The low-temperature Sherrington–Kirkpatrick free-energy limiting law

Proof. Equations (A1) and (A3) are the pressure, source, pinned-moment, and projection estimates of [19, Lemmas 2.1–2.3 and 3.5, equations (1) and (13)]. To match the terminal-mass-one normalization there, put

The conditional spin means and covariances are unchanged, while H_{au}=a\widetilde H_u. The projection denominator is unchanged as well, since (v/a)\sqrt{m a^2\Delta h}=v\sqrt{m\Delta h}. Appendix A records the pressure and moment conversions. The fixed-size path-limit result is PE, Proposition 2.14.

For (A2), we include the polynomial argument of PE, Appendix, paragraph “Source and pressure calculus; the scope of pinned moments.” Pin the terminal spin and write the Gaussian increments in terms of standard Gaussian drivers, as in the proof of PE, Lemma 2.3. Let \mathsf A_v be the Hessian of the continuation in these drivers and \mathsf B_v its mixed Hessian with the normalized spin source. After the transition densities telescope, the negative logarithm of the pinned driver density is the Gaussian quadratic potential minus aH_\sigma, which is linear, plus \int_0^a F_v\,dv, where F_v is the continuation at the cut v. Consequently

where \mathsf Q is the Hessian of the negative logarithm of the pinned driver density. The driver gradient of L_v is g_v=\mathsf B_v(y-X_v). Thus, for every driver direction z, |z\cdot g_v|^2\le(z^{\mathsf T}\mathsf A_vz)W_v. For the polynomial observable set

Cauchy–Schwarz in v gives |z\cdot G_P|^2\le(z^{\mathsf T}\mathsf Qz)R_P, hence G_P^{\mathsf T}\mathsf Q^{-1}G_P\le R_P. The inverse-Hessian Brascamp–Lieb inequality [6, Theorem 4.1] therefore gives (A2) for p=2. Applying the same variance inequality to the p/2-th absolute power of the centered integral, as in PE, Lemma 2.3, gives the assertion for every fixed p>2.

The zero-field spin gauge transformations act transitively on terminal spins and preserve both L_v and W_v. Because the coefficients of P_v and the weights k_v are deterministic, they also preserve the polynomial integral and R_P. Their conditional laws are consequently independent of the pinned spin, including the mean used for centering. Averaging over that spin yields the full single-path estimate. Bounded-weight approximation and then the fixed-m path limit give the stated range of clocks and weights.

Clocks are bounded for each N, possibly by a power of \log N; (A1) is clock independent. For (A3), we apply PE, equation (13), only in this bounded or polylogarithmic clock regime; together with the polynomial mass and field-length bounds, this gives the polynomial coefficient-ratio bounds needed in its moment iteration, as detailed in Appendix A. Our estimates can always be passed from affine finite-step interpolations, or from inserted fixed cuts, by the fixed-m limit assertion, keeping chosen revelation sides and the joint filling at observed intermediate jump cuts, and using the corresponding integrated inequalities. In particular no discretization rate in m is used.

The pinned inequalities in this lemma concern the full single-path law. When a tree contains additional branches, we first marginalize the branches not used by the test. Random factors retained on other branches are subsequently handled by unconditional Hölder inequalities; they are not made into random weights in (A2).□

Proof. By exchangeability, the observed overlap may be represented using one chosen site. Delete that site’s contribution from the other tested overlaps, retaining the denominator m, and interpolate its Gaussian kernels to those of an independent one-site scalar model with clock K. All other covariance kernels remain fixed. Both endpoints are positive semidefinite, as is their affine interpolation. In the expansion back from the independent endpoint, the covariance insertion is J_{cd}\tau_c\tau_d, where J uses the remaining-row overlap and the mask of the actual matrix clock. The omitted terms are O(m^{-1}) and constant diagonals. The row covariance change is bounded, so Taylor’s integral remainder has the form described above.

At the independent endpoint, averaging the removed spin gives \Psi_K when it is unobserved, and N_{ij,K} when its product on leaves i,j is observed. Expanding the former against the truncated quotient N_{ij,K}/\Psi_K cancels its factor \Psi_K to the retained order. Allocation order symmetry identifies the terms on each final topology and gives (A7). Products here multiply separately averaged scalar coefficients; they do not couple their scalar spins. In particular the linear cavity kernel on a pair cd is

This proof also gives the unestimated remainder representation using one interacting scalar row (the removed site) along the affine interpolation, if symmetries must be used before taking absolute values.□

Proof. PE covariance differentiation, equation (2), represents a derivative of a scalar coefficient by a new pair insertion (K'-K)_{cd}\tau_c\tau_d/2. Allocation order symmetry permits us to combine these with the insertions already present in the quotient. The apparent extra factors \Psi_K(K'-K) have no effect on an observed topology whose spin test is independent of the new labels: after allocating these labels last, every positive-degree term is a derivative of the unit scalar moment and sums to zero. For a standalone per-root sum the same terms can remain, but they are deterministic. Taylor’s integral remainder has bounded scalar coefficients and powers of K'-K; it can also retain ordinary polynomial factors of the original argument h+bQ^*-K'.□

Proof. Differentiating a scalar coefficient inserts \dot K_{ij}\tau_i\tau_j/2. This cancels the -\dot K part of {\cal D}J; terms with no previously attached labels are deterministic. Thus the logarithm contributes

Apply (A7) to replace the quotient by Q^*, with its stated error. The term bQ^*{\cal D}Q^*/2 then cancels the corresponding derivative of the quadratic term in (A6). In particular the choice of e disappears. The terms left are \frac12\sum\dot h Q^*+\frac14\sum\dot b(Q^*)^2, precisely the off-diagonal pressure insertions in (A1).

For the error, all tests have bounded degree and at most one differentiated weight. If \log\Psi is truncated after degree k, the residuals in the difference of the two pressure derivatives have at least k insertion factors. They also have one weight at a split, possibly on an additional pair, bounded by a constant times

the 1 is omitted when e=0. The O(m^{-1}) errors carry the same weights. This follows from the product rule, since differentiating one insertion can remove that factor, and from (A7). If a second reference K' is used, expand the quotient in this gradient first, then recombine by Lemma 2.3. This does not require differentiating K'.

There is one truncation convention to check. Differentiating an insertion lowers its insertion degree by one, whereas differentiating a scalar coefficient does not. Through degree k in the old J factors, the coefficient part of {\cal D}\Psi therefore inserts \frac12\sum(N_{ij,K})_{\le k}\dot K_{ij}, formally including its degree-zero term. In a product with old labels independent of the new placements, that term sums to zero when the new labels are allocated last: it differentiates a unit scalar observation. Standalone per root, its discrepancy is common and deterministic. Thus coefficient differentiation of the logarithm uses the reciprocal jet through degree k, while differentiation of an insertion uses the quotient through degree k-1. All these cancellations take place in the unrestricted signed sums, before imposing split-scale restrictions for estimates.□

Proof. For size changes at fixed clocks and s, compare (m+j)\psi_{m+j} to m\psi_m, with any fixed positive integer j. Interpolate the enlarged kernel from that of m old spins plus j independent scalar rows at K, to its actual value plus independent, spin-constant cumulative covariance j b q^2/2 (assume b q^2 nondecreasing). This addition gives the same pressure shift for both tests. The off-diagonal insertion before the factor 1/2 is

Indeed the normalization of the matrix feature changes from m to m+j while field terms are additive. This difference is bounded. Taylor through order k in the total pressure now gives a polynomial in j of degree \le k, constant term zero; its coefficient linear in j is the compared per-root expectation of (A6), \log\Psi truncated after degree k. For the linear coefficient, group the labeled insertions by their assigned scalar row. The scalar rows are independent on each fixed topology, so their series is \Psi_K^j. If C denotes the matrix counterterm \frac14\sum b(Q^{*2}-q^2), the combined series is \Psi_K^j e^{-jC}. Its coefficient of j is

The deterministic q^2 term cancels between the two tests, leaving (A6). This identity is taken at the same finite total Taylor order as the covariance expansion. Remainders after taking centered differences can be bounded by O(m^{-1}) and products of k+1 factors |\Delta|+|J| at the insertions, under the actual m-spin laws by the same Grönwall comparison. No (m+j) factor is to be put in front of this remainder.□

Proof. We use PE, Lemma 5.8, whose full-support input is [18, Theorem 1.1 and Lemma 2.2]. Its measure has support [0,\xi_*], \xi_*>0; on this support \Gamma_0(\xi)=\xi/T. Its hypotheses are precisely fixed T>1, zero field, and the Ising two-spin scalar objective above, with terminal mass 1. The lemma gives A_0(\xi)\sim \kappa_T\xi/T at 0, \kappa_T>0, and continuity of the quantile on the open mass interval. The measure therefore has support down to 0 with no interior gap or atom at 0.

At the minimizing z, vary z to z+tP_0(z) with smooth P_0 supported in (0,1) (both signs admissible for small t). By scalar (A1) the derivative of the objective is T\int (z-\Gamma_0(Tz))P_0(z)/2=0, giving self-consistency by continuity throughout the support. On [0,\xi_*] viewed from the active side, A_0 is smooth with bounded one-sided derivatives of all orders, A'_0\ge c>0 near zero. Indeed \Gamma_0''=0 in the interior gives A_0=\mathbf E (V_{zzz})^2/(2\mathbf E(V_{zz})^3) almost everywhere by PE, Section 5, Lemma 5.3 (in variance time).

The numerator and denominator are continuous moment functions. The positive-Hessian argument of the same lemma, restricted to a bounded field interval of uniformly positive probability, bounds the denominator away from zero throughout the compact active interval. Their ratio therefore gives a continuous representative of A_0, extending Lipschitzly to both endpoints. Differentiation with the scalar PDE and Itô generator improves this representative one derivative at a time; the positive denominator allows the iteration to every fixed order. The positive initial slope is the initial-density conclusion of PE.

In fact p_{\rm top}=A_0(\xi_*-) belongs to (0,1). Here is a useful argument. Extend by a mass-one annealed clock, and let \rho denote the law of Z_{\xi_*}. The finite measure w(dz)=\rho(dz)/\cosh z is even and log-concave. To see the sign of the potential, take a step approximation ending at \xi_* and telescope its transition weights. After division by \cosh z, the joint density of the Gaussian increments is their Gaussian density times the exponential of negative convex continuation potentials. Each coefficient is a nonnegative mass increment; the unused terminal mass similarly multiplies -\log\cosh z. Marginalizing the increments at fixed terminal field preserves log-concavity [24]. Fixed-clock approximation, as in PE, Proposition 2.14, passes it to w. The limiting field law has a density because its diffusion has bounded drift over positive time. At subsequent time \xi_*+d, the derivative \mathbf E\chi^2, \chi=V_{zz}, is

with G_d the centered Gaussian kernel. The inequality follows by even unimodality (translate overlaps of upper level intervals). At d=0 the two sides agree, so the inequality makes the right second derivative strictly negative. The left second derivative is zero by \Gamma_0(\xi)=\xi/T. The spatial moments in \Gamma''=\mathbf E(V_{zzz}^2-2A_0\chi^3) agree across the endpoint; only the mass changes. Consequently

which gives p_{\rm top}<1. Its positivity follows from the positive initial density and monotonicity of A_0. Thus the base quantile is constant on a nonempty terminal mass interval.□

Proof.

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],

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 (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 (P2) also with smoothing.

More explicitly, for the scalar conditional mean m_t and f(m)=\sqrt{1-m^2}, localization gives

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.□

Proof. For the direct reference, A_0'(0)>0 gives the inverse bounds. The perturbation of its inverse is confined to the initial interval, and differentiating its inverse relation gives the stated O(\lambda) bounds. The linearized PDE and coupled diffusion in the proof of Lemma 3.2 give the bound on \partial_{\log\lambda}q. That proof also differentiates the scalar gap only after subtracting its value at zero, which preserves the factor \xi^2.

For size variations, the normalized smoothed profile below M is independent of \ell; only its terminal datum changes through j=4/\ell. The diffusion representation of \partial_j\widehat V_j has terminal datum \partial_j[j^{-1}\log\cosh(jz)]. It differs from the constant \log2/j^2 by at most C\sqrt{1-\tanh^2(jz)}. The conditional martingale estimate in the preceding proof therefore makes its expectation exponentially small, uniformly in the starting field on a fixed compact time interval. Higher spatial derivatives are uniformly bounded by the step power rules with one bounded smooth terminal insertion and its bounded derivatives; the remaining slots are bounded spin derivatives controlled by PE, Corollary 2.10. If the uniform value error is \varepsilon, global finite-difference interpolation between orders zero and any fixed k>r bounds the derivative of order r by C_{r,k}\varepsilon^{1-r/k}. This proves the asserted exponential bound uniformly over all starting fields. The calculation differentiates only the terminal datum while the mass profile below M stays fixed.

To obtain (P2a), differentiate the scalar second-derivative identity at fixed normalized time. The spatial derivatives just bounded are exponentially small, \widehat V_z and \widehat V_{zzz} are odd, and the normalized mass is O(\xi) near zero. Differentiating the Brownian coupling gives a field change O_p(e^{-cM}\sqrt\xi). Hence \partial_j\widehat\Gamma''_j(\xi)=O(e^{-cM}\xi), and integration from zero proves (P2a). Repeated PDE and generator differentiation bounds the response time derivatives near zero, where the normalized mass profile is smooth with bounded derivatives. Applying the same calculation to the equations linearized in j bounds the mixed derivatives \partial_j\partial_\xi^r\widehat\Gamma_j there. Since A_0'(0)>0, composition with the normalized inverse and the exact rescalings \xi=\ell^2t, \widehat A(\xi)=v/\ell gives the stated one-sided mass derivatives and their size derivatives. Finally, \ell^{1-r}\partial_z^r V(0,0)=\partial_z^r\widehat V_j(0,0), and \partial_\theta j=-j/6. Thus treating the normalized initial spatial derivatives as constants during a size differentiation incurs only exponentially small errors.

It remains to control these derivatives after integration over all masses. Off the terminal plateau, the inverse relation is \widehat A(\xi)=v/\ell, with \xi=\ell^2t(v). Since \partial_\theta\ell=\ell/6, differentiation at fixed v gives

The second term can be large pointwise, but under dv=\ell\widehat A'(\xi)\,d\xi its inverse density cancels. Both resulting integrals are O(M). On the terminal plateau, t=M/\ell^2 and \partial_\theta t=-t/3. Its moving boundary creates no jump term because the clock is continuous there. In particular \int|\partial_\theta t(v)|\,dv\lesssim M^C, with no inverse power of \omega.

For q(v)=\widehat\Gamma_j(\xi), there is also dependence on the terminal mass j=4/\ell. We bound it at fixed normalized time \xi by representing \widehat\Gamma_j(\xi) as the expectation of the product of two terminal spins whose branches separate at that time. This test is bounded by 1 and does not depend on j. In a step approximation, differentiate the logarithm of its positive tree density, telescoped as in PE, Proposition 2.14. Below M all normalized masses are fixed. The derivative \partial_j[j^{-1}\log\cosh(jz)] is uniformly bounded, and the diffusion representation gives the same bound for interior continuation derivatives. Their coefficients have total variation bounded by the number of observed leaves times the terminal mass, independently of the number of steps. The remaining terminal terms are bounded by C(1+|Z_M^1|+|Z_M^2|). Hence the logarithmic density derivative has L^1 norm at most C(1+M), because bounded scalar drift gives \mathbf E|Z_M^i|\lesssim1+M. It follows that

uniformly in the step resolution and in \omega. The resulting uniform Lipschitz bound passes to continuous clocks by PE, Proposition 2.14; it supplies absolute continuity and the same derivative bound almost everywhere without requiring convergence of derivatives.

Off the plateau, the chain rule now reads

After integration in v, the first term again loses its inverse density. Its integral is bounded by C\int_0^M\widehat\Gamma_j'(\xi)\,d\xi\le C, since \widehat\Gamma_j is increasing and lies in [0,1]. The second term has integral O(1+M). On the plateau only this second term remains. Thus q, like t, has an absolutely integrable size derivative of norm O(M^C).□

Proof. Fix \lambda=x^{p_*}, \chi_{\rm lin}=1, and \chi_e=0, and keep \gamma,\ell the same in both systems. First extend the direct system to length 4 by a single field jump at mass \ell. For v>\ell, put K=K_{\rm post}=M/\ell^2,\ q=1,\ b=b_{\rm post}=b(\ell-),\ h=h_{\rm post}=K_{\rm post}-b_{\rm post}+H_{\rm lo}(\ell)+H_{\rm lin}(\ell). On this extension q=1 is a deterministic comparison function, rather than a claim about the finite-time scalar response. We first compare these four extended functions to their smoothed counterparts. For fixed 0<\epsilon<1/21, put

Then

Here and below the norms integrate over v\in(0,4). For the inverse clocks, the area between monotone quantiles equals the area between their mass profiles; the spread jump and density floor contribute O(\omega M^2). Below mass \ell, both scalar times remain in a fixed compact interval, so the earlier O_\epsilon(\omega^{1-\epsilon}) moment and initial-derivative comparisons control the response at a common scalar time and b_\circ. The remaining change in q, caused by evaluating the response at different scalar times, is bounded by \sup|\Gamma'|\,|K_{\rm hard}-K_{\rm smoothed}|. Its integral is therefore controlled by the inverse-clock estimate.

