Proof. For \(\mathcal T_m F=m^{-1}\log\mathbf E e^{mF}\), differentiation in any external source gives \[\nabla\mathcal T_mF=\mathbf E_m\nabla F,\qquad \nabla^2\mathcal T_mF=\mathbf E_m\nabla^2F +m\operatorname{Cov}_m(\nabla F),\] where \(\mathbf E_m\) is the normalized tilted expectation. At the terminal spin sum the same formulas hold with mass \(1\) and gradient \(\phi\). Iteration proves the claimed Hessian representation. More explicitly, if \(M_j=\mathbf E[\phi\mid\mathcal F_{m_j}]\) and \(C_L^\phi=\operatorname{Cov}(\phi\mid\mathcal F_{m_L})\), the Hessian at the prefix indexed by \(j\) is \[\mathbf E\left[C_L^\phi+ \sum_{k=j+1}^L m_k(M_k-M_{k-1})(M_k-M_{k-1})^{\mathsf T} \,\middle|\,\mathcal F_{m_j}\right].\] The same expression without the mass factors is the conditional covariance of \(\phi\). All remaining masses belong to \([u,1]\). This proves the matrix inequalities. Conditional independence proves the two overlap identities. Finally expand \(\|C_u+X_uX_u^{\mathsf T}\|_{\rm HS}^2\) and subtract \(S_u^2\). ◻

Proof. Consider first one feature map \(\phi\) of constant squared norm \(q\) and its cumulative covariance clock \(c_j\). Let \(A_j=\mathbf E|\mathbf E[\phi\mid\mathcal F_{m_j}]|^2\). Gaussian heat differentiation of the increment \(c_k-c_{k-1}\), followed by the tilted chain rule through all earlier transitions, gives \[\frac{\partial F_{-1}(0)}{\partial(c_k-c_{k-1})} =\tfrac12\mathbf E\{\operatorname{tr}H_k^\phi +m_k|M_k|^2\}.\] For \(k=0\) the second term is zero. The martingale decomposition in Lemma 2 gives the explicit telescoping sum \[\mathbf E\operatorname{tr}H_k^\phi =q-A_L+\sum_{j=k+1}^L m_j(A_j-A_{j-1}),\] so the last derivative equals \[\tfrac q2-\tfrac12\sum_{j=k}^L(m_{j+1}-m_j)A_j.\] Subtract \(qc_L/2\) from \(F_{-1}(0)\). On writing \(d(c_k-c_{k-1})=dc_k-dc_{k-1}\), the derivative becomes \(-\frac12\sum_j(m_{j+1}-m_j)A_j\,dc_j\). For the matrix feature \(A_j=nB_{m_j}/2\); for the field feature \(A_j=nS_{m_j}\). Dividing by \(n\) proves the pressure formula. All spin bounds used here remain true with an external source. Since \(0\leq S,B\leq1\), integration along the convex segment joining two monotone paths proves the asserted Lipschitz bound.

To prove the second line, fill one Gaussian increment continuously at its fixed mass \(m\). Its continuation solves the backwards heat equation with nonlinearity \(m|\nabla F|^2/2\). Under the normalized planted transition the driving Gaussian process acquires drift \(m\nabla F\). Differentiating the backwards equation cancels this drift in the stochastic differential of the source gradient. Thus \(X\) is a martingale throughout the filled increment. Its diffusion coefficient against the spin field of clock \(h\) is \(\sqrt n H\), since the actual field couples to \(\sqrt n y\). The independent matrix field contributes another positive semidefinite quadratic variation. Itô’s formula therefore gives \[d\mathbf E|X|^2 =n\mathbf E\operatorname{tr}H^2\,dh +\mathbf E\|J_b\|_{\rm HS}^2\,db,\] where \(J_b\) is the mixed Hessian for the matrix feature and normalized spin source. The last term is nonnegative. Concatenation proves the claim along the path and through filled jumps. The heat differentiations may first be made with positive variances and then passed to zero variance by dominated convergence of the finite Gaussian integrals. ◻

Proof. Write all independent Gaussian increments as linear images of a standard Gaussian vector \(\xi\). Multiply their normalized transition densities and the terminal Gibbs density, and then condition on the entire terminal configuration \(\sigma\in\{-1,1\}^n\). Terms at neighboring levels cancel. Up to an additive constant, the negative logarithm of the resulting density of \(\xi\) is \[U_\sigma(\xi)=\tfrac12|\xi|^2-\ell_\sigma(\xi) +\sum_{j=0}^L(m_{j+1}-m_j)F_j(G_j),\] where \(\ell_\sigma\) is linear. Each continuation is convex, by Lemma 2. If \(A_u\) is its Hessian in the underlying drivers, then \[Q:=\nabla^2U_\sigma=I+\int_0^1 A_u\,du\succeq I.\] Use a virtual normalized spin source \(\lambda\). The joint Hessian of the continuation in \((\xi,\lambda)\) has blocks \[\begin{pmatrix}A_u&B_u^{\rm mix}\\ (B_u^{\rm mix})^{\mathsf T}&H_u\end{pmatrix} \succeq0.\] With the spin fixed, the driver gradient of \(L_u\) is \(g_u=B_u^{\rm mix}(y-X_u)\). Positivity of this block matrix implies, for every driver direction \(v\), \[|v\cdot g_u|^2\leq(v^{\mathsf T}A_uv)W_u.\] Put \(G=\int_0^1k_ug_u\,du\) and \(J=\int_0^1k_u^2W_u\,du\). Cauchy–Schwarz in \(u\) gives \[|v\cdot G|^2\leq \left(\int_0^1v^{\mathsf T}A_uv\,du\right)J \leq(v^{\mathsf T}Qv)J.\] Taking \(v=Q^{-1}G\) proves \(G^{\mathsf T}Q^{-1}G\leq J\).

The classical Brascamp–Lieb variance inequality (Brascamp and Lieb 1976, Theorem 4.1) applies to this finite-dimensional density: \(U_\sigma\) is smooth, its Hessian is bounded below by \(I\), and it has Gaussian tails after a linear tilt. All required derivatives have at most the integrable growth allowed by this density; alternatively, one can first use smooth compactly supported approximations. It yields the conditional variance bound by \(\mathbf E[J\mid\sigma]\).

There is no change of centering on removing the pin. For \(\eta\in\{-1,1\}^n\), the transformation \[\sigma_i\mapsto\eta_i\sigma_i,\quad (Z_h)_i\mapsto\eta_i(Z_h)_i,\quad (Z_b)_{ij}\mapsto\eta_i\eta_j(Z_b)_{ij}\] is orthogonal in each Gaussian feature space, preserves the hierarchy, and sends \(X\) to \(\eta X\). Consequently it preserves \(L_u\) and \(W_u\). This group acts transitively on terminal spins. The conditional law of \(\int k_uL_u\,du\), as well as that of \(J\), is therefore independent of the pinned spin. In particular its conditional mean is its unconditional mean. Brascamp–Lieb proves the assertion for \(p=2\) with \(c_2=1\).

Here is an explicit extension to all real \(p>2\). Set \(A=\int k_u(L_u-\mathbf EL_u)\,du\). Apply the same variance inequality to \(|A|^{p/2}\), using smooth regularizations at zero when necessary. Hölder’s inequality gives \[\|A\|_p^p\leq\|A\|_{p/2}^p +\frac{p^2}{4}\|A\|_p^{p-2}\|J\|_{p/2}.\] If \(2<p\leq4\), the first term is bounded using \(\|A\|_{p/2}\leq\|A\|_2\). For \(p>4\) use the previously established bound at exponent \(p/2\) and \(\|J\|_{p/4}\leq\|J\|_{p/2}\). If \(\|J\|_{p/2}=0\), the variance bound already gives \(A=0\). Otherwise put \(t=\|A\|_p^2/\|J\|_{p/2}\) and \(\alpha=p/2\). Then \[t^\alpha\leq d^\alpha+(p^2/4)t^{\alpha-1}, \qquad d=1\ (p\leq4),\quad d=c_{p/2}\ (p>4).\] This implies \(t\leq d+p^2/4\): for \(t\geq d\) divide by \(t^{\alpha-1}\), and for \(t<d\) the claim is immediate. Induction down the finite sequence \(p,p/2,p/4,\ldots\) now gives \(c_p\leq1+p^2/3\). Finally, conditioning on the prefix gives \(\mathbf EW_u=\mathbf E\operatorname{tr}(H_uC_u)\). All integrals are justified first for bounded \(k\) and then by \(L^2\) approximation; \(|X|\leq1\) and \(0\preceq H\preceq C\preceq I\) suffice for this passage. ◻

Proof. It is enough to prove the formula for a finite hierarchy with positive masses and nondegenerate increments; degenerate cases follow by one-sided limits. At a node with incoming mass \(a\) and next child mass \(b\), let \(I\) denote the next child integral. Without observations the expression at this node is \(I^{a/b}\). If \(k\) distinguished children are prescribed, normalization of the planted transitions gives precisely \[I^{a/b-k}\prod_{i=1}^k I_i,\] where \(I_i\) is the corresponding tested child integral. A general spin test is a finite linear combination of products of single-slot tests, so this identity also determines its expectation by linearity. This finite expansion is used only for the identity, not to estimate its norm.

Gaussian differentiation at one covariance increment supplies two derivative slots and a factor \(1/2\). Above that increment ordinary chain rules propagate the integral derivative; below it product and power rules allocate the two field derivatives. A hit on an existing child has multiplicity one. If \(h\) distinct new children are used, the power-rule coefficient is \((a/b-k)^{\underline h}\). Each new child integral carries its incoming factor \(b\), so its effective coefficient is the polynomial \[b^h(a/b-k)^{\underline h} =\prod_{j=0}^{h-1}\bigl(a-(k+j)b\bigr).\] This identity holds for the actual real masses. In particular it has no division by a small mass. A double hit in the same child is retained as such. The final spin summation is the same product calculation with child mass \(1\).

More explicitly, give every derivative slot a distinct label in a finite set \(S\), even when directions agree, and write \(D_A=\prod_{s\in A}D_s\), \(D_\varnothing=1\), and \(\alpha=a/b-k\). The labeled product rule is \[D_S\!\left(I^\alpha\prod_{i=1}^k I_i\right) =\sum_{(S_1,\ldots,S_k,\pi)} \alpha^{\underline{|\pi|}}I^{\alpha-|\pi|} \prod_{B\in\pi}D_BI\prod_{i=1}^kD_{S_i}I_i.\] Here the \(S_i\) are disjoint subsets assigned to existing children, and \(\pi\) is an unordered partition of their complement in \(S\) into nonempty blocks, one per new child. Thus no additional \(|\pi|!\) occurs. Factoring the incoming \(b\) from every new-child derivative gives the polynomial coefficients above; zero masses use these polynomial coefficients and the one-sided limits of the normalized child laws. Recursing with these labeled partitions proves equality of each final labeled-topology coefficient for every order of allocating the slots, before any spin expectation is evaluated. Coincident labels occupy one terminal node. Consequently allocation order may be changed with any bounded function of the final topology carried along, before taking absolute variation.

For completeness, at the root perform the calculation on \(e^{a_0F_{\rm root}}\nu[P]\) with an auxiliary incoming power \(a_0\). Set \(a_0=0\) after differentiation. The derivative of the prefactor then vanishes and the normalized expectation remains. This is an algebraic identity in derivatives of positive real powers; there is no inference from values at integer masses. Summing the changes of independent covariance increments assigns to a pair the change accumulated along exactly its shared prefix, which is \(V_{cd}\). This proves [eq:2].

The one-label sum at the root has total weight \(a_0=0\), as can also be seen directly in Lemma 7. Hence a constant diagonal term multiplied by an old-label test sums to zero. A common source differentiation uses one derivative slot in the identical calculation. Auxiliary leaves representing conditional means may be marginalized at any point because all their transitions are normalized. ◻

Proof. At a node of mass \(a'\) having \(d\) occupied children of next mass \(b'\), the coefficient of a previously unoccupied child is \(a'-db'\), by the one-slot real-power rule. If \(d=1\), consecutive refinements telescope to the negative of the length of the occupied edge. When several occupied children first separate at \(u_v\), write the coefficient as its edge part plus \((1-d_v)u_v\). At a terminal singleton the existing-label hit has coefficient \(1\). These are exactly the three listed contributions.

To compute their sum, let the occupied edge lengths be \(\ell_e\). Summing endpoint differences over the finite tree gives \[\sum_e\ell_e+\sum_v(d_v-1)u_v=r-a.\] Indeed each terminal contributes \(1\), the initial point contributes \(-a\), and at a fork the endpoint contributions equal \(-(d_v-1)u_v\). All contributions except the \(r\) coincidences are nonpositive. Their signed sum is \(a\), and their total absolute sum is \(r+(r-a)\). If there is no occupied child, the one-slot coefficient is simply \(a\). Insert further labels sequentially and apply the variation bound to the enlarged occupied tree. The row-sum assertion follows before taking absolute values from the total signed mass calculation. Empty covariance steps merely subdivide edges and change none of these identities. ◻

Proof. Allocate the specified pair before the other new labels, using equality of the underlying finite product derivatives to reorder the allocations. If one label is old, its non-atomic separation from the new label occurs on an occupied edge and has density of absolute value \(1\); there are at most a number of edges depending on \(q\). Old forks give precisely the listed atoms. If both labels are new, first place one of them. A fork created by a placement on an edge already has a bounded mass density. The second label either separates on an edge, attaches at an old fork, or attaches at this newly created fork. The last coefficient is bounded by the number of occupied children, because its mass is at most \(1\). It therefore preserves the density bound in that variable. Coincidences are kept separate. All remaining allocations have bounded total variation by Lemma 7. Integration over an old fork variable with bounded density is the same argument followed by Fubini. The reordering was an exact identity before any absolute values were taken, so it introduces no cancellation assumption in the final variation estimate. ◻

Proof. Iterate [eq:2]. Affineness makes every covariance insertion fixed, so the \(j\)th derivative has \(2j\) additional labels and \(j\) insertions. Lemma 7 bounds the variation of the resulting measure by a constant depending only on \(q,j\). This proves the first two assertions, including Taylor’s integral remainder. For a nonnegative old-label test, each extended law has the same old-label marginal. Consequently \(\nu_\theta[|P V_{cd}|]\leq M\nu_\theta[P]\) for every attachment, giving the differential inequality. Integrating it, or using the integral form of Grönwall’s inequality when the expectation vanishes, proves the comparison. ◻

Proof. The first derivative is the conditional feature mean. Differentiate it with the one-slot version of Proposition 6, and repeat. Every additional derivative introduces one bounded feature and one summation label. Lemma 7 bounds each successive attachment measure. A continuation beginning at a prescribed prefix is treated as a rooted hierarchy by adjoining empty earlier steps; its prefix is a deterministic external source. This makes the same argument applicable without any lower bound on its first mass. ◻

Proof. Contributions departing before \(v\) are identical and cancel. There are at most \(q\) occupied subblocks at either cut. Inside an occupied subblock at \(w\), every deeper placement gives the same \(w\)-truncated expectation, so its signed sum is exactly \(w\) by Lemma 7. These collapsed tails have total absolute coefficient at most \(qw\leq2qv\). Departures on edges between \(v\) and \(w\) have total length at most \(q(w-v)\leq qv\). Fork atoms in this layer have total absolute weight at most \(w\sum(d_v-1)\leq2qv\). Treat forks lying exactly on a cut with the chosen revelation side; the same estimate covers them. Each of these grouped contributions is the difference between retaining its indicated coupling after \(v\) and making \(r\) independent at \(v\); the old marginal is preserved in both laws. Counting the two terms of these pairs gives at most \(10qv\), and \(12qv\) allows terminal coincidences with either cut convention. A terminal coincidence can occur in the layer only if \(w=1\), in which case \(v\geq1/2\) and its unit weight is covered by the same bound. Before that layer it is already included in a collapsed tail. The asserted restriction on coefficients is exactly what is needed for the identical-expectation grouping, and no further independence from the old tree is used. ◻

Proof. Normalize and integrate every branch not retained in the conditioning. Normalization of the transition kernels preserves the old subtree law. It also gives the same single-continuation marginal for \(r\), whether \(r\) follows an old branch for longer, coincides with an old terminal label, or is made independent immediately after \(v\). Conditional on \(\mathcal G\), the exterior information can change the distribution of the shared prefix but not the normalized future kernels in the affected subtree. This proves the two marginal assertions. Expand \(FP=(F-\bar F)(P-\bar P)+\bar FP+\bar PF-\bar F\bar P\); the last three terms have equal expectations under the two laws. Hölder’s inequality gives the displayed bound after averaging over \(\mathcal G\). If \(F'\) is an independent conditional copy, then \(F-\bar F=\mathbf E[F-F'\mid F,\mathcal G]\), so Jensen’s inequality gives the resampling bound; the argument for \(P\) is identical. Finally integrate against the coefficient measure of Lemma 12. Conditional bounds uniform in the realized prefix are not needed: bounds in the displayed unconditional \(L^p\) norms suffice. ◻

Proof. For a monomial use the exact identity \[\prod_{i=1}^d A_i-\prod_{i=1}^d A_i' =\sum_{j=1}^d(A_j-A_j') \prod_{i<j}A_i\prod_{i>j}A_i'.\] Factors supported in the exterior have \(A_j=A_j'\). A fixed scalar center also cancels from \(A_j-A_j'\). Apply Hölder’s inequality to each remaining term, at exponents whose reciprocal sum is the reciprocal of the desired exponent. Sum over the fixed finite collection of monomials. This proves the norm transfer without any assertion that internal overlaps are concentrated. The two stated operations introduce edges between the current target and another block; they create no edge with both endpoints in a single descendant block. The same argument therefore remains applicable after each operation. The coefficient and centering restrictions ensure that no unrecorded difference term is introduced. ◻

Proof. We give the details with \(n\) fixed; no quantitative estimate uniform in \(n\) is inferred from this approximation argument. A bounded monotone function has only countably many discontinuities. Choose step approximations converging at its continuity points and retaining each required side of the finitely many specified cuts. They converge in \(L^1\), and their endpoint clocks can be chosen to converge as well. Lemma 3 makes the self-corrected root values a Cauchy sequence, uniformly over every common external source.

The same calculation applies to a continuation beginning at a fixed cut \(u\). Its future increment clocks are the cumulative clocks above \(u\) minus their specified values at that cut. Heat differentiation and summation by parts give the same interior integral, together with an initial boundary term. The latter is bounded by a constant depending on \(n\) times the difference of the two clocks at \(u\), because all feature means have norm at most their feature bound. Thus continuation values converge uniformly in the external source whenever the clocks at the cut converge; the bound consists of their \(L^1\) differences above \(u\), their initial clock differences, and their terminal self corrections. This also proves uniqueness of these limits.

