Proof. For \(0<m<1\), a Poisson sum \(S=\sum_i s_iY_i\), with intensity \(m s^{-1-m}\,\mathrm ds\) and iid nonnegative marks satisfying \(0<\mathbb EY^m<\infty\), has Laplace transform \[\mathbb Ee^{-tS} =\exp\{-\Gamma(1-m)\mathbb E(Y^m)t^m\}.\] Indeed, the Poisson Laplace functional reduces the exponent to \(m\mathbb E\int_0^\infty(1-e^{-tsY})s^{-1-m}\,\mathrm ds\); integration by parts evaluates this integral. The resulting positive stable random variable has finite positive moments of every order less than \(m\) and finite negative moments of every order. The latter follows directly from \(x^{-a}=\Gamma(a)^{-1}\int_0^\infty t^{a-1}e^{-tx}\,\mathrm dt\); the former follows by applying the corresponding integral formula to \(x^a\), \(0<a<m\). Consequently \(\mathbb E|\log S|^2<\infty\). Starting at the last generation and using \(m_0<\cdots<m_{k-1}\) proves these assertions for every subtree total, and hence for \(T\).

Here is the change of measure at one vertex. Write \(\mu(\mathrm d\omega)\) for the conditional law of a child mark and put \(c(\omega)=e^{X_{vi}-X_v}\). Under the map \((s,\omega)\mapsto(sc(\omega),\omega)\), the marked intensity becomes \[m_{j-1}s^{-1-m_{j-1}}\,\mathrm ds\, c(\omega)^{m_{j-1}}\mu(\mathrm d\omega).\] The second factor is a probability measure by (12). Thus the transformed weights have their original Poisson law and are independent of marks with transition (13).

We spell out why this yields independence of the entire transformed weight array from the marked tree. At depth two, write the root mark as \(B\), a child mark as \(A_i\), and its descendant marks as \(C_{ij}\). Let \(s_i\) and \(t_{ij}\) be the edge processes of exponents \(m_0\) and \(m_1\). Set \[X_1(B,A)=m_1^{-1}\log\mathbb E_C e^{m_1X_2(B,A,C)},\qquad X_0(B)=m_0^{-1}\log\mathbb E_A e^{m_0X_1(B,A)}.\] The lower transformation \(t'_{ij}=t_{ij}e^{X_2(B,A_i,C_{ij})-X_1(B,A_i)}\) has, conditional on \((B,A_i)\), the original unmarked Poisson law, denoted \(\mathcal P_{m_1}\), independently of child marks with tilted transition \(K_1(\mathrm dC\mid B,A_i)\). In particular this law of the complete unmarked lower process is independent of \(A_i\) and \(B\). Before the upper transformation a decorated child therefore has law \[\mu_0(\mathrm dA\mid B)\,\mathcal P_{m_1}(\mathrm dt') \prod_jK_1(\mathrm dC_j\mid B,A).\] The upper map \(s'_i=s_i e^{X_1(B,A_i)-X_0(B)}\) replaces only \(\mu_0\) by its tilted transition \(K_0\); it leaves \(\mathcal P_{m_1}\) unchanged. The marked-Poisson theorem makes the ordered \(s'_i\) independent of these iid decorated children. Thus, after both reorderings, all the transformed weights are independent of the hierarchical marked tree, conditional on \(B\). The same argument inducts upward for any depth: replace \(\mathcal P_{m_1}\) by the already transformed descendant weight-array law, which is independent of its ancestor mark. A sampled genealogy depends only on the complete weight array and independent sampling uniforms. This includes the descendant total masses used in selecting a child. Conditioning on that genealogy therefore does not tilt its marks again. Replicas sharing a vertex reuse the one mark already drawn there; no additional power depending on the number of replicas is introduced.

Reordering preserves a bijection of leaves. Under that correspondence, products of the edge multipliers telescope: \[W'_\tau=W_\tau e^{X_\tau-X_\varnothing},\qquad \log\sum_\tau v_\tau e^{X_\tau} =X_\varnothing+\log T'-\log T.\] Conditionally on root marks, \(T'\) and \(T\) have the same marginal law; they need not be independent. Their integrable logarithms therefore cancel in expectation. The root factor cancels in normalized leaf probabilities, so the root marks are not tilted. Finally, exponential moments of a sum of independent families factor at each recursion step, and so do the Radon–Nikodym factors in (13). This proves the last assertion. ◻

Proof. We give the sampling computation, including the change caused by an ancestor. Let a Poisson process have intensity \(c\beta s^{-1-\beta}\,\mathrm ds\), \(0<\beta<1\), and total \(S\). Change its law by \(S^\eta/\mathbb ES^\eta\), where \(0\le\eta<\beta\). For a specified partition of \(n\) sampled indices into \(b\) blocks of sizes \(n_1,\ldots,n_b\), normalized Poisson weights give probability \[ \frac{\prod_{a=1}^{b-1}(a\beta-\eta)} {\prod_{a=1}^{n-1}(a-\eta)} \prod_{i=1}^b\prod_{a=1}^{n_i-1}(a-\beta). \tag{15}\] Empty products are one. To verify this, insert \[S^{\eta-n}=\frac1{\Gamma(n-\eta)} \int_0^\infty t^{n-\eta-1}e^{-tS}\,\mathrm dt\] into the expectation of the sum over distinct Poisson points. The Poisson factorial-moment formula gives one factor \(c\beta\Gamma(n_i-\beta)t^{\beta-n_i}\) for each selected point and the Laplace factor \(\exp\{-c\Gamma(1-\beta)t^\beta\}\). The remaining integral, after \(v=c\Gamma(1-\beta)t^\beta\), is a gamma integral. Dividing by \[\mathbb ES^\eta= [c\Gamma(1-\beta)]^{\eta/\beta} \frac{\Gamma(1-\eta/\beta)}{\Gamma(1-\eta)}\] (with value one at \(\eta=0\)) gives (15). Its ratios show that an existing block of size \(l\) receives the next sample with probability \((l-\beta)/(n-\eta)\); a new block has probability \((b\beta-\eta)/(n-\eta)\).

At a cascade vertex, multiply each outgoing edge by the total mass of its child subtree. The Poisson mapping calculation in the preceding proof makes the resulting child masses a stable Poisson process and tilts the child subtree law by its total mass raised to the parent parameter. More explicitly, if the outgoing parameter is \(\beta\), the masses \(r_i=s_iT_i\) are independent of child-subtree marks whose law is multiplied by \(T_i^\beta/\mathbb ET_i^\beta\). An incoming bias \((\sum_i r_i)^\eta\) changes only the mass process, so those child marks remain independent, also after conditioning on the choices of children. Thus (15) applies at the first level with \((\eta,\beta)=(0,m_0)\) and, inside a selected parent at later levels, with \((\eta,\beta)=(m_{j-2},m_{j-1})\). For two samples the conditional joining probabilities are \[1-m_0,\quad \frac{1-m_1}{1-m_0},\quad\ldots, \quad\frac{1-m_{k-1}}{1-m_{k-2}}.\] Their products give probability \(1-m_{j-1}\) of sharing the first \(j\) edges. Taking differences proves the first assertion.

More generally, multiply the conditional joining probabilities along the ancestral path to a cluster containing \(l\) of the first \(n\) samples at a level of parameter \(m\). The denominators telescope, giving \((l-m)/n\). These probabilities, and differences between probabilities of joining a parent and joining its existing children, determine all ways the next sample can enter a finite nested partition. They also show that deleting intermediate levels leaves the same rule at the retained levels. Construct nested partitions at all dyadic parameters by these consistent finite-dimensional laws, taking the universal partition at parameter zero and the singleton partition at parameter one, and put \(U_{ab}=\sup\{m:a,b\text{ belong to the same cluster at level }m\}\). The rule gives \(\mathbb P(U_{12}\ge m)=1-m\) at dyadic \(m\), hence uniformly on \([0,1]\). Nesting gives ultrametricity. The predictive probabilities for a finite nested partition are differences of \((l-m)/n\), and hence its probability is continuous in the finitely many level parameters. Approximating arbitrary real levels by dyadic levels, and using that no \(U_{ab}\) has an atom at a prescribed level, therefore recovers the same sampling law at every real level. Applying a step function \(q\) now recovers the finite cascade covariance law. Set \(q(1)=q_*\); nonnegative nondecreasing transforms of this nested array are positive semidefinite: for a step transform they are sums, with nonnegative coefficients, of block-indicator matrices, and the general case follows by limits. An additional nonnegative diagonal correction supplies any \(Q\ge q_*\). ◻

Proof. For a fixed number of replicas, the conditional characteristic function of the adjoined fields is \(\exp(-t^{\mathsf T}Ct/2)\). It is a bounded continuous function of the covariance matrix, including at singular matrices. Consequently the fields converge jointly with the arrays. The same argument with a product law adjoins the independent auxiliary variables.

Put \(W=e^X\), \(W_L=(W\vee L^{-1})\wedge L\), \(Z=G(W)\), and \(Z_L=G(W_L)\). For every fixed \(r\ge1\), Jensen’s inequality gives \[Z^{-r}\le G(e^{-rX}),\qquad Z_L^{-r}\le G(W_L^{-r})\le G(1+e^{-rX}).\] All these inverse moments are uniformly bounded in expectation. For arbitrary fixed \(r\ge1\) and \(A>0\), (16) and another application of Jensen give \[\sup_n\mathbb E|Z-Z_L|^r \le\sup_n\mathbb EG(|W-W_L|^r)=O(L^{-A}),\] after using a sufficiently large moment in (16). The elementary inequality \(|\log a-\log b|\le |a-b|(a^{-1}+b^{-1})\) and Hölder’s inequality show that replacing \(Z\) by \(Z_L\) changes its expected logarithm by a quantity tending to zero uniformly in \(n\).

At fixed \(L\), uniformly approximate \(\log z\) on \([L^{-1},L]\) by polynomials. Every expected polynomial of \(G(W_L)\) is a finite linear combination of replica expectations, which converge by the first paragraph. This proves the assertion for logarithms. For an \(r\)-replica observable \(B\), its tilted expectation is \[\mathbb E\frac{G_n^{\otimes r}(B\prod_{i=1}^rW_i)}{Z^r}.\] Replace the \(W_i\) and \(Z\) by their clipped versions. Hölder’s inequality, the inverse-moment bounds, and the displayed tail estimate make this error uniformly small for bounded \(B\). At fixed \(L\), approximate \(z^{-r}\) on \([L^{-1},L]\) by polynomials and again use finitely many replicas. A further truncation proves the unbounded version whenever the stated uniform integrability holds. Polynomial Gaussian observables have moments of all orders uniformly, because the covariance kernels are bounded; Hölder’s inequality with (16) and the inverse-moment bound then supplies this uniform integrability. ◻

Proof. We first prove the finite-profile identity, then use replica-array continuity to construct its extension and pass the derivative formula to the limit. The square of a bounded martingale is a submartingale. A zero-variance increment has identical incoming and outgoing derivative, proving all the monotonicity and plateau assertions. Given that two replicas share exactly \(j\) levels, their later marks are independent under the tilted transitions. Their terminal derivatives therefore have product expectation \(\mathbb EA_j^2\). The shared path itself has the single-path tilted law, because transformed weights and marks are independent, also conditional on the genealogy.

Put \(q\) and \(r\) on a common finite partition, first with no residual convolution. Gaussian covariance differentiation gives \[\frac{\mathrm d}{\mathrm dt}\Phi_u(q_t;Q) =\frac12\mathbb E\Big\langle [u''(z_1)+u'(z_1)^2]\dot C_{11} -u'(z_1)u'(z_2)\dot C_{12}\Big\rangle_t.\] The diagonal is fixed, so \(\dot C_{11}=0\); off the diagonal \(\dot C_{12}=(r-q)(U_{12})\). Lemmas 2 and 3, followed by the conditional product calculation, give (17). Bounded derivatives justify Gaussian integration by parts; one may truncate the number of leaves and then pass by dominated convergence and the exponential moments of the Gaussian marks. Integrating the derivative proves the Lipschitz bound for finite profiles.

Use the common rank array of Lemma 3. If bounded profiles \(q_n\to q\) in \(L^1\), then \(q_n(U_{ab})\to q(U_{ab})\) in probability for every \(a\ne b\); a union bound handles any finite subarray. Keep diagonal \(Q\) fixed. Since \(u\) has at most linear growth, (16) holds uniformly for \(X=u(z)\), as the base single-replica field has law \(\mathcal N(0,Q)\). Lemma 4 therefore constructs the limiting functional and tilted replica expectations independently of the step approximation. If an approximating profile is raised to \(Q\) on a last interval of length \(\varepsilon\), its final recursion is \[\frac1{1-\varepsilon}\log \mathbb Ee^{(1-\varepsilon)u(z+\sqrt{Q-q_k}\,G)}.\] As \(\varepsilon\downarrow0\) this is exactly the terminal convolution. The limiting covariance array has an independent residual Gaussian for each replica. This verifies both descriptions of the limit.

The \(D_{q_n}\) are bounded monotone functions and hence precompact in \(L^1\). The two-replica identity can be tested against the full rank \(U_{12}\): refine a step profile at the endpoints of any step test function, using zero Gaussian increments on the added levels inside each plateau. Lemma 2 preserves the entire refined genealogy, while the derivative martingale is unchanged across those zero increments. The finite shared-level calculation thus proves the identity for step test functions, and uniform approximation proves it for continuous ones. For each continuous \(\varphi:[0,1]\to\mathbb R\), this identity and Lemma 4 give convergence of \[\int_0^1\varphi(s)D_{q_n}(s)\,\mathrm ds =\mathbb E\langle\varphi(U_{12})u'(z_1)u'(z_2)\rangle_{q_n}.\] These limits identify every \(L^1\) subsequential limit, proving convergence of \(D_{q_n}\) and defining \(D_q\). The same argument proves continuity for any convergent sequence of general profiles. On an interval where \(q\) is constant, choose step approximations that have that value on every compact subinterval. Their derivative profiles are constant there, so the limit is constant almost everywhere there.

Pass to the limit in the integrated finite-profile version of (17), using boundedness and \(L^1\) continuity of \(D\). For fixed \(q,r\), the resulting integrand is continuous in \(t\). The fundamental theorem of calculus gives the claimed derivative, including one-sided endpoint derivatives, and completes the proof. ◻

Proof. The calculation has four steps. We evaluate a Gaussian spin integral, choose its quadratic multiplier so that one disorder-averaged spin has variance one, and compare the resulting typical shell with the sphere. The common Lipschitz bound then makes the limit uniform over bounded profiles.

First replace the spherical integral by \[Z_N^{\rm G}(b)=\sum_\tau v_\tau \int_{\mathbb R^N}\exp\{x\cdot z(\tau)-b\|x\|^2/2\} \frac{\mathrm dx}{(2\pi)^{N/2}}.\] The terminal logarithm factors into a sum of \(-\frac12\log b+z_i^2/(2b)\). The elementary Gaussian integral \[\frac1m\log\mathbb E\exp\!\left\{ m\left(c+\frac{(z+\sqrt d\,G)^2}{2\lambda}\right)\right\} =c+\frac1{2m}\log\frac\lambda{\lambda-md} +\frac{z^2}{2(\lambda-md)}\] is valid when \(\lambda>md\). Iterating it and averaging the root Gaussian gives (20), with \(\mathbb E\log Z_N^{\rm G}(b)=NX(b)\). The condition \(b>I\) ensures every required Gaussian exponential moment is finite.