Above \ell, the portion before the smoothed terminal plateau has mass length O(\omega M). Outside that strip, the martingale bound gives q=1-O(e^{-c'M}). Equations (P3) and (P6) then give the asserted L^1 estimates for b,h as well. Since \omega=x^{5+1/4}, the choice \epsilon<1/21 gives (5+1/4)(1-\epsilon)>5. With M=\log^3(1/x), this proves E_\epsilon=O(x^{5+c}) for a fixed c>0.

We next compute the pressure of this extension, denoted by f_{\rm hard}. Condition on arbitrary earlier Hamiltonian values (H_\sigma)_{\sigma\in\{-1,1\}^m}, and set D=h_{\rm post}-h(\ell-), p=\ell/4. If z has independent standard Gaussian coordinates and \Phi is the standard Gaussian distribution function, then

The upper bound is subadditivity for 0<p\le1. For the lower bound, retain the term indexed by \sigma on the orthant \sigma_i z_i\ge0 for every i. These orthants are disjoint, and exponential tilting of each Gaussian coordinate gives the factor \Phi(\ell\sqrt D). Thus the bound is uniform in the conditioned Hamiltonian. After logarithms and division by m\ell, the error lies between \ell^{-1}\log\Phi(\ell\sqrt D) and zero. Here D=M/\ell^2-O(1), so the error is O(e^{-c'M}), independently of m.

Restoring the normalized spin prior multiplies the quantity inside this expectation by 2^{-mp}, whereas the direct partition uses 2^{-m}. Their contribution to the per-site difference is \log2(1/\ell-1/4). The Gaussian expectation contributes \ell D/2. Combining this with the direct and length-four self corrections, and propagating the resulting uniform additive comparison through the earlier continuations, gives

The calculation also holds for the tilted initial revelation, so its explicit constant cancels in \psi. For the corrected untilted pressure, subtract the added tail (4-\ell)b_{\rm post}/4 of \int bq^2/4. The non-logarithmic correction becomes

which is independent of \gamma, as is the \log2 term. Every earlier power mean preserves a uniform additive comparison, including the tilted initial continuation. Finally, (A1) bounds a pressure change by the L^1 change of its clocks. Applying it to the hard and smoothed systems yields, uniformly,

For the scalar trial gap we also need stability of its quadratic integral. Although the smoothed K,h can be of order M, their difference is uniformly bounded: above the mask, K-h=bq-H_{\rm lo}-H_{\rm lin}, while below it K-h=K=O(x). The hard extension has the same bound. Since b\asymp1, the function (K-h)^2/b is uniformly Lipschitz on these paths. Thus its integral changes by O(E_\epsilon) under smoothing, just as the scalar and matrix pressures do.

It remains to compute the hard-extension change. The scalar trial gap f_{\rm spin}(K)+\int b^{-1}(K-h)^2/4-f_m^0, where f_{\rm spin} is the self-corrected one-site pressure, changes by

by the same hard-jump computation applied to the scalar pressure. Indeed K_{\rm post}-h_{\rm post}=b_{\rm post} -H_{\rm lo}(\ell)-H_{\rm lin}(\ell), so the displayed term is

At final strength,

and H_{\rm lin}(\ell)=O(x^3). Decreasing the fixed saving c if necessary incorporates this term in O(x^{5+c}). The parameters, in particular \ell, have been kept identical throughout.□

Proof. First take a finite hierarchy and include all future Gaussian drivers under the normalized positive law. Write the positive revelation masses as v_1<\cdots<v_L\le a, and put F_0=G_m. Let F_i be the continuation after the first i positive-mass revelations, as a function of that Gaussian prefix. In particular F_L is the terminal log partition function divided by a. The transition at v_i contributes \exp\{v_i(F_i-F_{i-1})\}, and the terminal Gibbs weight of a fixed spin \sigma contributes \exp\{aH_\sigma-aF_L\}. Multiplying these factors gives the Gaussian driver density times

Here H_\sigma is linear in the drivers and every continuation is convex in the corresponding Gaussian prefix, hence also in the full driver vector when extended independently of its later coordinates. This is the telescoping calculation in the proof of the single-spin pinned moment estimate in [19, Lemma 2.3]. All physical clocks are constant between their initial revelation and mass x. Thus v_1\ge x, or, if empty levels have been inserted, the identical initial-continuation terms combine to a coefficient at least x. Rescaling mass length does not change this coefficient: the product of a mass and its continuation is (v/a)(aF)=vF. In particular the smoothed length-four path has the same lower bound.

Before the pin, the future normalized transition kernels and terminal Gibbs probabilities integrate to one. The initial driver therefore has its original Gaussian law. The spin gauge transformations preserve G_m and act transitively on terminal spins, so the law of G_m is also unchanged by the pin. This uses the full joint positive law; no inequality under an arbitrary conditioning is required.

For z=G_m+\mathcal E_{x/2}, the independent exponential contributes \exp\{(x/2)(G_m-z)\} on the epigraph G_m\le z. At least x/2 of the negative convex-continuation coefficient remains. The epigraph is convex, and the joint density in the drivers and z is log-concave. Marginalization therefore gives a log-concave density in z, by the log-concave marginal theorem [24]. Fixed-size path approximation passes the claim to the paths in use: the laws converge weakly, their log-concavity inequalities pass on intervals, and the exponential convolution gives a density and zero mass to interval endpoints.

Let

The one-dimensional log-concave estimate of [19, Lemma 7.7] gives two-sided exponential tails on scale d_G, and constants c_1>0 such that \mathbf P(Z\ge c_1d_G)\ge c_1. On the stopped set,

The function e^t-1-t is nonnegative, and on the displayed right-hand event it is at least (s c_1d_G)^2/2. Since \mathbf EZ=0, it follows that

Consequently s d_G\lesssim x^{\varepsilon/2}. The two-sided exponential tails now bound \mathbf E e^{p sZ} for every fixed p. Conditional Jensen over the independent centered exponential transfers this bound to G_m-\mathbf E G_m. Since \mathcal K_m(s)\ge0, this proves the asserted bound for R_m.□

Proof. Move the stencil from Z to the weights. The coefficient of Z_m becomes \sum_jc_jw_{m-j}. Taylor expansion of the shifted weights, together with the defining polynomial identities of the stencil, leaves the derivative in n. Its remainder has size O_r(D^{-r-2}) for each weight and is summed over O(D) sites.

For (4.1), append a zero-input independent spin and interpolate to the completed (m+1)-site model with the adjusted matrix normalization. The change in total cumulative covariance is O(1+M), and the affine interpolation is positive semidefinite. Apply pressure differentiation to both tests in (A4). The spin prior is a probability measure, so the unused spin contributes no additional constant. This gives the displayed bound.□

Proof. Use a direct sweep at center N, the switch (P11), the size sweep to \theta=\log(N'/N), the reverse switch, and the reverse direct sweep at center N'. At both ends the regularization strength is x^{12}; the clocks differ from the bare data (T/\ell^2,0) by O_\circ(x^{12}). The self corrections cancel in the centered transform.

Apply Fubini to the finite sum of the nonnegative stopped derivative integrals, with the site weights above. Their total expectation in \gamma is O(x^{5+c}), so one choice of \gamma has this bound up to a fixed constant. The weights can be normalized by their sum, with an error smaller than any fixed power by increasing the sum–integral order. The weighted form of (I) expressly permits their dependence on \theta. The choice is made for this one dyad comparison and one test s; different tests may use different paths because the bare endpoint laws do not depend on \gamma.

Write

By (4.1), its active site values have oscillation O(sD(1+M))=x^{\varepsilon+1/2-o(1)}. Since \mathcal K_m\ge0 and \sum_mw_m=1+O_k(D^{-k}), whenever \overline{\mathcal K}\le x^\varepsilon, all contributing site counts, and the boundedly many neighboring stencil counts, lie in their individual stops \mathcal K_m\le3x^\varepsilon.

On a size segment use Z_m=m\psi_m(s) in Lemma 4.3. The derivative is

Indeed Z_m=O_\circ(N), by the pressure bounds in (A1). The dotted-clock L^1 bounds and (A1) make the middle term O_\circ(sD), after integration as well. The two auxiliary errors have sizes

up to subpower losses. Choose the fixed stencil order large enough to leave a positive saving. Multiplying the increment bounds by sn=O(x^{-5+\varepsilon}), and including the switches (P11), gives total drift O(x^{\varepsilon+c'}). The bare endpoint approximation costs O_\circ(sN x^{12})=O_\circ(x^{7+\varepsilon}). The savings can be decreased independently of \varepsilon.

At the bare initial endpoint the fixed-temperature exponent theorem of [19, Theorem 1.1], together with the augmented log-concave estimate, gives \mathcal K_m(s)=o(x^\varepsilon) uniformly in the active window. Indeed the variance scale is x^{-2-o(1)} before multiplication by x, and the small-positive-parameter bound gives O(x^{2\varepsilon-o(1)}). Continuity on each segment, and the small switch errors, exclude a first violation of \overline{\mathcal K}\le x^\varepsilon. Only stopped derivative estimates have entered this argument.

Remove the endpoint weights using (4.1); the cost is again O(sD(1+M)). At the final bare endpoint the terminal mass is \ell=(N'/N)^{1/6}, while the covariance is T/\ell^2. The terminal continuation is therefore distributed as F_{N'}^c/\ell, with the same physical inverse temperature. Since x/\ell=N'^{-1/6}, its x-scaled centered law is exactly Y_{N'}. The initial law is Y_N. This proves the claim, including its uniformity throughout the dyad.□

Proof. Fix k. The dyadic comparison gives values of the log Laplace transform at positive arguments. We recover its derivatives at zero by polynomial interpolation, choosing \varepsilon after k so that the interpolation loss is smaller than the comparison saving.

At bare endpoints, the exponent theorem, the augmented tail bound, and conditional Jensen give a scale L_n=x^{-o(1)}, uniform in the dyad, with \mathbb E e^{|Y_n|/L_n}\le C. Hence, for a sufficiently small constant a>0, M_n(z)=\mathbb E e^{zY_n} satisfies |M_n(z)-1|\le1/2 on |z|\le r_n^{\mathrm{an}}:=a/L_n, where its logarithm is analytic and bounded by an absolute constant. Cauchy’s estimate on this disk gives

which is x^{(k+1)\varepsilon-o(1)} at each fixed multiple of x^\varepsilon and uses only the preliminary subpower scale.

Interpolate at the k+1 nonnegative nodes 0,x^\varepsilon,\ldots,kx^\varepsilon. Proposition 4.4 bounds each nonzero-node difference; at zero the difference is zero. Division by the interpolation scale loses at most x^{-k\varepsilon}. Thus every cumulant through order k has dyadic difference bounded by

Take \varepsilon sufficiently small after fixing k, for example with k\varepsilon<c'/2. Both terms save a fixed positive power. Summing a geometric series over adjacent dyads proves convergence at a power rate on powers of two; the comparison with every integer in the next dyad proves the same conclusion for all integers. Moments are polynomials in finitely many cumulants, so they also have power rates. Enlarge the constants over the finitely many remaining sizes to obtain the bounds for every n\ge2.

Let v_n=\operatorname{Var}(Y_n), and let its finite limit be v. If v=0, its power rate would give v_n=O(n^{-a}) for some a>0. The lower bound in the PE exponent theorem gives v_n\ge n^{-2\eta} eventually for every fixed \eta>0, after a harmless adjustment of the exponent notation. Choosing 2\eta<a is a contradiction. Hence v>0.

The bounded variances improve the tail scale to a uniform constant. To see the normalization explicitly, put r_n=n^{-1/6} at a bare system of size n, and let E_n be independent exponential noise of rate r_n/2. The bare version of the pinning calculation above gives a log-concave density for F_n^c+E_n. At a bare path the negative continuation coefficient in that calculation is the terminal mass one, which exceeds r_n/2. Thus

The log-concave tail estimate applies with a common constant scale. Conditional Jensen, using the convex function z\mapsto e^{t|z|}, removes the centered independent noise. Consequently, for some t_0>0,

Finitely many small sizes can be included by decreasing t_0. These bounds give tightness and uniform integrability of every fixed power. They also imply moment determinacy, for example through the resulting factorial moment bounds. Every weak subsequential limit therefore has the same moments and is the same probability law. Thus weak convergence holds along the full sequence.

The probability spin prior changes the original free energy only by the deterministic n\log2. The completion is an independent centered Gaussian of variance T/2, whose scaled L^2 norm is O(n^{-1/6}) and whose scaled variance is T/(2n^{1/3}). It follows that the uncompleted scaled variance has the same positive limit v. Removing that Gaussian and dividing by the exact finite-size standard deviation is now justified by Slutsky’s theorem.□

Proof. Fix k, let m_k(l)=\mathbb EY_l^k, and denote its power-rate limit by m_k. Recall C=\sum_{l\ge2}l^{-2}, \mathbb P(J=l)=l^{-2}/C, and

where the completed systems are conditionally independent given J. Conditional on J=l and copy zero, the other k copies average the product to the k-th centered power of copy zero. Thus

The uniform moment bounds proved above also control every fixed absolute moment of P_k conditional on J=l, uniformly in l.

For \operatorname{Re}z>0, put

If |m_k(l)-m_k|\le C_k l^{-a} for some a>0, the difference series is holomorphic on \operatorname{Re}z>-a, locally absolutely convergent there. The constant part also extends across zero. Indeed, for \operatorname{Re}z>0, the difference between the sum and its integral is

On a compact subset of \operatorname{Re}z>-1, each summand is bounded by a constant times l^{-2-\operatorname{Re}z}. The discrepancy therefore converges locally uniformly and is holomorphic there. Since z\int_1^\infty r^{-1-z}\,dr=1, multiplication by z removes the integral’s pole and gives value one at zero. Combining this with the difference series, B_k extends to \operatorname{Re}z>-c_k for some c_k>0, with B_k(0)=m_k.

Its Taylor series about z=1 has radius strictly greater than one. To evaluate it at zero write z=1-w. Near w=0, differentiation under the expectation is legitimate: for every fixed h and 0<\delta<1, the conditional absolute-moment bound on P_k and \sum_{l\ge2}l^{-2+\delta}(\log l)^h<\infty give a dominating integrable function on |w|\le\delta. The coefficient of w^h in (1-w)e^{wt} is

with the second term omitted at h=0. Hence

and the scalar series is absolutely convergent. The expectation is taken before the infinite sum. Indeed, at each fixed J the bracket series equals e^t-e^t=0; averaging that pointwise sum would therefore fail to recover the positive second moment. The limiting law from Proposition 4.5 proves existence of the measure with these moments, its positive variance proves a_2>0, and its exponential moment bound proves uniqueness.

The final construction uses randomized single-term unbiased estimation with replication conditional on the sampled level. For this method, see [25, Section 2, equation (10), and Section 4]. Here the levels index Taylor coefficients; the integrability and almost-sure finiteness proved below do not assert finite expected work or import the SDE complexity guarantees of that reference.

For the final assertion, draw an independent integer H\ge0 with \mathbb P(H=h)=2^{-h-1}. Conditional on H=h, repeat the whole random-size experiment independently 256^h times, and let \overline A_{k,h} be the sample average of

Return W_k=2^{H+1}\overline A_{k,H}. The estimate t^h/h!\le4^h e^{t/4}, the prior proportional to l^{-2}, and the uniform conditional moments of P_k give \|A_{k,h}\|_2\le C_k4^h, including the shifted term. Independence of the repetitions yields

The means \mathbb EA_{k,h} are absolutely summable by the Taylor argument. Consequently

All priors are fixed and every experiment uses finitely many finite-size Gaussian systems almost surely. Neither a limiting random input nor a choice of subsequence enters the specification.□

Proof of Proposition 5.1. If there are no retained sheets, the assertion is immediate. Otherwise use the decomposition above. By Lemma 5.2, the gap hypothesis implies

on each retained sheet. We invert the two sectors, combine their block projections, and only then estimate their attachment coefficients.

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

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

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

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

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,

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

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

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:

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

The single differences cancel one block denominator. The coefficient of the two-attachment term is bounded pointwise by

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 (H2) passes to the limit with \mu^0=2\,dv.□

Proof of Lemma 5.2. We first calculate the row-zero action by conditional independence. The additive action then comes from a colored covariance variation. Finally, explicit scalar Hessian kernels give positivity and the weighted estimates used in the inverse.

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:

The constant and one-end terms vanish when summed against a row-zero input. The disconnected subtraction in (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 (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 (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

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

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)

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

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

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 (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

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

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

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.□

Proof. Replacing the matrix by the field bq gives a nonnegative square-error interpolation, which proves the first inequality. Throughout the proof normalize \ell to 1 as in (A1): multiply the pressure by \ell, the clocks by \ell^2, and use mass v/\ell. The size rescaling is held fixed; below \theta denotes a new interpolation parameter. We will remove the matrix in a constrained interpolation. The next two inputs control its endpoints: PE bounds the cost of an initial small removal, while scalar coercivity bounds the endpoint at which the matrix has been removed completely.

The finite-propagation comparison used below We use PE, Proposition 6.6, in its ordinary all-mass case I=[0,1] (PE, Definition 6.1 and equations (3)–(6)). Its parameters are a small fixed power slack m_{\rm PE}>0, \delta_{\rm PE}=m_{\rm PE}/10000, and scales satisfying

Here d is PE’s scale parameter, not our field defect. The bare clocks b_0,h_0 must be bounded, nonnegative and nondecreasing, with a fixed positive matrix margin after the comparison; d^2 has a fixed positive gap below T. At open-interval masses z\gtrsim a_{\rm PE}, with fixed comparison constants, the endpoint hypotheses are b_0(z)=T+o(z^2) and h_0(z)=o(z^3).

