The direct lift.

Work in normalized coordinates. Up to \(\xi_*\), positive-order derivatives of continuations change by \(O(\rho_1)\), \(\rho_1=\lambda\int P\). Indeed the PDE for their undifferentiated difference has linear drift with bounded smooth spatial coefficients on a fixed time interval and source bounded with derivatives by \(C\lambda P\); differentiate its diffusion representation (bounded drift derivatives). Coupling \(Z\)’s by common Brownian noise gives \(L^k\) change \(O_k(\rho_1\sqrt{\xi})\) for \(\xi\le z_0\), using oddness of the gradients, then \(O_k(\rho_1\sqrt{z_0})\) for \(\xi\in[z_0,\xi_*]\); continuation derivatives are unchanged on the latter interval. All constants here are uniform as \(z_0\) is decreased. Thus on \([0,z_0]\), \[\widehat\Gamma''(\xi)=-2\lambda P(\xi)\mathbf E\widehat\chi^3+O(\xi\rho_1)\] by oddness of third derivatives; \(\mathbf E\widehat\chi^3\) stays strictly bounded below. Near zero, \(P(\xi)=\xi\), so integration gives a drop of order \(\lambda\xi^2\) from \(\widehat\Gamma'(0)\). By time \(z_0\) the accumulated drop is of order \(\rho_1\), while the integrated error is \(O(\rho_1z_0^2)\). On \([z_0,\xi_*]\) the continuation itself is unchanged, and \(\widehat\Gamma'=1/T+O(\rho_1\sqrt{z_0})\). Choosing \(z_0\) small first makes this error smaller than the accumulated drop. This proves both the near-zero order and persistence of the gap. Beyond \(\xi_*\) use the annealed inequality just proved (valid also for \(A_\lambda\)). This argument differentiated once in \(\log\lambda\) gives \(O(\lambda\xi^2)\) for the derivative at fixed small \(\xi\) of the normalized gap (the linearized PDE and diffusion estimates are identical). Also \(\ell^2 b_\circ=T+O(\lambda)\).

Smoothing.

On any fixed compact time interval, continuing the direct PDE annealed to \(M\) with terminal mass \(j\) instead of 1 changes values, modulo spatial constants, by \(O(e^{-cM})\). Indeed at a fixed time above the support the expectation of \(\cosh^ {1/j}(j(z+B))\), over the remaining variance, lies between the sum of the two individual exponential contributions and the same sum times \(1-O(e^{-cM})\) (with the common factor \(2^{-1/j}\)), by subadditivity and restriction to disjoint half-lines for \(B\). Earlier nonlinear heat transitions preserve uniform additive comparison. Changing the normalized mass to the smoothed one now costs \(O(\omega M^2)\) in values, by PDE comparison and bounded gradients. Spatial derivatives on the compact interval thus change by \(O_\epsilon(\omega^{1-\epsilon})\), any fixed finite orders, via higher-order difference quotients and uniform spatial derivative bounds; diffusions couple there with the same power accuracy. Consequently the error in the active \(\widehat\Gamma''\) bound above is \(o(\lambda)\xi\) even at small \(\xi\) (use oddness, the added mass \(\omega\xi\), and Brownian coupling at small time with factor \(\sqrt{\xi}\)).

Beyond the support compare \(\widehat\Gamma'\) to the direct annealed case on a sufficiently long fixed interval. Thereafter use \(\widehat\chi\) between \(c(1-\widehat V_z^2)\) and \(C(1-\widehat V_z^2)\), by the scalar case of [eq:A1] since all remaining masses are bounded and bounded below. The martingale diffusion coefficient of \(\widehat V_z\) is \(\widehat\chi\); Itô’s formula for \(\sqrt{1-\widehat V_z^2}\) gives exponential decay of its expectation (localize inside \((-1,1)\) and use nonnegativity), so \(\mathbf E\widehat\chi^2\) eventually stays small. This proves [eq:P2] also with smoothing.

More explicitly, for the scalar conditional mean \(m_t\) and \(f(m)=\sqrt{1-m^2}\), localization gives \[\frac{d}{dt}\mathbf E f(m_t) =-\frac12\mathbf E\frac{\chi_t^2}{(1-m_t^2)^{3/2}} \le -c\,\mathbf E f(m_t)\] once the remaining masses are bounded below. The initial value is at most one, so this bound is uniform in the starting field. It also applies conditionally at the beginning of the long interval and supplies the exponential terminal-datum bound used next. ◻

The additive resolvent

Fix \(k\) and retain only the permitted rows and columns. Abbreviate \(B=B^k\) and \(w=\operatorname{diag}(\mu_t^k b_t/\ell_t^k)\). The diagonal matrix \(w\) is nonnegative. Consequently \(C=(I+B w)^{-1}B\) is symmetric and satisfies \(0\preceq C\preceq B\), as is seen by writing it as \(B^{1/2}(I+B^{1/2}wB^{1/2})^{-1}B^{1/2}\). The additive-sector inverse is \((\ell^k)^{-1}I+A^k\operatorname{diag}(\mu^k)\), where \(A^k_{rt}=-C_{rt}b_t/(\ell_r^k\ell_t^k)\). The bound \(C\preceq B\) controls the diagonal entries of \(C\), and positivity gives \(|C_{rt}|\le\sqrt{C_{rr}C_{tt}}\). Thus all entries are polynomially bounded. The identity \(C=B-BwC\), together with \(\sum_t w_{tt}=O_{\rm pol}(1)\), transfers the evaluation-time Lipschitz bounds for \(B\) to \(C\) and then to \(A\). For \(A\), the relevant distance is \(|\Delta K|+|\Delta b|\). Successive resolution indices satisfy the stronger estimate \[\|A^{k+1}-A^k\|_\infty\lesssim_{\rm pol} p_k(K_k-K_{k-1}+|b_k-b_{k-1}|+{\bf1}_{k=\min(\mathrm{retained})})\] where \(b_{-1}=b_0\). To prove this, note that changing \(k\) to \(k+1\) changes the sheet measure \(\mu\) by \(p_k(\delta_k-\delta_{k-1})\), with unretained atoms omitted. Use primes for the matrices at \(k+1\). Their resolvent identity is \[C'-C=(I-C'w')(B'-B)(I-wC)-C'(w'-w)C .\] If both atoms are retained, their contributions to \(C'(w'-w)C\) are subtracted before taking absolute values. The evaluation Lipschitz estimates then supply \(p_k(K_k-K_{k-1}+|b_k-b_{k-1}|)\). Changes in the kernel and in the diagonal gaps use the preceding bound with the same factor \(p_k\).

At the first retained sheet, the lower atom is absent. Its partner has mass exactly \(p_k\), which accounts for the indicator in the bound. There is only one such boundary term. It still has the factor needed to cancel the block-average denominator.

Absolute attachment domination

Fix an output pair \(ab\) on a retained sheet \(r\). First apply the diagonal multiplier \(1/\ell_r^{il}\) in the two child resolutions. Then add the row correction \[-\sum_{k,t}A^k_{rt}\{(E_k\,{\rm row}_t\,u)_a+(E_k\,{\rm row}_t\,u)_b\}, \qquad ({\rm row}_t\,u)_c=\sum_{d:\,cd\ {\rm on\ sheet}\ t}u_{cd}.\] This is the inverse on both parts of the decomposition. On a row-zero array, the correction vanishes. In each one-leaf summand of an additive array, the other endpoint is constant within its child and therefore has child index \(r+1\). The child multiplier on the summand is thus \(1/\ell_r^k\). The row sum contributes \(d_t^k=-\mu_t^k\), and the minus sign in the correction gives the remaining \(A^k\operatorname{diag}(\mu^k)\) part of the inverse.

We now combine the block projections before estimating them. Since \(E_0=P_0\), \(E_k=P_k-P_{k-1}\), and \(P_{R+1}=I\), the exact one-variable Abel identity is \[\sum_{k=0}^{R+1}A^k_{rt}E_k =A^{R+1}_{rt}I+\sum_{k=0}^{R}(A^k_{rt}-A^{k+1}_{rt})P_k .\] Thus the only terminal term is an identity operator. In every other term, the coefficient difference has a factor \(p_k\), canceling the denominator of \(P_k\).

Here is the corresponding pointwise comparison with attachment densities. Write \(c\sim_k a\) if \(c\) belongs to the \(p_k\)-block of \(a\), and let \(\mathcal S(dc)\) denote the ordinary signed attachment of \(c\) to the old tree, including \(ab\). By definition, \[(P_kF)_a=\frac1{p_k}\int {\bf1}_{\{c\sim_k a\}}F_c\,\mathcal S(dc).\] A subsequent row sum attaches \(d\) to the tree that now includes \(c\); denote that signed rule by \(\mathcal S_c(dd)\). Relative to these two successive attachment rules, the nonterminal correction at the endpoint \(a\) has coefficient \[\kappa_a(c,d)= -\sum_{k=0}^{R}\sum_{t\ {\rm retained}} \frac{A^k_{rt}-A^{k+1}_{rt}}{p_k} {\bf1}_{\{c\sim_k a\}} {\bf1}_{\{cd\ {\rm on\ sheet}\ t\}} .\] For a fixed allocation, the pair \(cd\) belongs to at most one sheet. Put \(\rho_k=K_k-K_{k-1}+|b_k-b_{k-1}| +{\bf1}_{k=\min(\mathrm{retained})}\). The weighted difference estimate therefore gives \[|\kappa_a(c,d)|\lesssim_{\rm pol} \sum_{k=0}^{R}\rho_k{\bf1}_{\{c\sim_k a\}} \le K_R+\operatorname{var}b+1 .\] This bounds the coefficient at each placement relative to the unrestricted absolute rule \(|\mathcal S|(dc)|\mathcal S_c|(dd)\). The endpoint \(b\) has the same bound. The terminal identity term requires just the row attachment of \(d\), with a polynomially bounded coefficient.

For the child multiplier, abbreviate \(D_{il}=1/\ell_r^{il}\), \(i,l=r+1,\ldots,R+1\). Set \(\Delta_iD_{il}=D_{il}-D_{i+1,l}\) and \(\Delta_lD_{il}=D_{il}-D_{i,l+1}\). Applying the same Abel identity in both positions gives all endpoint terms explicitly: \[\begin{split} \sum_{i,l=r+1}^{R+1}D_{il}\, \widetilde E_i^r\otimes\widetilde E_l^r ={}&D_{R+1,R+1}I\otimes I\\ &+\sum_{i=r+1}^{R}\Delta_iD_{i,R+1}P_i\otimes I +\sum_{l=r+1}^{R}\Delta_lD_{R+1,l}I\otimes P_l\\ &+\sum_{i,l=r+1}^{R}\Delta_i\Delta_lD_{il}P_i\otimes P_l . \end{split}\] There is no separate lowest-child average: it is included in the difference sums. With \(\Delta K_i=K_i-K_{i-1}\), the estimates for \(g_r^{il}\) and the gap \(\ell_r^{il}\ge e\) give \[\begin{split} |\Delta_iD_{i,R+1}|&\lesssim_{\rm pol}p_i\Delta K_i,\qquad |\Delta_lD_{R+1,l}|\lesssim_{\rm pol}p_l\Delta K_l,\\ |\Delta_i\Delta_lD_{il}| &\lesssim_{\rm pol}p_i p_l\Delta K_i\Delta K_l . \end{split}\] The single differences cancel one block denominator. The coefficient of the two-attachment term is bounded pointwise by \[\sum_{i,l=r+1}^{R} \frac{|\Delta_i\Delta_lD_{il}|}{p_i p_l} {\bf1}_{\{c\sim_i a\}}{\bf1}_{\{d\sim_l b\}} \lesssim_{\rm pol} \left(\sum_{i=r+1}^{R}\Delta K_i\right)^2 .\] The identity term uses no new label; the two edge sums use one; and the last sum uses at most two. All are thus bounded by the claimed unrestricted absolute attachment rules.

The allocation order identities in Section 2 allow the block and old-topology restrictions to be retained throughout these equalities. We discard them only in the pointwise inequalities above. The summable coefficients depend on total clock length, total variation of \(b\), and the single retained boundary, not on the number of steps. Existing-fork atoms and coincidences keep their ordinary attachment weights. This also explains why the argument controls new split densities, rather than only the total mass of the inverse.

Limits of finite resolutions

At a fixed finite resolution, let strictly separated clocks approach flat clocks and then let \(p_0\downarrow0\). The combined coefficients above stay bounded after their block denominators have been canceled. Taking subsequences of these finitely many coefficients preserves the inverse identity and its absolute bounds. A strict gap persists after an arbitrarily small weakening.

The same resolution operators can be applied to continuum placements: retain the indicators of membership in the blocks at each resolution cut, while keeping the exact old-label positions. To check this, insert empty steps between the cuts. Integer counting of additional slots depends only on their coarse genealogy with the old labels, so refining a group of children does not change that count. The real-mass coefficient identity then gives the same conclusion. Consequently the pointwise comparison with the unrestricted attachment measure remains valid for continuum placements, including atoms at old forks.

For a nonstep path, apply the inverse at finite resolutions and fixed system size. PE, Proposition 2.14 gives convergence of positive expectations at each fixed placement, preserving prescribed cuts and excluding only a null set of new placements. The inverse kernels need not themselves converge: their common absolute majorant bounds the error made by replacing a bounded step-path observable with its limit, and dominated convergence makes that error tend to zero. This is the form used in the applications.

Thresholds, prescribed levels, and specified sides of jumps can be included in every approximation. Sample simultaneous path values and, if needed, weaken the strict gap slightly; at fixed bounded clocks, the scalar PDE derivatives and moments are uniformly continuous under \(L^1\) approximation of the mass profile in common time. Finally, for sheet-only inputs the \(k=0\) formula [eq:H2] passes to the limit with \(\mu^0=2\,dv\). ◻

The row-zero action

Fix an input \(p_t\)-block on the integer tree and condition on its shared scalar prefix. Within a child define \(C_{ac}=\mathbf E_t[\tau_a\tau_c]-m_t^2\), and set \(C_{ac}=0\) when the two leaves are in different children. If the output leaves also lie in different children of the input block, conditional independence gives the following four-spin moment for an input pair \(cd\) on sheet \(t\): \[m_t^4+m_t^2(C_{ac}+C_{ad}+C_{bc}+C_{bd})+C_{ac}C_{bd}+C_{ad}C_{bc}.\] The constant and one-end terms vanish when summed against a row-zero input. The disconnected subtraction in [eq:A8] vanishes for the same reason: the input pair moment is constant on this block and sheet. The two remaining products give \(C\otimes C\); the two orderings of the input pair cancel the factor \(1/2\) in [eq:A8]. If the output does not occupy two different children of this same block, each kernel term instead depends on at most one input end and its sum is zero. This proves that different sheets do not mix.

The conditional matrix \(C\) is diagonal in the child resolutions. Its values at successive overlap levels are \(L_j=\mathbf E_t m_j^2-m_t^2\), with \(L_t=0\) and \(L_{R+1}=1-m_t^2\). The eigenvalue at index \(i\) is \(\sum_{j=i}^{R+1}p_j(L_j-L_{j-1})=\mathbf E_t\chi_{i-1}\), by the conditional source-Hessian formula. Multiplying the two conditional eigenvalues and averaging the prefix gives \(g_t^{il}\). Finally, \(C\) has the same constant row sum in each child. Its tensor action and the child projections therefore preserve the row-zero condition used above.