Differentiate this explicit expression to get (21). Its left-hand side is strictly decreasing and continuous on \((I,\infty)\) and tends to zero at infinity. At \(b\downarrow I\) it diverges: if \(p_0>0\), use \(p_0/\lambda_0^2\); otherwise use the summand corresponding to the first positive increment; if every \(p_j=0\), use \(1/b\). Hence the solution is unique. The equation also gives \(b\ge1\). If \(p_k\le B\), then \(I\le B\); at \(b=2(B+1)\) its left-hand side is at most \[\frac1{2(B+1)}+\frac{B}{(B+2)^2}\le\frac58<1.\] Thus the solution is below \(2(B+1)\).

We now justify replacing the Gaussian spin integral by the sphere. For one spin vector sampled from \(Z_N^{\rm G}(b)\) and then averaged over the disorder, the coordinates are iid. Indeed the terminal function is a sum of independent coordinate functions; Lemma 2 makes every path transition factor by coordinate, and the independent root Gaussian marks are not tilted. Conditional on the terminal fields, the spins are independent Gaussians of mean \(z_i/b\) and variance \(1/b\). Integrating the factorized tilted path law therefore preserves independence and identical distribution. Their common law can be computed directly. Write \(f_j(z)=c_j+z^2/(2\lambda_j)\) for the quadratic recursion. Completing the square in its tilted transition gives \[ Z_j\mid Z_{j-1}=z\ \sim \mathcal N\!\left(\frac{\lambda_j}{\lambda_{j-1}}z, \frac{\Delta p_j\lambda_j}{\lambda_{j-1}}\right). \tag{23}\] Consequently \(Y_j=Z_j/\lambda_j\) is a centered Gaussian martingale with independent increments of variances \(\Delta p_j/(\lambda_j\lambda_{j-1})\), starting with variance \(p_0/\lambda_0^2\). The terminal spin is \(Y_k\) plus an independent \(\mathcal N(0,1/b)\) variable, so its variance is \[\frac{p_0}{\lambda_0^2} +\sum_{j=1}^k\frac{\Delta p_j}{\lambda_j\lambda_{j-1}}+\frac1b=1\] by (21). Thus the averaged single-vector law is exactly \(\mathcal N(0,I_N)\). This assertion concerns a single disorder-averaged sample; it does not assert independence at fixed disorder or between replicas.

Let \(G_N^{\rm G}\) denote the \(x\)-marginal of the normalized Gibbs measure associated with the partition function \(Z_N^{\rm G}(b)\). Fix \(0<\delta<1\) and write \(A_{N,\delta}=\{|\|x\|^2/N-1|\le\delta\}\). The law of large numbers gives \(\mathbb EG_N^{\rm G}(A_{N,\delta}^c)\to0\), and Markov’s inequality gives \(G_N^{\rm G}(A_{N,\delta})\to1\) in disorder probability. To pass to logarithms, let \(Z_{N,\delta}^{\rm G}\) be the integral restricted to this shell. Lemma 2 expresses \(\log Z_N^{\rm G}\) as an iid coordinate sum at the root plus a difference of two log total cascade masses. More explicitly the root shift is \(\sum_{i=1}^N[c_0+Z_{0i}^2/(2\lambda_0)]\), where \[c_0=-\frac12\log b+ \sum_{j=1}^k\frac1{2m_{j-1}}\log\frac{\lambda_j}{\lambda_{j-1}}.\] At each vertex the edge multiplier has \(m\)th moment exactly one by (12), also when the marks comprise \(N\) coordinates. Hence the transformed weight processes have exactly their original intensities. Conditionally on root marks, \(T'\) has the same \(N\)-independent marginal law as \(T\); all scale changes have been included in the root shift. Its quadratic Gaussian summands have fixed second moments, and Lemma 2 bounds the second moments of the two log masses. Therefore \(\mathbb E|\log Z_N^{\rm G}|^2=O(N^2)\), with a constant allowed to depend on the fixed step profile. The restricted logarithm has a deterministic lower bound \(-C_{b,\delta}N\) for large \(N\): at any radius the sphere average of \(e^{x\cdot z}\) is at least one by Jensen’s inequality, and the Gaussian volume of the shell is at least \(e^{-C_{b,\delta}N}\). Consequently \[0\le\frac1N\log\frac{Z_N^{\rm G}}{Z_{N,\delta}^{\rm G}} \le\frac1N\log Z_N^{\rm G}+C_{b,\delta}\] is uniformly integrable. The left side tends to zero in probability, so \[ \mathbb E\log Z_{N,\delta}^{\rm G}=NX(b)+o(N). \tag{24}\]

For completeness the radial comparison includes the normalization of surface measure. With \(x=r\widehat x\), \(\widehat x\in\mathbb S_N\), polar coordinates give \[\frac{\mathrm dx}{(2\pi)^{N/2}} =c_N r^{N-1}\,\mathrm dr\,\mathrm d\sigma_N(\widehat x), \qquad c_N=\frac{2(N/2)^{N/2}}{\Gamma(N/2)}, \qquad \frac1N\log c_N\longrightarrow\frac12.\] For every fixed field the sphere integral of \(e^{r\widehat x\cdot z}\) is nondecreasing in \(r\ge0\), by symmetry and monotonicity of the hyperbolic cosine. On \(\sqrt{1-\delta}\le r\le\sqrt{1+\delta}\), the logarithm of the remaining radial factor, divided by \(N\), is \(1/2-b/2+O_b(\delta)+o(1)\). Its integral over this interval has the same estimate. Sandwich the spin integral between the values at the two shell endpoints. Scaling a field by \(r\) scales its covariance by \(r^2\), and (19), including the diagonal correction, gives \[|S_N(r^2p;r^2p_k)-S_N(p;p_k)|\le B|r^2-1|.\] Together with (24), the radial sandwich shows \(S_N(p;p_k)=X(b)+b/2-1/2+O_{b,B}(\delta)+o(1)\). First let \(N\to\infty\), then \(\delta\downarrow0\), to obtain (22).

Finally, nonnegative nondecreasing profiles bounded by \(B\) form a compact set in \(L^1\), and finite-step profiles are dense in it. The common Lipschitz estimate (19) transfers convergence from a finite net of step profiles to uniform convergence on this compact set. This proves the proposition for general bounded profiles. ◻

Proof. All the derivatives can first be taken with \(p_0>0\) and strictly positive increments, and then extended to boundary points by continuity. Differentiate (20) and (22). The coefficient of \(\mathrm db\) is \(X'(b)+1/2=0\) by (21). Substituting \(\mathrm d\lambda_j=\mathrm db-\sum_{i>j}m_{i-1}(\mathrm dp_i-\mathrm dp_{i-1})\) and collecting coefficients of \(\mathrm dp_j\) gives (26). The last equality there is precisely (21).

To check the Hessian sign without a sign convention ambiguity, invert this gradient map. Write \(\chi_j=1/\lambda_j\). The differences of (26) and the recursion for \(\lambda\) give \[r_j-r_{j-1}=\Delta p_j\chi_j\chi_{j-1},\qquad \chi_{j-1}-\chi_j=m_{j-1}(r_j-r_{j-1}),\qquad \chi_k=1-r_k.\] Telescoping yields the inverse formulas \[ \begin{split} \chi_j&=1-m_jr_j-\sum_{i>j}w_ir_i,\\ p_0&=r_0/\chi_0^2,\qquad p_j-p_{j-1} =\frac{\chi_j^{-1}-\chi_{j-1}^{-1}}{m_{j-1}} =\frac{r_j-r_{j-1}}{\chi_j\chi_{j-1}}. \end{split} \tag{28}\] Conversely every \(0\le r_0\le\cdots\le r_k<1\) gives positive \(\chi_j\) and a nonnegative nondecreasing \(p\) by these formulas; putting \(b=1/\chi_k\) recovers (21). Thus they are genuine inverse relations.

Here is the full quadratic-form calculation. Put \[T_j=\sum_{i\ge j}w_i\,\mathrm dr_i,\quad T_{k+1}=0, \qquad L_j=m_j\,\mathrm dr_j+T_{j+1}=-\mathrm d\chi_j; \qquad L_0=T_0.\] Differentiating (28) gives \[\mathrm dp_0=\frac{\mathrm dr_0}{\chi_0^2} +\frac{2r_0T_0}{\chi_0^3},\qquad \mathrm d(\Delta p_j)=\frac1{m_{j-1}} \left(\frac{L_j}{\chi_j^2}-\frac{L_{j-1}}{\chi_{j-1}^2}\right).\] Insert these into \(\sum_jw_j\mathrm dr_j\mathrm dp_j=T_0\mathrm dp_0+\sum_{j\ge1}T_j\mathrm d(\Delta p_j)\). Summation by parts gives exactly \[ \sum_{j=0}^k w_j\mathrm dr_j\mathrm dp_j =\frac{m_0(\mathrm dr_0)^2-T_1^2/m_0}{\chi_0^2} +\frac{2r_0T_0^2}{\chi_0^3} +\sum_{j=1}^k\frac{w_jL_j^2}{m_{j-1}m_j\chi_j^2}. \tag{29}\] Since \(\chi_j\le\chi_0\), replace \(\chi_j^{-2}\) in the last, nonnegative sum by \(\chi_0^{-2}\). The identity \[\frac{w_jL_j^2}{m_{j-1}m_j} =w_j(\mathrm dr_j)^2+\frac{T_j^2}{m_{j-1}}-\frac{T_{j+1}^2}{m_j}\] then telescopes and proves \[ \sum_jw_j\mathrm dr_j\mathrm dp_j \ge\frac1{\chi_0^2}\sum_jw_j(\mathrm dr_j)^2\ge0. \tag{30}\] For \(k=0\) the same calculation has \(m_0=1\) and empty sums. Along a line in \(p\), (26) and (30) give \(\mathrm d^2\mathcal S\le0\). Continuity proves concavity on the boundary and, by step approximation, on the full domain.

For general \(p\le B\), choose step profiles \(p_n\le B\) converging to \(p\). Their gradients have nonnegative nondecreasing representatives \(r_n\) and, by Proposition 6, \[(r_n)_*=1-1/b_n\le1-[2(B+1)]^{-1}.\] Pass to an \(L^1\) convergent subsequence. The finite-step tangent inequality holds against every step competitor after refining to a common partition. Letting \(n\to\infty\), and then approximating an arbitrary bounded competitor, gives (27). The displayed uniform bound passes to the limit.

This same construction gives the asserted inverse relation. For a step profile \(v\), \(\chi_v\) is constant below its lowest value and affine of slope \(-m_{j-1}\) between successive values. Integrating \(\chi_v^{-2}\) therefore gives \(v_0/\chi_0^2\) and increments \((v_j-v_{j-1})/(\chi_j\chi_{j-1})\). Thus the finite inverse formulas are exactly \(p_n=\mathcal I(r_n)\). To pass to the limit, Fubini gives \[ \chi_v(t)=1-t-\int_0^1(v(s)-t)_+\,\mathrm ds, \qquad \sup_t|\chi_v(t)-\chi_{\widetilde v}(t)| \le\|v-\widetilde v\|_1. \tag{31}\] For \(v,\widetilde v\le Q<1\), their integrated distributions are at least \(1-Q\) on \([0,Q]\). Comparing the integrands and the upper integration limits in (25) gives \[\|\mathcal I(v)-\mathcal I(\widetilde v)\|_1 \le C_Q\|v-\widetilde v\|_1.\] Consequently \(p=\mathcal I(r)\). This identifies the supergradient obtained from the step approximations; no assertion about uniqueness of all supergradients on the monotone cone is needed. ◻

Proof. For a step profile \(q\), set \(r=q\) in (28) and obtain \(p\). The tangent inequality in Proposition 7 gives, for every competitor \(\widetilde p\), \[\mathcal S(\widetilde p)+\tfrac12\int q\widetilde p \le\mathcal S(p)+\tfrac12\int qp;\] thus \(p\) attains the supremum. In these inverse formulas \(\chi_j=\chi_q(q_j)\), \(b=1/(1-q_k)\), and \[p_0=\frac{q_0}{\chi_0^2},\qquad \Delta p_j=\frac{q_j-q_{j-1}}{\chi_j\chi_{j-1}}.\] The algebra in the value can be made explicit. Abel summation and the inverse formulas give \[\begin{align*} p_k-\sum_jw_jq_jp_j &=p_0\chi_0+ \sum_{j=1}^k\Delta p_j(\chi_{j-1}+m_{j-1}q_{j-1})\\ &=\frac{q_0}{\chi_0}+ \sum_{j=1}^k\left(\frac{q_j}{\chi_j} -\frac{q_{j-1}}{\chi_{j-1}}\right) =\frac{q_k}{\chi_k}=b-1. \end{align*}\] Substitution into (20) and (22) therefore yields \[2E(q)=\log(1-q_k)+\frac{q_0}{\chi_0} +\sum_{j=1}^k\frac1{m_{j-1}} \log\frac{\chi_{j-1}}{\chi_j}.\] On \([0,q_0]\), \(\chi_q=\chi_0\); on each interval \([q_{j-1},q_j]\) it is affine of slope \(-m_{j-1}\). Integrating its reciprocal on these intervals gives exactly (32), also when an interval has zero length.

For passage to general profiles, fix \(Q<1\) with \(q\le Q\) and approximate by step profiles \(q_n\le Q\) in \(L^1\). Their inverse profiles satisfy \[(p_n)_*=\frac{(q_n)_0}{(\chi_n)_0^2} +\sum_j\frac{(q_n)_j-(q_n)_{j-1}} {(\chi_n)_j(\chi_n)_{j-1}} \le\frac{Q}{(1-Q)^2},\] because every \(\chi_n\ge1-Q\). Compactness gives a subsequence \(p_n\to p\) in \(L^1\). The finite-step tangent inequalities, first against step competitors and then against arbitrary bounded competitors, pass to the limit by the Lipschitz continuity of \(\mathcal S\) and the uniform bounds. They prove attainment for \(q\) and show that its value is the limit of the finite-step values. This argument does not interchange an uncontrolled supremum with a limit.