At strength 0<\lambda_{\rm PE}\le d^2, this comparison removes e(r)=\lambda_{\rm PE}f(\lambda_{\rm PE}r/d^3) from the matrix, adds re(r) to the field, and adds \int e(r)r^2/4 to the objective. It minimizes over onto quantiles of [0,1] whose Lipschitz inverses satisfy

The fixed smooth taper f equals 1 near zero and is of order (1+\log t)/t at infinity. Its logarithmic decay slope p(t)=-tf'(t)/f(t) increases from 0 towards 1, with 1-p\gtrsim(1+\log(1+t))^{-1}. Provided the changed paths are admissible, the minimum exceeds the bare pressure by at most m^{20\delta_{\rm PE}}E, and each minimizing radius is comparable to mass above m^{\delta_{\rm PE}}\sqrt m E/\sqrt{\lambda_{\rm PE}}. In our use, the matrix margin and taper monotonicities ensure admissibility.

A common varying base temperature The preceding conclusion remains valid when the endpoint hypotheses use a common T_m\to T>1, with T_m in a fixed compact subset of (1,\infty) and held fixed throughout each parent/child tree at size m. The scale restrictions and propagation slack stay the same. This sequential extension is proved in [19, Appendix: Companion interfaces, paragraph “A common prelimit temperature”]. We recall its argument to specify how it applies here without a rate of convergence for T_m.

First, the constraints and conditional moment estimates in PE, Section 3 use bounded clocks and fixed comparability constants; its endpoint Taylor hypothesis enters only the subsequent profile argument. At the regular calibration cut in PE, Proposition 4.1, use b_*=T_m+O(W^2), K_*=T_mS_*+O(W^3), and set J_0=T_m\chi^2-1 in PE (26)–(27). The proof divides by \chi K_*\asymp W and absorbs \eta_n|J_0|x_{\rm test}^2/W^2. These operations use the actual T_m, so they introduce no T_m-T error; the conclusion PE (22) likewise uses the actual scalar clock.

In the small-scale proof in PE, Section 5 take time K/T_m, subtract K/T_m in the scalar compactness and uniform scalar linearization lemmas, and put ((b_0-T_m)S+h_0)/x_{\rm test}^3 for the endpoint remainder at test scale x_{\rm test}. All prelimit identities and bounds there are then identical (scalar derivative and strict curvature bounds use only compact temperature and total time bounds); in the rescaled ODE/potential limits T_m converges to T.

In the macroscopic direct result (PE, Section 5, Proposition 5.9) the actual endpoint convergence and limiting variational problem are at T, so its fixed-temperature scalar-minimizer input needs no uniform version. This gives the same positive upper/lower bounds on limiting slopes for the profile contradictions.

Child data, covariance margins (choose with room at the limit), and moving barriers in PE, Section 6 then use PE (4) with that common T_m, and the fixed finite depth subsequence proof applies unchanged. Thus the parameter replacement invokes no theorem uniform up to the critical temperature or change of the fixed propagation slack.

Scalar coercivity We first establish a coercive comparison. If K'=h_*+b z is an admissible scalar path with 0\le z\le C, then

where Q_0 is a fixed clock bound, after annealed extension of the time distributions. Use the base optimal feedback martingale u_s=V_z(s,Z_s) as control in both scalar problems, as verified in PE, Section 5. This gives the lower first variation \frac12\int\Gamma(s)(\alpha_{K'}-\alpha_t)(s)ds, together with the terminal \log\cosh convexity remainder.

Write P(s)=\int_s^{Q_0}(\alpha_{K'}-\alpha_t) and A=\int_0^{Q_0}P(s)^2ds. The terminal displacement is Y=\int(\alpha_{K'}-\alpha_t)u=\int P(s)\chi_s\,dB_s, where \chi_s=V_{zz}(s,Z_s). On bounded clocks, \mathbf E\chi_s^2\ge c, while \chi_s is bounded, so \mathbf EY^2\ge cA and \mathbf EY^4\le CA^2. Also Y is deterministically bounded from its original integral. Choose a fixed compact set for the base terminal argument with complement probability at most c^2/(4C). Cauchy–Schwarz leaves at least cA/2 of \mathbf EY^2 on this set. The second derivative of \log\cosh is bounded below along every displaced segment there. Its convexity remainder is therefore at least c'A, as required in (G2).

The first variation plus penalty change cancels linearly, leaving \tfrac12\int dv\, b(z-q)^2\int_0^1(1-y)[1-b\Gamma'(t+y b(z-q))]dy\ge0 by (P5), without needing monotonicity of z. The lower bound on \mathbf E\chi_s^2 here follows also by comparison to the base T scalar path on a fixed compact including its annealed extension as in (P2).

A constrained interpolation Choose e_0=x^{d_g}, where the small exponent d_g>0 will be fixed below, and set d_{\rm PE}=e_0^{1/4}. Apply PE with d=d_{\rm PE}, strength e_0, and error parameter E=x^{4-d_g/16}d_{\rm PE}=x^{4+3d_g/16}. Taking its fixed power slack sufficiently small relative to d_g satisfies PE (3) and makes m^{20\delta_{\rm PE}}E\le x^{4+c}. The endpoint scale is a_{\rm PE}\asymp x^{1-d_g/16}\gtrsim x. Also e_0/d_{\rm PE}^3\to0, so the taper is identically 1: this comparison removes the constant e_0 and inserts e_0r. Equations (P2)–(P4) and (P9) verify the endpoint and admissibility hypotheses with the common base T_m\to T just discussed. The optimized value, including \int e_0r^2/4, is consequently within x^{4+c} of the bare pressure.

We can pass to onto inverse paths \alpha:[0,2]\to[0,1] with constant bounds \kappa=x^{4+1/4}\le\alpha'\le M_c=x^{-10}, at upper-cost loss O(\kappa+1/M_c). Indeed any radius law on [0,1] can be approximated in first transport moment to that accuracy, by a slight bounded-density spreading within the enlarged interval and mixing a uniform floor. Cost follows by (A1). Keep these latter constraints fixed and minimize

The density box is compact at fixed size, so a minimizer exists. Equation (A1) gives monotonicity and Lipschitz continuity of \Phi, as well as its envelope upper derivative at a minimizer. All paths are admissible. At the two endpoints, \Phi(0)\le f_m(b,h_*)+x^{4+c} by the PE comparison and \Phi(1)\ge F_* by (G2). At every intermediate \theta, trying a constrained approximation to q and replacing the remaining matrix by the field (b-e)q gives \Phi(\theta)\le F_*+O(\kappa+M_c^{-1}).

Fix \theta and a minimizing r. The objective is an averaged pressure, so this choice, its inverse \alpha, and all intervals chosen from them are deterministic before taking the positive-law expectation. Evaluate S,B,D at this changed path. A subscript in radius s means the cut at mass \alpha(s). Put

This empirical scalar clock replaces each remaining matrix overlap by its mean S. It is nonnegative and nondecreasing.

It remains to show that \Phi increases by at most O(x^{4+c}). Its envelope derivative is bounded by the integral of D^2+(S-r)^2. We first use the minimizing density constraints to control S-r, then compare K^{\rm emp} to t. Deterministic radius buffers will give high moments of Q-S; the pair inverse will improve these to the required integrated bound on D^2. The final envelope estimate combines these four steps.

First variation gives a potential H with H'(s)=e(\alpha(s))(s-S_{\alpha(s)}). Indeed the variation against \delta\alpha is -\int H'\delta\alpha/2, by changing variables in (A1). The inverse is bi-Lipschitz at fixed size, and the required path moments are continuous under these perturbations by PE, Section 2. After integration by parts this is a linear functional of the density variation. Exchanging density between two locations, while keeping total mass fixed, gives a constant shift of H for which \alpha'=\kappa on H>0 and \alpha'=M_c on H<0.

A component of H<0 has radius width at most M_c^{-1}. Its endpoint signs and monotonicity put S between its two radius endpoints; at exterior boundaries use 0\le S\le1. On H=0, S=s almost everywhere. We call the complement of the positive components nonfloor. Thus |S-s|\le M_c^{-1} there, while the floor has total mass at most 2\kappa. Integrating H' on a negative component also gives H\ge-C(e_0+\theta)M_c^{-2}.

We will use the following local consequence. Within distance O(h) of a nonfloor point s_0, where M_c^{-1}\ll h, S=s_0+O(h), provided one further interval of length h fits on each required side. For if S exceeded this range, monotonicity would keep s-S of one sign and of size \gtrsim h on such an interval. Since e\asymp e_0+\theta, integrating H' to or from s_0, where H(s_0)\le0, would force a negative value of size \gtrsim(e_0+\theta)h^2. This contradicts the lower bound for H.

A quantitative coarse clock comparison The empirical scalar clock is a full mass path. Its time distribution will be denoted by \alpha_{K^{\rm emp}}, with field time as its argument. We first need the coarse estimate

where c_1>0 is independent of sufficiently small d_g. Here and below these are scalar time profiles, not optimizer inverse paths. There is a monotone 0\le z\le1 with

for some absolute c_0>0. If \sup\widetilde b\lesssim x^{1/10}, take z=0 and use the pressure Lipschitz bound in (A1). Otherwise \widetilde b=(1-\theta)(b-e_0) has b_*:=\inf\widetilde b\gtrsim x^{1/10}, since b-e_0\asymp1. We use the quantitative bounds in the proof of PE, Section 5, Lemma 5.7, as follows.

In that scalar-trial interpolation, denote its interpolation time by \tau, its inverse floor and cap by \eta_m=m^{-1/8} and Q_m=m^{1/8}, and its optimizer by z_\tau. In this paragraph the observables refer to this inner optimizer, with clocks ((1-\tau)\widetilde b,h_*+er+\tau\widetilde b z_\tau). The mismatch satisfies \int(S-z_\tau)^2dv\le\eta_m+Q_m^{-2}. For \tau\ge\varepsilon, after also discarding masses below \varepsilon, the covariance trace satisfies

and the variance of an unnormalized L-interval integral is at most C Q_m/(m b_*\varepsilon^2). Partition the mass interval where S varies by at most \Delta, and discard end strips of width \rho. There are at most C(1+\Delta^{-1}) strips. The omitted masses and times therefore cost O(\varepsilon+\rho(1+\Delta^{-1})). On every retained mass, the earlier and later L-averages give

Finally use D_v^2=\operatorname{Var}(|X_v|^2)+\mathbf E\operatorname{tr}C_v^2 +2\mathbf E X_v^{\mathsf T}C_vX_v, with the last term bounded in integrated mass by 2R^{1/2}. The integrated envelope derivative is consequently at most

With \varepsilon=\Delta=x^{1/100} and \rho=x^{2/100}, each term saves a fixed power of x: for example, R\lesssim x^{128/25} and A\lesssim x^{1/100}+x^{509/200}. These estimates use only bounded monotone actual clocks and the lower bound on b_*, so they are uniform over our minimizing r. The scalar endpoint of this interpolation supplies the asserted trial. We now return to the observables of the outer minimizing r.

Put K'=h_*+er+\widetilde b z and \bar z=(er+\widetilde b z)/b. The exact penalty decomposition is

Apply (G2) to K'=h_*+b\bar z. The trial inequality and \Phi(\theta)\le F_*+O(\kappa+M_c^{-1}) then bound both the squared primitive in (G2) and \int e\widetilde b(r-z)^2dv by O(x^{c_0}), after decreasing c_0.

To pass from the primitive to the profiles, write g=\alpha_{K'}-\alpha_t and P(s)=\int_s^{Q_0}g. The total variation of g is bounded. Averaging over intervals of length a gives \|g\|_1\le C(a+a^{-1}\|P\|_1); boundary intervals have the same O(a) cost. Since \|P\|_1\lesssim x^{c_0/2}, choosing a=x^{c_0/4} gives a power-small profile error. Moreover the cap/floor mismatch and Cauchy–Schwarz give

For monotone clocks, the L^1 distance between their quantiles equals that between their time distributions. Thus this is also the cost of replacing K' by K^{\rm emp}. Choose d_g sufficiently small relative to c_0; a fixed smaller exponent c_1>0 now proves (G3) independently of d_g. Decrease c_1 once more, if needed, to cover the bounded-clock scalar derivative comparison used at the end of the proof; all remaining exponents are chosen after this choice.

Integrated pinned moments and rough concentration Choose \delta=x^{a_g} with 0<a_g\ll c_1, put h=\delta^2, and take d_g sufficiently small relative to a_g. Near a nonfloor point s_0\ge\delta, the local potential bound gives S=s_0+O(h), and hence K^{\rm emp}\ge c\delta. Such a clock cannot occur at mass \ll\delta: monotonicity and the regular initial slope of t would give an area of order \delta^2 between the two time distributions, contradicting (G3), since c_1>2a_g. Thus these neighborhoods and all later cuts have mass \gtrsim\delta. They have room for the required buffers, since s_0\le1+M_c^{-1}, while the radius interval extends to 2.

For fixed p\ge2, on a radius window I there (within [0,5/4]) we have

A reusable residual estimate. We isolate the deterministic blocking calculation also used in Sections 7 and 8. It applies to the full single-path positive law and does not require an optimizing radius.

Application to the integrated pinned moment. We return to the normalized terminal mass a=1 and prove (G4). Here H_s, the source Hessian, is distinct from the optimizer potential H(s). The lemma also applies after an increment on branch \mathsf b is averaged back to its fork, by (6.3). We use \mu'_j\le1, or directly H_s\preceq C_s, giving

For the residual decomposition in (G4), fix a small constant L_*>0 with s+L_*<2 for s\in I. Choose l_0=2^{-J_*}L_* of arbitrarily small inverse-polynomial size. Use the initial block X_{s+l_0}-X_s, the blocks X_{s+2l}-X_{s+l} for l=l_0,2l_0,\ldots,L_*/2, and the terminal residual y-X_{s+L_*}. There are k=O(\log(1/x)) blocks and their sum is exactly y-X_s.

For any pair other than two initial blocks, project the block with the later starting distance, denoted l_{\max}. The anchor A is the other branch’s increment. Write A=|A|\widehat A, with \widehat A=0 when A=0, and apply (A3) to \widehat A. Conditional on the common prefix, the whole other branch is exterior data, so fixing this anchor leaves the projected branch’s normalized future kernel unchanged. Equal starting distances may be resolved in either order. The gap after the fork has field length at least e_0l_{\max} and mass at least c\delta. For a finite block, conditional Jensen converts the residual estimate at its start into the same estimate for its increment.

Fix a small \epsilon>0. At a sufficiently high fixed even moment, Markov’s inequality gives

for any prescribed fixed K. Indeed (A3) gives the same square scale with x^6\log^4N; taking the moment order large relative to K/\epsilon supplies this tail bound. The factor |A|^2 remains in the subsequent integral. On the exceptional event the unnormalized squared product is bounded, since |y|,|X|\le1. Jensen before and after conditional averaging shows that its integral has L^{p/2} norm O(x^{2K/p}). Choose K large enough to make this negligible.

By (6.2)–(6.3), the earlier squared anchor integrated over I has L^{p/2} norm at most C_p l_{\max}. If both blocks are terminal, boundedness gives the same assertion since l_{\max}=L_* is fixed. Thus each projected pair contributes at most C_p x^{6-\epsilon}/(e_0\delta^2). For two initial blocks, bound one by a constant and integrate the other’s square; the cost is O_p(l_0), negligible by the choice of l_0. Summing the logarithmically many pairs in (6.1) proves (G4).

Earlier and later buffers. We next turn the integrated pinned estimate into control at individual cuts. The following bracket argument also records what is needed when the later applications use different radius windows.

Apply the lemma with normalized radius averages over buffers of length comparable to h, before and after cuts near s_0. Their measures are deterministic because the optimizer was fixed before sampling. For an interval of radius length h, the mass weight in (A1) is k(v)=1/[h\alpha'(r(v))]. Consequently (G4) gives

The last estimate holds by taking the fixed exponents small before choosing the subpower loss. The local potential bound gives S_v=s_0+O(h) throughout these buffers. Lemma 6.3 therefore applies with \bar S=s_0, \eta=O(h), and \varepsilon_p=O_p(h). It bounds both \||X_w|^2-s_0\|_p and the averaged past squared increments by O_p(h). Using the same earlier buffer for two nearby cuts gives \||X_w-X_{w'}|^2\|_p=O_p(h).

Now take two branches splitting at s_0, and project their terminal spins to cuts a distance comparable to h later. Their squared norms are s_0+O_{L^p}(h), and their increments from the common ancestor have squared norms O_{L^p}(h). Polarization therefore puts their projected inner product within O_{L^p}(h) of s_0. The two projection errors in (A3) are at most O_{p,\circ}(x^3/(\delta\sqrt{e_0h})), so in particular \|Q-S\|_p=O_p(\delta) at this nonfloor split.

For splits below radius \delta, choose instead a nonfloor cut at radius comparable to a sufficiently large fixed multiple of \delta. To find it, take a mass band of that scale. Equation (G3) and monotonicity put K^{\rm emp} between comparable multiples of \delta on an interior band; the discarded floor has mass at most 2\kappa\ll\delta. At a nonfloor point of this band, h_*=o(\delta) and r=S+O(M_c^{-1}), so K^{\rm emp}=br+o(\delta) and r\asymp\delta, as needed. The same buffers give \||X|^2\|_p=O_p(\delta) there. Project both branches to this cut, using its last comparable subinterval for (A3); Cauchy–Schwarz bounds their projected product by O_p(\delta). Since S\lesssim\delta at the original cut, we obtain, after weakening the exponent, \|Q-S\|_p\lesssim_p x^{a_g/2} at almost every nonfloor split, for every fixed moment order.