This calculation transfers to real masses by the coefficient rule above: the entries of \(C\) are differences of pair moments, and products under a shared prefix are four-leaf moments with independent descendant branches. Keeping the complete old-label genealogy in the input variables preserves the formula for arbitrary pair functions.

A colored covariance calculation

To compute the matrix in [eq:H2], it suffices to compute its quadratic form on an \(E_k\) sector. For \(k\ge1\), color half of the children at depth \(k\) in each parent \(+\) and half \(-\), and give every descendant the sign \(\epsilon\) of its child. This leaf function lies in the range of \(E_k\). For real powers the corresponding expression has two child integrals, each raised to \(p_{k-1}/(2p_k)\). For \(k=0\), use the constant \(+\) function.

Multiply each Gaussian increment on a leaf’s path by \(1+\theta\epsilon h_j\) at depth \(j\), including increments inherited from above the colored fork. Choose \(h_j(K_j-K_{j-1})=f_j-f_{j-1}\), where \(f_{-1}=0\), and choose any terminal value \(f_{R+1}\). Strictly separated clocks make this choice possible. Write \(G_j=\sum_{s\le j}h_s^2(K_s-K_{s-1})\), \(G_{-1}=0\), and \(s_t=p_{t+1}-p_t\).

Differentiate the log root partition divided by \(p_0\) twice at zero. Covariance acceleration contributes \[G_{R+1}-\sum_{t\ge k}s_t G_t\mathbf E m_t^2-p_k G_{k-1}\mathbf E m_{k-1}^2\] with the last term omitted for \(k=0\). The remaining curvature is \(\sum_t d_t^k f_t({\bf M}^k f)_t\). Indeed, on sheet \(t\) the first covariance variation is \((\epsilon_a+\epsilon_b)f_t\), and the coefficient of \(\theta^2\) is \(\epsilon_a\epsilon_bG_t\). The spin-square diagonal is constant and contributes no covariance Hessian. On an integer complete tree, ordinary differentiation with \(\sum_a\epsilon_a^2=p_0\) gives the claimed contraction, pattern by pattern in the moments.

For real powers, carry the inherited \(+\) and \(-\) inputs as a two-component Gaussian field until the color separation, where the two child powers are multiplied. Gaussian covariance differentiation and the labeled product rule assign two hits to the same or different children with the same falling-power coefficients as in the integer calculation. Only shared increments enter a covariance. At \(\theta=0\), every normalized transition has the unperturbed scalar law, so the moment factors are the ones required above.

The disconnected product comes from differentiating the root logarithm. Its two pair sums are independent symbolic sums; on a common extended topology, each moment is evaluated on its own restriction. The allocation identity from Section 2 preserves their product. All coefficients therefore agree by the real-mass argument, with singular Gaussian covariance matrices handled by continuity. This identifies the derivative with the contraction before we evaluate it.

Evaluation of the colored Hessian

We now use the increasing real masses and evaluate the derivative under the unperturbed positive scalar law. Continue to write \(A=K_{k-1}\), and temporarily put \(u=m_{k-1}\), \(F=f_{k-1}\), \(p=p_k\), and \(\chi=\chi_{k-1}\). When \(k=0\), take \(A=0\), \(F=0\), and evaluate \(u,\chi\) at the initial state. Let \(\mathcal Z\) be the sum of the Gaussian inputs before color separation, each weighted by \(h_j\), \(j<k\).

For a positive-color child, let \(V_\theta\) be the continuation whose times, including its starting and ending times, have moved to \(K_t+2\theta f_t+\theta^2G_t\). Dots below mean derivatives at \(\theta=0\). For \(k>0\), the first derivatives from the two opposite colors cancel pointwise before the fork. The second derivative of the normalized root value is consequently \[\mathbf E[\ddot V+2\mathcal Z\dot V_z+\mathcal Z^2\chi] .\] For \(k=0\) it is the root continuation derivative with \(\mathcal Z=0\).

Set \(\eta=2\sum_{t\ge k}s_tf_t\delta_{K_t}\), \(J=D_\alpha V_A[\eta]=\sum_{t\ge k}s_tf_t\mathbf E_A m_t^2\). Differentiating the moved times gives (at fixed starting field) \[\begin{split} \dot V&= f_{R+1}-F(\chi+p u^2)-J,\\ \ddot V&=G_{R+1}-G_{k-1}(\chi+p u^2)-\sum_{t\ge k}s_t G_t\mathbf E_A m_t^2 -2\sum_{t\ge k}s_t f_t^2\mathbf E_A\chi_t^2\\ &\qquad+4F^2\partial_{AA} V-4F\partial_A J+ \operatorname{Hess}_{\alpha\alpha}V_A[\eta,\eta]. \end{split}\] To account for the terms, moving a step location changes the profile by minus its jump times the displacement. Its first response is \(\mathbf E_A m_t^2/2\), whose time derivative is \(\mathbf E_A\chi_t^2/2\). These give \(-J\) in the first line and the two sums over \(t\ge k\) in the second. The remaining terms differentiate the starting time, the annealed terminal constant, and the profile twice.

The impulses can also be obtained by integrating over the short intervals swept out by the moving steps. The Hessian kernels below are continuous when the impulse times meet. At fixed bounded clocks, the continuation and its needed derivatives are continuous under \(L^1\) profile perturbations, by their bounded-coefficient diffusion equations. This justifies the impulse limit. Extend the initial constant-mass interval slightly when differentiating the starting time; strict clock separation keeps all other steps outside that interval.

The prefix is sampled under its normalized positive law, rather than under the original Gaussian measure. Multiplying its successive density factors gives a telescoping exponential. Gaussian integration by parts therefore differentiates both the endpoint test and this density. For \(Y\) depending on the endpoint field, the first two identities are \[\begin{split} \mathbf E[\mathcal ZY]&=\mathbf E(FY_z+HY),\\ \mathbf E[(\mathcal Z^2-G_{k-1})Y]&=\mathbf E[F^2Y_{zz}+2FHY_z+(H^2+N')Y],\\ H&=pF u-\sum_{t<k}s_t f_t m_t,\qquad N'=pF^2\chi-\sum_{t<k}s_t f_t^2\chi_t . \end{split}\] Here \(H\) and \(N'\) are the first and second contributions from differentiating the tilted density. The starting-time PDE gives \(4V_{AA}=\chi_{zz}+2p\chi^2+4p u\chi_z+4p^2u^2\chi\) and \(-4J_A=2J_{zz}+4p uJ_z\). Insert these and the two integration-by-parts identities into the second derivative. After subtracting the covariance acceleration displayed above, the contraction becomes \[-p F^2\mathbf E\chi^2-\sum_{t<k}s_t f_t^2\mathbf E(\chi\chi_t) -2\sum_{t\ge k}s_t f_t^2\mathbf E\chi_t^2 +\mathbf E\left[\operatorname{Hess}_{z,\alpha}V_A[\xi,\xi]\right],\] where \(\xi=(pF u+\sum_{t<k}s_t f_t m_t,\eta)\), and the Hessian is evaluated as a bilinear form. For \(t<k\), \(\mu_t^k=s_t+p_k{\bf1}_{t=k-1}\); for \(t\ge k\), \(\mu_t^k=2s_t\). Thus \(\xi=\sum_t\mu_t^k f_t\beta_t\), and the three negative sums are \(-\sum_t\mu_t^k g_t^k f_t^2\). Using \(d_t^k=-\mu_t^k\), this is the quadratic form of [eq:H2]. Polarization recovers the matrix action, which is symmetric with respect to the sheet inner product.

Positivity and weighted kernel estimates

The estimates must retain the mass factors that will cancel the denominators in \(P_k\). First, \(B^k\) is positive semidefinite. The scalar continuation is jointly convex in its starting field and its subsequent mass profile. Indeed, for a fixed bounded adapted control, the controlled terminal field is affine in these variables, the terminal value is convex, and the control cost is linear in the profile. Taking the supremum preserves convexity; completing the square verifies this control representation (PE, Section 5). The same calculation defines the scalar PDE and control formula for every bounded nonnegative profile, without imposing monotonicity. A strictly positive step profile therefore admits both signs of every bounded profile direction, so its full Hessian is nonnegative. Approximate impulses by such directions, and only then average the quadratic form over the prefix. The coefficients \(m_t\), \(t<k\), in \(\beta_t\) are held fixed during this differentiation.

Here are explicit kernels for the impulse limit and its estimates. Put \(u(y)=V_z(y,Z_y)\), \(L^{\,y}(s)=\mathbf E_s[u(y)^2]/2\) for \(s\le y\), and let \(L_z^{\,y}(s)\) denote its derivative with respect to the field at time \(s\), evaluated at \(Z_s\). The subscript \(z\) is a field derivative; the separate symbol \(z'\) below is a time. Order the two evaluation times as \(y\le z'\), and keep the starting time \(A=K_{k-1}\) visible. If both times precede \(A\), the kernel is \(\mathbf E[u(y)u(z')\chi(A)]\). If \(y\le A\le z'\), it is \(\mathbf E[u(y)L_z^{\,z'}(A)]\). If both times follow \(A\), it is \[\mathbf E[u(y)L_z^{\,z'}(y)] +\int_A^y \alpha(s)\mathbf E[L_z^{\,y}(s)L_z^{\,z'}(s)]\,ds .\] These formulas come from the second derivative of the diffusion PDE. For bounded directions \(\eta,\eta'\), its mixed source is \(\eta u D_zD_\alpha V[\eta']+ \eta' u D_zD_\alpha V[\eta]+ \alpha D_zD_\alpha V[\eta]D_zD_\alpha V[\eta']\). Letting the directions concentrate at the two indicated times gives the kernels above. When the two times coincide, the first term has the common limit \(\mathbf E[u(y)^2\chi(y)]\). The formulas also agree at \(y=A\) and \(z'=A\), since \(L_z^{\,A}(A)=u(A)\chi(A)\). Thus no additional boundary term appears when an evaluation time crosses \(A\). For \(k=0\), the initial field is \(Z_0=0\). Field parity, bounded derivatives, and the Brownian moment bound at the earlier time give \(|B^0_{rt}|\lesssim C(1+K_{R+1})^C K_{\min(r,t)}\) when that time is small.

The kernels \(B^k,g^k\) are bounded and Lipschitz in the evaluation times, with polynomial constants. The spatial derivatives needed for this claim are uniformly bounded: the source and attachment rules express \(u,\chi\), their products, and the marked conditional expectations as finite signed spin moments, even after any fixed number of common-field derivatives.

For the kernel with both times after \(A\), varying the later time uses \(\partial_{z'}L^{\,z'}(s)=\mathbf E_s\chi(z')^2/2\). Varying the earlier time also differentiates \(u(y)\) and \(L_z^{\,z'}(y)\); the latter has drift \(-\alpha\chi L_z^{\,z'}\). For an evaluation time before \(A\), condition the later factors on that time and apply Itô’s product rule with \(u(y)\). The bounded derivatives just described control all terms. The same argument applies to \(g^k\), using the drift \(-\alpha\chi^2\) of \(\chi\).

Moving the starting time from \(K_{k-1}\) to \(K_k\) gives the stronger estimate \[\|B^{k+1}-B^k\|_\infty+\|g^{k+1}-g^k\|_\infty \lesssim_{\rm pol} p_k(K_k-K_{k-1}).\] On this interval \(\alpha=p_k\). In the three cases above, the derivative with respect to \(A\) is respectively \(-\alpha\mathbf E[u(y)u(z')\chi(A)^2]\), \(-\alpha\mathbf E[u(y)\chi(A)L_z^{\,z'}(A)]\), and \(-\alpha\mathbf E[L_z^{\,y}(A)L_z^{\,z'}(A)]\). They remain bounded when a marker crosses \(A\), where the formulas match. Integration over the interval supplies the factor \(p_k(K_k-K_{k-1})\).

For the child kernels, set \(a_i=\mathbf E_t\chi_{i-1}\), and write \(\chi(s)=V_{zz}(s,Z_s)\) for the continuous scalar-time process. Its supermartingale drift gives \[a_i-a_{i+1} =p_i\int_{K_{i-1}}^{K_i}\mathbf E_t\chi(s)^2\,ds.\] Thus a mixed difference of \(g_t^{il}=\mathbf E[a_i a_l]\) is the expectation of the product of the two differences. This proves the stated single and mixed bounds. The same product bound holds for \((1-b_tg_t^{il})^{-1}\), as used above: interpolate the two factors \(a_i,a_l\) linearly and differentiate \(F(z)=(1-b_tz)^{-1}\) twice. The interpolated kernel is bounded above by \(\mathbf E\chi_t^2\), so the gap bounds both derivatives of \(F\). This includes \(i=l\); flat clock intervals follow by continuity. Positivity and the same supermartingale drift give \(0\le g_t^k,g_t^{il}\le\mathbf E\chi_t^2\), completing the scalar estimates. ◻

Second moments in radius and in mass.

For the second bound, over each box we have \[\int\mathbf EW\,dr\lesssim u^3 A^3d_0,\qquad \int\mathbf E\operatorname{tr}C_v^2\,dr\lesssim u^2 A^3d_0.\] These estimates follow from [eq:A1], the \(S\) oscillation bound, and constant removal strength. Use earlier and later windows of length \(h=t'd_0\). The sliding variance estimate proved at [eq:C10] is now \[\int_{J_h}\operatorname{Var}(\overline L^{\,\pm}_{r,h})\,dr \lesssim\frac1{\kappa_jh}\int_{J^+}\mathbb EW\,dr.\] Here \(J_h\) omits endpoint strips of width \(h\), so the windows stay inside \(J\); \(J^+\) merely provides the available integral upper bound. Discard those strips with the pointwise moment bound. Apply the absorption calculation at [eq:C10] on height scale \(d_0\), now with trace term \({\cal T}\lesssim A^3/L^2\) and variance term \((A/L)^3/(\kappa_jt')\). It gives \[\int D^2dr/d_0^3\lesssim t'+(A/L)^3/(\kappa_j t')+A^3/L^2.\] Take \(t'=(A/L)^{3/2}/\sqrt{\kappa_j}\le\lambda^{0.18}\). This proves the radius estimate and also the mass estimate on the floor, whose density is bounded above. To treat the remaining mass, enlarge endpoint caps to whole caps. Total mass and \(S\) oscillation remain \(O(d_0)\). Equation [eq:C6] and PE (20), with \(p=0\), give the same integrated \(W\) and trace bounds on caps and contact.

Order this measure by the deterministic coordinate \(s(v)=\int^v\mathbf1_{\rm nonfloor}(t)\,dt\), and take windows of \(s\)-length \(h=t'd_0\). The optimizer and its floor set were fixed before sampling; hence the window indicators are deterministic weights in [eq:A1]. Fubini now bounds the integrated centered variance by \(h^{-1}\int\mathbb EW\,dv\), with no floor-density denominator. The ordered identity and monotonicity control the noise and mean changes as in radius. If the whole nonfloor measure has length less than two windows, the endpoint-strip bound alone suffices. This proves the second assertion of [eq:C15] in both measures.

The alias error.