Corollary 11 bounds all source derivatives uniformly over the approximations. Uniform convergence of values and the bound on the next derivative imply convergence of any fixed derivative by finite differences: approximate a derivative by a fixed small difference quotient, pass to the limit in the quotient, and then let its step tend to zero. Iterating this argument proves the claim for mixed derivatives as well. In particular the limiting gradient and Hessian are the conditional mean and weighted covariance limits.

To verify convergence of the probability laws, use independent standard Brownian drivers for the finitely many features on each branch of the fixed observed tree. Evaluate them at the deterministic \(b,h\) clocks. Continuity of Brownian paths gives convergence at every cut whose clocks converge, including its prescribed side. Multiplication of normalized transition densities telescopes on each edge exactly as in the pinned calculation: the logarithm of the joint density is a finite sum of endpoint and fork continuation terms and terminal Hamiltonians, minus integrals of continuations along the edges in the mass coordinate. The coefficients of endpoint terms are bounded in terms of the number of observed leaves.

Here is a uniform integrable bound for this density. For fixed \(n\) and bounded clocks, terminal log partition functions are Lipschitz in the feature source with fixed constant. The same is true of continuations by the first derivative formula. Their values at source zero are nonnegative and bounded by the annealed value, which is at most \(n(b(1-)/4+h(1-)/2)\). It follows that the absolute value of the telescoped log density is at most \[C_{n,q}\left(1+\sum_{\text{branches and features}} \sup_{0\leq t\leq M}|B_t|\right),\] where \(M\) bounds all clocks and \(q\) is the fixed leaf count. The exponential of this expression is integrable for a finite collection of Brownian motions. Continuation convergence and dominated convergence in the mass integrals therefore give almost sure convergence of the densities and convergence in \(L^1\). The same domination, together with the source derivative bounds, applies to the differentiated observables under consideration. This proves convergence of positive planted expectations and their fixed-order source versions.

The attachment measure consists of finitely many fork and coincidence atoms and Lebesgue measures on finitely many occupied edges, with uniform total variation. At a Lebesgue placement, the clocks converge outside a countable exceptional set. At an existing fork they converge on the specified side by construction. Dominated convergence and Lemma 7 therefore pass every fixed finite sequence of signed allocations to the limit. The same argument on an affine interpolation segment passes fixed-order differentiation and Taylor’s integral remainder, since their variation bounds are uniform. Exact permutation and row-sum identities survive because they held before passage to the limit.

Finally \(|L_u|\leq3/2\) and \(0\leq W_u\leq4\). Thus the moment estimate passes first for bounded deterministic weights and then for all \(L^2\) weights. For the clock identity, choose refinements that retain the prescribed joint filling of every jump whose total clock size exceeds a cutoff, and then let the cutoff decrease to zero. After the finitely many retained jumps are separated, remove the remaining jumps temporarily. Their total clock size tends to zero; the continuation and source-derivative estimates above control this perturbation. The resulting clocks are continuous on each interval between retained jumps, so their monotonicity gives uniform clock convergence there. These estimates and Brownian continuity give convergence of the quadratic-variation integrands, including within the vanishing increments of the refinements, and hence of their Stieltjes integrals. The uniform integrand bounds control the direct contribution of the omitted jumps as well. At each retained jump the prescribed joint filling fixes the separate clock integrals. Their sum is the total nonnegative increment of \(S\), independently of the filling. Discarding this increment gives the susceptibility inequality on the continuous portions. This proves the stated clock assertions without claiming that the separate jump contributions are independent of their joint filling. ◻

Proof. We have \(k\le n^{-3\delta}\) and \(M\ge n^\delta\), while \[\int_0^1 k(r)\,dr\le n^{-\delta}u_* =o(1),\qquad \int_{u_0}^1\frac{du}{M(u)} \le n^{-\delta}L\log\frac{1+u_*}{u_0+u_*}=o(1).\] The floor alone does not reach the prescribed terminal mass. The cap flow reaches it before the prescribed terminal radius. Interpolating controls in \(\alpha'=k+w(M(\alpha)-k)\), \(0\le w\le1\), gives a feasible endpoint by continuity of the scalar flow; equivalently, interpolate the travel time until the cap flow reaches mass \(1\). At fixed \(n\) the derivatives are uniformly bounded by \(\max_{[u_0,1]}M\). Uniform limits preserve both differential inequalities in their integrated form, so Arzelà–Ascoli gives compactness. The positive floor preserves invertibility. Continuity of the pressure follows from the bounded variation formula [eq:1]; this proves attainment.

For a fixed quantile, interpolate the amount of covariance removed from \(0\) to \(e\). The derivative of the pressure plus the corresponding penalty is the integral of \([B-2rS+r^2]/4\) times the nonnegative strength increment. Since \(B=S^2+D^2\), it is nonnegative. This proves the upper-bound assertion and monotonicity, including after minimizing. At positive strength use an old minimizer as a competitor for a positive increment of strength. The same differentiation and \(\partial_{\log\lambda}e=(1-p)e\le e\) give [eq:7]. Bounded spin moments give local Lipschitz continuity of the objective uniformly over the compact admissible class, and hence of its minimum. Integration of its almost-everywhere derivative is therefore legitimate. The whole interval \(0\le\lambda\le n^{-10}\) costs \(O(n^{-10})\). ◻

Proof. Differentiating the objective with respect to its forward quantile gives \(\frac14\int G\,\Delta r\,du\). The inverse variation is therefore \(-\frac14\int G\,\Delta\alpha\,dr\). Write \(\alpha'=k+w(M(\alpha)-k)\) with \(0\le w\le1\). Its linearized equation has fundamental factor \[J(r)=\exp\left(\int_{r_{\min}}^r w(t)M'(\alpha(t))\,dt\right).\] Terminal preservation says \(\int(M-k)\Delta w/J=0\). Integrating the first variation in the opposite order gives a box-constrained linear functional with this one linear constraint. A scalar multiplier gives a function with derivative \(JG\), which must be nonpositive where \(w=1\) and nonnegative where \(w=0\). Both signs of correcting endpoint variation exist: a feasible path cannot be the pure floor or pure cap, as their travel times above show. Thus the multiplier can equally be obtained by adding a correcting variation and taking one-sided limits. This proves the asserted rule. Directional derivatives follow by the segment variation formula; the fixed-\(n\) inverse functions have dominated difference quotients. Values at jumps affect no integrals. Moreover \[\frac{M'(u)}{M(u)}\le\frac2{u+u_*},\qquad wM\le\alpha',\] which proves bounded distortion of \(J\) on the stated bands.

The bracket in \(G\) decreases with \(p\), since its \(p\) derivative is \(-[(r-S)^2+D^2]/r\). At a noninitial left cap endpoint its limiting value is nonpositive, and at the right endpoint it is nonnegative. At radius \(1\) the latter remains true because \(S,B\le1\). The right inequality implies \(S\le r\) there and \(pB\le2r^2\). Use the right-end slope parameter at both endpoints. If their radii are comparable, subtraction of the inequalities, monotonicity of \(B\), and \(pB\le2r_{\mathrm{right}}^2\) give \(2(1-p)\Delta S\le C\ell\). If the radii are not comparable, \(S\le r_{\mathrm{right}}\lesssim\ell\) suffices. At an initial component use \(S\ge0\), introducing only \(r_{\min}\). Since \((1-p)^{-1}=O(\log n)\) on the polynomial scales under consideration, this proves the first line of [eq:10].

For the mismatch bound, the case \(S>r\) follows from the right endpoint. In a noninitial component with comparable endpoint radii, the left inequality and monotonicity, followed by replacement of its radius by the current radius, give \[(r-S)\left[(2-p)+pS/r\right]\le C\ell+pD^2/r.\] When \(S<r\), this gives \(r-S\le C(\ell+D^2/r)\); splitting into \(D\ge r\) and \(D<r\) proves the squared estimate. Noncomparable and initial components are covered by their length and wall terms. The same calculation at \(G=0\) proves \(S\le r\) and \((S-r)^2\lesssim D^2\) on contact. At density points of the zero set, difference quotients between adjacent zero-set points yield \(S'\lesssim(1-p)^{-1}\): the derivatives of \(B\) and \(p\) have favorable signs, and \(pB/r^2\le2\) there. This proves [eq:9]. The exceptional contact set has zero radius length and, since \(\alpha\) is Lipschitz at fixed \(n\), zero mass.

On a cap, \(dr=du/M(u)\), so the two length bounds follow respectively from the two positive terms of \(M\). If \(p=0\), the endpoint signs say \(S_{\mathrm{left}}\ge r_{\mathrm{left}}\) and \(S_{\mathrm{right}}\le r_{\mathrm{right}}\), proving the sharper mismatch statement. Finally, floor mass is bounded by \(\int_0^1 k(r)\,dr\le n^{-\delta}u_*\). ◻

Proof. Condition on the outside data so that \(V\) is fixed for the branch calculation. Put \(Y=V\cdot y_b\) and \(Y_t=V\cdot X_{b,t}\). In field-clock time, the spin Brownian driver has planted drift \(t\sqrt n X_{b,t}\). This follows on each filled increment from the normalized exponential tilt and its logarithmic gradient; subdivision gives the Brownian formulation. Gaussian differentiation of the telescoping planted density gives, for a subinterval \(I\) and \(P=(Y-Y_{\mathrm{start}})^{p-1}\), \[\mathbb E\left[P\int_I V\cdot(dG-t\sqrt n X_{b,t}\,dh(t))\right] =\sqrt n\int_I\mathbb E\left[ P\left\{t(Y-Y_t)+\int_t^1(Y-Y_z)\,dz\right\}\right]dh(t).\] The terminal tilt contributes \(Y\); the subsequent continuation terms and the subtracted innovation drift contribute \(-\int_t^1Y_z\,dz-tY_t\). Prefix-dependent parts of the test are fixed before the subinterval, so no derivative of \(V\) or \(Y_{\mathrm{start}}\) is taken. This identity is first an ordinary finite Gaussian integration by parts and then passes to filled clocks by the path approximation already established.

For \(z\) after the subinterval’s start, conditional centering gives \[\mathbb E[P(Y-Y_z)] =\mathbb E\left[ \bigl((Y-Y_{\mathrm{start}})^{p-1} -(Y_z-Y_{\mathrm{start}})^{p-1}\bigr)(Y-Y_z)\right] \ge c_p\mathbb E|Y-Y_z|^p.\] This is the uniform monotonicity inequality for the odd power \(p-1\). Also \(\|Y-Y_{\mathrm{end}}\|_p\le2\|Y-Y_t\|_p\) for earlier \(t\). The innovation on the left, divided by \(\sqrt{h(I)}\), has norm at most \(C_p\|V\|_p\). Hölder’s inequality therefore gives the recursion \[\|Y-Y_{\mathrm{end}}\|_p^p \le \frac{C_p\|V\|_p}{v_-\sqrt{nh(I)}} \|Y-Y_{\mathrm{start}}\|_p^{p-1}.\] Divide the given clock into \(N=\lceil(\log n)^2\rceil\) equal pieces. The coefficient is \[\frac{C_p\sqrt N\|V\|_p}{v_-\sqrt{nD_h}}.\] Iteration raises the ratio of the initial bound \(2\|V\|_p\) to this coefficient to the power \((1-1/p)^N\). If the ratio is larger than one it is at most a fixed power of \(n\), so its final power tends to one. If it is smaller than one the assertion is already immediate. This proves [eq:13], with its harmless logarithmic allowance. Conditional Jensen proves the assertion for later projections. ◻

Proof of Proposition 19. Integrated susceptibility bound. Fix an integer \(l\ge1\) and let \[Q=\max\left(1,\sup_u \frac{\|X_u\|_{4l}}{\sqrt{x+u}}\right),\] including intermediate revelations and the final spin. This is finite at each fixed \(n\). We first bound the integrated weight in [eq:1] on a small regular band of mass and radius comparable to \(v\). Uniformly on a bounded enlargement of this band, choose \(\rho\) so that \[ M(O(v))\rho\le n^{-c},\qquad ne(v)v^4\rho\ge n^c \tag{14} \] for a fixed \(c>0\) independent of \(l\). For example take the geometric mean between the reciprocal scales; [eq:11] gives the required room, and \(\rho=o(1)\).

We give the layer estimate first on a finite filled hierarchy, appending the terminal spin as a martingale step of mass \(1\). In this paragraph parentheses denote radius coordinates: \(X(s)=X_{\alpha(s)}\), \(\mathcal F(s)=\mathcal F_{\alpha(s)}\), and likewise \(H(s),W(s)\); write \(X(\dagger)=y\) and \(\mathbb E_s=\mathbb E[\,\cdot\mid\mathcal F(s)]\). Enclose the band in a radius interval \(I=[r_-,r_+]\), with \(r_\pm\asymp v\) and \(r_+<r_c/2\) for a fixed small \(r_c>0\). Its fixed enlargement is regular. Put \[\mathsf A_v=M(Cv),\quad \mathsf B_v=ne(v)v^4,\quad \rho=(\mathsf A_v\mathsf B_v)^{-1/2},\quad \eta=\rho v.\] By [eq:11], with \(\kappa=3m-\delta>0\), \(\mathsf A_v/\mathsf B_v\lesssim n^{-\kappa}\); hence \(\mathsf A_v\rho=(\mathsf B_v\rho)^{-1}\lesssim n^{-\kappa/2}\), as required in [eq:14]. Also \(\mathsf A_v\ge n^\delta\), so \(\rho=o(1)\).

For each \(s\in I\), fork an independent continuation \(b\) from the original branch \(a\). On both branches use cuts \(t_0=s\), \(t_1=s+\eta\), \(t_2=2s\), then doubled radii up to the first \(t_J\in[r_c,2r_c)\), and \(t_{J+1}=\dagger\). Set \(\Delta_i^a=X^a(t_{i+1})-X^a(t_i)\), and similarly for \(b\). There are \(K=J+1=O(\log n)\) layers. With \(w_0=w_1=s\), \(w_i=t_i\) for \(2\le i<J\), and \(w_J=1\), their largest masses \(\mu_i\) satisfy \(\mu_i\lesssim w_i\) and \(\|\Delta_i^{a,b}\|_{4l}\lesssim_l Q\sqrt{w_i}\). For elementary martingale increments \(d_\nu^b\) after the fork, conditional orthogonality gives the indexed positive-semidefinite bound \[H(s)=\mathbb E_s\sum_{\nu\text{ after }s} m_\nu d_\nu^b(d_\nu^b)^{\mathsf T} \preceq\sum_{j=0}^J\mu_j \mathbb E_s[\Delta_j^b(\Delta_j^b)^{\mathsf T}].\] Indeed cross terms between distinct elementary increments in each block have zero conditional expectation. Since \(y^a-X(s)=\sum_i\Delta_i^a\), scalar Cauchy–Schwarz yields \[W(s)\le K\sum_{i,j=0}^J\mu_j\, \mathbb E_b[(\Delta_i^a\cdot\Delta_j^b)^2\mid a].\] Here conditioning on \(a\) includes the common prefix; no coupling of the auxiliary branches for different \(s\) is needed.

For \((i,j)\ne(0,0)\) choose the branch carrying layer \(k=\max(i,j)\), taking either branch on a tie. Condition on the entire opposite branch and use its increment as the exterior anchor in Lemma 20. The clock preceding the selected layer is \[[s,s+\eta]\quad(k=1),\qquad [3t_k/4,t_k]\quad(2\le k\le J).\] Its masses are bounded below by \(cw_k\), also for the terminal layer because \(t_J\asymp1\). The field-clock length is at least \(c\rho ve(v)/\log n\): for \(k=1\) this follows from \(\partial_r h\gtrsim e(r)/\log n\), and for \(k\ge2\) from the same bound and monotonicity of \(re(r)\), with no factor \(\rho\) needed. All these scales are bounded below by inverse powers of \(n\). The opposite increment has norm at most \(C_lQ\sqrt{w_k}\), even when its endpoint is later than the fork, since it belongs to an independent branch. Equation [eq:13] and conditional Jensen for later projections therefore give \[\mu_j\|\Delta_i^a\cdot\Delta_j^b\|_{2l}^2 \le \frac{C_l(\log n)^5Q^2}{\rho nve(v)}.\] Conditional Jensen also bounds the \(L^l\) norm of the corresponding conditional square by this squared \(L^{2l}\) norm. Integrating over physical mass \(O(v)\) and summing the \(K^2\) pairs, including the factor \(K\) above, bounds their total by \(C_l(\log n)^8Q^2v^4/(\mathsf B_v\rho)\).

For \((i,j)=(0,0)\) put \(q_s=\mathbb E_s|X(s+\eta)-X(s)|^2\) and \(Z=\sup_{r_-\le t\le r_++\eta}|X(t)|\) on the original path. The integrand is at most \(CvZ^2q_s\); changing coordinates costs only \(du\le \mathsf A_v\,ds\). Doob’s inequality gives \(\|Z^2\|_{2l}\le C_lQ^2v\). To estimate the remaining integral, for \(\xi\in[0,\eta)\) let \(r_j=r_-+\xi+j\eta\), retaining precisely the \(N=N(\xi)\) starting points in \(I\) and their last endpoint \(r_N\le r_++\eta\). Set \[U_j=|X(r_j)|^2,\quad A_0=0,\quad A_{j+1}=A_j+q_{r_j},\qquad q_{r_j}=\mathbb E[U_{j+1}-U_j\mid\mathcal F(r_j)].\] Thus \(A_{j+1}\) is \(\mathcal F(r_j)\)-measurable. For \(p=2l\), \(A_N^p\le p\sum_jA_{j+1}^{p-1}q_{r_j}\), and the exact identity \[\begin{split} \sum_{j=0}^{N-1}A_{j+1}^{p-1}(U_{j+1}-U_j) ={}&A_N^{p-1}U_N-A_1^{p-1}U_0\\ &-\sum_{j=1}^{N-1}(A_{j+1}^{p-1}-A_j^{p-1})U_j\\ \le{}&A_N^{p-1}U_N \end{split}\] gives \(\mathbb E A_N^p\le p\mathbb E[A_N^{p-1}U_N]\). Consequently \(\|A_N\|_{2l}\le2l\|U_N\|_{2l}\le C_lQ^2v\), independently of the grid cardinality. The shifted-grid identity \[\int_Iq_s\,ds =\int_0^\eta\sum_{j=0}^{N(\xi)-1}q_{r_-+\xi+j\eta}\,d\xi\] and Minkowski give \(\|\int_Iq_s\,ds\|_{2l}\le C_l\eta Q^2v\). Hölder therefore bounds the integrated first-pair term by \(C_lv\mathsf A_v(Q^2v)(\eta Q^2v)=C_l\rho \mathsf A_vQ^4v^4\).