Centered inversion and absorption At the sampled clock K^{\rm emp}, the matrix multiplier \widetilde b has the gap required in (H1), of size at least ce_0. To check this, (G3) and the bounded-clock PDE estimates compare scalar derivatives at a common time with error O(x^{c_1}). For v\ge\delta, monotonicity and the same area argument give K^{\rm emp}(v)\gtrsim v. The positive part of b(v)-b_\circ is at most Cc_b\lambda v^2, which is absorbed by (P2) when c_b is small. Below \delta this positive part is o(e_0). Finally, replacing b by b-e gains e\Gamma': wherever this is not comparable to e_0, \Gamma' is small and the original gap is bounded below by a constant. The common-time error is o(e_0), so the gap persists in both cases.

Fix the old pair ij and use (A7) to first order about K^{\rm emp}, with test T_{ij}=Q_{ij}-S_{ij}. The row mismatch is exactly

The test and the deterministic centers S remain fixed throughout this cavity interpolation; the actual row and the independent scalar row at K^{\rm emp} are its positive-semidefinite endpoints. The unknown pair function is the deterministic, topology-indexed expectation u_{ab}=\nu[T_{ij}(Q_{ab}-S_{ab})]. It is unchanged on adding unused labels, by positive-law marginalization. The scalar degree-zero term, including any deterministic offset on a given input topology, vanishes because \nu[T_{ij}]=0. The linear terms give (H1) with multiplier \widetilde b. Its errors are O(x^6) and products of T_{ij} with at least two Q-S insertions.

Apply the inverse, evaluate at ab=ij, and then integrate the old fork mass. Absolute attachment domination bounds new fork densities by a polynomial in e_0^{-1}; a fork reused at ij is now integrated as well. Truncate one error insertion at x^{a_g/4}. Its higher fixed moments, just proved off the floor, make the exceptional probability O(x^6) by Markov’s inequality, and all factors are bounded there. To charge a floor placement, follow its fork to the first time that vertex was introduced. It is either the original ij fork, whose mass is now integrated, or a fresh fork with bounded density. Later reuses retain the ordinary bounded atom weights and introduce no new fixed mass. The floor mass is O(\kappa), so the finite union of these placements also costs O(\kappa) under the dominated allocation measure. On the retained event this leaves x^{a_g/4}|T_{ij}(Q_{ab}-S_{ab})|. Cauchy–Schwarz bounds its expectation by x^{a_g/4}D_{v_{ij}}D_{v_{ab}}; the bounded fork measures and 2D_vD_w\le D_v^2+D_w^2 give the integrated estimate

For nonstep paths, perform this inversion first at fixed m on step approximations, using each approximation’s own S and K^{\rm emp} in the cavity formula. Their moments converge at almost every placement and in mass L^1. The common-time scalar comparison and monotone lower clock bound persist, so the sheet gap remains positive when b,e are sampled from their paths. Pass to the limit inside the dominated integrated error bounds, as in the fixed-resolution limit argument of Proposition 5.1.

Now choose d_g so small that Cd_g<a_g/4 and Cd_g<1/4. The first inequality absorbs the variance term on the right; the second makes e_0^{-C}\kappa smaller than x^4 by a fixed power. At a minimizing r, (A1) gives the envelope upper derivative

Here the mismatch integral is O(\kappa+M_c^{-2}), by the floor mass and the nonfloor cap bound. Integrate over 0\le\theta\le1, and use F_*\le\Phi(1) and \Phi(0)\le f_m^0+x^{4+c}. Decreasing the fixed saving exponent if necessary proves (G1).□

Reduction to localized comparisons.

We first obtain the size-path bounds from the global comparison. Fix a direct endpoint and an unseeded, unmasked path whose global parameter is smaller by a fixed \delta>0. To reach the endpoint, remove \delta c_g from the matrix clock and add \delta c_gq+d to the field clock. Both paths are admissible. Completing the square along their affine interpolation and using (G1) bounds

below by f_{\rm spin}(K)-O_\delta({\cal B}); the corresponding upper bound follows by the trial replacing the remaining matrix to reach K. Here, with subpower losses harmless,

The lower comparison costs \int d^2/(\delta c_g), and the upper comparison costs \int d^2/b. Omitting the mask costs O(\lambda U^2x^3). The linear restoration is independent of \gamma, so (P10) can be integrated between the global parameter endpoints. For some fixed b_s>0, this gives

also when the global interval of averaging is any fixed interior interval leaving crossing room. We can choose b_s,\tau_0 sufficiently small relative to \eta, with further room to absorb errors x^4u\lambda^{k_c/3-1-o(1)} on the right. At final strength with edge off this argument and (P11) give both parts of (C2), with savings independent of further decreases in p_*: even the differentiated estimate uses only O_\circ(x^{4+c_G}/\lambda+\lambda^{2\eta}U^5+x^6/\lambda^2) plus the switch cost from (P11) divided by \lambda.

For (C1), the remaining task is a direct-path dyad u of (P3) with U\lesssim u\le\lambda^{b_\ell}, where b_\ell>0 is sufficiently small relative to b_s. Bounded overlaps already suffice when U is large. Fix all parameters other than \gamma_0,\gamma_u, and start at \gamma_u=0. At this endpoint we minimize over an auxiliary quantile r: the matrix removal is C(r)>0, the field addition is C(r)r, and the penalty is \int C(r)r^2/4. We will specify its density constraints below. Writing X=x/u, it is enough to prove the following two assertions for the final constraints:

1. (i)

   averaged excess of this optimized value over f_m^0 at most O(x^4 u\lambda^{1-2s_0});

2. (ii)

   for the used optimizers, on \operatorname{supp}(c_u), c_u=-\partial_{\gamma_u}b, one has C(r)\gg c_u and

   \mathbf E_{\gamma_0}\left[\sup |r-q|^2\right] \lesssim (u X^2\lambda^{-s_0})^2 . \tag{C3}

To see this, compare a minimizing auxiliary path to the actual endpoint \gamma_u=1. Starting at that actual path, the square interpolation removes C(r)-c_u and adds C(r)r-c_uq. The increase of the restored pressure R between the two endpoints is therefore at most the optimizer excess plus \int C(r)c_u(r-q)^2/[4(C(r)-c_u)]. Averaging this bound controls the nonnegative derivative in (P10); summing over dyads costs at most a logarithm. Before differentiating on a smoothed size path, transfer the same pressure difference by (P11). Its cost is O(x^{5+c}), since only the restoration -\int bq^2/4 affects the difference. Fixed rescalings of mass change these bounds only by bounded factors.

The second requirement is an estimate for every optimizer outside one common null set of the global parameter. It is enough to construct a measurable scalar majorant for all these optimizers; no choice of an optimizer needs to be integrated. The remaining subsections establish these two local assertions and hence complete (C1). Both parts of (C2), with a saving independent of the local strengthening depth, have already been proved above.□

Proof.

The first variation minimizes a linear functional of \alpha', subject to its prescribed lower and upper bounds and total mass one. The multiplier for total mass adds a constant to the primitive {\cal H}. With this choice, the minimizing density equals M_j where {\cal H}<0, and \bar k where {\cal H}>0. These are the cap and floor regions, respectively; the remaining points are contacts. Differentiation of inverse monotone maps is justified as at (G1); the radii at which a jump is crossed form a null set.

The monotonicities of S,B,p, together with the zero initial wall, give the endpoint signs, almost-everywhere contact relations, and the cap bounds in PE (9)–(10). Each cap has radius width at most M_j^{-1}. We call cap and contact points nonfloor points, discarding the null set of contact exceptions. There |S-r|\lesssim_\circ D+O(x^{90}); on a component in the constant part of C, the stronger bound is |S-r|\le M_j^{-1}. Also S\le r at almost every contact. On a negative cap, S cannot exceed its right endpoint radius: otherwise B\ge S^2 and monotonicity make {\cal H}'<0 on a terminal portion of the cap, contradicting its endpoint sign. At the outer endpoint the same argument uses S\le1.□

Proof.

Raw moments under outer regularity. At regular dyads of scale v (mass and radius comparable beyond a sufficiently large fixed multiple of w) one has field slope at least C(v)/O(\log(1/x)) in radius, and

diverges polynomially. We have S=O(v+w) by looking ahead to a nonfloor point (a band of mass comparable to regular v has floor mass o(v)) or by monotonicity below regular scales. We will prove the raw bounds stated above by an absorption argument, following PE (12).

Fix the optimizer, all path parameters, and l\ge1. Define

where t ranges over deterministic revelation cuts, including the specified intermediate cuts, and v(t) is their mass. At fixed size this is a finite deterministic number: w>0 and |X_t|\le1. We will bound it uniformly, without assuming an upper bound on the mass of a cap. On a regular small radius band J\asymp v, we claim

Fork an independent continuation at each radius r. Decompose both residuals into future martingale increments, using the weighted Hessian representation in Lemma 6.2. Retaining its mass weight is necessary here. First use dyadic distances from a sufficiently small inverse power of size up to a fixed fraction of v. Then use doubling scales up to a fixed positive radius and the terminal layer. For a pair of layers with largest scale z\gtrsim v, the Hessian weight is O(z); for two short layers it is O(v).

For short layers, let h' be the starting distance of the later one. Unless both are the first increment, (A3) bounds the squared product by the earlier squared anchor times x^{6-\epsilon}/(v^2C(v)h'), apart from a negligible tail. Here one first uses arbitrarily high fixed moments for the normalized anchor, then truncates; boundedness of the actual increments controls the discarded tail. The shifted-grid and dual conditional-expectation estimates used for (G4) bound the integral of the earlier squared increment by O_l(h'Q_{\rm raw}^2v) in L^l. Thus the three factors in a short-layer contribution are

Two first increments are negligible by the square-sum estimate and the choice of their inverse-power length.

For a larger layer of scale z, there is a preceding regular interval of length and mass comparable to z. Projection costs x^{6-\epsilon}/(z^3C(z)), and the squared opposite anchor has L^l norm O_l(Q_{\rm raw}^2z). Its integrated contribution is therefore at most

The last step uses |J|=O(v) and zC(z)\gtrsim vC(v), also at the fixed outer cutoff. Summing the logarithmically many layer pairs proves (C9).

By (A1) and the floor lower bound, the centered average of L over a future radius window of length comparable to v has L^{2l} norm at most \varepsilon_m vQ_{\rm raw}, where \varepsilon_m\to0 with a power saving. At scale v, its bound is O_\circ([mC(v)k(v)v^4]^{-1/2}); the polynomial divergence proved above absorbs the subpower loss. Conditional expectation back to an earlier cut bounds |X|^2/2 by this average. Its deterministic mean is O(v). For a cut inside the nonregular region, choose the window at a fixed larger regular scale comparable to w; for a regular small cut, choose it at a fixed multiple of that cut’s scale. Macroscopic cuts use |X|\le1. Taking the supremum over these deterministic cuts gives

Finiteness of Q_{\rm raw} now allows absorption and proves its uniform bound. For a small fork, project both branches into the same type of later window and telescope the remaining synchronized layers, as in PE (16). If v is the window scale, the residual is O_{p,\circ}(x^3/(v\sqrt{C(v)}))\ll v. This proves (C8), and the PE (16) postprojection bounds whenever the required preceding regular intervals fit.□

Proof.

Second moments and the enhanced floor.

For the nonfloor contribution to (C10), use the whole-cap trace and bracket estimates PE (20)–(21), with cap width \le M_j^{-1}, (C6) giving the same \Delta S and contact hypotheses, and the admissible changed clocks giving (A1). Enlarge bands to include complete caps with comparable masses by regularity and the small radius width. This gives the first two terms. Floor mass from k uses simply (C8).

For a nonzero extra floor, use sliding radius windows on I, where \alpha'\ge\kappa and S\lesssim u. Equation (A1), integrated in radius, gives

Here C_v denotes the spin covariance and the masses are comparable to u. Let \overline L^{\,+}_{r,h} and \overline L^{\,-}_{r,h} be normalized averages on the following and preceding radius windows of length h=t'u. Applied in mass coordinates, their deterministic weight is 1/(h\alpha'). Let I_h omit radius strips of width h at the ends of I, so both windows remain where \alpha'\ge\kappa. Equation (A1) and Fubini give

The single inverse power of h comes from integrating the moving window, whose multiplicity is at most h.

For two ordered cuts i<j, use the identity

Its noise term has second moment at most

Average this identity to the right and left of the tested cut. The radius-window kernels

are supported in I\times I, and each marginal has density at most one. Monotonicity and \operatorname{osc}_I S=O(u) give \iint K_h^\pm(S_j-S_i)^2\,di\,dj=O(hu^2)=O(t'u^3). Also \iint K_h^\pm(D_i+D_j)^2\le4\int_I D^2. Define

Cauchy–Schwarz therefore bounds the integrated noise, divided by u^3, by C\sqrt{{\cal T}(t'+{\cal D})}. Combining the side-average variance bound, the raw endpoint strips, and D^2\le4\operatorname{Var}L+\mathbb E\operatorname{tr}C_v^2, we obtain

Young’s inequality absorbs the part involving {\cal D}: C\sqrt{{\cal T}(t'+{\cal D})} \le({\cal D}+t')/2+C'{\cal T}. Thus the two-sided sliding estimate is

Boundary strips of width h use the raw moments. Choose t' to balance its first two terms, or use the raw bound if that choice exceeds the allowed interval. After multiplying by the extra density \kappa, its contribution is O(A^{3/2}\sqrt\kappa)\le O(B_f^2). This supplies the final term of (C10).□

Proof.

Inner mass and radius estimates.

For v\gtrsim W the right side of (C10) is polynomially smaller than w^2/W (v C(v)\gtrsim W^3); for the extra floor this uses

On regular bands between a large fixed multiple of w and scale W, (C10) saves a power relative to w^2/(W^2v)^{1/3}. The extra floor is absent there. We also need

on bands with mass between fixed positive multiples of w and radius O(w), separately in mass and radius. In radius, apply the sliding estimate with S=O(w), field slope C_0, and floor lower bound k(O(w)). Its errors vanish because mC_0k(O(w))w^4\gg1; narrow domains and boundary strips use (C8).

In mass, the floor contributes o(w^3), since its mass is o(w) and the raw scale is O(w). On caps and contact, use PE (20)–(21) with that same raw scale. If a cap crosses the lower cutoff cw from below cw/2, use its portion above cw/2. This portion is a fixed fraction of its total mass, because outer comparability puts its upper endpoint at O(w). The whole-cap bound on \Delta S and the constant cap density give the trace estimate on that portion. Thus the trace bound is always used with a positive lower mass cutoff. The condition mC_0w^4\gg1 finishes the mass estimate. Summing these bands and the regular intermediate estimates, and using (C8) below them, gives \int_0^{C'W}D\,dv=o(w^2) for every fixed C'.

A good calibration cut.

Choose a regular band of scale W, above any required fixed multiple of w. The constant removal, (C6), and (C10) give a nonfloor cut * there with S_*\asymp W and D_* polynomially smaller than w^2/W. This selection uses deterministic moments of the fixed optimizer and parameters. It is made before sampling spins and independently of a moment order.

At this same cut, \|Q_*-S_*\|_p\ll W for every fixed p, with a saving exponent independent of p. To prove this, take earlier and later radius buffers of length h'=x^{c_\theta}W, with fixed c_\theta>0 sufficiently small. Integrating (C6) to or from * shows that S varies by O(h') on fixed enlargements: use {\cal H}(*)\le0, the local lower bound {\cal H}\ge-O(C_0/M_j^2), and monotonicity of S. Equations (C9) and (A1) make the centered averages on these buffers polynomially smaller than W. The past/future brackets and common-past comparison in Lemma 6.3 then give the same improvement for \||X|^2-S_*\|_p and for squared path increments in L^p. Finally project both leaves to a central cut a fixed fraction of h' beyond their fork. Equations (A3) and (C8), with synchronized residual layers, bound its residual by O_{p,\circ}(x^3\sqrt{W/h'}/(W\sqrt{C_0})). The two normalized errors just used are bounded, up to subpower losses, by

respectively. Both save a power when c_\theta is small enough, independently of the fixed moment order. This radius-window argument uses the floor lower bound mC(v)k(v)v^4\gg1; it does not require the upper-cap bound used in PE’s original mass-window proof.□

Proof. The constraints (C5) differ from those in the statement of PE’s closure proposition. We therefore use its proof, and verify the replacement inputs below. A block here consists of leaves sharing through a fixed continuity cut of mass comparable to W. A boundary edge joins two such blocks. The tests used in that proof are the fixed finite products of centered overlap factors and projected factors in [19, Definition 4.2]; their natural raw size is denoted by p_0. The constants may depend on that fixed degree. All centers are the actual moments S, held fixed during removal of one site.

1. (i)

   Projection estimates and row changes. Use the selected star * and a regular continuity cut at a large fixed multiple of W to form the blocks. For each projection at scale v, outer comparability supplies a preceding radius interval of length comparable to v, after the low fork, with masses comparable to v. Its field length is at least C(v)v/O(\log m). Above a sufficiently large fixed multiple of W, vC(v)\gtrsim W^3. Define z_v=w^3/(W^{3/2}\sqrt v). The postprojection bound and the star-anchor version of (A3) then read

   \frac{x^3}{v\sqrt{C(v)}} \lesssim (x/w)^3 z_v, \qquad \frac{x^3\sqrt W}{\sqrt{v^3C(v)}} \lesssim (x/w)^3\frac{w^3}{Wv}.