For the final bound of [eq:C15], it remains by [eq:A1]’s covariance convention to bound \(\mathbf E(X_i\cdot(y-X_i))^2\). Telescope \(X_i\) into past layers ending at dyadic distances from the current radius, down to \(x^{40}\). For an increment ending at distance \(t'\) after starting at distance \(\asymp t'\), with \(x^{40}\lesssim t'\ll u\), [eq:A3] costs for its squared product with \(y-X_i\) at most \(O_\circ(x^6/(u^2 C_0 t'))\) times the layer’s squared norm in expectation. That norm has box average \(O(t'\lambda^{-\rho})\). In radius this follows for \(t'\le d_0\) by sliding the \(S\) variation, for \(d_0\le t'\ll b_{\rm buf}\) pointwise, and otherwise by [eq:C8].

In mass, the floor uses the radius estimate, and around a nonfloor evaluation the \(S\) oscillation at distances \(O(t')\) within the buffered patch is \(O(t')\), by [eq:C6] and the lower potential bound \(-O(C_0/M_j^2)\), as at the good cut. The unprojected last past increment thus has negligible squared norm on average by the same facts, and the initial anchor, projected from fixed comparable distance within constant strength, costs \(O_\circ(u^2 A^3)\). Summing proves the last assertion. ◻

Two insertions pinned at the observed fork.

It remains to treat both degree-two pairs pinned. Write each centered overlap as \(m_i+\eta_{\rm pair}\) with \(m_i=|X_i|^2-S_i\) on the common prefix and \(\eta_{\rm pair}=Q_i-|X_i|^2\). Single \(\eta\) terms vanish by conditioning.

For two \(\eta\)’s the coefficients individually gain \(O(u)\): either a third child has entered at the fork (coefficient \(O(u)\)), or both pairs run across the two occupied children. In the latter case the total number of scalar spins, including old spins, in each child is odd; at least one scalar moment factor has a sign cancellation on turning off scalar revelation through this cut (keep later cumulative clock via a jump after the observation, or take its immediate right limit). Indeed the two children then have independent sign symmetry. If using cuts immediately to the right for a fixed placement with the same children, they separate from all further distinct fork vertices eventually; the restoring bound by PE (2) is uniform \(O(u)\). Thus [eq:C15]’s final bound and [eq:C18] suffice.

The factor \(\mathbb E m_i^2\) is common to all pinned topologies, so extract it before summing coefficients. We claim that the remaining signed scalar coefficient, before the common \((b-C)^2\), is \(\Gamma_s''(K_s(i))/2\) at almost every buffered cut.

To identify it, compress a small neighborhood of the observed mass into a clock jump at that mass, retaining the observation at its original intermediate clock value. At a continuity cut there is room to move this value in both directions: \(K_s\) is strictly increasing locally because its field contains \(C_0r\). On an already extended old tree, increase the common subincrement at the distinguished vertex by variance \(\epsilon\), and decrease the following subincrement in every occupied child by the same amount. The covariance direction is then one for pairs entering different children of that vertex and zero for same-child pairs, diagonals, and pairs outside its subtree. Total leaf variances and all other split covariances stay fixed. Thus this is exactly the pinned direction, even when earlier insertions have occupied more than two children.

In the quotient expansion of [eq:A5], the degree-two scalar terms are \[N_2-N_1\Psi_1+N_0(\Psi_1^2-\Psi_2).\] In every term with positive denominator degree, allocate one whole such \(\Psi\) moment last. Previously placed factors pull back from the induced old genealogy, and the extracted prefix expectation is independent of placement. The complete signed sum of that denominator moment is therefore a covariance derivative of the constant-one observation on this genealogy, hence zero. This cancellation requires the full signed sum; it is not a row-sum identity for a nonconstant restricted test. Coincidences within the permitted children remain part of that sum.

Only \(N_2\) remains. Its coefficient includes the Taylor factor \(1/2!\). The two subincrements at the same positive mass compose into the original scalar transition, so moving their intervening split changes only the evaluation time of \(\Gamma_s\). The coefficient is consequently \(\Gamma_s''/2\). This calculation introduces no additional split atom: a new attachment at a same-mass subdivision without old divergence has zero coefficient, because that edge has zero length and one continuing child. A different vertex with the same numerical mass likewise creates no atom at the observed fork.

Finally shrink the compressed neighborhood. Bounded-clock path limits with the cut retained pass the aggregated coefficients to the original path. Other freely varying fork masses in the shrinking interval have vanishing absolute weight. The scalar identity \(\Gamma_s''=\mathbb EV_{zzz}^2-2\alpha_t\mathbb EV_{zz}^3\) uses exactly the fork mass at the intermediate observation. At the limit this remains its mass, since \(K_s\) grows strictly locally and the spatial derivatives and path moments converge. Null contact and jump exceptions do not affect either box integral. This proves the claimed coefficient identity.

By the drift estimates at [eq:C14], \(\Gamma_s''(K_s(i))=O(d_0+\lambda u)\). The \(m_i^2\) average by [eq:C15] is at most \(O(d_0^2\lambda^{0.14})\); their product suffices since \(\lambda^{0.14}L(L+\lambda)\ll G_1\). This completes [eq:C19].

In particular the error budget remains relative to the removal strength: \[\frac{u^3P_1}{d_0h_1^2} =\frac{G_1}{\max(L^2,\lambda)} \asymp\min(1,R_n)\] with the notation used next. This observation will be needed when \(R_n\) tends to zero; an absolute \(o(1)\) error would not suffice. ◻

Unbounded rescaled removal.

Suppose first \(R_n\to\infty\). Equation [eq:C20] gives \(Y_n=o(1)\) in box-averaged radius, hence uniformly on compacts (a violation persists over a fraction of a box on one side, by monotonicity of \(S\)). Then \(\widetilde m_n-m_n\to0\) in local \(L^1\): \(\widetilde r(\alpha(r))-r=o(d_0)\), and by Fubini the area between distributions on each such interval of length \(O(h_1)\) costs \(o(d_0)\) times \(O(h_1)\) mass (crossings are local by monotonicity). Write \[J_n(z)=\Gamma_*'(K_*(\alpha(r_0)))\,{\cal H}(r)/(2d_0 h_1^3),\qquad \ell_1=\lim c_q D_n .\] The variational sign gives \(J_nm_n'\le0\). Equation [eq:C20] gives \(J_n'=-R_n(0)Y_n=-g_n-\ell_1m_n+o(1)\), where the error tends to zero in the moving radius-box integrals. Nearby nonfloor calibrations give \(J_n=O(\theta)\), with a common local bound; the normalized \(M_j^{-2}\) potential error is \(o(\theta)\). Take a locally uniform limit \(J\) and a weak-star limit \(m_n\rightharpoonup m_\infty\). Then \(J'=-g-\ell_1m_\infty\) and \(g''=-A_*m_\infty\).

For clarity, the energy estimate follows by two integrations by parts. On a fixed interval, integrate \(J_nm_n'\le0\), substitute the equation for \(J_n'\), and use \(m_n=-g_n''/A_*+o(1)\). This gives \[0\ge[J_nm_n]+\int g_nm_n+\ell_1\int m_n^2+o(1), \qquad \int g_nm_n =-[g_ng_n'/A_*]+\int g_n'^2/A_*+o(1).\] Consequently \[\int (g_n'^2/A_*+\ell_1m_n^2)\le [g_n g_n'/A_* -J_n m_n]_{\rm left}^{\rm right}+o(1).\] All boundary factors are uniformly bounded, independently of the fixed interval. Passing to the limit and then enlarging the interval first shows that \(g'\in L^2(\mathbb R)\). Bounded curvature makes \(g'\) uniformly continuous, hence it tends to zero at both ends. If \(\theta\to0\), then \(J=0\), so both boundary terms vanish as the endpoints tend to infinity. If \(\theta\) has positive limit, the measure \(d(c_\beta z+(\lim\theta)m_\infty)\) has support unbounded in both directions. Nonfloor approximants give \(J=0\) on that support, because the floor density vanishes. Send the endpoints to infinity through its support; again both boundary terms vanish. The energy estimate now gives \(g'=0\), and \(g''=-A_*m_\infty\) gives \(m_\infty=0\). Finally \(J'=-g\) and boundedness of \(J\) give \(g=0\). For positive limiting \(\theta\), monotonicity of \(\alpha\) then gives the pointwise contradiction \(|m_n(0)|=1\).

Unbounded removal: the shrinking-box subcase.

If \(\theta\to0\), we have \(\ell_1>0\) since \(h_1^2=\lambda u^2\). Put \(K_n(t)=J_n(\theta t)/\theta,\ M_n(t)=m_n(\theta t)\). They are bounded on compacts with uniform limiting bounds, \(K_n'=-\ell_1 M_n+o_{L^1_{\rm loc}}(1)\) by the moving box estimate and \(g=0\), \(K_n M_n'\le0\). \(M_n\) converge on subselection in local \(L^1\) and at continuity points by monotonicity with the deterministic trend added. Now \(\ell_1\int M_n^2\le-[K_n M_n]+o(1)\) on fixed intervals. The limit thus has finite global squared norm; sending endpoints to infinities through continuity points where it tends to zero makes that norm zero. Again monotonicity gives the pointwise contradiction.

Bounded rescaled removal, including a vanishing limit.

Suppose now that \(R_n\) is bounded. Then \(\theta=1\). After adding their deterministic trends, \(m_n,Y_n\) are monotone; subselection therefore gives almost-everywhere and local \(L^1\) limits. The transport estimate also gives \(\widetilde m_n-m_n\to0\) in local \(L^1\). Indeed on nonfloor points the displacement \(\widetilde r(\alpha(r))-r\) is \(o(d_0)\); floor crossings travel only \(O(d_0)\) and have density \(o(1)\). Fubini bounds the area between the inverse distributions on a compact interval by \(o(d_0h_1)\). It follows that \(g''=-A_*m_\infty\). The normalized potential \(\mathcal H(r)/(2C_0d_0h_1)\) has derivative \(-Y_n\). Its lower bound and nearby nonfloor references give a bounded nonnegative limiting function \(\bar J\).

We next calibrate at the support of \(\mu=d(c_\beta z+m_\infty)\). The measures \(\mu_n=d\alpha/h_1\) in this coordinate converge vaguely to \(\mu\). Every fixed neighborhood of a support point has positive limiting mass. Its floor mass is \(o(1)\), while the mass integral of \(|e_n|/R_n\) tends to zero. Hence it contains a nonfloor point with \(e_n/R_n\to0\). At this point \(Y_n=o(1)\) and \(D_n/R_n\to0\), so [eq:C20] gives \(g_n(\varphi_n)=o(R_n)\). Shrink the fixed neighborhoods and choose a diagonal subsequence. This produces approximants to each support endpoint used below. At a contact the potential is zero; in a cap its nearest zero is within \(M_j^{-1}\), whose normalized error vanishes. We conclude that \(\bar J=g=0\) on \(\operatorname{supp}\mu\), even if \(R_n\to0\).

A floor interval can still have positive rescaled radius length; only its mass has vanished. We therefore retain the separate radius estimate for \(e_n/R_n\) on any such support gap. After a further subsequence, for almost every radius coordinate \(z\), \[s(z)=\lim\varphi_n(z)=z+Y(z),\qquad g(s(z))=(\lim R_n)Y(z),\qquad \bar J'=-Y.\] The function \(s\) is nondecreasing. On a support gap \((a',b')\), monotonicity and the endpoint approximants give \(s\in[a',b']\). The gap is bounded, and \(g''\) is affine there with slope \(A_*c_\beta>0\).

Suppose \(g'(a')>0\). If \(\lim R_n>0\), then \(g(s)=(\lim R_n)(s-z)\) forces \(s(z)>z\) almost everywhere just to the right of \(a'\): this holds if \(s\) is near \(a'\), where \(g\) is positive to its right, and also if \(s\) is farther inside the gap. Thus \(\bar J'=-Y<0\), contradicting \(\bar J(a')=0\) and \(\bar J\ge0\).

If \(R_n\to0\), then \(g(s)=0\), and the sign requires the relative error. Fix an almost-everywhere point \(z>a'\) where the radius error divided by \(R_n\) tends to zero. If \(s(z)=a'\), take nonfloor approximants \(z_n^a\to a'\) as above. Eventually \(z_n^a<z\), so \(\varphi_n(z_n^a)\le\varphi_n(z)\). Both images lie near \(a'\), where \(g_n'>0\). Since the value at the approximant is \(o(R_n)\) and \(R_n\) is asymptotically constant on compact intervals, \[\liminf_n\frac{g_n(\varphi_n(z))}{R_n(z)}\ge0.\] Divide [eq:C20] by \(R_n\). It gives \(Y(z)\ge0\), contrary to \(Y(z)=a'-z<0\). Every other zero of \(g\) is bounded away from \(a'\) in this neighborhood, and hence gives \(s(z)>z\). Again \(Y>0\) almost everywhere just to the right, contradicting the potential sign. Reflection excludes \(g'(b')>0\) in both cases.

But \(g(a')=g(b')=0\) and the affine curvature imply \[g'(a')+g'(b')=A_*c_\beta(b'-a')^2/6>0.\] At least one endpoint derivative must be positive. Thus no support gap exists.

Finally, the boundedness of \(m_\infty\) makes the support of \(d(c_\beta z+m_\infty)\) unbounded in both directions. With no gaps it is all of \(\mathbb R\), so \(g=0\) and \(m_\infty=0\). In each regime where the normalized discrepancies converge locally in \(L^1\) to zero, the contradiction at the originally selected point is pointwise: monotonicity of the inverse distribution, after adding back its deterministic trend, gives for fixed continuity points \(\pm h\) \[-C h+o(1)\le m_n(0)\le C h+o(1).\] Here one may take \(C=c_\beta/\theta_\infty\) when \(\theta\to\theta_\infty>0\), and \(C=c_\beta\) in the bounded removal case \(\theta=1\). Letting \(h\downarrow0\) contradicts \(|m_n(0)|=1\). For the second-rescaling sequence the identical argument is applied to \(M_n\), whose deterministic trend has slope tending to \(c_\beta\). No pointwise relative error at the selected point is used.

This proves [eq:C12], completing the finite induction. ◻

The intrinsic width.

The map \(l\mapsto l^2\int_{z-l}^{z+l}(G_{s,\mathrm{refl}}+x^K)ds\) is strictly increasing. Ball inclusion therefore compares its solution at neighboring centers. Together with the cap \(b_\epsilon r\), this proves the stated quasi-Lipschitz bound for \(H_0\) and comparability within a small fraction of \(H_0\). Boundedness of \(G\) gives \[H_0\gtrsim\min(b_\epsilon r,U^2/r).\] The adjacent-interval comparison at this radius also gives \(G_zH_0^3\lesssim U^6/r^3\), up to an arbitrarily small additive error. Since \(G\) is bounded, the interpolation \[G_zH_0^2=(G_zH_0^3)^{2/3}G_z^{1/3}\lesssim U^4/r^2\] proves the squared width budget after integration over a band of \(z\)-length \(O(r)\) and division by \(r^3\).

The deviation cone.