Combining the pairs, including their logarithmic multiplicities, gives \[\left\|\int_{\mathrm{band}}W_u\,du\right\|_l \le C_l(\log n)^8 \left[\mathsf A_v\rho Q^4+(\mathsf B_v\rho)^{-1}Q^2\right]v^4.\] All estimates use moments at most \(4l\). Include deterministic jumps and intermediate cuts in the finite hierarchy. An after-revelation endpoint assigns its jump to the preceding block; a before-revelation endpoint assigns it to the following block. Equivalently insert both ordered nodes, counting the jump exactly once. The constants are independent of the mesh and grid cardinality, so the established bounded-path approximation passes to filled clocks at fixed \(n\); the countable set of jump radii does not affect the radius integral. Absorbing logarithms, for example with \(c'=\kappa/4\), proves \[ \left\|\int_{\mathrm{band}}W_u\,du\right\|_l \lesssim_l n^{-c'}(Q^4+Q^2)v^4, \tag{15} \] where \(c'>0\) is independent of \(l\). The argument also bounds integrals over subsets, since \(W\ge0\).

Closing the moment estimate. For a regular small mass \(u\), choose a future window of mass comparable to \(u\) at radii comparable to \(u\). Its means satisfy \(\mathbb EL_j=S_j/2\le Cu\). Indeed this follows on nonfloor points from Lemma 18; the floor has total mass \(o(u)\), so monotonicity permits looking ahead to a nonfloor point. For \(j\ge u\), \(2\mathbb E_uL_j=\mathbb E_u|X_j|^2\ge|X_u|^2\). Apply [eq:1] and [eq:15] to the normalized window average. Conditional Jensen yields \[\||X_u|^2\|_{2l}\le Cu+C_l n^{-c''}Q^2u.\] For smaller masses, including \(u<u_0\), choose a deterministic future regular cut \(t\asymp x\). The identity \(X_u=\mathbb E[X_t\mid\mathcal F_u]\) and conditional Jensen give \[\|X_u\|_{4l}^2\le\|X_t\|_{4l}^2 \le Cx+C_l n^{-c''}Q^2x.\] Macroscopic masses are bounded directly by \(|X|\le1\). Taking the supremum gives \(Q^2\le C_l+C_l n^{-c''}Q^2\), which closes for all sufficiently large \(n\). This proves the first bound of [eq:12] for exponents divisible by four, and hence for every fixed exponent. It also removes \(Q\) from [eq:15].

The two-leaf overlap. If \(u+x\) is bounded below, the overlap bound is immediate from \(|R|\le1\). Otherwise choose a regular radius \(q=C(u+x)\), with \(C\) fixed and sufficiently large that \([q/2,q]\) is regular and lies after the fork. Outer comparability and monotonicity permit this choice even when \(u<u_0\). Write \(X^a(q)=X^a_{\alpha(q)}\) and similarly for branch \(b\). The projected inner product has norm \(\|X^a(q)\cdot X^b(q)\|_p=O_p(q)\) by the bound just proved.

Decompose each leaf into its projection at \(q\) and subsequent doubling layers, ending with the terminal spin as in the susceptibility estimate. Every product other than the two initial projections contains a later layer of scale \(w\ge q\). An innovation interval before that layer has mass at least \(cw\) and field-clock length at least \(ce(w)w/\log n\). The opposite branch supplies an exterior anchor of norm \(O_p(\sqrt w)\). Lemma 20, now without a mass weight, bounds the product by \(C_p(\log n)^C/\sqrt{ne(w)w^2}\). Since \(e(w)w^2\) increases, summing the logarithmically many layer pairs gives \[\|R-X^a(q)\cdot X^b(q)\|_p \le\frac{C_p(\log n)^C}{\sqrt{ne(q)q^2}} =q\,\frac{C_p(\log n)^C}{\sqrt{ne(q)q^4}} =o_{\rm poly}(q),\] where [eq:11] supplies the last saving. This proves the second bound in [eq:12]. All innovation intervals lie in the regular region, including when the original fork is below the working interval. ◻

Proof. Use the synchronized two-branch layer decomposition in the last part of the preceding proof. Every term involving the changed endpoint contains an increment after its projection. At the later layer of scale \(w\), the projection estimate gives \(C_p(\log n)^C/\sqrt{ne(w)w^2}\). Monotonicity of \(we(w)\) controls the sum from its first scale \(v\), up to logarithms. For a fixed projected anchor replace its raw \(\sqrt w\) bound in [eq:13] by \(\sqrt{x_1}\). Conditional Jensen covers replacement by conditional projection, and two such replacements cover independent resampling. ◻

Proof. Write \(\theta=n^{-c_1}\), \(\eta=n^{-c_1/2}\). Among order \(\theta^{-1}\) disjoint buffered windows of mass length comparable to \(\theta v\), at most order \(\eta^{-1}\) have \(S\) oscillation larger than \(\eta v\), because the total increase is \(O(v)\). Choose one of the remaining windows. This is a deterministic selection involving only \(S\). Fix past and future buffers of length comparable to \(\theta v\) surrounding a central region.

After the raw bootstrap, [eq:15] takes the form \(\|\int W\,du\|_q\le C_q n^{-\gamma}v^4\) for a \(\gamma>0\) independent of fixed \(q\). Positivity of \(W\) and [eq:1] give, for the normalized \(L\) average \(A_J\) over either buffer, \[\|A_J-\mathbb EA_J\|_p \le C_p n^{-\gamma/2+c_1}v=:C_p\zeta v.\] Let \(t\) be a deterministic cut in the central region. The exact conditional identities give \[2\mathbb E_t A_-\le|X_t|^2\le2\mathbb E_t A_+, \qquad |X_t|^2-2\mathbb E_t A_- =\operatorname{avg}_{s\in J_-}|X_t-X_s|^2\ge0.\] Since \(2\mathbb EA_\pm\) differs from \(S_t\) by at most \(\eta v\), conditional contraction implies \[\||X_t|^2-S_t\|_p+ \left\|\operatorname{avg}_{s\in J_-}|X_t-X_s|^2\right\|_p \le C_p(\eta+\zeta)v.\] For two central cuts \(t,t'\) on the same branch, average the triangle-square inequality through the common past buffer to obtain \[\|X_t-X_{t'}\|_{2p} \le C_p(\eta+\zeta)^{1/2}\sqrt v.\] These estimates hold uniformly over deterministic cuts, including cuts inside filled increments. No supremum over random cuts is asserted. Let \(a_0=\min(c_1/2,\gamma/2-c_1)>0\).

Take the allowed star cuts in a smaller central fraction and project the two branches at a later central cut separated by a fixed fraction of \(\theta v\). This leaves a field clock at least \(c\theta e(v)v/[M(O(v))\log n]\) after every allowed fork. For an outside anchor of norm \(O_p(\sqrt v)\), [eq:13] bounds the residual by \[C_p(\log n)^C\frac{\sqrt{M(O(v))}}{\sqrt{n\theta e(v)}\,v} = C_p(\log n)^C v \sqrt{\frac{M(O(v))}{n\theta e(v)v^4}}.\] Decompose both leaves into the central projection and later layers. The first later layers still have norm \(O_p(\sqrt v)\), so the displayed short clock applies. For subsequent dyads use Lemma 21; the later scales only improve its bound. Thus the same bound, up to logarithms, controls the difference between \(R_*\) and the inner product of the two central projections. This two-branch decomposition is necessary: using a terminal spin of norm one as the first anchor would lose a factor \(\sqrt v\).

By [eq:11], \(M(O(v))/(ne(v)v^4)\le Cn^{-\kappa}\) for some \(\kappa>0\) independent of the moment order. The preceding projection distance estimates and Hölder give an error at most \(C_pn^{-a_0/2}v\) between the two central projections’ inner product and \(|X_*|^2\). Their square-norm concentration supplies the remaining centering. Choose \(c_1\) sufficiently small. For example any sufficiently small \(c_2<\frac14\min(c_1,\gamma-2c_1,\kappa-c_1)\) works after absorbing logarithms. If extra fixed buffers cost a fixed power of \(\theta\), replace the last \(c_1\) by that fixed multiple when making the choice. All choices are independent of fixed \(p\). ◻

Proof. In radius coordinates [eq:1] gives \(dS/dr\ge n \partial_r h\mathbb E\operatorname{tr}H^2\) on continuous parts, with nonnegative contributions through jumps. On contact apply [eq:9]. On a cap of radius length \(\ell\), integrate and use \(\Delta S\le n^{o(1)}\ell\). The density \(M(u)\) varies by at most a constant factor within a unit with comparable endpoint masses, and its mass is comparable to \(M(u)\ell\). Multiplication by this density and division by unit mass give [eq:20]. The inequalities \(H\ge uC\) imply \(\operatorname{tr}C^2\le u^{-2}\operatorname{tr}H^2\) and \(\operatorname{tr}HC\le u^{-1}\operatorname{tr}H^2\), proving the last claims. These trace comparisons follow directly by taking traces against positive semidefinite matrices, without a commutativity assumption. ◻

Proof. We give the sampling construction explicitly. Order the units in radius, and assign a cap unit of physical mass \(\mu\) an effective interval of length \(\mu\). At every effective position in that interval, sample its physical point uniformly in the cap’s normalized mass. When taking a side average, use the physical average on that cap, weighted by the fraction of its effective interval that the side occupies. Contact elements retain their physical order. For a finite-cap approximation, retain each included cap as a whole sampling unit and keep the contact part as a separate integral; omitted cap mass tends to zero before taking the limit in \(n\), and floor mass is treated separately. All auxiliary samples choose locations on the same planted path and use its same terminal spin. The effective measure has exactly the original mass. Physical samples within a cap may be taken independently, even for partial effective intervals. Side averages themselves are deterministic weighted averages of the random variables \(L_u\), not random weights.

Bin units according to their end value of \(S\), into at most \(k_0\) successive height intervals of width \(Cy_0/k_0\). A cap stays in one bin; by [eq:10], its internal variation of \(S\) is at most \(n^{o(1)}\ell_0\). Within each bin discard effective strips of length \(n^{-10}\) at both ends from the set of evaluation positions only. For each remaining effective position \(i\), both side averages use the entire portion of the bin on the corresponding side, including the discarded endpoint strips; thus both available side lengths are at least \(n^{-10}\). Their deterministic means differ from \(\mathbb EL_i=S_i/2\) by at most \(C(y_0/k_0+n^{o(1)}\ell_0)\).

If one side has length \(m_1\), its physical weight on a unit of mass \(\mu\) is \(q/(m_1\mu)\), where \(0\le q\le\mu\) is the effective mass of that unit included in the side. By [eq:1] and the last assertion of Lemma 26, its variance is at most \[\frac{n^{o(1)}}{ne_gv\,m_1^2} \sum_{\mathrm{units}}\frac{q^2}{\mu} \le\frac{n^{o(1)}}{ne_gv\,m_1}.\] The corresponding integral formula applies on contact. Integrating over \(i\) costs \(O(k_0\log n)\) from the reciprocal side length.

We next compare \(L_i\) to its side averages. For physically ordered points \(j>i\), \[L_j-L_i=(y-X_j)\cdot(X_j-X_i)+\frac12|X_j-X_i|^2.\] The first term has second moment at most \[(\mathbb E\operatorname{tr}C_j^2)^{1/2} [S_j-S_i+C(D_i+D_j)].\] Indeed the conditional covariance at \(j\) bounds its square by \(\mathbb E[(X_j-X_i)^{\mathsf T}C_j(X_j-X_i)]\) and then Cauchy–Schwarz applies. The identity \[D_u^2=\operatorname{Var}|X_u|^2 +2\mathbb E X_u^{\mathsf T}C_uX_u +\mathbb E\operatorname{tr}C_u^2\] bounds the \(L^2\) deviations of \(|X_u|^2\) and of \(X_i\cdot X_j=\mathbb E_j(X_i\cdot y)\) by the indicated \(D\) terms. This proves the displayed estimate for \(\mathbb E|X_j-X_i|^4\) needed in the Cauchy–Schwarz step.

For the upper deviation of \(L_i\), use the right side: the quadratic term has favorable sign. For its lower deviation use the left side and the same identity with the larger physical index last. Different units are physically ordered. If two samples in the same cap have the reverse physical order, retain also the quadratic term. Since \(\mathbb E|X_j-X_i|^2=|S_j-S_i|\le n^{o(1)}\ell_0\), interpolation with an arbitrarily high fixed raw moment gives \[\mathbb E|X_j-X_i|^4\le C_{\epsilon'}n^{\epsilon'}y_0\ell_0.\] To see the power accounting, at moment \(2q\) the interpolation loss is at most \((y_0/\ell_0)^{1/(q-1)}\) times logarithms. Both scales are bounded by fixed powers of \(n\), so a sufficiently large fixed \(q\) gives the stated \(n^{\epsilon'}\) loss. A polynomial upper bound \(\ell_0\) may be used even for a much shorter individual cap.

Integrate the squared one-sided estimates and use Jensen for each side average. The side kernels have logarithmic marginal-density loss at either position: in effective coordinates this is the integral of the reciprocal available side length, cut off at \(n^{-10}\). A partial unit is spread uniformly across its physical mass. Consequently a trace term at either position is bounded by its whole-unit average in [eq:20], with at most this logarithmic loss. This remains true when an ordering indicator is present, because it can be discarded in an upper bound for a nonnegative term.

Put \(I_D=\int D^2\,du\) and \(\tau_C=n^{o(1)}/(ne_gv^2)\). The terms involving one \(D\) and one square root of a trace are bounded by \(n^{o(1)}(I_Dv\tau_C)^{1/2}\). Young’s inequality, with a sufficiently small inverse-logarithmic coefficient, absorbs the \(I_D\) part. Terms involving the deterministic height difference are bounded by \(n^{o(1)}v[(y_0/k_0)^2+\ell_0^2+\tau_C]\). The reverse-order quadratic term adds \(n^{O(\epsilon')}vy_0\ell_0\). The side-average variances add \(n^{o(1)}k_0/(ne_gv)\). Finally \[D^2\le4\operatorname{Var}L+\mathbb E\operatorname{tr}C^2,\] because \(\operatorname{Var}L =\mathbb E X^{\mathsf T}CX+\tfrac14\operatorname{Var}|X|^2\). This proves [eq:21], including the discarded-strip error by bounded spin observables. Optimizing the integer \(k_0\) proves the last assertion: use \(k_0\) comparable to \((y_0^2ne_gv^2)^{1/3}\) when this exceeds one, and otherwise use one. In the applications \(y_0\) may always be capped by a fixed constant, so this choice is below \(n^2\). ◻

Proof. Apply [eq:2]. The row kernel is bounded and the total variation of its two added-label measure is bounded in terms of \(N\). Marginalizing the new leaves leaves the original test, so \(|\partial_t\nu_t[A]|\le C_N\nu_t[A]\). Grönwall in both directions proves the assertion. Centering constants are those of the actual path and are fixed. When the row is scalar, its recursion factors from the rest-spin recursion. The latter retains all original external-field increments on the remaining coordinates. The proofs of [eq:13] and [eq:16] use these increments and raw moments, so apply to this rest-only law. No identification of interpolated and actual conditional means is needed. ◻

Proof. The exact collapse is Lemma 7, and the layer estimate is Lemma 12. Their coefficient restriction holds because the old topology and earlier test expectations are fixed before \(c\) is introduced. For two old leaves splitting at \(v_B\) inside a block of mass \(U\), for example, the total is \[2-2(1-v_B)-v_B-(v_B-U)=U.\] Here \(-v_B\) is the old-fork atom. For coincident leaves the corresponding identity is \(1-(1-U)=U\). These are signed identities; replacing the coefficients by their absolute values before this grouping would lose the factor \(U\). ◻

Proof. Apply Lemma 13 to each grouped layer of Lemma 32. Its conditioning consists of the complete prefix of the affected occupied subblock and all branches that have already departed. The new leaf supplies \(F_c\); the affected old subtree supplies \(P\). The other endpoints of the two factors in \(F_c\), and of every varying edge in \(P\), lie outside that subtree. Before an old fork we retain both old children; after it the other child belongs to the exterior. The normalized transition kernels therefore give exactly the two conditional marginals required by the coupling lemma, including for coincidence. Its centered product estimate applies after averaging the prefix.

Lemma 14, the resampling estimates above, and the raw moments now give \[\|F_c-F_c'\|_p\le C_{N,p}n^{-\kappa}Wz_v,\qquad \|P-P'\|_p\le C_{N,p}n^{-\kappa}(z_v/x)p_0.\] The second bound uses the smallest factor scale \(x\); star factors give the stronger denominator \(W\). Lemma 32 bounds each layer by a polynomially saved multiple of \[v(Wz_v)(p_0z_v/x)=p_0x^5/W^2\le p_0Wx^2.\] There are \(O(\log n)\) layers on these polynomial scales. Their number is absorbed by the saving. This argument is uniform in the old topology and its split values, so bounded-variation integration preserves the estimate. ◻

Proof. Regard cut blocks as vertices and overlap factors as edges. Each edge joins different vertices. A projection adds fresh teeth at \(*\), hence boundary edges. A switch adds an edge from a new vertex to a high relative in the former target vertex. It adds no overlap edge between that relative and the former target: their relation enters only through a deterministic scalar coefficient and the old topology. The new target is unique in its block and absent from previous factors. Centering replaces monomials by earlier expectations, independent of subsequent leaf placements.

Two switches from \(\Delta_{ao}\), for example, give \[(\Delta_{ao}\Delta_{bc}-\nu[\Delta_{ao}\Delta_{bc}]) \Delta_{de}\] with old scalar coefficient \(\gamma_{ac}\gamma_{be}\). In the block containing \(b,e\) the varying edges are \(bc,de\); there is no \(be\) overlap factor. Before their old fork resample \(b,e\) jointly, retaining the fork. Test variation is \(O_p(xWz_v)\), at most \(O_p(p_0z_v/x)\). The coefficients are fixed by the old topology, so Lemma 32 applies. Induction gives the same graph property for every fixed number of switches. Finite-label total-variation and split-density bounds from [eq:2] depend on this degree, not the mesh or the least mass. ◻

Proof. We first control the third remainder in the low revelation parameter. For the first and second low coefficients we then freeze the actual low kernel \(Q\) while expanding only the high law; this is a separate expansion from the mixed derivative with its varying kernel \(L_t\). It isolates the retained linear correction in [eq:24] and a second-order term passing through a third block. The latter is treated separately for an unobserved and an observed block. First delete the distinguished site from every test. Let \(\mathcal I_\mu[J;V_1,\ldots,V_k]\) denote \(k\) signed covariance insertions, including \(2^{-k}\) from [eq:2]. Kernels include site spins. Write \[\delta_{cd}=b_j\Delta^-_{cd},\quad L_t=(s_cs_d(K_j+t\delta_{cd}))_{\rm low},\quad E_l=(s_cs_d\delta_{cd})_{\rm high} +l(s_cs_d\delta_{cd})_{\rm low}.\] Here low means distinct cut blocks. Set \[F(l,t)=\nu_{l,t}[s_as_bP]-g(l)\nu_{l,t}[P],\qquad g(l)=\nu_l^s[s_as_b].\] In particular, \(g(1)=\Gamma(K_*)\) for the full scalar path. Then \(F(0,t)=F(l,0)=0\). Since the low covariance is \(l(K+t\delta)\), the exact differentiation rule, for an integer \(k\ge0\), is \[\partial_t\partial_l^k\nu_{l,t}[J] =\mathcal I_{\nu_{l,t}}[J;E_l,L_t^{\otimes k}] +k\mathcal I_{\nu_{l,t}} [J;\delta_{\rm low},L_t^{\otimes(k-1)}],\] the second term being absent for \(k=0\).