   The factor (x/w)^3 is power small. These are the two boundary estimates in PE (13), (16), and its Section 4, “Analytic input and finite-degree tests”. They also hold for conditional replacements. The raw bounds (C8), the good-star bound, and the second-moment estimates above supply its remaining inputs (12), (17)–(19), with m,w,W in place of n,x,W.

   The clock \widehat K is monotone, bounded, and O(v+w). Above the regularity threshold this follows from outer comparability, (C8), and (P9); below it use monotonicity. Consequently the scalar row sum \chi in that proof is bounded above and below by positive constants, and its two-spin coefficient at distinct high mass v is O(v). The bounded-row comparison of [19, Lemma 4.3] applies: all clocks remain bounded and genuine, and the ordinary field increments on the remaining sites do not change. Their raw projection norms also follow by conditional Jensen from positive overlap tests.

2. (ii)

   The finite-degree expansion and calibration. The row interpolation replaces the cavity-spin kernel by

   h+C(r)r+(b-C(r))Q^{-},

   where Q^{-} is the overlap with the deleted site omitted but with the original normalization. Its difference from \widehat K is (b-C(r))(Q^{-}-S), up to the O(m^{-1}) common-noise convention. Every row interpolation preserves positive covariance. Thus the row expansion in [19, Lemmas 4.5, 4.7 and 4.8], (23)–(25), has the same algebra and the same centers. Here is how its errors are bounded after substitution.

   A high error integrated with bounded split density, including repeated vertices in one dyad, costs a power-small multiple of w^2/W. The error at * has the same bound. Ordinary low factors cost O_p(W), or O_p(w) if their split is at the target scale. A varying low error costs \int_0^{C'W}D=o(w^2); an error at the old target costs D_i+O(m^{-1}), with positive powers of D_i/w after moment interpolation. Deleting the row costs (mw)^{-1}=o(w^3). The observed-block truncation step uses variations power-smaller than Wz_v and p_0z_v/w, which were supplied in (i). These estimates justify the truncation before any signed sums are bounded absolutely.

   For completeness, suppress the common raw test factor p_0. In the terms with three low insertions, an integrated error is o(w^2) and its other low factors cost O(W^2) or O(W^3). If the error is at the old target and w\ll W, one additional factor has scale w. When necessary, this factor is obtained by turning off revelation below a smaller block cut of scale w, then restoring it once; the zero-field sign symmetry cancels the term before restoration. High remainders contain either one high error and two low factors, or two high errors and one low factor. Their bounds are power-small multiples of Ww^2 and w^4/W, respectively. Moments slightly above two suffice for the integrated errors; their interpolation loss is paid by the strict power saving.

   If a third block is already observed, first project the fresh endpoint in the other target block to the star and collapse its scalar row to \chi. Next replace the factor at the second leaf of the third block by its value at the first leaf. The postprojection estimate in (i) gives a power-saved w^3/(Wv) for this replacement. Its scalar coefficient is O(v), and the other low factor costs O(W), so the total is o(w^3)\le o(Ww^2). After this replacement that leaf’s scalar row also collapses to \chi. These are the two row reductions in [19, Lemma 4.7, observed-third-block case].

   Let v_{\rm blk}\asymp W denote the continuity cut defining the blocks. The tests have no overlap factor internal to an old block; its old internal relationships occur only in fixed scalar coefficients and the prescribed topology. Only the unweighted attachment of the first leaf in the third block remains. At its fresh-at-cut value the summand is independent of the deeper attachment, so the signed row-sum rule, including all old-fork atoms and coincidences, gives the factor v_{\rm blk}=O(W). At the star the baseline product difference is O(WD_*); its contribution is therefore

   O(v_{\rm blk}WD_*)=O(W^2D_*)=o(Ww^2).

   Restoring the deeper attachment uses the boundary comparison of [19, Lemma 4.5]; its high-layer bound is

   \varepsilon_m v(Wz_v)(z_v/w) =\varepsilon_m w^5/W^2\le\varepsilon_m Ww^2, \qquad \varepsilon_m\to0,

   where \varepsilon_m has the fixed power saving supplied by (i). If the third block is unobserved, its varying low error costs W\int_0^{O(W)}D=o(Ww^2); a star attachment has the same entry factor O(W). At the old target the two low factors have scale w, with the positive-power modulus in D_i/w described above. Thus both observed and unobserved third blocks obey the required remainder bound.

   For the leading scalar-row replacement, we spell out the test difference in [19, Lemma 4.8, proof of the first replacement]. By linearity take an admissible monomial

   P_a=H Y_a^k E_{oa},\qquad P_c=H Y_c^k E_{oc},\qquad Y_d=X_*\cdot y_d-S_*,\qquad E_{od}=Q_{od}-S_j,

   where H uses only exterior leaves, the incoming edge has split j\in\{i,*\}, and the edge factor is omitted if absent. The leaf c is a fresh relative of a in its block. Let \mathcal F be the complete target-trunk prefix through *, excluding exterior continuations and subsequent target innovations. Define \Delta P_{ac} by averaging P_a-P_c over the entire exterior joint continuation, conditional on \mathcal F and the joint future of a,c. The signed exchange identity is

   \sum_c\gamma_{ac}\nu[(Y_c-Y_a)P_a] =\frac12\sum_c\gamma_{ac}\nu[(Y_c-Y_a)\Delta P_{ac}],

   where \gamma_{ac} is the scalar two-spin coefficient with revelation below the block cut moved to that cut, and the sum uses the signed attachment rule within the block. This is the source’s first symmetrization, followed by conditional expectation.

   Put \mathfrak s_i=w, \mathfrak s_*=W, and let p_H be the raw scale of H. Then p_0=p_HW^k\mathfrak s_j with the incoming edge and p_0=p_HW^k without it. The exterior anchor V=\mathbb E[Hy_o\mid\mathcal F] obeys

   \|V\|_p\le\|H\|_{2p}\|Q_j\|_p^{1/2} \lesssim_p p_H\sqrt{\mathfrak s_j}.

   This is the conditional Cauchy–Schwarz estimate in that same source proof. It uses the ordinary split-j marginal; no exterior terminal spin is conditioned upon. The target projection powers stay outside this averaging and outside the anchor. For a regular innovation interval before a high split of scale v, write the unit-anchor bound as

   \pi_v=O_{p,\circ}\left(\frac{x^3}{\sqrt{v^3C(v)}}\right) \le\varepsilon_m\frac{w^3}{W^{3/2}v},

   with \varepsilon_m power small. Then \|Y_c-Y_a\|_p\lesssim\sqrt W\pi_v. Telescoping P_a-P_c, a changed projection power costs p_0\pi_v/\sqrt W; the changed incoming edge, after the exterior average, costs p_0\pi_v/\sqrt{\mathfrak s_j}. Apply projection before Hölder with the remaining powers, as in PE’s test-difference estimate. Thus

   \|\Delta P_{ac}\|_p \lesssim\frac{\varepsilon_m p_0}{v} \left(\frac{w^3}{W^2}+\frac{w^{5/2}}{W^{3/2}}\right).

   The norm is of the exterior-averaged difference, not of the original incoming-overlap difference. At a distinct split, the attachment variation per dyad and \gamma_{ac} each contribute O(v). Consequently the symmetrized error per dyad is at most

   C v^2\left(\frac{\varepsilon_m w^3}{Wv}\right) \frac{\varepsilon_m p_0}{v} \left(\frac{w^3}{W^2}+\frac{w^{5/2}}{W^{3/2}}\right) \lesssim\varepsilon_m^2p_0Ww^2,

   since w\le W. The saving pays the dyadic sum. For a connected high insertion, one O(v) placement factor suffices, because its integrated high error supplies the additional saved w^2/W; this remains true when the vertices share a dyad.

   This verifies the remainder estimates in PE (24)–(25). No variation of the optimizer occurs in this finite-tree argument. We spell out how the finite recursion preserves their uniform target modulus. For a block \mathscr A at the chosen continuity cut v_{\rm blk}, let \nu^{\rm sc} be the scalar law with revelation below v_{\rm blk} moved to that cut, and put \gamma_{ac}=\nu^{\rm sc}[\tau_a\tau_c]. The connected insertion of PE (23), paired with a test P, is

   \mathcal L(P,a)=\frac12 \sum_{\substack{c,e,f\in\mathscr A\\ e\ne f}} \bigl(\nu^{\rm sc}[\tau_a\tau_c\tau_e\tau_f] -\gamma_{ac}\gamma_{ef}\bigr) (b-C)_{ef}\,\nu[P(Q_{ef}-S_{ef})].

   Here S_{ef} is the actual moment at the split of e,f, and all three added labels use the signed within-block attachment rule. The coefficients and expectations are evaluated on the same final topology. This formula defines the insertion through its pairing; no positive probability interpretation of the signed sum is used.

   At the star, \widehat K_*=T_mS_*+O(W^3) and the changed matrix is T_m+O(W^2). Put J_0=T_m\chi^2-1. The recursion of [19, Lemma 4.9] divides by \chi\widehat K_*\asymp W. The remaining pairing of the same degree at the partner root is removed by applying the row identity to two fresh interchangeable roots absent from P. Their equal leading pairings have total coefficient 2\chi\widehat K_*; every remaining term adds a star factor at cost O(W^{-1}). At depth k, accumulated coefficients are O_k(W^{-k}) and every unresolved term has k new centered star factors. Lemma 7.6 therefore bounds its terminal contribution by O_k(p_0x^{kc_*}). Choose k fixed so that kc_*>2, using w\ge w_c\gg x; then this is o(w^2p_0). Only after choosing k do we fix the finite degrees and Hölder orders. The intermediate errors give

   |\mathcal L(P,a)|\le Cw^2p_0 \left\{\varepsilon_m+(D_i/w)^\vartheta +\varepsilon_m|J_0|/W^2\right\}, \qquad \varepsilon_m\to0,

   with the target term absent for star-only tests. There are only finitely many target exponents, so their positive minimum gives one \vartheta for the whole fixed target band.

   Return to the finite-degree row identity with P=1, as in [19, Lemma 4.10]. Its surviving high corrections are pairings against one star factor, of raw size p_0=W. Apply the preceding bound to those pairings and use \Gamma_{\widehat K}(\widehat K_*)=\chi^2\widehat K_*+O(W^3). Indeed, the second scalar coefficient is a two-edge path through an unobserved third block: its O(W) entry factor multiplies two low factors of size O(W), and bounded finite-label variation controls the third-order remainder. This gives

   S_*=\chi^2T_mS_*+O(W^3) +O\!\left(\varepsilon_m Ww^2(1+|J_0|/W^2)\right).

   Dividing by S_*\asymp W gives

   |J_0|\le CW^2+\varepsilon_m w^2 +\varepsilon_m|J_0|w^2/W^2.

   Since w\le W, the last term is absorbed. Hence J_0=O(W^2), and the calibrated bound is

   |\mathcal L(P,a)|\le Cw^2p_0 \{\varepsilon_m+(D_i/w)^\vartheta\}.

   The common base T_m has remained fixed throughout the finite tree; no rate for T_m-T enters these estimates.

3. (iii)

   Return from the calibration scale to the target. Repeat the mean expansion with the block cut at a large fixed multiple of w. The two regular-dyad bounds make high errors power-smaller than w, and all low factors now have raw scale w. An old-target error contributes a positive power of D_i/w, while the remaining errors tend to zero uniformly over the target band. The surviving mean equation at this smaller block cut is

   S_i-\Gamma_{\widehat K}(\widehat K_i) =2(b_i-C(r_i))\chi_w\, \mathcal L_w(Y_a^{(i)},a)+\operatorname{Rem}_i, \qquad Y_a^{(i)}=X_i\cdot y_a-S_i.

   Here \mathcal L_w is the same connected pairing with block cut of scale w, and \chi_w its scalar row sum. The remainder is bounded by Cw^3\{\varepsilon_m+(D_i/w)^\vartheta\}, uniformly in the fixed target band. The changed matrix coefficient is b_i-C(r_i). To use the calibrated insertion, first remove insertion edges between this cut and the calibration block cut v_{\rm blk}. Their cost is

   O(w)\int_0^{C'W}D=o(w^3).

   Next change the high scalar kernels; above a sufficiently large fixed c'W, the cost is

   O(wW)\int_{c'W}^1D=o(w^3).

   These are precisely the two cutoff changes in PE’s “Return to the target cut”. Allocate the error pair first relative to the sole old root a in its block, and then the summed leaf c; this gives the bounded split density used in these integrals without a high-fork atom. After the scalar-kernel change, independent block flips force c into a’s block, while the error endpoints share a block; if that block is different, the connected scalar covariance vanishes. The remaining pairing is therefore \mathcal L(Y_a^{(i)},a) at the calibration-scale block cut v_{\rm blk}. The test Y_a^{(i)} has raw scale p_0=w and is admissible by its fresh-branch representation. Its calibrated bound is therefore Cw^3\{\varepsilon_m+(D_i/w)^\vartheta\}; b-C and \chi_w are bounded. This proves (7.1). The already proved target-band D^2 bounds then give both integrals in (7.2).

For example, when 0<\vartheta\le2, Hölder bounds its discrepancy term by a constant times (w^{-3}\int D^2)^{\vartheta/2}; a larger exponent may first be decreased using the raw bound D=O(w).□

Proof of Proposition 7.3. We first exclude a failure of comparability, using a PE child for weak removal and Proposition 7.7 for strong removal. The final step identifies the initial slope by a macroscopic transfer.

Profile arguments allow subselection. If loose fixed comparability fails, choose an offending mass w\ge w_c comparable to the supremum of offending masses. This supplies the outer regularity above a fixed multiple of w. Choose the loose constants strictly outside the bounds of the PE direct profile limits. The scale estimates displayed before Lemma 7.4 follow from w\ge w_c.

Step 1: weak removal and feasible barriers.

First suppose C(w)=o(w^2). Apply PE finite propagation to a child acting from the changed path of this optimizer, with the data d',E' of [19, Lemma 6.2] at w, and ordinary PE constraints. The data verification in that lemma uses (3) and the parent strength, cutoff and outer bound (not the parent’s optimization constraints); hence (3)–(4) hold in the child. It has nonnegative monotone bare paths with bounded clocks and a fixed positive all-mass matrix margin, choosing its fixed d'/w small. Its final error is at most m^{20\delta}E'; write z for a final minimizing quantile and \mathcal D(z)=d'^2 f(z/d'). The movement inequality of [19, Lemma 6.3] applies with parent-removal C to any trial satisfying (C5): its proof only uses feasibility, optimality, the slope bound on C and the square interpolation. We explain the corresponding barriers to handle the modified floor. Here d'=c_0 w,\ E'=\min(m^{-1/2}d'^2,m^{-100\delta}C(w)w^3), c_0>0 sufficiently small fixed. In particular the parent changed endpoint still tests against the common T_m by the estimates of [19, Lemma 6.2], since r=O(y) on comparable mass bands of scale y\gtrsim w, r=O(w) below that.

Suppose that r and z are separated by fixed thresholds whose gap is comparable to y, on a mass interval of length comparable to y, where y\gtrsim w. Macro intervals lie strictly inside (0,1). Failure of agreement in normalized measure produces such an interval by monotonicity, and outer regularity gives r\le C'y there. Here agreement means that, on each fixed comparable mass band, the mass of \{|r-z|>\epsilon y\}, divided by y, tends to zero for every fixed \epsilon>0.

The forward constraints are l\le r'\le1/\bar k(r), with l=M_j^{-1}. We have \sup\bar k=o(1) and \int_0^1\bar k=o(w), the latter using w\gtrsim u\sqrt\Lambda. Reserve fixed short portions at the two ends of the separation interval for the following barriers. If z is above r, choose an intermediate threshold H and an early subinterval [a,b], with b-a\asymp y, and set

Choose H\asymp y in the threshold gap, leaving fixed distance from its ends. The rise has fixed slope s=l+H/(b-a), and eventually s\sup\bar k\le1. Also l+H<1, so \max(r,b_\uparrow) preserves both endpoints and both slope bounds.

If z is below r, use b_\downarrow(v)=H+lv until a late point a. Thereafter solve

until this curve meets 1-(1-v)l, and follow that line after the meeting. The ramp has derivative 1/\bar k. Since l\sup\bar k<1/2, its mass length is at most 2\int_0^1\bar k=o(w), so it fits in the reserved part of the interval. The trial \min(r,b_\downarrow) again preserves both endpoints and both bounds. In both constructions choose the threshold and reserved portions as in [19, Lemma 6.4], so every moved point stays distance \gtrsim y from z. Upward motion has radii O(y), with r\asymp y if y/w\to\infty; downward motion has both parent radii comparable to y.

If w\ge\lambda^{k_c/4}u, full clipping already satisfies the strict movement fraction (displaced distance less than |z-r|\mathcal D(z)/(\mathcal D(z)+C(r))). For y/w\to\infty, y\mathcal D(y)\gtrsim w^3\gg y C(y) since y C(y)\lesssim_\circ u^3\Lambda; at y\asymp w, C(0)\ll w^2. Thus for z\le O(y) the ratio of strengths is sufficient with our placement margin, and for z/y\to\infty use monotonicity of z\mathcal D(z) to permit any fixed O(y) displacement.