For each source point \(s\), the function \([\min(b_\epsilon r(s),|f(v(s))|/Z^2)-A|s-z|]_+\) is \(A\)-Lipschitz. Their supremum is therefore Lipschitz too. At its source, a height of order \(l\) implies \(|f|\gtrsim Z^2l\). If \(f\) is positive, monotonicity of \(S\) and Lipschitz continuity of \(q(v(z))\) preserve half this deviation on a forward interval of length \(cl\); for a negative deviation use the backward interval. Thus each source of height \(l\) supplies a one-sided interval \(J_s\) with \[Z^4l^2\int_{J_s}G_z\,dz \lesssim \int_{J_s}f(v(z))^2G_z\,dz.\] Its cone is supported within \(O(l)\) of \(s\). The adjacent-interval comparison bounds the mass of that support by the mass of \(J_s\), with negligible additive error. At each dyadic height \(l\ge x^K\), choose separated source points to cover the cone superlevel set. The corresponding intervals have bounded overlap at that height. Summing the preceding inequality, and then the logarithmically many heights, gives \[\int_{\rm band}G_zH_f(z)^2\,dz \lesssim_\diamond Z^{-4}\int_{\rm enlarged\ band}f(v)^2\,dv \lesssim_\diamond U^4/r.\] Sources of height below \(x^K\) contribute negligibly. The lower boundary causes no difficulty because \(l\le b_\epsilon r\). If a positive source’s forward interval reaches the upper active endpoint, its deviation persists through the plateau. The macroscopic version of [eq:D1] then gives \(Z^4l^2\lesssim Y^4\), or \(l\lesssim U^2\); these sources already obey the same macroscopic squared width budget.

The plateau and its active edge.

The reference \(q\) is constant on the plateau. A positive deviation of \(S-q\) persists towards its upper end; a negative deviation persists towards its lower end. Integrate [eq:D1] over the corresponding endpoint strip, whose mass is comparable to the distance to that endpoint. Away from the capped strips, the result is \(|f|\lesssim Z^2\mathfrak h\). Inside the upper \(d_+\) strip, \(\mathfrak h\asymp1\) and the bound follows from boundedness of \(S,q\). For a negative deviation inside the lower \(d_-\) strip, the backward interval must instead use the active side. The last active \(z\)-interval of length \(E=E_b\) has mass comparable to \(d_-=U^4/E^2\). Either \(|f|=O(Z^2E)\) already, or the Lipschitz variation of \(q\) over that interval is too small to remove half the negative deviation. In the latter case [eq:D1] gives \[|f|^2\frac{U^4}{E^2}\lesssim Y^4, \quad\hbox{hence}\quad |f|\lesssim Z^2E,\] which is the required lower-edge bound.

The explicit plateau formula also gives \(\int_{\rm plateau}\mathfrak h^2\,dv\lesssim_\circ U^4\) by integrating \(U^4/(d+d_\pm)\) from each endpoint. The first extra active function has height \(O(E)\) on a region of mass \(O(U^4/E^2)\), so its squared cost is \(O(U^4)\). The second has height at most \(2\min(E,b_\epsilon)\); its part inside the last \(E\) costs the same. For its additional layer of length \(2\min(E,b_\epsilon)\), compare adjacent active intervals at that length. Their mass is controlled by the preceding last-\(E\) region, giving again \(O_\diamond(U^4)\). These two functions also majorize the active ramp defect and its parameter derivative divided by \(\lambda\), with at most a \(b_\epsilon^{-1}\) loss. The bounds in [eq:P6]–[eq:P8] then give the claimed defect estimates. Below \(Y\), those estimates use only the low and mask bounds from [eq:P6], [eq:P8].

Matching the two coordinates at the join.

Suppose first that \(E\lesssim b_\epsilon\), and let \(d_*=z_*-z\) on the active side. We claim, for \(d_*\le Cb_\epsilon\), that \[|f(v(z))|/Z^2\lesssim E+d_*.\] For \(d_*\lesssim E\), use the last active interval of length \(E\) or the adjacent plateau, in the direction in which the deviation persists. The preceding mass calculation proves the claim. For \(d_*\gtrsim E\), an interval of length comparable to \(d_*\) fits on the active side in either direction. Unless the claimed bound already holds, the deviation persists on that interval. Adjacent interval comparison bounds below its mass by that of the last active length \(E\), up to negligible error. Applying [eq:D1] as above would then give \(|f|\lesssim Z^2E\), proving the claim in this case too.

With \(A\) sufficiently large, sources of the cone obeying this linear bound imply \(H_f(z)\lesssim E+d_*\). The same bound holds for \(H_0\): a ball of radius \(C(E+d_*)\) contains the last active length \(E\), and its squared radius times that mass is at least a constant times \(U^4\), which exceeds the intrinsic target \(U^6/r^3\) on this macroscopic band. When \(E>b_\epsilon\), the corresponding upper bounds follow from the caps in \(H_0,H_f\). The explicit endpoint functions provide the lower bound near the join. Thus at distances much smaller than \(E\), both the active and plateau widths are comparable to \(E\). Their separate quasi-Lipschitz bounds consequently remain valid across the join.

Windows and tiles.

Consider first an active point with \(d_*\ge E\) for which \(Z^2\mathfrak h<b_\epsilon r\) up to the fixed threshold in (ii). The intrinsic cap is then inactive. Its defining equation and \(\mathfrak h\ge H_0\) give \[\mathfrak h^2\int_{z-C\mathfrak h}^{z+C\mathfrak h}G_s\,ds \gtrsim U^6/r^3,\] with negligible additive error. Choose a past interval and a future interval, each a small fixed fraction of \(\mathfrak h\) and within \(O(\mathfrak h)\) of \(z\). Since \(\mathfrak h\lesssim E+d_*\lesssim d_*\), sufficiently small fixed fractions fit wholly on the active side. Near zero the cap \(\mathfrak h\lesssim b_\epsilon r\) also leaves room for the past window. Adjacent-interval comparison gives, for each chosen window, \(\sup G\lesssim G_*\) and \(G_*\mathfrak h^3\gtrsim U^6/r^3\), which is [eq:D4].

For active points with \(0\le d_*\le E\), the endpoint functions and the nontrivial accuracy requirement force \(E\lesssim b_\epsilon\) and \(\mathfrak h\asymp E\). The plateau width is likewise comparable to \(E\) in a sufficiently small fixed \(E\)-neighborhood of the join. The mass of the last active interval of length \(E\) is comparable to \(U^4/E^2\), so comparable active windows there have average density \(G_*\asymp U^4/E^3\). The adjoining plateau windows in that small neighborhood have the same bound. Hence \(G_*\mathfrak h^3\asymp U^4\gtrsim U^6\), as required on this macroscopic band. Outside that neighborhood on the plateau, the explicit width formula allows small fixed fractional windows wholly within the plateau and makes \(G\) comparable on each of them. The identity \(G\mathfrak h^3\asymp U^4\) gives the same conclusion. Finally the terminal \(d_+\) strip has macroscopic \(z\)-length and width comparable to one. A point requiring nontrivial accuracy must precede that strip, leaving the fixed forward length needed for longer projection layers.

Tiling can for instance use equal bounded small steps in the integral of \(dz/\mathfrak h\) on each band (resizing the steps by bounded factors to fit). Width comparability holds by the local property. On an active tile longer than \(b_\epsilon\), subdivide into \(b_\epsilon\)-size pieces for [eq:D5]; otherwise use the adjacent-interval bound directly. For a tile \(I\) of length comparable to \(\mathfrak h\), use \(S=q+f\), the Lipschitz bound on \(q(v(z))\), and \(|f|\lesssim b_\epsilon^{-1}e\) on its enlargement. This gives \[\Delta_I S\lesssim b_\epsilon^{-1}\min(r,Z^2|I|).\] Finally, with \(a_I=\sup_I e/r\), width comparability and \(e=\min(r,Z^2\mathfrak h)\) imply \[a_I^2\lesssim \frac{Z^2|I|}{r},\qquad \sum_I a_I^2\lesssim Z^2,\] since the disjoint tiles fill an interval of length \(O(r)\). The same argument applies to fixed comparable enlargements. Combining the squared budgets for the four defining widths also gives \[\int e(v)^2\,dv/r \le Z^4\int\mathfrak h(z)^2G_z\,dz/r \lesssim_\diamond Y^4/r^2=m_r^2.\] The lower intrinsic width yields \(e\gtrsim b_\epsilon Y^2/r\) for \(r\gtrsim Y\), including when either cap is active.

The low-region construction.

Choose a small fixed working interval and a larger fixed buffer interval on which the size-path inverse is regular. Use \(z=v\) there, so \(G=1\) and [eq:D3] holds. Define \[\mathfrak h=\max\{\min(b_\epsilon r,Y^2/r),H_f\}, \qquad e=\min(r,\mathfrak h),\] where the cone now uses \(|f|\) in place of \(|f|/Z^2\). The preceding cone argument gives squared width budget \((Y/r)^4\) on the working interval. The intrinsic lower width is now \(\min(b_\epsilon r,Y^2/r)\), which is at least the direct-path lower width because \(Y\ge U\). Consequently the window bound [eq:D4] remains valid. The pointwise and variation bounds use \(e=\min(r,\mathfrak h)\), and the tile sum is now \(\sum_I\sup_I(e/r)^2\lesssim1\); these are precisely the later bounds in which \(Z_h=1\).

If more room is needed for buffers, start a fixed Lipschitz ramp only beyond the working interval, increasing \(\mathfrak h\) to a macroscopic value before the end of the buffer interval. No integrated width estimate is used on that auxiliary ramp. Longer layers stop after a fixed small macroscopic length within the regular region. The defect bounds are the low size-path bounds of [eq:P8].

Measurability of the construction.

All windows can be chosen measurably and deterministically conditional on the path parameters. For example, fix the fractional lengths and casewise windows constructed above, using the prescribed comparable band enlargements for buffers, and tile by equal increments of \(\int dz/\mathfrak h(z)\). The cone supremum may be formed over a countable dense set together with the specified one-sided values of the monotone function \(S\). Its value is unchanged, since \(q\) is continuous and the one-sided limits of \(S\) exist. The implicit intrinsic radius is the monotone solution of its displayed integral equation. These constructions are measurable functions of the deterministic path data. In particular, none of the windows is selected after observing a spin sample. ◻

Raw-moment bootstrap.

Fix a large even exponent \(p\) and set \[B_0=\max\left(1,\sup_v\frac{\||X_v|^2\|_p}{v+Y}\right).\] This number is finite at each fixed size because \(|X_v|\le1\) and \(Y>0\). At a small scale \(r\gtrsim Y\) in the regular inverse region, let \(J\) be a fork window of \(z\)-length comparable to \(r\). We claim \[ \left\|\int_J W_z\,dz/r\right\|_{p/2} \lesssim_{p,\circ}(z_r^0)^2r^3B_0. \tag{D6} \]

Use an initial block of arbitrarily small inverse-polynomial length, then blocks starting at doubled distances from the fork, and finally a terminal residual. Stop the doubling at a fixed small macroscopic distance, with a full preceding interval still inside the regular region. Consider a block pair other than two initial blocks, and let \(lr\) be its later starting distance. If \(l\lesssim1\), the projection gap has mass at least \(cr\) and field length at least \(c\lambda_1r^3l\). Its squared projection coefficient, before the mass weight, is consequently \(O_\circ(x^6/(\lambda_1r^5l))\).

The earlier squared anchor has the integral bound \[\left\|\int_J|A_z|^2\,dz/r\right\|_{p/2} \le C_p lB_0r.\] Indeed, (11) integrates translated grids of disjoint increments, and the raw bootstrap bounds the terminal squared norm on the enlarged window by \(CB_0r\). If that anchor belongs to the independently resampled continuation, apply (12) to its nonnegative squared increment to average it back to the fork. Thus the contribution from this block pair is at most \[\underbrace{Cr}_{\text{mass weight}}\, \underbrace{C_{p,\circ}\frac{x^6}{\lambda_1r^5l}} _{\text{squared projection coefficient}}\, \underbrace{C_p lB_0r}_{\text{integrated squared anchor}} \lesssim_{p,\circ}(z_r^0)^2r^3B_0.\] The two initial blocks instead use the sliding estimate without projection. Choose their length so small that the resulting term is negligible after the polynomial prefactors.

For a later starting distance comparable to \(r'\gtrsim r\), use a full gap at masses comparable to \(r'\), with field length at least \(c\lambda_1(r')^3\). This includes a macroscopic \(r'\) for the terminal block. The mass weight, squared projection coefficient, and raw squared anchor now cost respectively \[Cr',\qquad C_{p,\circ}\frac{x^6}{\lambda_1(r')^5},\qquad C_pB_0r'.\] Their product is \(O_{p,\circ}((z_{r'}^0)^2(r')^3B_0)\), no larger than the right side of [eq:D6]. The normalized fork window has bounded length. Summing the logarithmically many block pairs in (10) proves [eq:D6]. The sliding and conditional integration bounds are uniform in the inserted grid sizes, so this argument uses no size-uniform path approximation rate and no derivative of \(S\).

Let \(L_J=|J|^{-1}\int_JL_u\,du\) for a future mass window \(J\) of width comparable to \(r\). The future bracket identity in Lemma 25 gives \(|X_v|^2\le2\mathbf E_vL_J\), while \(\mathbf EL_J=O(r)\) by the deterministic mean bound. Equations [eq:A1] and [eq:D6] give \[\|L_J-\mathbf EL_J\|_p^2 \lesssim r^{-1}\left\|\int_JW_u\,du/r\right\|_{p/2} \lesssim_{p,\circ}(z_r^0)^2r^2B_0.\] Conditional Jensen therefore yields \(\||X_v|^2\|_p\lesssim_pr(1+z_r^0\sqrt{B_0})\). For \(v\lesssim Y\) use a future window at scale \(Y\); at macroscopic masses use \(|X_v|\le1\). Taking the supremum in the definition of \(B_0\) now gives \(B_0\le C_p(1+g_0\sqrt{B_0})\), including the harmless subpower factors. The fixed power saving in \(g_0\) absorbs these factors, and this quadratic inequality bounds \(B_0\) uniformly.

For \(L,Q\), project endpoints into a future interval at comparable scale \(v+Y\) and then telescope both branches in future doubled layers with terminal block, as above. Each tail product uses [eq:A3] on the later layer (with comparable preceding clock fitting strictly after the fork); the opposite anchor has norm \(O_p(\sqrt{r'})\) at worst at scale \(r'\), so these tails cost \(O_{p,\circ}(z_{r'}^0 r')\). For \(L\) the anchor \(X_v\) already precedes the gap. This gives the first line of [eq:D2]. It suffices here too to telescope at small scales.

Refined-moment bootstrap.

For this argument write \(Z_h=Z\) on direct paths and \(Z_h=1\) in the low-region size-path variant. Let \(B_1\ge1\) be the least constant for which \[\begin{equation*} \||X_{z'}-X_z|^2\|_p \le B_1Z_h^2(|z-z'|+\mathfrak h(z)) \end{equation*}\] holds for comparable nearby cuts in the bands \(r\gtrsim Y\) and their buffers. The supremum defining \(B_1\) is finite at fixed size because \(\mathfrak h\ge x^C\); take it simultaneously over the bands to avoid artificial interior boundaries. At \(z\asymp Y\) the raw bound already suffices with a \(b_\epsilon^{-1}\) loss, as it does on the large-width auxiliary ramp. Cuts whose separation is not a small fraction of their scale can likewise use the raw bound.