To identify the limiting expression, use (31). For \(q\le Q\), \(\chi_q(t)\ge1-Q\) on \([0,Q]\), and \(\chi_q(t)=1-t\) on \([Q,1]\). Consequently the right side of (33) is continuous under these approximations, with the explicit bound (34). It agrees with (32) because \(\int_0^{q_*}(1-t)^{-1}\,\mathrm dt=-\log(1-q_*)\). Approximating with \(Q=q_*\) gives the stated optimizer bound; when \(q_*=0\), \(p=0\) suffices. This completes the proof. ◻

Proof. Construction and curvature bounds. For a step coefficient \(m\), use the displayed Gaussian operators. Their integrals exist since \(u\) is bounded above. Jensen’s inequality and iteration give \[-A(1+z^2+Q-t)\leq F(t,z)\leq B.\] The differentiated equation, with \(\mathcal D=\partial_t+\frac12\partial_{zz}+mF_z\partial_z\), is \[\mathcal D F_z=0,\qquad \mathcal D F_{zz}=-m F_{zz}^2.\] Backward parabolic comparison, or forward comparison after the change of time \(\tau=Q-t\), bounds the second derivative below by \(-C\) and above by the solution of \(K'=K^2\), \(K(0)=\kappa\). These comparisons can first be made for bounded-slope terminals and on bounded spatial intervals, with quadratic barriers, and then passed to expanding intervals. Here is an explicit bounded-slope approximation preserving the constants needed for that passage. Keep \(u\) on \([-R,R]\). If \(p=u'(R)\leq0\), continue affinely to the right. If \(p>0\), continue with derivative \(p-C(z-R)\) until that derivative reaches zero, and then continue constantly. The maximum of this continuation is \(u(R)+p^2/(2C)\leq B\): the inequality follows by applying \(u(R+h)\geq u(R)+ph-Ch^2/2\) at \(h=p/C\) to the original terminal. On the left, continue affinely if \(u'(-R)\geq0\); otherwise continue with second derivative \(-C\) until the derivative reaches zero, and then constantly. The same argument bounds its maximum by \(B\). If \(C=0\), the original terminal is convex and bounded above, hence constant, so no continuation is needed. The resulting \(C^{1,1}\) function has curvature in \([-C,\kappa]\) distributionally. Its value and derivative at zero agree with those of \(u\), so the lower curvature bound supplies a common negative quadratic envelope on the whole line. Convolution with a compact nonnegative mollifier preserves both curvature bounds and the upper bound; let its width tend to zero as \(R\to\infty\). The preceding quadratic barriers and Gaussian integrability now permit passage to the original terminal. The two-sided bound on \(F\) near \(z=0\), together with the bound on \(F_{zz}\), bounds \(F_z(t,0)\); integration of \(F_{zz}\) then gives the linear-growth bound in (38).

Higher spatial derivatives. These estimates do not require bounds on derivatives of \(m\). For example, \[\mathcal D F_{zzz}=-3mF_{zz}F_{zzz}.\] The diffusion representation bounds this derivative by \(\|u'''\|_\infty\exp(3KQ)\), where \(K\) bounds \(|F_{zz}|\). Each further spatial differentiation gives a linear equation whose zeroth-order coefficient is bounded and whose source contains only already bounded derivatives. Induction gives the assertion.

General coefficients. Approximate a continuous \(m\) uniformly by step functions. If \(F\) and \(F'\) are the two step solutions, their difference solves a linear backward equation with diffusion coefficient \(1/2\), drift \(m(F_z+F'_z)/2\), and source \(-(m-m')(F'_z)^2/2\). Its drift is globally Lipschitz and has linear growth. The diffusion representation, with the terminal difference zero, gives on each compact spatial interval \[|F-F'|\leq C_R\|m-m'\|_\infty.\] The spatial derivative bounds and the equation give local compactness also for the derivatives. Thus the limit solves (36); the same difference equation proves uniqueness. This constructs the claimed classical solution and identifies it with the Gaussian recursion. The argument also works with step approximations of bounded measurable \(m\), using convergence in measure and dominated convergence in the diffusion representation whenever needed below.

Dependence on the covariance profile. Work in rank time and keep \(\dot q\) positive in a neighborhood of the parameter under consideration. Subtracting two equations gives a linear equation with drift \(\dot q_{\varepsilon}s(f_{\varepsilon,z}+f_z)/2\) and source \(-(\dot q_{\varepsilon}-\dot q)(f_{zz}+s f_z^2)/2\). Its diffusion representation and its spatially differentiated equations bound the difference and each fixed number of its spatial derivatives by \(C\|\dot q_{\varepsilon}-\dot q\|_\infty(1+z^2)\). To see that differentiation does not increase this polynomial degree, keep the linear-growth drift in the transport operator: every spatial derivative of that drift of order at least one is bounded. The commutator terms are therefore bounded coefficients times derivatives already estimated, while the source and all its fixed-order spatial derivatives have at most quadratic growth. Induction and the diffusion moment bounds preserve the same quadratic envelope. The constant here may depend on the fixed profile and on finitely many terminal derivatives. Applying the same bounds to difference quotients, and then to the difference between those quotients and their linearized equations, proves first and second parameter differentiability. This also justifies differentiation under the diffusion expectations below. Indeed the diffusions have moments of every fixed order, and stopping at \(|Z|=R\) followed by \(R\to\infty\) removes all spatial localizations. ◻

Proof. For \(Mg=g_{zz}+2sag_z\), direct differentiation gives \[\mathcal D(Mg)=M(\mathcal Dg)+(2a-\dot q s U_z)g_z.\] Integrate the first term on the right side of the second equation in (40) by parts, using \(\frac{\mathrm d}{\mathrm ds}\mathbb EMg(s,Z_s)=\mathbb E\mathcal D(Mg)(s,Z_s)\). Substitute \(v_z=l-\eta U_z/2\) and complete the square. Apart from \(\dot q s l^2-2\eta a l\), the remaining integrand is \[-\frac14\dot q s\eta^2 U_z^2+\eta^2aU_z +\frac12\eta\dot\eta\,MU.\] The second identity \[\mathcal D(MU)=2b^2+4aU_z-\dot q s U_z^2\] and one further integration by parts turn this integrand into \(-\eta^2b^2/2\). There are no boundary terms because \(\eta(0)=\eta(1)=0\). Finally \(\mathcal D\bar v=\eta a^2/2\) and spatial differentiation, taking account of the derivative \(\dot q s b\) of the drift, give the equation for \(l\). ◻

Proof. First suppose \(u''\leq-\rho<0\). The stochastic flow in (39), started from \(z\) at time \(s\), has spatial Jacobian \(J_{s,r}=\exp\{\int_s^r\dot q(t)t b(t,Z_t)\,\mathrm dt\}\). Differentiating the martingale representation \(a(s,z)=\mathbb E_{s,z}u'(Z_1)\) gives \[ b(s,z)=\mathbb E_{s,z}\left[u''(Z_1) \exp\left\{\int_s^1\dot q(t)t b(t,Z_t)\,\mathrm dt\right\}\right]. \tag{42}\] Thus \(k=-b\) is bounded and bounded away from zero. The product and quotient rules for \(\mathcal D\) give \[\mathcal D(l^2/k)=-2\eta a l+\dot q s l^2 +\frac{\dot q}{k}(l_z-lk_z/k)^2.\] Since \(l(1,\cdot)=0\), substitution into (41) yields the exact identity \[ \begin{split} w(0,0)={}&-\frac{l(0,0)^2}{k(0,0)} -\mathbb E\int_0^1\frac{\dot q}{k}(l_z-lk_z/k)^2\,\mathrm ds\\ &-\frac12\int_0^1\eta(s)^2\mathbb Eb(s,Z_s)^2\,\mathrm ds\leq0. \end{split} \tag{43}\] For a merely concave \(u\), apply this argument to \(u-\lambda z^2\) and let \(\lambda\downarrow0\). The parabolic comparison and moment bounds in Lemma 9 give convergence on every smooth profile and its segments. Concavity passes to that limit. Finally approximate two arbitrary profiles and their segment together by smooth increasing profiles with the prescribed endpoint values. The \(L^1\) continuity in (17) proves this extension for Lipschitz terminals already in the analytic class.

To cover a general concave globally \(L\)-Lipschitz terminal, convolve it with a compact nonnegative mollifier of width \(\delta\). The resulting \(u_\delta\) is smooth and concave, has bounded derivatives of every order at least two, and satisfies \(\|u_\delta-u\|_\infty\leq L\delta\). For each \(\lambda>0\), \(u_\delta-\lambda z^2\) belongs to (37), so concavity holds for smooth profiles. Let \(\lambda\downarrow0\) on each of the three profiles of a fixed chord. This limit needs no bound on \(\sup u_\delta\) uniform in \(\lambda\). Indeed, for \(0<\lambda\leq1\), Jensen gives a common negative quadratic lower bound for the solutions, while \(u_\delta(z)-\lambda z^2\leq u_\delta(0)+L|z|\) and \(\mathcal T_{a,T}v\leq\mathcal T_{1,T}v\) for \(a\in[0,1]\) give the upper bound \[F_{\delta,\lambda}(t,z) \leq u_\delta(0)+L|z|+ \log\mathbb E\exp\{L\sqrt Q\,|G|\}.\] The curvature bounds, hence the linear-growth drift and its diffusion moments, are uniform for \(\lambda\leq1\) at fixed \(\delta\). The linear equation comparing two such solutions therefore bounds their difference by \(C_\delta|\lambda-\lambda'|(1+z^2)\). It identifies the limit with the Gaussian recursion for \(u_\delta\); alternatively the same conclusion follows by monotone convergence of its terminal weights, with the displayed upper and lower integrable bounds. Thus concavity holds for \(u_\delta\) on smooth profiles. Profile continuity from (17) extends it to all profiles, and the \(1\)-Lipschitz dependence of \(\Phi\) on the terminal uniform norm lets \(\delta\downarrow0\). The quadratic hinge and its other required limits are treated below. ◻

Proof. Condition on the Brownian bridge and write the Brownian path over the interval as \(\mathsf b_t+(t/T)\xi\), where \(\xi\sim\mathcal N(0,T)\) is independent of the bridge. The endpoint \(X_T=\mathcal F(\xi)\) is an increasing function of \(\xi\). Differentiation of the integral equation gives \[\mathcal F'(\xi)=\frac1T\int_0^T \exp\left\{\int_r^T\partial_zA(v,X_v)\,\mathrm dv\right\}\mathrm dr \in[e^{-KT},e^{KT}].\] The conditional change-of-variables formula proves the upper density bound, which remains true after averaging over the bridge. For the lower bound, restrict the bridge to an event on which its supremum is at most a fixed constant. Such an event has uniformly positive probability for \(Q\in[1/2,2]\), by Brownian scaling. Gronwall’s inequality then bounds \(|\mathcal F(0)|\) uniformly. The inverse image of \(I\) is an interval of length at least \(e^{-2K}|I|\) contained in a fixed bounded interval. Gaussian densities with variance \(Q\in[1/2,2]\) have a positive common lower bound there. Multiplication by the bridge-event probability proves the claim. ◻

Proof. In variance time the drift is \(m(t)F_z(t,z)\). Its Lipschitz and growth bounds are uniform by Lemma 9. Thus Lemma 12 gives \(\mathbb P(Z_1\in I)\geq p_\beta>0\), where \(Z_1\) denotes the rank-time endpoint. From (42), the bound on \(|b|\), and (44), for every \(s\), \[\mathbb E[-b(s,Z_s)]\geq\beta e^{-KQ}\mathbb P(Z_1\in I) \geq\beta e^{-2K}p_\beta.\] Jensen’s inequality supplies a positive uniform lower bound on \(\mathbb Eb(s,Z_s)^2\). Equation (43) proves (45). The lower bound does not depend on \(\lambda\), so it survives \(\lambda\downarrow0\).

To prove stability, we compare slopes, then curvatures, then the modified variation \(l\) under a common Brownian coupling. These three comparisons control every term in the Hessian identity. The curvature comparison uses the total variation of \(u''\), so its constants remain uniform as the mollifier is removed.

Slope comparison. A tilde will denote the terminal \(u+d\psi\). Choose \(|d|\) initially so that \(|d|\|\psi''\|_\infty<1/4\); then its positive curvature bound is strictly less than \(1/(2Q)\) for \(Q\in[1/2,2]\), and all solution and drift bounds remain uniform. In this proof only, use variance-time notation \(a=F_z\), \(b=F_{zz}\) and its tilde analogue. At the same spatial point, \[\widetilde{\mathcal D}(\widetilde a-a) =-m b(\widetilde a-a),\qquad (\widetilde a-a)(Q,\cdot)=d\psi'.\] The diffusion representation implies \[ \sup_{t,z}|\widetilde a-a|\leq C|d|. \tag{46}\] Couple both diffusions using the same Brownian motion. Starting from a common position at time \(t\), Gronwall gives \(|\widetilde X_r-X_r|\leq C|d|(r-t)\) for \(t\leq r\leq Q\).

Curvature comparison. We use estimates that depend on the total variation of \(u''\), and not on \(\|u'''\|_\infty\). The equation \(\mathcal D b_z=-3mbb_z\) and its diffusion representation give \[ \int_\mathbb R|b_z(t,z)|\,\mathrm dz\leq C, \qquad \sup_z|b_z(t,z)|\leq\frac{C}{\sqrt{Q-t}}\quad(t<Q). \tag{47}\] For the first bound integrate the representation over starting positions; the pathwise spatial flow has Jacobian between \(e^{-KQ}\) and \(e^{KQ}\). For the second use the upper density bound of Lemma 12 and \(\int|u'''|\leq2\beta\).

Compare the flow representations (42) for \(b\) and \(\widetilde b\), started from the same point at time \(t\). The terminal perturbation contributes at most \(C|d|\). For the displacement in the base terminal, use the elementary inequality \[|u''(y)-u''(x)|\leq \int_{x-h}^{x+h}|u'''(v)|\,\mathrm dv\qquad(|y-x|\leq h).\] With \(h=C|d|(Q-t)\) and the transition-density bound, its expectation is at most \(C|d|\sqrt{Q-t}\). The difference of the two exponential Jacobians is at most a constant times the integral of the difference of their curvatures along the paths. Putting \(A(t)=\sup_z|\widetilde b(t,z)-b(t,z)|\) and using (47) therefore gives \[A(t)\leq C|d|+C\int_t^Q \left(A(r)+\frac{|d|(r-t)}{\sqrt{Q-r}}\right)\mathrm dr.\] The singularity is integrable, and backward Gronwall yields \[ \sup_{t,z}|\widetilde b(t,z)-b(t,z)|\leq C|d|. \tag{48}\]

For paths coupled from the root, their displacement at variance time \(t\) is at most \(C|d|t\). The same elementary inequality, now applied to \(b(t,\cdot)\), and its total variation and the density bound at time \(t\), give \[\mathbb E|b(t,\widetilde X_t)-b(t,X_t)|\leq C|d|\sqrt t.\] At \(t=0\) the positional difference is zero. Combining this estimate with (48) and the uniform boundedness of both curvatures gives, for every fixed \(p\geq1\), \[ \sup_t\|\widetilde b(t,\widetilde X_t)-b(t,X_t)\|_{L^p} \leq C_p|d|^{1/p}. \tag{49}\] The coupled slopes differ pointwise by \(C|d|\) and have uniformly bounded moments of every fixed order.

Comparison of the modified variations. Return to rank time and write \(a_s=a(s,Z_s)\), \(b_s=b(s,Z_s)\) and \(l_s=l(s,Z_s)\). On the common Brownian filtration \(\mathcal F_s\), the second equation of (41) gives \[ l_s=-\mathbb E\left[ \left.\int_s^1\eta(r)a_rb_r \exp\left\{\int_s^r\dot q(v)v b_v\,\mathrm dv\right\}\mathrm dr \right|\mathcal F_s\right]. \tag{50}\] This same filtration is used for the tilde representation; no comparison of unrelated conditional laws is involved. Both exponential factors are uniformly bounded since \(\int_0^1\dot q\,\mathrm ds=Q\leq2\). Conditional expectation is a contraction in \(L^p\), so \[\sup_s\|l_s\|_{L^p}+\sup_s\|\widetilde l_s\|_{L^p} \leq C_p\|\eta\|_1.\] Hölder’s inequality, (49) with \(p=4\), the slope estimates, and Minkowski’s inequality for the exponential integrals give \[ \sup_s\|\widetilde l_s-l_s\|_{L^2} \leq C|d|^{1/4}\|\eta\|_1. \tag{51}\] For example, the difference of the exponential factors has \(L^4\) norm at most \(C\int_s^r\dot q(v)\|\widetilde b_v-b_v\|_{L^4}\mathrm dv \leq C|d|^{1/4}\); multiplying by \(a_rb_r\), whose \(L^4\) norm is bounded, proves the relevant \(L^2\) bound.

