact middle subarc and eventually lie in its fixed saturation neighborhood \(U\).

Fix the Gaussian pinning circle outside that neighborhood and all the late candidate supports, as in the equality-tangent proof. Every deterministic recipe is then a Cameron–Martin direction of this same actual Gaussian law. Its real shift is constant on the complete preservation patch and vanishes on the preserved exterior patch; the larger imaginary buffer is unchanged. Resample the local area marks with their shifted intensity. The conditional product kernel and standard Borel completion in Definition 63 give exactly the unselected construction of Lemma 77. Its finite Gaussian pairing cutoffs are part of the catalogue event before annulus coverage or a geodesic hit is selected.

For clarity, the required output count can use a shell where the shift is identically zero. Include among the local reference tests a lower bound \(\kappa w\) for every crossing of \(B_{3r}(z)\setminus\overline{B_{2r}(z)}\). Every forced reference geodesic enters the compartment inside \(B_{2r}(z)\) from anchors outside \(B_{4r}(z)\), and crosses this unchanged shell. Its output length inside \(B_{4r}(z)\) is therefore at least \(\kappa w\). Define the output event by this length bound, the inherited deterministic anchor-length cutoff, and the alignment condition on the outer circle average. That average is unchanged by the recipe supported in \(B_{2r}(z)\), so every generated output satisfies the event. These are functions only of the output field, its fixed-anchor geodesic and the deterministic candidate index.

For each fixed ball, alignment allows at most \(2+\lceil2L/\log\lambda\rceil\) volume labels, and there are only finitely many recipes. The center mesh has bounded overlap at each radius, and Lemma 73 uses \(O(N)\) radii. Charging the output event’s weight to its geodesic length in \(B_{4r}(z)\) thus bounds the total output weight by a constant times \(N\) times the output anchor length. The source length cutoff and uniform shift amplitude bound this last length deterministically. This proves (d) of Proposition 89; the preceding paragraphs prove (a)–(c) and its completion hypothesis. Uniform stability follows from Lemma 83. ◻

Proof. If the lower optimal constant \(c\) were smaller than \(C\), the common annulus lemma, comparison transport, and amplification would give the equality tangent of Proposition 76. Lemma 90 supplies every additional hypothesis of Proposition 89, whose weighted output argument is a contradiction. Thus \(c=C>0\). The intrinsic lower bound is Lemma 68; the upper bound follows from continuum joining and port comparison, as in Equation (44). Countable compact exhaustion gives the simultaneous assertion on open charts. The constants in Equation (57) are deterministic properties of the whole law; proving them equal is what excludes an additional sample-dependent multiplier. ◻

Proof. Take \(T>4(1+M/m)^2\), increasing it for any fixed geometric enlargement. If at least one quarter of the labels in arbitrarily late windows satisfy \(b_{i+\ell}>Tb_i\), their passage lower bounds are \(mTw_\ell\). The reference upper bounds are \(Mw_\ell\). Annulus transfer and Corollary 67 give the deterministic lower comparison \(G_i\ge(mT/M)D\). The base shell test has positive probability of a passage crossing of cost at most \(M\) and reference distance at least \(m\), contradicting \(m^2T/M>M\).

If one quarter instead satisfy \(b_{i+\ell}<b_i/T\), the same argument in the other direction gives upper comparison \(M/(mT)\). A base reference crossing of length at most \(M\), whose passage crossing cost is at least \(m\), gives the same contradiction. Thus each of these two exceptional sets has fewer than one quarter of the labels in all sufficiently late windows. Lemma 93 concludes the proof. Almost-sure comparisons are obtained by choosing a subsequence of annulus windows with summable failure probabilities, and then exhausting countably many buffered tests. ◻

Proof. For large \(N\), the windows based at \(i,j\) have \(N-O(|i-j|)\) common indices. In their overlap, the conditions \(b_k\le Tb_i\) and \(b_k\ge b_j/T\) have a common index, since their excluded sets together have fewer than \((N+1)/2\) elements. Hence \(b_j\le T^2b_i\). Interchange \(i,j\). ◻

Proof. Sample one word \(W\sim P\). Conditional on that word, draw the entire countable family \((Z^k)_{k\ge0}\) once from the product of the fixed kernels \(P_k(dZ\mid W)\). Every offset array reuses the coordinate \(Z^{j-i}\) from this family. Use the previously fixed definition of \(X_{j-i}\) on that same word and auxiliary contour. Its marginal is exactly the one defining the quantile, even though different offset coordinates are dependent through \(W\).

For \(v_i=4^{-i}\), write \((\mathsf T_i Z)(t)=v_i^{-1/2}Z(v_it)\). Both \(Z^{j-i}\) and \(\mathsf T_i Z^j\) approximate the same resolved word in offset units. Thus their compact-uniform difference tends to zero in probability for every fixed \(i\). A finite union handles any fixed offset family. Both arguments of each canonical readout have the same fixed Brownian marginal. For a measurable readout into a Polish space, Lusin’s theorem makes it uniformly continuous on a compact set of marginal probability at least \(1-\epsilon\). Consequently readouts of these two close arguments differ by a quantity tending to zero in probability, after a loss at most \(2\epsilon\). Let \(\epsilon\downarrow0\), apply the argument to finite subarrays, and diagonalize over the countable chart family. This retains the joint field/metric readout, rather than only its separate marginal laws. At this stage its arbitrary rotational sections are identified by quantum-surface equivalence.

The standard residual rotation is now sampled once, uniformly, independently of the common contour and the entire observation array. This restores the standard embedded cone law. The concentric winding tests, and the Euclidean-advance tests in the concentric compact annulus used below, are unchanged pointwise; consequently the exact finite circuit laws and the spatial-advance test laws used in Lemma 97 are preserved. An imaginary-field completion not determined by the contour is likewise adjoined once, conditional on the entire common limiting contour and this rotation, independently of the entire actual and reference observation array under the convention of Proposition 54. Its copies at other offsets are transported from that same draw. More explicitly, let \(\phi_i\) map the circle-average representative at offset \(i\) to the common base representative, choosing its orientation by transport, and put \(c_i=\gamma^{-1}\log v_i\). It is a positive dilation after this common choice of orientation. The actual readouts satisfy \[h_i=h_0\circ\phi_i+Q\log|\phi_i'|-c_i,\qquad \eta_i(t)=\phi_i^{-1}\eta_0(v_it),\qquad D_i=v_i^{-p}\phi_i^*D_0.\] The same identities hold for every internal metric on its transported domain. Intrinsic affine covariance of the local kernels then gives the indexed kernel \(\mathcal K_i\) in these units. These are transported annuli, which need not coincide as Euclidean sets in the base chart. In unchanged numerical cost units, if \(F_0=CD_0\), the pulled-back rule satisfies \(\phi_i^*F_0=Cv_i^pD_i\). Thus only the deterministic quantum-unit factor enters the offset comparison; the random coordinate dilation supplies no additional normalization. Unrelated rotations or imaginary-field draws at different offsets are not used to infer coherence.

The finite variables \(Z^k\) serve only to define the quantile and its projection readouts. The exact raw-chart resampling identities are proved before the limiting spatial readout, and no conditional independence of finite microscopic observations given \(Z^k\) is asserted. In the limit the contour-determined readouts are the measurable functions just identified; the additional variables have the stipulated conditional independence. These are precisely the auxiliary-conditioning conditions used by Proposition 54. ◻

Proof. Inside its normalization disk the cone field is \(G_1-\gamma\log|z|\), where \(G_1\) has zero unit-circle average (Duplantier et al. 2021, Definition 4.10 and the following discussion). Under the deterministic coordinate map \(z\mapsto z/R\), this is \[G_R(z)-\gamma\log|z|+(\gamma-Q)\log R, \qquad G_R(z)=G_1(z/R).\] On a compact ordinary patch in \(B_R\setminus\{0\}\) the displayed drift has a smooth finite-energy extension supported away from both \(0\) and the pinning circle. Cameron–Martin gives mutual absolute continuity with the actual \(G_R\) restriction. To put a prescribed buffered shell in such a patch, first map it by a fixed positive affine map into \(B_1\setminus\{0\}\). The pulled-back pinning circle remains outside the complete buffer; the drift, including \(Q\log|\phi'|\), again has a Cameron–Martin extension. The pin need not be centered on the shell.

Any two fixed exterior pins give equivalent restrictions on a smaller common patch. Indeed, on an intervening domain \(W\) their domain Markov decompositions have an independent zero-boundary GFF and a harmonic part. On \(K\Subset W\) each harmonic part admits a finite-energy cutoff in \(W\). Conditional Cameron–Martin equivalence with the zero-boundary restriction, followed by integration, proves the assertion. No circle average is reset in this comparison. The independent stationary-phase imaginary field has the same local law under these affine maps. Adjoining the same local kernels, with their genuine order and correct local area marks, preserves each field density. If \(Z=d\nu/d\mu\), then for any indexed kernel and event \(E\), \[(\nu\otimes\mathcal K_i)(E) \le T(\mu\otimes\mathcal K_i)(E) +\mathbb E_\mu[Z\mathbf1_{\{Z>T\}}].\] The tail is independent of \(i\). Thus uniform probability bounds transfer by truncating this fixed density; a quantitative exponential rate under an arbitrary equivalent field law is not asserted.

For the full atlas use the actual common offsets of Lemma 94, including each fixed negative offset. Put \(c_v=\gamma^{-1}\log v\) and \[r_v=\sup\{r>0:h_r(0)+Q\log r=c_v\},\qquad h^{(v)}=h(r_v\,\cdot)+Q\log r_v-c_v.\] The circle-average cone law and its quantum scaling (Duplantier et al. 2021, Proposition 4.13(i)) give the original joint background law at every fixed \(v\). The imaginary field is pulled back from the same sample; its stationary phase and independence of the real field preserve this joint law. The matching raw time, height, cost, and reference arrays were transformed together before extraction. Their intrinsic readouts therefore obey \[F^{(v)}_{U}=v^{-p}\phi_v^*F_{\phi_v(U)},\qquad D_{h^{(v)}}(\cdot,\cdot;U) =v^{-p}\phi_v^*D_h(\cdot,\cdot;\phi_v(U)), \quad \phi_v(z)=r_vz,\] and the local kernel in the new units is \(\mathcal K^{(v)}\). These are identities for the transported observations, with fresh reference kernels identified by Proposition 54. They are not field-shift identities for an individual passage sample.

Set \(H(r)=h_r(0)+Q\log r\). The radial cone construction gives \(H(r)\to-\infty\) as \(r\downarrow0\), and \(H\) is continuous. Hence \(\sup_{0<r\le R}H(r)<\infty\) for each finite \(R\). Since \(c_{4^m}\to\infty\), it follows that \(r_{4^m}\to\infty\). For each fixed \(m\), the comparisons under the indexed Gaussian kernel transfer to every ordinary patch inside the normalized cone disk. Pulling them back cancels the two factors \(4^{-mp}\) and covers \(B_{r_{4^m}}\setminus\{0\}\). A countable intersection over \(m\) and a countable chart exhaustion gives the whole ordinary cone atlas. This uses an exact joint quantum-unit identity, without conditioning on a random dilation and then applying local absolute continuity. For the reverse assertion, transfer a local comparison on ordinary patches of the unit disk to the Gaussian reference by mutual absolute continuity and intrinsic affine transport. Lemma 74 then preserves its numerical constant at every fixed quantum index. ◻

Proof. Cover a reference surrounding circuit by finitely many transverse port tests in the narrow band. Zero upper comparison gives certificates of arbitrarily small total cost; robust transverse intersections stitch them into a primal winding circuit. Exhaust its finite number of ports, reference length, and strict margins. For every \(\delta>0\), this gives, with limiting probability at least \(1-\delta\), a middle-band circuit of cost less than \(A_j/2\) for all sufficiently large \(j\). Choose \(\delta<1-\alpha\). Its probability is then strictly greater than \(\alpha\), contradicting \(\mathbb P[X_j<A_j]<\alpha\). ◻

Proof. Keep the three concentric winding bands fixed above, and choose a concentric compact annulus \(K\), bounded away from the cone mark, containing the widest band in its interior. Finite disk covers of a slightly larger annulus supply all required local chart patches. Choose \(\delta>0\) smaller than the Euclidean diameter forced by winding around the inner hole. A deterministic polygonal loop in the narrowest band has an approximate primal lift by the projection topology. Its lift is a finite graph walk, stays in the middle band, and has nonzero winding. Extract a simple surrounding subcircuit. Thus, on the events where the projection error is smaller than the fixed margins, \(X_j\) is finite. These events have probability tending to one. Time windows are first enlarged to contain the preimages of the annulus; transience gives this with arbitrarily high probability. No cost bound for the lift is asserted.

Define \(Y_j\) to be the infimal number of edges of a primal path whose projected range lies in \(K\) and whose projected endpoints are at Euclidean distance at least \(\delta\). Use \(\infty\) for an empty collection. Slightly larger compact confinements and slightly smaller separations provide strict versions. The contour coupling and the closed chord relation make the largest projected diameter of a single edge tend to zero, uniformly on the relevant compact time windows. A path of at most \(k\) edges has diameter at most \(k\) times this quantity. Therefore, for each fixed \(k\), \(\mathbb P[Y_j\le k]\to0\). Every winding circuit tested by \(X_j\) contains an admissible path for \(Y_j\), so \(Y_j\le X_j\) and \(A_j\to\infty\).

At any proposed cost unit \(b_j\), failure of the test \(\mathbb P[Y_j\ge b_j]\ge1-\eta\) retains, with probability at least \(\eta\), a path of normalized cost at most one and endpoints separated by at least \(\delta\). The endpoints and range lie in fixed compact sets. Marked extraction retains a certificate with that separation and finite cost. First choose a sufficiently large time window that its lost probability is less than \(\eta/2\), then pass to the extraction and exhaust windows. The fixed spatial margins retain the separated endpoints and compact confinement.

Choose a buffered annulus inside the interior of \(K\) with radial crossing separation greater than \(\delta\). Its primal traversal tests are included among these spatial-advance tests. Translations and the fixed affine changes needed to place the annulus use a finite collection of slightly enlarged versions of \(K\). The common local kernels transfer their high-probability tests to the fixed GFF reference: the field change of density is fixed, so Equation (50) allows \(\eta\) to be chosen small enough for the annulus threshold. Uniform bounded-edge advance gives the required initial runs, and the retained finite-cost witness gives the required first-failure contradiction. These are precisely the test conditions of Proposition 98. ◻

Proof. Fix \(0<\varepsilon<p\). Start at \(j_0\) with denominator one and use \(b_j=4^{(p-\varepsilon)(j-j_0)}\) thereafter. Choose the finite lower test’s failure probability smaller than the annulus threshold, \(\alpha\), and \(1-\alpha\), distributing it among the finitely many crossing geometries. By (b), every fixed initial run passes as \(j_0\to\infty\).

Suppose there are arbitrarily late starts for which a first failure \(f(j_0)\) occurs. Then \(f(j_0)-j_0\to\nobreak\infty\). Extract at the failed scale in units \(b_{f(j_0)}\). At each fixed preceding offset \(i\) the lower crossing threshold, expressed in reference metric units, is \[\frac{b_{f(j_0)-i}}{b_{f(j_0)}4^{-ip}} =4^{\varepsilon i}.\] For any \(T<\infty\), all sufficiently large labels therefore have barriers of at least \(Tw_i\), with reference surrounding costs at most \(A_0w_i\) on a sufficiently high probability regularity event. Annulus transfer and Corollary 67 force every nonconstant retained certificate to have cost at least \((T/A_0)D\) between its endpoints. Take the countable intersection over integer \(T\). This excludes every finite-cost certificate with positive spatial advance, contrary to the failed-test witness from (a) and (c).

Consequently a sufficiently late start passes forever. With the chosen probability margins, (c) gives \(\mathbb P[X_j<b_j]<\alpha\) at every subsequent level, and hence \(A_j\ge b_j\) (integer rounding changes this by at most one). This yields the stated lower growth after letting \(\varepsilon\downarrow0\). ◻

Proof. For every fixed \(t<p\), the sequence \(4^{-tj}A_j\) is unbounded and has arbitrarily late record maxima. At a record \(r\), \(4^{-t(r-i)}A_{r-i}\le4^{-tr}A_r\). Choose successively larger records for a sequence \(t_k\uparrow p\). ◻

Proof. In the fixed narrow reference annulus choose an almost surely finite rectifiable surrounding circuit, with a finite length bound \(Y\). The upper comparison on this ordinary annulus, finite transverse port tests, and small primal separators at its ordinary points realize a winding primal circuit in the middle annulus, with cost at most \(CY+1\). Only this fixed annular construction is used here.

Apply it in each of the exact units \(v=4^{-m}\) retained in Lemma 94. In base units the circuit has cost at most \(v^p(CY_v+1)\) and lies in a fixed multiple of \(B_{r_v}(0)\), where \(Y_v\) has the same finite marginal law as \(Y\). The radial construction in (Duplantier et al. 2021, Definitions 4.5 and 4.10, Remark 4.4) gives \(H(r)\to+\infty\) as \(r\to\infty\) and \(H(r)\to-\infty\) as \(r\downarrow0\), since \(Q-\gamma>0\). Consequently \(\inf_{r\ge\varepsilon}H(r)>-\infty\) for every \(\varepsilon>0\), and \(r_v\to0\) as \(v\downarrow0\). Select a deterministic subsequence \(m_k\) so that the probabilities of a radius larger than \(2^{-k}\) and of a cost larger than \(2^{-k}\) are each summable in \(k\). Borel–Cantelli gives the asserted circuits, with strict winding margins and simultaneous realizations. No independence of the different offsets is required.

For a compact reference path meeting \(0\), use the first and last crossings of one of these circuits. The exterior initial and final subpaths are compactly away from \(0\), so their port upper bounds are already known. Join their realizations along the primal circuit, whose extra cost tends to zero. If an endpoint is \(0\), choose the circuit inside its prescribed open endpoint port. This proves the same upper port comparison for all paths. The circuits at \(0\) and the existing circuits at ordinary points now give Lemma 65 on the whole chart.

For the lower comparison, retain the ordered prefix and suffix outside \(B_\varepsilon(0)\) of a realizing certificate. The omitted middle cost is nonnegative. The known lower bounds for the two exterior pieces give the lower bound for its endpoints up to \(c\operatorname{diam}_D(B_\varepsilon(0))\), which tends to zero by Lemma 62. The same argument with just one exterior piece treats an endpoint at \(0\); if both endpoints are \(0\) the bound is zero. First extract these markings with open margins, then let \(\varepsilon\downarrow0\). This proves the ambient lower bound everywhere, and Lemma 68 makes it intrinsic. This is a statement about continuum certificates. ◻

Proof. At label \(i\) the wide-band upper circuit test has, with probability at least \(\alpha\) and arbitrary cost slack, a surrounding circuit of cost at most \(K4^{-ip}\). Include a reference lower crossing bound \(a_0 4^{-ip}\), with failure probability chosen before \(\alpha\). The common coordinate kernels make this the corresponding fixed-index test under the local reference field law; fixed changes of density are handled by Equation (50). Lemma 73 supplies arbitrarily fine circuits at every point of each compact set. Apply the actual-primal-circuit version of Lemma 66, charging a reference path’s complete traversals. It gives upper factor \(K/a_0\), up to the fixed geometric constants. The circuit costs also tend uniformly to zero, so Lemma 65 applies.