Let \(J\) be one of the windows in [eq:D4] near \(z_0\), and put \(h_0=\mathfrak h(z_0)/r\), so \(|J|\asymp rh_0\). We claim \[\left\|\int_J W_z\,dz/r\right\|_{p/2} \lesssim_{p,\diamond}(z_r^0)^2r^3h_0B_1Z_h^2.\] For a block pair whose later starting distance is \(lr\) with \(l\ll1\), the same mass and projection factors as above apply. The integrated earlier anchor satisfies, in both ranges of \(l\), \[\left\|\int_J|A_z|^2\,dz/r\right\|_{p/2} \le C_p lB_1Z_h^2rh_0.\] When \(l\ge h_0\), the bootstrap and quasi-Lipschitz width bound give \(\||A_z|^2\|_{p/2}\le CB_1Z_h^2lr\) pointwise; integration contributes \(|J|/r\asymp h_0\). When \(l<h_0\), apply (11) after subtracting the initial martingale value on an enlargement of \(J\). Its terminal squared increment is bounded by \(CB_1Z_h^2rh_0\), and the sliding estimate contributes \(l\). The conditional integration inequality handles a resampled anchor in exactly the same way as in the raw argument. Hence each such pair costs \[\underbrace{Cr}_{\text{mass weight}}\, \underbrace{C_{p,\diamond}\frac{x^6}{\lambda_1r^5l}} _{\text{squared projection coefficient}}\, \underbrace{C_p lB_1Z_h^2rh_0}_{\text{integrated squared anchor}} \lesssim_{p,\diamond}(z_r^0)^2r^3h_0B_1Z_h^2.\] The pair of initial blocks again has negligible cost. Longer layers use the already closed raw bound. Their product at scale \(r'\gtrsim r\) is \(O_{p,\diamond}((z_{r'}^0)^2(r')^3)\) pointwise, and the fork window contributes \(h_0\). Since \(B_1Z_h^2\ge1\) and \((z_{r'}^0)^2(r')^3\le(z_r^0)^2r^3\), these layers obey the same bound. The forward length in Lemma 42 supplies their full projection gaps, including the terminal one. Summing the blocks proves the claim.

The mass of this window is comparable to \(G_*rh_0\), and \(\sup G\lesssim G_*\). Apply [eq:A1] with the constant weight \(1/(G_*rh_0)\) on the window. The squared centered norm of its mass average is at most \[\begin{equation*} C_{p,\diamond} \frac{G_*r\,(z_r^0)^2r^3h_0B_1Z_h^2}{(G_*rh_0)^2} = C_{p,\diamond}\frac{(z_r^0)^2r^2B_1Z_h^2}{G_*h_0} \lesssim_{p,\diamond}r^2B_1Z_h^2h_0^2. \end{equation*}\] In the last step we used \(z_r^0=g_0(U/r)^3\) and \(G_*h_0^3\gtrsim(U/r)^6\) from [eq:D4]. We now apply the bracket identities of Lemma 25. Earlier and later averages bracket \(|X_z|^2\). To see also how they control increments, let \(s\le z\le z'\) and use \[2\mathbf E_zL_s=|X_z|^2-|X_z-X_s|^2.\] Average \(s\) over one common past buffer before \(z\). A future buffer after each of \(z,z'\) bounds its squared norm from above; the common past identity then bounds the averages of \(|X_z-X_s|^2\) and \(|X_{z'}-X_s|^2\). For every such \(s\), \[|X_{z'}-X_z|^2 \le2|X_{z'}-X_s|^2+2|X_z-X_s|^2.\] The means of these buffer differences cost at most \(Cb_\epsilon^{-1}Z_h^2(|z'-z|+\mathfrak h(z))\) by the deterministic variation bound. Their centered parts have the bound just proved, with conditional Jensen where needed. Consequently \[\||X_{z'}-X_z|^2\|_p \le C_{p,\diamond}(1+\sqrt{B_1}) Z_h^2(|z'-z|+\mathfrak h(z)).\] If either cut has \(Z_h^2\mathfrak h\gtrsim b_\epsilon r\), the raw bound already gives this inequality, using the quasi-Lipschitz width bound to compare the two cuts. The lower band and the auxiliary ramp are handled in the same way. The supremum defining \(B_1\) therefore satisfies \(B_1\le C_{p,\diamond}(1+\sqrt{B_1})\), so it is bounded with the permitted loss. Applying the same two-sided brackets at a single cut gives \(\||X_v|^2-q(v)\|_p\lesssim_{p,\diamond}e(v)\).

Finally project the endpoints of an overlap, or the terminal endpoint in \(L\), to a cut a distance comparable to \(\mathfrak h\) later. Polarization about the common ancestor expresses the projected inner product through the squared norms and squared differences just bounded. The remaining tails, estimated by [eq:A3] and longer layers, cost at most \(O_{p,\diamond}(z_r^0r/\sqrt{h_0})\). The lower width \(h_0\gtrsim b_\epsilon(U/r)^2\) gives \[\frac{z_r^0r}{\sqrt{h_0}} =rh_0\frac{g_0(U/r)^3}{h_0^{3/2}} \lesssim b_\epsilon^{-3/2}g_0rh_0 \lesssim_\diamond e(v)\] at every nontrivial cut. This completes [eq:D2]. ◻

The mixed covariance term.

Telescope \(X_z\) backwards over dyadic distances \(lr\), down to an arbitrarily small inverse-polynomial length, and retain an oldest anchor a fixed small fraction of \(r\) earlier. For each nonterminal increment, the gap after its end allows [eq:A3] against \(y-X_z\). Its squared projection coefficient is at most \(C_\circ(z_r^0)^2r/l\), multiplied by the increase of \(S\) over \([z-2lr,z-lr]\).

We need the integral of that increase against \(G_z\,dz/r\). If \(lr\gtrsim\mathfrak h(z)\), the deterministic variation estimate bounds it pointwise by \(C_\diamond Z_h^2lr\). If \(lr\) is shorter than a tile \(I\), Fubini for the positive measure \(dS\) gives \[\int_I[S(z-lr)-S(z-2lr)]\,dz \le lr\,\Delta_{I^+}S,\] where \(I^+\) is a fixed enlargement. Multiply by \(\sup_I G/r\), use \(\Delta_{I^+}S\lesssim_\diamond Z_h^2|I|\), and sum the tile mass bounds from [eq:D5]. In both cases the weighted integral is \(O_\diamond(rZ_h^2l)\). Multiplication by the projection coefficient therefore costs \(O_\diamond((z_r^0)^2r^2Z_h^2)\). The last tiny increment is treated without projection by the same sliding estimate, choosing its length to dominate all polynomial prefactors. The oldest anchor uses a comparable field gap from [eq:D3] and its raw moment from [eq:D2]. Only logarithmically many layers occur. Summing proves the claimed mixed covariance bound.

Mass averaging.

Let \(h=rz_r^0Z_h\) be a width in the mass coordinate. If \(h\) is comparable to or larger than \(r\), the raw bound \(D_v=O_p(r)\) already proves the desired estimate. Otherwise bracket \(|X_v|^2\) by mass averages of \(L\) on its two sides. The points within distance \(O(h)\) of a band edge can be discarded at cost \(O(r^2h/r)\) using [eq:D2]. For the remaining points, monotonicity and \(\Delta S\lesssim r\) give the same cost for the integrated squared mean error. More explicitly, a sliding increment of \(S\) is at most \(Cr\), and its integral in \(dv/r\) is \(O(h)\); its squared integral is therefore \(O(rh)\). By [eq:A1] and Fubini, the centered averages contribute at most \(C h^{-1}\int\mathbf EW\,dv/r\). Thus \[\int\operatorname{Var}(|X_v|^2)\,dv/r \lesssim_\diamond r h+ \frac{(z_r^0)^2Z_h^2r^3}{h} \lesssim_\diamond r^2z_r^0Z_h.\] The first two parts of the lemma control the other terms in the exact conditional-covariance identity \[D_v^2=\operatorname{Var}(|X_v|^2) +2\mathbf EX_v^{\mathsf T}C_vX_v +\mathbf E\operatorname{tr}C_v^2.\] This proves the last assertion of [eq:D7]. ◻

The large-\(Y\) event.

Choose \(\kappa>0\) small relative to the power gains below. Markov’s inequality and \(\mathbf E_\gamma Z^4\lesssim_\circ1\) give \[\mathbf P_\gamma(Y>x^\kappa)\lesssim_\circ U^4x^{-4\kappa}.\] We first bound the contribution at each parameter value on this event. Here \(U\) retains its cap \(\min(1,x\lambda^{-s_*/2})\). Since \(v\,dh/dv\gtrsim\lambda_1U^3\) for \(v\ge x\), the source estimate and \(0\le S\le1\) give \[\begin{equation*} \int_{v\ge x}\mathbf EW\,dv\lesssim e_1^2, \qquad e_1=\frac{x^3}{\sqrt{\lambda_1}U^{3/2}}. \end{equation*}\] With unit raw bounds, mass slides of width \(e_1\) then cost \(O(e_1)\) in both their mean and centered parts. The trace term on \(v\ge\max(e_1,x)\) costs \(O(e_1)\) by division by \(v\); below \(e_1\) use the unit raw bound. Cauchy–Schwarz on dyads handles the mixed covariance term. Hence \[\int_{v\ge x}D_v^2\,dv\lesssim_\circ e_1.\]

The dotted clocks in the strength derivative are \(O_\circ(\lambda)\). In the pressure difference from [eq:A1], the portion \(v<x\) therefore costs \(O_\circ(\lambda x)\). At the other splits expand \(Q^2\) around the common untilted mean \(S_v\). The quadratic centered term uses the preceding \(D_v^2\) estimate. For the linear term, conditional averaging replaces \(Q-S_v\) first by \(|X_v|^2-S_v\) and then by \(2L_v-S_v\) exactly; there is no alias error in its expectation. The centered integral estimate from [eq:A1] costs \(O_\circ(e_1)\). Transfer these estimates to the stopped tilted law by [eq:A4]. The stopped derivative is consequently \(O_\diamond(\lambda(x+e_1))\) at each parameter value.

After parameter averaging, apart from \(x^{-4\kappa}\), the two costs are bounded by \[\lambda xU^4\le x^5\lambda^{1-2s_*},\qquad \lambda e_1U^4=x^3\lambda^{(1-\eta)/2}U^{5/2} \lesssim x^{3+(1-\eta)/s_*}.\] For the second bound, if \(U=1\) then \(\lambda\le x^{2/s_*}\) and the claim follows directly. If \(U<1\), substitution gives \(x^{11/2}\lambda^{(1-\eta)/2-5s_*/4}\). The exponent of \(\lambda\) is negative for \(s_*\) sufficiently close to \(1/2\); using \(\lambda\ge x^{2/s_*}\) gives the same bound. Since \(s_*<1/2\) and \(\eta\) is chosen small, both terms leave a fixed power saving beyond \(x^5\). Choose \(\kappa\) to preserve it after \(x^{-4\kappa}\), subpower losses, and strength integration.

The multiplier changes.

In varying \(\chi_{\rm lin}\) and then \(\chi_e\) at final strength, \(U\) is power small. Use the direct active and plateau coordinates and tiles of [eq:D5] on macroscopic bands, also when \(Y>x^\kappa\). The deterministic width properties there only use [eq:D1], \(S,q\) bounded, and do not require \(Y=o(1)\). The plateau lower slope in [eq:D3] comes from \(\chi_eH_e\); for \(\chi_e>0\) its use therefore costs \(\chi_e^{-1}\). The active part has at least the same weaker bound. Consequently for \(H=H_{\rm lin}\) or \(H_e\), respectively, \[\begin{equation*} \mathbf E\int H^2 W\,dv\ \lesssim_\diamond\ \chi_e^{-1}x^6\lambda_1^{-1} Z^2 \begin{cases} x^6,\\ \lambda_1^2 U^4. \end{cases} \end{equation*}\] Indeed both are supported at masses bounded below, the first kernel is uniformly \(O(x^3)\), the second bounded by \(O_\diamond(\lambda_1\mathfrak h)\); apply the tile expectation bound of [eq:D7] with that same loss \(\chi_e^{-1}\) and the integrated squared width estimate. By [eq:A1], [eq:A4] the derivative difference along each multiplier costs the square root of this bound on the stopped set (again exact linear conditional averaging); \(\chi_e^{-1/2}\) is integrable. Both \(x^6/\sqrt{\lambda_1}\) at final strength and \(x^3\sqrt{\lambda_1} U^2\) allow power saving beyond \(x^5\). This handles the multiplier changes. ◻

Freezing beyond the wall.

At \(t\ge t_M/2\) and thereafter \(1-q\) and \(1-S\) are smaller than every power of \(x\). In fact for the scalar spin the conditional mean martingale above a fixed macroscopic time has diffusion coefficient between \(c(1-V_z^2)\) and \(C(1-V_z^2)\), giving exponential decay of \(\Gamma'\) as after [eq:P2]. Thus \(dh/dt\ge c>0\) above a sufficiently large fixed time (with \(b\) constant there). For each actual spin, the conditional mean \(m_i=\sqrt m (X_v)_i\) has martingale bracket at least \(c(1-m_i^2)^2dh\) there, by the \(i\)-th diagonal source-Hessian bound. Indeed the field contributes diffusion matrix \(m H_v\) to all these coordinates (the source-Hessian and clock identities in (OpenAI 2026, sec. 2)); matrix innovations are independent. More explicitly, since the masses here are bounded below, \(H_{ii}\ge c(1-m_i^2)/m\). The field contribution to quadratic variation is at least \(m^2H_{ii}^2dh\ge c(1-m_i^2)^2dh\); the independent matrix innovations only increase it. With \(dh/dt\ge c\), localized Itô calculus for \(F(u)=\sqrt{1-u^2}\) gives \[\frac d{dt}\mathbf EF(m_i(t)) \le -c\mathbf E \frac{(1-m_i(t)^2)^2}{(1-m_i(t)^2)^{3/2}} =-c\mathbf EF(m_i(t)).\] Thus its expectation decays exponentially, uniformly in \(i,m\). The calculation works first across filled finite increments at masses bounded below, then by approximation. Averaging proves the assertion; since \(Q\le1,\ \nu Q=S\), it gives [eq:D11] in this region already.

Integrated fluctuation control.

Let \(\mathfrak a=1/200,\ V_0'=x^{\mathfrak a}\), with \(p_*<\mathfrak a/100\). On \([V_0',v_M]\), [eq:A1] gives \[\begin{equation*} \int \mathbf E W\,dt\lesssim x^6/(\lambda (V_0')^3). \end{equation*}\] Let \(R=x^6/(\omega\lambda(V_0')^3)\). The bounds on the two coordinate derivatives give \[\int\mathbf EW\,dv =\int\mathbf EW\frac{dv}{dt}\,dt\lesssim R, \qquad \int\mathbf EW\Big(\frac{dt}{dv}\Big)^2dv =\int\mathbf EW\frac{dt}{dv}\,dt\lesssim R.\] The second integral is the one required by [eq:A1] for a \(t\)-average, since that average has weight \(dt/dv\) when expressed in mass. Mass slides and \(t\)-slides can therefore each use width \(R^{1/2}\), with unit raw bounds. They give \[ \int_{V_0'}^{v_M} D_v^2 dv+ \int_{t(V_0')}^{t_M} D_{v(t)}^2 dt\le x^{1/4}. \tag{D12} \] Use the bracket identities of Lemma 25, with raw unit bounds, monotone \(S\), and width \((x^6/(\omega\lambda (V_0')^3))^{1/2}\) in each coordinate, using \(dt/dv\) in the \(t\)-averages in [eq:A1]. The \(\operatorname{tr} C^2\) terms cost at most \(C x^6/(\omega\lambda(V_0')^4)\) and Cauchy suffices for \(X^{\mathsf T}CX\) (coordinate lengths at most \(O_\circ(1)\)). All these bounds have the indicated slack.

A convex mass variation.