Hessian comparison. Compare the three terms of (41). The \(\dot q s l^2\) term changes by at most \(C|d|^{1/4}Q\|\eta\|_1^2\); the \(2\eta a l\) term by at most \(C|d|^{1/4}\|\eta\|_1^2\); and the \(\eta^2b^2/2\) term by at most \(C|d|\|\eta\|_2^2\). Since \(\|\eta\|_1\leq\|\eta\|_2\), \[ |\widetilde w(0,0)-w(0,0)| \leq C|d|^{1/4}\|\eta\|_2^2. \tag{52}\] Choosing \(d_0\) so that \(Cd_0^{1/4}\leq c_\beta/2\) proves the proposition. ◻

Proof. The profile inequalities above are uniform in \(\dot q\): their constants use only \(Q=\int\dot q\). To approximate two arbitrary profiles \(q,r\), smooth their monotone extensions, add a small increasing linear function, and insert short ramps at \(0\) and \(1\) to impose the values \(0,Q\). These approximations can be chosen to remain in \([0,Q]\) and converge in \(L^1\), hence also in \(L^2\); their convex combinations approximate the whole segment. For a fixed capped and mollified terminal, first let \(\lambda\downarrow0\) on these smooth segments. Then (17) passes the strong concavity inequalities to all profiles. Integrating the Hessian bound gives (53) for each capped and mollified terminal, with coefficient \(c_\beta/2\) in the unperturbed case and \(c_\beta/4\) in the perturbed case.

We justify the remaining terminal limits uniformly in the profile. If \(\mu\) is any probability measure on \(\mathbb R\), \(w\) is nonnegative and increasing with \(0<\int w\,\mathrm d\mu<\infty\), and \(T\) is nonnegative and decreasing, then \[ \frac{\int Tw\,\mathrm d\mu}{\int w\,\mathrm d\mu}\leq\int T\,\mathrm d\mu. \tag{54}\] First take \(T\) bounded. The covariance of \(T\) and \(w\) equals half the integral of \((T(x)-T(y))(w(x)-w(y))\) against \(\mu\otimes\mu\), and is nonpositive. Apply this to \(T\wedge K\) and let \(K\to\infty\) to obtain the general case by monotone convergence. To apply this inequality, condition first on the cavity measure and the new Gaussian field, and let \(\mu\) be the pushforward of the cavity measure under that field. Insertion of an increasing terminal \(u\) replaces \(\mu\) by its tilt with weight \(w=e^u\). Only after applying the inequality do we average over the independent field and the cavity randomness. Before insertion, one sampled field value then has law \(\mathcal N(0,Q)\), because its diagonal covariance is \(Q\). For the RPC row functional the same construction includes any independent residual Gaussian in the base space. Adding \(d\psi\) increases the right side of (54) by at most the factor \(\exp(2|d|\|\psi\|_\infty)\).

The tangent cap satisfies the exact identity \[0\leq u_L(z)+\beta V(z)=\frac\beta2(z+L)_-^2.\] Every terminal between \(u_L\) and \(-\beta V\) is increasing. Interpolating between them and using (54) proves \[ \sup_q\big|\Phi_{u_L+d\psi}(q;Q) -\Phi_{-\beta V+d\psi}(q;Q)\big| \leq e^{2|d|\|\psi\|_\infty} \mathbb E\frac\beta2(\sqrt Q G+L)_-^2\longrightarrow0. \tag{55}\] The convergence is also uniform for \(Q\in[1/2,2]\). For each fixed \(L\), mollification converges uniformly in terminal value, since \(u_L\) is Lipschitz; the row functional is \(1\)-Lipschitz for the uniform norm on its terminal. Thus the limits are taken in the order \(\lambda\downarrow0\), \(\delta\downarrow0\), and \(L\to\infty\). Equation (53), including its perturbed form, survives them.

The derivative functional also converges uniformly in the profile. Here are the necessary uniform-integrability details. The derivatives of \(u_L+d\psi\) agree with that of \(u_d\) outside \((-\infty,-L)\) and their absolute values there are bounded by a constant times \(1+|z|\). Equation (54), applied to higher negative-tail moments, bounds their tilted moments of every fixed order uniformly. Let \(\nu_L\) denote a row measure tilted by \(u_L+d\psi\) and let \(\nu\) be the corresponding uncapped measure. Both measures are defined on the same marked base space: the cascade leaf, its full Gaussian field, and an independent residual Gaussian integrated inside the terminal weight, when present. Include the common refined rank genealogy, inserting zero-variance levels as in Section 2, or its equivalent auxiliary rank variables. In a two-replica expectation the continuous rank is then a common measurable observable even when both replicas select the same leaf of the original finite cascade. Thus comparison of the full product measures also controls every bounded rank test. Their unnormalized density ratio is \(e^{-\beta V-u_L}\leq1\) and equals one on \([-L,\infty)\). If \(\varepsilon_L=\nu_L(( -\infty,-L))\), normalization and the triangle inequality give \[\|\nu-\nu_L\|_{\mathrm{TV},1}\leq4\varepsilon_L,\] where \(\|\cdot\|_{\mathrm{TV},1}\) is total variation without the factor \(1/2\). The averaged right side tends to zero uniformly by (54). The same is true for two replicas. Test the product of the two terminal derivatives against any bounded function of rank, first truncating the derivatives and then using their uniform higher moments. The two-replica formula for \(D_q\) therefore gives uniform \(L^1(\mathrm ds)\) convergence of the capped derivative profiles to \(D^{u_d}_q\). Mollification is handled similarly at each fixed cap. In particular \(D^{u_d}_q\) is continuous in \(q\) in \(L^1\), by the corresponding capped assertion and this uniform approximation. Moreover its \(L^\infty\) norm is uniformly bounded: the derivative martingale and the terminal second-moment bound give \(D^{u_d}_q(s)\leq\mathbb E_{\mathrm{tilted}}(u_d')^2\leq C_{\beta,\psi}\). Thus the integrated derivative formula (17) and its one-sided derivative interpretation extend to \(u_d\). ◻

Proof. The terminal is concave and increasing. Its derivative is Lipschitz, has at most linear growth, and vanishes on \([-1,\infty)\). Choose \(L\) so large that \(-L<\inf\mathop{\mathrm{supp}}\chi\), with the condition omitted when \(\chi=0\), and extend \(u\) tangentially below \(-L\). The resulting terminal \(u_L\) is concave, increasing, and globally Lipschitz. Proposition 11 applies after mollification and its removal. Because the cutoff lies below the entire support of \(\chi\), its error is exactly \[u_L(z)-u(z)=\frac{\beta c}{2}(z+L)_-^2.\] Every intermediate terminal is increasing, so (54) bounds the difference of the row functionals by \(\frac{\beta c}{2}\mathbb E(\sqrt QG+L)_-^2\), uniformly in the profile. The same comparison applies to a row insertion in the microscopic Gaussian model. The derivatives agree outside the cutoff region and are bounded by \(C_{\beta,W}(1+|z|)\) within it. Higher Gaussian negative-tail moments and the marked-space total-variation comparison above prove uniform convergence of \(D_q\) and its differential formula. This proves all assertions, including passage of the interpolation bound to the uncapped loss. ◻

Proof. Enumerate pairs \((j,\lambda)\), where \(j\ge1\) and \(\lambda\) belongs to a countable dense subset of \([0,\infty)\) containing \(0\). Let \(h_{j,\lambda}\) be independent unit-variance Gaussian fields with covariance \[K_{j,\lambda}((x,\tau),(x',\tau')) =\left(\frac{R(x,x')+\lambda Q(\tau,\tau')}{1+\lambda}\right)^j.\] These are positive semidefinite kernels: \(R+\lambda Q\) is a Gram kernel and integer powers preserve positive semidefiniteness by the Schur product identity. The binomial expansion separates powers of \(R\) and \(Q\). Each power of \(Q\) has nonnegative increments along the cascade, so the fields can be realized using independent child fields conditional on their ancestor fields. Choose positive weights \(w_{j,\lambda}\) with \(\sum w_{j,\lambda}^2j<\infty\), and set \[H_N^{\rm pert}=s_N\sum_{j,\lambda} w_{j,\lambda}c_{j,\lambda}h_{j,\lambda}, \qquad c_{j,\lambda}\in[1,2].\] In the sphere-only construction use just covariance kernels \(R^j\). The Gaussian covariance metric is continuous on each sphere, and the fields are taken in their separable versions. Gaussian interpolation bounds the pressure change by \(Cs_N^2/N\).

Here is the concentration estimate needed for the perturbation argument: \[ \mathbb E|\log Z-\mathbb E\log Z|\le C(\sqrt N+s_N). \tag{58}\] Without a cascade, Gaussian concentration gives this directly. Indeed the gradient of \(\sum_{\mu\le M}u(g_\mu(x))\) with respect to the row entries has squared norm at most \(M\|u'\|_\infty^2\), uniformly on the sphere. Any linear Gaussian spin terms have squared reproducing-kernel norm \(O(N)\), and the perturbation has squared norm \(O(s_N^2)\).

For a fixed finite cascade apply Lemma 2 to the spin-integrated leaf partition marks. Conditional on shared root Gaussian fields, the log partition is a recursive root shift plus the difference of two log cascade masses. The latter have uniformly bounded second moments, with a constant depending on the fixed cascade. Recursive logarithmic moments preserve a uniform Lipschitz constant: if \(|F(g,z)-F(g',z)|\le L\|g-g'\|\) for every child mark \(z\), the same bound holds after applying \(m^{-1}\log\mathbb E_z e^{mF}\). Thus the root shift has Lipschitz constant \(C(\sqrt N+s_N)\) in its shared Gaussian fields. All child moments needed here are finite, by bounded slope and Gaussian moments; at fixed dimension one can also bound them by the supremum of the corresponding Gaussian sphere process. This proves (58), uniformly in interpolation time and after deletion of a fixed number of rows.

For completeness we derive the identities rather than invoke a separate perturbation theorem. Fix one field \(h\) with coefficient \(swc\), where \(s=s_N\) and \(w>0\) is fixed, and put \(F(c)=\log Z(c)\). Then \[F'(c)=sw\langle h\rangle,\qquad F''(c)=(sw)^2\langle(h-\langle h\rangle)^2\rangle.\] Gaussian integration by parts gives \[\mathbb E\langle h\rangle=swc\,\mathbb E\langle K_{11}-K_{12}\rangle.\] Consequently integration over \(c\in[1,2]\) bounds the mean thermal variance of \(h\) by a constant. The resulting integrated thermal absolute deviation is \(O(1)=o(s)\).

For the disorder deviation, the convex difference-quotient inequalities for \(F\) and \(\mathbb EF\), at distance \(d\in(0,1/4)\), give \[\int_1^2\mathbb E|\langle h\rangle-\mathbb E\langle h\rangle|\,\mathrm dc \le C_w\left\{\frac{\sqrt N+s}{sd}+sd\right\}.\] To justify the second term, integrate the increments of the monotone function \((\mathbb EF)'\) over the enlarged interval \([1-d,2+d]\); their total is at most \(C_w d s^2\), before division by \(sw\). The concentration bound applies on this enlarged interval as well. Taking \(d=N^{1/4-\rho}\) makes the last display \(o(s)\). In Gaussian integration by parts against a bounded replica test \(f\), one may therefore replace \(h(x^1,\tau^1)\) by its disorder mean at an error \(o(s)\) after coefficient averaging. Division by \(swc\) gives a vanishing integrated absolute error in (57), first with \(\psi=K\).

For each total degree \(j\), the polynomials \((R+\lambda Q)^j\) at \(j+1\) distinct values of \(\lambda\) span the homogeneous polynomials of degree \(j\) in \((R,Q)\). Polynomial approximation gives all continuous \(\psi\) and all continuous finite-array tests; a monotone-class argument gives measurable \(f\). Choose countable determining families. The sum of their integrated errors, with summable weights, tends to zero. A subsequence with summable weighted errors has pointwise vanishing errors for almost every coefficient vector. Including interpolation time and a countable list of deleted-row systems in this extraction proves (iii). Compactness of the array space then gives the asserted limiting identities. ◻

Proof. Apply ultrametricity separately to \(R\), \(Q\), and \(R+Q\), using (57). First, if \(\mathbb P(R_{12}\le a)=m>0\) for some \(a<0\), ultrametricity makes \(R_{ij}>a\) an equivalence relation on distinct replicas. On the event that \(n\) sampled replicas occupy distinct classes, GG gives probability \((1-m)/n\) for the next replica to join any specified class. These joining events are disjoint, so the probability of avoiding them all is \(m\). Induction gives positive probability of arbitrarily many pairwise overlaps at most \(a\). Positive semidefiniteness forbids this, since on that event the sum of the entries of the \(n\times n\) Gram matrix is at most \(n+n(n-1)a<0\) for sufficiently large \(n\).

Let \(\mu\) be the law of \((R_{12},Q_{12})\). If two points in its support were strictly incomparable coordinatewise, choose disjoint neighborhoods \(A,B\) with \(R_A<R_B\) and \(Q_A>Q_B\). GG with two old replicas gives \[\mathbb P\big((R_{12},Q_{12})\in A, (R_{13},Q_{13})\in B\big) =\tfrac12\mu(A)\mu(B)>0.\] On this event scalar ultrametricity forces \(R_{23}=R_{12}\) and \(Q_{23}=Q_{13}\). Thus \(R_{23}+Q_{23}\) is strictly smaller than both \(R_{12}+Q_{12}\) and \(R_{13}+Q_{13}\), contradicting ultrametricity of the sum. The support of \(\mu\) is therefore a chain. Each coordinate is a nondecreasing function of the sum on that support, and the pair has comonotone quantiles.

It remains to identify the whole array, since a pair marginal alone would otherwise be insufficient. At any threshold with lower mass \(m\), GG says, conditional on the old array, that the new replica joins an existing ultrametric class containing \(l\) old replicas with probability \((l-m)/n\). For finitely many thresholds, the classes form a nested partition. These joining probabilities, and their differences for a parent and its children, determine the distribution of the new branch. Induction in the number of replicas determines the law of the discretized array. By Lemma 3 it is exactly the finite RPC sampling law. Refining thresholds at continuity points identifies the full scalar array. Apply this to the sum and then to its two coordinate functions to obtain the common rank representation.