The same bound holds for every fixed forward index \(v_0=4^m\). Indeed, its sufficiently fine indexed laws satisfy \[(\mathcal K^{(4^m)})^{(4^{-\ell})} =\mathcal K^{(4^{m-\ell})},\qquad \ell\ge m.\] This tail belongs to the backward family with the same shell-test bounds. Deleting finitely many labels does not change the favorable density in arbitrarily late windows or the resulting comparison ratio. Thus one finite bound works for every fixed \(m\), without a rate uniform in \(m\). The same fine annuli provide the small primal circuits at ordinary points at every fixed index. Lemma 95 transfers these assertions to the ordinary cone atlas, and Lemma 100 extends it across the mark. The upper optimum cannot be zero by Lemma 96. Its optimal constants have the same numerical values in the indexed laws by the reverse transfer in Lemma 95. Theorem 91 gives equality on ordinary charts, and the same marked extension gives the stated intrinsic conclusion. ◻

Proof. In the wide reference band, every winding circuit has two points at a fixed positive Euclidean separation. By \(D\)-continuity on the compact band, the infimum \(Y_-\) of the distances of all such pairs is strictly positive almost surely. In the narrow reference band there is an almost surely finite reference rectifiable surrounding circuit; let \(Y_+<\infty\) be any measurably chosen upper length bound. Choose \(m_*>0\) sufficiently small and \(M_*<\infty\) sufficiently large that \[\mathbb P[Y_->2m_*]>1-\alpha/2,\qquad \mathbb P[Y_+<M_*/2]>(1+\alpha)/2.\] The reference laws of these variables are the same at every fixed offset in their own quantum units.

If \(A_{j-i}/A_j\) had a subsequential limit below \(m_*C4^{-ip}\), the probability-\(\alpha\) quantile upper event would retain a wide-band certificate below this cost with probability at least \(\alpha\). The lower comparison and the \(Y_-\) bound permit this with probability less than \(\alpha/2\), a contradiction. This argument also rules out a zero limiting ratio.

Conversely, on the event \(Y_+<M_*/2\), upper comparison and transverse port stitching realize a middle-band circuit below \(M_*C4^{-ip}A_j\), with strict cost and probability margins. The quantile definition forces \(A_{j-i}\) below this bound for all sufficiently large \(j\) along the extraction. It also rules out an infinite limiting ratio. All costs and probabilities were separated from their thresholds, so no continuity at a quantile atom was used. Set \(i=0\) to obtain the bounds on \(C\). ◻

Proof. For indices beyond the finite exceptional range write \(j-q=am+b\) with \(0\le b<m\). Iteration gives \(A_q/A_j\le4^{-sam}R^b\le(R4^s)^{m-1}4^{-s(j-q)}\). The finitely many earlier indices can be joined through \(J+m\) and absorbed in the constant. If \(q=\lfloor\log_4 n'\rfloor\) and \(j=\lfloor\log_4 n\rfloor\), then \(4^{q-j}\le4n'/n\). This proves Equation (63). Taking \(n'=1\) gives \(B(n)\ge K^{-1}B(1)n^s\). ◻

Proof. A neighborhood of the rigid laws. We describe the required law neighborhood, since a first-failure argument needs more than a pointwise quantile bound. Use the product topology of the full marked certificate records, the contours and the reference internal metrics on countably many buffered domains. Include the auxiliary reference experiments that ensure closure of the common-kernel identities by Proposition 54. The cost coordinates are compactified; the reference marginal is fixed and tight. Thus these joint laws are precompact under the assumed framework.

Let \(\mathcal R\) be the closure of the quantile-normalized rigid laws arising in these extractions. Lemma 102 puts their deterministic factors in a fixed compact subset of \((0,\infty)\). The closure still consists of rigid laws. Indeed, along a convergent subsequence their deterministic factors converge. Marked extraction retains the intrinsic lower comparison. Finite port realizations and the uniform reference moduli retain the upper comparison and its small-circuit splicing. These arguments are the same two directions used in Proposition 76. Hence \(\mathcal R\) is compact and its laws all have the same uniform comparison bounds.

Choose coarse shell constants \(0<m_0<M_0<\infty\) so loosely that every law of \(\mathcal R\) passes the tests in Proposition 92 with strict cost, probability, and confinement margins. Such constants are obtained from the fixed reference law and the compact interval of factors. Upper tests ask for finitely many strict port or winding realizations in an open margin; lower tests exclude a closed set of bounded-cost certificates with separated endpoints in compact confinement. Retaining a slightly wider band and using a slightly weaker cost threshold makes these tests stable under nearby laws. More precisely, after the geometric slack has been fixed their good events are open in the retained data topology. For an upper test this is the existence of a certificate satisfying strict cost, port, and open-confinement conditions. An upper winding test uses a finite marked recipe of robust transverse port crossings; it does not ask for equality of two continuously perturbed loop endpoints. For a lower test the good event is disjointness from a closed bounded-cost certificate set with compact confinement and separated endpoints. For each \(\mu\in\mathcal R\), choose a bounded continuous function \(f\) below the indicator of each such good event, with \(\int f\,d\mu>1-\epsilon/2\). The neighborhood \(\{\nu:\int f\,d\nu>1-3\epsilon/4\}\) has closure contained in the laws assigning that good event probability at least \(1-3\epsilon/4>1-\epsilon\). Finite time and chart cutoffs lose arbitrarily little probability, since their continuum marginal is fixed. Intersect the finitely many neighborhoods for the tests. Compactness then gives a finite union \(\mathcal O\) covering \(\mathcal R\), whose closure passes all the weaker coarse tests.

By Lemma 102, every rigid joint limit has all fixed ratios in \[\frac{m_*}{M_*}4^{-ip} \le A_{j-i}/A_j\le \frac{M_*}{m_*}4^{-ip}.\] Choose \(R\) strictly larger than both adjacent ratio bounds and their reciprocals. Then choose \(m\) large enough that \((M_*/m_*)4^{-mp}<\tfrac12 4^{-ms}\), with \(s=p/2\). Call a finite scale successful if its own-unit law belongs to \(\mathcal O\) and its adjacent and \(m\)-step ratio inequalities hold with these thresholds.

Record scales start long successful runs. Initialization and Lemma 99 supply record indices \(r_k\to\infty\) with complete preceding tails \(A_{r_k-i}/A_{r_k}\le4^{-t_ki}\), where \(t_k\uparrow p\) and \(t_k\ge s\). Proposition 101 and rigidity apply to every joint record extraction. Apply the same argument on a fixed finite forward enlargement, initially retaining the record’s cost units. At forward offset \(a\), its fine past offsets satisfy \(A_{r_k+a-\ell}/A_{r_k}\le4^{-t_k(\ell-a)}\) for fixed \(\ell\ge a\), whose limit is \(4^{pa}4^{-p\ell}\). These bounds first give the enlarged comparison in the old record units. A zero upper factor would contradict the retained old \(X_{r_k}\) quantile by Lemma 96, before any bound on the new quantile is needed. Lemma 102 makes its own quantile ratios positive and finite, so changing to those units gives laws in \(\mathcal R\). All numerical inequalities pass with strict margins. Therefore every fixed finite run after a sufficiently late record is successful. Diagonalization gives successful initial runs of lengths tending to infinity.

The successful tail and the first-failure extraction. Before a first failure, the full record tail remains useful. If all levels from \(r\) to \(j\) are successful and \(r\) has its record tail, iteration of the \(m\)-step inequalities and the remaining at most \(m-1\) adjacent inequalities gives \[\frac{A_q}{A_j}\le K_0 4^{-s(j-q)},\qquad 0\le q\le j, \qquad K_0=(R4^s)^{m-1}.\] For \(q<r\), combine the same bound at \(r\) with the record tail.

Suppose unsuccessful levels occur arbitrarily late. Let \(f_k\) be the first failure after \(r_k\), and set \(j_k=f_k-1\). The initial success runs imply \(j_k-r_k\to\infty\). Extract in units \(A_{j_k}\). Every fixed preceding own-unit law lies in \(\overline{\mathcal O}\), and the adjacent bounds ensure that each fixed relative quantile has a positive finite limit. Write \(b_i=4^{ip}\lim_k A_{j_k-i}/A_{j_k}\), and let \(G_i\) denote the limiting passage rule at offset \(i\) in its own cost units. For fixed \(i,\ell\ge0\), viewed from base \(i\), the cost unit at offset \(i+\ell\) contributes the ratio \(A_{j_k-i-\ell}/A_{j_k-i}\). Dividing by the reference length unit \(4^{-\ell p}\) gives \[4^{\ell p}\lim_k\frac{A_{j_k-i-\ell}}{A_{j_k-i}} =\frac{b_{i+\ell}}{b_i}.\] The limiting denominators \(b_i\) are positive by the preceding adjacent bounds. The common-word coupling of Lemma 94 therefore makes \((b_{i+\ell}/b_i)G_{i+\ell}\) the actual physical offset family of each base \(i\). The assumed common-kernel framework and its annulus transfer, Lemma 73, supply the physical compatibility hypothesis of Proposition 92. Proposition 92 makes all \(b_i\) uniformly comparable. Consequently the past ratios satisfy an exact-power upper bound. Proposition 101 and rigidity make the base law rigid.

The failed scale in preceding units. Examine the failed level in the preceding units before normalizing it. It is only a fixed time enlargement. Its cost records are still available, and the fine past offsets just used give a finite upper comparison there. This upper optimum is positive: a zero upper comparison would restrict to the retained old winding annulus and contradict Lemma 96 for the old \(X_{j_k}\) quantile. Rigidity is therefore available before any division by the failed scale’s own normalization. Lemma 102 then makes \(A_{f_k}/A_{j_k}\) positive and finite in the limit. The failed own-unit law therefore belongs to \(\mathcal R\), and all its ratio tests lie strictly within the successful bounds. Since \(\mathcal O\) is open, it is successful for large \(k\), a contradiction. All large scales are successful. Finally Lemma 103 gives the stated polynomial bound. ◻

Proof. Initial tests and local kernels. Lemma 94 fixes the exact quantile marginals and their joint offset readouts before any normalization extraction. It also verifies the contour-determined and independent-augmentation conventions required by Proposition 54. Lemma 25 supplies full marked records, realizability, and additive cost subdivisions. Proposition 36 supplies primal winding and intersection, the small projected advance of a bounded number of edges, and the approximate mesh lifts. Lemma 97 therefore gives finite circuit quantiles, their divergence, and the concrete separated-endpoint lower tests. Its nested compact confinements and the three winding bands give the strict transfer rules at every fixed time offset.

Theorem 56 supplies restriction-compatible spatial kernels with arbitrarily small real-field buffers, a possibly larger imaginary buffer, and finite genuine traversal order. The fixed-chart reference experiments and Lemma 55 are retained jointly for all countable chart refinements and fixed quantum offsets. Proposition 54 proves closure for the resulting admissible laws, including the further varying-law and changing-unit extractions used in the equality tangent and first-failure argument. Its diagonal discrete approximation retains every prescribed finite array and sends every fixed rescaled time unit to infinity in lattice units. The exact density calculation and local limit theorem apply anew to that diagonal array and identify its kernels with its own reference kernels. Its retained actual joint field/internal-metric marginal is unchanged. This verifies the closure hypothesis without inferring conditional independence from weak convergence alone. Intrinsic coordinate transport is part of Theorem 56; Equations (45) and (47) specify all changes of cost and volume units. Corollary 57 supplies their density transfer and the correct resampling of local area marks. The independent real and imaginary GFF inputs and their local equivalences come from Theorem 8.

In particular, the indexed family in Equation (45) uses actual discrete offset experiments at times \(Nv_i\) and cost denominators \(B_Nv_i^p\), retained jointly before applying the fixed local field-density changes. This checks admissibility of the indexed laws required by Proposition 54; it is not deduced solely from the algebraic kernel formula.

Annulus comparison. Consequently Lemma 73 applies. It proves the annulus transfer needed for initialization, exact-power comparison, and amplification. Small circuits give port splicing before any endpoint continuity of \(F_U\) is used. The exact tail reindexing in Proposition 101 gives one numerical upper bound at every fixed quantum index. The actual atlas in Lemma 95, followed by Lemma 74, then identifies the optimal constants of the whole joint law with those of its indexed members. The metric and confluence inputs for equality tangents are exactly Theorem 11.

The common interface and forcing. We have verified Definition 63(i)–(iv) using marked compact certificate records, the spatial product kernels, actual indexed experiments, and the diagonal closure theorem. For the FK records the compactified cost and marked path coordinates provide the required tightness before comparison; after comparison, reference metric moduli and small-circuit joining give the uniform continuum moduli used in its further extractions. The edge-cost condition follows from the divergence of the quantile units. The finite genuine order satisfies (vi) by Lemma 85. Once the annulus argument has provided the finite positive upper comparison, small circuits and Lemma 65 supply (v), before any theorem about a prescribed discrete endpoint is used.

Lemma 95 supplies the actual outside-pinned Gaussian law and the countable cone atlas. In particular, all recipes in the forcing argument are Cameron–Martin directions for one fixed Gaussian law, and the same local kernels are used with correctly resampled area marks. The reference inputs are Theorem 11. Therefore Lemma 90 supplies the complete catalogue, its deterministic gate budgets, unselected completion followed by likelihood clipping, source weight and observable output count. Its disk-profile bound is Lemma 72; it retains the real preservation patch inside the denominator collar and allows a larger unchanged imaginary buffer. Theorem 91 applies to every equality-tangent family, and Lemma 100 gives the conclusion across the cone mark.

Rigidity and propagation. Rigidity is therefore available for every family used at records or at a proposed first failure. The remaining inputs of Proposition 104 were verified above, so its first-failure argument applies to the actual FK normalizations. ◻

Proof. Proposition 105 permits application of Proposition 104; every sufficiently late fixed-offset law passes its tests. The same common-word coupling and intrinsic indexed-kernel identities identify these offsets with the physical annular family, as required by Proposition 92. Apply Proposition 92, then Proposition 101 and rigidity, to any subsequential family. The quantile bounds give the stated interval for \(C\).

For arbitrary time units \(n_r\to\infty\), put \[k_r=\lfloor\log_4 n_r\rfloor,\qquad m_r=4^{k_r},\qquad u_r=n_r/m_r.\] Pass to a subsequence with \(u_r\to u\in[1,4]\). Before extracting the \(m_r\)-time arrays, add the countable family of fixed \(u4^m\) transforms, \(m\in\mathbb Z\), of their charts and reference experiments. Retain also the actual and reference experiments in \(n_r\) time units, sampling their reference duration and shift densities in those target units as in Proposition 54. Retain the full joint continuum input and its fixed-\(u\) transform, including the actual fields, curves, coordinate dilation and internal reference metrics. Lemma 94, the exact joint transport in Lemma 95, and Proposition 54 permit this enlargement with one common residual rotation and one transported imaginary-field completion. The joint background law is fixed along the extraction; no continuity of its measurable reembedding as a function of \(u_r\) is required. If \(Z_r\) is the retained normalized base contour, the contour in \(n_r\) units is approximated by \(u_r^{-1/2}Z_r(u_r\,\cdot)\). The contour modulus identifies its limit with the fixed-\(u\) Brownian rescaling.

Keep graph costs in the units \(A_{k_r}=B(n_r)\). In the fixed-\(u\) coordinates, let \(\phi_{u,r}\) map back to the base coordinates and let \(\eta_r\) be the retained base curve. The projection at \(n_r\)-time label \(t\) is \[\phi_{u,r}^{-1}\eta_r(u_rt) =\eta_r^{(u)}\bigl((u_r/u)t\bigr),\qquad \eta_r^{(u)}(t)=\phi_{u,r}^{-1}\eta_r(ut).\] The fixed-\(u\) curve law has a tight local uniform modulus, so this projection differs uniformly on compact time windows from \(\eta_r^{(u)}(t)\) by a quantity tending to zero in probability. Retain the raw marked path arrays on fixed enlarged windows. Their graph paths and costs are unchanged; only their time labels and spatial projections have been relabeled. Strict open confinement and port margins preserve upper realizations, and the reverse marked extraction retains every bounded-cost confined path. Thus the limiting passage collection is the fixed-\(u\) transport of the power-of-four extraction. This step uses geometric margins for certificates, not a bound for joining prescribed discrete endpoints.

If the latter extraction has multiplier \(C\), its transported passage rule, with the graph costs still in the base units, satisfies \[\widetilde F_U(x,y) =C D_h(\phi_u(x),\phi_u(y);\phi_u(U)) =Cu^pD_{h^{(u)}}(x,y;U).\] The multiplier \(Cu^p\) is deterministic and remains in a fixed compact subset of \((0,\infty)\) because \(u\in[1,4]\). Intrinsic scope follows from Lemma 68 and Equation (44). ◻

Proof. Discard any loops of \(G\), which do not change its connected components. We construct a finite metric tree directly from the minima of \(f\). This also explains the contour order used in the proof.

Let \([A,B]\) contain all the intervals, and let \(m_*=\min_{[A,B]}f\). For each \(I_v=[a_v,b_v]\), choose a point \(r_v\) at which \(f\) attains its minimum on \(I_v\). Let \(\mathcal T\) be the finite set consisting of \(A,B\), all \(a_v,b_v,r_v,t_v\), all chord endpoints, and a point attaining \(m_*\). Coincident marked times occur only once in this set. Write \[M(s,t)=\min_{[s\wedge t,s\vee t]}f\qquad(s,t\in\mathcal T).\] For each \(t\in\mathcal T\) take a vertical segment \(\{t\}\times[m_*,f(t)]\). Identify \((s,z)\) with \((t,z)\) exactly when \(z\le M(s,t)\). This is an equivalence relation: the only nontrivial check follows from \[M(s,u)\ge\min\{M(s,t),M(t,u)\},\] because the interval between \(s\) and \(u\) is contained in the union of the intervals between \(s,t\) and between \(t,u\).

The quotient, with distance measured along the vertical segments, is a finite rooted metric tree \(T\). To verify this explicitly, adjoin the marked times in increasing order. The segment for a new time \(t_j\) shares with the previously constructed tree precisely the initial segment up to height \(M(t_{j-1},t_j)\): indeed \[\max_{i<j}M(t_i,t_j)=M(t_{j-1},t_j).\] Its remaining segment is therefore attached at a single point. This induction constructs a tree, allowing a new segment to have length zero. Declare all marked tips and all attachment points to be vertices of \(T\). All its segments start at the common root of height \(m_*\). Denote the top of the segment for \(t\) by \(\pi(t)\). The common part of the two root paths for \(\pi(s),\pi(t)\) ends at height \(M(s,t)\), so \[ d_T(\pi(s),\pi(t))=f(s)+f(t)-2M(s,t). \tag{67}\] In particular, the endpoints of every minimum chord have the same image in \(T\).