We use the trial gap in [eq:C2] to control the integral of \(|S-q|\) in mass. Let \(\phi\) be a smooth bump of either sign, supported between \(V_0'\) and \(\ell\), and set \[\frac1{m_\theta(v)}=\frac1v-\theta\frac{\phi(v)}{v^2}, \qquad 0\le\theta\le1.\] The bumps used below have small slope and \(|\phi|/v\ll1\), making \(m_\theta\) an increasing bi-Lipschitz change of mass. Reassign each clock increment formerly at \(v\) to \(m_\theta(v)\), keeping the increment itself, the terminal length, and the terminal self correction fixed. Let \(F(\theta)\) be the resulting pressure.

The exponential variational formula expresses each step power mean as a conditional supremum of expectation minus relative entropy divided by the step mass. After iteration, the complete pressure is a supremum over the choices of transition laws; the entropy coefficients are the reciprocal masses \(1/m_\theta(v)\), which are affine in \(\theta\). Thus \(F\) is convex. The initial mass-zero expectations are unchanged. Passing from step clocks to the fixed-size limit preserves convexity. The segment formula [eq:A1] gives \[F'(0)=\frac14\int(Bb'+2Sh')\phi\,dv.\] The clocks are Lipschitz on the bump at fixed size, so this chain rule also follows directly by approximating with continuous observation cuts.

In the new mass coordinate \(w=m_\theta(v)\), compare to the fixed scalar trial \(K(w)\). Square completion along the affine interpolation to that trial gives \[F(\theta)\le f_{\rm spin}(K)+P(\theta),\qquad P(\theta)=\int\frac{(K(w)-h(v))^2}{4b(v)}\,dw, \quad v=m_\theta^{-1}(w).\] At \(\theta=0\), [eq:C2] bounds this trial excess by \(x^{4+c}\). Convexity therefore yields \[F'(0)\le F(1)-F(0) \le x^{4+c}+P(1)-P(0).\] Differentiating the penalty and returning to the old coordinate gives \[\begin{equation*} P'(\theta)=\tfrac14\int(z_\theta^2b'+2z_\theta h') \partial_\theta m_\theta(v)\,dv, \qquad z_\theta=\frac{K(m_\theta(v))-h(v)}{b(v)}. \end{equation*}\] If the oscillation of \(K\) on the swept support is at most \(x^{3\mathfrak a}\), then \(z_\theta-z_0=O(x^{3\mathfrak a})\). Also \(\partial_\theta m_\theta=\phi(1-\theta\phi/v)^{-2} =\phi[1+O(\sup|\phi/v|)]\). These are the two errors in comparing \(P'(\theta)\) to \(P'(0)\). Integrating that comparison in \(\theta\) gives \[ \int [(B-z_0^2)b'+2(S-z_0)h']\phi \le 4x^{4+c}+C(x^{3\mathfrak a}+\sup|\phi/v|)\int(b'+h')|\phi|. \tag{D13} \] Here \(z_0=q-d/b\) is bounded and \(|d|\lesssim_\circ x^3\) throughout, by [eq:P6].

Partition \([V_0',\ell]\) into cells of mass width \(l=x^{5\mathfrak a}\). Give each cell a fixed enlargement, with uniformly bounded overlap. Discard the boundary cells and every cell whose enlargement has \(K\)-oscillation greater than \(x^{3\mathfrak a}\). Since \(K\) is nondecreasing and its total increase is \(O(M)\), \[\#\{\hbox{discarded interior cells}\}\,x^{3\mathfrak a} \lesssim M, \qquad |\hbox{discarded mass}|\lesssim l+Ml/x^{3\mathfrak a}.\] On each retained enlargement, Lipschitz continuity of \(\Gamma\) gives \(q\)-oscillation \(O(x^{3\mathfrak a})=o(x^{\mathfrak a})\). If \(|S-q|\ge Cx^{\mathfrak a}\) somewhere in its cell, monotonicity of \(S\) preserves that sign, at half the threshold, on a one-sided interval of length comparable to \(l\) in the enlargement: use the forward side for a positive deviation and the backward side for a negative one.

Choose a smooth bump \(\phi\) on this one-sided interval, with that sign, height \(cl\), slopes bounded by a sufficiently small fixed constant, and \(\int|\phi|\asymp l^2\). Its swept support remains in the good enlargement. Moreover \(|\phi|/v\lesssim l/V_0'=x^{4\mathfrak a}\); the small slope and this bound make all \(m_\theta\) increasing and bi-Lipschitz. Since \(|d|\lesssim_\circ x^3\), \(b\asymp1\), and \(q\gtrsim V_0'\) on this support, \(z_0=q-d/b\ge0\) and \[(S-z_0)\phi\gtrsim x^{\mathfrak a}|\phi|.\] Put \(I_\phi=\int h'|\phi|\). The error on the right of [eq:D13] is at most \[C(x^{3\mathfrak a}+x^{4\mathfrak a})(1+x^{-\mathfrak a})I_\phi =O(x^{2\mathfrak a})I_\phi =o(x^{\mathfrak a}I_\phi),\] because \(b'\lesssim h'/V_0'\).

To estimate the remaining term on its left, use the exact identity \[B-z_0^2=(S-z_0)(S+z_0)+D_v^2.\] The first summand has favorable sign after multiplication by \(b'\phi\). The second can have unfavorable sign only for a negative bump. If \(b'\) meets the bump, its support lies in the regular low region, where \(b'\lesssim\lambda\) and \(h'\gtrsim\lambda(V_0')^2\). By [eq:D12], \[\int D_v^2b'|\phi|\lesssim\lambda l x^{1/4},\qquad x^{\mathfrak a}I_\phi \gtrsim\lambda x^{3\mathfrak a}l^2.\] Their ratio is \(O(x^{1/4-8\mathfrak a})=o(1)\). If the bump lies outside that low region, \(b'=0\) and there is no variance error. Finally \(h'\gtrsim x^3\) everywhere on the bump: use the regular low inverse before \(4v_b\), and \(H_{\rm lin}\) thereafter. Thus \[x^{\mathfrak a}I_\phi\gtrsim x^{3+11\mathfrak a} \gg x^{4+c},\] contradicting [eq:D13]. Every retained cell therefore has \(|S-q|\lesssim x^{\mathfrak a}\). Since \(S,q\) are bounded, the low interval of length \(V_0'\) and the discarded cells contribute \(O(x^{\mathfrak a})\) and \(O(l+Ml/x^{3\mathfrak a})\), respectively. The strip between \(\ell\) and \(v_M\) has mass \(O(\omega M)\), so its contribution is \(O(\omega M)=o(x^{\mathfrak a})\), including its part before the frozen region. The remaining tail is controlled by the freezing estimate for \(t\ge t_M/2\). This proves \[ \int |S-q|\,dv\lesssim x^{\mathfrak a}. \tag{D14} \]

From integrated means to pointwise means.

Set \(K_S=h+b S\), nondecreasing, and write \(\Gamma_S\) for its scalar response (time variable), extending both profiles as needed by terminal annealed clocks. Use one common terminal extension for the two monotone clocks. The area between their graphs can be integrated first in mass or first in time, giving \[\int|\alpha_{K_S}(t)-\alpha_K(t)|\,dt =\int|K_S(v)-K(v)|\,dv \lesssim x^{\mathfrak a}\] by [eq:D14] and \(K_S-K=bf+d\). Thus \[\begin{equation*} \sup_t|\Gamma_S(t)-\Gamma(t)|\lesssim (1+M)^C x^{\mathfrak a} \end{equation*}\] on the needed times. To see the polynomial constant, interpolate inverse profiles with evaluation time fixed; \(\Gamma(y)/2\) is the root impulse derivative in the profile. Its derivative uses the post/post Hessian kernel in the proof of Lemma 22, which is uniformly \(O(1+M)\) for mixtures as well: the spatial derivative \(L_z^{\,y}(s)\) of the conditional scalar pair moment divided by two is bounded by the labeled source derivative rule (also directly by telescoping the conditional transition log-density when shifting the field, with bounded gradient of each scalar continuation). Impulse formulas pass from step profiles as there. Extend to one common terminal time \(O(M+1)\).

At reference \(K_S\), the returning mismatch is exactly \(h+bQ-K_S=b(Q-S)\). Apply [eq:A7] with \(P=1,k=1\). Its absolute attachment measure has only one prescribed fork: the tested pair’s split at \(v\). An insertion tied there costs \(D_v\); a freely placed insertion has bounded mass density and costs \(\int D_u\,du\). The row-deletion contact contributes \(O(m^{-1})\). Consequently \[\begin{equation*} |S_v-\Gamma_S(K_S(v))|\lesssim D_v+\int D_u du+O(m^{-1}). \end{equation*}\] All row changes are bounded uniformly. Integrate over \(0<t<t_M\) at \(v=v(t)\), using [eq:D12], discarding \(v<V_0'\) at cost \(O((1+M)V_0')\) since the low inverse is regular, and using the frozen estimate for post-wall attachments. Combine with the response comparison and \(|d|\lesssim_\circ x^3\); we get \(\int |S_{v(t)}-\Gamma(t+b(v)(S_v-q(v)))|dt\lesssim(1+M)^C x^{\mathfrak a}\). To use this residual estimate, write \(f=S-q\) and integrate the scalar derivative along the segment from \(t\) to \(t+bf\): \[S-\Gamma(t+bf) =f\int_0^1[1-b\Gamma'(t+ybf)]\,dy.\] The average in brackets is at least \(c\lambda v^2\) by [eq:P5]. On \(v\ge x^{\mathfrak a/10}\), division by this gap therefore gives, allowing the preceding subpower factors, \[\begin{equation*} \int_{v(t)\ge x^{\mathfrak a/10}}|f(v(t))|\,dt \lesssim_\circ x^{4\mathfrak a/5-p_*} \le x^{\mathfrak a/2}. \end{equation*}\] Here \(p_*<\mathfrak a/100\) leaves room for the weakened exponent. The function \(S_{v(t)}\) is nondecreasing, whereas \(\Gamma(t)\) is Lipschitz. A deviation of height \(d\) therefore persists at height at least \(d/2\) on a one-sided \(t\)-interval of length comparable to \(d\). The integral bound forces \(d^2\lesssim x^{\mathfrak a/2}\), giving \(|f|\lesssim x^{\mathfrak a/4}\) above mass \(2x^{\mathfrak a/10}\). Both-sided buffers fit until the frozen region, which was treated separately. Below this cutoff, monotonicity and \(q(v)\asymp v\) near zero give \(S_v+q(v)\lesssim x^{\mathfrak a/10}\). Consequently \(\sup_v|S_v-q(v)|\lesssim x^{\mathfrak a/10}\).

The all-moment upgrade.

Fix a finite moment order \(p\). First take a cut of mass \(v\ge4x^{\mathfrak a/10}\) and scalar time \(t\le t_M/2\). Its small \(t\)-neighborhoods stay above \(V_0'\). Apply Lemma 24 with unit raw bounds. For a later block at \(t\)-distance \(l\), the gap has field length at least \(c\lambda(V_0')^2l\) and mass at least \(V_0'\). Its squared projection coefficient and the integrated earlier anchor cost are \[\frac{C_{p,\circ}x^6}{\lambda(V_0')^4l} \quad\hbox{and}\quad C_pl,\] respectively. Stop the doubling layers at a fixed positive \(t\)-distance and then use a terminal block. Summing gives \[\begin{equation*} \left\|\int W\,dt\right\|_{p/2} \lesssim_{p,\circ}\frac{x^6}{\lambda(V_0')^4} \end{equation*}\] on each bounded \(t\)-window under consideration.

Put \(\rho=x^{\mathfrak a/4}\). Choose a past buffer and a future buffer of width comparable to \(\rho\) on the two sides of a smaller neighborhood of the cut. They remain at mass at least \(V_0'\) and below \(t_M\); any part past \(t_M/2\) is already controlled by freezing. The global mean bound and Lipschitz continuity of \(\Gamma\) give, for two buffer times \(t_1,t_2\), \[|S_{v(t_2)}-S_{v(t_1)}| \lesssim x^{\mathfrak a/10}+|t_2-t_1| \lesssim x^{\mathfrak a/10},\] since \(|t_2-t_1|=O(\rho)\) and \(\rho=x^{\mathfrak a/4} =o(x^{\mathfrak a/10})\). Applying [eq:A1] to a normalized \(t\)-average, with \(dt/dv\lesssim1/\omega\), gives \[\|L_{\rm buffer}-\mathbf EL_{\rm buffer}\|_p \lesssim_{p,\circ} \frac{x^3}{\sqrt{\omega\lambda}(V_0')^2\rho}.\] This is smaller than \(x^{\mathfrak a/10}\) for our parameter choices. The two-sided brackets and the common-past estimate in Lemma 25 consequently give \[\||X_t|^2-q(v(t))\|_p +\||X_t-X_{t'}|^2\|_p \lesssim_{p,\circ}x^{\mathfrak a/10}\] for cuts \(t,t'\) in the smaller neighborhood.

For an overlap split there, project both endpoints to a cut a fixed fraction of \(\rho\) later. The two projected branches have the preceding norm and increment bounds, so polarization about their common ancestor controls their inner product. Equation [eq:A3], now even with unit anchors, bounds the residual projection errors using that \(\rho\)-length gap. For a smaller split, choose instead a central cut at mass a sufficiently large fixed multiple of \(x^{\mathfrak a/10}\). Its lower comparable mass interval lies strictly after the fork and supplies a full projection gap; it has the buffers just constructed. Its squared posterior norm has \(L^p\) size \(O_p(x^{\mathfrak a/10})\). Together with the frozen region, these estimates prove [eq:D11], for example with \(\delta_*=\mathfrak a/50\). The stopped transfer [eq:A4] gives the same coarse accuracy for shifted active overlaps. ◻

The region below \(Y\).

At \(v<x\), a stochastic insertion costs \(O_p(Y)\). If it is the only low factor, the required coefficient is \(O(v)\), and the integral is \(O(x^2Y)\). A second factor at the same vertex gives \(O(xY^2)\) instead. At a distinct low vertex its integrated envelope is at most \(O_\diamond(Y^2)\), giving \(O_\diamond(xY^3)\). All three are bounded by \(O_\diamond(x^3\lambda^{-C}Z^2)\), since \(Y=x\lambda^{-s_*/2}Z\le1\). Additional factors are bounded after truncation.

For \(x\le v\lesssim Y\), retain one stochastic factor in second norm. In the sole-factor case its coefficient is at most \(CY\); in the tied case the other envelope is \(O(Y)\); in the distinct-vertex case its integrated envelope is \(O_\diamond(Y^2)\le O_\diamond(Y)\). All three cases are therefore bounded by \[Y\int_{x\le v\lesssim Y}D_v\,dv \lesssim_\circ x^3\lambda^{-C}Z^{7/4} \le x^3\lambda^{-C}Z^2,\] using [eq:D9] and summing dyads. Cauchy–Schwarz with \(P\) supplies its factor \(D_{ab}\); the remaining factors are truncated at their deterministic envelopes. Tilted active moments transfer on the stop as already specified.

Alias errors above \(Y\).