Finally, at finite \(N\) the cascade genealogy retains its prescribed law after the spin-integrated tilt: all Gaussian fields in the model, including the perturbation, have conditional child independence. This is precisely Lemma 2. Hence the limiting second marginal is the prescribed trial profile. ◻

Proof. Conditionally on the deleted-row Gibbs measure, the new row values sampled on finitely many replicas are centered Gaussian vectors with the displayed covariance arrays. They therefore converge jointly with the base arrays. For every fixed \(a<\infty\), their bounded-slope potentials obey \(\mathbb E\langle e^{a|u(g)|}\rangle\le C_a\) before insertion. The replica truncation argument of Lemma 4 consequently passes both the finite-row tilt and its observables to the limit. For a finite RPC, claims (i)–(iii) are the marking transformation and additive-mark factorization of Lemma 2, together with the derivative-martingale definition of \(D_C\). In particular, the marking transformation makes the transformed weight tree independent of the separate row-mark families. Sampling replicas from those weights therefore preserves their conditional factorization given the genealogy. This is the depth-two calculation in the proof of Lemma 2, applied to the sum of the inserted-row potentials; it uses no independence between a stable total and its normalized jumps.

For the passage to general profiles, retain the canonical rank array itself in the replica convergence, even on plateaus of the covariance profile. The following conditional-kernel version of the truncation argument makes the factorization passage precise. For a bounded continuous observable \(F\) of the row marks at the first \(n\) replicas, its conditional expectation given their ranks is approximated in \(L^1\) of the fixed rank law by the finite-replica polynomial approximants to the truncated tilt numerator and denominator. The truncation error is uniform over bounded profiles, by the exponential moment bounds. At finite profile, invariance of the tilted rank marginal lets us interpret these approximants as conditional kernels relative to the same canonical rank law. A polynomial approximant uses a fixed finite number of additional replicas. Integrating its Gaussian marks gives a bounded function \(K_j\) of all the ranks of those replicas. Under \(C_j\to C\) in \(L^1\), the covariance entries converge almost surely along a subsequence; Gaussian covariance continuity and dominated convergence give \(K_j\to K\) in \(L^1\) of the fixed rank law. Averaging the additional ranks conditional on the first \(n\) ranks preserves this convergence, since \[\big\|\mathbb E[K_j-K\mid(U_{ab})_{a,b\le n}]\big\|_1 \le\|K_j-K\|_1.\] The conditional law of those additional ranks is independent of the covariance profile. Thus the exact bounded conditional kernels converge in \(L^1\). Products of such kernels also converge in \(L^1\). The finite-profile factorization identities against any bounded cylinder test of the ranks therefore pass to the limit. A monotone-class argument extends them to the entire countable genealogy. This proves (i)–(iii) without presuming that conditional independence is closed under weak convergence.

In particular, for two inserted rows \(A,B\) with profile \(C_j\) and for any bounded cylinder test \(H\) of the full rank array, the identity needed below is \[\mathbb E\langle H a_A^1a_A^2a_B^1a_B^2\rangle =\mathbb E[H D_{C_j}(U_{12})^2].\] Its limit also follows directly from \(D_{C_j}\to D_C\) in \(L^1\) and bounded slope. Single-replica analogues have constant right-hand factors. Polynomial observables follow by truncation and the same exponential moment bounds. The last assertion follows by first taking a joint subsequence of the rowful and deleted-row laws, then representing the former as the tilt of the latter. Claim (i) identifies their complete overlap-array laws. ◻

Proof. For a continuous function \(F\) of one overlap, remove a specified row and reinsert it using Lemma 18. Row exchangeability gives \[\lim_N\mathbb E\langle p^N_{12}F(R_{12})\rangle =\alpha\int_0^1D_q(s)F(q(s))\,\mathrm ds.\] On expanding \((p^N_{12})^2\), the terms with equal row indices have total size \(O(N^{-1})\), since \(u'\) is bounded. For the distinct indices, delete both rows and use conditional factorization of their tilted mark families. Hence \[\lim_N\mathbb E\langle(p^N_{12})^2\rangle =\alpha^2\int_0^1D_q(s)^2\,\mathrm ds.\] The function \(D_q\) is constant almost everywhere on each plateau of \(q\). Thus there is a bounded Borel function \(d_q\) such that \(D_q(s)=d_q(q(s))\) almost everywhere. Approximate \(d_q\) in \(L^2\) of the overlap marginal by continuous functions. The preceding two displays imply that any joint limit \(p_{12}\) satisfies \(\mathbb E(p_{12}-\alpha d_q(R_{12}))^2=0\). Exchangeability gives all off-diagonal entries simultaneously.

The same one-row and two-row calculation for the single-replica mark \(a_\mu(x)^2\) identifies the diagonal limit. With only one sampled replica there is no variable pair genealogy, so its conditional mean is the scalar \(D_{q,d}\). Apply the calculation also to \(u''(g_\mu)-g_\mu u'(g_\mu)\). This observable has at most linear growth. Its moments after inserting one or two rows are uniformly bounded by reciprocal Jensen, Gaussian exponential moments, and bounded slope; this justifies truncation in the preceding argument and gives a deterministic \(L^2\) limit for \(b_N\).

To compute that limit, Gaussian integration by parts in a single row gives \[\mathbb E\langle g_\mu u'(g_\mu)\rangle =\mathbb E\langle u''(g_\mu)+u'(g_\mu)^2\rangle -\mathbb E\langle R_{12}u'(g_\mu^1)u'(g_\mu^2)\rangle.\] Multiplication by \(M_N/N\) and the already identified row limits give exactly \(b=-p_d+\int q p_q\). All deletions and limits take place at a fixed admissible coefficient vector; there is no mixture-of-profiles step. ◻

Proof. Let \(q\) minimize \(\mathcal P_u\). By Proposition 7, \(\mathcal S\) has at \(p_q\) a supergradient \(-r/2\), where \(r\) is a nonnegative nondecreasing profile bounded strictly below \(1\) and \(p_q=\mathcal I(r)\) for the inverse map (25). Put \(q_t=q+t(r-q)\) and \(p_t=p_{q_t}\). The supporting inequality for the concave row functional and the supergradient inequality for \(\mathcal S\) give \[\begin{split} \alpha\big[\Phi_u(q_t)-\Phi_u(q)\big] &\le-\frac t2\int(r-q)p_q,\\ \mathcal S(p_t)-\mathcal S(p_q) &\le-\frac12\int r(p_t-p_q). \end{split}\] Adding the exact difference of the coupling terms yields \[\mathcal P_u(q_t)-\mathcal P_u(q) \le-\frac{1-t}{2}\int(r-q)(p_t-p_q).\] By (56), for \(0<t<1\) the integral on the right is at least \(c_{\alpha,\beta}t\|r-q\|_2^2\). Minimality therefore forces \(r=q\). In particular \(q_*<1\) and \(p_q=\mathcal I(q)\), proving the asserted spin inverse relation for the original profile \(p_q\). The equality of this spin supergradient with \(-q/2\) means that \(p_q\) attains the entropy conjugate: \[E(q)=\mathcal S(p_q)+\frac12\int q p_q.\] For any competing profile \(q'\) with \(q'_*<1\), the definition of \(E\) gives \(\mathcal P_u(q')\le\alpha\Phi_u(q')+E(q')\). Equality at the minimizing \(q\) proves (65). ◻

Proof. The support of \(\psi(1+z)\) is a compact subset of \(z<-1\), where \(V''=1\), \(V>0\), and \(V'<0\). Taking \(|w|\) sufficiently small preserves all three asserted inequalities, including at the boundary of that support. The threshold for \(w\) is a property of the loss and does not depend on \(\beta\). For the fixed concave terminal \(-\beta L_w\), take tangent caps and concave mollifications. Ordinary concave-terminal profile concavity from Section 3 gives the interpolation upper bound for each such cap. The entropy conjugate bounds its spin terms by \(E(q)\), so the right side is simply \(\alpha\Phi(q)+E(q)\) at this fixed trial. Mollification errors vanish uniformly in terminal value for a fixed cap. Beyond the compact perturbation support the cap error is the original negative Gaussian quadratic-tail error, controlled by monotone row insertion both microscopically and in the cascade. Sending the cap to infinity proves the displayed bound. Only pressure and row-functional values are passed to the limit; no derivative-profile uniformity in \(w\) and no complete pressure formula for the perturbed loss are required. ◻

Proof. For the first inequality put the variance scale into the terminal, so the Gaussian variance profile is \(q\) on the fixed time interval \([0,1]\). Let \(A_v\) and \(A_1\) be the derivative solutions for the terminals \(u(\sqrt v z)\) and \(u(z)\), respectively. Their difference \(H=A_v-A_1\) satisfies \[H_t+\tfrac12H_{zz}+m_q A_vH_z+m_q(A_1)_zH=0, \qquad H(1,z)=\sqrt v\,\beta(-1-\sqrt v z)_+ -\beta(-1-z)_+\le0.\] The zeroth-order coefficient lies in \([-\beta,0]\), and the drift is Lipschitz with at most linear growth. The maximum principle gives \(H\le0\). To handle the hinge, convolve the two terminal derivative functions with the same nonnegative mollifier, preserving their order, and pass to the limit. When \(q(0)>0\), \(m_q=0\) before that root variance, so the same comparison includes the initial heat interval. The derivative martingale at \((0,0)\) identifies \(A_v(0,0)\) with the expectation on the left of (76).

For the second inequality use variance time, writing the tilted diffusion as \[\mathrm dZ_s=\mathrm dB_s+m(s)a(s,Z_s)\,\mathrm ds,\qquad Z_0=0, \qquad 0\le s\le v.\] The base derivative \(a=f_z\) is nonnegative, nonincreasing, and convex in position. Nonnegativity and monotonicity follow from the terminal and parabolic comparison. Convexity follows by differentiating once more: \(c=f_{zzz}\) has nonnegative terminal value (a positive measure for the hinge) and solves the linear equation \((\partial_s+\frac12\partial_{zz}+ma\partial_z)c=-3mf_{zz}c\). Smooth concave approximations with convex first derivative and no far cap justify the claim; their diffusion limits give the hinge. Thus the drift is nonnegative, nonincreasing, and convex.

Condition on the Brownian bridge \(\omega_s=B_s-(s/v)B_v\) and write \(b=B_v\sim\mathcal N(0,v)\), independent of the bridge. The endpoint map \(T_\omega(b)=Z_v\) is increasing, convex, and \(1\)-Lipschitz. Indeed its first variation \(J_s\) solves \(J_s'=1/v+d_z(s,Z_s)J_s\), \(J_0=0\), for \(d=ma\); hence \(0\le J_s\le s/v\) and, more precisely, \[T_\omega'(b)=\frac1v\int_0^v \exp\left\{\int_r^v d_z(s,Z_s)\,\mathrm ds\right\}\mathrm dr \ge\frac{1-e^{-\beta v}}{\beta v}>0.\] Its second variation solves \(K_s'=d_zK_s+d_{zz}J_s^2\), \(K_0=0\), and is nonnegative. These deterministic integral-equation variations remain valid for every continuous bridge path by smooth approximation. Also \(T_\omega(b)\ge b\) by nonnegative drift. The strictly positive lower slope bound makes \(T_\omega\) onto, so it has a unique crossing \(b_0\le-1\) of \(-1\). Set \(R(s)=-1-T_\omega(b_0-s)\), \(s\ge0\). Then \(R\) is increasing and concave, \(R(0)=0\), and \(0\le R(s)\le s\). Consequently \(R(s)/s\) is nonincreasing. Under the Gaussian left-tail measure with density proportional to \(\exp(-(b_0-s)^2/(2v))\mathbf 1_{s\ge0}\), decreasing-likelihood-ratio comparison gives \[\frac{\mathbb E[R(s)^2]}{\mathbb E[R(s)]} \le \frac{\mathbb E[sR(s)]}{\mathbb E[R(s)]} \le \frac{\mathbb E[s^2]}{\mathbb E[s]} \le\sqrt{\pi/2}.\] The middle inequality follows from the same covariance identity used above, now relative to the probability density proportional to \(s\exp(-(b_0-s)^2/(2v))\). Relative to the density proportional to \(s e^{-s^2/2}\), the logarithmic derivative of its likelihood ratio is \(b_0/v-(1/v-1)s\le0\). The last ratio is therefore at most \(\int_0^\infty s^2e^{-s^2/2}\mathrm ds/\int_0^\infty se^{-s^2/2}\mathrm ds =\sqrt{\pi/2}\). Multiply by the conditional first moment and integrate over the bridge to obtain the second assertion of (76). The lower slope bound is used only for the crossing and approximation; its dependence on \(\beta\) does not enter the moment constant. Mollified drifts retain all three shape properties and the common Lipschitz bound \(\beta\). Approximate a general monotone \(m\) by steps in \(L^1\); the linear PDE difference equation and uniform diffusion moments give local convergence of solutions, and the flow maps converge on compact endpoint sets. Finally \((-1-Z_v)_+\le(-1-B_v)_+\) supplies uniform integrability for all residual moments. These observations justify the hinge and coefficient limits in the bridge argument. ◻

Proof. First, the normalized random log partition concentrates for the uncapped potentials \(u+d\psi\). On the convex operator-norm ball \(\|A\|_{\mathrm{op}}\le K\sqrt N\), differentiation gives \[\|\nabla_A\log Z\|_F \le \beta(\|A\|_{\mathrm{op}}+\sqrt M) +|d|\|\psi'\|_\infty\sqrt M=O(\sqrt N).\] Extend its restriction from this ball with the same Lipschitz constant and use Gaussian concentration, through \(T\) for mixture entries. The complement has probability exponentially small in \(N\) for a large fixed \(K\), by (69). Jensen’s lower bound \(\log Z\ge-C_\beta(M+\|A\|_{\mathrm{op}}^2)-M|d|\|\psi\|_\infty\) and the corresponding upper bound show that the extension changes its expectation by \(o(N)\). Hence \(N^{-1}\log Z-P_{N,\varepsilon}(u+d\psi)\) converges to zero in probability.

For a fixed test \(\psi\), put \(S(d)=\alpha\Phi_{u+d\psi}(q;1)+E(q)\). Theorem 22 gives \(S(0)=P(u)\) and \(S'(0)=\alpha\int\psi\,\mathrm d\mu_{\beta,\alpha}\). If \(b>S'(0)\), choose a sufficiently small \(d>0\) and \(c>0\) such that \(S(d)-S(0)<db-2c\). The expected upper support and base pressure limit, together with the preceding concentration, imply with disorder probability tending to one that \[\log Z(u+d\psi)-\log Z(u)\le N(db-c).\] This uses only a limsup bound at \(d\); a thermodynamic pressure limit for the perturbed terminal has not been assumed. Exponential Markov under the Gibbs measure gives \[\left\langle\mathbf 1_{\{N^{-1}\sum_\mu\psi(g_\mu)\ge b\}}\right\rangle \le\exp\{\log Z(u+d\psi)-\log Z(u)-Ndb\}.\] On that event the right side is at most \(e^{-Nc}\). The same argument with \(d<0\) gives the lower bound. Therefore \(\int\psi\,\mathrm dL_N\to\int\psi\,\mathrm d\mu_{\beta,\alpha}\) in joint probability for every \(\psi\in C_c^\infty\). A countable determining class, together with tightness obtained by smooth cutoffs whose \(\mu_{\beta,\alpha}\)-masses tend to one, proves weak convergence. The same limits for every base minimizer prove their equality of row marginals.