Third low remainder. The third low Taylor remainder in \(F(1,1)\) is \[\frac12\int_0^1(1-l)^2\int_0^1 F_{lllt}(l,t)\,dt\,dl .\] Its product term uses the full Leibniz rule \[\partial_t\partial_l^3(g\nu[P]) =\sum_{r=0}^3\binom3r g^{(r)} \partial_t\partial_l^{3-r}\nu[P].\] Thus each measure-derivative term has one error and three low factors, and each mixed-covariance derivative has one low error and two low factors. A scalar \(g^{(r)}=O_N(W^r)\) counts as \(r\) low factors. The mixed terms have multiplicity \(\binom3r(3-r)\), in addition to insertion normalizations.

Allocate an error pair before allocating the other new labels. A noncoincident high error has a bounded split density, including old high splits after their old integration, and costs a polynomially saved \(x^2/W\). So does an error at \(*\). A new low error has bounded split density and costs \(\int_0^{C_1W}D=o(x^2)\). This order of allocation prevents a newly created fork from becoming a fixed atom. The only other old low atom is \(i\).

At \(i\), an ordinary measure-side edge crossing a regular cut of scale \(Cx\) above \(i\) and below \(*\) costs \(O_p(x)\). If there is none, turn off site revelation below this smaller cut. The \(i\)-error is odd in its two blocks; the target factor, if present, is even and other measure-side insertions remain inside blocks. The expectation vanishes. Restoring revelation by one derivative supplies an \(O_p(x)\) edge. Scalar \(g^{(r)}\) factors cannot repair this parity. In particular the term \(3g''\mathcal I[P;\delta_{\rm low}]\), which has no ordinary low edge in its measure, costs at most \[C_NW^2x(D_i+1/n)p_0\le C_NWx(D_i+1/n)p_0.\] Every old-\(i\) term therefore has the latter bound, up to fixed-moment interpolation of \(D_i/x\). If \(W=O(x)\), its two original low factors suffice. Diagonal errors vanish. This treats every Leibniz and mixed-covariance term of the third remainder.

Expansion with the actual low kernel fixed. For the first and second low coefficients, freeze the actual low kernel \(Q\) while changing only the high law. Put \(\mu_t=\nu_{0,t}\), \(c_j=g^{(j)}(0)\), and \[\begin{split} A(t)&=\mathcal I_{\mu_t}[s_as_bP;Q_{\rm low}] -c_1\mu_t[P],\\ B(t)&=\mathcal I_{\mu_t}[s_as_bP;Q_{\rm low},Q_{\rm low}] -2c_1\mathcal I_{\mu_t}[P;Q_{\rm low}]-c_2\mu_t[P]. \end{split}\] In particular \(A(t)\ne F_l(0,t)\) in general. The exact decomposition is \[\begin{split} F(1,1)={}&A(0)+A'(0)+\tfrac12B(0) +\int_0^1(1-t)A''(t)\,dt +\tfrac12\int_0^1B'(t)\,dt\\ &+\frac12\int_0^1(1-l)^2\int_0^1F_{lllt}(l,t)\,dt\,dl . \end{split}\] The \(A''\) terms have two high errors and one low factor; the \(B'\) terms have one high error and two low factors. Their bounds are respectively \[C_Nn^{-\kappa}W(x^2/W)^2p_0 \le C_Nn^{-\kappa}Wx^2p_0,\qquad C_Nn^{-\kappa}W^2(x^2/W)p_0.\] For two saved \(L^2\) errors use Hölder exponents just above 2, interpolating with bounded moments; the polynomial saving pays the fixed loss. A lone low error uses Cauchy–Schwarz and raw higher moments, requiring no rate for its little-oh.

The retained first-order terms. Parity restricts the low edge in \(A(0)\) to \(c\in A,d\in B'\), and \(c_1=\chi^2K_*\). Define \[C(t)=b_*\sum_{c\in A,d\in B'}\gamma_{ac}\gamma_{bd} \mu_t[\Delta^-_{cd}P].\] Then \(A(0)=C(0)\). Subtracting \(C'(0)\) from \(A'(0)\) leaves the high insertion coefficient \[\nu^s[s_as_cs_bs_ds_es_f] -\gamma_{ac}\gamma_{bd}\gamma_{ef}\] times \(Q_{cd}P\). Scalar independence cancels it unless the high edge lies in \(A\) or \(B'\). There it is \((G_{ac}\gamma_{bd}+\gamma_{ac}G_{bd})Q_{cd}\). Replacing \(C(0)+C'(0)\) by \(C(1)\) adds two high errors and a low factor. Restoring the rest-only measure in the connected insertion adds one further high error. Both have the \(A''\) bound above.

Restoring full low revelation in \(C(1)\), the first derivative at zero vanishes by block parity; the second costs \(C_ND_*W^2p_0=o(Wx^2p_0)\). For the connected high insertion an absolute first derivative costs one high error and two low factors, covered by the \(B'\) bound. These are the actual-measure terms in [eq:24].

The second low coefficient. It remains to bound \(B(0)/2\). Its single-low insertion vanishes by parity. Two low edges paired with \(s_as_b\) must form a path \(A-D-B'\) through a third block \(D\). Two edge orders and two orientations per edge, times \(1/4\) from the insertions, give exactly the coefficient 2. The scalar \(c_2\) cancels this same sum with \(QQ\) replaced by \(KK\). The Taylor factor \(1/2\) leaves exactly \[\sum_{D\ne A,B'}\ \sum_{\substack{f\in A,\ c,e\in D\\d\in B'}} \gamma_{af}\gamma_{ce}\gamma_{db} \mu_0[P\{Q_{fc}Q_{ed}-K_{fc}K_{ed}\}].\] This includes \(c=e\), \(f=a\), and \(d=b\); none is a diagonal low edge.

Unobserved third block. If \(D\) is unobserved, its first entry leaves the old tree below \(U\), with bounded departure density or old-fork coefficient \(O(W)\). Use \(QQ-KK=\delta Q+K\delta\). A varying low error costs \(o(x^2)\) after integration, while the other factor costs \(W\). At \(*\) the costs are \(o_{\rm poly}(x^2/W)\), \(W\), and the entry coefficient \(O(W)\). If an edge splits at \(i\), both do; the product difference is \(O(x^2(D_i/x)^{\vartheta_N})\), before entry. A departure along a terminal trunk between \(*\) and \(U\) has one varying split and the other fixed at \(*\), covered by the same bounds. These exhaust a new third block’s locations relative to the target pair.

Observed third block. If \(D\) is observed, its two exterior splits are both \(*\) or both \(i\). Average the fresh endpoint \(d\) to its projection at \(*\), and sum its scalar row to \(\chi\). This remains valid in the latter case. Evaluate the factor depending on \(e\) at \(c\). For distinct \(c,e\) at high split \(v\), its error is a saved \(x^3/(Wv)\). The factor \(\gamma_{ce}\le Cv\) pays this denominator, including at old high atoms; the other low factor costs at most \(W\). The result is \(o(x^3p_0)\le o(Wx^2p_0)\). At coincidence the error is zero. The resulting rest observable is independent of \(e\)’s deeper placement, so its scalar row sums to \(\chi\).

Only unweighted allocation of \(c\) in \(D\) remains. Its fresh-at-cut baseline gains \(U\). The baseline product difference is \(O(WD_*)= o(x^2)\) at \(*\), or \(O(x^2(D_i/x)^{\vartheta_N})\) at \(i\). Restoring deeper attachment is precisely Lemma 33. By Lemma 34 its hypotheses hold also for several old high relatives and earlier centering constants.

For completeness the exhaustive remainder ledger, with \(p_0\) suppressed, is

All derivatives occur in the exact decompositions above. Allocating an error pair first gives only diagonal errors, old low atoms, integrated old high splits, or new splits of bounded density. Remaining allocations have bounded total variation at fixed degree. Interpolate \(\Delta_i/x\) between its \(L^2\) norm and bounded higher moments if a higher Hölder exponent is required. The finitely many resulting positive exponents have a positive minimum \(\vartheta_N\), proving the claimed uniform modulus. Finally restore deleted overlaps and use permutation symmetry. ◻

Proof. The partner operation is exact: its entire block is fresh, \(P\) does not use its leaves, and its attachment measure is invariant under exchanging root and summed leaf. Fresh-block averaging is exact as well. Since \(a\) is unique in its observed block, symmetry gives \[\sum_c\gamma_{ac}\nu[(Y_c-Y_a)P_a] =\frac12\sum_c\gamma_{ac} \nu[(Y_c-Y_a)(P_a-P_c)].\] The diagonal vanishes. At a distinct split of scale \(v\), the allocation has total variation \(O(v)\) per dyad, and \(\gamma_{ac}=O(v)\). The projected difference costs a saved \(x^3/(Wv)\).

For the test difference, fix its deterministic replica tree and write the factors not involving \(a,c\) as \(H\). Let \(o\) be the exterior endpoint of an incoming edge, whose split from the target trunk is \(j\in\{i,*\}\), and put \(s_j=x\) or \(W\), respectively. Set \[\mathcal F=\mathcal F_*^a,\] where \(\mathcal F_*^a\) is the sigma-field generated by the complete planted prefix on the target trunk through \(*\), including its common initial field. No exterior continuation after its departure from this trunk, no exterior terminal spin, and no target innovation after \(*\) is included in \(\mathcal F\). In particular \(X_*\) is \(\mathcal F\)-measurable.

All leaves used by \(H\) lie outside the current target block and leave its trunk by \(*\). The product of normalized planted transition kernels therefore makes their joint continuation, together with \(y_o\), independent of the joint future of \(a,c\), conditional on \(\mathcal F\). Average this entire exterior joint law and define \[V=\mathbf E[Hy_o\mid\mathcal F], \qquad A=\mathbf E[y_oy_o^{\mathsf T}\mid\mathcal F].\] The projection powers at \(a,c\) are kept outside this average. Explicitly, for any integrable scalar function \(Z\) of their continuations and of \(\mathcal F\), \[\nu[ZH\,(y_a-y_c)\cdot y_o] =\nu[Z\,(y_a-y_c)\cdot V].\] This identity applies with \(Z\) containing \(Y_a-Y_c\) and the projection powers produced by telescoping \(P_a-P_c\).

If \(j=*\), the unweighted marginal of \(y_o\) conditional on \(\mathcal F\) is its ordinary continuation from that prefix. If \(j=i<*\), the extra target-trunk revelation between \(i\) and \(*\) is independent of the exterior continuation given \(\mathcal F_i^a\), so the same marginal is the ordinary continuation from \(i\). Consequently, for two independent copies \(o,o'\) of this conditional marginal, \[\|A\|_{\rm op}\le\|A\|_{\rm HS} =\bigl(\mathbf E[R_{oo'}^2\mid\mathcal F]\bigr)^{1/2},\] and their unconditional overlap has the split-\(j\) marginal. Directional conditional Cauchy–Schwarz gives \[\|V\|^2\le\mathbf E[H^2\mid\mathcal F]\,\|A\|_{\rm op}.\] Thus conditional Jensen and Hölder imply, for fixed \(p\ge2\), \[\|V\|_p\le\|H\|_{2p}\|R_j\|_p^{1/2} \le C_p p_H\sqrt{s_j},\] where \(p_H\) is the raw scale of \(H\). Correlations among \(H\) and the exterior spin are permitted; none of their terminal observations have been conditioned upon.

The vector \(V\) is measurable before the regular innovation interval, which is chosen after \(*\) and before the high split of \(a,c\). Their common projection at the interval’s end cancels in \(V\cdot(y_a-y_c)\), so [eq:13] applies to its two residuals with this fixed anchor. Apply that estimate before Hölder with the remaining target projection powers; those powers are neither part of the anchor nor conditioned upon. Terms where a projection power changes instead use the raw incoming-edge bound and telescope the powers of \(Y\). Together these operations bound the test difference, in units of \(p_0\), by a saved \[\frac1v\left(\frac{x^3}{W^2} +\frac{x^{5/2}}{W^{3/2}}\right).\] The first term covers projection powers and star incoming edges; the second covers an incoming \(i\) edge. Multiplying by the projected difference and the two factors \(v\) gives, per dyad, \[n^{-\kappa}p_0 \left(\frac{x^6}{W^3}+\frac{x^{11/2}}{W^{5/2}}\right) \le Cn^{-\kappa}p_0Wx^2.\] The saving absorbs the logarithmic number of dyads.

For the \(G_{ac}\) replacement one projection estimate suffices. Allocate \(c\) first; its \(O(v)\) weight cancels the denominator in \(x^3/(Wv)\). The high centered error in \(G\) adds a saved \(x^2/W\). If its split equals that of \(a,c\), use [eq:18] over this same dyad; otherwise allocate that error pair next and use Lemma 9. Reordering these allocations retains the full scalar topology coefficient, by the labeled product rule in Proposition 6; that coefficient is bounded here and is not collapsed as a free row. The result is a saved \(p_0x^5/W^2\le p_0Wx^2\). All bounds are uniform in the target band. ◻

Proof. Equations [eq:4] and [eq:6] give \[b_*=T+O(W^2),\qquad K_*=TS_*+O(W^3),\qquad \chi K_*\asymp W.\] The good-split bound and raw higher moments yield \[\frac{|(1-b_*\chi^2)\nu[Y_aP^\circ]|}{\chi K_*} \le\eta_nx^2p_0(1+|J_0|/W^2).\] Any fixed interpolation loss in the polynomial \(D_*\) saving is included in \(\eta_n\). To display one generation of the recursion, put \[\mathcal C(P;a,b)= \frac{b_*}{K_*}\mathcal L(P^\circ Y_a,a) +\frac{b_*}{\chi K_*} \sum_c\gamma_{ac}\mathcal L(P^\circ\Delta_{bc},b).\] After division by \(\chi K_*\), Equation [eq:25] reads \[\mathcal L(P,a)+\mathcal L(P,b) =\varepsilon(P;a,b)-\mathcal C(P;a,b),\] where, uniformly over the allowed target band, \[|\varepsilon(P;a,b)|\le C_Nx^2p_0 \{\eta_n+(D_i/x)^\vartheta+\eta_n|J_0|/W^2\}.\] The target term is absent for star-only tests. Every term in \(\mathcal C\) adds one star factor, with coefficient \(O(1/W)\).

Now choose two fresh roots \(u,v\) at \(*\), neither in \(P\). Pruning the unused fresh root preserves the normalized planted law; relabeling therefore identifies each of \(\mathcal L(P,u)\) and \(\mathcal L(P,v)\) with \(\mathcal L(P,b)\). Apply the same identity to \(u,v\), and subtract half of it from the identity for \(a,b\). This gives \[\mathcal L(P,a) =\varepsilon(P;a,b)-\tfrac12\varepsilon(P;u,v) -\mathcal C(P;a,b)+\tfrac12\mathcal C(P;u,v).\] Every insertion on the right now has one more star factor than the insertion on the left. This is the recursion we iterate.

Fix a depth \(k\), independent of \(n\). At depth \(j\) the natural scale is \(p_0W^j\), and accumulated coefficients are at most \(C_jW^{-j}\). Lemma 34 ensures admissibility after every switch, even with old high subtrees. Ancestor coefficients are fixed during each new interpolation. Row weights have total variation bounded in terms of \(j,N\). Thus intermediate errors are bounded by \[C_{k,N}x^2p_0\{\eta_{k,N,n}+(D_i/x)^{\vartheta_{k,N}} +\eta_{k,N,n}|J_0|/W^2\}.\] There are finitely many such errors.

At a terminal node there are exactly \(k\) new marked star factors, centered overlaps or projections. Hölder at its required fixed order and marginalization identify each with a star pair; [eq:17] bounds each by \(C_kn^{-c_2}W\), with \(c_2\) independent of that order. Initial factors use raw scales. The high insertion has bounded fixed moments and finite-label total variation. Including the \(W^{-k}\) normalization, the terminal bound is \(C_{k,N}p_0n^{-kc_2}\). Centering can send marked factors into earlier separate expectations; applying this argument to each expectation preserves their total number \(k\). Choose \(k\) fixed with \(kc_2>1/3\). Since \(x\ge n^{-1/6+m}\), the terminal bound is \(o(x^2p_0)\). Only then fix the degrees and Hölder orders. Summing the finite recursion proves the lemma. ◻

Proof. Use [eq:24] with \(P=1\). The linear centered term and the constant \(K_*\) parts of the high corrections vanish by marginalization. By Lemma 36, remaining terms are insertions against one star factor. Equation [eq:26] with \(p_0=W\) bounds them by \(o(Wx^2(1+|J_0|/W^2))\).

The scalar low expansion is \(\Gamma(K_*)=\chi^2K_*+O(W^3)\). Its second coefficient is a two-edge path through a third block. With no other observations the block is unobserved, gaining \(O(W)\), and its two deterministic low factors are \(O(W^2)\). The third derivative is \(O(W^3)\) by bounded finite-label variation. Consequently \[S_*=\chi^2TS_*+O(W^3)+o(Wx^2(1+|J_0|/W^2)).\] Divide by \(S_*\asymp W\) to obtain \[|J_0|\le CW^2+\eta_nx^2+\eta_n|J_0|x^2/W^2.\] Since \(x\le W\), absorb the last term. Lemma 37 now gives [eq:27] and its uniform version. ◻

Proof of Proposition 29. Choose a regular continuity cut \(U_x\asymp C_2x\) above the entire fixed target band. Repeat the mean expansion of Lemma 35 with split \(i\) and \(P=1\), also allowing \(W\) bounded below. The scalar-law and block conventions in this repetition use \(U_x\) in place of \(U\). All low factors cost \(O_p(x)\). Above \(U_x\), the dyadic bounds in [eq:18] are polynomially smaller than \(x\). Indeed, on the intermediate dyads and the high dyads, respectively, \[\frac{x^2}{(W^2v)^{1/3}}\lesssim x \quad(v\ge C_2x,\ W\ge x), \qquad \frac{x^2}{W}\le x.\] These are bounds after split integration; no pointwise high-split estimate is needed. A third-remainder atom at \(i\) costs \(O(x^2D_i)\); an integrated low error costs \(x\int_0^{C_2x}D=o(x^3)\). High Taylor remainders cost two saved high errors times \(x\), or one saved high error times \(x^2\), and hence are \(o(x^3)\). The centered first-coefficient low restoration has zero first derivative and second cost \(O(x^2D_i)\). At second low order every third block is unobserved; its departure density or \(O(x)\) entry coefficient gives the same product difference bounds. This exhausts the earlier ledger at scale \(x\), with remainder \[Cx^3\{\eta_n+(D_i/x)^\vartheta\}.\]

The linear centered term and deterministic high means vanish. In the remaining \(G_{ac}\) term exchange \(a,c\) in the sum and average the opposite fresh branch. Let \(M_a^{(x)},\chi_x\) denote the insertion and scalar row at \(U_x\), and \(Y_a^{(i)}=X_i\cdot y_a-S_i\). The partner term is identical. The surviving form is therefore \[S_i-\Gamma(K_i)= 2b_i\chi_x\,\nu[Y_a^{(i)}M_a^{(x)}] +\text{remainder}.\] No test depending on \(a\) preceded this mean expansion, so this exchange is exact.