For \(I_v=[a_v,b_v]\), put \(m_v=f(r_v)\) and define the subtree \[L_v=[\pi(a_v),\pi(b_v)]_T \mathbin{\cup}[\pi(r_v),\pi(t_v)]_T.\] Here and below \([x,y]_T\) denotes the unique tree arc between \(x\) and \(y\). The point \(\pi(r_v)\) lies on the first arc: it is the common ancestor of \(\pi(a_v)\) and \(\pi(b_v)\) at height \(m_v\). It is also an ancestor of \(\pi(t_v)\). Thus \(L_v\) is connected. If \(\ell\) denotes the length of a finite metric tree, then \[\begin{align*} \ell(L_v) &\le f(a_v)+f(b_v)-2m_v+f(t_v)-m_v\tag{68}\\ &\le 3\omega_v. \end{align*}\] Overlapping arcs only reduce the first quantity.

Every chord contact in \(I_v\) with a time outside \(I_v\) belongs to \([\pi(a_v),\pi(b_v)]_T\). To see this, let \(u\in I_v\) be the contact and let \(w<a_v\) be the other endpoint. The chord condition implies \[M(a_v,u)=f(u),\qquad m_v\le f(u)\le f(a_v).\] Hence \(\pi(u)\) is the point of height \(f(u)\) on the root path to \(\pi(a_v)\), between \(\pi(r_v)\) and \(\pi(a_v)\). If instead \(w>b_v\), the same argument uses \(b_v\). This proves the claim, including contacts at interval endpoints.

For each edge of \(G\) joining \(v\) to \(w\), its common chord image therefore belongs to \(L_v\cap L_w\). Graph connectivity shows that \[U_C=\bigcup_{v\in C}L_v\] is connected. A connected subset of a tree contains the unique arc between any two of its points: removing an omitted point of that arc would separate the two points. Consequently \(U_C\) is a subtree and contains the finite hull \(H\) of \(\pi(t_{v_1}),\ldots,\pi(t_{v_k})\). Moreover, \[ \ell(H)\le\ell(U_C)\le3\sum_{v\in C}\omega_v. \tag{69}\]

We next justify the contour tour bound for the chronologically ordered representatives. Subdivide \(H\) at every vertex of \(T\) lying in \(H\), including all representative images; there are finitely many resulting edges. For a point \(x\) in the interior of such an edge, let \(\lambda\) be its absolute height. Among the components of \(T\setminus\{x\}\), call the one not containing the root the distal component. The representative indices whose images belong to this component form a consecutive block. Indeed, if the images at times \(s<u\) belong to it, then their common ancestor has absolute height strictly greater than \(\lambda\). Thus \(M(s,u)>\lambda\). For any marked time \(t\in[s,u]\), \[M(s,t)\ge M(s,u)>\lambda,\] which puts \(\pi(t)\) in the same distal component. The strict inequality is valid because \(x\) is in an edge interior and all marked images are vertices of the finite tree after subdivision.

The sequence of consecutive representative arcs can therefore cross any edge of \(H\) at most twice: it can enter and leave the consecutive block of indices on its distal side only once each. Summing edge lengths gives \[ \sum_{i=1}^{k-1} d_T\bigl(\pi(t_{v_i}),\pi(t_{v_{i+1}})\bigr) \le2\ell(H). \tag{70}\] This is the outer-face tour inequality, with its ordering verified directly from interval minima. It also constructs the relevant planar order: root \(H\) at its point nearest the root of \(T\), order each child subtree by its consecutive block of marked indices, and put repeated marks of an internal vertex in the intervening corners. The boundary tour of this ordered tree encounters the marks chronologically and uses each positive-length edge twice. The preceding edge-count argument does not require choosing any of those corners.

Equations (67), (69), and (70) give Equation (66). Finally, \[E_i=(h_{v_i}-\mu_i)+(h_{v_{i+1}}-\mu_i)\] is a sum of two nonnegative numbers. It dominates their difference in absolute value and each summand separately, proving Equation (65). A singleton component has an empty sum and needs no separate argument. ◻

Proof. Sum \(\eta\mathbf 1_{\{E_i>\eta\}}\le E_i\) and apply Equation (66). ◻

Proof. On the finite set of original chord endpoints, put \(s\sim t\) when \(f(s)=f(t)=M(s,t)\). This is an equivalence relation, either directly from the interval-minimum rule or from Equation (67). For each class, add the minimum chords between its consecutive times. They connect all times of that class, so they connect the endpoints of every original chord.

These new chords can be drawn without crossings in the upper half-plane. In fact, suppose two minimum chords have strictly interleaving endpoints \(a<b<c<d\), with pairs \((a,c)\) and \((b,d)\) at heights \(h\) and \(h'\). Since \(b\in[a,c]\) and \(c\in[b,d]\), the minimum conditions give respectively \(h'\ge h\) and \(h\ge h'\). Both heights agree, and all four times belong to the same equivalence class. Consecutive pairs of one class do not interleave. Chords from different classes therefore do not interleave either. Open semicircular arcs with nested or disjoint time intervals as diameters do not intersect. Shared endpoints belong to the same class and cause no crossing.

Contract each interval \(I_v\) in the boundary line to a single point. This preserves an embedding with all vertices on the outer boundary. One explicit way to see this is to choose a continuous proper nondecreasing function \(\varphi\) on the line that is constant precisely on the nondegenerate \(I_v\) and strictly increasing elsewhere. For \(y>0\), the map \[(x,y)\longmapsto \left(\frac{\varphi(x)+yx}{1+y},y\right)\] is a homeomorphism of the open upper half-plane, extends continuously to the boundary, and there makes exactly the desired contractions. Thus distinct chord interiors remain disjoint. Discard loops and keep one edge from each collection of parallel edges. In every resulting component choose a spanning tree, retaining its plane embedding. The earlier connectivity observation proves the first two assertions. Using only endpoints from a single original connected component gives the final assertion as well.

For completeness, the number of simple noncrossing forests on a fixed ordered vertex set is exponential in its size. Place the vertices on a convex \(m\)-gon in that cyclic order. Any noncrossing forest extends to a triangulation after missing polygon sides are added and the remaining faces are triangulated. For \(m\ge3\) there are fewer than \(4^m\) triangulations, by the Catalan recursion, and each has fewer than \(3m\) edges including the polygon boundary. Its edge subsets number at most \(2^{3m}\). Their product is at most \(32^m\). The cases \(m=1,2\) satisfy the same bound directly. ◻

Proof. Include the projected jump edges temporarily. Every occupied vertex then has a chain to a base: a marker arises in a fully retained chain, or is itself a base. The enlarged graph therefore has at most \(b_j\) components. Removing the \(D_j\) jump edges increases the component count by at most \(D_j\). For the rank inequality, take a spanning forest of the remaining union graph. Its edges of either color form a forest in that monochrome graph, so their number is bounded by that color’s rank. Summing the two bounds gives \(m_j-c_j\leq e_j\). ◻

Proof. Call a vertex critical if it is a root, has a number of children different from one, or carries a terminal entry. There are at most \(M\) leaves and at most \(M\) terminating vertices. For a finite forest, \[\sum_{v:\,\deg^+(v)\geq2}(\deg^+(v)-1) =\#\{\text{leaves}\}-\#\{\text{roots}\},\] so there are at most \(M\) branching vertices as well. Thus at most \(4M\) vertices are critical. Fix an integer \(g=4\) and discard all vertices at forest distance at most \(g\) from a critical vertex. Since the degree is at most \(R+1\), this discards at most \[C_RM,\qquad C_R=4\sum_{k=0}^g(R+1)^k.\]

For a remaining vertex \(v\in V_j\), its ancestor \(Q_v\) at depth \(j-g\) exists. The intervening path is unary. In particular, \(v\) is the only occupied depth-\(j\) descendant of \(Q_v\). Additionally discard \(v\) if its cell is at distance less than \(3r_j\) from \(\partial Q_v\). Call this a boundary deletion. We first prove disjointness for the vertices left by these two deletion rules.

Take distinct retained vertices \(v\in V_j\) and \(w\in V_k\), with \(k\geq j\). If \(k=j\), the cell of \(w\) is outside \(Q_v\), since \(v\) is its sole occupied depth-\(j\) descendant. Hence \(|t_w-t_v|\geq3r_j\), whereas the sum of outer slot radii is \(2r_j\). If \(k>j\) and \(w\) descends from \(v\), then \[|t_w-t_v|\leq\delta_j,\qquad r_k\leq r_j/R, \qquad \delta_j+r_j/R =r_j(R^{-2}+R^{-1})<2r_j/R.\] Both slots at \(w\) therefore lie strictly inside the inner gap of the slots at \(v\). If \(k>j\) and \(w\) does not descend from \(v\), its occupied ancestor at depth \(j\) lies outside \(Q_v\). Thus \(t_w\) is outside \(Q_v\), \(|t_w-t_v|\geq3r_j\), and the sum of outer radii is at most \(r_j+r_j/R<2r_j\). This proves disjointness in every case. The two slots at a single vertex are disjoint by their definition.

It remains to count boundary deletions without imposing any distribution on the marker positions. Condition on all these positions and terminal levels. At each depth \(j\), let \(s_j\) be the occupancy count for a fixed auxiliary grid of width \(\delta_j\), with no random displacement, and let \(S=\sum_j s_j\). Each cell of either of two grids of the same width meets at most two cells of the other grid. It follows, for every \(U\), that \[ S/2\leq T(U)\leq2S. \tag{75}\] Choose one active marker in each occupied auxiliary cell. If an occupied primary cell at depth \(j\geq g\) is boundary-deleted, a chosen representative in an auxiliary cell meeting it lies within \(3r_j+3\delta_j\) of a depth-\((j-g)\) boundary. The displacement modulo \(\delta_{j-g}\) is uniform. The probability of this event for a fixed representative is at most \[\frac{2(3r_j+3\delta_j)}{\delta_{j-g}} \leq12R^{2-g}.\] An auxiliary cell meets at most two primary cells. If \(B(U)\) is the number of boundary deletions, linearity of expectation gives \[\mathbb E_U B(U)\leq24R^{2-g}S.\] By Equation (75) and Markov’s inequality, \[ \mathbb P_U\{B(U)>\varepsilon T(U)\} \leq\frac{48R^{2-g}}{\varepsilon}. \tag{76}\] For \(g=4\), choose \(R\) large enough that this is at most \(1/8\). Together with the structural deletions this proves Equation (73). ◻

Proof. Two fixed times at distance \(d\) lie in different uniformly displaced cells of width \(\delta_j\) with probability at most \(\min\{1,d/\delta_j\}\). Therefore a jump of length at most \(K\delta_\ell\) has expected total separated projections at most \[\sum_{j=0}^\ell\min\{1,K R^{j-\ell}\} \leq \sum_{a=0}^\infty\min\{1,K R^{-a}\} =:C_{K,R}<\infty.\] Linearity proves Equation (77). For Equation (78), list the bases in increasing order and join consecutive bases by artificial edges for counting only. At every depth, \(b_j-1\) is bounded by the number of these edges that cross a cell boundary. Their total Euclidean length is at most \(CHe^{-N}\). Hence \[\mathbb E_U(b_j-1)\leq C e^{-N}R^j, \qquad \sum_{j=0}^{j_N}C e^{-N}R^j\leq \frac{CR}{R-1}.\] For fixed \(K\), both \(M\) and \(M_{\mathrm{jump}}\) are bounded by a fixed multiple of \(P+h_0\). Choose \(d_N=o(N)\) so slowly that \((P+h_0+1)/d_N\longrightarrow0\). Markov’s inequality gives Equation (79); the bound \(C_RM=o(N)\) follows from \(P=o(N)\). The last assertion follows by intersecting this event with the event of Lemma 114. ◻

Proof. If \(M=0\), all partitions are empty and the assertions are immediate. Keep the labels and incidences of coincident entries when grouping their positions. As \(j\) increases, entries and hence distinct active positions can only disappear. Initially there are at most \(M-1\) consecutive-position pairs; deleting one position creates at most one new pair. Thus at most \(2M-1\) distinct consecutive pairs occur over all levels. Deleting all positions with a common terminal level one at a time only enlarges this bound.

Every grid partition refines \(\mathcal P_j^*\): positions in the same half-open cell have total span strictly less than \(\delta_j\), and so do all their successive gaps. The partitions can differ only if a consecutive active pair with gap \(d<\delta_j\) is split by a cell boundary. For a fixed pair this has probability \(d/\delta_j\), since \(U\) modulo \(\delta_j\) is uniform. For each positive gap, \[\sum_{j:\,\delta_j>d}\frac d{\delta_j} \le \frac R{R-1}.\] The same bound holds after restricting to the levels at which the pair is consecutive and active. A union bound over pairs and levels proves Equation (80).

All active edge incidences and base labels were fixed before the grid, so equal marker partitions give identical projected graph data. Choose a deterministic \(q_N=o(N)\) with \(M_N/q_N\to0\). Markov’s inequality bounds the conditional probability of \(\Delta(U)>q_N\) by \(2RM_N/((R-1)q_N)=o(1)\). Apply this to each of two independent displacements. Outside that exceptional event they both agree with \(\mathcal P_j^*\) except at at most \(2q_N\) levels. The estimate is conditional on arbitrary marker data, so it also holds under any selection law sampled before the displacements. ◻

Proof. An ordered forest with \(T\) vertices and at most two roots has at most \(C4^T\) possible shapes. One way to see this is to attach a new root above its roots and use the depth-first parenthesis word of the resulting ordered tree; its length is \(2(T+1)\), and a balanced parenthesis word determines the tree. Each nonroot cell has one of \(R\) child offsets in its parent, so the shape has at most \(R^T\) geometric assignments. The occupied root cells have a bounded presence mask among the cells meeting \(I\), whose names are already fixed by \(I,U\). This proves the bound \(C(4R)^T\) for the forest before labels. Its vertices’ depths follow from its parent relation.

At probe depth \(j_p\), name a position by its ordered index among the \(m_{j_p}\) occupied cells; use one extra index for an absent list entry. Thus the position entries have at most \((1+m_{j_p})^K\) possibilities. Repeated indices record coincident positions. Names in different probe lists do not require global forest indices: each local index specifies a forest vertex, whose ancestors are already determined by the forest. A correspondence to a distinguished base uses that base’s fixed name, and its depth-\(J\) address determines its coarser addresses. The list’s chain order, colors, endpoint incidences, and base names have at most \(\exp(CK\log(K+1))\) possibilities. The count includes absent entries and shorter lists. It never specifies an exact endpoint within its terminal cell.

Multiply these choices and use the distinctness of probe depths to obtain \(\sum_p m_{j_p}\leq T\). Concavity gives \[\sum_{p=1}^P\log(1+m_{j_p}) \leq P\log(1+T/P),\] which proves Equation (81). The same estimate bounds the maximum number of list decorations over all forest shapes, so variation of the individual \(m_j\) between shapes causes no extra factor.

For \(0<\eta\leq K\) and \(z\geq0\), \[K\log(1+z)\leq\eta z+K\log(K/\eta).\] Indeed the maximum of \(\log(1+z)-(\eta/K)z\) occurs at \(z=K/\eta-1\) and is at most \(\log(K/\eta)\). Hence \[ \log\#\{\text{codes of occupancy }T\} \leq(\lambda+C_0+\eta)T +KP\log(K/\eta)+CPK\log(K+1)+O(1). \tag{84}\] For fixed \(K\), let \(\eta\) tend to zero sufficiently slowly while \(P=o(N)\). This yields Equation (82) uniformly in \(T\). Under Equation (83), for example \(\eta_N=1/\log L\) for large \(L\) makes both remainders in Equation (84) equal to \(o(N)\). No condition \(L\gg\log N\) is used. If the probe depths themselves must be encoded among the \(J+1\) levels, their extra cost is at most \(\log\binom{J+1}{P}\leq P\log(e(J+1)/P)=o(N)\) for \(P=O(N/L)\); alternatively their independent mesh displacement can be conditioned on.

Finally give \(T\) a weight proportional to \((T+1)^{-2}\) and distribute that weight uniformly among its finite enumeration of codes. The weights sum to at most one, and the negative logarithm of each weight is the code count bound plus \(2\log(T+1)+O(1)\). ◻

Proof. Cut the union of the intervals at all their endpoints. There are at most \(2|\mathcal I|\) nonempty atoms \(J\). Proposition 13 gives independent dominating variables \(Y_J=2M_{|J|}/\sqrt{|J|}\), with a common exponential-moment bound, for the oscillations on these disjoint fresh raw pieces. Subadditivity of oscillation gives \[\sum_{I\in\mathcal I}\frac{\operatorname{osc}_I Z}{\sqrt{|I|}} \le\sum_J a_JY_J, \qquad a_J=\sqrt{|J|}\sum_{I\supset J}|I|^{-1/2}.\] If \(I_0\) is the shortest interval containing \(J\), then \[a_J\le K\sqrt{|J|/|I_0|} \sum_{v\ge0}R^{-v/2} \le \frac K{1-R^{-1/2}}.\] Apply the exponential-moment bound with a parameter divided by this last constant, and use Markov’s inequality. This proves the claim. The bound on \(a_J\) is uniform for \(R\ge2\), proving the asserted uniformity. ◻

Proof. Choose a deterministic time before all the code’s representative points and sampling slots, and anchor both contour coordinates at zero there. A common coordinate translation changes none of the chord constraints or slot tests. With this convention the heights observed while exposing the finite word window are measurable in its chronological raw filtration, even when the original bilateral contours had been anchored at time zero inside that window.

For each coordinate and each projected chord component, apply the sorted comparison lemma to the occupied intervals of that component. It gives a comparison between consecutive occupied intervals, one per unit of monochrome rank, with a total height-discrepancy bound of at most six times the sum of the relevant box oscillations. A box is used in at most one component of each color. Therefore the sum of all normalized comparison discrepancies is controlled by a fixed multiple of the normalized box oscillations.

Include also the intervals between each regular-node representative midpoint and its adjacent sampling-slot endpoint. Their lengths are comparable to \(r/R\) at a node of sampling radius \(r\). The same-scale overlaps are bounded independently of \(R\): regular centers are separated at the scale \(r\), whereas the box intervals themselves have disjoint interiors. Thus Lemma 120 applies, separately to a bounded number of geometric length families. For fixed \(0<\epsilon<\delta\), outside an event of probability at most \[\exp\{-cR^\epsilon T+CT\},\] the sum of all these normalized oscillations is at most \(R^\epsilon T\). Discard each comparison whose assigned discrepancy exceeds \(R^\delta\sqrt{r/R}\), and each node-side with an excessive local oscillation. At most \(C R^{\epsilon-\delta}T\) comparison incidences are lost. The originally irregular nodes lose at most \(C\epsilon_0T+C_RM\) incidences: after sorting, a node has at most one predecessor and one successor per color. Choosing \(R\) large makes the first two relative losses as small as required.