For stochastic insertions at \(v\gtrsim Y\), first replace \(Q-S_v\) by \(|X_v|^2-S_v\). On a dyad \(r\), the second line of [eq:D7], with \(Z_h=1\), gives the following integral of the alias’s second norm: \[A_r:=\int_{v\asymp r}\|Q-|X_v|^2\|_2\,dv \lesssim_\diamond z_r^0r^2 =\frac{x^3}{\sqrt{\lambda_1}r}.\] The sole-factor coefficient cancels this denominator: \(rA_r\lesssim x^3/\sqrt{\lambda_1}\). If a second factor is tied to the same fork, Cauchy–Schwarz in mass uses its integrated envelope budget and gives \[\int_{v\asymp r}e(v)\|Q-|X_v|^2\|_2\,dv \lesssim_\diamond z_r^0r^2m_r =\frac{x^3Y^2}{\sqrt{\lambda_1}r^2} \le\frac{x^3}{\sqrt{\lambda_1}}.\] For a second distinct vertex, integrate its envelope separately; the cost is at most \(A_r\int_{v<V_i}e(v)dv \lesssim_\diamond x^3Y^2/(\sqrt{\lambda_1}r)\), which is sufficient because \(r\gtrsim Y\) and \(Y\le1\). Truncation handles any further factors, and Cauchy–Schwarz with \(P\) again retains \(D_{ab}\).

Centered integration in a fixed marginal.

Process the remaining stochastic vertices in decreasing time order. At the current vertex, replace every tied overlap by the same ancestor value \(|X_v|^2-S_v\), and then replace its power by the corresponding power of \(2L_v-S_v\) on one continuation through that vertex. The alias estimates just proved bound these changes. Next replace this polynomial by its deterministic untilted mean. The mean is bounded by the corresponding envelope power.

The centered error in this last replacement is estimated by [eq:D10]. Its bound has the same three forms as the alias bounds: a sole linear factor uses the extra coefficient \(v\); a tied product uses the \(m_r\) gain; a factor at a distinct vertex uses the integrated envelope there. Earlier random ancestor factors are truncated at their deterministic envelopes before this estimate. To apply [eq:D10], we must realize the entire moving-\(v\) integral and its retained random tests under a single fixed marginal law. The next construction verifies that requirement.

More explicitly, fix the tree shape, its time order, every other distinct fork coordinate, and every projection cut. If an old leaf descends through the moving vertex, retain that leaf as the continuation defining \(L_v\). Both leaves of the old pair then lie in the same child until their prescribed higher split. The retained old pair consequently has its original, fixed split law. If no old leaf descends through the moving vertex, the chosen continuation separates from the retained old tree at a different, already fixed vertex. In both cases normalized continuations of all other children integrate to one. Earlier random ancestor factors are evaluated before the moving coordinate; later stochastic factors have already been replaced by deterministic means. Projected old endpoints are evaluations of the same retained paths at fixed later cuts. Auxiliary labels with no remaining spin observable can be deleted by the same consistency rule.

Thus the entire varying-\(v\) integral lives under one marginal law. We do not condition the centered estimate on the retained random test \(P\). Let \(I\) denote the centered integral, and truncate the product \(R\) of the earlier factors at its deterministic envelope product \(M_R\). Then \[|\mathbf E[PRI]|\le M_R\|P\|_2\|I\|_2 =M_R D_{ab}\|I\|_2.\] Equation [eq:D10] is applied to \(I\) under the fixed marginal, before this Cauchy–Schwarz inequality. The discarded tails are negligible by the previously specified moment truncation. In the tilted comparison, transfer and Hölder require a norm only slightly above \(L^2\), with the already permitted small power loss. A stochastic tie at the moving vertex is one polynomial power of its ancestor value; it does not add an independent coordinate or create a second atom. In the comparison version all moving stochastic times here are active; the same fixed marginal reasoning and [eq:A4] suffice. After successive replacements (split ties as one power), the deterministic product vanishes against centered \(P\), or cancels across ensembles. This proves [eq:D16]. ◻

The covariance equation.

Fix an old pair \(ab\) splitting at or above \(V_i\), where \(i\in\{1,2\}\), and put \(P=Q_{ab}-S_{ab}\). For every tested pair \(jk\) at or above the same threshold, define \[u_{jk}=\nu[P(Q_{jk}-S_{jk})].\] These are pair functions on extensions of the old labels \(ab\); unused labels are ignored by marginalization. Apply [eq:A7] at the reference \(K\), expanding \(J=b\delta+bf+d\), where \(\delta=Q-S\). Choose a fixed truncation degree large enough that [eq:D11] makes the true Taylor remainder \(O(x^K)\) for any needed fixed \(K\). The row-deletion error remains \(O(m^{-1})=O(x^6)\). Every all-deterministic term vanishes against \(P\).

Retain the high stochastic linear term on the left. The equations have the form \[u_{jk}-\sum_{cd:\ v_{cd}\ge V_i}{\bf M}_{jk,cd}b_{cd}u_{cd} =R_{jk}^{\rm low}+R_{jk}^{\rm nonlinear} +O(x^K)+O(x^6).\] The first source includes the stochastic linear term below \(V_i\); the second contains at least two insertions. This left side is exactly [eq:H1], because the positive-law expectation of each inserted pair with \(P\) depends only on the indicated labels. By [eq:P5], its gap on retained levels is at least \(c\lambda V_i^2\). Proposition 21 therefore inverts it with cost \(O(x^{-Ca_i})\), once \(p_*\ll a_i\) is chosen. Evaluate the result at \(jk=ab\). The inverse adds only a bounded number of freely attached labels. Their attachment densities and existing-fork weights are bounded after removing the inverse cost; they do not turn a new low fork into a prescribed atom.

As explained in the fixed-resolution limit argument, these operations can first be made on resolutions keeping the tested cut and threshold. At fixed system size gaps persist on fine approximations; inverse domination with block restrictions allows replacing path expectations and coefficients outside that inverse by continuous-path ones with errors tending to zero (empty step intervals can be refined for attachment). Our bounds apply to bounded measurable topology weights from that inverse after its cost factor is removed.

Consider a nonlinear term containing a high stochastic factor \(\delta_{cd}\). Keep \(P\) and that factor untruncated. There is at least one further small factor: truncate it at \(x^{\delta_*/2}\), and bound all other factors uniformly. Equation [eq:D11] controls the discarded tails in arbitrarily high fixed moments. Cauchy–Schwarz on the two untruncated factors gives \[|\nu[P\delta_{cd}\,\hbox{remaining factors}]| \lesssim x^{\delta_*/2}\|P\|_2\|\delta_{cd}\|_2+O(x^K) \le x^{\delta_*/2}D_{ab}D_{H,i}+O(x^K).\] This preserves the exact second-moment factors; no higher norm proportional to \(D_{ab}\) is assumed. Every remaining source contains only low stochastic factors, so Lemma 52 applies, including to the low linear term.

The coefficient of a sole low factor.

If just one low insertion occurs at \(v\), its scalar coefficient indeed costs \(O(v)\): in every term of the finite quotient it belongs to a spin pair moment (possibly with other pairs), and all other pairs in that moment, including the tested pair if present, are high. Immediately across the low branching its two ends thus give two different groups of odd counts. Conditional on prefixes up to that cut, group expectations on independent continuations are odd smooth bounded functions (for the odd groups) with bounded gradients by the scalar source rules. Their product costs \(O(K(v))=O(v)\) using Brownian moments and bounded drift. More explicitly, the two odd group functions \(F_1(z),F_2(z)\) vanish at zero and have uniformly bounded derivatives, so \(|F_1(z)F_2(z)|\le C|z|^2\). The shared scalar prefix field at time \(K(v)\) has second moment \(O(K(v))\) by its Brownian representation with bounded drift. Hence this scalar coefficient is \(O(K(v))=O(v)\). Additional labels and coincidences within the groups do not change the odd parity or the derivative bound.

Combining these estimates after inversion gives \[\begin{equation*} D_{ab}^2\lesssim_\diamond x^{-Ca_i} \bigl(x^{\delta_*/2}D_{ab}D_{H,i} +x^3\lambda^{-C}Z^2D_{ab}+x^6\bigr). \end{equation*}\] The final \(x^6\) is the row-deletion contact error. Increasing the Taylor degree does not remove it. Take the supremum over \(ab\) and write \(D=D_{H,i}\). Choose the power losses so that \(Cx^{\delta_*/2-Ca_i}<1/2\). Absorption then leaves \[D^2\lesssim_\diamond x^{3-Ca_i}\lambda^{-C}Z^2D +x^{6-Ca_i}.\] If \(D^2\le AD+B\) with \(D\ge0\), then \(D\le A+\sqrt B\) up to a fixed constant. Since \(Z\ge1\) and \(p_*\ll a_i\), the factors \(\lambda^{-C}\) and the permitted small losses can be included in a fixed \(C_Ha_i\). This proves \(D_{H,i}\le x^{3-C_Ha_i}Z^2\) for \(i=1,2\).

The longitudinal equation.

For \(r\ge V_1\), apply [eq:A7] without a random test. The longitudinal identity [eq:H2] separates the linear term into its diagonal and its integral part. The resulting scalar equation is \[[1-b(r)\Gamma'(K(r))]f(r) =\Gamma'(K(r))d(r)+R_{\rm int}(r)+R_{\rm nonlinear}(r)+O(x^6).\] The diagonal is in field units; returning from normalized mass changes only bounded constants. Its lower bound is \[1-b(r)\Gamma'(K(r))\ge c\lambda V_1^2.\] For the integral term, the post kernel bound in Section 5 gains the earlier time \(K(u)\lesssim u\) when \(u\) is small. It gives \[|R_{\rm int}(r)|\lesssim(1+M)^C\left( \int|d|\,du+\int_{u<V_0}u|f(u)|\,du +\int_{u\ge V_0}|f(u)|\,du\right).\] The low dyadic envelope budget and the raw bound below \(Y\) imply \(\int_{u<V_0}u|f(u)|du\lesssim_\diamond V_0Y^2\). For the last integral use [eq:C2] and the majorant \(Z_g\): Cauchy–Schwarz, followed by a harmless weaker cutoff bound, gives \(\int_{u\ge V_0}|f(u)|du\lesssim x^{2+c/2-a_0}Z_g\). Thus the three terms on this right side are \[O_\circ(x^3),\qquad O_\diamond(x^{2+a_0-p_*s_*}Z^2),\qquad O(x^{2+c/2-a_0}Z_g).\]

For a nonlinear allocation with a factor at the prescribed fork \(r\), keep that factor and truncate one of the others at \(x^{\delta_*/2}\). Its contribution is \[O(x^{\delta_*/2})\bigl(|f(r)|+x^3+D_{H,1}\bigr).\] If no factor is pinned at \(r\), integrate one factor in second norm. The low envelope and the high comparison budget give \[\int\|Q-q(u)\|_2\,du+\int|d(u)|\,du \lesssim_\diamond Y^2+x^{2+c/2-a_0}Z_g+x^3.\] A further factor contributes \(x^{\delta_*/2}\) after truncation, even when it is tied to the same vertex. The negligible tails use a fixed higher moment from [eq:D11].

Divide by \(c\lambda V_1^2\). The coefficient of \(|f(r)|\) is at most \(Cx^{\delta_*/2-p_*-2a_1}\), up to the fixed polynomial and small power losses, and is absorbed. The remaining leading exponents before division are \[3,\qquad 2+a_0-p_*s_*,\qquad 2+c/2-a_0,\qquad 2+\delta_*/2-p_*s_*.\] The already proved variance bound controls the extra \(x^{\delta_*/2}D_{H,1}\) term. Choose \(a_0\) small relative to \(c,\delta_*\), then choose \(b_H\) to be a sufficiently small fixed fraction of \(a_0\). Next take \(a_1\) sufficiently small, choose \(a_2\ll a_1\), and only then take \(p_*\) sufficiently small that the loss from the gap, \((1+M)^C\), and every permitted small power loss is less than the remaining margins above \(2+b_H\). This also ensures \(b_H\gg C_Ha_1\). We obtain uniformly for \(r\ge V_1\) \[|f(r)|\le x^{2+b_H}(Z^2+Z_g),\] as required. This choice of \(a_2\ll a_1\), followed by \(p_*\), preserves both this bound and the covariance absorption. ◻

The objects in the compression formula.

For a degree \(j\) term, partition its \(2j\) labeled endpoint slots into blocks at \(c\). Let \(D\) be the number of blocks, ordered by their first slots. The \(j\) ordered pairs form a graph on these blocks; every edge has two distinct ends. The skeleton allocation measure is the signed per-root measure for \(D\) ordered distinct end labels in the hierarchy ending at \(c\), divided by \(c^D\). Its absolute weight on each of finitely many shapes is bounded by \[C_jc^{-1}\prod_{s\ {\rm skeleton\ fork}}dv_s/c.\] Indeed the first label has unit per-root weight, and each of the remaining \(D-1\) placements contributes scale \(c\). Division by \(c^D\) leaves \(c^{-1}\). After rescaling mass to \([0,1]\), the remaining densities are bounded. Attachments at existing forks may reuse their mass coordinates, but block identities remain distinct. There are no prescribed old leaves or forks. Thus distinct free fork coordinates are unequal almost everywhere, while a fork may have more than two children.

Next let \(\pi\) be a partition of the \(j\) edges into cumulant parts, keeping repeated edges as separately labeled slots. For a part \(P\in\pi\), let \(k_{A,P}\) be the number of its endpoints in block \(A\), and put \[D_\pi=\sum_{P\in\pi}\#\{A:k_{A,P}>0\}.\] Thus \(D\) counts blocks in the full skeleton, whereas \(D_\pi\) counts a block again each time it is used by another part. Neither counts endpoint multiplicities.

On the fixed skeleton, let \(\zeta^A_u\), \(0\le u\le c\), be the scalar prefix fields under the positive scalar law. They share their Brownian driver until their skeleton separation, and thereafter use independent transitions. Each part \(P\) uses a separate copy of this law restricted to its blocks. The normalized source sum for \(k\) slots inside one block is \[s_k(z)=c^{-1}e^{-cV(t(c),z)}\partial_z^k e^{cV(t(c),z)}.\] With the explicit edge weights of [eq:E3] kept outside, the coefficient multiplying the skeleton measure is \[\sum_{\pi}(-1)^{|\pi|-1}(|\pi|-1)!\ c^{D_\pi} \prod_{P\in\pi}\mathbf E_{\rm low} \prod_{A\ {\rm in}\ P} s_{k_{A,P}}(\zeta^A_c). \tag{E5}\] The expectation \(\mathbf E_{\rm low}\) is probabilistic; the preceding grouping of allocation coefficients by skeleton is a signed sum.

For example, take two copies of the edge \(AB\) between cut blocks. Their original terminal labels may differ; projection makes their spin-model pair tests identical. Abbreviate \(s_1(\zeta^A_c)=\mathsf M_A\), \(s_2(\zeta^A_c)=\mathsf P_A\). The one-part partition uses two blocks, whereas the two-part partition uses each block twice. Their combined internal coefficient is therefore \[c^2\mathbf E_{\rm low}(\mathsf P_A\mathsf P_B) -c^4\bigl(\mathbf E_{\rm low}(\mathsf M_A\mathsf M_B)\bigr)^2.\] The skeleton measure contributes \(c^{-1}\), giving net powers \(c\) and \(c^3\). There are two orientations of the second edge for a fixed first edge, so its factor in [eq:E3] is \(2/(2^2 2!)=1/4\). The square counterterm uses this same skeleton, with internal coefficient \(c^2\) and weight \(-b/4\). These normalizations will give the quadratic cancellation below.

The finite signed identity.

First expand each cumulant into products of positive scalar moments on its indicated edge sets. Once the skeleton assignments are fixed, there is no restriction on the internal placements within a block. A part observes only its own induced genealogy, so the internal signed sums for different parts can be performed separately, even when their actual terminal leaves coincide.