If \(W\) is bounded below along a subsequence, [eq:18]–[eq:19] give \(\int_{U_x}^1D_u\,du=o(x^2)\): up to \(C_3x\) use [eq:19], from \(C_3x\) to a fixed multiple of \(W\) use the intermediate dyadic bound of [eq:18], and above use its high-band bound. Take \(C_3\) sufficiently large and fixed. Allocate the error pair first, relative only to \(a\) in its block, giving a bounded density. The insertion costs \(O(x)\int D=o(x^3)\).

If \(W=o(1)\), choose the calibration cut \(U\) above \(U_x\), adjusting its fixed multiple also when \(W\asymp x\). Remove error edges splitting between the cuts at cost \[Cx\int_{U_x}^{U}D_u\,du=o(x^3).\] For remaining edges replace the scalar connected kernel at \(U_x\) by the one with revelation off up to \(U\). Each bounded-spin scalar moment changes by \(O(W)\): its covariance change below \(U\) is \(O(W)\), and [eq:2] has bounded finite-label variation. The insertion error is at most \[CxW\int_U^1D_u\,du=o(x^3)\] by the saved \(x^2/W\) high bound. No arbitrary high-atom estimate is needed: allocate the error pair first relative to the sole old root \(a\) in this block, and then allocate the summed leaf \(c\).

For the new kernel, independent block flips force \(c\) into \(a\)’s calibration block. The error endpoints share a block because their split is above \(U\). If it is different from the block of \(a,c\), the connected scalar covariance vanishes. Thus the remaining operator is exactly \(M_a\) of [eq:23]. The test \(Y_a^{(i)}\) belongs to our family by its fresh-tooth representation and has one initial factor of scale \(x\). Equation [eq:27] proves \(\nu[Y_a^{(i)}M_a]=o(x^3)\), with its uniform target modulus.

Cuts and integrated estimates are chosen once for the full fixed target band. After fixing the recursion depth only finitely many Hölder orders are used. Target dependence is a finite collection of positive powers of \(D_i/x\); their minimum is a common \(\vartheta>0\). All remaining errors give a common \(\eta_n\to0\).

For omitted lower cuts suppose \(W=O(x)\), hence \(W\asymp x\), and choose all regular cuts at the required fixed multiples of \(x\), uniformly in the omitted target. First perform the star-only calibration: it gives \[J_0=O(x^2),\qquad \chi K_*\asymp x,\] using [eq:4] only in the regular calibration region. At every omitted \(i\), the global raw bounds and conditional Jensen give \[\|R_i\|_p+|S_i|+D_i+\|Y_a^{(i)}\|_p\le C_px, \qquad \|X_i\|_p\le C_p\sqrt x .\] Monotonicity from the regular region also gives \(h_i,K_i\le Cx\).

Enlarge the admissible tests by allowing their one initial edge to split at any omitted \(i\), retaining its natural scale \(x\). The row-identity proof remains uniform for this class: an old-\(i\) error costs at most \(C_Nx^2(D_i+1/n)p_0\); an integrated low error costs \(C_Nx p_0\int_0^{Cx}(D_u+1/n)\,du\). The observed-third-block baseline at \(i\) has product difference \(O_p(x^2)\) and gains a cut mass \(O(x)\). The other remainder classes retain their previous bounds. Thus the remainder in [eq:24] is \(O_N(x^3p_0)\). All projection and resampling intervals lie in the regular region at masses at least \(Cx\). For an initial incoming edge the weighted-anchor bound uses only its split-\(i\) raw moments and is \(O_p(p_H\sqrt x)\). Its projection denominator is the regular innovation mass, never \(u_i\). Finite-label attachment variation is bounded independently of \(u_i\); zero splits follow by the same degenerate-tree conventions and continuity.

Use [eq:25] with this enlarged class, after the star-only calibration, and divide by \(\chi K_*\asymp x\). Its left side and remainder are \(O_N(x^2p_0)\). The two-fresh-root equation removes the same-degree nuisance as before. At depth \(j\), the test scale is \(p_0x^j\) and the accumulated coefficient is at most \(C_jx^{-j}\), so each intermediate error is \(O_j(x^2p_0)\). The \(k\) marked star factors bound terminal terms by \(C_kp_0n^{-kc_2}\), also when centering places some marks in separate expectations. Choose fixed \(kc_2>1/3\); then \(n^{-kc_2}=o(x^2)\). The finite recursion proves \[|\mathcal L(P,a)|\le C_Nx^2p_0.\]

In particular \(P=Y_a^{(i)}\) has scale \(x\), so \(|\nu[Y_a^{(i)}M_a]|\le Cx^3\). The preceding cutoff and scalar-kernel changes still cost \(o(x^3)\), uniformly in \(i\). The target mean identity therefore gives \[S_i-\Gamma(K_i) =2b_i\chi_x\,\nu[Y_a^{(i)}M_a]+O(x^3)=O(x^3),\] since \(b_i,\chi_x\) are bounded. Here \(K_i=h_i+b_iS_i\) is always the actual changed clock: it is never replaced by \(TS_i\), so no term \((b_i-T)S_i\) occurs. No endpoint hypothesis is used at the omitted cut. Finally, if the omitted mass is \(u_0=o(x)\), its contribution to the integrated \(D\) bound is at most \(Cx u_0=o(x^2)\).

Finally let \(d\mu_n=du/x\) on a mass band of [eq:19], or \(d\mu_n=dr/x\) on its radius band, restricted in each case to the indicated set. These measures have uniformly bounded total mass. Equation [eq:19] and the raw bound imply \[\int(D_i/x)^2\,d\mu_n(i)\longrightarrow0,\qquad D_i/x\le C.\] Decrease \(\vartheta\) to at most 2. Jensen gives \[\int\frac{|S_i-\Gamma(K_i)|}{x^3}\,d\mu_n(i) \le C\eta_n+ C\left(\int(D_i/x)^2\,d\mu_n(i)\right)^{\vartheta/2} \longrightarrow0.\] This proves both normalized \(L^1\) claims. ◻

Proof. The concentration assertions are [eq:19]; their radius version is a direct estimate, rather than a change of variables in a mass estimate. For clarity, the estimate that pays the inverse-density loss is \[\frac{\operatorname{Var}(x^{-1}\int_I L\,dr)}{x^2} \lesssim\frac{n^{o(1)}}{ne(x)x^4k(Cx)} \lesssim\frac{n^{1-\delta+o(1)}E^2}{e(x)x^2} \lesssim n^{-\delta+o(1)}\] for radius subintervals of the band. Here one applies [eq:1] with mass weight \(\mathbf1_I(r(u))/(x\alpha'(r(u)))\), and uses the clock inequality, \(H_u\ge cxC_u\), and \(k(Cx)\gtrsim n^\delta u_*^2/x^2\). The ordered side-average argument in Proposition 24 converts this estimate and the integrated trace estimate to the asserted radius mean-square bound. Thus the conclusion remains valid even when \(\alpha'=k\) throughout most of the radius interval.

The raw bounds [eq:12] give \(D_i=O(x)\) uniformly here. Proposition 29 supplies a fixed \(\vartheta>0\) and \(\eta_n\to0\) such that \[|\rho_n(i)|\le C\{\eta_n+(D_i/x)^\vartheta\}\] at every split in the band. This gives the uniform bound on \(\rho_n\), including moving splits. For either normalized measure \(du/x\) or \(dr/x\), the band has bounded measure and \(D/x\) tends to zero in \(L^2\) while remaining uniformly bounded. Hölder’s inequality therefore makes the integral of \((D/x)^\vartheta\) tend to zero. The displayed modulus proves both asserted \(L^1\) limits, including the radius limit on floor intervals. ◻

Proof. Corollary 11, applied to one Ising spin, gives the spatial derivative bounds independently of the partition. The finite-step formulas pass to general clocks by Proposition 15. The PDE expresses the time derivatives in terms of bounded spatial derivatives and \(0\le\alpha_K\le1\). Convexity and evenness are preserved by every scalar Gaussian continuation.

Itô’s formula gives \[dV_z(q,Z_q)=\sqrt T\,V_{zz}(q,Z_q)\,dB_q,\] and \[dV_{zz}(q,Z_q)=-T\alpha_K(q)V_{zz}(q,Z_q)^2\,dq +\sqrt T\,V_{zzz}(q,Z_q)\,dB_q.\] Taking expectations of the squares proves the two displayed derivative identities. Absolute continuity is sufficient for these applications of Itô’s formula; alternatively one first works with step clocks and passes to the limit using the stated derivative bounds.

For the strict lower bounds, let \(Q\) be terminal scalar time. Differentiating the equation once more shows that \(w=V_{zz}\) solves a linear backward equation with drift \(T\alpha_KV_z\), potential \(T\alpha_Kw\), and terminal value \(\operatorname{sech}^2z\). Since the drift is uniformly bounded and the potential is nonnegative, its Feynman–Kac representation bounds \(w(0,0)\) below by a positive constant uniformly for bounded \(Q\). One can see the uniformity directly by restricting the terminal diffusion to a fixed bounded interval, which has uniformly positive probability.

Set \(v=V_{zzz}\). It satisfies the linear equation \[v_q+\frac T2v_{zz}+T\alpha_KV_zv_z +3T\alpha_KV_{zz}v=0.\] On \(z>0\), the drift and potential are nonnegative, \(v(q,0)=0\), and the terminal value is \(-2\tanh z\,\operatorname{sech}^2z<0\). Use the killed-diffusion representation on the positive half-line. For \(Q\) in a fixed compact subinterval of \((0,\infty)\) and small initial \(z>0\), the event that \(z+\sqrt T B\) stays positive and ends in a fixed interval \([a,b]\subset(0,\infty)\) has probability at least \(c_1z\), by the reflected Gaussian kernel. On this event the diffusion with nonnegative bounded drift also survives, and its terminal value lies in \([a,b+TQ]\). Its negative terminal datum is therefore bounded away from zero in absolute value, while the Feynman–Kac weight is at least one. Hence \(-v(0,z)\ge c_2z\), and division by \(z\) gives \(-V_{zzzz}(0,0)\ge c_2\). If \(Q\) tends to zero, the bounded time derivative and \(V_{zzzz}(Q,0)=-2\) give the same conclusion. Upper bounds follow from the spatial derivative bounds. ◻

Proof. The limiting mass distribution. Give a Borel radius set \(J\) rescaled mass \(x^{-1}|\{u>u_0:r(u)/x\in J\}|\). Its cumulative function differs from \(\alpha(xt)/x\) by at most \(u_0/x=o(1)\). Outer comparability bounds its mass on each compact set and bounds its cumulative function above and below by multiples of \(t\) for sufficiently large \(t\). Helly selection gives a locally finite weak limit \(\nu\).

We next compare the scalar clock. The total floor mass is bounded by \(\int k(r)\,dr=o(x)\). On a positive compact mass band, outer comparability and monotonicity give \(r=O(x)\), while cap widths and the initial wall are \(o(x)\). Since \(x=O(d)\) and \(\lambda\le d^2\), the taper argument \(\lambda r/d^3\) is bounded on these bands, including their cap endpoints. Thus \(1-p\) has a fixed positive lower bound. The proof of Lemma 18 then gives [eq:10] with a fixed constant in place of \(n^{o(1)}\) on these bands. Together with [eq:9] and the mass estimate in [eq:19], this proves \(S-r=o(x)\) in normalized mass measure off a negligible set. Since \(b_0-T=o(x^2)\), \(h_0=o(x^3)\), and \(S=O(x)\) there, \[K/T-r=(1-e/T)(S-r)+((b_0-T)S+h_0)/T=o(x)\] in the same measure. Mass below \(\varepsilon x\) contributes at most \(\varepsilon\) to the rescaled measure. Monotonicity and an outer regular point at a sufficiently large fixed multiple of \(x\) prevent mass above a sufficiently large multiple of \(x\) from contributing to a fixed compact clock interval. Letting \(\varepsilon\downarrow0\) proves the same local weak limit for \(K/(Tx)\), including possible atoms at zero. In particular \(\alpha_K(xt)/x\) is locally bounded and converges to \(U(t)\) at its continuity points.

The second derivative of the scalar defect. Write \(a_{2,n}=V_{n,zz}(0,0)\) and \(a_{4,n}=V_{n,zzzz}(0,0)\). Select convergent subsequences of these two numbers. On compact \(t\)-intervals the diffusion has \[\frac{\mathbb E Z_{xt}^2}{x}\longrightarrow Tt, \qquad \mathbb E|Z_{xt}|^{2j}=O_j(x^j).\] These statements follow from bounded drift, or directly from its bound \(T\alpha_K|V_z|\le Cx|Z|\) on the relevant clocks. Oddness and the derivative bounds give \[\frac{\mathbb E V_{n,zzz}(xt,Z_{xt})^2}{x} \longrightarrow T a_4^2t, \qquad \mathbb E V_{n,zz}(xt,Z_{xt})^3\longrightarrow a_2^3.\] For example, \(V_{n,zzz}(q,z)=a_{4,n}z+O(q|z|+|z|^3)\) uniformly on bounded \(z\)-intervals, and the moment bounds control the complement. By Lemma 41, \[\begin{aligned} g_n''(t)&=\frac{\widehat\Gamma_n''(xt)}x\\ &=T^2\left[ \frac{\mathbb E V_{n,zzz}(xt,Z_{xt})^2}{x} -2\frac{\alpha_K(xt)}x\mathbb E V_{n,zz}(xt,Z_{xt})^3\right]. \end{aligned}\] These second derivatives are locally bounded and converge almost everywhere, and in local \(L^1\), to \[At-BU(t),\qquad A=T^3a_4^2,\qquad B=2T^2a_2^3.\] Both coefficients have the asserted strict bounds.

Calibration of the linear term. It remains to bound \(g_n'(0)\); a bound on \(g_n''\) alone would leave this coefficient undetermined. Choose a nonfloor split in an outer band with \(u,r\asymp x\) and \(D=o(x)\), using the mass version of [eq:19]. Its scalar clock \(K/(Tx)\) is bounded above and below by positive constants. If \(e(x)/x^2\) is bounded, then \(e(0)=O(x^2)\) on these compact radius bands and \((S-K/T)/x^3=O(1)\). If \(e(x)/x^2\to\infty\), then \(x/d\to0\) and \(e=\lambda\), \(p=0\) exactly on every compact radius band. On contact \(S=r\); on a cap, \[|S-r|\le \ell+r_{\min}\mathbf1_{\mathrm{initial}},\qquad \frac{\lambda(\ell+r_{\min})}{x^3} \lesssim n^{-\delta}\log n\, \frac{\lambda}{d^2}\left(\frac ax\right)^3=o(1).\] Here we used \(L\le a^3/d^2\), the cap-width bound, and \(r_{\min}\le n^{-\delta}L\). Thus the same scalar value is bounded also in this regime. Closure [eq:22] bounds \(g_n\) at this positive clock. As \(g_n(0)=0\) and \(g_n''\) is locally bounded, it bounds \(g_n'(0)\). Subsequence selection and integration of \(g_n''\) prove [eq:29].

Zeros on the support. The same argument at almost every selected mass point gives \(g=0\) on the support of \(\nu\). More explicitly, in the bounded-strength regime \(e(S-r)=o(x^3)\) on the full-measure selections above; in the constant-profile regime the displayed cap estimate gives this conclusion. The bare-clock error is \(o(x^3)\), and Lemma 40 supplies little-oh closure at such points. Therefore \(g_n(K/(Tx))\to0\) on full normalized-mass selections. Every positive support point can be approached by these selections from neighborhoods carrying positive limiting mass, including an approximating atom. Local uniform convergence of \(g_n\) then gives \(g=0\) there. At zero use \(g(0)=0\). ◻

Proof. Choose the outer calibration band in the preceding proof beyond the fixed target interval. At its selected split, \(K_c\asymp x\) and \(K_c>Cx\). Monotonicity gives \(\alpha_K(q)=O(x)\) for every smaller clock; the omitted mass contributes only \(o(x)\). The second-derivative identity then gives \(\widehat\Gamma''(q)=O(x)\) for \(q=O(x)\). The positive-clock calibration gives \(\Gamma(K_c)-K_c/T=O(x^3)\), and \(\Gamma(0)=0\). Taylor integration therefore yields the derivative estimate and the uniform bound down to zero, including clock gaps. Equivalently, they follow from the uniform local bounds on \(g_n\) and \(g_n'\) in the preceding proof. If uniformity along the allowed sequences failed, selecting the violating sequence and repeating that proof would be a contradiction. No lower-group stationarity or lower-group closure is used here. ◻

Proof. On each compact interval above \(l\), masses \(\alpha(xt)\) lie between fixed positive multiples of \(x\). The raw estimates give \(S=O(x)\), \(B=O(x^2)\), and \(K=O(x)\) there. Helly selection supplies \(s\). Introduce the bare-clock remainder \[\zeta_n(r)=\frac{(b_0(\alpha(r))-T)S(r)+h_0(\alpha(r))}{x^3}.\] By [eq:4], it is uniformly \(o(1)\) on these compact mass bands. The exact closure identity is \[\frac{e(r)(S(r)-r)}{Tx^3} =g_n\!\left(\frac{K(r)}{Tx}\right)+\rho_n(r)+\frac{\zeta_n(r)}T.\] In particular the closure error is additive in this identity. Lemma 40 gives both its uniform bound and its strong radius \(L^1\) convergence to zero.

If \(e(x)/x^2\to\infty\), then \(\lambda r/d^3\le Cx/d\to0\) on a compact radius band. Thus \(e=\lambda\) and \(p=0\) exactly there. The last identity and the boundedness of \(g_n\) imply \[\sup\frac{|S-r|}{x}\lesssim\frac{x^2}{e}\longrightarrow0, \qquad \frac{K(r)}{Tx}-\frac rx\longrightarrow0\] uniformly on the band. Therefore \(G=2e(r-S)\) is uniformly of order \(x^3\) and \(G(xt)/x^3\to-2Tg(t)\) strongly in \(L^1\).