All remaining required slot conditions now have fixed deterministic tolerance \(C R^\delta\sqrt{r/R}\). On a left slot the target wall is the already observed representative height in the previous occupied box of the same component, minus that tolerance. The comparison bound makes survival above this wall and landing near it necessary. The previous box lies strictly before the slot, by regular-node separation. On a right slot use the current midpoint height minus the same tolerance; that height is already observed before this slot. Its initial discrepancy is small by the retained local oscillation condition. We do not choose a random wall using the unobserved actual comparison error.

Expose all slots in chronological order. Lemma 114 makes them disjoint, and the intervening raw words can also be exposed in chronological order. Each slot is fresh conditional on this history. If \(k\) colors are incident on a retained left slot, Lemma 18 gives logarithmic charge at least \[(k/2+d_k-O(\delta)-o(1))\lambda.\] For a right slot it gives \((d_k-O(\delta)-o(1))\lambda\). The \(o(1)\) is uniform once the minimum radius is sufficiently large for the fixed \(R\). Summed incoming incidences give \(E/2\) before deletions. There are \(2E\) side incidences in total, and \(d_k\ge k d_2/2\) for \(k=0,1,2\). The undeleted leading survival charge is therefore at least \(d_2E\). Deletions lose only a bounded charge per incidence. The choices of deleted incidences have at most \(\exp(CT)\) possibilities, which are summed rather than conditioned upon.

Choose \(\delta\) sufficiently small in terms of \(\zeta\), then choose \(R\) large enough that \(C/\lambda\), the deletion fractions, and the fixed Brownian constants divided by \(\lambda\) fit within the remaining error budget. The oscillation-exception exponent can simultaneously be made larger than \(3\beta\lambda T\) in absolute value, since \(R^\epsilon/\log R\to\infty\). As \(E\le2T\), that exception is absorbed into Equation (86) after an arbitrarily small enlargement of its error budget. Finally take the minimum radius large enough for the uniform lattice-slot estimates. Chronological multiplication and the finite sum over discard patterns give the proposition. ◻

Proof. Fix \(U\). Use Equation (84) with a small fixed \(\eta>0\) and then let the sparse terminal error be \(o(N)\). For every favorable code of total occupancy \(T\), combine Equations (87) and (85). Summing over its possible codes gives \[\begin{align*} &\mathbb P\{\text{some favorable realized code has occupancy }T\}\\ &\quad\leq \exp\bigl\{ -[(\beta-1-\zeta)\lambda-C_0-\eta]T +\beta\lambda h_0(J+1) +\beta\lambda d_N+C_{R,\zeta}M+o(N) \bigr\}. \end{align*}\] Choose fixed \(R\) large enough and then the error tolerances small enough that \[\delta:=\beta-1-\zeta-(C_0+\eta)/\lambda>0.\] Since \(\lambda(J+1)=bN+O(1)\), \(d_N=o(N)\), and \(M=o(N)\), there is a fixed \(B<\infty\) such that the last probability is at most \[ \exp\{-\delta\lambda T+BN\}. \tag{90}\] Here \(B\) and \(\delta\) are independent of the fixed list cutoff: its contribution was confined to the sparse \(o(N)\) terms. Geometric summation over \(\lambda T\geq x\) shows, uniformly in \(U\), \[\mathbb P\{\text{favorable and }A\geq x\} \leq C_2\exp(BN-\delta x).\] For \(C_*=(B+2)/\delta\), this gives \[ \mathbb P\{\text{favorable and }A\geq C_*N+u\} \leq C_2\exp(-2N-\delta u). \tag{91}\] The same bound holds after integrating over the independent \(U\).

The event of a favorable displacement has only a positive conditional probability; we do not give its complement an exponential estimate. Instead compare all shifts of the same realized witness system. The same-width cell intersection argument gives \[T(U)\leq2T(V)\qquad\text{for all displacements }U,V.\] Let \(A_{\max}=\max_U\lambda T(U)\). This maximum is measurable: there are finitely many markers and levels, so only finitely many marker–boundary crossings can change the occupancy as \(U\) varies in \([0,H)\). If \(A_{\max}\geq x\), every displacement has \(A\geq x/2\), and at least half of the displacements are favorable. Conditional on the complete witness system, this proves \[\mathbf 1_{\{A_{\max}\geq x\}} \leq2\mathbb P_U\{\text{favorable and }A\geq x/2 \mid\text{witness system}\}.\] Integrate and apply Equation (91). Increasing the constant multiplying \(N\) proves Equation (88) for \(A_{\max}\), and therefore for every particular grid. All word-dependent choices have been included in the finite code enumeration. The auxiliary favorable shift has never been selected as unencoded sample-dependent information.

Under Equation (71), the tail is bounded above by \[\mathbb Q_N\{A\geq CN+u\} \leq\min\{1,C_1\exp(k_N-cN-cu)\}.\] Since \(k_N=o(N)\), integration in \(u\) gives \[\mathbb E_{\mathbb Q_N}A \leq CN+C_1c^{-1}\exp(k_N-cN)=O(N).\] Equation (84), including its optional integer-size code, then gives the stated deterministic \(O(N)\) code cutoff outside an event of probability tending to zero. Finally, for an independently uniform level, \[\mathbb E_{\mathbb Q_N}m_j =\frac{\mathbb E_{\mathbb Q_N}T}{J+1}=O(1),\] since \(J+1\) is comparable to \(N\) for fixed \(R,b\). ◻

Proof. At a level with \(m\) occupied boxes, \(c\) union components, and total monochrome rank \(e\), one has \[e\ge m-c,\qquad m-\beta e\le\beta c-\varepsilon m.\] The integrated component excess caused by projected jumps is \(o(N)\) on favorable grids. Before level \(N\), all base cells can be charged as one component up to \(o(N)\); hence the possible positive credit there is at most \(C_0N+o(N)\), independently of \(h\). On \([N,aN]\), there are at most \(h\) base components, so positive credit is at most \(C_1(a-1)hN+o(N)\). The constants include fixed endpoint-rounding and grid-step changes.

At every level on which all base components have rank cycles, \(e\ge m\) after the negligible connectivity errors. The separated bases occupy at least \(h\) boxes on \([aN,bN]\), so the credit there is at most \(-\varepsilon h\) per unit log length. At exceptional levels the positive credit can still be bounded by a constant times the number of acyclic base components; a bound on the maximum \(m\) is not needed. Proposition 122 absorbs all small relative errors in the integrated occupancy into an arbitrarily small multiple of \(N\).

Choose \(a-1\) such that \(C_1(a-1)<\varepsilon(b-a)/4\), and then choose \(h\) with \(C_0<\varepsilon(b-a)h/4\). The total leading address budget minus arm charge is then at most \(-cN\) for some \(c>0\), even after the permitted errors. Summing Equation (86) over the length-weighted codes gives an exponentially small probability for the stated cycle event and a favorable displacement, under \(\mathbb P\). Multiplying by \(e^{o(N)}\) gives the same vanishing conclusion under \(\mathbb Q_N\).

We now remove favorability without assuming that cyclicity is invariant under a change of grid. Let \(\eta_N\downarrow0\) and denote by \(E_N(U)\) the cycle event with at most \(\eta_NN\) exceptional generations for the independent displacement \(U\). Generation count and logarithmic length differ by the fixed factor \(\lambda\). Lemma 117 supplies \(q_N=o(N)\) and a uniform conditional error \(\alpha_N=o(1)\) such that \(\Delta(U)\le q_N\) outside that error. Let \(E_N^*\) be the corresponding cycle event for the grid-independent partitions \(\mathcal P_j^*\), with at most \(\eta_NN+q_N\) exceptions. Then \[\mathbb Q_N(E_N(U))\le\mathbb Q_N(E_N^*)+\alpha_N.\] Sample a second independent displacement \(V\). Conditional on the marker system, \(V\) is favorable and has \(\Delta(V)\le q_N\) with probability at least \(1/2-\alpha_N\). On \(E_N^*\) this gives the projected cycle event \(E_N'(V)\) with at most \(\eta_NN+2q_N=o(N)\) exceptional generations. Consequently \[(1/2-\alpha_N)\mathbb Q_N(E_N^*) \le \mathbb Q_N(E_N'(V)\cap\{V\text{ favorable}\}).\] The preceding favorable-grid estimate applies to this right side. Indeed, on each added exceptional level the positive credit is at most \(\beta(h+D_j)\), by Equation (72); the extra levels and the favorable-grid sum \(\sum_jD_j\) are both \(o(N)\). No maximum-occupancy bound is needed for this error. The right side therefore tends to zero, proving the claim for \(U\). The conditional grid estimates are uniform in the preselected data; their errors are not multiplied by the selected density. ◻

Proof. Let \(\pi_c=e^{-\ell(c)}\) and place any missing mass at an unused code. Compare the joint law of \((C,W)\) under \(\mathbb Q\) to the law in which \(C\) has probabilities \(\pi_c\) and \(W\) independently has law \(\mathbb P\). The chain rule gives \[D(\mathbb Q_{C,W}\Vert\pi\otimes\mathbb P) =D(\mathbb Q_W\Vert\mathbb P) +\mathbb E_{\mathbb Q_W} D(\mathbb Q_{C\mid W}\Vert\pi) \le\kappa+\mathbb E_{\mathbb Q}\ell(C),\] because the conditional Shannon entropy of \(C\) is nonnegative. Disintegrate instead first over \(C\) and then over the chronological raw pieces. Product structure under the reference law makes the reference conditional law in every slot a fresh iid block. Discarding the nonnegative entropy terms for the gaps, past, and other pieces leaves the asserted upper bound for the slots. This proof also applies to finite restrictions of an infinite past, followed by monotone increase of the observed sigma fields.

For an independent grid displacement \(U\), apply the argument conditionally on \(U\) and integrate. The original word law is independent of \(U\), and the selected law before grid sampling has the same bounded-density property. We do not condition on a sample-dependent favorable displacement before this calculation. ◻

Proof. Replace each \(I_v\) by the deterministic interval \[\widehat I_v= [\inf v-A r_v/2,\ \sup v+A r_v/2],\] where \(v\) denotes the code’s location cell. This interval contains \(I_v\) regardless of the chosen in-cell center. Its length is exactly \((A+R^{-2})r_v\). At a fixed level, location cells have common length \(r_v/R^2\), so these enclosures have overlap multiplicity at most \(C_A R^2\). Across levels their lengths are geometric. The proof of Lemma 120 gives constants \(\theta,c>0\) such that, for every fixed code, \[\mathbb E_{\mathbb P} \exp\left\{\theta\sum_v S_v-c\sum_jm_j\right\}\le1.\] The same holds under the product reference in which the code has its Kraft weights and is independent of the raw word. The entropy variational inequality and Lemma 125 imply \[\theta\mathbb E_{\mathbb Q}\sum_vS_v \le c\mathbb E_{\mathbb Q}\sum_jm_j+ \kappa+\mathbb E_{\mathbb Q}\ell(C)=O(N).\] Division by the number of intermediate levels proves the last claim. ◻

Proof. First fix \(d\). Start with a bounded macroscopic time interval \(I_0\) containing every permitted base. Enlarge it by \(d\) on each side to allow a time jump of length \(dn_j\). On this enlarged interval the limiting Brownian coordinates have finite minima. In each coordinate choose one time to its left and one to its right at which that coordinate is below its minimum on the enlarged interval by more than three. Such times exist almost surely by scalar Brownian recurrence. They need not be the same times for the two coordinates. Let \(I_1\) contain the enlarged interval and all four chosen times. A scalar partner of a source in the enlarged interval cannot lie beyond its corresponding lower barrier: the intervening minimum would then be strictly smaller than the source height. Repeat this enlargement and barrier construction \(d+1\) times. It gives an almost surely finite random interval containing every such chain, uniformly over all its choices of source, color, and endpoint.

Choose a deterministic \(C_d\) large enough that these finitely many strict barriers lie in \([-C_d,C_d]\) except with probability at most \(\epsilon2^{-d-2}\). Increase the constants to make them nested. Compact-uniform contour convergence preserves all the barrier inequalities with a margin exceeding one for all sufficiently large \(j\) on this event. A discrete scalar link changes the tested endpoint level by at most one lattice step, whose normalized size tends to zero. Thus the same argument confines the actual discrete chains. The barriers also confine their last-upcrossing and first-downcrossing searches whenever these are used to realize such a source chord. This is simultaneous containment over the continuum of source times, not a separate probability estimate for each seed.

For (ii), keep \(d\) fixed and use its deterministic enclosing interval. If uniform spatial collapse failed, some subsequence would have a base and a chain of at most \(d\) steps with a projected separation bounded below. Pass to a further subsequence on which all their rescaled times converge in that compact interval, repeating vertices to keep the chain length fixed. The endpoints of a time jump have the same limit because \(d\rho_j\to0\). The endpoints of an actual scalar link limit to a scalar horizontal pair, by compact-uniform contour convergence and its endpoint/minimum test. They have the same curve image by Theorem 8(iii). Continuity of the curve then makes every successive pair of projected chain points have the same limiting image, contradicting the claimed separation. This compactness argument is uniform over all bases, all seed levels, and all choices of a bounded chain. In particular it uses only \(r_j(x)\le\rho_jn_j\), not the number of tested levels.

For each fixed finite collection \(d\le k\), the preceding convergence and barrier events imply their simultaneous conclusions with spatial error at most \(1/k\) and failure at most \(\epsilon+o_j(1)\); the sum of the displayed barrier budgets is less than \(\epsilon/2\). Choose deterministic thresholds \(j_k\) increasing so that for \(j\ge j_k\) the remaining error is at most \(\epsilon\), and put \(D_j=k\) between successive thresholds. Let \(\mathcal T_j\) be that finite intersection and \(\alpha_j=1/D_j\). This defines a master event with the asserted probability before any subsequent selection. Additional deterministic slow-growth requirements on \(D_j\) may be imposed by increasing \(j_k\).

If a selected law is supported on \(\mathcal T_j\), all these assertions hold under it by support, without transferring a small ordinary probability through its density bound. Every later fixed exploration or composition bound is eventually below \(D_j\). A diagonal chosen after that law may only decrease this cutoff, retaining each earlier window \(C_d\) separately. For a prescribed open spatial neighborhood with a fixed positive margin about the permitted bases, (ii) gives the required confinement once \(\alpha_j\) is below that margin. ◻

Proof. Apply Lemma 125 to the selected law, the code \(C_j\), and its chronological disjoint slots. It bounds the sum of their expected conditional relative entropies by \(\kappa_j+E_{Q_j}\ell_j(C_j)\le\kappa_j+KN_j\). If only a coarser preceding history is retained, data processing gives the same bound for that history. Since \(I_j\) is uniform after sampling and \(T_j\ge cN_j\), its mean conditional relative entropy is at most \((\kappa_j+KN_j)/(cN_j)=(K+o(1))/c\). No independence of the resolved walk from its past is asserted or used. ◻

Proof. Data processing to the indicator of \(E\) gives \(D(Q\Vert P)\ge q\log(1/p)-\log2\), where \(q=Q(E)\) and \(p=P(E)\). For conditional laws apply this inequality to each section and integrate. ◻

Proof. Choose \(L=L(H)\to\infty\) slowly enough that \(He^{-L}>2c\). At each seed \(y\) take a maximal \(H\)-packing, truncated at \(M\). Throughout its coarser cell \([y-L,y]\), the chosen terminals remain in different projected boxes, since the time unit grows by at most \(e^L\). Thus their count contributes at least \(L\) times itself to \(\int m_j\), apart from the two end cells. Averaging the independent mesh translation converts the sum of these seed counts into the uniform-level integral. On the exceptional event bound the truncated packing by \(M\), without attempting to bound \(m_j\). Divide by \(T_j\), let \(j\to\infty\), remove the prescribed probability loss, and then let \(H\) and \(L(H)\) tend to infinity. The end-cell loss is at most \(2ML/T_j\) and vanishes in the first limit. ◻

Proof. Take a countable joint subsequential limit of all truncated packing counts for integer \(k,H,M\). On this common product probability space write \(P_{k,H}\) for the extended-integer limits. Their monotonicities are preserved. Set \(A_k=\lim_{H\to\infty}P_{k,H}\) and \(A=\lim_{k\to\infty}A_k\). Equation (96), first with finite truncations, gives \(EA\le C\). Thus \(A\) is finite, and the increasing integer sequence \(A_k\) equals \(A\) eventually on each sample.

At a candidate seed level, construct an \(H_j\)-separated greedy set by processing \(\mathcal R_{j,1},\ldots,\mathcal R_{j,k_j}\) in that order. Within each stage add points until maximal, using a fixed measurable ordering of the finite lattice set. Let \(G_{j,k}\) be its size after stage \(k\). Maximality and the nearest-center assignment imply \[P_{j,k,3H_j}\le G_{j,k}\le P_{j,k,H_j},\qquad k\le k_j.\] Choose \(H_j\) and \(k_j\) sufficiently slowly that \(G_{j,k}\Rightarrow A_k\) for each fixed \(k\), while the final count \(K_j=G_{j,k_j}\Rightarrow A\), jointly with the original fixed-cutoff array. Explicitly, at stage \(d\) choose a finite height cutoff large enough to approximate the first \(d\) limiting counts and their threefold cutoffs, then choose \(j_d\) so their joint laws for \(j\ge j_d\) are within \(1/d\) of those limits; let the stage index tend to infinity only through these thresholds. A further slow stage choice handles the depth cutoff. No convergence rate is required.

Let \(D_j\) be the largest access depth of a selected center. The greedy construction gives \(\{D_j>k\}=\{G_{j,k}<K_j\}\). In the joint integer limits the two counts are ordered and have marginals \(A_k\) and \(A\). Since these marginals approach the same finite law, their probability of inequality tends to zero. Thus \(D_j\) is tight.

In a further joint limit write \(K\) for the limit of \(K_j\) and retain the entire fixed packing array. On \(D_j\le d\) the selected centers are an \(H\)-packing in \(\mathcal R_{j,d}\) for every fixed \(H\) and all large \(j\). Taking \(j\), then \(H\), then \(d\) to infinity shows \(K\le A\) almost surely. Their laws agree, so \(K=A\) almost surely. For fixed \(k,d,H\), a point of \(\mathcal R_{j,k}\) at distance exceeding \(Hr_j(Y_j)\) from all centers, on \(D_j\le d\), adds one point to their packing. Therefore \[Q_j[\text{coverage fails at }H] \le Q_j[D_j>d]+Q_j[P_{j,\max(k,d),H}(Y_j)\ge K_j+1].\] In the joint limit the second probability tends to zero as \(H\to\infty\), because its packing count decreases to \(A_{\max(k,d)}\le A=K\). Let \(d\to\infty\). This proves the seed version of Equation (97).