In the remaining case w<\lambda^{k_c/4}u we need separation tests only up to y\le m^{2\delta}w\ll u, since the child is already comparable above m^\delta d' by finite propagation. All active radii are now below the enhanced patch. Use the partial interpolation of [19, Lemma 6.4] with fraction a small constant times \min(1,\mathcal D(y)/C(y)), forward for upwards, affine in \int_0^r k for downwards. On the active portions \bar k=k; feasibility and the strict fraction proof in that lemma use just l as lower slope, convexity of the PE k in its integrated coordinate, C(r)\asymp C(y) for r\asymp y, C(0)=o(w^2) at y\asymp w, and monotonicity of z\mathcal D(z). Hence they are unchanged.

For either trial write \widehat r=r+\omega(z-r). The movement inequality of [19, Lemma 6.3] states

The constructions just verified make its left side at least c y^3\min(C(y),\mathcal D(y))\gg m^{20\delta}E'. Thus no fixed threshold separation can occur. The resulting agreement in normalized measure gives the child outer regularity above w when w=o(1), and agreement on compact positive bands of scale w. Its direct-test hypotheses hold there with \mathcal D(w)\asymp w^2 and mE'^2\lesssim w^4. Apply [19, Theorem 5.1 and Proposition 5.9] to the child. Its continuous limiting profile transfers to the parent pointwise: bracket any tested mass between two nearby continuity masses, use agreement in measure to select comparison points on either side, and then use monotonicity of r,z. Shrinking the bracket gives the child’s loose bounds also at the offending point. Near mass one use r\le1 as well. This excludes a weak-removal violation.

Step 2: strong removal and the profile contradiction.

Suppose C(w)\gtrsim w^2. Then w\le W=\max(w,\sqrt{C(w)})\asymp u\sqrt\Lambda=o(u), so the conditional analytic hypotheses hold. The bare endpoints satisfy (P9) on the regular exterior. Lemma 7.6 and Proposition 7.7 therefore provide the common calibration cut and the closure in both coordinates required next.

We use this closure to identify the scaled optimizer profile, following PE (28)–(29). On each compact band in the scaled radius, p=0 and C=C_0. Cap widths remain negligible after multiplication by C_0/w^3, and the floor mass on each such band is o(w). Equation (C6) has no additional state-dependent factor there.

Subselect the distribution of r/w under mass divided by w. It converges locally to a Radon measure on [0,\infty), whose cumulative function we denote by U_1. Outer regularity gives U_1(t)\asymp t for large t. The scalar clock \widehat K/(T_mw) has the same limiting distribution. Indeed on compact positive bands of scaled mass, r,S=O(w), |S-r|\le M_j^{-1} almost everywhere off the negligible floor, and ((b-T_m)S+h)/w^3=o(1). Very small scaled masses contribute at most their length. Monotonicity and the comparison on large fixed bands exclude far scaled masses from any fixed clock band.

The profile contradiction.

Set

extending the scalar clock if necessary. Let a_2,a_4 be subsequential limits of its second and fourth spatial continuation derivatives at zero time and field. The scalar derivative formula and [19, Lemma 5.3], applied to the bounded normalized clocks, give

Both constants have fixed positive lower and upper bounds, and the second derivatives are locally bounded. To see the limit, oddness and bounded time and space derivatives give V_{zzz}(T_mwt,Z)=a_{4,m}Z+O_p(w^{3/2}), while \mathbb EZ^2/w\to Tt. The rescaled clock mass converges at continuity points of U_1.

Choose a nonfloor point of regular scale comparable to w. Such points exist because the local floor mass is small. There S\asymp w and \widehat K-T_mS=o(w^3), since C_0|S-r|/w^3=o(1). Equation (7.1) bounds g_m at an argument bounded above and below by positive constants. Together with g_m(0)=0 and the curvature bound, this bounds its initial slope. Pass to a C^1_{\rm loc} limit g. The mass estimate in (7.2) allows nonfloor approximants to every support point, giving g=0 on the support; at zero use g(0)=0.

Above the support infimum, subselect further so that S_{\alpha(wt)}/w\to s(t) almost everywhere. On compact intervals there, \alpha(wt)\asymp w. The closure identity is

The error is bounded and tends to zero in local L^1(dt), by the radius estimate in (7.2) and the bare-clock bound. If C_0/w^2\to h_1\in(0,\infty), it gives

If C_0/w^2\to\infty, it instead gives S_{\alpha(wt)}=wt+o(w) and \widehat K/(T_mw)=t+o(1); the derivative limit is then -2Tg(t). These derivative sequences are locally bounded and converge in local L^1.

On an interior support gap, the potential \mathcal H(wt)/w^4 therefore has a nonnegative limiting primitive that vanishes at both ends. Nonfloor approximants pin the endpoint values to zero; for a negative cap, use its nearest right zero within M_j^{-1}. The same cap-width bound excludes a negative excursion inside the gap. If the left endpoint is a first atom, choose approximants after a fixed positive amount of scaled mass has accumulated. The derivative bound then holds between them and the gap, even when the atom is at zero. All these estimates are used only where scaled radii are bounded and scaled masses are bounded above and below.

Monotonicity gives s(t) in the closed gap interval almost everywhere. A positive derivative of g at the left endpoint would force the limiting primitive to decrease immediately to its right. For finite h_1, the identity g(s(t))=h_1(s(t)-t)/T forces s(t)>t, whether s(t) is near that endpoint or farther into the gap. For infinite strength, the conclusion follows from the derivative -2Tg(t). Both contradict nonnegativity and the zero endpoint value. Reflection excludes a positive derivative at the right endpoint as well. But on a gap of length L, the curvature is g''(t)=At-\mathrm{const} and the endpoint values are zero, so the sum of endpoint derivatives is AL^2/6>0. No gap is possible.

Above the support infimum l, we thus have g=0 and U_1(t)=At/B. If l>0, then g''=At below l and g(l)=g'(l)=0, contradicting g(0)=0. Consequently U_1(t)=At/B throughout. This is the linear quantile limit PE (28), with the same fixed slope bounds. It contradicts our choice of loose comparability constants and proves the first part of (C7).

The first part of (C7) has now been proved for every optimizer. In particular Lemmas 7.4 and 7.5 can henceforth be used with w=w_c and fixed uniform comparability constants. This is the version of (C8)–(C10) used in the strengthening argument below; it does not invoke the strong-removal closure again.

Step 3: identification of the initial slope.

Repeat the weak-removal transfer at w=u, now without assuming a violation. We identify the child’s linear slope by its macroscopic clock limit. First, S_{\rm child}-z\to0 in mass, by PE (18), geometry, and regularity on each fixed interior band. The required error estimates are

Compare z to a further ordinary PE minimizer at final strength, with fixed small removal parameter d''>0 and E''=m^{-1/2}d''^2. Its macroscopic endpoint hypothesis PE (4) follows from (P9) and the vanishing preceding removals; all-mass admissibility and a positive covariance margin remain. Finite propagation gives error m^{20\delta}E''\ll u^3, and [19, Proposition 5.9] gives the further minimizer’s limiting quantile q_T.

The ordinary PE barriers transfer this limit to z. At a fixed interior mass scale, full clipping is feasible with the child’s minimum slope 1/M(v); both forced lengths are o(1). The movement fraction holds because \sup\mathcal D=o(1), whereas the newest removal is bounded below. A fixed threshold separation would cost \gtrsim\inf\mathcal D\gtrsim u^3, contradicting the propagation error. Thus h_{\rm child}+b_{\rm child}S_{\rm child}\to Tq_T in macroscopic L^1.

Bounded-clock PDE stability, as at (G3), now gives convergence of the scalar spatial derivatives at zero. Equivalently one can use drift comparison after L^1 convergence of the inverse-clock distributions. The slope formula A/B in PE (29), together with [19, Lemma 5.8], therefore identifies the child’s inverse-profile slope as \kappa_T. Agreement transfers its linear quantile limit to r; meanwhile q(v)/v\to1/\kappa_T by (P2) and the normalized inverse construction. This proves the second part of (C7). On compact positive radius bands of scale u inside the constant-removal region, it also gives S-r=o(u) uniformly, by (C6), monotonicity, and the local o(u) floor mass.□

Proof.

Let the old optimizer have radius law k\,dr+\nu. On a fixed enlargement of I_j, convolve the fraction c'\kappa_j of \nu with the symmetric uniform kernel of radius \sigma=\min(c_{\rm small}u,C_{\rm big}u a_{j-1}Z). Choose the fixed constants, depending on the stage, so that all motion stays inside the preceding controlled band and the constant part of C. Every radius interval of length 2\sigma used here carries mass \gtrsim\sigma. This follows from (C12) and the positive lower slope of \beta_0; if the fixed cutoff determines \sigma, use (C7) instead. Subtracting the density k does not affect this lower bound. Taking c' sufficiently large therefore supplies density at least \kappa_j throughout I_j. The convolved fraction is o(1), so the new upper density bound is at most twice the old one.

Write r_{\rm old},r_{\rm new} for the two quantiles. Conditional Jensen shows that the new radius law dominates the old in convex order and has the same mean. For every a'\in[0,1], the upper-tail quantile formula

therefore gives a nonnegative upper-tail integral of r_{\rm new}-r_{\rm old}. Integration by parts against the nondecreasing moment S_t at each forward interpolation time yields

Since all motion occurs where C=C_0, (A1) makes the pressure change nonpositive. Only the penalty must be paid; it is O(C_0\kappa_ju\sigma^2)\lesssim x^4u\Lambda^{1/3}Z^2. Averaging and starting from the bound before (C5) gives the asserted cost at every stage.

We next obtain a norm bound common to every new minimizer. Write g=\gamma_0, let F_j(g,\alpha) be the objective with the inverse profile fixed, and put V_j(g)=\min_\alpha F_j(g,\alpha). The admissible density box is independent of g. Equation (A1), followed by its fixed-size path limit, makes each frozen objective differentiable and bounds its derivative uniformly. The minimum V_j is therefore Lipschitz.

At a differentiability point of V_j, fix any minimizing \alpha. The function g'\mapsto F_j(g',\alpha)-V_j(g') is nonnegative and vanishes at g. Its two one-sided derivatives give

The exceptional set is the nondifferentiability set of the single function V_j; it is independent of the minimizer.

Restore the scalar terms by setting

For each minimizer define

The aligned matrix and field variation in (P10) gives \mathcal R_j'(g)\ge c\lambda\mathcal N^2. Above the mask this is the completed square c_g[D^2+(S-q)^2]/4; below it the field stays fixed and the extra term is nonnegative. Thus the same derivative bounds \mathcal N for every minimizer at g.

Choose an upper parameter endpoint outside the common null set in a fixed positive-length margin beyond the next averaging interval, where the optimized excess is at most a fixed multiple of its average on that margin. At the lower endpoint use the nonnegativity of the excess. Integrating \mathcal R_j' and using the direct pressure comparison gives

This notation denotes a bound by the common measurable derivative, not integration of a chosen optimizer. The endpoint depends only on the stage and the fixed path data.

Choose 0<h_e\ll\min(1,c_G), with p_*\ll h_e. The bound on \mathcal B, with \zeta sufficiently small, now permits the following explicit common majorant. Off the common null set take

using the lower-bound constant c above, and set Z=1 on that null set. If \lambda=x^{p_*}, u\le x^{h_e}, and \chi_e\le x^{h_e}, use the smaller denominator c\lambda x^{4+h_e/2} instead. The integrated derivative bound gives \mathbf E Z^2\lesssim1 in both cases. Thus Z\ge1, and simultaneously for every minimizer outside that set,

Only direct paths occur here. The majorant may change at the next stage. Restricting to slightly smaller deterministic parameter intervals leaves room beyond [1/3,2/3], since there are only finitely many stages. The only use of the preceding-stage accuracy was to pay for the new floor; no estimate at the new stage was assumed.□

Proof. Let Q_0 be a fixed bound for the extended clocks. At time t\asymp u, PDE drift comparison bounds the difference of continuation derivatives in sup norm by O(\int_t^{Q_0}|\delta\alpha_v|\,dv); odd derivatives gain an additional factor |z|. For a bounded smooth even observable, changing the forward diffusion costs at most

Indeed the drift difference is odd and bounded by the indicated coefficient times |Z_v|. The propagated spatial gradient is also odd with bounded derivative, and \mathbb EZ_v^2=O(v). These comparisons require only bounded spatial drift derivatives on the bounded time interval.

Now use \Gamma''=\mathbb EV_{zzz}^2-2\alpha_t\mathbb EV_{zz}^3. The two masses are O(u), and their difference is O(d_0). Oddness in the first term and the preceding transport estimates leave, to order d_0, only the mass-distribution difference in the second term. Multiplying by k_*'^2 gives the displayed coefficient A_*. The chain-rule term involving k_*''=O(\lambda u) uses the same comparison for \Gamma_s'-\Gamma_*' and is o(d_0). This proves (C14).□

Proof.

Fix the optimizer and the moment order p. Choose a finite sequence of nested buffers B_0\supset B_1\supset\cdots\supset B_{n_{\rm buf}}, with successive boundaries separated by fixed fractions of b_{\rm buf}. Their number will be fixed below, before taking the size limit. Define

The cuts in this supremum are deterministic and lie on one path. Boundedness gives finiteness at fixed size, and (C8) gives {\cal Q}_0=O_p(L^{-1}). To estimate cuts in B_{k+1}, use only windows lying in B_k. If such a window J has radius length \Delta\gtrsim d_0, the weighted-layer calculation for (C9) gives

For layers ending within radius distance O(u) and still at comparable scales, projection after distance t' costs O_\circ(x^6/(C_0ut')) times the earlier squared increment, including the Hessian weight. The normalized-anchor truncation in Paragraph 6 retains that increment; it requires no higher moment of {\cal Q}_k. When t'\lesssim\Delta, (6.2) bounds its integral by O_p(t'{\cal Q}_k\Delta), using terminal increments on an interval of length O(\Delta) inside B_k. When t'\gtrsim\Delta, its pointwise norm is O_p(t'{\cal Q}_k), giving the same integrated bound. Layers leaving B_k start at distance t'\gtrsim u\lambda^\rho and use (C8); this accounts for \lambda^{-\rho}. The first inverse-power increments are negligible by the square-sum estimate. Outer layers at scale v\gtrsim u cost O_{p,\circ}(x^6/(vC(v))) per unit radius length, which also fits the displayed bound.

Using \alpha'\ge\kappa_j, normalized L averages on such intervals have centered norm relative to \Delta at most O_{p,\circ}((A/L)^{3/2}\kappa_j^{-1/2})({\cal Q}_k+\lambda^{-\rho})^{1/2}. The prefactor, before subpower losses, is at most \lambda^{0.18} by (C11). For two cuts in B_{k+1}, take earlier and later windows of length comparable to \Delta=\max(d_0,\text{their radius separation}). The brackets in Lemma 6.3 then control both centered square norms and their squared increment. The deterministic oscillation of S costs O(\Delta). After absorbing subpower losses and the \lambda^{-\rho} term, this is the recurrence

There is a fixed K with {\cal Q}_0\lesssim_p\lambda^{-K}. After n_{\rm buf} iterations its potentially growing term is bounded by a constant times \lambda^{0.2(1-2^{-n_{\rm buf}})-K2^{-n_{\rm buf}}}. Choose the fixed n_{\rm buf} so that this exponent is positive. This proves bounded ratios on B_{n_{\rm buf}}, and the same brackets give \||X_i|^2-S_i\|_p\lesssim_p d_0.

Project both leaves to radius distance comparable to d_0 after their fork. The synchronized later layers cost \lesssim_{p,\circ}uA^{3/2}/\sqrt L\ll d_0. Polarization controls the projected inner product, since the squared distance of the two branches through their common prefix is O_p(d_0). This proves the first assertion of (C15).

Second moments in radius and in mass.

For the second bound, over each box we have

These estimates follow from (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 (C10) is now

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 (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

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 (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 (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 (C15) in both measures.

The alias error.

For the final bound of (C15), it remains by (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, (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 (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 (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.□

Proof.

Regular bands use (C10), with y_c\gtrsim\sqrt A; the inner region v\lesssim w_c uses (C8) for the first line, also on floor (of mass O(u A)) for the second.

For contact and comparable-endpoint caps in inner dyads v/u\asymp t'\le O(y_c), (20)–(21) with constant field slope lower bound C_0 and raw scale u y_c give for the dyadic mass average of D^2/u^2 at most

apart from negligible cap-width and strip errors. Clip also by O(y_c^2); summation weighted by t' costs O_\circ(A^{3/2}y_c).

For noncomparable-endpoint caps meeting the region the upper masses are O(u y_c) by (C7) and the width bound. Slice each such cap geometrically into units at masses comparable to their lengths, of scale u t'. On one unit the whole-cap \Delta S\le M_j^{-1} trace proof and constant cap density cost an additional factor O(y_c/t') in both trace bounds of (20). Use one height bin there: in (21)’s proof the squared mean-oscillation term costs negligible since S varies by at most M_j^{-1}, including for reverse ordering (its fourth moment estimate uses (C8)). Thus the same average is bounded by O_\circ(A^3 y_c/t'^3), again clipped by O(y_c^2). There are boundedly many such units per mass dyad by disjointness; summation gives O_\circ(A y_c^{5/3}). All band calculations can stop at v\sim x^{10} with negligible cost. This proves (C16).