In the other case, the fixed smooth strength profile has a locally convergent subsequence with positive limit \(h_1\); the same is true of its logarithmic slope. Now \(e=O(x^2)\), so \(K/(Tx)-S/x\to0\). Pass to the limit in the exact closure identity to obtain the displayed equation for \(g(s)\). Since \(B=S^2+D^2\) and the radius version of [eq:19] makes \(D^2/x^2\) tend to zero in \(L^1\), the limit of \(G/x^3\) is \[F(t)=h_1(t)\left((2-p(t))t-2(1-p(t))s(t) -p(t)\frac{s(t)^2}{t}\right).\] The bracket factors as \[(t-s(t))\left(2-p(t)+p(t)\frac{s(t)}t\right),\] and its second factor is positive. Raw bounds give a uniform bound on \(G/x^3\) on compact positive radius intervals. If a band extends to zero, \(p=0\) on a fixed scaled neighborhood of zero because \(x=O(d)\) and \(\lambda\le d^2\); this removes the apparent \(1/t\) singularity.

We construct the potential, including its endpoint values. On any band with masses comparable to \(x\), normalize the first-variation factor \(J\) at one reference radius. Its distortion is bounded above and below: indeed \[\int wM'(\alpha)\,dr \le\int \frac{M'(\alpha)}{M(\alpha)}\alpha'\,dr \le2\log\frac{\alpha_{\mathrm{right}}+u_*} {\alpha_{\mathrm{left}}+u_*}.\] Thus \(J/J_{\mathrm{ref}}\) has a weak-star convergent subsequence in \(L^\infty\) with limit \(j\) bounded above and below by positive constants. Normalize the potential by \(x^4J_{\mathrm{ref}}\). Its derivative in scaled radius is \((J/J_{\mathrm{ref}})G/x^3\), which is uniformly bounded. Strong \(L^1\) convergence of \(G/x^3\) identifies the derivative of every uniform potential limit as \(jF\).

A support point is approached by nonfloor mass points, since a neighborhood with positive limiting rescaled mass cannot consist of the \(o(x)\) total floor mass. At contact the potential is zero. At a point in a negative cap its right zero lies at radius distance \(o(x)\); the derivative bound makes the normalized potential there \(o(1)\). Outer comparability bounds the masses above by \(Cx\) after a fixed enlargement of the band. If the point is the first support point and begins a gap, it must be an atom. Choose the reference points inside its approximating atom after a fixed positive amount of rescaled mass has accumulated. The mass is then bounded below by \(cx\) all the way from that reference to the interior of the gap; no estimate on smaller masses is needed. The normalized potentials are controlled on intervals starting at these moving reference points. Their uniform Lipschitz bound extends the limit to the atom with endpoint value zero. This also covers an atom at radius zero, without requiring a bound at the exact prelimit radius zero.

These reference points both bound the additive constant of the normalized potential and pin its limiting endpoint values to zero. Negative cap components have radius width \(o(x)\), so their negative excursions disappear under the same Lipschitz bound. The limit is therefore nonnegative. This uses small floor mass only to select a reference point inside positive support mass, never to discard a floor interval in radius.

Finally, in the bounded-strength case, monotonicity bounds the virtual values of \(S/x\) between active mass selections approaching the two support endpoints. At those selections \(S-r=o(x)\). Consequently \(a_1\le s(t)\le a_2\) on the gap, as claimed. ◻

Proof of Theorem 39. We show that the limiting support is all of \([0,\infty)\). On a bounded gap, the variational potential forces both endpoint slopes of \(g\) to be nonpositive, while its scalar equation forces their sum to be positive. Consider such a nontrivial gap \((a_1,a_2)\) with positive cumulative mass. We claim \(g'(a_1)\le0\) and \(g'(a_2)\le0\). If \(g'(a_1)>0\), then \(g>0\) immediately to the right of \(a_1\). In the diverging-strength case this makes \(F<0\) there, contrary to \(\mathcal H\ge0\) and \(\mathcal H(a_1)=0\). In the bounded-strength case the closure equation excludes \(s=a_1\) for \(t>a_1\). If \(s\) is close to \(a_1\), the inequality \(g(s)>0\) forces \(s>t\); if \(s\) is farther from \(a_1\), it is again greater than \(t\) for all sufficiently nearby \(t\). Thus \(F<0\) almost everywhere immediately to the right, giving the same contradiction. If \(g'(a_2)>0\), the analogous argument on the left gives \(F>0\) there, which contradicts \(\mathcal H\ge0\) and \(\mathcal H(a_2)=0\).

On the gap, \(U\) is constant, \(g(a_1)=g(a_2)=0\), and \(g''(t)=At-\text{constant}\). Direct integration gives \[g'(a_1)+g'(a_2)=\frac A6(a_2-a_1)^2>0,\] a contradiction. The support is unbounded because \(U(t)\asymp t\) for large \(t\). Hence, with \(l=\inf\operatorname{supp}\nu\), it has no gaps on \((l,\infty)\). There \(g=0\), so [eq:29] implies \(U(t)=(A/B)t\), first almost everywhere and then everywhere by right continuity. If \(l>0\), then \(g(l)=g'(l)=0\) and \(g''(t)=At\) for \(0<t<l\). These imply \(g(0)=Al^3/3>0\), a contradiction. Thus \(l=0\) and \(U(t)=\kappa t\), where \(\kappa=A/B\).

This cumulative function is continuous and strictly increasing. Local weak convergence of the rescaled measures therefore implies convergence of their inverse quantiles at every positive mass, and monotonicity makes this convergence locally uniform. The strict bounds on \(A,B\) give the bounds on \(\kappa\). This proves [eq:28]. ◻

Proof. Write \(\eta_n=n^{-1/8}\) and \(Q_n=n^{1/8}\), and first minimize over onto quantiles whose inverse \(\alpha:[0,1]\to[0,1]\) satisfies \[\eta_n\le\alpha'\le Q_n,\qquad \alpha(0)=0,\quad\alpha(1)=1.\] This is a nonempty compact class of inverse paths at fixed \(n\). For \(0\le t\le1\), put \[\Phi_n(t)=\min_z\left\{ f_n((1-t)b,h+tbz)+\frac t4\int_0^1 bz^2\,du\right\}.\] All the interpolated paths are admissible. The first identity in [eq:1] and the envelope argument of Lemma 16 give local Lipschitz continuity and, at almost every \(t\), the derivative formula at a minimizing path, \[0\le\Phi_n'(t)=\frac14\int_0^1 b\bigl[D^2+(S-z)^2\bigr]\,du.\] An upper bound by the right side would also suffice. All these derivatives are bounded uniformly in \(n,t\).

For \(t>0\), first variation in the quantile has bracket \(t b(u)(r-S)/2\), where \(r=z(u)\). The inverse-density constraints are now constant. The proof of Lemma 18 gives a potential with derivative a positive constant times \(b(u)(r-S)\): the inverse density equals \(Q_n\) on its negative components and \(\eta_n\) on its positive components. On its zero set \(S=r\) almost everywhere. A negative component has radius width at most \(Q_n^{-1}\), since its mass is at most one. Its endpoint signs and monotonicity of \(S\) place \(S(r)\) between its radius endpoints. This also holds for an initial or final negative component, using \(0\le S\le1\) at the exterior endpoint. Consequently \[\int_0^1(S-z)^2\,du\le Q_n^{-2}+\eta_n.\] Indeed the total mass on the floor is at most \(\eta_n\), and off the floor the pointwise error is at most \(Q_n^{-1}\).

We give a concentration argument uniform over these minimizers for \(t\) bounded away from zero. In radius coordinates the field clock satisfies \(\partial_r(h+tbz)\ge tb_*\) in the sense of measures; the matrix clock is also nondecreasing. The second identity in [eq:1], with nonnegative jump contributions discarded, yields \[\int_0^1\mathbb E\operatorname{tr}H_r^2\,dr \le\frac1{ntb_*},\qquad \int_0^1\mathbb E\operatorname{tr}H_u^2\,du \le\frac{Q_n}{ntb_*}.\] Fix \(\varepsilon>0\). Since \(H_u\ge uC_u\), the trace and Brascamp–Lieb bounds in [eq:1] imply \[\int_\varepsilon^1\mathbb E\operatorname{tr}C_u^2\,du \le\frac{Q_n}{ntb_*\varepsilon^2},\qquad \operatorname{Var}\left(\int_I L_u\,du\right) \le\frac{C Q_n}{ntb_*\varepsilon}\] for every interval \(I\subset[\varepsilon,1]\). For the second bound use \(\mathbb E W_u=\mathbb E\operatorname{tr}(H_uC_u) \le u^{-1}\mathbb E\operatorname{tr}H_u^2\).

Here are the details converting interval concentration to pointwise concentration in integrated mass. Partition \([\varepsilon,1]\) into at most \(1+\lceil\Delta^{-1}\rceil\) intervals on which the nondecreasing deterministic function \(S\) varies by at most \(\Delta\), allowing empty intervals. Remove a strip of mass \(\rho\) at each end of each interval, and discard any interval shorter than \(2\rho\). The discarded mass is at most \(C\rho(1+\Delta^{-1})\). At a retained mass \(u\), both \(I_-=[u-\rho,u]\) and \(I_+=[u,u+\rho]\) lie in the same interval. The martingale property of \(X\) gives, for \(v\le u\le w\), \[\mathbb E_u L_v =\frac{|X_u|^2}{2}-\frac{|X_u-X_v|^2}{2} \le\frac{|X_u|^2}{2} \le\frac{\mathbb E_u|X_w|^2}{2} =\mathbb E_u L_w.\] After averaging over \(I_-\) and \(I_+\) and subtracting \(S_u/2\), conditional Jensen’s inequality and the interval variance bound give \[\bigl\||X_u|^2-S_u\bigr\|_2 \le C\Delta+\frac C\rho \left(\frac{Q_n}{ntb_*\varepsilon}\right)^{1/2}.\] On discarded masses use \(0\le|X_u|^2,S_u\le1\). Thus, first letting \(n\to\infty\) for fixed \(t\ge\varepsilon,\Delta,\rho\), then \(\rho\downarrow0\) and \(\Delta\downarrow0\), proves \[\sup_{t\in[\varepsilon,1]} \int_\varepsilon^1\operatorname{Var}(|X_u|^2)\,du\longrightarrow0.\] This order of limits is legitimate because every bound above is uniform over the minimizing paths and over \(t\ge\varepsilon\). The identity \[D_u^2=\operatorname{Var}(|X_u|^2) +\mathbb E\operatorname{tr}C_u^2 +2\mathbb E X_u^{\mathsf T}C_uX_u\] and \(\mathbb E X_u^{\mathsf T}C_uX_u \le(\mathbb E\operatorname{tr}C_u^2)^{1/2}\) now show that the same integrated convergence holds for \(D^2\). Masses \(u<\varepsilon\) and interpolation times \(t<\varepsilon\) cost \(O(\varepsilon)\). Sending \(\varepsilon\downarrow0\) in the integrated envelope bound proves \(\Phi_n(1)-\Phi_n(0)\to0\).

At \(t=0\) the value is \(f_n(b,h)\), independently of \(z\). At \(t=1\) it is exactly the scalar objective in the statement, restricted to the inverse-density class. Every probability distribution on \([0,1]\) can be approximated weakly by smooth strictly positive densities bounded above and below; their quantiles converge in \(L^1\). Each such approximating density belongs to the constrained class for all sufficiently large \(n\). Conversely, monotone quantiles form a compact set in \(L^1\). The variation formula gives \[|f_n(b,h)-f_n(\widetilde b,\widetilde h)| \le\frac14\|b-\widetilde b\|_1 +\frac12\|h-\widetilde h\|_1\] between any two admissible pairs, uniformly in \(n\); the same bound holds for \(f_{\rm spin}\). These continuity bounds pass both the constrained minima and the varying endpoint paths to their limits. They also prove attainment of the scalar minimum. ◻

Proof. Existence follows from compactness and the continuity just proved. We include the uniqueness argument to specify the needed form, using the stochastic-control and strict-convexity method of (Auffinger and Chen 2015b, Theorems 2–3). The scalar control representation is \[V_\alpha(0,0)=\sup_{|m|\le1} \mathbb E\left[ \log\cosh\left(\sqrt T B_1+T\int_0^1\alpha(s)m_s\,ds\right) -\frac T2\int_0^1\alpha(s)m_s^2\,ds\right],\] where controls are progressively measurable. To verify it, apply Itô’s formula to \(V_\alpha(t,Z_t)\) for \(dZ_t=T\alpha(t)m_t\,dt+\sqrt T\,dB_t\). The resulting difference between the initial PDE value and the displayed payoff is \(\frac T2\mathbb E\int\alpha(t)(m_t-V_{\alpha,z}(t,Z_t))^2\,dt\). It vanishes for the bounded feedback control \(m_t=V_{\alpha,z}(t,Z_t)\).

Suppose \(\alpha_1\) and \(\alpha_2\) differ on a set of positive Lebesgue measure. Use the optimal control \(m\) for their midpoint \(\bar\alpha\) in all three functionals. For this control, \(m_0=0\) by symmetry and \[dm_t=\sqrt T\,V_{\bar\alpha,zz}(t,Z_t)\,dB_t, \qquad V_{\bar\alpha,zz}(t,Z_t)>0.\] With \(a=\alpha_1-\alpha_2\), stochastic Fubini gives \[\int_0^1a(t)m_t\,dt =\sqrt T\int_0^1 \left(\int_s^1a(t)\,dt\right) V_{\bar\alpha,zz}(s,Z_s)\,dB_s.\] Its second moment is strictly positive by Itô isometry: a vanishing continuous primitive would imply \(a=0\) almost everywhere. Consequently the terminal arguments for \(\alpha_1\) and \(\alpha_2\) differ with positive probability. Strict convexity of \(\log\cosh\) makes the midpoint payoff strictly smaller than the average of the two payoffs for this control, which are bounded by the corresponding optimal values. The integral penalty in the Parisi functional is linear in \(\alpha\). Strict convexity of the full functional therefore proves uniqueness. Distribution functions equal almost everywhere describe the same probability measure on \([0,1]\).

Lopatto’s support theorem (Lopatto 2026, Theorem 1.1) states that, at zero field and \(T=\beta^2>1\), the minimizing measure has support \([0,q_\beta]\) with \(q_\beta>0\), no atom at zero, and a smooth density near zero. Its scalar martingale satisfies \(\widehat\Gamma(s)=s\) on that support by (Lopatto 2026, Lemma 2.2, equation (2.3)). We now compute the initial density and prove its positivity. Since \(\widehat\Gamma''(s)=0\) for \(0<s<q_\beta\), Lemma 41 gives, for sufficiently small positive \(s\), \[\alpha_T(s)= \frac{\mathbb E V_{zzz}(s,Z_s)^2} {2\mathbb E V_{zz}(s,Z_s)^3}.\] Since \(\alpha_T(s)\to0\), the diffusion has \(\mathbb E Z_s^2=Ts+o(s)\) and moments of order \(2j\) of order \(s^j\). Oddness, the uniform spatial derivative bounds, and bounded time derivatives imply \[\mathbb E V_{zzz}(s,Z_s)^2 =T V_{zzzz}(0,0)^2s+o(s),\qquad \mathbb E V_{zz}(s,Z_s)^3\longrightarrow V_{zz}(0,0)^3.\] The strict bounds in Lemma 41 yield \[\kappa_T=\frac{T V_{zzzz}(0,0)^2}{2V_{zz}(0,0)^3}>0.\] Thus \(q_T(u)\sim u/\kappa_T\) at zero. Full interval support makes the distribution strictly increasing on its support and hence makes the quantile continuous on \((0,1)\), even if the measure has atoms. Away from zero, monotonicity, positivity, and \(q_T\le1\) extend the two-sided comparison with \(u\) to the entire open mass interval. ◻

Proof. Pass to a subsequence on which \(\lambda_n,d_n\) converge. The assumed positive strength bounds force both limiting parameters to be positive. Consequently \(e_n\) converges uniformly on \([0,1]\) to a continuous strictly positive function \(e\), with \(T-e\ge c\). The endpoint hypothesis [eq:4] implies \(b_0\to T\) and \(h_0\to0\) in \(L^1\) on the working masses. Indeed \(a=O(1)\), so each fixed positive mass satisfies \(a\lesssim u\) with a fixed constant; boundedness then gives dominated convergence. The omitted mass tends to zero and cannot affect these limits.

The constraints approximate every limiting quantile: their floor is at most \(n^{-3\delta}\) and their cap is at least \(n^\delta\). First approximate a desired distribution by a fixed smooth density bounded above and below; its slight affine adjustment to the endpoints \((r_{\min},u_0)\) satisfies the constraints for all sufficiently large \(n\). Here \(r_{\min}\to0\) by [eq:5]. The same construction supplies competitors on the working interval. All changed paths used by the optimization and these competitors are admissible by hypothesis.

Extract an \(L^1\) limit \(r\) of the minimizing quantiles. The uniform pressure continuity bound in Lemma 45 and that lemma’s variational limit, applied to the actual changed paths, identify the limiting minimization as the joint minimum in quantiles \(r,z\) of \[f_{\rm spin}\bigl(e(r)r+(T-e(r))z\bigr) +\frac14\int_0^1\bigl[e(r)r^2+(T-e(r))z^2\bigr]\,du.\] This reasoning applies both to the selected minimizing sequence and to every approximating competitor, so it identifies a minimum and its minimizers, not just an upper bound. It never evaluates a possibly inadmissible bare endpoint.

Set \[q=\frac{e(r)r+(T-e(r))z}{T}.\] This is a \([0,1]\)-valued quantile: \(re(r)\) is nondecreasing, \(T-e(r)\) is nonnegative and nondecreasing, and \(z\) is nonnegative and nondecreasing. Completing the square gives \[e(r)r^2+(T-e(r))z^2 =Tq^2+\frac{e(r)(T-e(r))}{T}(r-z)^2.\] The joint objective is therefore at least the ordinary scalar minimum. Equality is attained by \(r=z=q_T\). The coefficient of \((r-z)^2\) is bounded below positively, so every minimizing pair has \(r=z=q\) almost everywhere. Lemma 46 forces this quantile to be \(q_T\).