Here are direct bounds ensuring that threshold tests and normalization are permitted. Remove one row and let \(\langle\cdot\rangle_0\) be its cavity measure. Its reweighting denominator is \(D=\langle e^{u(g)}\rangle_0\le1\), and Jensen gives \(D\ge\exp\langle u(g)\rangle_0\). Since the row is independent of this measure, \(\mathbb E[-\langle u(g)\rangle_0]\le C_\beta\). Thus \(\mathbb P(D<e^{-K})\le C_\beta/K\). For an interval \(J\), conditional on a unit cavity spin the fresh projection has density bounded by an absolute constant, and \(e^u\le1\). Consequently its reweighted marginal satisfies \[\mathbb E\langle\mathbf 1_{\{g\in J\}}\rangle \le C e^K|J|+C_\beta/K.\] This estimate is uniform in \(N\) and passes to the deterministic limiting law by continuous tests. Sending \(|J|\downarrow0\) and then \(K\to\infty\) proves that the law has no atoms. Conversely, for a fixed nonempty interval \(J\Subset(-\infty,-1)\), \(D\le1\) gives \[\mathbb E\langle\mathbf 1_{\{g\in J\}}\rangle \ge \inf_{g\in J}e^{u(g)} \inf_{v\in[1/2,3/2]}\mathbb P(\sqrt v Z\in J)>0.\] Hence its limiting negative-gap probability is positive. Finally the single-row covariance inequality used above gives, for every fixed \(p>0\), \[\mathbb E\langle(-1-g)_+^p\rangle \le\sup_{v\in[1/2,3/2]}\mathbb E(-1-\sqrt v Z)_+^p<\infty.\] Using an exponent larger than \(p\) gives uniform integrability, including for empirical averages.

Set \[p_-=\mu_{\beta,\alpha}(( -\infty,-1)),\qquad m_-=\int(-1-g)_+\,\mu_{\beta,\alpha}(\mathrm dg),\qquad \bar r=m_-/p_-.\] Both \(p_-\) and \(m_-\) are strictly positive. Weak convergence, atomlessness, and uniform integrability imply in joint probability \[\frac{|C(y)|}{M}\longrightarrow p_-,\qquad \frac1M\sum_{\mu\in C(y)}(-1-g_\mu)\longrightarrow m_-.\] For the second assertion, the second-moment bound gives \(\mathbb E\int r\mathbf 1_{\{r>K\}}\,\mathrm dL_N\le C/K\), where \(r=(-1-g)_+\). Markov’s inequality makes this truncation error small in joint probability. The bounded continuous truncated residual converges by weak convergence, and then \(K\to\infty\) gives the assertion. Thus the configuration-dependent conditional mean residual converges to \(\bar r\). For every \(s>0\), sandwich its random threshold between \(s(\bar r-\delta)\) and \(s(\bar r+\delta)\), apply weak convergence to these fixed thresholds, and send \(\delta\downarrow0\). Atomlessness at \(g=-1-s\bar r\) justifies the last step. Since the observables lie in \([0,1]\), convergence in probability also gives convergence of their expectations. Explicitly, \[\begin{align*} \lim_{N\to\infty}G_{N,\beta,\alpha}(\delta,u_+) &=\mu_{\beta,\alpha}((\delta-1,u_+-1]),\qquad 0<\delta<u_+,\\ \lim_{N\to\infty}F_{N,\beta,\alpha}(s) &=\frac{\mu_{\beta,\alpha} (\{-1-s\bar r\le g<-1\})}{p_-}. \end{align*}\] The convention when \(C(y)\) is empty is immaterial, since its probability tends to zero. These formulas prove the proposition with exactly the force normalization specified in the problem. ◻

Proof. The positive-part map is \(1\)-Lipschitz, so for unit \(y,y'\), \[\|r_A(y)-r_A(y')\|\leq\|A\|\,\|y-y'\|, \qquad \|r_A(y)\|\leq\sqrt M+\|A\|.\] It follows that \(H_A\) has Lipschitz constant at most \(L_A=\|A\|(\sqrt M+\|A\|)\) on the sphere. A spherical cap of Euclidean radius \(\delta\) about a minimizer has surface probability at least \((c\delta)^N\), for an absolute \(c>0\). For completeness, rotate its center to the last coordinate vector and parametrize the upper hemisphere by \(z\mapsto(z,\sqrt{1-\|z\|^2})\). The ball \(\|z\|\leq\delta/2\) maps inside the cap, and its surface Jacobian is at least one. The ratio of the \((N-1)\)-ball volume to the \((N-1)\)-sphere area is bounded below by \(c_1^N\), as follows directly from their gamma-function formulas; this proves the stated bound. Integrating over this cap and also using \(H_A\geq\min H_A\) gives the deterministic estimate \[0\leq-\frac1{\beta N}\log\int e^{-\beta H_A}\,\mathrm d\sigma -\frac{\min H_A}{N} \leq\frac{L_A\delta}{N}+\frac1\beta\log\frac C\delta.\] Equation (77) gives \(\mathbb E\|A\|^2\leq C(B_0)N\); hence \(\mathbb EL_A/N\leq C(B_0)\). This proves (78).

At fixed \(N\), the quantity \(-P_{N,\alpha}^{\nu_0}(\beta)/\beta\) is nonincreasing in \(\beta\) and nonnegative. Indeed, \(\beta\mapsto\log\int e^{-\beta H_A}\mathrm d\sigma\) is convex and vanishes at zero. Its limiting pressure inherits these properties, so its zero-temperature limit exists. First let \(N\to\infty\) in the inequality, then let \(\beta\to\infty\) with, for example, \(\delta=\beta^{-1/2}\). The liminf and limsup of the expected ground energy coincide with this limit. The identical argument applies to any law with the same fixed-temperature pressures.

Couple matrices of different row counts by appending independent rows. Appending rows cannot decrease the minimum energy. Conversely, evaluate the new rows at a measurable approximate minimizer of the old matrix and then let its approximation error tend to zero. For every fixed unit \(y\), a fresh projection \(g=A_\mu y\) has mean zero and variance one, and \[\mathbb EV(g)\leq\tfrac12\mathbb E(1+g)^2=1.\] Thus the expected minimum energy increases by at most one per appended row. After dividing by \(N\) and passing to the limit this proves the monotonicity and Lipschitz assertions. Approximate minimizers can be selected measurably by taking the first adequate point in a fixed countable dense subset of the sphere, since \(H_A\) is continuous. If \(M<N\), the row kernel contains a unit vector, at which every gap is one. Hence \(e(\alpha)=0\) for \(\alpha<1\); continuity gives \(e(1)=0\). ◻

Proof. Conditioned on the variance marks of a mixture matrix, a fixed unit-spin projection is Gaussian with variance in \([1/2,3/2]\). Thus \[\mathbb P(A_\mu y<-2)\geq p, \qquad p=\mathbb P(Z<-2\sqrt2)>0, \quad Z\sim\mathcal N(0,1).\] At a deterministic unit \(y\), row independence and the elementary Chernoff bound give \[\mathbb P\bigl(\#\{\mu:A_\mu y<-2\}<pM/2\bigr) \leq e^{-c_pM}.\] Outside this event, \(\|r_A(y)\|\geq\sqrt{pM/2}\). There is an absolute \(K_0\) such that \(\|A\|\leq K_0(\sqrt N+\sqrt M)\) with probability tending to one, uniformly in these entry laws. Choose a fixed \(\delta>0\) so small that \(2K_0\delta\leq\frac12\sqrt{p/2}\), and take a \(\delta\)-net of the unit sphere of cardinality at most \((1+2/\delta)^N\). For \(M\geq N\), the Lipschitz bound for \(r_A\) transfers the residual lower bound on the net to \[R(A)\geq\tfrac12\sqrt{pM/2}.\] The probability that a net point fails is at most \(\exp\{N\log(1+2/\delta)-c_pM\}\). Choosing \(B\) sufficiently large and setting \(M=\lfloor BN\rfloor\) makes this probability tend to zero. Therefore \(\min H_A/N\geq pM/(16N)\) with probability tending to one, and \(e(B)\geq pB/16>0\). ◻

Proof. Choose \(K=K(B_0)\) large enough in (77) that the operator-norm failure probability is exponentially small, and work on this event for the two net transfers below.

For a fixed \(I\), \(k=|I|\), and fixed \(v\in\mathbb R^k\) with \(\|v\|=1\), the coordinates of \(A_I^{\mathsf T}v\), conditional on all variance marks, are independent centered Gaussians with variances at least \(1/2\). Their joint density is at most \(\pi^{-N/2}\). The Euclidean ball-volume formula therefore gives \[\mathbb P\bigl(\|A_I^{\mathsf T}v\|\leq2c\sqrt N\bigr) \leq(C_1c)^N.\] Use a \((c/K)\)-net of the unit sphere in \(\mathbb R^k\), of cardinality at most \((3K/c)^k\), taking \(c<K\). If (80) fails while \(\|A\|\leq K\sqrt N\), some net point has image norm at most \(2c\sqrt N\). Summing over dimensions and row sets bounds the failure probability by \[N\,2^{B_0N}(3K/c)^{(1-\rho)N}(C_1c)^N.\] Its logarithm divided by \(N\) is bounded by a constant depending on \((B_0,K)\) plus \(\rho\log c+o(1)\). Choose \(c\) sufficiently small to make this bound negative.

For the gap-count estimate take a net of radius \(\eta=u_0\sqrt\rho/(4K)\). If \(y\) and its net representative \(y_0\) satisfy \(\|y-y_0\|\leq\eta\), then \[\#\{\mu:|A_\mu(y-y_0)|>u_0\} \leq\frac{\|A(y-y_0)\|^2}{u_0^2} \leq\rho N/16.\] If (81) fails, more than \((1+7\rho/16)N\) rows at \(y_0\) therefore satisfy \(|1+A_\mu y_0|\leq2u_0\); in particular at least \(\lceil(1+\rho/4)N\rceil\) do. For fixed \(y_0\), each row projection has density at most \(1/\sqrt\pi\), conditionally on its variance marks and also unconditionally. Hence its probability of this interval is at most \(C_2u_0\). Row independence, a union bound over row subsets, and then a union bound over the spin net give, for \(C_2u_0<1\), \[\mathbb P(\text{gap-count failure},\ \|A\|\leq K\sqrt N) \leq (1+8K/(u_0\sqrt\rho))^N\,2^{B_0N} (C_2u_0)^{(1+\rho/4)N}.\] Its logarithm divided by \(N\) is at most a constant depending on \((B_0,K,\rho)\) plus \((\rho/4)\log u_0\). Choose \(u_0\) sufficiently small. These choices establish all three assertions with exponential error. ◻

Proof. If \(L\) has a nonzero kernel, take a unit kernel vector. Otherwise the set \(Q=\{w\in W:\|Lw\|_\infty\leq1\}\) is compact and has nonempty interior. Let \(w_*\) maximize \(\|w\|\) over \(Q\). The active linear functionals, those satisfying \(|(Lw_*)_j|=1\), span \(W^*\). Indeed, if they had a common nonzero annihilator \(h\), both \(w_*+th\) and \(w_*-th\) would remain in \(Q\) for sufficiently small \(t>0\): active inequalities would be unchanged and the finitely many inactive inequalities have positive slack. But at least one of these two points has squared norm at least \(\|w_*\|^2+t^2\|h\|^2\), a contradiction. There are consequently at least \(D\) distinct active rows, and \[D\leq\|Lw_*\|^2\leq\|L\|^2\|w_*\|^2.\] The vector \(v=w_*/\|w_*\|\) satisfies the required bound. ◻

Proof. We first correct the deficient gaps in the ambient vector space, keeping all uncontrolled rows a fixed distance above zero. We then return the corrected vector to the unit sphere.

Correct the gaps. Choose \(d_*>0\) sufficiently small that, for \(d\leq d_*\), \[ \frac dc\leq\frac12,\qquad L_d:=\frac{16K^2d^2}{c^2u_0^2}\leq\frac\rho4,\qquad K\sqrt{\frac{4d}{c\rho}}\leq\frac{u_0}{2}. \tag{83}\] Write \(h=\mathbf 1+Ay\) and \(r=(-h)_+=r_A(y)\); these vectors remain fixed throughout the construction. For each imposed set \(T\supseteq S\), consider the equations \(A_Sz=r_S\) and \(A_{T\setminus S}z=0\). When they are solvable, let \(z_T\) be their minimum-norm solution. Whenever \(|T|\leq(1-\rho)N\), the surjection bound gives existence and \[ \|z_T\|\leq\frac{\|r_S\|}{c\sqrt N}\leq\frac dc. \tag{84}\] The empty-set case is understood as \(z_\varnothing=0\).

Start at \(T=S\). If \[J=\{j\notin T:|A_jz_T|>u_0/4\}\] is nonempty, add all of \(J\) to \(T\), impose \(A_Jz=0\), and solve again. Here is a count that verifies that every solve is permitted. If \(z'\) is the new solution, then \(z'-z_T\in\ker A_T\) while \(z_T\) belongs to the row span of \(A_T\). Thus \[\|z'\|^2-\|z_T\|^2=\|z'-z_T\|^2 \geq\frac{\|A_Jz_T\|^2}{K^2N} >\frac{|J|u_0^2}{16K^2N}.\] Summing over completed additions and using (84) shows that they contain at most \(L_dN\) rows altogether. The next, not yet imposed, set also has at most \(L_dN\) rows, since \[|J|u_0^2/16<\|A_Jz_T\|^2 \leq K^2N\|z_T\|^2\leq K^2N(d/c)^2.\] Consequently, before every proposed solve, \[|T\cup J|\leq(1-2\rho)N+2L_dN \leq(1-3\rho/2)N<(1-\rho)N.\] This proves the induction without assuming existence of the proposed solution. Each nonterminal step adds at least one row, so the procedure terminates after finitely many steps. At termination write \(T\) for the imposed set and \(z=z_T\). In particular \(|T|\leq(1-\rho)N\) and \(\|z\|\leq d/c\).

Put \(y'=y+z\). On \(S\) its gaps are \(1+A_Sy'=h_S+(-h_S)_+=(h_S)_+\geq0\). On \(T\setminus S\) the correction has zero image and the original gaps exceed \(u_0\). Every remaining row has original gap greater than \(u_0\) and \(|A_jz|\leq u_0/4\), so its new gap is at least \(3u_0/4\). Thus \(y'\) is feasible in the ambient space.

Restore the norm. If \(n=\|y'\|\geq1\), set \(\widehat y=y'/n\). Each gap becomes \[1+A_\mu\widehat y=(1-n^{-1})+n^{-1}(1+A_\mu y')\geq0.\] If \(n<1\), take \(W=\ker A_T\cap(y')^\perp\), whose dimension satisfies \(D\geq N-|T|-1\geq\rho N-1\geq\rho N/2\) for large \(N\). Apply Lemma 28 to the remaining rows restricted to \(W\). There is a unit \(v\in W\) such that \[\|A_{T^c}v\|_\infty\leq K\sqrt{N/D} \leq K\sqrt{2/\rho}.\] Here the infinity norm of an empty vector is zero. Set \(\widehat y=y'+\sqrt{1-n^2}\,v\). Orthogonality gives \(\|\widehat y\|=1\), and imposed gaps are unchanged. Since \(n\geq1-\|z\|\geq1-d/c\), we have \(1-n^2\leq2d/c\). The absolute change of every remaining gap is at most \(K\sqrt{4d/(c\rho)}\leq u_0/2\), leaving it at least \(u_0/4\). This completes the repair. ◻

Proof. Deleting rows serves one precise purpose: it reduces the number of small or negative gaps from at most slightly more than \(N\) to strictly fewer than \(N\). The remaining rows can then be repaired using the surplus dimension in Lemma 29.