Choose \(L_j\to\infty\) with \(L_j=o(T_j\wedge\log H_j)\). The old centers remain separated by \(H_je^{-L_j}\to\infty\) in the new units, and their access depth stays tight by Equation (95). Independence of the seed level from the selected environment gives \[ \bigl\|\mathcal L(\omega_j,X_j)- \mathcal L(\omega_j,Y_j)\bigr\|_{\mathrm{TV}} \le L_j/T_j. \tag{98}\] To verify the inequality, condition on \(\omega_j,U_j\) and translate the uniform interval for \(Y_j\). Use an outer buffer of length \(L_j\), or discard its vanishing relative length. The full limiting packing count \(B\) at \(X_j\) consequently has the law of \(A\). Tight access and preserved separation give \(K\le B\). Equality of the finite laws gives \(K=B\). The preceding extra-point argument now proves coarse coverage. It also covers bounded explorations started in compact ranges around centers, by the composition property of reachability.

We detail the simultaneous edge selection. Label centers chronologically. For each fixed \(d\), assign points of \(\mathcal R_{j,d}(Y_j)\) to their nearest center and record all color-and-unordered-label-pair bits witnessed by actual links with both endpoints in that set. Call the graph \(G_{j,d}\). With centers fixed, these graphs increase with \(d\). Coverage and diverging separations make each endpoint’s containing label unique after a tight coordinate truncation. A countable joint extraction, first truncating \(K_j\), gives increasing graphs \(G_d\) on \(K\) labels. Their union \(G\) stabilizes because it has at most \(2\binom K2\) bits. A slow cutoff attains this union in law. For each bit choose its witness at its first available depth. Monotone finite stabilization implies that the maximum of these first depths is tight, exactly as for \(D_j\). Include all endpoints and access chains in the same list; they occur in one environment, so this is simultaneous realization. Seed coverage makes all their within-line offsets tight. Since \(U_j\to\infty\) in probability, those offsets and every bounded seed jump vanish in coarse units.

Finally recompute the identical greedy-center and earliest-edge recipe at \(X_j\). Equation (98) gives the same limiting joint profile of line and edge counts. The recomputed centers and the old centers match uniquely in chronological order: each system has tight access, covers the other by coarse coverage, and has divergent separations; the equal saturated counts turn the resulting injection into a bijection. All preserved seed edges belong to the recomputed full coarse graph. The two finite edge counts have the same law, so inclusion forces equality. Any additional bounded-reach actual coarse edge could be united with the existing witnesses and would add a bit, a contradiction. All auxiliary graph cutoffs are incorporated into the earlier slow diagonal. After any fixed probability truncation their lists are bounded and eligible for Lemma 131. ◻

Proof. Seed recipes and one code. Fix the address ratio \(R\) of Definition 112, and put \(\lambda=\log R\). Round \(L_j\) to an integer multiple \(m_j^*\lambda\), retaining \(L_j\to\infty\) and \(L_j=o(T_j\wedge\log H_j)\). Sample an independent logarithmic mesh displacement \(V_j\sim\operatorname{Unif}[0,L_j)\). Retain the complete cells \([y_p-L_j,y_p]\) in \(I_j\), where the endpoints \(y_p\) run through this translated mesh. Their number \(P_j^*\) satisfies \(P_j^*L_j=T_j+O(L_j)\).

At every \(y_p\), run exactly the deterministic greedy-center and earliest-edge recipe in Proposition 132, with the same fixed orderings and diagonal cutoffs. Temporarily truncate at an integer \(d_j\): retain the recipe only if its center count, all access depths, all earliest-edge witness depths, and all witness-endpoint distances from their assigned centers in seed units are at most \(d_j\). Otherwise replace that seed’s list by the first distinguished base. Include every access chain, every chosen actual colored link, and a time-jump entry associating each endpoint of an earliest-edge witness with its assigned seed center. The latter jump has length at most \(d_jr_j(y_p)\). These entries record the same center labels as the saturated graph, rather than merely their separate access from bases. There are at most \(2\binom{d_j}{2}\) colored bits, and each access chain has at most \(d_j\) entries. Thus a deterministic bound \(\Lambda_j=C(d_j+1)^3\) bounds the total marker and edge entries per seed and its jump tolerance. It includes the distinguished base names.

The enclosing chronological interval is part of the master event in Lemma 128. Use its window indexed by a deterministic composition bound \(b(d_j)\) for all the retained access chains, association jumps, and the finitely many current band requests. Choose the actual cutoff slowly enough that \(b(d_j)\le D_j\). Write \(\widehat C_j=C_{b(d_j)}\). A seed leaving \([-\widehat C_jn_j,\widehat C_jn_j]\) is also replaced by the base. Every fixed recipe is eventually covered, since the selected laws are already supported on the master event. No ordinary high-probability chronological bound is transferred through the growing density bound in Equation (93).

All lists have now been chosen, including the association jumps. Only next sample the independent chronological grid displacement of Definition 112. Form one projected forest from their union, marking every chain through its seed probe depth and the distinguished bases through the final depth. The seed depth is rounded to the nearest address generation; its time unit differs by a factor at most \(R\), and different seeds still have distinct depths. Round origins to the fixed even representatives inside their coded terminal cells. This is a function of the resulting code and grid; it makes no new exploration or witness choice. All projections of the original chains and association jumps are already present at every ancestor generation.

Code and occupation budgets. Choose \(d_j\to\infty\) sufficiently slowly that \[ \frac{\Lambda_j\log(\Lambda_jL_j)}{L_j}\longrightarrow0. \tag{99}\] Also require \(\Lambda_j\log(2\widehat C_j)/L_j\to0\), by slowing \(d_j\) further. At each fixed cutoff the address root interval is a fixed multiple of \(n_j\), as required in Definition 112. Along the diagonal its enlarged root interval adds at most \(O(1+\log_R(2\widehat C_j))\) ancestors per marker. The crude total extra occupancy and code cost is therefore \(O(\Lambda_j(N_j/L_j)\log(2\widehat C_j))=o(N_j)\). For every fixed ancestor padding \(g\), the total number of markers and the structural deletion error are then \[M_j\le \Lambda_jP_j^*+O(1) =O(\Lambda_jN_j/L_j)=o(N_j),\qquad C_{R,g}M_j=o(N_j).\] Lemma 116 bounds the integrated jump error by \(O_R((1+\log_R\Lambda_j)\Lambda_jN_j/L_j)=o(N_j)\). This count includes the endpoint-association jumps. Write \(\mathsf T_j\) for the forest’s total occupied-node count. The exact code-count bound of Lemma 119 is \[(\lambda+C_0)\mathsf T_j+ \Lambda_jP_j^*\log(1+\mathsf T_j/P_j^*)+ C P_j^*\Lambda_j\log(\Lambda_j+1)+O(\log(\mathsf T_j+1)).\] Equation (99) and Equation (84) bound this by \(C_R\mathsf T_j+o(N_j)+O(1)\). Proposition 122 applies to these simultaneous lists before the grid displacement; its constant is independent of the fixed list cutoff. Its exponential tail and Equation (93) give \[ E_{Q_j}\mathsf T_j=O(N_j),\qquad E_{Q_j}\ell_j(C_j)=O(N_j). \tag{100}\] The Kraft code includes its occupancy-size cost. Enlarging the slow diagonal keeps these estimates and every prescribed fixed-cutoff convergence. No event requiring all seed recipes to pass their truncation has been conditioned on.

The selected view law. Choose \(p\) uniformly among the \(P_j^*\) complete cells and, independently, \(U_j\sim\operatorname{Unif}[0,L_j]\). Set \(Y_j=y_p\) and \(X_j=Y_j-U_j\). Averaging the independent mesh displacement and the uniform cell makes the law of \(Y_j\) differ from the uniform law on \(I_j\) by \(O(L_j/T_j)\) in total variation, jointly with the word and all its single-seed recipe data. To see this, extend a bounded test function by zero outside \(I_j\) and average its mesh sum; every interior point receives density \(1/(P_j^*L_j)\), and the discarded end intervals have total length at most \(2L_j\). The denominator differs from \(T_j\) by at most \(2L_j\). The mesh is independent of the word. Tightness of the single-seed center count, first witness depths, and centered endpoint offsets therefore makes the probability of fallback at this uniformly selected seed tend to zero as the cutoffs are exhausted. This proves the claimed equality of selected laws up to vanishing error. Choosing only among successful seeds would not have this property.

One trial at each generation. Write \[U_j=h\lambda+\tau,\qquad h\in\{0,\ldots,m_j^*-1\},\quad \tau\in[0,\lambda).\] The phase \(\tau\) is independent of the entire code and is sampled after it. Conditional on this phase and the mesh, the pair \((p,h)\) is uniform on \(P_j^*m_j^*\) possibilities. The levels \(y_p-h\lambda\) are all distinct consecutive address generations, up to their common fixed fractional phase. Consequently \[ J_j^*=P_j^*m_j^*=\Theta(N_j/\lambda). \tag{101}\] At such a generation use the frozen origins belonging to its unique seed cell. Distinct selected centers are separated by \(H_je^{-L_j}\to\infty\) in viewing units, so they occupy distinct location nodes for all large \(j\). A fallback uses only one base. Thus the selected center occurrences inject into occupied forest nodes at each generation. One does not count a new copy of a node for each other seed whose chain also projects there.

Central oscillations. Corollary 126 and Equation (100) give \(E_{Q_j}\sum_vS_v=O(N_j)\) for the same forest. The injection of the selected centers into its nodes and division by \(J_j^*\) give the asserted bounded first moment of the selected central amplitudes. The estimate holds on every fixed enlargement of the selected time neighborhoods. In particular, normalized start-height differences between blocks on the same line are tight before any passage transfer or continuity conclusion is used.

Disjoint slots and entropy transfer. For a fixed finite band request, use the enlarged bands of Remark 127. The bounded phase factor \(e^\tau\) and the bounded seed-depth rounding factor only change their fixed inner and outer radii. Remove a fixed number of generations near each seed endpoint; their selected probability is \(O(1/L_j)\). The terminal rounding and association offsets then vanish in view units: for example \(Q_j(d_je^{-U_j}>\epsilon)\le \log^+(d_j/\epsilon)/L_j\to0\). For each fixed residue class with sufficiently spaced generations, the regular-node construction gives disjoint enlarged chronological slots. Use an arbitrarily large fixed ancestor depth \(g\) in that construction. Its boundary-deletion proof gives, for a constant \(C_{A,R}\) depending on the requested bands, \[\frac{E_{Q_j}\{\text{deleted selected center occurrences}\}}{J_j^*} \le o_j(1)+C_{A,R}R^{-g} \frac{E_{Q_j}\mathsf T_j}{J_j^*}.\] Indeed, conditional on the marker list, boundary loss is bounded by a constant times \(R^{-g}\) times the occupancy of a fixed reference grid, which is within a factor two of every displaced-grid occupancy. The structural loss is \(C_{R,g}M_j=o(N_j)\). Sending \(j\to\infty\) first and then \(g\to\infty\) makes the selected deletion probability arbitrarily small, with \(R\) and the underlying view law unchanged.

Given the code and the independent side information, all surviving slot boundaries are fixed. For one fixed residue, group these disjoint raw slots by their trial \((p,h)\), assigning no slots to a deleted request. Let \(h_i\) be the expected relative entropy of the \(i\)th chronological raw-slot kernel, conditional on the code and its entire preceding raw history, relative to a fresh iid block. Averaging over the code, Lemma 125 gives \[\sum_i h_i\le B_j:=\kappa_j+E_{Q_j}\ell_j(C_j)=O(N_j).\] There is a direct comparison retaining incoming stacks. For a trial \(t=(p,h)\) with slot set \(S_t\), keep the selected law’s code marginal and all its chronological gap and nonselected-slot kernels, but replace precisely the kernels in \(S_t\) by fresh iid blocks. Call this comparison \(R_{j,t}\). If \(H_i(C_j,\mathcal H_i)\) is the conditional relative entropy before slot \(i\), so that \(h_i=E_{Q_j}H_i\), the chronological chain rule gives \[D(Q_j\Vert R_{j,t}) =E_{Q_j}\sum_{i\in S_t(C_j)}H_i(C_j,\mathcal H_i).\] Keep the code-dependent trial membership inside this expectation. Use finite observations first and then increase the observed history. Under \(R_{j,t}\), each requested slot is fresh conditional on its actual incoming history, so the reference domination of Proposition 59 applies on each fixed compatible-instruction subevent. No joint conditioning on future gap histories is made; such histories could encode the earlier target slots.

Sum over trials and then the fixed finite residue classes, and divide by \(J_j^*\) in Equation (101). Every slot belongs to at most one trial within its residue, so the mean comparison entropy is \(O(1)\). The denominator counts all seed-generation trials, including deleted or empty requests. Averaging over trials counts each generation once, regardless of its number of lines. This comparison includes any fixed finite family of the selected lines. Lemma 130 and the reference-slot domination transfer null tests on every fixed compact range of retained shifts, on those compatible-instruction subevents. Whenever these shifts are tight under the selected law, the compact restriction can then be removed. The separate continuum argument in Lemma 134 below requires no flexible-order instruction. First send the reference failure probability to zero, and then remove the arbitrary deletion probability. This divides the all-seed code budget by order \(N_j\) levels.

Stationarity. All seed recipes, the code, and its rounded origins precede \(U_j\); conditioning on them leaves \(U_j\) uniform. The fixed-origin total-variation estimate of Lemma 135 is therefore unchanged. ◻

Proof. Fix the forest pattern. In each of its components replace the tree differences by differences from its chronologically earliest block. This is an invertible integer linear change of coordinates with determinant of magnitude one. At each later block there is now at most one new difference per coordinate, relative to an already observed block. A pattern chosen later can be handled by proving the assertion for each fixed pattern and then restricting the limiting measures.

Anchor the contour before all slots. Write \(a_i^N\) for the beginning of slot \(i\), \(u_i^N\) for its retained-block start, and \(K_i\subseteq \{1,2\}\) for the coordinates with a new difference. If \(p(i,c)<i\) is the earlier block used in coordinate \(c\), set \[A_{i,N}^c= \frac{Z_c(a_i^N)-Z_c(u_{p(i,c)}^N)}{\sqrt{r_N}}, \qquad G_{i,N}^c= \frac{Z_c(u_i^N)-Z_c(a_i^N)}{\sqrt{r_N}} .\] The first vector is measurable before the fresh slot. The new retained difference satisfies the exact identity \[D_{i,N}=A_{i,N}+G_{i,N}.\] Let \(X_{i,N}\) be the retained increment path, parametrized on \([0,1]\). Proposition 14, applied to the entire fresh safety band and retained block, gives conditional bounded-Lipschitz convergence \[(G_{i,N},X_{i,N})\ \Longrightarrow\ (G_i,B_i), \qquad G_i\sim N(0,\tau_i\Sigma|_{K_i}),\quad B_i\sim\nu_{t_i}.\] Here \(\Sigma\) is the Brownian covariance matrix, \(\tau_i\) and \(t_i\) are the limiting safety and retained durations, and \(\nu_{t_i}\) is the Brownian increment-path law of duration \(t_i\). The Gaussian \(G_i\) and the path \(B_i\) are independent. Convergence is uniform over incoming stacks and compact duration ranges: the arbitrary-typing error is uniform, and deterministic restrictions and time changes of the whole-slot limit give the displayed pair. No compatible flexible-order instructions are needed.

The density of \(G_i\) is bounded uniformly because \(\tau_i\) is bounded away from zero. Give the \(K_i\)-coordinate difference an independent positive smooth reference density \(p_i\). For a nonnegative test \(f\) supported on \(|d|\le M\) and any translation \(A_i\), \[\mathbb E f(A_i+G_i,B_i) \le C_M\int f(d,x)p_i(d)\,dd\,\nu_{t_i}(dx).\] For bounded-Lipschitz tests the prelimit inequality has a uniform \(o(1)\) error. Indeed translation by \(A_i\) changes neither the bound nor the Lipschitz constant of the test. Empty \(K_i\) requires only the Brownian increment-path comparison.

Apply this bound to one nonnegative bounded-Lipschitz function of all retained outputs, supported on a compact set of their differences. Start at the last fresh slot and work backwards. Earlier outputs are fixed at each conditional step. Integrating the current output against its independent reference law leaves a test with the same bounded-Lipschitz control in the earlier outputs. The intervening causal gap kernels integrate to one; they may depend on earlier target slots, but their future realizations are never conditioned on jointly. This backward iteration bounds the limiting expectation by a fixed multiple of the product Brownian and smooth-height reference expectation, including the common parameter marginal. Bounded-Lipschitz approximation then gives local domination of the joint continuum measures.

If a retained raw record includes a left extension before its used start, place the fresh safety band before the entire record. The used-start difference is then exactly \(D=A+G+\Phi(X)\), where \(\Phi\) is the retained extension’s endpoint evaluation. Conditional on the retained Brownian path this is another translation of the same Gaussian. Endpoint evaluation is Lipschitz in the uniform path metric, so the preceding argument is unchanged.

The comparison does not assume tightness of incoming heights. Explicitly, on \(|D_{i,N}|_\infty\le M\) and \(|A_{i,N}|_\infty>K\), the fresh safety fluctuation is at least \(K-M\). Proposition 13 and its deterministic oscillation domination therefore give, uniformly over the incoming history, \[R_N\bigl[|D_{i,N}|_\infty\le M,\ |A_{i,N}|_\infty>K\bigr] \le C e^{-c(K-M)},\qquad K>M.\] With a used-start extension, apply the same bound to the safety band and retained prefix together. Thus escaped incoming heights produce no unaccounted mass at finite retained differences.

Finally compactify the recorded differences to \([-\infty,\infty]\). The retained path laws are tight under \(R_N\) by the uniform fresh-slot limit, and under \(Q_N\) by the binary entropy bound. First adjoin the sampled trial under both laws; their common trial marginal turns the averaged bound into joint entropy. Then apply data processing to the finite duration and forest parameters, retained paths, and compactified differences, projecting away the trial identifier. A joint weak extraction of these two observation laws gives, by lower semicontinuity of relative entropy, \[D(Q^{\rm obs}\Vert R^{\rm obs})\le H<\infty, \qquad Q^{\rm obs}\ll R^{\rm obs}.\] The finite-difference part of \(R^{\rm obs}\) has the local domination proved above. Exhausting finite compact difference sets proves the selected assertion. For a restriction by an additional event, adjoin its indicator in the extraction; its limiting observation submeasure is bounded by \(Q^{\rm obs}\) and inherits the same absolute continuity on its finite-difference part. Undo the forest coordinate change. Countably many buffered requests give the simultaneous off-origin null-event assertions used below. This reasoning uses weak limits of contour paths; it supplies no prelimit domination for discontinuous lattice passage costs. ◻

Proof. Condition on everything sampled before \(U_j\). Replacing the time unit by \(e^v\) times itself replaces \(U_j\) by \(U_j+v\), with the same origins. Uniform measures on \([0,L_j]\) and \([v,L_j+v]\) differ in total variation by at most \(|v|/L_j\). Pushforward cannot increase total variation. Integrate the conditional inequality and remove the boundary exceptional events. For a weak limit, test bounded continuous functions and their compositions with \(\mathcal D_v\). First take rational \(v\); continuity gives all real \(v\). ◻

Proof. For (i), the law of \(H\) equals the law of \(e^{-v/2}H\). The disjoint geometric annuli \((e^m,e^{m+1}]\), \(m\in\mathbb Z\), have equal probability. Each therefore has probability zero. Apply the same argument to \(-H\), and to each normalized minimum, which transforms by the same scalar multiplier.