Every subsequential limit is thus \(q_T\), proving \(L^1\) convergence of the full sequence. Its continuity at each interior mass and monotonicity of the approximating quantiles give local uniform convergence. The artificial endpoint condition \(r_n(1)=1\) is irrelevant for this conclusion on the open mass interval. ◻

Proof. The data conditions imply, with fixed implicit constants, \[d\ge a\gtrsim n^{-1/6+m},\qquad E\gtrsim n^{-5/6+2m},\qquad nEa\gtrsim n^{3m}.\] The strength profile obeys \(e(t)\gtrsim \min(\lambda,d^3/t)\), while \(te(t)\) is increasing. Since \(x\ge A(\lambda)\) and \(\sqrt\lambda\sqrt n E\le d^3\), \[x\ge n^\delta a,\qquad e(x)x^2\gtrsim \min(\lambda x^2,d^3x)\gtrsim n^{1+\delta}E^2.\] These estimates are valid whether \(x\) tends to zero or is macroscopic.

Put \(E'_{\mathrm c}=n^{-1/2}c_0^2x^2\) and \(E'_{\mathrm s}=n^{-100\delta}e(x)x^3\). For the lower bound in [eq:3], separately for the two candidate values, we have \[\frac{E'_{\mathrm c}}{n^{-2/3+m}d'} =c_0 n^{1/6-m}x\gtrsim c_0n^\delta, \qquad \frac{E'_{\mathrm s}}{n^{-2/3+m}d'} \gtrsim c_0^{-1}n^{3m-99\delta}.\] Both tend to infinity; the required upper bound holds by definition. The same calculation gives \[\frac{E'_{\mathrm s}}E \gtrsim n^{1-99\delta}Ex\gtrsim n^{3m-99\delta}.\] Because \(e(x)=o(x^2)\), the preceding lower bound on \(e(x)x^2\) also gives \(x^4/(n^{1+\delta}E^2)\to\infty\). Consequently \[\frac{E'_{\mathrm c}}E =c_0^2\frac{x^2}{\sqrt n E} \gg n^{\delta/2}.\] Taking the smaller candidate proves \(E'/E\ge n^{\delta/4}\) eventually. In the ceiling case \(a'=c_0x\ge a\) eventually. In the other case, \[\frac{a'}a =c_0^{-1}n^{-100\delta}\frac{e(x)x^2d}E \gtrsim c_0^{-1}n^{1-99\delta}Ed \gtrsim n^{3m-99\delta},\] which proves the asserted cutoff inheritance.

It remains to check the actual covariance perturbations in [eq:4]. The parent outer bound and monotonicity give \(r(u)=O(y)\) when \(u\asymp y\gtrsim x\), and \(r(u)=O(x)\) when \(u=O(x)\). Also \(e(0)=o(x^2)\). Indeed, along any subsequence on which \(d/x\to0\) this follows from \(e(0)\le d^2\); if \(d/x\) stays bounded below, the argument \(\lambda x/d^3\le x/d\) of \(f\) stays bounded, so \(e(0)\asymp e(x)=o(x^2)\). Thus at every scale \(y\gtrsim x\) the new matrix change is \(o(y^2)\) and the new field change is \(o(y^3)\).

Consider now a test sequence \(a'\lesssim y\ll x\). This can only occur in the nonceiling case. Here \[\frac{a'^3}{e(x)x} =c_0^{-3}n^{3/2-300\delta}e(x)^2x^5 \gtrsim n^{7/2-300\delta}E^4x \gtrsim n^{9m-300\delta}.\] In particular the ratio has a polynomial divergence. If \(\lambda x/d^3\) stays bounded, \(e(0)\asymp e(x)\) and, for \(r\le Cx\), \[e(r)\le O(e(x))=o(a'^2),\qquad re(r)\le O(e(x)x)=o(a'^3).\] The first conclusion uses also \(a'\le d'=c_0x\). Otherwise \(e(x)x\gtrsim d^3\), and the global profile bounds give \[e(r)\le d^2=o(a'^2),\qquad re(r)\lesssim d^3\log n=o(a'^3).\] The polynomial divergence absorbs the logarithm. These two subsequence alternatives prove the required little-oh estimates at every \(y\gtrsim a'\). The old bare endpoint already satisfies [eq:4] there, since \(a'\ge a\). The omitted interval assertion is immediate. The same reasoning applies after a change from high-only to all-mass comparisons: positive comparable test masses at the new cutoff still lie above the original omitted interval. ◻

Proof. Write \(e=e(r)\), \(\widehat e=e(\widehat r)\) and \(D=\mathcal D(z)\) in this proof only, and set \[a=e\mathbf1_I+D\mathbf1_J-\widehat e\mathbf1_I, \qquad c=er\mathbf1_I+Dz\mathbf1_J-\widehat e\widehat r\mathbf1_I.\] At a moved point the log-slope bound \(0\le-re'/e\le1\) implies \(\widehat e\le e/(1-\omega)\). This follows from monotonicity when \(\widehat r\ge r\), and otherwise from \(\widehat r\ge(1-\omega)r\) and the monotonicity of \(te(t)\). Hence \(a>0\) there. At unmoved points of \(J\), \(a=D\) and \(c=Dz\); outside \(J\), \(a=c=0\).

The ratio needed below is integrable even if \(a\) approaches zero. Indeed put \(A=e+D\), \(\mu=D/A\), and \(\Delta=z-r\) at a moved point. The preceding bound gives \[a\ge\frac{A(\mu-\omega)}{1-\omega},\qquad \frac ca=r+\left(\omega+\frac{A(\mu-\omega)}a\right)\Delta.\] The coefficient of \(\Delta\) lies in \([\omega,1]\), so \(c/a\) lies between \(\widehat r\) and \(z\), both in \([0,1]\). Thus \(c^2/a\le a\le e+D\). At unmoved points of \(J\) the ratio is \(Dz^2\), and outside \(J\) it is zero. These bounds also cover radius zero and require no uniform lower bound on \(a\).

The convex segment from \(Q\) to \(P_{\widehat r}\) consists of admissible covariance paths. Its pointwise variation is \((a,-c)\); the variation itself need not be monotone. By [eq:1] and \(B\ge S^2\), at every point of that segment, \[-\frac a4B+\frac c2S\le\frac{c^2}{4a}.\] With the ratio interpreted as zero when \(a=c=0\), integration gives \[f_n(P_{\widehat r})\le f_n(Q)+\frac14\int\frac{c^2}a\,du.\] Combining this with parent minimality and the displayed child bound proves \[\int\left[er^2\mathbf1_I+Dz^2\mathbf1_J -\widehat e\widehat r^2\mathbf1_I-\frac{c^2}a\right]du \le4\,\mathrm{err}.\] Any fixed low-group penalty has canceled. At every unmoved point of \(J\) the integrand is zero, including points below \(u_0\) when \(J=[0,1]\). At a moved point its value is \[\left\{\frac{eD}{e+D} -\frac{\widehat e(e+D)}{e+D-\widehat e} \left(\omega-\frac D{e+D}\right)^2\right\}(r-z)^2.\] The expression in braces decreases as \(\widehat e\) increases. Substitution of its upper bound \(e/(1-\omega)\) reduces it exactly to \(e\omega\), proving the assertion. Only the three admissible paths \(P_r,Q,P_{\widehat r}\) have been evaluated. ◻

Proof. In forward coordinates the constraints [eq:5] read \[l(u)\le r'(u)\le\frac1{k(r(u))},\qquad l(u)=\frac1{M(u)}.\] Their forced lengths satisfy \[\int_{u_0}^1l(u)\,du \le n^{-\delta}L\log\frac{1+u_*}{u_0+u_*}=o(x), \qquad \int_0^1 k(t)\,dt\le n^{-\delta}u_*=o(x).\] Also \(l\le n^{-\delta}\), \(k\le n^{-3\delta}\), and \(r_{\min}\le n^{-\delta}u_*=o(x)\). The estimates follow directly from [eq:5] and \(u_*/a\lesssim n^{-3m}\). We may shorten the separation interval by a fixed proportion to leave room for ramps and strict threshold inequalities. Put \[t_y=\min\{1,\mathcal D(y)/e(y)\}.\]

First suppose \(z\) is above the upper threshold and \(r\) below the lower threshold. Starting from \(r_{\min}\), construct a barrier \(w\) with slope \(l\) until an early portion of the separation interval. On that portion give it a fixed sufficiently large slope to reach an intermediate threshold, and thereafter slope \(l\) again. The ramp needs only a fixed small fraction of the interval, since its rise is \(O(y)\). Its fixed slope is eventually less than \(n^{3\delta}\) and greater than \(l\). The \(o(x)\) total forced rise leaves \(w<1\) and keeps a fixed distance of order \(y\) between \(w\) and \(z\) wherever \(w>r\). Before the ramp the minimum slope condition gives \(w\le r\); after it, monotonicity of \(z\) preserves the separation. Outer regularity shows that clipping can occur only at masses \(O(y)\).

Let \(v=\max(r,w)\) and set \(\widehat r=(1-t)r+tv\), where \(t=\eta t_y\) and \(\eta>0\) is a sufficiently small fixed constant. Both endpoints of the quantile are unchanged. The lower slope bound is preserved. Where clipping occurs, \(v'=w'\le n^{3\delta}\le1/k(r)\); hence \[\widehat r'\le\frac1{k(r)}\le\frac1{k(\widehat r)}.\] Thus the upper bound is preserved as well.

We verify the movement fraction even if \(z/y\) is unbounded. At clipped points \(r<w\le C_1y\) and \(z-w\ge c_1y\), so \(z-r\ge c_2z\). Monotonicity of \(t\mathcal D(t)\) and the fixed-scale bounds for \(\mathcal D\) give \[\mathcal D(z)(z-r)\ge c_3y\mathcal D(y), \qquad \mathcal D(z)\le C_2\mathcal D(y).\] If \(y/x\to\infty\), outer regularity gives \(r\asymp y\) at the clipped masses, and \(e(r)\le C_3e(y)\). Therefore the largest allowed absolute displacement is bounded below by \[\frac{\mathcal D(z)(z-r)}{\mathcal D(z)+e(r)} \ge c_4yt_y.\] If instead \(y\asymp x\), then \(e(r)\le e(0)=o(x^2)\) and \(\mathcal D(y)\asymp x^2\). The same bound holds, with \(t_y=1\) eventually, without a lower bound on \(r/y\). Every sequence has a subsequence in one of these two cases. Choosing \(\eta\) small makes the movement strictly admissible. On a remaining interval of length \(cy\), both \(v-r\) and \(z-r\) are at least \(cy\), and \(e(r)\ge ce(y)\). Since \(\omega(z-r)=\widehat r-r=t(v-r)\), the gain there is at least \(c e(y)t_y y^3\).

For a downward separation, start an upper barrier \(w\) at an intermediate positive threshold plus the accumulated slope \(l\). Keep that slope until a late portion of the separation interval. There let \(w'=1/k(w)\) until \(w\) meets \[U(u)=1-\int_u^1l(v)\,dv,\] and follow \(U\) thereafter. Every feasible \(r\) satisfies \(r\le U\). The ramp takes \(o(x)\) mass: while \(w<U\), the speed of \(U\) is at most \(n^{-\delta}\) and that of \(w\) is at least \(n^{3\delta}\), so its length is bounded by twice \(\int_0^1k=o(x)\) for large \(n\). Place this entire ramp inside the reserved late portion of the separation interval; its length is \(o(x)=o(y)\), so it fits there. Where \(v=\min(r,w)\) differs from \(r\), both \(v\) and \(r\) lie in \([cy,Cy]\), while \(z\) is a fixed distance below \(v\). The assertion before the separation interval follows from monotonicity of \(z\); the barrier is no longer active after its ramp. It agrees with \(r\) at the two endpoints, because \(r_{\min}=o(x)\) and \(r\le U\).

Set \(\psi(q)=\int_0^qk(t)\,dt\) and now interpolate by \[\psi(\widehat r)=(1-t)\psi(r)+t\psi(v),\qquad t=\eta t_y.\] Both \(r\) and \(v\) satisfy the two slope bounds. Therefore \[k(\widehat r)\widehat r' =(1-t)k(r)r'+tk(v)v'\le1.\] For the other bound write \(k(q)=A_0/(q+B_0)^2\), with \(A_0=n^\delta u_*^2\) and \(B_0=n^{2\delta}u_*\). Then \[k\bigl(\psi^{-1}(s)\bigr) =\frac1{A_0}\left(\frac{A_0}{B_0}-s\right)^2\] is convex. Jensen’s inequality gives \[k(\widehat r)\widehat r' \ge l\{(1-t)k(r)+tk(v)\}\ge l k(\widehat r).\] This proves feasibility. On the clipping region the radii are comparable to \(y\), so the distortion between the interpolation fraction in \(\psi\) and the fraction in radius is bounded above and below by fixed constants. Here \(e(r)\asymp e(y)\) and \(\mathcal D(z)\gtrsim\mathcal D(y)\), since \(z\le Cy\). Taking \(\eta\) smaller if necessary gives the required strict movement fraction. On an interval of length \(cy\), the radius displacement is at least \(ct_yy\), and \(r-z\ge cy\). The same gain bound follows.

Both constructions preserve the parent’s lower wall. Hence their changed paths are admissible in every family of Definition 48. All moved masses lie above both lower working boundaries, since \(u_0=o(a)\le o(x)\). ◻

Proof. First, on every test scale \(y\gtrsim x\), the parent and final child quantiles agree to \(o(y)\) in normalized mass measure on compact positive mass bands of scale \(y\). At macroscopic scales take bands inside \((0,1)\). If this failed, monotone subsequence compactness (truncating \(z/y\) first if necessary) would produce an interval of length comparable to \(y\) on which the quantiles are separated by fixed thresholds of distance comparable to \(y\). The parent outer bound and monotonicity give \(r\le Cy\) there. Thus Lemmas 50 and 51 imply \[c\min\{e(y),\mathcal D(y)\}y^3\le4n^{20\delta}E'.\] Both radius times strength profiles are increasing. Moreover \(\mathcal D(x)=c_0^2f(1/c_0)x^2\asymp x^2\) and \(e(x)=o(x^2)\). Consequently, for every \(y\gtrsim x\), \[\min\{e(y),\mathcal D(y)\}y^3 \gtrsim e(x)x^3, \qquad n^{20\delta}E'\le n^{-80\delta}e(x)x^3,\] a contradiction. For \(y/x\) bounded below by a fixed number smaller than one, the same inequalities follow from the fixed-scale profile bounds; thus the conclusion includes any fixed positive multiple of \(x\).

The child’s numerical direct-test conditions now hold at \(x\): \[a'\le c_0x,\qquad nE'^2\le c_0^4x^4\lesssim\mathcal D(x)x^2, \qquad \mathcal D(x)\asymp x^2.\] Its changed paths and feasible competitors are admissible, and \(d'^2\le T-\rho\) by Definition 48.

Suppose \(x\to0\). This agreement gives the child a regular outer region above \(x\), with fixed, possibly enlarged constants. Indeed, a violation at masses whose ratio to \(x\) tends to infinity gives, by monotonicity and a neighboring interval of comparable masses, a threshold separation from the parent’s regular outer profile. A violation at bounded multiples is included by enlarging the lower cutoff of that outer region. 39 therefore applies to the child, giving [eq:28]. Agreement in measure and monotonicity transfer this limit to the parent at every positive scaled mass: bracket a point by nearby continuity points of the strictly increasing linear limit and then shrink the bracket. If \(x\) stays bounded below, apply 47 to the child instead and use the same agreement. Near mass one, fixed loose comparison bounds follow from monotonicity and \(r\le1\). This proves the assertion. ◻

Proof. Fix the bounds and recursion depth. Enlarge the initial total-clock bounds by one to allow for descendant field additions. Use these enlarged bounds when choosing the constants in the limiting profiles. Choose \(c,C\) strictly outside the bounds of all the limiting profiles described at the start of this section, with additional room near mass one. Fix \[N=\left\lceil\frac{10}{\delta}\right\rceil.\] Before taking \(n\to\infty\), choose \(c_0>0\) small enough that \(Nc_0^2<1\) and \(Nc_0^2\) is less than half each initial fixed residual matrix lower bound or gap. It may also be decreased to meet the fixed bounds needed in the direct tests, including \(c_0^2\le T-\rho\). Every child has \(d'=c_0x\le c_0\), and hence \(d'^2\le T-\rho\). The sum of all further matrix removals along a chain of length \(N\) is therefore at most \(Nc_0^2\). All field additions are increasing and have total size at most \(Nc_0^2\). Thus the total clock bounds and the covariance alternatives of Definition 48 persist along this chain. In particular every child endpoint, every feasible child path, and every segment used in Lemma 50 are admissible.

A failed parent profile forces a failed child error. Assume first that the profile assertion fails along a sequence. At its selected strength choose an offending mass \(x\ge A(\lambda)\) within a factor two of the supremum of all offending masses. There are no offending masses above \(2x\), so the path has a regular outer region above \(x\). The numerical direct-test conditions \(x\gtrsim a\) and \(e(x)x^2\gtrsim nE^2\) follow from [eq:8]. If \(e(x)/x^2\) has a positive lower bound along a subsequence, Theorem 39 or Proposition 47, according to the scale, immediately contradicts the choice of the loose constants \(c,C\). After subselection the only remaining case is \(e(x)/x^2\to0\). Lemma 49 constructs a child from the changed parent endpoint. If its error estimate were successful, Lemma 52 would again contradict the parent violation. Hence a profile failure forces a child error failure.

A failed error selects the next profile failure. Every such child has an admissible bare endpoint. If all its minimizers at all relevant strengths had the asserted profile, Proposition 25 and the optimized envelope of Lemma 16 would give \[\mathcal V(d'^2)-f_n(b'_0,h'_0) \le O(\log n)n^{10\delta}E'+O(n^{-10}) <n^{20\delta}E'\] eventually. Strengths below \(n^{-10}\) contribute \(O(n^{-10})\) by bounded overlaps; when \(A(\lambda)\) is macroscopic the conditional error bound is unconditional. The derivative bound uses an old minimizer for positive increments, or equivalently the Lipschitz envelope inequality, and needs no monotone selection of minimizers. It follows that a child error failure selects a strength and a minimizing quantile with a profile failure. The same argument applies if the initial failure was an error failure at an ordinary endpoint. The lower bound in the proposition is the nonnegative segment interpolation in [eq:6].

The error parameter cannot grow for the whole fixed chain. We may therefore repeat the construction: each bad profile forces a bad error for a valid child; that bad error selects the next bad profile. Lemma 49 preserves all asymptotic endpoint hypotheses and multiplies \(E\) by at least \(n^{\delta/4}\) at each stage. All choices of constants have already been made. At most \(N\) successive subsequence extractions are involved, so the eventual inequalities can be imposed together. On the other hand [eq:3], at every stage, confines the error parameter to \[c_j n^{-5/6+2m}\le E_j\le C_j n^{-1/2},\] where \(c_j,C_j\) are fixed constants for that stage. A chain of \(N\) increases would imply \(E_N\ge n^{N\delta/4}E_0\), contradicting these bounds since \(N\delta/4\ge5/2>1/3-2m\). This proves both assertions.