The constants are chosen in the following order. First fix \(\eta\in(0,1-\alpha/\alpha')\), so that for all sufficiently large \(N\) the retained row count \(m=\lfloor\alpha N\rfloor\) is at most \((1-\eta)M\), where \(M=\lfloor\alpha'N\rfloor\). Next choose \(0<\rho<\min(1/8,\eta/8)\) and obtain \(K,c,u_0\) from Lemma 27 for the parent matrix. Finally fix \(d>0\) satisfying (83) and \(d^2/u_0^2\leq\rho/2\).

Choose a measurable approximate minimizer \(y\) of the parent matrix, with energy at most \(\min H_A+N^{-1}\). Markov’s inequality and the hypothesis show that \(\|r_A(y)\|\leq d\sqrt N\) with probability tending to one. On the uniform geometric event, the set of problematic parent rows \(S_{\mathrm{par}}=\{\mu:1+A_\mu y\leq u_0\}\) has size at most \((1+\rho)N\). Indeed, the rows with \(|1+A_\mu y|\leq u_0\) number at most \((1+\rho/2)N\), while those with \(1+A_\mu y<-u_0\) number at most \(\|r_A(y)\|^2/u_0^2\leq\rho N/2\).

Independently of the parent matrix, retain a uniformly chosen set \(I\) of exactly \(m\) rows. Conditional on the parent and \(y\), the random variable \(X=|I\cap S_{\mathrm{par}}|\) is hypergeometric, with \[\mathbb E[X\mid A,y]\leq(1-\eta)(1+\rho)N\leq(1-7\rho)N.\] Writing \(X\) as a sum of inclusion indicators gives, with \(k=|S_{\mathrm{par}}|\) and \(r=m/M\), \[\mathop{\mathrm{Var}}(X\mid A,y)=kr(1-r)\frac{M-k}{M-1}\leq\frac{k}{4} \leq\frac{(1+\rho)N}{4}.\] Here the covariance of two distinct inclusion indicators is \(-r(1-r)/(M-1)\); the formula is also immediate when \(k=0\). Chebyshev’s inequality therefore gives \[\mathbb P\{X>(1-2\rho)N\mid A,y\} \leq\frac{1+\rho}{100\rho^2N}.\] Thus \(X\leq(1-2\rho)N\) with probability tending to one, uniformly over the parent on the stated event.

The retained matrix inherits the parent’s operator-norm and surjection bounds, and its residual at \(y\) is no larger than the parent residual. Lemma 29 makes it exactly feasible on the unit sphere. Finally, an independently chosen subset of rows of an iid matrix, listed in increasing order, has precisely the law of an iid \(m\times N\) matrix. This proves the assertion for the desired row count. ◻

Proof. If \(\alpha<\alpha_J\), choose a fixed \(\alpha'\in(\alpha,\alpha_J)\). The expected parent ground energy is \(o(N)\) for both laws by Lemma 25 and pressure universality. Proposition 30 proves satisfiability with probability tending to one.

For \(\alpha_J<\alpha\leq B_0\), the same ground-energy limit is strictly positive. The map \(A\mapsto R(A)\) is \(1\)-Lipschitz in Frobenius norm: for every fixed unit \(y\), \[\bigl|\|r_A(y)\|-\|r_{A'}(y)\|\bigr| \leq\|(A-A')y\|\leq\|A-A'\|_{\mathrm F},\] and taking minima preserves this bound. Gaussian concentration gives \(\mathop{\mathrm{Var}}R(A)\leq C\) for Gaussian entries. For mixture entries use the coordinatewise Gaussian quantile representation from Section 5, whose Lipschitz constant is at most \(2\sqrt{1+\varepsilon}\); the same variance bound follows with an absolute constant for \(\varepsilon\leq1/2\). Since \[\frac{\mathbb ER(A)^2}{N}=2\frac{\mathbb E\min H_A}{N}\longrightarrow2e(\alpha),\] we obtain \(\mathbb ER(A)/\sqrt N\to\sqrt{2e(\alpha)}>0\) by subtracting the bounded variance. Chebyshev’s inequality now implies \[\mathbb P(\mathrm{SAT})=\mathbb P(R(A)=0) \leq\frac{\mathop{\mathrm{Var}}R(A)}{(\mathbb ER(A))^2}\longrightarrow0.\] For larger densities, retain the first \(\lfloor\alpha_*N\rfloor\) rows at any fixed \(\alpha_*\in(\alpha_J,B_0)\) and use monotonicity of satisfiability. ◻

Proof. The fixed-temperature row-marginal result in Section 5 gives equality of the inner \(N\to\infty\) limits for every fixed \(\beta\), argument, and \(\alpha\in(\alpha_J,B_0]\). Limits of these identical functions, taken successively in the prescribed order, have identical existence and values. This assertion makes no exchange of limits; existence of the Gaussian zero-temperature and jamming limits is established in the following sections. ◻

Proof. The entropy \(E(q)\) is nonnegative. Since \(P(-\beta V)/\beta\to-e\), at a minimizer \(\alpha g_\beta(0,0)\le-e/2\) for large \(\beta\). Jensen’s inequality in the row recursion gives \(g_\beta(0,z)\ge-\mathbb EV(z+G)\), where \(G\sim\mathcal N(0,1)\). Choose \(R_e>0\) so that \(\mathbb EV(R_e+G)<e/(4\alpha)\). Concavity then implies \(g_{\beta,z}(0,0)\ge e/(4\alpha R_e)\). After dividing (66) by the appropriate powers of \(\beta\), \[\alpha g_{\beta,z}(0,0)^2\bar\chi_\beta(0)^2\le1,\] which proves the upper bound for \(\bar\chi_\beta(0)\). Conversely \(E(q_\beta)\le C\beta\), since the pressure is nonpositive and \(\Phi_{-\beta V}\ge-\beta\mathbb EV(G)\). The integral version of (32), restricted to \([0,1/2]\), gives \[E(q_\beta)\ge\frac1{4\chi_{q_\beta}(0)}-\frac{\log2}{2},\] and hence the lower bound.

We shall also use the following consequences of the spin inverse relations: \[ \frac1{1-q_{\beta,*}} =\frac1{\chi_{q_\beta}(0)}+\int_0^1 s\,\mathrm dp_{q_\beta}(s), \qquad q_{\beta,*}\le\chi_{q_\beta}(0)^2p_{q_\beta}(1-). \tag{86}\] These follow directly from the continuous inverse relation (25). For brevity write \(q=q_\beta\), \(p=p_q\), and \(\chi=\chi_q\). Stieltjes integration by parts and Fubini’s theorem give \[p(1-)=\int_0^{q_*}\frac{\mathrm dt}{\chi(t)^2},\qquad \int_0^1s\,\mathrm dp(s) =p(1-)-\int_0^1p(s)\,\mathrm ds =\int_0^{q_*}\frac{m_q(t)}{\chi(t)^2}\,\mathrm dt.\] Since \(\chi'=-m_q\) and \(\chi(q_*)=1-q_*\), the last integral is \((1-q_*)^{-1}-\chi(0)^{-1}\). Also \(\chi(t)\le\chi(0)\) in the first integral, so \(p(1-)\ge q_*/\chi(0)^2\). This proves both relations, including general profiles and their top limits.

Let \(Z_t\) solve \(\mathrm dZ_t=m_{q_\beta}(t)f_{\beta,z}(t,Z_t)\,\mathrm dt+\mathrm dB_t\), \(Z_0=0\), and put \(\mathcal{P}_\beta(t)=\alpha\mathbb Ef_{\beta,z}(t,Z_t)^2\). The derivative martingale and Itô’s isometry imply \(\mathcal{P}_\beta'(t)=\alpha\mathbb Ef_{\beta,zz}(t,Z_t)^2\le\alpha\beta^2\), and \(p_{q_\beta}(s)=\mathcal{P}_\beta(q_\beta(s))\). The Stieltjes chain rule, including jumps by integration over the intervening variance interval, yields \[\int s\,\mathrm dp_{q_\beta}(s) \le\alpha\beta^2\int s\,\mathrm dq_\beta(s) \le\alpha\beta^2\chi_{q_\beta}(0).\] Thus the first relation in (86) gives \[\frac1{\beta(1-q_{\beta,*})} \le\frac1{\bar\chi_\beta(0)}+\alpha\bar\chi_\beta(0).\] The reverse endpoint bound follows from \(1-q_{\beta,*}\le\chi_{q_\beta}(0)\).

Finally, bounded mass and monotonicity give subsequential weak compactness of \(\gamma_\beta\,\mathrm dt\). On every compact interval before 1, monotonicity bounds the densities uniformly and gives \(L^1\) compactness. Any singular mass is therefore at 1. The terminal interval has mass \(\beta(1-q_{\beta,*})\ge c_e\), so the endpoint atom has mass at least \(c_e\). ◻

Proof. We give the endpoint comparison explicitly. For \(t<1\), set \[U_\beta(t,z)=\sup_{w\in\mathbb R} \left\{-V(w)-\frac{(z-w)^2}{2\bar\chi_\beta(t)}\right\} =-\frac{V(z)}{1+\bar\chi_\beta(t)}.\] At \(t=1\) use the continuous extension \(U_\beta(1,z)=-V(z)\). This solves \(U_t=-\gamma_\beta U_z^2/2\) and has curvature in \([-1,0]\). Subtraction from the viscous equation for \(g_\beta\) produces a linear equation with drift \(\gamma_\beta(g_{\beta,z}+U_{\beta,z})/2\) and a source of absolute value at most \(1/2\). Its diffusion representation, or comparison after smoothing the hinge, gives \[ \sup_z|g_\beta(t,z)-U_\beta(t,z)|\le(1-t)/2, \qquad \sup_z|g_{\beta,z}(t,z)-U_{\beta,z}(t,z)|\le C\sqrt{1-t}. \tag{88}\] For the gradient bound use the value bound and the common Lipschitz bound on the two gradients, comparing difference quotients with step \(\sqrt{1-t}\). The identical comparison holds for \(g\), with \(\bar\chi\) in place of \(\bar\chi_\beta\).

On \([0,T]\), subtraction of two viscous equations again gives a linear equation. Its source is bounded by the coefficient difference times \(C(1+z^2)\), and its drift has a spatial Lipschitz constant bounded by an integrable function with uniformly bounded integral. The associated diffusions have uniform moments of every fixed order, by Gronwall’s inequality. The diffusion representation therefore proves stability under \(L^1\) coefficient convergence and local uniform terminal convergence with a common quadratic growth bound. Applying this at \(T<1\), using (88), and then sending \(T\uparrow1\), proves local uniform convergence of values and gradients on compact time intervals before 1. The same argument constructs, and proves uniqueness of, the stated solution for an integrable \(\gamma\).

For the entropy, (32) gives \[\frac{E(q_\beta)}\beta =\frac12\int_0^{q_{\beta,*}}\frac{\mathrm dt}{\bar\chi_\beta(t)} +\frac{\log(1-q_{\beta,*})}{2\beta}.\] On this integration interval the denominator is at least \(\beta(1-q_{\beta,*})\ge c_e\). Dominated convergence applies, and the logarithmic term is \(O_e(\log\beta/\beta)\). This proves (87).

For an arbitrary trial pair, use the finite-temperature CDF \(m_\beta(t)=\min\{\gamma(t)/\beta,1\}\) until \(1-\Delta/\beta\), and set \(m_\beta=1\) thereafter. This is nondecreasing and its scaled measure converges to \(\gamma\,\mathrm dt+\Delta\delta_1\); integrability of \(\gamma\) ensures that its mass on the shrinking terminal interval vanishes. The comparisons just proved show convergence of its row term and entropy. The finite-temperature minimum is at most this trial value, so \(-e\le\mathcal E_\alpha(\gamma,\Delta)\). ◻

Proof. On each constant-CDF interval the row tilt is the Gaussian Doob transform. Composing these transforms, then approximating a general CDF, identifies its physical gap with \(1+Z_1^\beta\), where \[Z_t^\beta=B_t+\int_0^t \gamma_\beta(v)g_{\beta,z}(v,Z_v^\beta)\,\mathrm dv.\] Couple \(Z^\beta\) and \(Z\) with the same Brownian motion. Before a fixed time \(T<1\), the coefficient and gradient convergence above and Gronwall’s inequality imply convergence in expected supremum norm. On \([T,1]\), replace the gradient by \(U_{\beta,z}\). The error has expected size at most \(C_e\sqrt{1-T}\), by (88) and the uniform bound on \(\int\gamma_\beta\); the Brownian increment has the same bound. The resulting deterministic flow fixes positive gaps and multiplies a negative gap between times \(T\) and \(t\) by \((1+\bar\chi_\beta(t))/(1+\bar\chi_\beta(T))\). First send \(\beta\to\infty\), and then \(T\uparrow1\). The limiting integrable drift on \([T,1]\) has vanishing total mass, whereas the total mass concentrating near time \(1\) in \(\gamma_\beta(t)\,\mathrm dt\) is \(\Delta\). Since \(\bar\chi_\beta(1)=0\) and \(\bar\chi_\beta(T)\to\bar\chi(T)\), the displayed flow factor tends to \((1+\Delta)^{-1}\) as \(\beta\to\infty\) and then \(T\uparrow1\). This proves (89). Since the drift is nonnegative, the negative part of the gap is bounded by \((1+B_1)_-\); this gives uniform integrability of every moment of the negative part.

For atomlessness, condition on the Brownian bridge \(\widetilde B_t=B_t-tB_1\). The map from \(b=B_1\) to \(Z_1\) is increasing and has slope between \(\exp(-\int\gamma)\) and 1. Indeed its variation satisfies \(J'(t)=1+\gamma(t)g_{zz}(t,Z_t)J(t)\), \(J(0)=0\). Approximation proves the same bounds when derivatives exist only almost everywhere. The conditional endpoint law is thus the pushforward of a Gaussian under a globally bi-Lipschitz map. That map is onto \(\mathbb R\), so the law has a bounded density and assigns positive probability to every nonempty open interval. The piecewise linear map in (89) preserves these properties. ◻

Proof. We perturb the loss in a compact negative-gap interval. For either sign of the perturbation, the pressure trial gives a lower support for the ground energy. Evaluating the perturbed energy at an unperturbed minimizer gives the opposite inequality, and the two one-sided slopes identify its empirical test average. The perturbation interval must remain fixed as \(\beta\to\infty\).

Let \(\psi\) be smooth with support in a compact negative-gap interval \([-R,-a]\), \(a>0\), and put \(W_w(z)=V(z)+w\psi(1+z)\). Choose \(w_0>0\) so that \[w_0\|\psi''\|_\infty\le\tfrac12,\qquad w_0\|\psi'\|_\infty\le a/2,\qquad w_0\|\psi\|_\infty\le a^2/4.\] For \(|w|\le w_0\), \(W_w\) is convex, nonnegative, and nonincreasing, and agrees with \(V\) outside that compact interval. In particular \(-\beta W_w\) is concave for every \(\beta>0\). This is a fixed interval of loss perturbations, independent of \(\beta\).