Consider the positive half-line and put \(\tau=\inf\{t>0:Z(t)<0\}\), with the infimum of the empty set equal to \(\infty\). Its law is invariant under multiplication by \(e^v\), so \(\tau\in\{0,\infty\}\) almost surely by the same argument. If \(\tau=\infty\), the path is nonnegative. Any zero at positive time is then a local minimum. There can be at most one such zero, by the distinct-minima hypothesis. The time of that sole zero, if it exists, again has a multiplicatively invariant distribution on \((0,\infty)\), so it cannot exist. This proves (ii). For (iii), apply the argument to \(\sup\{t>0:Z(t)\le0\}\), with the empty-set convention zero. Eventual strict positivity makes this supremum finite, so invariance forces it to be zero. Reflect time for the other half-line. Truncate the line count, apply the argument to each coordinate and each label, and then remove the truncation. ◻

Proof. Amplitude and off-origin control. Division by the order-\(N_j\) trial count in Equation (101), using the selected-node injection of Lemma 133, bounds the expected sum over lines of the normalized oscillations on every fixed enlargement. The projected boxes contain the rounded origins and the requested bounded time intervals. Thus, for every fixed \(A<\infty\), the centered amplitudes on \([-A,A]\) are tight. Relative start heights between blocks on the same line are differences of these amplitudes and are tight as well. This is the source of the compact height truncation; it is not an inference about the entire selected past.

Truncate the line count and fix finitely many bands with rational endpoints avoiding zero. Coarse rounding errors are covered by slightly enlarging these bands. Along each unary part of the projected address tree, choose a fixed residue class of logarithmic levels sufficiently spaced that their enlarged bands are disjoint. Increase the ancestor padding by the fixed amount required for these bands. The regular-slot estimate has arbitrarily small deletion probability by the fixed-padding exhaustion in Lemma 133; the residue is chosen uniformly and independently, preserving the level marginal. Conditional on the code and independent phase, the surviving raw-word slots have fixed disjoint boundaries. Give each tested block a deterministic left safety band of positive rescaled duration within its enlarged off-origin slot. Apply Lemma 134 to the freshened trial experiment of Lemma 133. It gives punctured compact-uniform tightness and joint continuum absolute continuity without choosing flexible-order instructions. The Gaussian safety increment smooths each new forest difference; the following retained increment path is Brownian in the reference limit. The bounded average trial entropy transfers its null events. Tightness of the tested finite differences allows their compact ranges to be exhausted. The same argument applies to every additional fixed forest whose differences are tight on the restricted subprobability under study. Exact lattice observations will instead use Proposition 59 on compatible protected charts later in the argument.

Control at the origins. For the fixed unit enlargement write the preceding expectation bound as \(C'\), and put \(M_j(\delta)=\sum_i\sup_{|u|\le\delta}|Z_j^i(u)|\). Apply Lemma 135 before taking a limit. For fixed \(\delta,\epsilon>0\), scaling the frozen view by \(\delta\) gives \[Q_j[M_j(\delta)>\epsilon] \le Q_j[M_j(1)>\epsilon/\sqrt\delta] +O(|\log\delta|/L_j)+o_j(1) \le C'\sqrt\delta/\epsilon+O(|\log\delta|/L_j)+o_j(1).\] Thus \(\lim_{\delta\downarrow0}\limsup_j Q_j[M_j(\delta)>\epsilon]=0\). Together with tightness on punctured compacts, this is full compact-uniform tightness, including the origins. It excludes narrow prelimit spikes which punctured convergence alone would miss. In a joint subsequential limit, Fatou’s lemma gives \[E\sum_{i=1}^K\sup_{0<|u|\le1}|Z^i(u)|\le C'.\] Lemma 135 applies on the punctured-line path space. Consequently, for \(0<\delta<1\), \[E\sum_{i=1}^K\sup_{0<|u|\le\delta}|Z^i(u)| =\sqrt\delta\,E\sum_{i=1}^K\sup_{0<|u|\le1}|Z^i(u)| \le C'\sqrt\delta.\] The left-hand suprema decrease as \(\delta\downarrow0\), so their limits are zero almost surely. Set \(Z^i(0)=0\). This proves continuity at every origin, together with the already obtained continuity elsewhere. Countably many bands, increasing line truncations, and the original packing and graph statistics can all be included in the same diagonal. ◻

Proof. Write \(Z_j\) for the chosen scalar contour, \(c_{j,i}\) for a center, and \(r_j\) for the viewing time unit. For block starts at coordinates \(u\) and \(v\) on lines \(i\) and \(\ell\), their normalized height difference is \[r_j^{-1/2}\bigl(Z_j(c_{j,\ell})-Z_j(c_{j,i})\bigr) +Z_j^\ell(v)-Z_j^i(u).\] The first term tends to zero within a finite relative-height group by Lemma 136(i). The remaining terms are tight by the compact centered-profile bounds. For a random group pattern, these assertions concern the corresponding restricted subprobability law; stationarity is used under the original law before this restriction. After truncating the line count there are finitely many group and forest choices; applying the off-origin transfer to each gives the joint assertion. No finite relative height between different groups is needed. ◻

Proof. Consider a positive right half-line of center \(i\) and fix a departure time \(d>0\). Choose deterministic \(\epsilon_j\downarrow0\) sufficiently slowly to dominate contour errors on every currently fixed compact and the vanishing differences of tied center heights. Round \(Z_j(c_{j,i})+\epsilon_j\sqrt{r_j}\) to an attainable scalar lattice level \(h_j\). Its rounding error divided by \(\sqrt{r_j}\) vanishes. Let \(a_j\) be the last visit to \(h_j\) before \(c_{j,i}+dr_j\), and let \(b_j\) be the first visit thereafter. The first exists because the center height is below \(h_j\) and the departure height is above it; the second exists by bilateral recurrence. The last/first convention gives \[Z_j(a_j)=Z_j(b_j)=h_j, \qquad \min_{[a_j,b_j]}Z_j=h_j.\] This is an actual scalar chord. Strict positivity on each compact \([\eta,d]\) implies \((a_j-c_{j,i})/r_j\to0\).

The first endpoint is reached by one bounded jump, and the second by one chord. Universal coverage in Equation (97) therefore puts \(b_j\) within a tight coordinate range of an enrolled line. Choose a joint convergent subsequence of its label and coordinate, denoting the latter by \(y\). The receiving line cannot be \(i\), by positivity on all its positive compacts. It is a later line, say \(\ell\). The equality at \(b_j\) shows that the center-height difference is finite; Lemma 136(i) makes it zero, and the receiving profile satisfies \(Z^\ell(y)=0\). Every finite receiving coordinate \(u<y\) occurs between the departure and the return for large \(j\), so \(Z^\ell(u)\ge0\).

We claim \(y=0\). If \(y<0\), nonnegativity on \((-\infty,y)\) and distinct off-origin minima imply strict positivity sufficiently far to the left; Lemma 136(iii) then gives strict positivity on the entire left half-line, contradicting \(Z^\ell(y)=0\). If \(y>0\), nonnegativity on \((0,y)\) rules out the negative-near-zero alternative on the right half; strict positivity there again contradicts the zero. Hence \(y=0\) and the receiving left half is positive. The chord inequalities also give Equation (102), so \(i\) and \(\ell\) are in the same scalar class. Reflect time for a positive left half.

Conversely a center contact makes the corresponding inward half-lines nonnegative, hence positive by Lemma 136. A positive outward half would, by the preceding return construction, produce an additional class member beyond the extreme one. The outward halves must therefore have negative values arbitrarily near zero. This proves (i).

For consecutive class members \(i<\ell\), choose the errors \(\epsilon_j\) also to exceed all their vanishing center-height differences. The first return from \(i\) occurs before \(c_{j,\ell}\), whose height is below \(h_j\). The preceding argument places any limiting return at an intermediate center in the same class. Consecutiveness forces it to be \(\ell\), and its coordinate tends to zero. These are actual finite-reach witnesses. Their finite union can be included in the edge extraction of Proposition 132. This proves (ii), including the rank \(|A|-1\) from consecutive pairs of a scalar class \(A\).

For (iii), suppose instead that a whole omitted gap has normalized minimum approaching zero with probability bounded below. Record that event in a joint extraction, retaining the original unconditioned stationary law; do not condition the stationary law on the event. Take \(\epsilon_j\downarrow0\) more slowly than this minimum and the convergence errors. Start the preceding chord search at a fixed positive departure lying inside the retained interval before that gap. Its first return must occur before the low point in the omitted gap. By (ii), and by choosing the start after the last near-origin crossing of \(h_j\), every subsequential return approaches the next class center. If the search encounters a retained intervening line, equality at the return and the minimum inequality put its center into the same class; consecutiveness excludes it. More precisely, the offending low point precedes \(c_{j,\ell}-\eta r_j\) for the fixed radius \(\eta>0\) of the removed neighborhood of the next member \(\ell\). Hence \(b_j\le c_{j,\ell}-\eta r_j\), whereas the return argument in (ii), applied with these same slowly decreasing heights, gives \((b_j-c_{j,\ell})/r_j\to0\). This is a contradiction. Universal coverage has already excluded escape between the enrolled lines or to an unbounded coordinate on one of them. This proves that the whole-gap minimum is bounded away from zero after a probability truncation. There are finitely many classes and gaps on each truncation, so their minimum margin has the same property. Remove the truncations and use a countable sequence of origin neighborhoods. ◻

Proof. The equality of endpoint heights and bounded local contour oscillations make the center-height difference finite, hence zero by Lemma 136. The chord lower bound, with the bounded end pieces between endpoints and centers, bounds the normalized whole-center-gap minimum from below. Its limit is finite, hence zero by the same Lemma. This is Equation (102). ◻

Proof. Handle the two coordinates separately for the height assignment. A scalar partition of ordered contour visits is noncrossing: two interleaving scalar contacts are at the same height and are one class. Consequently class spans are nested or disjoint. If \(B\) lies between consecutive members of \(A\), impose \(A\prec B\). These finitely many nesting inequalities have no cycle. They also agree with the original strict height ordering: the minimum inequality for \(A\) makes a center inside its span no lower than \(A\); equality would make it a member of \(A\), so a distinct intervening class is strictly higher. By stationarity its original relative height is in fact \(+\infty\) in view units.

Choose rational cuts on every retained half-profile. On each outward half retain a definite negative dip before its cut. The two coordinates need not have their negative dips at the same time; take the common cut after both required dips. Every inward half is positive at its cut. Let \(K_0\) bound all retained centered oscillations. For each coordinate choose a linear ordering of its classes extending nesting, with ranks \(s(A)\in\{1,\ldots,k\}\), and assign \[H_A=(4K_0+10)s(A)+u_A,\qquad |u_A|<1.\] Give every profile of class \(A\) this common added height. All pieces are positive. Every enclosed class lies above an enclosing level by more than \(2K_0+8\). The two coordinates have distinct height variables, so these choices impose no cross-coordinate compatibility equation. Equal heights of unrelated classes need not be retained.

Choose \(\delta>0\) smaller than one quarter of the finitely many inward cut heights, chosen outward dip depths, and original protected-gap margins from Lemma 139. Shrink the working neighborhoods of the centers so all centered profile heights there have magnitude less than \(\delta\). At a gap between consecutive retained intervals, the classes whose spans cross that gap form a nested chain. Both cut values exceed every active class height by more than \(2\delta\): a cut in that class lies on its inward side, and every other relevant cut belongs to a strictly enclosed class. If no class spans the gap, positivity is the only lower requirement. Coordinatewise linear interpolation joins the endpoint vectors while preserving these strict bounds. A small uniform tube about it does likewise. Exterior tails join to zero with positive interior values; ordinary quadrant-excursion entrance and exit pieces can be used in place of deterministic tails.

Inward positivity and connector bounds join all members of a prescribed scalar class at their common level. Outward dips block any contact leaving its span. No additional zero-level contact occurs on a retained half: inward halves are strictly positive and every outward interval starting at the center contains a negative value. Distinct scalar classes have distinct assigned heights. Thus the prescribed center class is saturated under both scalar relations. All inequalities use strict margins, so the height variables may have a density supported in the indicated open feasible box. Brownian bridges have positive probability of satisfying the connector tubes by Lemma 143. Countably many rational cuts and margins suffice for the choices. ◻

Proof. For a chosen filling, finitely many strict continuous inequalities have a common positive margin. A sufficiently small open box of component heights and a sufficiently small uniform neighborhood of the chosen continuous gap paths preserve that margin. The height box has positive probability. A Brownian bridge with nondegenerate covariance has full support on the continuous paths with its endpoints: approximate the chosen path by a polygonal path, constrain the bridge at its finitely many vertices using the strictly positive Gaussian density, and constrain its bridge fluctuations on the remaining subintervals to a small uniform ball. Each of these probabilities is positive. The same statement holds uniformly on a sufficiently small box of endpoints, although no uniform lower bound over all openly fillable data is asserted. Independent filling of finitely many gaps proves positivity. Rational approximations to the height boxes, polygonal paths, and margins give the last assertion. ◻

Proof. The assumed null event says \(\int f(v)K(v,E_v)\nu(dv)=0\). The integrand is nonnegative and \(f\) is positive on the set in question. For instance, restrict to \(f\ge1/k\) and let \(k\to\infty\). This gives the assertion. The sequential version follows from the definition of tight absolute continuity, or from Lemma 130 together with the required iid reference domination. ◻

Proof. First fix the countable choices of center labels, scalar center classes, rational cuts, and inequality margins. Restricting to the event of this choice preserves every null assertion. For blocks in retained profiles, record a spanning tree of block-start differences within each scalar center class, separately in each coordinate. Their union is a forest. Each such class lies in a finite relative-height group, so Corollary 138 supplies joint absolute continuity of these differences and the block increments. Discard differences between distinct scalar center classes, even when their center heights belonged to the same finite relative-height group before completion.

For each scalar center class, adjoin the absolute start height of one reference block. Conditional on the retained profiles, this anchor is a translate of that class’s continuously randomized height and hence has a density. The within-class forest differences and one anchor per class parameterize all block-start heights by an invertible linear map with determinant of magnitude one. Fubini’s theorem gives absolute continuity of all starts and increments jointly. No Brownian conditional law given a singular center profile is required.

Conditionally on the retained profiles and their heights, each filler is a Brownian bridge restricted to an event of positive conditional probability. Its law is absolutely continuous with respect to the unrestricted bridge with those endpoints. Marginalizing any unused central data preserves that absolute continuity. Brownian bridge disintegration across a deterministic cut identifies the resulting reference measure on a neighborhood crossing the cut with a Brownian restriction having the same start density. Thus the assertion holds also for intervals meeting seams. The entrance and exit pieces have the same local property away from the common endpoint. Finally take the countable union over recipe choices and rational windows covering the requested compact intervals. ◻

Proof. Realize \(\mathcal C\) as the unit circle by \(z(t)=\exp(2\pi i t/T)\). The scalar relation is closed, since the interval minimum is a continuous function of its endpoints. Distinct scalar classes are unlinked: if \(s<t<u<v\), \(s\sim_c u\), and \(t\sim_c v\), the minimum inequalities imply \(c(s)\le c(t)\le c(u)=c(s)\), and all four points belong to one scalar class. Hence the closed convex hulls of distinct scalar classes are disjoint.

We check that these hulls cover their closed disk; no unstated maximality of a lamination is needed. For a circular interval \([a,b]\) write \(K[a,b]=\operatorname{conv}(z([a,b]))\). Suppose an interior point \(w\) is in no scalar-class hull. For \(h>0\), components \((a,b)\) of \(\{c>h\}\) have scalar-equivalent endpoints. If \(w\in K[a,b]\), it is in the interior of that cap, since its chord edge is already in a class hull. There is such a cap at some positive \(h\): a sufficiently long compact arc avoiding the common endpoint contains \(w\) in its cap and has positive minimum height. Let \(H\) be the supremum of these heights. Along heights increasing to \(H\) the relevant components are nested. Their limiting endpoints \(a,b\) satisfy \(c(a)=c(b)=H\), \(c\ge H\) on \([a,b]\), and \(w\in K[a,b]\). The finite set \(A=\{t\in[a,b]:c(t)=H\}\) is contained in one scalar class. The cap \(K[a,b]\) is the union of \(\operatorname{conv}(z(A))\) and the caps over successive complementary intervals of \(A\). Their separating edges belong to the former hull. Thus \(w\), being outside that hull, lies in the interior of a cap over an interval \((u,v)\) where \(c>H\). There are \(u<u'<v'<v\) with \(w\in K[u',v']\); the positive minimum of \(c-H\) on \([u',v']\) gives a component cap containing \(w\) at a height above \(H\). This contradiction proves coverage. Singleton hulls cover any remaining boundary points.

The generated relation is also closed. If \(s_n\sim t_n\) and \((s_n,t_n)\to(s,t)\), join each pair by a simple monochrome chain of length at most \(M-1\). Pad it with constant steps. There are finitely many color patterns; after choosing one pattern, compactness of the finite product \(\mathcal C^M\) supplies limits for all intermediate points. Closedness of the scalar relations gives a chain from \(s\) to \(t\).

Glue an \(L\)-disk and an \(R\)-disk along \(\mathcal C\). For a combined class \(E\), let \(H_E\) be the union of its scalar hulls in the respective disks. Coverage and disjointness show that these sets partition the sphere. Each is a compact connected finite union of polygons, segments, and points. The decomposition relation on the sphere is closed: if \(x_n,y_n\in H_{E_n}\) converge, choose their containing hemispheres and express each as a convex combination of at most three hull vertices. After a subsequence all six vertices and their coefficients converge. Each triple remains scalar-equivalent, and all six vertices remain combined-equivalent by the closedness just proved. The limits therefore belong to one \(H_E\).

Each \(H_E\) is nonseparating. A polygon can be strongly deformation retracted, fixing its boundary vertices, onto the star from an interior point to those vertices: triangulate from the interior point and collapse each triangle onto its two radial sides. These collapses agree where triangles and hulls meet. Segments and singleton hulls need no change. The resulting finite graph is the subdivided bipartite incidence graph, with the harmless contractions due to singleton hulls. Hypothesis (ii) makes it a tree. Thus \(H_E\) is contractible. It is a finite planar polyhedron, so Alexander duality gives \(\widetilde H_0(S^2\setminus H_E;\mathbb Z)=0\); its complement is connected.