For completeness, the additional broad-cap cost can be seen directly. Writing y=y_c, its normalized mass sum is bounded by

The two terms meet at t'_0=A y^{-1/3}. Below this scale the geometric sum is at most Cy^2t'_0; above it the sum is at most CA^3y/(t'_0)^2. Both equal O(Ay^{5/3}). If the crossover is outside the finite summation range, the corresponding one-sided geometric bound is only smaller. The original-floor raw contribution is at most O(Ay^2), absorbed in this same term since y=o(1). The analogous crossover for comparable caps is t'_0=A^{3/2}/y, giving O(A^{3/2}y).

The trace conversion on a sliced cap uses the lower mass bound of that slice. The whole-cap numerator is divided by its smaller slice mass, which is precisely the factor y/t' already paid above. No trace estimate is applied below its lower mass cutoff. These are mass estimates; the radius estimates used in Lemma 7.11 and in the coarse profile proof were proved separately by radius slides.□

Proof. The first option in (C17) is the postprojection bound obtained after (C9). The actual field gives the second option: dh\gtrsim\lambda w^2\,dv on comparable active intervals by (P4) and bounded active density; the edge ramp loss is absorbed there. Even at the plateau or a macroscopic band, a preceding active interval is available at a comparable scale. Telescope both branches in regular layers to obtain the stated minimum.

We next estimate the centered primitive. On the active part, (A1) gives

For the first bound integrate \mathbb EW using PE (20) on comparable caps and contact, and in radius on the floor. Complete a partial cap before estimating it. This option also holds on the plateau. The second option uses the actual-field slope just given; balancing the two in w/u gives the last expression. Its ratio to u^2G_1 is power small by (C18) below.

For the plateau, first suppose \chi_e>0. From (C13), {\cal N}\le x^2\lambda^{-s_0}Z\le U^2Z. On a plateau strip of width d', Cauchy–Schwarz and monotonicity compare S to the constant reference q on neighboring intervals of length comparable to d'. The denominator is \sqrt{d'}, so the resulting oscillation bound is

On the capped left strip, use also the preceding active interval of mass comparable to d_-. Its reference varies by at most O(E_b)=O(U^2/\sqrt{d_-}), which fits the same bound. On the capped right strip, boundedness suffices, because its width is d_+=U^4 and U^2/\sqrt{d_+}=1.

The edge slope on this strip is dh/dv\gtrsim\lambda_1\chi_e U^2/(d')^{3/2}. Thus (A1) bounds the squared primitive norm there by

Summing the dyadic strips gives the edge bound O_\circ(x^3\lambda_1^{-1/2}\chi_e^{-1/2}\sqrt Z). When \chi_e=1, the auxiliary-removal bound at w\asymp1 is O_\circ(x^3u^{-3/2}\Lambda^{-1/2}). Taking the smaller bound, and using \min(a,b)\le a^{2/3}b^{1/3}, gives

Equation (C18) again makes its ratio to u^2G_1 power small, including the dependence on Z.

Other values of \chi_e occur only at \lambda=x^{p_*}, with \chi_{\rm lin}=1. The violation bound supplies the useful lower bound

There are three cases.

1. (a)

   If \chi_e\ge x^{h_e}, the edge bound divided by this lower bound is at most

   O_\circ\left( x^{1-h_e/2}\lambda_1^{-1/2}\lambda^{s_0-1/2}/\sqrt Z\right)=o(1).
2. (b)

   If \chi_e<x^{h_e} but u\ge x^{h_e}, the linear seed gives primitive norm O(x^{3/2}) by (A1) and boundedness of S. Since G_1\ge\lambda, its ratio to the target is at most O(x^{3/2-2h_e}\lambda^{-1})=o(1).

3. (c)

   If both \chi_e<x^{h_e} and u<x^{h_e}, remove strips of width \delta=x^{2+h_e/8} at the two plateau ends. Their primitive norm is O(\delta) by boundedness, and their ratio to the target is

   O\left(x^{h_e/8}\lambda^{s_0-1/2}/Z\right)=o(1).

   On the remaining plateau, the stronger part of (C13) and the same neighboring-interval argument give

   \operatorname{osc}S \lesssim x^{2+h_e/4}Z/\sqrt\delta =x^{1+3h_e/16}Z.

   The linear slope is dh/dv\gtrsim x^3, so (A1) gives primitive norm at most O(x^{2+3h_e/32}\sqrt Z). Its ratio to the target is O(x^{3h_e/32}\lambda^{s_0-1/2}/\sqrt Z)=o(1).

All displayed ratios have fixed positive power margins because Z\ge1, h_e is small and fixed, and p_*\ll h_e. This proves the plateau assertion in every case.□

Proof.

Write X=\lambda^\xi, with \xi\ge s_*/2-o(1). For terms whose numerator does not depend on L, monotonicity of G_1 and P_1=LG_1 reduces the estimate to L=a_{\rm goal}. For LT_1/G_1, also check L=\sqrt\lambda and L=1; these are the remaining possible extrema of the piecewise ratio, and the former requires \xi\ge s_0/2+1/4. The Z-dependent term is treated separately below. Finally AT_1\lesssim T_2; inserting the powers of A shows that the X^3A and \lambda^{-2\rho}A^3 ratios have larger margins than the principal ratios computed next, for sufficiently small \rho.

The tables below give the three principal ratio calculations. At s_*=s_0=1/2, \eta=0, k_c=0.03, and with the arbitrarily small \zeta-loss suppressed, write e(F) for the power of \lambda in F. At L=a_{\rm goal}, e(P_1)=2\xi-1/2+e(G_1), and the required piecewise functions are

The first two pieces for T_2 come from Ay_c^{5/3}, and the last from B_f^2. Subtracting the denominator exponents shows that all three minima occur at \xi=1/2:

On the final intervals the relative exponents have positive slopes. Thus strictly positive margins persist for the prescribed small \tau,\eta,\tau_0, and for sufficiently small \rho, \zeta/p_*, and fixed-order moment interpolation losses, chosen in the stated parameter order. The number of strengthening stages is already fixed at this point. None of these choices requires decreasing p_* again.

The dependence on an unbounded Z in the primitive estimate can also be checked directly. Put Y=X\sqrt Z=\lambda^\xi and L_0=Y^2\lambda^{-s_0}\le L\le1, so \xi\ge s_0/2. Since G_1 is nondecreasing in L, the same piecewise calculation gives

The minimum occurs at \xi=s_0/2+1/4. Thus a large majorant only improves this ratio; no pointwise upper bound on Z was used.□

Proof.

Apply (A7) through degree two at K_s, with the untilted law, the two observed labels a,b, and test 1. Each mismatch insertion is (b-C)\delta. Linear terms vanish termwise after averaging spins under the actual law. For the remainders, expand the cavity-spin test to order two, then each quotient term of degree j\le2 to order 2-j, from the independent-row endpoint to the actual row. Taylor’s integral remainder has three mismatch factors on a fixed finite extension of the tree. It retains \tau_a\tau_b, either in the interpolated test or in one deterministic scalar coefficient; every additional cavity spin occurs in an inserted pair.

After deleting the row, nonnegative spin-only bounds transfer by the bounded-covariance Grönwall rule, also during the bounded positive semidefinite row changes below. Deletion itself costs O(m^{-1})\ll u^3P_1, since P_1\ge\sqrt\lambda\,a_{\rm goal}^2. Keep deleted overlaps during sign arguments, so the other mismatch factors use only the remaining rows. Changing this convention costs another O(m^{-1}). Moving a revelation cutoff means an affine positive semidefinite interpolation of the row before applying PE (2). It does not change the conditioning in a term already being estimated.

The allocation rule after (A1) supplies all fork measures. Distinct free vertices away from the observed fork have bounded joint densities in their mass coordinates, by sequential attachment. Repeated attachments at one vertex use its single variable. Ordering restrictions are deterministic and may be retained or dropped when taking bounds. An inserted pair is pinned if its two leaves split at the observed fork. These give the only split atoms, since diagonal pairs are omitted. Every other pair is called low or high according as its split mass is below or above c_Hu.

1. (i)

   Terms with no pinned pair: two low errors cost O_\circ(\int_0^{c_Hu}D^2) by marginalization and Cauchy–Schwarz, whether at the same vertex or not. One low and one high cost O_\circ(u^3 T_1 A) using distinct-vertex densities. Any extras here can be dropped.

   If all errors are high one gains an extra O_p(u) low covariance. Indeed turn off the relevant cavity-row revelation below c_Hu, retaining its later cumulative covariance by a jump there. This includes the scalar row law containing \tau_a\tau_b if that factor occurs in a deterministic coefficient: restoring that law costs O(u) by PE (2) since K_s=O(u) there. Products can be treated factor by factor. With low revelation off, the term containing those two spins vanishes by sign flips of a whole branch component (negating the attached row Gaussians if interacting); they are in distinct components and all other pairs are internal.

   In an interpolated interacting row this is an admissible PSD interpolation (move the cumulative row-specific field and rest-row couplings, or their affine mixture with the scalar row, to the jump). Spin-only tests on remaining rows are invariant. The new pair for restoring this row costs O_p(u) at each low split by (C8) and bounded row transfer, since its coefficient, apart from O(1/m), is bounded by O(|Q_v^{\rm deleted}|+u). Take large fixed moments here and slight higher-than-second moments on the two high errors, costing arbitrarily small power losses by boundedness and (C16). Old topology and weights remain fixed during restoration. Thus all-high terms cost O_\circ(u^3 A^2).

2. (ii)

   At degree at least three with pinned errors: for three pinned factors use one (C15) averaged RMS saving and pointwise higher orders on two, paying O(\lambda^{0.07}d_0^3) (or with arbitrarily small extra losses); \lambda^{0.07} L^2\ll G_1.

   For two pinned and one free factor use the pointwise costs d_0^2 times O_\circ(u A+u^2 T_1). With one pinned and two free, use a high fixed moment for the pinned error and second moments with slight power loss for the others. The latter two integrated cost at most O_\circ(u^2 A^2+u^3 T_2+u^3 A T_1). These bounds and the preceding ones are sufficient by (C18).

3. (iii)

   At degree two with one pinned error, a low free error costs O_\circ(d_0\lambda^{0.07}u^2T_1), using second moments and the average at the observed fork. For a high free error, fix its dyad v\asymp w and project the pinned overlap as in (C17). The error, including the free-vertex mass length, is

   O_\circ(wz_w\,uA),\qquad wz_w\lesssim u^2X^3\Lambda^{-1/3}\lambda^{-1/6}.

   It is o(u^3P_1) by (C18).

   For the projected term we must keep its joint law fixed as v varies. Fix the placement topology and all other free vertex coordinates. Use a projection cut fixed throughout the dyad, prune the pinned branches at that cut, and retain one path through the high vertex to its terminal spin. The sharing of these retained data is independent of v: any sharing through the high fork already extends beyond the projection cut. All discarded branches have normalized continuation kernels. Let F be the projected centered pinned factor; it is measurable with respect to the retained earlier data. Conditional on those data and the high prefix, the high pair has independent continuations with conditional mean X_v. Hence

   \mathbb E[F(Q_v-S_v)] =\mathbb E[F(|X_v|^2-S_v)] =2\mathbb E[F(L_v-\mathbb EL_v)].

   The second equality conditions the retained terminal spin on its prefix. The scalar coefficients and allocation restrictions are bounded deterministic weights k_v, and \|F\|_2=O(d_0). Lemma 7.13 therefore gives

   \left|\mathbb E\left[F\int k_v\,2(L_v-\mathbb EL_v)\,dv\right]\right| \le2\|F\|_2 \left\|\int k_v(L_v-\mathbb EL_v)\,dv\right\|_2 =o(d_0u^2G_1)=o(u^3P_1).

   This uses unconditional Cauchy–Schwarz on the fixed marginal law. If other slots attach at the same high vertex, there is still only one integration in its mass. Fubini and the power margins permit summation over topologies and dyads.

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 (C15)’s final bound and (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 (A5), the degree-two scalar terms are

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 (C14), \Gamma_s''(K_s(i))=O(d_0+\lambda u). The m_i^2 average by (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 (C19).

In particular the error budget remains relative to the removal strength:

with the notation used next. This observation will be needed when R_n tends to zero; an absolute o(1) error would not suffice.□

Proof of Lemma 7.8. The ratio R_n defined below compares the removal strength with the curvature scale of the rescaled box. We first treat R_n\to\infty, with a second rescaling if the box shrinks, and then bounded R_n, including R_n\to0. The closure error is o(\min(1,R_n)); its relative form is essential when calibration at support points requires division by R_n.

Use r=r_0+h_1 z,\ \theta=d_0/h_1 (local notation, unrelated here to size-sweep time), and indices n along the growing systems. Put

The first three functions are bounded on compact sets with a common bound, and we can take \beta_0'\to c_\beta>0,\ c_q=1/c_\beta in these local ranges. We have on compacts

the curvature statement a.e. (also distributionally, since the scalar first derivatives of \Gamma are absolutely continuous) with A_* now its positive limiting value. Here \varphi_n is a nondecreasing map. Indeed Taylor expand \widetilde r(v) at K_*(v),\ v=\alpha(r); the argument increment is b(S-q)-C_0(S-r)+d and \Gamma_*''=O(\lambda u), so the quadratic error is O(\lambda u d_0^2)=o(u^3P_1). Also q(v)-r=d_0(c_qm_n+o(1)). Hence (C19), with d=o(u^3P_1), gives

where e_n=o(\min(1,R_n)) in box-averaged absolute value (both measures as before) on all moving boxes of length comparable to \theta in compact z-ranges. By (P2)–(P5), D_n\asymp\lambda u^2/h_1^2\lesssim1,\ D_n=o(R_n); the pointwise lift pays the small v^2 offset in (P3), and U\lesssim u. On compacts D_n varies by o(1), R_n by o(R_n), by the active derivative bounds (b'=O(\lambda u),\ \Gamma_*''=O(\lambda u)). We subselect repeatedly.

We have uniform limiting bounds on g_n,g_n' independent of compact interval. Indeed intervals of length a sufficiently large fixed multiple of \theta contain nonfloor physical mass \gtrsim d_0 by the buffer bounds on m_n; there R_n|Y_n|\le O(R_n/(d_0 M_j))=o(1). Thus (C20) calibrates bounded values at bounded spacings, sufficient by the curvature bound. Take a C^1_{\rm loc} limit g. Recall {\cal H}'=2C_0(r-S) in radius and

locally since \beta_0' lies strictly between floor and cap. On nonfloor {\cal H}\le0.

Unbounded rescaled removal.

Suppose first R_n\to\infty. Equation (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

The variational sign gives J_nm_n'\le0. Equation (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

Consequently

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 (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,

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,

Divide (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

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

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 (C12), completing the finite induction.□

Completion of Proposition 7.1. At its last stage the radius assertion converts by (C7) to (C3); the averaged excess O(x^{4-2\zeta}u\Lambda^{1/3}) suffices there. This completes the local comparison required for (C1).

Here are the final exponent checks. The local pressure derivative has scale \lambda u when it is used to bound u\int_{v\asymp u}A_v\,dv. The optimized excess therefore contributes

The parenthesized factor is power small because k_c/3>1-2s_0 and \zeta is chosen after p_*. The discrepancy term in (C3) contributes u^4a_{\rm goal}^2=x^4\lambda^{-2s_0}. Since U^4=x^4\lambda^{-2s_*}, the remaining factor \lambda^{2\tau_0} pays dyadic logarithms and the permitted small losses. At the smallest dyad u\asymp U, the switch error from (P11), divided by \lambda u and then by U^4, is at most

which has a strict saving. Larger dyads only improve this estimate. The global budget handles the remaining dyads u\ge\lambda^{b_\ell}, as in the initial reduction. Both parts of (C2) were already obtained from the global comparison, with a final-strength power independent of subsequent sufficiently small choices of p_*. Thus the local induction does not enter the choice of the later size Taylor order.□

Proof.

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

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

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

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

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 (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 (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 (D1) gives

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 (P6)–(P8) then give the claimed defect estimates. Below Y, those estimates use only the low and mask bounds from (P6), (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

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 (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

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 (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 (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

Finally, with a_I=\sup_I e/r, width comparability and e=\min(r,Z^2\mathfrak h) imply

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

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 (D3) holds. Define

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 (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 (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.□

Proof of Proposition 8.1. All estimates in this proof use the untilted positive law with fixed path data. Lemma 6.2 supplies the conditional block decomposition (6.1) and its integrated-increment bounds. At each deterministic fork cut, partition the residual on each of two conditionally independent continuations into future martingale increments, including a last residual ending at the terminal Gibbs spin. A block pair contributes its squared inner product, weighted by the largest mass in the Hessian block. We may bound that weight by the largest mass in either block and project the block with the later starting point.

For the opposite increment write A=|A|\widehat A, setting \widehat A=0 when A=0. Conditional on the common prefix, its whole branch is exterior data for the projected continuation. Applying (A3) to \widehat A therefore preserves the projected branch’s normalized future kernel. The factor |A|^2 remains in the subsequent integral. Fix a permitted loss \varepsilon>0. At each fixed fork, declare an exception when the squared projection exceeds x^{-\varepsilon} times its squared scale in (A3). Choose the projection moment sufficiently high after \varepsilon and the required negligible exponent K are fixed. Markov’s inequality then makes this exception have probability O(x^K). The factor x^{-\varepsilon} is included in the permitted losses below. Bounded spins, integration over the fork coordinate, and conditional Jensen make its contribution negligible in the integral norm. This does not require an event simultaneous over all real forks, or a higher-moment bound for |A| proportional to the bootstrap constant. These operations can first be performed with inserted deterministic cuts and then passed to the fixed-size path limit.