This factorization can be checked directly at finitely many levels including \(c\). In the labeled product rule of (OpenAI 2026, Proposition 2.5), the coefficients choosing distinct children are falling factorials of incoming-to-child mass ratios, and every terminal source slot carries its terminal mass factor. Keep the induced genealogies as discrete marks and first take the successive ratios, including an auxiliary root ratio, to be sufficiently large positive integers. The coefficients then count child identities. After distinct block identities have been assigned at \(c\), the choices for disjoint sets of slots inside a block are independent counts, with terminal slots chosen with replacement. Their counts therefore multiply. The resulting rational coefficient identity extends by polynomial identity to the real masses, keeping terminal mass fixed and dividing by the root mass before its zero limit. This argument concerns only the coefficients; it does not extend spin tests or probability laws from integer ratios.

Evaluation of the internal source sums.

For one part, the positive scalar prefix law on its restricted skeleton is independent of the internal placements after \(c\). Conditional on these prefix fields, the source product rule sums \(k\) internal slots in a block to \(e^{-cV}\partial_z^k e^{cV}=c s_k\). Different blocks use independent future kernels. Thus every used block contributes one factor \(c\), giving \(c^{D_\pi}\), and the cumulant partition formula gives [eq:E5]. Signed path limits and bounded fixed-order derivatives, as supplied by PE, pass this finite-level identity to the continuous paths. Equivalently, one may fix the prefix fields and approximate only the future clocks within the blocks. There are no prescribed internal split times to preserve in that approximation.

Differentiation at a fixed skeleton.

The cut and skeleton measure stay fixed. Differentiating the scalar moments adds one covariance pair with weight \(\dot K\). By [eq:P8], its absolute integrated weight is finite; its non-atomic placements have bounded density for fixed cut and label count. The same bound holds along affine comparison chords of the clocks, so the finite signed differentiation rule and Fubini apply to the internal sums.

To pass from chords to ordinary derivatives, fix \(N\). The clock difference quotients are uniformly bounded in mass: for size paths this follows from the smoothing floor and clipping, and for direct paths from the localized smooth inverse perturbation. For almost every parameter they converge at every mass in the low region and at almost every other mass. The scalar kernels converge by fixed-size path continuity. Dominated convergence therefore passes the divided covariance insertions to the dotted insertions, also for almost every fixed skeleton. A reused split is either one of these integration coordinates or a prescribed low skeleton mass, where convergence is pointwise; constant diagonals still cancel. This proves that internal summation and a.e. coefficient differentiation commute. We may choose the low-region partition separately at each parameter when estimating the result: the coefficient has already been differentiated, so no moving indicator or cut is being differentiated.

Scalar jets and common-prefix estimates.

Write \(\chi=V_{zz}(0,0)\), \(g=V_{zzzz}(0,0)\), \(k_0=t'(0+)>0\), and \(G(u,z)=V(t(u),z)\). The notation \(O_*(A)\) controls both the value and the parameter derivative: on size paths the derivative is \(\partial_\theta\), and on direct paths it is \(\lambda^{-1}\partial_{\log\lambda}\). The same convention applies to fixed \(L^p\) norms when fields are coupled by common Brownian drivers in mass coordinates. Derivatives of the explicit weights \(b,d\) will be handled separately.

By [eq:P1]–[eq:P2], the inverse \(t\) is smooth here, with \(t'\) bounded above and below and with the required higher derivatives bounded. The positive-order spatial derivatives of \(G\) also have bounded mass and spatial derivatives of every required fixed order, allowing polynomial growth in \(z\) after parameter differentiation. To see the mass derivatives, use \[\partial_uG=-\tfrac12t'(G_{zz}+uG_z^2).\] For direct derivatives, differentiate the scalar PDE in \(\lambda\) at fixed normalized time, as in the lift proof, and then compose with the regular inverse. For size derivatives, use the rescaled scalar jets, including the terminal-mass derivative already estimated in Section 3. These operations preserve evenness in the field. In particular, on size paths, \[\partial_\theta k_0=-3 k_0/6,\qquad \partial_\theta\chi=\chi/6+O(e^{-\Omega(M)}),\qquad \partial_\theta g=3g/6+O(e^{-\Omega(M)}). \tag{E6}\] The normalized inverse slope and initial spatial derivatives are constant under this rescaling, up to the exponentially small terminal effect. This gives [eq:E6]. Moreover, \(\chi\) is uniformly bounded above and away from zero. At every designated split, \[b=b_\circ+O(\lambda c^2),\qquad b_\circ=\chi^{-2},\] and the error has the same bound under unscaled parameter differentiation. By [eq:D2], each \(d\) or \(\dot d\) can likewise contribute \(O_\diamond(\lambda c^2)\) to the coefficient, leaving its ordinary envelope factor.

The low prefix equation is \[d\zeta=\sqrt{t'}\,dB+u t'G_z(u,\zeta)du.\] Starting from zero, this equation gives \(\zeta_u=O_*(\sqrt u)\) and \(\zeta_u-\sqrt{k_0}B_u=O_*(u^{3/2})\) in every fixed moment. For a conditional evolution starting at \(z\), the field and its parameter derivative have norms at most \(C_p(|z|+\sqrt c)(1+|z|)^C\). Its spatial flow derivatives have bounded moments, allowing polynomial growth in \(z\). These bounds follow by differentiating the additive-noise equation: the drift is smooth with bounded spatial derivatives, the differentiated coefficients have polynomial growth, and each odd differentiated drift vanishes at zero.

Evaluate the block observables at their fields at \(c\), unless another time is specified. Taylor expansion, together with even/odd parity and the definition of \(s_k\), gives \[\begin{split} \mathsf M_A := s_1(\zeta^A_c)&=G_z(c,\zeta^A_c)=\chi\zeta^A_c+O_*(c^{3/2}),\\ \mathsf P_A:=s_2(\zeta^A_c)&=G_{zz}(c,\zeta^A_c)+c(G_z(c,\zeta^A_c))^2 =\chi+O_*(c),\\ s_3(\zeta^A_c)&=g\zeta^A_c+O_*(c^{3/2}),\qquad s_4(\zeta^A_c)=g+O_*(c). \end{split} \tag{E7}\] For blocks \(A,B\) splitting at \(v_{AB}<c\), the martingale property of \(G_z\) reduces their product expectation to the common prefix at \(v_{AB}\). Thus \[\mathbf E_{\rm low} \mathsf M_A\mathsf M_B=\chi^2 k_0 v_{AB}+O_*(v_{AB}^2).\] We also need \[\mathbf E_{\rm low} \mathsf P_A\mathsf P_B =\chi^2+\chi^3 k_0 c^2+\tfrac12 g^2 k_0^2 v_{AB}^2+O_*(c^3). \tag{E8}\] To obtain [eq:E8], keep the terminal cut \(c\) fixed and set \(H_u=G_{zz}(u,\zeta_u)+cG_z(u,\zeta_u)^2\). The PDE and Itô’s formula give drift \((c-u)t'G_{zz}^2\) and diffusion coefficient \(\sqrt{t'}(G_{zzz}+2cG_zG_{zz})\). In the product of the two block processes, the two drift integrals sum to \(\chi^3k_0c^2+O_*(c^3)\), by [eq:E7]. Their common quadratic variation stops at \(v_{AB}\); its expected rate before that time is \(g^2k_0^2u+O_*(cu)\). Its integral is the term \(\tfrac12g^2k_0^2v_{AB}^2\), with an error of order \(O_*(c^3)\).

For three blocks we use \[\mathbf E_{\rm low} \mathsf P_A \mathsf M_B\mathsf M_C=\chi^3 k_0 v_{BC}+O_*(c^2). \tag{E9}\] When the designated splits are distinct, this estimate retains the earlier mass. Suppose \(v_{AB}=h'<v_{AC}\), so that \(v_{BC}=h'\). Condition at \(h'\) on the common field \(z\). After subtracting \(\chi\mathsf M_B\mathsf M_C\), the conditional expectation on the \(A,C\) side is that of \((\mathsf P_A-\chi)\mathsf M_C\). It is an odd function of \(z\), with spatial derivative bounded by \(C(c+z^2)(1+|z|)^C\), also after parameter differentiation at fixed \(z\). Indeed \(\mathsf P-\chi=O(c+|\zeta_c|^2)\), its spatial derivative is \(O(|\zeta_c|)\), and \(\mathsf M=O(|\zeta_c|)\) with bounded derivative; apply the conditional flow bounds and Hölder to their product.

The \(B\)-side conditional mean is \(G_z(h',z)\), of size \(O_*(|z|(1+|z|)^C)\). Substituting the common prefix field \(z=O_*(\sqrt{h'})\) therefore bounds the error in [eq:E9] by \(O_*(ch')\). The other ordering is identical after exchanging \(B,C\). Without a designated earlier split, [eq:E7] gives the stated \(O_*(c^2)\) error.

The coefficients canceled by size rescaling.

For a pure degree \(j\) term with no defect factors, the formal rule \(\partial_\theta\Delta=-\Delta/6\) makes the size operator on its coefficient equal to \(1-j/6+\partial_\theta\). Since \(\ell=e^{\theta/6}\), \[\left(1-\frac j6+\partial_\theta\right) \ell^{-(6-j)}=0.\] Equation [eq:E6] and \(b_\circ=\chi^{-2}\) identify exactly the coefficients with this scaling: \[\begin{array}{c|c|c} j&\text{coefficient monomials}&\text{size scaling}\\ \hline 2&b_\circ^2\chi^3k_0,\quad b_\circ^2g^2k_0^2&\ell^{-4}\\ 3&b_\circ^3g^2k_0,\quad b_\circ^3\chi^3&\ell^{-3}\\ 4&b_\circ^4g^2&\ell^{-2}. \end{array}\] The masses multiplying these monomials are held fixed. Exponentially small terminal errors will be retained until the end. On direct paths, the operator is only \(\partial_{\log\lambda}\), and the \(O_*\) convention already supplies the factor \(\lambda\). For a term with a defect, use the product rule and extract at least one factor \(\lambda c^2\) as above. In the remaining case analysis we include the net factor \(c^{D_\pi-1}\); the normalized skeleton fork measure in [eq:E4] is left outside.

The low-degree cases.

1. Suppose \(j=2\) and \(D=2\). The two edges join the same blocks, so use the repeated-edge calculation following [eq:E5]. Its coefficient combines with the square counterterm because both multiply the same projected pair square. After extracting the common skeleton measure and \(c^2/4\), their coefficient before differentiation is \[b^2\left\{\mathbf E_{\rm low}\mathsf P_A\mathsf P_B -c^2(\mathbf E_{\rm low}\mathsf M_A\mathsf M_B)^2\right\}-b.\] The constant term \(b^2\chi^2-b\), and its unscaled derivatives, are \(O(\lambda c^2)\) because \(b_\circ\chi^2=1\) and \(b-b_\circ=O(\lambda c^2)\). The next two terms from [eq:E8], after replacing \(b\) by \(b_\circ\), are \[b_\circ^2\chi^3k_0c^2, \qquad \tfrac12b_\circ^2g^2k_0^2v_{AB}^2.\] Their direct derivatives are \(O(\lambda c^2)\); their size contributions cancel by the \(j=2\) row of the table. Substitution errors from \(b-b_\circ\) have the required bound. The remaining \(O_*(c^3)\) errors, including the smaller product-of-means term, contribute \(O(c^4)\) after the net compression factor \(c\). On size paths this is \(O(\lambda c^3)\) because \(c\lesssim\lambda\); on direct paths the derivative supplies \(\lambda\) itself. Thus [eq:E4] holds. A defect factor supplies \(\lambda c^2\) directly and gives the same bound, with the allowed \(O_\diamond(1)\) loss.

2. Suppose \(j=2\) and \(D=3\), with edges \(AB,AC\). For the one-part partition, [eq:E9] gives the leading coefficient \(b_\circ^2\chi^3k_0v_{BC}\), with net compression factor \(c^2\). Its size contribution cancels by the same row of the table; its direct derivative costs \(O(\lambda c^3)\), or \(O(\lambda c^2h')\) when \(v_{BC}=h'\) is the earlier split. The error is \(O_*(c^2)\), or \(O_*(ch')\) when the designated vertices are distinct and their earlier mass is \(h'\). Multiplication by \(c^2\), followed on size paths by \(c\lesssim\lambda\), gives respectively \(O(\lambda c^3)\) and \(O(\lambda c^2h')\). The errors from \(b-b_\circ\) and any defect factors satisfy these same bounds. For the two-part partition the product of pair moments is \(O_*(v_{AB}v_{AC})\), with net factor \(c^3\), which is smaller still. Since \(h'\lesssim\min_s r_s\), the refined bound gives the extra ratio required in [eq:E4].

3. Suppose \(j=2\) and \(D=4\). Both partitions have net factor \(c^3\), and their products of moments are \(O_*(c^2)\). When the designated vertices are distinct with earlier mass \(h'\), these products are also \(O_*(h')\). For a product of two pair moments this follows at once from their individual bounds. For the four-spin moment, condition on the prefixes at \(h'\). The earlier edge has separated ends, while the ends of the later edge are still together. Hence at least two of the independent future groups have odd parity. Each such group’s conditional expectation vanishes at zero starting field and has a polynomially bounded spatial derivative, also after parameter differentiation. The common prefix fields have size \(O_*(\sqrt{h'})\), so the product of these two odd expectations gives \(O_*(h')\).

   On size paths the refined contribution is therefore \(O(c^3h')\le O(\lambda c^2h')\), exactly the required smaller-scale bound. On direct paths the derivative supplies \(\lambda\). The unrefined \(O_*(c^2)\) estimate gives [eq:E4] without the ratio. Extracting defect factors preserves both conclusions.

4. Suppose \(j=3\), for which the target is \(O_\diamond(\lambda c^2)\). Terms with defects or with \(D_\pi\ge4\) already have enough powers of \(c\): on a size path one extra power is bounded by \(\lambda\), and on a direct path the derivative supplies \(\lambda\). It remains to consider one-part partitions with two or three blocks. For \(D=2\), each block has degree three and [eq:E7] gives \(g^2k_0v_{AB}+O_*(c^2)\). For \(D=3\), an odd block degree gives \(O_*(c)\); otherwise all three degrees are two, giving \(\chi^3+O_*(c)\). The two leading monomials after multiplication by \(b_\circ^3\) are precisely those in the \(j=3\) row of the table, so their size contributions cancel. With net factors \(c\) and \(c^2\), respectively, the remaining errors are \(O(c^3)\), which is at most \(O(\lambda c^2)\). Direct derivatives give the same target without this cancellation.

5. Suppose \(j=4\), for which the target is \(O_\diamond(\lambda c)\). Only size terms with no defect and \(D_\pi=2\) lack the required power before cancellation. Such a term has one part, on two blocks of degree four each. Its moment is \(g^2+O_*(c)\). Multiplication by \(b_\circ^4\) gives the last row of the table, so the leading size contribution cancels. The remainder with net factor \(c\) is \(O(c^2)\), at most \(O(\lambda c)\). All other terms have an additional compression power, a defect factor, or the direct derivative factor \(\lambda\).

The exponentially small terminal errors retain the mass factors of their corresponding main terms. Since \(M=\log^3(1/x)\), they are smaller than every fixed power of \(x\). This proves [eq:E4]. ◻