We have a closed decomposition into compact connected nonseparating sets. It is not the one-set decomposition, because its intersections with the infinite circle have at most \(M\) points. Moore’s sphere theorem in this form, as stated in (Timorin 2010, Theorem 1.1), gives a sphere quotient. Every hull meets the circle, so the induced map from \(\mathcal C/\sim\) is a continuous bijection onto that sphere. A continuous bijection from a compact space to a Hausdorff space is a homeomorphism. ◻

Proof. By Lemma [arm:finite-diagram-polarity], a separated off-center diagram is polar when \(\beta e>m\). A rank cycle has \(e\ge m\) and is therefore polar. Choose an integer \(m_0\) with \(\beta(m_0-1)>m_0\). A connected diagram with \(m_0\) distinct visits has \(e\ge m_0-1\) and is also polar. An equivalence class with at least \(m_0\) visits contains such a finite connected diagram: start at one point and add new points successively along a monochrome connection. All its points lie in some finite family of disjoint rational windows avoiding the distinguished times. A countable union of the polarity assertions therefore bounds every other class by \(m_0-1\) and excludes its cycles. The distinguished set is saturated by assumption, and the common endpoint is a singleton by strict positivity. Apply Theorem 146 with \(M=\max\{|S|,m_0-1,1\}\). ◻

Proof. The compact set \(\eta(\mathcal C\setminus J)\) excludes \(z\). Choose the nested disks inside its open complement using a sphere chart at \(z\). For the last assertion, the annular preimage is compact and disjoint from the finite fiber \(S\). ◻

Proof. A scalar contour identification is the equality of the endpoint heights together with the statement that every intervening height is at least their common value. A step match is the corresponding adjacent-level version. Condition (i) preserves the exact endpoint equality and adjacent levels. Condition (ii) preserves every intervening retained comparison, and Condition (iii) supplies the remaining comparisons across omitted gaps. Condition (iv) excludes all false partners. Apply this argument in both coordinates to the resolved sewing instructions. These instructions specify the triangle sides, their orientations, and the cyclic incidence around vertices. The full-star assumption therefore gives an isomorphism of the relevant planar cell complexes, and of their primal map edges. It preserves the disk orientation and hence the indicated interior of each circuit. Only actual primal map edges are counted in its length. ◻

Proof. Truncate the line count and the centers’ access depth. All times within one view unit of the relevant centers are reached by one more bounded jump. Every one of their scalar partners is reached by a further link. Equation (97) therefore puts all these partners, simultaneously, within a deterministic range \([-M,M]\) of enrolled centers, except with any prescribed small probability. Interline partners must belong to the same scalar center class by Corollary 140. This assertion is uniform: if it failed with positive probability, select a violating pair, apply the uniform range bound, and extract its finite coordinates; the Corollary would give the excluded center contact. Thus partners cannot lie on a line outside the chosen combined component.

Retain intervals extending beyond \([-M,M]\) on all its lines. Choose their rational cuts after the required outward dips. The construction in Lemma 141 remains valid for these larger retained pieces; increase \(K_0\) and the assigned class heights if needed. Use Lemma 139(iii) to truncate the reciprocals of all whole-gap margins. Choose a smaller positive \(\delta\) below these margins and the outward dip depths, and shrink the working intervals about the origins until both centered coordinates there have magnitude less than \(\delta\). This is possible uniformly in the approximants by the compact-uniform tightness proved in Lemma 137.

Complete the discrete contours using the actual resolved words on the retained intervals. In each coordinate translate a whole tied group by one common integer. Choose these translations in a fixed parity class to approximate the assigned heights; their diffusive rounding errors vanish. In particular, different original center heights in a tied group are not forced equal at the lattice level. Fill each gap by an axial walk approximating its chosen continuous connector and staying within its open tube. This can be done with the required parity: approximate the connector first by a polygonal path, realize its finitely many displacements in sufficiently long time slots, and fill the unused steps with unit backtracks. The displacements have order \(\sqrt{r_j}\) whereas each fixed positive-duration slot has order \(r_j\) steps. Adjust gap lengths by at most a bounded number of steps to satisfy parity. Refine the polygonal approximation only along a sufficiently slow diagonal. The positive margins keep all interior fillers nonnegative; axial entrance and exit pieces complete a finite excursion. These fillers are used to construct the auxiliary planar map, not to charge paths in its artificial parts.

We verify the exact incidence criterion for links beginning in the working intervals. Their tested levels differ from the relevant class height by at most \(\delta\sqrt{r_j}+1\). Every original partner is retained, by the uniform coverage just proved, and belongs to the same translation group. All intervening retained pieces of that group have their exact relative heights preserved. Any other retained class inside an active gap has original relative height \(+\infty\), and the auxiliary height assignment puts it strictly above the active level as well. Original omitted gaps and auxiliary connectors both lie above the tested levels by the chosen margin. Finally the retained outward dips block links escaping the extremities of a class in either sewing. Candidates in unrelated high classes also fail the endpoint-height comparison. In the auxiliary sewing, a low-level partner cannot lie in a filler or an unrelated high class. These checks apply to full scalar pairs and to adjacent-level step matches in both coordinates. They are exactly Conditions (i)–(iv) of Lemma 150.

By Lemma 145, the off-center polarity argument in Corollary 148 applies to the completed contours. The central class is saturated by construction and acyclic by assumption. Its quotient is a sphere. Lemma 149 gives nested disks and collars whose time preimages are contained in the working intervals.

The link-approximation hypothesis of Proposition 41 also holds at the distinguished times. For each scalar class, use the consecutive-member return witnesses of Lemma 139(ii). The last departure up-step and first return down-step at their slowly vanishing attained level form a matched pair, with endpoints approaching the two center visits up to one mesh step. The common group translations and strict gap comparisons just checked preserve these matches in the auxiliary sewing. The tree/cotree dictionary supplies a primal identification or one genuine primal edge for each pair. These finitely many pairs generate the distinguished combined class.

For a generator outside that class, saturation excludes a distinguished time at its tested minimum level anywhere in its span. Its endpoints and all visits at that level are therefore off-center, where joint Brownian absolute continuity gives strict local minima of distinct heights. The proof of Lemma 33 applies to this generator: nearby levels can avoid the countable off-center minimum heights and the finitely many distinguished heights. Thus all required links have approximations. This verifies (T3) without a metric bound on chronological connectors.

Apply Proposition 41 to the discrete approximations of these disks and collars. Its core disk has the full stars needed for incidences inside a larger retained collar; it does not assert that a proper subcomplex is closed under stars of all of its vertices. Lemma 150 now transfers this patch to the original map. The compact annular preimage avoids the central finite fiber, hence is bounded away from all its singular times. All truncation losses were arbitrary and may be removed in the stated order. ◻

Proof. It suffices to use Lebesgue measure on the compact anchor range; a positive smooth density is bounded below there. For a bad joint event \(E_j\), let \(p_j(c,y)\) be its anchor-sectional Lebesgue measure. If its reference product mass is \(u_j\to0\), Markov’s inequality gives reference probability at most \(\sqrt{u_j}\) to \(p_j>\sqrt{u_j}\). The selected probability of this set tends to zero by tight absolute continuity. On its complement, the conditional density bound gives joint selected mass at most \(C\sqrt{u_j}\). The density bound given the entire sample also holds given \((c_j,y_j)\) by conditional expectation. Replace Lebesgue measure by the covolume-normalized counting measure for the lattice statement. Truncations of the finite anchor count and height ranges can subsequently be removed. ◻

Proof. Treat a gap \(G=[a,b]\) and sources to its left first. Let \(A_G\) consist of \(t\in P\) with \(t<a\) such that all retained values between \(t\) and \(a\) are at least \(f(t)\), ignoring the omitted gaps. This set is compact: \(P\) stays a positive distance from the retained cuts, and finitely many continuous retained-interval minimum inequalities define eligibility. For \(t\in A_G\) write \(h=f(t)\). If any omitted gap before or including \(G\) reached height at most \(h\), take the first such gap. All preceding retained values are at least \(h\) and all earlier gaps are above \(h\). Its entrance cut must be strictly above \(h\): equality would itself give a scalar partner of \(t\) at a retained cut, outside \(\operatorname{int}J\), contrary to the hypothesis. Continuity then gives a first hit of \(h\) inside this omitted gap or at its far cut. The minimum from \(t\) to that hit is \(h\), so it is again a forbidden partner. This contradiction proves \(\min_G f>f(t)\) for every \(t\in A_G\). By compactness, \(\min_G f>\max_{t\in A_G}f(t)\). Choose a rational \(q_G\) strictly between these quantities. Do the same from the right, and take the larger lower bound if both source sets are nonempty. An empty eligibility set imposes no bound. Exterior gaps are treated in the same one-sided way.

Now take an endpoint pair with its source in \(P\). If its endpoint heights differ, it is not a relation. If an intervening retained value is below the source height, it is not a relation in either completion. Otherwise the source is eligible at every intervening omitted gap, and each replacement gap has minimum above its source height by the bounds just chosen. Thus the pair is a relation in both completions exactly when its retained comparisons permit it. Similarly a new partner in an omitted gap would require a first hit of the source height in an eligible gap, which its strict lower bound forbids. This proves both preservation and absence of new external partners. The all-partners hypothesis is essential: without it, raising individual gaps could remove a genuine omitted obstruction and create a false relation. ◻

Proof. Full-fiber protection. Choose a compact annular core inside larger protected collars, all avoiding the central point. Their full time preimages avoid the finitely many singular centers. Let \(M\) bound the off-center fiber sizes, as provided by Lemma [arm:finite-diagram-polarity]. Use \(M+2\) nested compact annular collars and choose rational closed time unions \(J_0\subset\operatorname{int}J_1\subset\cdots \subset\operatorname{int}J_{M+1}\) so that the core preimage lies in \(\operatorname{int}J_0\) and every full scalar partner of every time of \(J_\ell\) belongs to \(\operatorname{int}J_{\ell+1}\). Such choices follow from the finite-fiber quotient: choose the image of \(J_\ell\) inside the next compact collar and put its entire time preimage in the interior of \(J_{\ell+1}\). Uniform continuity permits rational time covers, and only finitely many layers are needed.

Apply Lemma 153 in both coordinates to \(P=J_M\) and \(J=J_{M+1}\). Record a rational lower bound for every omitted gap in each coordinate, including gaps whose original eligibility sets are empty. In the latter case choose any rational number below the original gap minimum. For other gaps use the strict bounds supplied by the lemma. Validity on any new retained input requires each recorded bound to exceed all its eligible source heights, with supremum of an empty set interpreted as \(-\infty\). These finitely many bounds preserve every scalar relation involving \(J_M\), including all rejected candidates, and exclude new omitted partners. The proof explicitly uses full source-fiber closure and exclusion of the cuts. It does not replace an arbitrary inactive gap by a higher path. If one instead uses the more general charts of Lemma 32, the equivalent recipe keeps the combined untested-minimum bands of Lemma 31, preserving both accepted and rejected comparisons; a responsible omitted piece for an upper minimum bound is just another finite recipe choice. Keep all included minima exact. Refine flexible instructions and left safety bands at the same time.

Any new combined chain starting in \(J_0\) has its first \(M\) scalar steps preserved: after \(\ell\) steps it is in \(J_\ell\). A simple chain with \(M\) steps would have \(M+1\) original visits, impossible by the original fiber bound. Thus the full combined fibers meeting the core, and not only their individual scalar edges, are unchanged in every allowed completion. This argument also excludes a new path through an omitted singular neighborhood.

The retained input. Record one absolute first-block anchor for each scalar center class as well. In Lemma 141 its height has the form of a sample-dependent translate of a uniform variable on a fixed open box. The uniform variables are sampled independently after the original sample and its fixed pre-anchor retained observation vector \(y\). Their translations may depend on that sample, but the conditional density of the resulting anchor vector is still bounded. The vector \(y\) is the countable consistent family of fixed-recipe passage observations, not an observation selected to encode the anchors. Lemma 152 therefore permits adjoining these anchors to the retained input. Together with the forest differences they determine every retained absolute block-start height. All guard and gap inequalities are now functions of this augmented input and the finitely many prescribed fill constraints. No original singular center profile is retained as an additional argument of the crossing test.

A common open quotient. Let \(T=J_{M+1}\), viewed as the finite disjoint union of its retained closed time blocks. For each coordinate draw a pair \((t,u)\) exactly when \(t,u\in T\), at least one endpoint belongs to \(J_M\), their anchored heights agree, and every intervening retained value is at least that height. Include the diagonal everywhere else and take the generated equivalence relation \(\mathcal R_c\) of the two colors. Lemma 153 proves that these explicit retained tests are equivalent to genuine scalar pairing in every allowed completion. No relation is inferred in an unprotected outer part of \(T\). The preceding \(M\)-step argument makes the combined fibers meeting \(J_0\) identical in every such completion. The generator relations are closed, since there are finitely many closed block pairs, \(J_M\) is closed, and the retained minimum tests are continuous. Their classes are bounded by Lemma [arm:finite-diagram-polarity] on these off-center intervals, since all their generators are genuine in a compatible completion. Hence their generated relation is closed by the finite-chain argument in Theorem 146. The quotient \(X_c=T/\mathcal R_c\) is therefore compact Hausdorff. Put \(O=X_c\setminus q_c(T\setminus\operatorname{int}J_0)\), where \(q_c\) is its quotient map. Every fiber in \(O\) lies wholly in \(\operatorname{int}J_0\) and is its full true fiber in every compatible completion, by the \(M\)-step argument. Hence \(q_c^{-1}(O)\) is open in \(T\), is contained in \(\operatorname{int}T\), and is open on the entire completed contour circle. It is saturated for the true equivalence relation of every completion. The quotient property therefore makes its image an open sphere patch in each completion. Identical saturated open sets give a homeomorphism from \(O\) to each such patch. The core is covered by these sets, since all its full fibers lie in \(\operatorname{int}J_0\). Thus identity of time labels identifies the local ambient topology, not just a finite list of incidences. Free class anchors only enforce the recorded separation conditions; they do not change a retained scalar relation.

Countable annular recipes. There is a useful countable basis for this quotient. If \(q_c:T\to X_c\) is the projection and \(J\subset T\) is a finite union of relatively open rational time intervals, put \[O_J(c)=X_c\setminus q_c(T\setminus J).\] This is open, since \(q_c\) is a closed map on a compact space. These sets form a basis: if \(x\in V\) with \(V\) open, the compact fiber \(q_c^{-1}(x)\) lies in the open set \(q_c^{-1}(V)\), so finitely many rational intervals give \(q_c^{-1}(x)\subset J\subset q_c^{-1}(V)\) and then \(x\in O_J(c)\subset V\). For compact containment use regularity of the compact Hausdorff quotient: choose \(V'\) with \(x\in V'\subset\overline{V'}\subset O_J(c)\), and then choose a rational time union \(J'\) with \(q_c^{-1}(x)\subset J'\subset q_c^{-1}(V')\). It follows that \(\overline{O_{J'}(c)}\subset O_J(c)\). Containment of time-interval closures alone would not justify this quotient-space containment. The recipe records the latter containment explicitly.

Place finitely many quadrilateral tubes cyclically in the annular core, with overlapping transverse passages and strict port margins. Choose an inner route corridor in each tube and a slightly larger outer quadrilateral, both within the common protected patch. Refine by finitely many basis sets to arrange \[\text{inner compact route}\ \subset\quad \text{recorded open confinement}\ \Subset\quad \text{outer quadrilateral},\] with the recorded port closures inside its separated end-port neighborhoods. All time unions used here lie inside \(\operatorname{int}J_0\), so their full fibers are preserved. The complete recipe records only their rational time unions; the finite incidence, orientation, and cyclic-order pattern; inner and outer collars; rational inequality margins; and the earlier block and instruction lists. There are countably many such records. Include a retained-data validity event \(V_k(c)\): each recorded gap bound is strictly above the supremum of its eligible source heights computed from \(c\); the recorded scalar pairs send \(J_\ell\) into \(\operatorname{int}J_{\ell+1}\); and their generated classes have at most \(M\) visits. These checks are measurable compact candidate tests. For example, failure of the last check has a connected diagram of \(M+1\) distinct points, and one takes a countable union over positive separation cutoffs; all its finite retained equality and minimum tests are closed. The analogous compact source tests handle the first two checks. For arbitrary reference input satisfying \(V_k\), the strict gap bounds alone give exact scalar relations and no outside partners for \(J_M\); the recorded nesting and \(M\)-step argument then give the same full core fibers in every compatible completion. Thus the common-quotient claim is not restricted to inputs originally sampled from the synthetic surface.

Feasibility from retained data. Let \(F_k(c)\) assert \(V_k(c)\), positive compatible fill weight, and that the specified sets in the retained local quotient satisfy these inclusions for an oriented cyclic quadrilateral pattern, with strict transverse-overlap and separation margins. In particular the entire permitted confinement sets lie inside the witness tubes; it is insufficient that they merely contain some correctly arranged tubes. Thus every allowed port-to-port path, including a minimizing one, has the prescribed topological crossing behavior. The inner routes ensure that the required port pairs can be connected inside the recorded open confinements. This is a property of \(X_c\) and the recorded sets and is therefore the same in every allowed completion. Its measurability can also be seen without choosing a coordinate chart for the singular synthetic surface. Sample an ordinary positive-weight Brownian completion as in Lemma 143. In that ordinary surface the pattern event is measurable, by a countable polygonal approximation in its coordinate charts with strict margins. More explicitly, for \(J\subset\operatorname{int}J_0\) its recorded open set is \(S^2\setminus\eta(S^1\setminus J)\) in that completion. The complementary compact image is measurable from its continuous quotient curve. The nested route, confinement, port, and outer-quadrilateral conditions are therefore a countable union of measurable rational polygonal witness tests. Its indicator is constant over all compatible completions, by the local quotient identification. More explicitly, let \(Q_c\) be the measurable ordinary filling kernel, \(G_k\) its compatible strict-fill event, and \(H_k\) the marked planar pattern event. On positive fill weight, constancy on compatible fills gives the measurable definition \[F_k(c)=\mathbf 1_{V_k(c)}\mathbf 1\left\{ \int\mathbf1_{G_k}\mathbf1_{H_k}\,dQ_c =\int\mathbf1_{G_k}\,dQ_c>0\right\}.\] No uniform positive lower bound on this fill weight is asserted. The polygonal witnesses and their positive spatial margins are existential in the pattern event, not fixed Euclidean coordinates in the recipe. Only its time unions, order, contour inequalities, and topological inclusions are fixed. Homeomorphisms of compatible completed patches preserve this existence statement even when their Euclidean coordinates differ.