For a fixed strength root, the argument begins directly with its bad profile. There is no root sweep. Its first child starts from the admissible changed path \(P_r\), and every subsequent error sweep has an admissible bare endpoint. Lemma 50 shows explicitly that no bare root pressure is needed. ◻

Proof. The low matrix covariance is nonnegative because \(c_L\le(T-\varepsilon)/2\). At the high boundary, \(e(r_{\min})=d^2\) for large \(n\), since \(r_{\min}/d\to0\). The low field is at most \[c_Lr_0\le\frac14d^2r_{\min},\] whereas the high field is at least \(d^2r_{\min}\). The high matrix gap is at least \(\varepsilon-d^2>0\). Thus every changed parent trial is an admissible path. The low penalty, if included, is independent of the high quantile and cancels in Lemma 50.

If \(\varepsilon<T/4\), the low matrix covariance is greater than \(5T/8\); for large \(n\) the high covariance has a fixed positive lower bound as well. We therefore use all-mass descendants. If \(\varepsilon\ge T/4\), the high-to-low gap is at least \(T/4-d^2\), so we use high-only descendants with zero wall. Choose the recursion budget as a fixed small fraction of \(T\). These alternatives remain valid even when the low covariance tends to zero, since in that case descendants leave it unchanged. Above \(s\) the original bare coefficients are exactly \(T,0\); [eq:4] is therefore uniform there. Lemma 49 preserves it in every descendant, and \(s=o(a')\) persists.

Any failure of uniformity would yield a sequence of wall parameters and, after subselection, one of these two fixed-margin alternatives. Proposition 53 excludes that sequence. Only changed paths are used, even when the displayed bare field has a downward jump at \(s\). ◻

Proof. The segment \((b+ta,h-tc)\) consists of admissible paths. By [eq:1], its derivative is \[-\frac14\int aB\,du+\frac12\int cS\,du \le\frac14\int(-aS^2+2cS)\,du \le\frac14\int\frac{c^2}{a}\,du,\] where \(B\ge S^2\) and completion of the square give the two inequalities. Integration proves the result. The integrated variation identity [eq:1], established by step approximation, applies to the bounded segments used here. ◻

Proof. Direct substitution in [eq:5] gives \[a=n^{-1/6+10m},\quad u_*=L=n^{-1/6-25m},\quad M(s)\asymp n^{30m+\delta},\quad r_{\min}\asymp n^{-1/6-55m-\delta}.\] In particular, \(s=o(a)\), \(d^2=o(\varepsilon_0)\), and \(r_{\min}/d\to0\). Since \(f=1\) near zero, the initial high field is exactly \(d^2r_{\min}\). On the low group the field is at most \[c_Lr_0\le\frac\varepsilon2 \frac{d^2r_{\min}}{2\varepsilon} =\frac{d^2r_{\min}}4.\] There is therefore no downward field jump at \(s\). Both field paths are increasing on their own intervals. The low matrix coefficient is at least \((T-\varepsilon)/2\ge0\), and its gap to the high coefficient at the wall is \(\varepsilon+\theta c_L-d^2>0\). The high matrix path is increasing because \(e\) decreases.

These are the wall data of Corollary 54, with \(\varphi=\psi\). Total clocks are uniformly bounded, and \(d^2<T/2\) for large \(n\). The fixed covariance margin required there is also uniform: when \(\varepsilon<T/4\), the low covariance is at least \(5T/8\) and the high covariance is at least \(T-d^2\); when \(\varepsilon\ge T/4\), the high-to-low gap is at least \(T/4-d^2\). These are respectively the all-mass and high-only alternatives in the corollary. Above \(s\), the bare coefficients are exactly \(T,0\), so [eq:4] holds uniformly.

The low penalty in \(J_\theta\) is independent of the high quantile. It therefore leaves its minimizers unchanged and cancels in the movement inequality used by the corollary. Applying the corollary at this fixed positive strength gives the profile uniformly along all wall-parameter sequences, including \(\theta\to0\) and \(T-\varepsilon\to0\), without evaluating the bare low-field pressure or sweeping the root from zero strength. Here its final-strength cutoff is \[A_{d^2}=n^\delta a=n^{-1/6+10m+\delta}=o(d).\] Thus the resulting outer regularity applies at scale \(d\), as claimed. ◻

Proof. At scale \(x=d\), the final strength satisfies \(e(d)\asymp d^2\), so \(W\asymp d\). The cutoff just proved supplies outer regularity there, and the other direct-test inequalities follow from \[d/a=n^{5m},\qquad \frac{nE^2}{e(d)d^2}\asymp n^{-10m}.\] Also \(ne(d)d^4\asymp n^{90m}\to\infty\). Projection from a regular high band and [eq:12] give \(K_u=O(d)\) and \(D_u=O(d)\) on the omitted low group. The omitted-group big-O version of [eq:22] therefore gives \(S_u=\Gamma(K_u)+O(d^3)\). Its hypotheses require an admissible changed path and an omitted mass \(s=o(d)\); they do not require [eq:4] at the omitted splits.

Choose a fixed \(C\) containing all low-group clocks in \([0,Cd]\). Lemma 43, applied at \(x=d\) with omitted mass \(u_0=s=o(d)\), gives \[\sup_{0\le K\le Cd}|\Gamma(K)-K/T|=O(d^3)\] uniformly in the wall parameters. Its high-clock calibration covers this interval down to zero, including clock gaps, and requires no version of [eq:4] on the omitted group. Combining this bound with the preceding omitted-group cavity estimate proves the first assertion of [eq:31].

On the low group, \(b_u=T-\varepsilon-\theta c_L\) and \(0\le h_u\le d^2r_{\min}/4=O(d^3)\). Therefore \[(\varepsilon+\theta c_L)S_u=h_u+O(d^3)\le C d^3,\] which proves the second assertion. Uniformity follows either from the uniform constants just used or by applying the same argument to an arbitrary sequence of wall parameters and low splits. ◻

Proof. We first take \(\theta\ge\tau=n^{-10}\). For the dyadic values \(v=s/2,s/4,\ldots\) with \(v\ge4\tau\), put \[I_v^-=[s-v,s-v/2],\qquad I_v^+=[s-v/2,s-v/4].\] The intervals \(I_v^-\) cover \([s/2,s)\) except for a terminal strip of length at most \(4\tau\). Each \(I_v^+\) lies after \(I_v^-\), has length \(v/4\), and stays at distance at least \(\tau\) from \(s\). On either interval, the field derivative is at least \(c\theta c_Lr_0/(v\log n)\). Integrating the susceptibility inequality in [eq:1] and using [eq:31] gives \[\int_J\mathbb E\operatorname{tr}H_u^2\,du \le\frac{C a_Lv\log n}{n\theta c_Lr_0}.\] Here \(J\) is either \(I_v^-\) or \(I_v^+\). Since \(H_u\ge u C_u\) and \(u\ge s/2\) on both, it also gives \[\int_J\mathbb E\operatorname{tr}C_u^2\,du \le\frac{C a_Lv\log n}{n\theta c_Lr_0s^2}.\]

For \(u\in I_v^-\), average the single-path observable over \(I_v^+\): \[\overline L_v=\frac1{|I_v^+|}\int_{I_v^+}L_j\,dj,\qquad |X_u|^2\le2\mathbb E_u\overline L_v.\] Indeed \(2\mathbb E_u L_j=\mathbb E_u|X_j|^2\ge|X_u|^2\) for \(j\ge u\). Moreover \(\mathbb E\overline L_v\le a_L/2\). Apply the pinned moment inequality in [eq:1] with \(p=2\) and the deterministic weight \(k_j=|I_v^+|^{-1}\mathbf1_{I_v^+}(j)\). Since \[\mathbb E W_j=\mathbb E\operatorname{tr}(H_jC_j) \le\frac2s\mathbb E\operatorname{tr}H_j^2,\] the variance under the full planted law satisfies \[\operatorname{Var}(\overline L_v) \le\frac1{|I_v^+|^2}\int_{I_v^+}\mathbb E W_j\,dj \le\frac{C a_L\log n}{n\theta c_Lr_0sv}.\] Conditional Jensen is used only after this unconditional estimate: \[\mathbb E|X_u|^4 \le4\mathbb E\overline L_v^2 \le C a_L^2+ \frac{C a_L\log n}{n\theta c_Lr_0sv}.\] Also \[B_u=\mathbb E\|X_uX_u^{\mathsf T}+C_u\|_{\mathrm{HS}}^2 \le2\mathbb E|X_u|^4+2\mathbb E\operatorname{tr}C_u^2.\] On integrating over \(I_v^-\), the fourth-moment error loses its factor \(v^{-1}\). The covariance term is no larger, because \(v\le s\): \[\int_{I_v^-}B_u\,du \le Cv a_L^2+ \frac{C a_L\log n}{n\theta c_Lr_0s}.\] Summing the \(O(\log n)\) dyads proves \[\int_{s/2}^{s-C\tau}B_u\,du \le Csa_L^2+ \frac{C a_L(\log n)^2}{n\theta c_Lr_0s}.\] The omitted terminal strip has cost at most \(C\tau\), since \(B_u\le1\). For \(u<s/2\), monotonicity gives \(\int_0^{s/2}B_u\,du\le2\int_{s/2}^{3s/4}B_u\,du\). Hence the same bound, with a \(C\tau\) remainder, holds over the whole low group.

The feasible high-quantile class is fixed, so the pressure Lipschitz bound makes \(J_\theta^*\) Lipschitz in \(\theta\). At an optimizing high quantile, the upper envelope derivative per unit \(\log\theta\) is at most \[\frac{\theta c_L}{4}\int_0^s [D_u^2+(S_u-r_0\psi(u))^2]\,du.\] This is [eq:7] for the low variation, with the high optimizer held fixed when taking an upper difference quotient. The estimate is uniform over all minimizers, so it also bounds the derivative of the minimum at almost every parameter. Dropping the nonpositive cross term and using \(D_u^2+S_u^2=B_u\) bounds it by \[C\theta c_Ls(a_L^2+r_0^2) +\frac{C a_L(\log n)^2}{nr_0s}+C\theta c_L\tau.\] Integration over \(\tau\le\theta\le1\) gives the claimed three-logarithm term. On \(0\le\theta\le\tau\) the trivial bounds on the overlap and the fixed low coefficient give total cost \(C\tau c_Ls\le C\tau\). This proves the first assertion.

At \(\theta=0\), the high comparison begins from the admissible endpoint defining \(p(\varepsilon)\) and can be swept from strength zero. Hence \(J_0^*\le p(\varepsilon)+n^{20\delta}E\). At \(\theta=1\), the chord to \(p(\varepsilon+q)\) has residual matrix covariance \(e\) on the high group and \(c_L-q=c_L/2\) on the low group. Its residual field is \(er\) high and \(c_Lr_0\psi\) low. Relative to the penalties included in \(J_1^*\), the excess chord cost is \[\frac14\int_0^s \left(\frac{c_L^2}{c_L-q}-c_L\right)r_0^2\psi^2\,du =\frac{c_L}{4}\int_0^s r_0^2\psi^2\,du \le Csc_Lr_0^2.\] Both endpoints and their connecting segment are admissible. Finally, \(a_L=Cd^3/\varepsilon\), \(c_L\le\varepsilon/2\), and \(r_0/a_L\le C r_{\min}/d\to0\). These imply \[c_Ls(a_L^2+r_0^2)\le Csd^6/\varepsilon_0, \qquad \frac{a_L}{r_0}\le C d/r_{\min},\] and prove the second assertion. ◻

Proof. Represent all disorder, including the common Gaussian, by a standard Gaussian vector \(g\in\mathbb R^N\). There are linear forms \(a_\sigma\cdot g\) such that \[F_n^c(g)=\log\left(2^{-n}\sum_\sigma e^{a_\sigma\cdot g}\right).\] For each \(\tau\in\{-1,1\}^n\), the orthogonal gauge transformation \(g_{ij}\mapsto\tau_i\tau_jg_{ij}\), leaving the common coordinate fixed, preserves both the Gaussian law and \(F_n^c\). These transformations act transitively on the spin labels. The density of \(G_n\) is \[q(z)=e^{-z}\mathbb E\left[e^{F_n^c(g)} \mathbf1_{\{F_n^c(g)\le z\}}\right].\] Upon expanding \(e^{F_n^c}\), gauge invariance shows that all terms have the same integral. Thus, for one fixed spin label \(\sigma_*\), \(q\) is the marginal, up to a constant, of \[\exp\{-|g|^2/2+a_{\sigma_*}\cdot g-z\} \mathbf1_{\{F_n^c(g)\le z\}}.\] The function \(F_n^c\) is convex, being a logarithm of a sum of exponentials of linear functions. Its epigraph is convex, and the displayed integrand is therefore log-concave jointly in \((g,z)\). It is integrable: integrating first in \(z\) leaves a Gaussian density times \(e^{a_{\sigma_*}\cdot g-F_n^c(g)}\le2^n\). Prékopa’s marginal theorem (Prékopa 1973, Theorems 6 and 8) implies that \(q\) is log-concave. Independence and \(\mathbb E e^{t(\mathcal E-1)}=e^{-t}/(1-t)\) give the two identities. ◻

Proof. The survival functions of \(X\) and \(-X\) are log-concave by Prékopa’s theorem. Chebyshev’s inequality bounds the survival probability at \(-2\sigma\) below by \(3/4\), and that at \(2\sigma\) above by \(1/4\). Concavity of its logarithm and the secant slope between these two points give, for \(t\ge2\), \[\mathbb P(X\ge t\sigma)\le\frac{\sqrt3}{4} \exp\left(-\frac{\log3}{4}t\right).\] Apply the same argument to \(-X\); the trivial bound for \(t\le2\) then gives the first assertion with \(h=(\log3)/4\) in place of \(c\). Integration of the tails yields \[\mathbb E|X|^k\le2k!(\sigma/h)^k,\qquad k\ge1.\] For \(r=t\sigma/h\le1/2\), the exponential series converges absolutely and, since \(\mathbb EX=0\), \[\log\mathbb E e^{tX} \le\log\left(1+2\sum_{k\ge2}r^k\right) \le\frac{2r^2}{1-r}\le4r^2.\]

The fourth-moment bound and Hölder’s inequality give \[\mathbb E|X|\ge \frac{(\mathbb EX^2)^{3/2}}{(\mathbb EX^4)^{1/2}} \ge c_1\sigma.\] Centering implies \(\mathbb EX_+=\mathbb E|X|/2\), whereas \(\mathbb EX_+^2\le\sigma^2\). The Paley–Zygmund inequality, or its elementary proof by Cauchy–Schwarz on the event \(\{X_+\ge\mathbb EX_+/2\}\), now gives \(\mathbb P(X\ge c_2\sigma)\ge c_2\) for a universal \(c_2>0\). Restricting the exponential moment to that event proves its lower bound.

For the density estimate, write \(M=\sup f_X\) and choose a mode \(x_0\). An integrable one-dimensional log-concave density has a finite positive maximum, with one-sided limits used at a support endpoint. On either side of the mode the density must fall to \(M/e\) within distance at most \(e/M\), or reach the support endpoint: otherwise its integral on that side would exceed one. Concavity of the log density then bounds the tail beyond that point by its exponentially decreasing secant. Together with \(f_X\le M\) before that point, this gives \[f_X(x_0+y)\le eM\exp(-M|y|/e),\] with zero outside the support. Consequently \[\sigma^2\le\mathbb E(X-x_0)^2 \le2eM\int_0^\infty y^2e^{-My/e}\,dy =\frac{4e^4}{M^2}.\] Rearranging proves \(M\le2e^2/\sigma\). ◻

Proof of Theorem 1. Let \(\sigma_n^2=\operatorname{Var}G_n\). For the fixed \(m\) used in Proposition 59, \(s\to0\). The exponential correction in Lemma 60 is \(O(s^2)\), and hence \[c_{T,m}n^{20m}\le \log\mathbb E e^{s(G_n-\mathbb EG_n)} \le C_{T,m}n^{100m}\] for sufficiently large \(n\), after changing constants. If \(s\sigma_n\) were smaller than the universal small-tilt constant in Lemma 61, its upper quadratic bound would make this log moment uniformly bounded. The divergent lower bound therefore gives \(s\sigma_n\ge c\) eventually. Its positive-side mass bound, on the other hand, gives \[cs\sigma_n+\log c \le\log\mathbb E e^{s(G_n-\mathbb EG_n)} \le C_{T,m}n^{100m}.\] It follows that \[c_{T,m}n^{1/6-5m}\le\sigma_n \le C_{T,m}n^{1/6+95m}.\] Given any desired power slack \(\eta>0\), first choose a fixed \(m<\min(1/300,\eta/200)\). The constants are absorbed by the remaining strict power slack as \(n\to\infty\). Thus \(\sigma_n=n^{1/6+o(1)}\). Since \[\operatorname{Var}G_n=\operatorname{Var}F_n+T/2+1,\] the same standard-deviation exponent holds for \(F_n\), and its variance is \(n^{1/3+o(1)}\).

We also prove the asserted typical magnitude. Let \(X_n=G_n-\mathbb EG_n\). The density estimate in Lemma 61 implies, for every \(a>0\), \[\mathbb P(|X_n|\le a)\le 2Ca/\sigma_n.\] Using the standard-deviation bounds with any fixed slack smaller than \(\eta\), this tends to zero at \(a=n^{1/6-\eta}\). Chebyshev’s inequality, or the exponential-tail bound of that lemma, shows that \(\mathbb P(|X_n|\ge n^{1/6+\eta})\to0\). Hence the smoothed centered magnitude lies between these two powers with probability tending to one.

To transfer this conclusion, write \[X_n=(F_n-\mathbb EF_n)+N,\qquad N=D+(\mathcal E-1),\qquad \mathbb EN^2=T/2+1.\] First suppose \(0<\eta<1/6\). At the lower threshold \(a_n=n^{1/6-\eta}\), the event \(|F_n-\mathbb EF_n|\le a_n\) is contained in \(\{|X_n|\le2a_n\}\cup\{|N|>a_n\}\); both probabilities tend to zero. At the upper threshold \(b_n=n^{1/6+\eta}\), the event \(|F_n-\mathbb EF_n|>b_n\) is contained in \(\{|X_n|>b_n/2\}\cup\{|N|>b_n/2\}\), again with vanishing probability. Fixed numerical factors are harmless because the standard-deviation estimates are available with arbitrarily smaller fixed slack. For larger \(\eta\), use inclusion from any smaller positive slack less than \(1/6\). This proves the typical-magnitude assertion for every fixed \(\eta>0\).

Finally, division by \(n\) changes the standard-deviation and typical magnitude exponents to \(-5/6\). The physical free-energy convention introduces the fixed nonzero factor \(-1/\beta\), and the counting spin prior introduces only a deterministic constant. These changes do not alter either conclusion. All constants and thresholds may depend on the fixed \(T>1\); no assertion uniform at either temperature endpoint is used or obtained. ◻