Raw-moment bootstrap. Fix a large even exponent p and set

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

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

Indeed, (6.2) 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 (6.3) to its nonnegative squared increment to average it back to the fork. Thus the contribution from this block pair is at most

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

Their product is O_{p,\circ}((z_{r'}^0)^2(r')^3B_0), no larger than the right side of (D6). The normalized fork window has bounded length. Summing the logarithmically many block pairs in (6.1) proves (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 6.3 gives |X_v|^2\le2\mathbf E_vL_J, while \mathbf EL_J=O(r) by the deterministic mean bound. Equations (A1) and (D6) give

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 (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 (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

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 (D4) near z_0, and put h_0=\mathfrak h(z_0)/r, so |J|\asymp rh_0. We claim

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,

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 (6.2) 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

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 8.2 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 (A1) with the constant weight 1/(G_*rh_0) on the window. The squared centered norm of its mass average is at most

In the last step we used z_r^0=g_0(U/r)^3 and G_*h_0^3\gtrsim(U/r)^6 from (D4). We now apply the bracket identities of Lemma 6.3. Earlier and later averages bracket |X_z|^2. To see also how they control increments, let s\le z\le z' and use

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,

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

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 (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

at every nontrivial cut. This completes (D2).□

Proof. On a tile I in a band v\asymp r, the inequalities v\mathbf EW\le\mathbf E\operatorname{tr}H^2 and dS\ge m\mathbf E\operatorname{tr}H^2\,dh from (A1), together with (D3), imply

Here Z_h=Z for the direct construction and Z_h=1 for the low-region construction, exactly as in their respective variation bounds. Multiply by the supremum of G on I and use (D5) to obtain

There are only polynomially many tiles because their widths have a polynomial lower bound; increasing K absorbs all additive errors. Summing gives the first line of (D7) when k=0. When k\ge1, e is comparable on each tile and e\lesssim r, so

This proves the weighted version. Since H_v\succeq vC_v, \mathbf E\operatorname{tr}C_v^2\le\mathbf EW_v/v; the same bound also proves the trace part of the second line.

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 (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

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 (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 (D3) and its raw moment from (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 (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 (A1) and Fubini, the centered averages contribute at most C h^{-1}\int\mathbf EW\,dv/r. Thus

The first two parts of the lemma control the other terms in the exact conditional-covariance identity

This proves the last assertion of (D7).□

Proof. Retain every tile meeting \{e\ge x^{\delta'}\}. Such a tile has r\gtrsim x^{\delta'}, and width comparability makes e at least a constant times x^{\delta'} throughout it. Write a_I=\sup_I e/r and recall that its mass is r\mu_I.

Bracket within a retained tile using mass windows of width h. The pointwise error bound in (D2) gives squared accuracy O_\diamond(r^2a_I^2) near the tile edges. Since their mass is O(h), this costs O_\diamond(ra_I^2h) in dv/r. Monotonicity and \Delta_IS\lesssim_\diamond ra_I give that same bound for the interior mean errors. The tile estimate for W in the preceding proof gives centered cost O_\diamond((z_r^0)^2r^3Z^2\mu_I/h). Balance these two terms by taking

If h does not fit in the tile, use the raw squared error on the whole tile instead. Its cost r^2a_I^2\mu_I is no larger than the displayed bound when h\gtrsim r\mu_I. This also handles overlapping buffers. Tiles of negligible mass may first be discarded so the additive x^K errors in the W bound do not enter the denominator.

The trace contribution is bounded by the tile W estimate divided by r. For the mixed term use Cauchy–Schwarz and the raw fourth moment of X on the tile. These two contributions together cost at most O_\diamond(r^2z_r^0Z\mu_I). To sum them and the variance bound, the squared envelope budget gives on the retained tiles

The second inequality uses (D5). Converting dv/r back to dv multiplies a band bound by r, and r^3z_r^0=z_1^0. Since Y=o(1) and Z\ge1, the sum of the mixed and trace terms is also bounded by x^{-C\delta'}z_1^0Y^2Z^2. Summing the logarithmically many retained bands gives (D8), with a fixed exponent C_0 independent of \delta'.□

Proof. On these dyads, v\,dh/dv\gtrsim\lambda_1\max(v,U)^3 and S\le CY. The source inequality in (A1), integrated against the total increase of S, therefore gives

If w<\epsilon_w/Y, the raw bound gives directly \int D_v^2dv\lesssim Y^2w\le Y\epsilon_w. In the other case use mass slides of width h=\epsilon_w/Y. The mean and boundary errors cost O(Y^2h), while the centered error costs O(\epsilon_w^2/h); both equal O(Y\epsilon_w). Also \int\mathbf E\operatorname{tr}C_v^2dv\lesssim\epsilon_w^2/w \le Y\epsilon_w. Cauchy–Schwarz with the raw bound for |X|^2 bounds the mixed term by CY\epsilon_w. This proves (D9).

The largest \epsilon_w is E_0, so summing dyads shows that \int_{x\le v\lesssim Y}D_v^2\,dv/Y\lesssim_\circ E_0. We shall also use the following first-moment consequence:

The expression in w is largest at w\asymp U; substitution of Y=UZ and U=x\lambda^{-s_*/2} proves the last inequality.□

Proof. Write P_v(t)=\sum_{i=0}^j c_i(v)t^i, with |c_i(v)|\le Ce(v)^{j-i}. On the event |2L_v-S_v|\le x^{-\epsilon'}e(v), differentiation gives |P'_v(2L_v-S_v)|\lesssim x^{-C_j\epsilon'}e(v)^{j-1}. Equations (A2) and (D7), with k=j-1, therefore give

The factors of two in the argument of P_v only change constants. Taking square roots proves (D10). For the truncation, first fix the allowed loss \epsilon'>0, then a finite moment order large enough that (D2) makes the tail negligible; W is bounded. Choose the geometric slack in (D2) last, small enough for that moment order. The same truncation of the other factors at a fork, followed by Cauchy–Schwarz in dv/r, bounds its alias errors by the same expression.

At x\le v\le C Y a centered linear integral of 2L_v-S_v with bounded coefficients instead costs O_\circ(E_0) before any mass normalization, by (D9) and (A1). When there are no other random factors in the expectation of that linear term, no alias error is required even with an initial density tilt: conditional averaging reduces the overlap to |X_v|^2 and then 2L_v exactly.□

Proof. Use future doubling layers up to a small macroscopic mass before the long terminal block, hitting the later layer in each product via (A3) with a full gap after the fork (also for projected rather than terminal ends). This is the two-branch raw argument of (D6).□

Proof.

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

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

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

The dotted clocks in the strength derivative are O_\circ(\lambda). In the pressure difference from (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 (A1) costs O_\circ(e_1). Transfer these estimates to the stopped tilted law by (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

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 (D5) on macroscopic bands, also when Y>x^\kappa. The deterministic width properties there only use (D1), S,q bounded, and do not require Y=o(1). The plateau lower slope in (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,

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 (D7) with that same loss \chi_e^{-1} and the integrated squared width estimate. By (A1), (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.□

Proof. Use both coordinates v,t=t(v). Up to t_M=M/\ell^2, before the terminal constant piece beginning at v_M, both coordinate derivatives are bounded by C/\omega; the two are inverse, with v_M=\ell+O(\omega M). For v>x we have dh/dt\gtrsim\lambda v^2 by (P4).

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 (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 [19, Section 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

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 (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], (A1) gives

Let R=x^6/(\omega\lambda(V_0')^3). The bounds on the two coordinate derivatives give

The second integral is the one required by (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

Use the bracket identities of Lemma 6.3, 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 (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 (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

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 (A1) gives

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

At \theta=0, (C2) bounds this trial excess by x^{4+c}. Convexity therefore yields

Differentiating the penalty and returning to the old coordinate gives

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

Here z_0=q-d/b is bounded and |d|\lesssim_\circ x^3 throughout, by (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),

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

Put I_\phi=\int h'|\phi|. The error on the right of (D13) is at most

because b'\lesssim h'/V_0'.

To estimate the remaining term on its left, use the exact identity

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 (D12),

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

contradicting (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

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

by (D14) and K_S-K=bf+d. Thus

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 5.2, 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 (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

All row changes are bounded uniformly. Integrate over 0<t<t_M at v=v(t), using (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:

The average in brackets is at least c\lambda v^2 by (P5). On v\ge x^{\mathfrak a/10}, division by this gap therefore gives, allowing the preceding subpower factors,

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 6.2 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

respectively. Stop the doubling layers at a fixed positive t-distance and then use a terminal block. Summing gives

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,

since |t_2-t_1|=O(\rho) and \rho=x^{\mathfrak a/4} =o(x^{\mathfrak a/10}). Applying (A1) to a normalized t-average, with dt/dv\lesssim1/\omega, gives

This is smaller than x^{\mathfrak a/10} for our parameter choices. The two-sided brackets and the common-past estimate in Lemma 6.3 consequently give

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 (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 (D11), for example with \delta_*=\mathfrak a/50. The stopped transfer (A4) gives the same coarse accuracy for shifted active overlaps.□

Proof.

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

using (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 (D7), with Z_h=1, gives the following integral of the alias’s second norm:

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

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 (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 (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

Equation (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 (A4) suffice. After successive replacements (split ties as one power), the deterministic product vanishes against centered P, or cancels across ensembles. This proves (D16).□

Proof of Proposition 8.11.

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

These are pair functions on extensions of the old labels ab; unused labels are ignored by marginalization. Apply (A7) at the reference K, expanding J=b\delta+bf+d, where \delta=Q-S. Choose a fixed truncation degree large enough that (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

The first source includes the stochastic linear term below V_i; the second contains at least two insertions. This left side is exactly (H1), because the positive-law expectation of each inserted pair with P depends only on the indicated labels. By (P5), its gap on retained levels is at least c\lambda V_i^2. Proposition 5.1 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 (D11) controls the discarded tails in arbitrarily high fixed moments. Cauchy–Schwarz on the two untruncated factors gives

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 8.12 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

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

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 (A7) without a random test. The longitudinal identity (H2) separates the linear term into its diagonal and its integral part. The resulting scalar equation is

The diagonal is in field units; returning from normalized mass changes only bounded constants. Its lower bound is

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

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 (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

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

If no factor is pinned at r, integrate one factor in second norm. The low envelope and the high comparison budget give

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 (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

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

as required. This choice of a_2\ll a_1, followed by p_*, preserves both this bound and the covariance absorption.□

Proof. Set a_s=r_s/Y, A=w/Y, and m_{r_s}=Y/a_s. The ratios a_s and A are at least one. At a vertex carrying k factors, (D2) and the integrated envelope bound give

after integration in dv_s/r_s. Thus one or two factors use the first or second envelope moment, and each further factor costs r_s. Changing dv_s/w to (r_s/w)dv_s/r_s in (E2) shows that the multiplier relative to \lambda Y^5 is

with the additional factor \min_s a_s/A when j=l=2. The exponent of an individual a_s is 0 for a single factor, -1 for a double factor, and k_s-3 for k_s\ge3. Consequently

If an assigned vertex lies below x, its integration uses only the fraction x/Y of the lower band. In this case at least one a_s=1, and every entry in the table is at most a constant. Raw bounds in the two ensembles therefore give the first contribution in (E3).

We may now assume that all assigned vertices are at least x. Fully deterministic products cancel, so there is a stochastic factor. Suppose first that all stochastic factors occur at one vertex s. At that vertex, replace the overlaps by the common ancestor norm squared and then by 2L_v on a path through the fork. The alias errors are controlled by (D7). Center the resulting polynomial at its untilted expectation and apply (D10), treating the other times and restrictions as deterministic coefficients. After truncating the other factors at their envelopes, the cost is O_\diamond(z_{r_s}^0r_sZ) times 1 for k_s=1, or times m_{r_s}r_s^{k_s-2} for k_s\ge2. Relative to P_{k_s}, this gives at least the gain g_B/a_s. On the lower band the same gain follows from conditional averaging and the linear bound after (D9) when there is one stochastic power, and from the integrated D_v^2 bound when there are two or more. Multiplying by the appropriate entry for F gives O_\diamond(g_B).

If stochastic factors occur at several vertices, select two of them at scales aY,bY. Their integrated second moments about S, from (D7) and (D9), are bounded by O_\diamond(g_BY^2/a) and O_\diamond(g_BY^2/b). First apply unconditional Cauchy–Schwarz to the two selected stochastic factors at fixed masses, and truncate the remaining factors at their deterministic envelopes. At a selected vertex of scale r=aY and multiplicity k\ge2, Cauchy–Schwarz in mass then gives

Dividing by the raw budget P_k=m_r^2r^{k-2} gives the factor O_\diamond(\sqrt{g_Ba}); for k=1, division of (\int D_v^2\,dv/r)^{1/2} by P_1=m_r gives the same factor. Thus the two selected vertices improve the raw product by O_\diamond(g_B\sqrt{ab}). Since \sqrt{ab}\le A, this proves the claim whenever F\lesssim A^{-1}.

There are two exceptional multiplicity patterns. For (1,2), let a_d be the duplicate scale and a_s the singleton scale. If both factors at the duplicate are stochastic, selecting them uses \int D_v^2\,dv/r at that vertex. Relative to its raw budget m_r^2, this gains O_\diamond(g_Ba_d), so F g_Ba_d=g_B. If only one is stochastic, selecting it together with the singleton gives

which suffices when a_s\lesssim a_d. In the remaining case the later singleton is linear and stochastic. Integrating it as a centered path integral gives g_B/A, so the product with F=1/a_d is again at most g_B. For (1,1), both factors must be stochastic. The lower-band case has a_1=a_2=1 and follows directly from Cauchy–Schwarz. Otherwise the later linear factor has scale A and gives g_B/A; multiplying by F=\min(a_1,a_2) gives at most g_B.

For completeness, the last centered integration has a fixed marginal law even though its fork moves. There is only one earlier stochastic pair to retain. At most one of its leaves descends through the later fork, so choose that leaf for the L-path when it is present. After the alias replacement the other child of the moving fork has no observable and integrates to one. The retained pair always splits at its fixed earlier time. A prior projection merely evaluates these same two paths at fixed later cuts. If neither endpoint descends through the moving fork, choose a continuation whose separation from the retained tree is at another fixed vertex. In either case the moving time occurs only in the L-observable and in deterministic kernels. Apply (A2), or its consequence (D10), unconditionally under this marginal and then apply Hölder with the earlier random factor. This is the construction in Remark 8.13; it does not require an inequality conditioned on that factor. Deterministic ties belong to the kernels, and stochastic ties form one polynomial power. The truncation and tilted-law transfers follow the conventions after (D10). This proves (E3).□

Proof of Proposition 9.2. We first evaluate the internal signed sum in (E1) before applying {\cal O}, and then justify commuting that operator with the sum. Absolute values are taken only after the resulting cancellations. Fix a cut c in the regular inverse region with all designated pair splits strictly below it and c\gtrsim w\ge Y\ge U; on size paths also c\lesssim\lambda. Throughout coefficient differentiation, hold this cut, the skeleton shape, its mass coordinates, and all mass restrictions fixed.

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

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

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

With the explicit edge weights of (E1) kept outside, the coefficient multiplying the skeleton measure is

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

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 (E1) 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 [19, 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 (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 (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 (P1)–(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

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,

The normalized inverse slope and initial spatial derivatives are constant under this rescaling, up to the exponentially small terminal effect. This gives (E6). Moreover, \chi is uniformly bounded above and away from zero. At every designated split,

and the error has the same bound under unscaled parameter differentiation. By (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

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

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

We also need

To obtain (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 (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

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 (E9) by O_*(ch'). The other ordering is identical after exchanging B,C. Without a designated earlier split, (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},

Equation (E6) and b_\circ=\chi^{-2} identify exactly the coefficients with this scaling:

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 (E4) is left outside.

The low-degree cases.

1. 1.

   Suppose j=2 and D=2. The two edges join the same blocks, so use the repeated-edge calculation following (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 (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 (E4) holds. A defect factor supplies \lambda c^2 directly and gives the same bound, with the allowed O_\diamond(1) loss.

2. 2.

   Suppose j=2 and D=3, with edges AB,AC. For the one-part partition, (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 (E4).

3. 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 (E4) without the ratio. Extracting defect factors preserves both conclusions.

4. 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 (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. 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 (E4).□

Proof of Corollary 1.4. Abbreviate F_n^{\mathrm{CW}}=F_n^{\mathrm{CW}}(\beta,\gamma) and couple it with F_n(\beta) using the same disorder. For \Delta_n=F_n^{\mathrm{CW}}-F_n(\beta), [13, Proposition 1.5] gives \Delta_n\ge0 and, for every u\ge0 and n,

Integrating the tail yields the uniform bounds

In particular the perturbed mean and variance are finite. Let t_n=\sqrt{\Var F_n^{\mathrm{CW}}}. The reverse triangle inequality in L^2 gives

Since s_n(\beta)\sim\sqrt{a_2(\beta)}n^{1/6}\to\infty, we have t_n/s_n(\beta)\to1, which proves the variance asymptotic and makes t_n positive for all sufficiently large n. For those n, using \|X_n(\beta)\|_2=1,

Theorem 1.1 and Slutsky’s theorem now give the asserted law.□

Proof of Corollary 1.5. For the Gaussian field in (1.1),

Writing a single subscript for one Gibbs average, set

These Gibbs means are smooth with all derivatives bounded. Applying [7, Lemma 3.3] to them at its two-sided time t/2 shows that q is nonnegative and nonincreasing. The variance identity [7, Theorem 3.8], applied to F_n/\beta, gives