Crossing tests and transfer. At the lattice level the recipe defines port vertex sets and confinement subgraphs from its fixed time-cell labels and the exact retained chart incidences. One may require that all representatives of a vertex in the reconstructed chart lie in the chosen inner time union, and use a slightly larger union for the allowed confinement. Full-fiber protection makes this an entirely local test. Record the infimal genuine all-edge cost of a path for each specified port pair inside that subgraph, with the convention \(+\infty\) if no path exists. These are measurable functions of the finite raw blocks, exact shifts, and fixed instructions. Define \(E_{k,A}(c,y)\) as \(F_k(c)\) together with failure of one of these costs to be at most \(A\) in view units, and define \(E_{k,\infty}\) as \(F_k(c)\) together with at least one infinite limiting recorded cost. These tests have no argument for the omitted geometry. Open inner/outer margins permit the usual certificate limits and prevent a boundary equality from deciding the test. In every compatible completion the same recorded tubes have the same transverse-intersection pattern; their successful crossings force a circuit. In the original synthetic surface the indicated hole is the one containing the center, because that is how this feasible recipe was chosen. This last geometric fact is used only after the local crossing test has succeeded.

Finally, distinguish the two reference laws in the entropy argument. The Kraft-product reference makes the code independent of the whole raw word. The trial law \(R_{j,t}\) instead keeps the selected code marginal and chronological nontrial kernels; each requested slot is fresh conditional on that code and its actual incoming history. For fixed \(k\), the code and recipe fix the rounded origins, interval endpoints, and guard/noise search anchors before the requested letters are sampled. Additional marks have the independent laws required by Proposition 59, whose bound is uniform in the incoming history. Use unnormalized compatible subprobabilities in Proposition 59; do not condition on a rare successful guard atom. If the successful recipe was found using omitted geometry, its index is still one of the countable records above. The null assertion for every fixed \(E_{k,\infty}\) can therefore be intersected over all \(k\). A finite threshold failure need not be null; its probability tends to zero as \(A\to\infty\). For such vanishing-probability statements first restrict to a finite number of successful recipes carrying all but a prescribed small probability. This proves the assertion even for adaptive recipe selection. ◻

Proof. In an annulus choose a finite cyclic collection of quadrilateral tubes, with successive passages transverse in their overlaps. Any crossings of their prescribed ports meet by planar separation. Their union contains a closed walk with nonzero winding around the inner disk. Removing inessential loops gives a simple surrounding circuit of no greater cost. Condition (i) transfers this circuit and its interior to the original map.

For any fixed \(\eta>0\), with loss at most \(O(\eta)\) first bound \(C_j,M_j,\delta_j^{-1}\) and the number of intervals by deterministic constants. The exceptional event \(bN_j-X_j\le\sqrt{N_j}\) has probability \(O(N_j^{-1/2})\). Outside it, \(\delta_j r_j(X_j)/(n_j e^{-bN_j})\to\infty\), which gives the lower time bound. The circuit and its closed interior lie in the closed outer disk, whose full time preimage is covered in (ii). Hence every uninterrupted visit in the closed circuit disk, including a boundary visit or a run along its edges, lies in the union of those retained intervals. Its length is at most a constant times \(r_j(X_j)\); even if the intervals touch, their total length has this bound. Since \(X_j\ge aN_j\), \(r_j(X_j)/(n_j e^{-a_0N_j})\le e^{-(a-a_0)N_j}\to0\). Finally \[\frac{C_jB(r_j(X_j))}{B(n_j)e^{-s'N_j}} \le K C_j\exp\{-(sa-s')N_j\}\longrightarrow0\] in probability. Here \(sa-s'>0\). Remove the fixed tightness truncations by letting \(\eta\downarrow0\). Condition (iv) gives confinement. ◻

Proof. The area-\(T\) field is the unit-area field plus \(\gamma^{-1}\log T\). Lemma 9 applies also to this actual field: after its stated domination, add this deterministic constant on a neighborhood of the tested compact patch and remove it from the Gaussian reference by a Cameron–Martin cutoff. Thus, in the affine chart of a first auxiliary area pair, for every bounded \(U\) with \(\overline U\subset\mathbb C\setminus\{0,1\}\), its actual real-field law \(\mu_{T,U}\) is absolutely continuous with respect to the local cone law \(\mu_{{\rm cone},U}\) used in that Lemma. The cone normalization circle is chosen outside \(\overline U\).

Retain the entire independent stationary-phase imaginary field and its unparameterized directed curve, with common law \(\nu\). Let \(\Pi_h\) be the common conditional Poisson area-mark law in \(U\). Then \[\mu_{T,U}(dh)\,\nu(d\vartheta)\,\Pi_h(d\pi) \ll \mu_{{\rm cone},U}(dh)\,\nu(d\vartheta)\,\Pi_h(d\pi).\] In particular the complete traversal order is retained; the finite order \(O\) required by a protected chart is a measurable function of this input. Theorem 56 gives the same local observation kernel on both sides, including the same spatial interpretation and all strict port and confinement tests. Appending that common kernel preserves the displayed domination. The upper comparison of Corollary 106 is a probability-one event for the cone input and observations, with its chosen deterministic multiplier. The reference internal metric on \(U\) is measurable from \(h|_U\), so this event transfers to the sphere. No unrestricted metric depending on the omitted field is transported. The sphere observations are strict-interior observations because the full preimage of a compact patch avoiding the exploration root is a compact subset of \((0,T)\).

Intersect over bounded rational-box domains, their buffered protected charts, and rational open losses. Adjoin a second independent area pair and repeat for its three-marked marginal, as in Remark 10. No comparison is conditioned on the other pair. Their off-mark covers exhaust the punctured sphere. The affine covariance of the kernels and of the internal metrics makes both conclusions statements in the same quantum units, with the same bilateral multiplier and the same exploration root.

For completeness, an open-port crossing does not require joining independently extracted endpoints from these covers. In either affine representation a connected open surface confinement remains connected after removal of the two auxiliary points. Choose points of its two open ports away from those points and a compact topological route between them. The local length-metric and internal connection inputs of Theorem 11, applied along a finite cover of the route, give a finite-length \(D_h\) path compactly inside the punctured confinement. Enclose that path in a bounded connected finite union of rational coordinate boxes \(U\) whose closure is still inside the confinement and avoids \(0,1\). Both Lemma 9 and Theorem 56 permit this domain \(U\); they do not require \(U\) to be a disk. The transferred upper comparison supplies one finite-cost crossing certificate in \(U\), with its endpoints strictly inside the prescribed ports. Lemma 25(i) realizes it in the retained primal chart with the prescribed open losses. The full-fiber inner port and outer confinement margins in the protected atlas ensure that these records obey the recipe’s time-label restrictions as well. Hence the recorded infimal port cost is finite. Apply this argument to a finite list simultaneously and sum its finite costs. This uses endpoint slack only, with no arbitrary-vertex joining input. ◻

Proof. Choose a maximal \(\ell\)-separated set of bad times in \(I\). It has at most \(h-1\) elements. The union of their chronological \(\ell\)-neighborhoods covers every bad time. Merge overlapping neighborhoods and intersect with \(I\). Their total length is at most \(2(h-1)\ell+2h\), and the remaining set has at most \(h\) integer intervals, all of whose times are good.

Consider one such interval \([a,b]\cap\mathbb Z\). Choose good times \(a=t_1<\cdots<t_k=b\) with successive differences at most \(d\), using at most \(2+(b-a)/d\) times. For each one choose an available circuit. The interiors of consecutive circuits overlap: the walk at each time between \(t_u\) and \(t_{u+1}\) belongs to both interiors on a nonempty chronological subinterval.

Among this finite family retain the disks maximal for inclusion. Every discarded disk is contained in a retained disk. The intersection graph of retained interiors is connected: a chain of overlapping original disks maps to a chain of overlapping containing maximal disks. Two distinct maximal Jordan disks which overlap have intersecting boundaries. Indeed disjoint Jordan boundaries with intersecting interiors would force one disk to contain the other. Thus the union of the retained circuit boundaries is a connected subgraph. Its number of edges is at most the sum of the lengths of the original circuits.

Select a retained disk containing the first disk, so that it contains \(v(a)\) in its interior. Condition (ii) implies that the interpolated walk first reaches its boundary at an integer time in \([a,a+\ell+1]\). The intersection is a graph vertex, since both the walk and the circuit use edges of the same planar graph. Similarly a retained disk containing the last disk has a boundary vertex represented at a time in \([b-\ell-1,b]\). The connected union of retained boundaries joins these two vertices at a cost no greater than the sum of the chosen circuit lengths. We have made no estimate of the cost of reaching either boundary from \(v(a)\) or \(v(b)\).

Do this for every remaining good interval, and concatenate the resulting boundary paths in the order of those good intervals. Within a short good interval the forward hit may follow the backward hit; orient its boundary path from the forward hit to the backward hit regardless of their time order. Each gap between these paths is represented by a pair of times separated by a removed bad interval and at most \(2\ell+2\) extra steps. Initial and final gaps obey the same bound. There are at most \(h+1\) gaps; allowing \(h+2\) covers endpoint and rounding conventions. Every gap has length at most \(K_h\ell\), and every time used lies in the \(K_h\ell\) enlargement of \(I\), for a constant depending only on \(h\). The total number of circuits is at most \(2h+T/d\), which gives the stated cost after increasing \(K_h\).

If no good interval remains, the length of \(I\) is at most the total length of the bad neighborhoods. Use the single unresolved pair \((i,j)\). This proves all claims, including Equation (105). ◻

Proof. Mark the \(h\) selected bad bases and finite witness diagrams before choosing the independent shifted address grids. The address estimate of Section 8 bounds the integrated box count under Equation (109) and charges \(\beta\) times the integrated monochrome rank, up to arbitrarily small relative errors. Proposition 157 shows that a positive probability of an acyclic base component at a uniform viewing level in \([aN,bN]\) would supply an available circuit at one of the bad bases.

We relate these center components to the projected graphs charged by the address estimate. The bases are separated by \(\ell/r(X)\ge e^{(a-a_0)N}\) at a view level \(X\in[aN,bN]\), so their enrolled centers are distinct. On each fixed finite truncation, Lemma 139(ii) supplies actual links witnessing a spanning tree of every scalar center class. Proposition 132 identifies these links with the graph attained by the seed recipe. The all-seed code includes its actual witnesses and the jumps associating their endpoints to their frozen centers. Outside the levels charged by the integrated jump error, these endpoints and centers occupy the same projected cells; distinct center lines occupy distinct cells. Bases also have tight seed offsets from their assigned centers. Apply the independent-grid boundary estimate to the original centers, chosen before that grid: conditional on a bounded seed offset, the crossing probability is at most a fixed multiple of \(\min\{1,e^{-U_j}\}\), whose uniform-coarsening average tends to zero. Rounding an origin keeps it in the same terminal cell, and hence in the same cell at every coarser generation. Thus each base and its frozen origin occupy the same projected cell with probability tending to one.

Thus a rank cycle of a limiting base component is witnessed inside the corresponding projected base component. Adding connected vertices cannot destroy it: extend monochrome spanning forests by a union spanning tree rooted at the old component, gaining at least one unit of total rank per new vertex. All other code entries may therefore be kept. The acyclic-circuit estimate makes the probability of an acyclic base component at a uniform view level tend to zero. Averaging over the seed-generation trials and applying Markov’s inequality now shows that every projected base component has a rank cycle outside \(o(N)\) generations, in probability. Fixed truncations are removed along the sparse diagonal of Lemma 133.

The separation hypothesis of Proposition 124 holds for every displacement. At a fixed chronological cutoff, the largest charged cell width is \(O(ne^{-aN})=o(\ell)\). Along the permitted enlarging cutoff, the all-seed condition \(\Lambda_j\log(2\widehat C_j)/L_j\to0\), together with \(L_j=o(N_j)\), gives \(\log\widehat C_j=o(N_j)\), so the same comparison remains true.

The rank-credit estimate gives negative credit at least \((d_2-1/2-o(1))(b-a)hN\) on this range, whereas the earlier possible positive credit is at most \(C_0N+C_1(a-1)hN+o(N)\). The choices in Equation (107), followed by the choice of \(h\), make the former strictly greater. The selected density costs only \(o(N)\), so this contradicts the address probability bound and proves the proposition. ◻

Proof. Suppose the probability of some bad level is bounded below by \(\delta>0\) along a subsequence on a fixed compactness and clearance cutoff. Choose \(\epsilon<\delta/8\) and apply Lemma 128 with \(\rho_n=e^{-N_0}\) to a fixed slight enlargement of the tested window. This gives one master event \(\mathcal T_n\) for every applied level, before any level or bases are selected. The union of the bad-level events, intersected with \(\mathcal T_n\) and the fixed cutoff, still has probability at least \(\delta/2\) for all large \(n\). Since \(\sum_{j\ge0}(j+1)^{-2}<\infty\), there is a deterministic index \(j=j(n)\) for which the intersection of that level’s bad event with these same preselected events has probability at least \(c\delta(j+1)^{-2}\). Condition on this intersection and then choose the bad bases by the finite lattice ordering. The logarithm of the density bound is at most \(C_\delta+2\log(j+1)\), whereas \(N_j=(a'_0)^jN_0\); thus the density bound is \(\exp(o(N_j))\).

At every applied level, \[n e^{-bN_j}\ge n^{1-b(1-s/2)}\longrightarrow\infty,\] since \(0<s<1\) and \(b-1<s/8\) imply \[b(1-s/2)<(1+s/8)(1-s/2)<1.\] All interval-window enlargements fit in a fixed slightly larger window: their total size is bounded by \(\sum_j C_h e^{-a_0(a'_0)^jN_0}\), which tends to zero. The resulting conditional laws contradict Proposition 160. Letting the arbitrary preselection loss tend to zero and exhausting the compactness and clearance cutoffs proves the claim. ◻

Proof. The claim is immediate when \(n\delta_n<1\), since only identical integer times are tested. Otherwise set \(N_0=-\log\delta_n\). If \(N_0>(1-s/2)\log n\), chronological steps give a bound \(C n^{-s/2}\) after normalization. Otherwise iterate Equation (110) until the first new level \(N_{j_*}>(1-s/2)\log n\). Lemma 162 shows that all recursions are valid with probability tending to one. Their total normalized payment is bounded by \[\sum_{j\ge0}H^j C_h e^{-c(a'_0)^jN_0},\] which tends to zero. For example, after increasing \(N_0\), the ratio of successive terms is at most \(1/2\), so the sum is at most twice its first term.

Each terminal unresolved pair has chronological separation at most \(n e^{-N_{j_*}}\le n^{s/2}\). There are at most \(H^{j_*}\) such pairs, and \(j_*\le C_1+C_2\log\log n\). Equations (104) and (106) bound their total normalized cost by \[C(\log n)^{C_3}n^{-s/2}\longrightarrow0.\] Their chronological paths stay in \(U\) with probability tending to one by the uniform contour coupling and the stated clearance. The same is true of all earlier circuit paths by their definition. This proves the assertion, including confinement. ◻

Proof. If uniformity failed, select a violating pair with positive probability and then a further coupled subsequence. First restrict to a compact sewing; in an exhaustion, truncate the stipulated tight chronological representatives. This truncation is part of the hypothesis; it is not deduced from a passage comparison. Property (T1) gives limiting representative times \(t\) and \(u\) with \(\eta(t)=\eta(u)\). Their equivalence is generated by a finite chain of horizontal coordinate identifications by (T3). For each ordinary generator, strict-height approximations yield actual primal identifications or genuine primal edges whose time representatives approach its two endpoints. At a branch visit one first splits into the finitely many consecutive ordinary generators. This is a topological statement about the discrete sewing; it estimates no chronological connector cost.

Choose closed rational time neighborhoods of all the finitely many visits whose curve images lie in the prescribed open spatial neighborhood. Truncate the number of visits and then their countable rational neighborhood choices to a finite list, with arbitrarily small probability loss. The finitely many time errors in the link approximations are then bounded, with probability tending to one, by a deterministic sequence \(\delta_n\downarrow0\), chosen sufficiently slowly. Proposition 163 joins the selected original representatives to the approximating link endpoints, and joins successive approximating endpoints at each visit, at cost \(o(B(n))\). The finitely many genuine edge costs are also \(o(B(n))\) because \(B(n)\to\infty\), and their projected diameters tend to zero, so they stay inside the prescribed neighborhood. Concatenation contradicts the selected violation after the arbitrary truncation losses vanish. This proves the uniform sequential assertion. Notice that a second-coordinate scalar identification is not itself declared a unit-cost primal edge: it is the strict-height matched-step approximation, together with the chronological modulus just proved, that supplies the primal connection. ◻

Proof. The strong confluence theorem is (Bhatia and Kavvadias 2026, Theorem 2). More specifically, its Propositions 5–6 say that a strictly interior geodesic segment is uniquely minimizing, and that sufficiently small perturbations of its endpoints force every connecting geodesic to contain a prescribed shorter interior segment. Given an open subarc of \(P:[0,T]\to\mathbb C\), choose \(s<t\) strictly inside its parameter interval and \(0<\varepsilon<(t-s)/2\). Choose points of \(\mathcal Q\) in the endpoint neighborhoods from Proposition 6. Their geodesic contains \(P([s+\varepsilon,t-\varepsilon])\). There are countably many pairs in \(\mathcal Q\), so fixed-pair uniqueness holds for all of them on one event of probability one.

For a local chart first shorten the subarc so that it lies in a smaller domain with positive metric distance to the boundary of a larger chart. The internal version of strong confluence, proved in (Bhatia and Kavvadias 2026, sec. 6.1, proof of Proposition 50, Equation (93)), applies there. The field absolute continuity in Proposition 169 transfers this local event. By shortening again, connecting geodesics between nearby endpoints are too short to reach the chart boundary, so their relevant segments are also ambient geodesics. Additive constants rescale every length equally and do not affect the conclusion. ◻

Proof. For \(0\leq r\leq j\), let \(M_{j,r}\) be the maximum of \(X_J\) over the \(2^r\) consecutive dyadic intervals \(J\) of length \(2^{j-r}\) in \([0,2^j]\). For nonnegative random variables, the \(k\)th power of their maximum is at most the sum of their \(k\)th powers. Thus \[ \|M_{j,r}\|_{L^k}\leq2^{r/k}A_{j-r}. \tag{126}\] Decompose \([0,u]\), for any integer \(u\in[0,2^j]\), into dyadic intervals with at most one of each length. Their internal endpoint paths concatenate to an all-edge path in the full map. Thus, with \(\pi\) denoting the full-map projection, \[\max_{0\leq u\leq2^j}d_{\mathcal M}(\pi(0),\pi(u)) \leq\sum_{r=0}^jM_{j,r}.\] Suppose (125) fails. Since \(1/k<p\), we may choose \(1/k<\alpha<\beta<p\) and a finite \(C\) such that \(A_j\leq C2^{\alpha j}\) for every \(j\). By Minkowski’s inequality and (126), the \(L^k\) norm of the last display’s right-hand side is at most \[C2^{\alpha j}\sum_{r=0}^j2^{-r(\alpha-1/k)} \leq C'2^{\alpha j}.\] Since \(\alpha<\beta\), Markov’s inequality places every vertex visited during \([0,2^j]\) in the ambient ball of radius \(\lceil2^{\beta j}\rceil\) about \(\pi(0)\), with probability tending to one.