Here only a trial upper bound is needed. The Gaussian interpolation with smooth concave tangent caps, followed by the spin conjugate inequality (Corollary 21), gives for every \(q_*<1\) \[ \limsup_{N\to\infty}P_N(-\beta W_w) \le\alpha\Phi_{-\beta W_w}(q)+E(q). \tag{91}\] This uses Corollary 15, namely ordinary profile concavity for a concave terminal, not the small-terminal-perturbation radius in (67). The tangent caps have the required concavity, and removal of the cap is valid at each fixed \(\beta,w\), because \(W_w\) has the same quadratic tail as \(V\). Neither strict profile concavity nor identification of a perturbed minimizer is required for (91).

For a base limiting pair \((\gamma,\Delta)\), let \(g_w\) solve the PDE with coefficient \(\gamma\) and terminal value \[g_w(1,z)=\sup_{v\in\mathbb R} \left\{-W_w(v)-\frac{(z-v)^2}{2\Delta}\right\}, \qquad S(w)=-\alpha g_w(0,0)-\frac12\int_0^1\frac{\mathrm dt}{\bar\chi(t)}.\] Write \(\widehat e_N(w)=N^{-1}\min_{\|y\|=1}\sum_iW_w(A_i y)\) and \(e_N(w)=\mathbb E\widehat e_N(w)\). The surface-cap pressure–energy estimate applies to \(W_w\), whose curvature and growth bounds are uniform for \(|w|\le w_0\). Use the recovery profiles of Proposition 34 in (91), take \(N\to\infty\), and then \(\beta\to\infty\). For completeness, the endpoint comparison for this nonquadratic loss uses \[U_{\beta,w}(t,z)=\sup_v\left\{-W_w(v) -\frac{(z-v)^2}{2\bar\chi_\beta(t)}\right\},\qquad t<1,\] with continuous terminal value \(U_{\beta,w}(1,z)=-W_w(z)\). Both \(U_{\beta,w}\) and the scaled viscous solution \(g_{\beta,w}\) have curvature in \([-C,0]\), uniformly for \(|w|\le w_0\), and \(U_{\beta,w,t}=-\gamma_\beta U_{\beta,w,z}^2/2\). The difference equation has forcing bounded by \(C/2\), so \[\sup_z|g_{\beta,w}(t,z)-U_{\beta,w}(t,z)|\le C(1-t),\qquad \sup_z|g_{\beta,w,z}(t,z)-U_{\beta,w,z}(t,z)|\le C\sqrt{1-t}.\] Recovery has \(\int\gamma_\beta=L+o(1)\); hence its drift Lipschitz constants have bounded integrals and its diffusions have uniform moments. Compact-time \(L^1\) coefficient stability, followed by \(t\uparrow1\), gives exactly the terminal supremum defining \(g_w\), and the entropy converges as before. We obtain \[ \liminf_{N\to\infty}e_N(w)\ge S(w),\qquad S(0)=e. \tag{92}\] No existence of a perturbed thermodynamic ground-energy limit is being asserted or used.

The function \(S\) is differentiable at zero. The quadratic penalization in the endpoint supremum is strictly concave in \(v\), so the optimizer is unique; its envelope derivative is minus \(\psi\) at the proximal gap. Linearization of the PDE propagates this bounded terminal derivative along the tilted diffusion. The coefficient stability follows from the difference equation used above. Hence \[ S'(0)=\alpha\mathbb E\psi(h^*). \tag{93}\]

For every base ground minimizer \(y_N\), let \(T_N=N^{-1}\sum_i\psi(1+A_i y_N)\). Direct evaluation gives \[\widehat e_N(w)\le\widehat e_N(0)+wT_N.\] For each fixed \(w\), the centered ground energy \(\widehat e_N(w)-e_N(w)\) tends to zero in probability. Indeed on \(\|A\|\le K\sqrt N\) its matrix Lipschitz constant is \(C_w/\sqrt N\), uniformly over spins; extend Lipschitz and use Gaussian concentration, and then control the complement by the quadratic growth bound. Thus (92) and the known convergence \(\widehat e_N(0)\to e\) imply, for fixed \(w>0\) and \(v<0\), uniformly over all base minimizers, \[\frac{S(w)-e}{w}-o_{\mathbb P}(1) \le T_N\le \frac{S(v)-e}{v}+o_{\mathbb P}(1).\] Sending \(w\downarrow0\) and \(v\uparrow0\) proves \(T_N\to S'(0)\) in probability. Apply this argument to any two base limiting pairs: the same empirical tests have the two asserted limits, so their derivatives agree. Smooth compact negative tests determine the restriction of a measure to \((-\infty,0)\), proving the asserted uniqueness and, at the same time, microscopic negative-test convergence.

For the energy identity use \(W_w=(1+w)V\), \(|w|<1/2\). The same trial-support argument applies, while now \(\widehat e_N(w)=(1+w)\widehat e_N(0)\) identically. The derivative squeeze therefore gives \(e=S'(0)=\alpha\mathbb EV(h^*-1)\), proving (90). Quadratic growth in this differentiation is justified by the uniform diffusion moments. ◻

Proof. Let \(Z^\beta\) denote the finite-temperature tilted diffusion. At finite temperature vary the overlap distribution, with CDF \(m\), by moving mass to a point below 1. Linearization of the PDE and (32) gives, for variations supported below a cutoff \(T<1\), \[\delta(\alpha\Phi+E)=\frac12\int_0^T\delta m(t)J(t)\,\mathrm dt, \qquad J(t)=\mathcal{P}_\beta(t)-\int_0^t\frac{\mathrm dv}{\chi_{q_\beta}(v)^2}.\] Let \(\nu\) be the minimizing overlap distribution, choose \(q_{\beta,*}<T<1\), and fix \(s\in[0,T)\). The feasible direction \(\nu_\varepsilon=(1-\varepsilon)\nu+\varepsilon\delta_s\) has \(m_\varepsilon(t)-m(t)=\varepsilon(\mathbf 1_{\{s\le t\}}-m(t))\) for \(t\le T\), and vanishes for \(t>T\). With \(K(s)=\int_s^T J(t)\,\mathrm dt\), Fubini’s theorem makes its directional derivative \[\frac12\left(K(s)-\int K(r)\,\nu(\mathrm dr)\right)\ge0.\] Therefore \(K\) is at least its \(\nu\)-average everywhere on \([0,T)\), and equals that average on the support, by continuity. Thus every support point in \((0,T)\) is a local minimum of \(K\). Since \(K'=-J\) and \(K''=-J'\), at each such point \[\alpha\mathbb Ef_{\beta,zz}(s,Z^\beta_s)^2\chi_{q_\beta}(s)^2\le1.\] The derivatives \(J'\) and \(K''\) are continuous before time 1, including across jumps of \(m\). Indeed the mild heat equation and the local spatial Lipschitz bound on \(f_{\beta,z}^2\) give joint continuity of \(f_{\beta,zz}\) away from the terminal time; the second derivative of the heat kernel applied to a Lipschitz function has an integrable time singularity. The diffusion moment formula then gives continuity of \(\mathcal{P}_\beta'\), and explicitly \[J'(s)=\alpha\mathbb Ef_{\beta,zz}(s,Z_s^\beta)^2 -\chi_{q_\beta}(s)^{-2}.\] No continuity of \(f_{\beta,t}\) at a jump of \(m\) is asserted.

Select continuity times \(t_n\uparrow1\) with \((1-t_n)\gamma(t_n)\to0\); this follows from integrability and monotonicity. A diagonal choice \(t_\beta\uparrow1\) ensures \(t_\beta<q_{\beta,*}\), \(\bar\chi_\beta(t_\beta)\to\Delta\), and \((1-t_\beta)\gamma_\beta(t_\beta)\to0\). Let \(s_\beta\) be the next overlap-support point at or above \(t_\beta\). The CDF is constant between these points, so \[\bar\chi_\beta(t_\beta)-\bar\chi_\beta(s_\beta) =\gamma_\beta(t_\beta)(s_\beta-t_\beta) \le(1-t_\beta)\gamma_\beta(t_\beta)\longrightarrow0.\] Consequently \(\bar\chi_\beta(s_\beta)\to\Delta\). Moreover, for fixed \(T<1\), \[\int_T^{s_\beta}\gamma_\beta(t)\,\mathrm dt \longrightarrow\int_T^1\gamma(t)\,\mathrm dt.\] The drift over this interval is negligible after \(T\uparrow1\), so \(Z^\beta_{s_\beta}\) converges to the pre-atom endpoint \(Z_1\).

By (88), \(g_{\beta,z}(s_\beta,z)\to(-1-z)_+/(1+\Delta)\) locally uniformly. These gradients are convex in \(z\): differentiating the PDE three times gives a linear equation preserving the nonnegative third spatial derivative of a smoothed hinge terminal, and passage to the limit preserves convexity. Secant bounds for convex functions therefore give locally uniform convergence of their derivatives on each compact interval avoiding \(z=-1\): the limit derivative is continuous there, and a finite secant mesh gives a uniform bound. Since these derivatives are bounded and \(Z_1\) is atomless, the finite-temperature stability inequality passes to \[\alpha\frac{\Delta^2}{(1+\Delta)^2}\mathbb P(Z_1<-1)\le1,\] as asserted. ◻

Proof. For a fixed \(C\), Gaussian genericity gives simple nonzero singular values of \(A_C\) and a nonzero component of \(A_C^T\mathbf1\) in each corresponding right singular direction. These properties hold almost surely, since their exceptional sets are proper algebraic sets. Stationarity implies the first identity in (95); its scalar product with \(y\) gives \(\lambda>0\). The genericity just noted precludes \(\lambda\) from being an eigenvalue of \(A_C^TA_C\). Thus the stationary vector is obtained by inversion. Its unit-norm equation is rational in \(\lambda\), is not identically satisfied (it tends to zero at infinity), and has finitely many roots. Conditional on \(A_C\), any row outside \(C\) almost surely avoids gap zero at all these finitely many vectors. A union over the finitely many sets \(C\) proves the first assertion, even if such a row was initially allowed to have zero gap. The energy is now smooth near the minimizer, so its constrained Hessian is nonnegative on the tangent space. Finally stationarity with \(\lambda>0\) puts \(y\) in the row span of \(A_C\). A nontrivial kernel of \(A_C\) would be tangent and would have Hessian value \(-\lambda\), a contradiction. Hence \(\operatorname{rank} A_C=N\), and \(k\ge N\). ◻

Proof. Set \(t=\sqrt e\). Since \(H_{\min}/N\to e\), with probability tending to one, \[\frac{3e}{4}\le\frac{H_{\min}}N\le\frac{5e}{4},\qquad t\sqrt N\le\|r\|_2\le2t\sqrt N.\] Thus a single residual annulus suffices at every fixed density. Moreover \(t\le\sqrt{B_0}\), since \(e(\alpha)\le\alpha\le B_0\). All exponential bases in the counting estimates can consequently be chosen independently of the density; the net mesh may still depend on its fixed value of \(t\). If \(\|r\|_1\le c_1tN\), transfer to a net of sufficiently small mesh so that the same active set \(C\) has \(\sum_{i\in C}|1+A_i y_0|\le2c_1tN\). The joint projection density is bounded by an absolute constant to the power \(k\), and the \(k\)-dimensional cross-polytope volume is \(2^k(2c_1tN)^k/k!\). Since \(k\ge N\), the projection probability is at most \((Cc_1t)^k\). Combining with (97) and the net gives \(a^{-1}C^N c_1^k t^{k-N+1}\), which is exponentially small for a uniformly small \(c_1\), because \(t\le\sqrt{B_0}\) and \(N\le k\le B_0N\). Thus \(\|r\|_1\ge c_1tN\); the previously established first-moment convergence proves the bound for \(\mu\).

Suppose at least \(\eta N\) active residuals are at most \(st\), where \(0<\eta<1/2\). Choose \(a\ll st\sqrt\eta/K\). By the squared projection displacement bound, at least \(j=\lfloor\eta N/2\rfloor\) corresponding net gaps have absolute value at most \(2st\). The remaining coordinates of \(C\) lie in a Euclidean ball of radius \(Ct\sqrt N\). Its volume, together with the \(j\) intervals, bounds the projection probability by \(C^N t^k s^j\); here \(k-j\ge N/2\) bounds the ball’s dimension factor. Equation (98) excludes this event for \(s\le\exp(-C'/\eta)\), with \(C'\) uniform. At fixed \(e>0\), this estimate and convergence of compact negative tests show that no extensive active mass is lost at zero, proving \(k/N\to\alpha\mathbb P(h^*<0)=z_c\).

The mean residual among active rows is at most \(\|r\|_2/\sqrt k\le2t\). A normalized force at most \(s\) therefore has residual at most \(Cst\). The preceding estimate, divided by \(k\ge N\), gives \(F_{\alpha,\infty}(s)\le C/|\log s|\). The mean-one laws are tight by Markov’s inequality; in fact their second moments are uniformly bounded, since \(k\|r\|_2^2/\|r\|_1^2\le C\). Their weak limits therefore retain mean one. The displayed logarithmic bound prevents any subsequential limit from putting mass at zero.

For the contact density, use the uniform all-spin near-gap estimate in Lemma 27. For each fixed \(\rho>0\), it supplies \(u_0>0\) such that every ground minimizer satisfies \[k\le(1+\rho)N+2H_{\min}/u_0^2.\] Take \(N\to\infty\), then \(\alpha\downarrow\alpha_J\), and finally \(\rho\downarrow0\). This gives \(\limsup z_c\le1\), while Lemma 38 gives \(z_c\ge1\). ◻

Proof. Fix \(\varepsilon>0\). For each aspect ratio in \([1,B_0]\), choose a nonnegative Lipschitz function of the singular value, supported strictly below \(\sqrt z-1+\varepsilon\), with positive integral under the limiting singular-value law. By continuity of the spectral edges and density, each fixed test stays below the cutoff and retains a positive limiting integral throughout the closure of a sufficiently small ratio neighborhood. A finite collection of these neighborhoods covers the compact ratio interval, giving a common positive lower bound for the relevant limiting integrals. Convergence of their expectations is uniform within the chosen neighborhoods: otherwise a violating sequence of ratios would have a convergent subsequence and contradict the stated spectral limit. The Frobenius singular-value inequality gives \[A\longmapsto\frac1N\sum_{i=1}^N \varphi\bigl(s_i(A/\sqrt N)\bigr) \quad\hbox{Lipschitz constant at most }\mathop{\mathrm{Lip}}(\varphi)/N.\] Gaussian concentration therefore bounds a fixed positive downward deviation by \(2\exp(-c_\varepsilon N^2)\). Union over at most \(2^{B_0N}\) subsets still has probability tending to zero. A positive normalized test average implies a positive multiple of \(N\) singular values below the cut, hence at least two for large \(N\).

The span of two corresponding right singular vectors intersects \(y^\perp\). The Hessian inequality (95) therefore yields \[\frac\lambda N\le(\sqrt{k/N}-1+\varepsilon)^2.\] Since \(\lambda/N=2H_{\min}/N+\|r\|_1/N\), first take \(N\to\infty\), then \(\varepsilon\downarrow0\), and use the contact and moment convergence. The upper bound \(\Delta^{-2}\) follows from (94) and \(z_c\ge1\). The estimate for \(\mu\) in Lemma 39 then gives \(\Delta\le Ce^{-1/4}\).

The second inequality in (86), together with \(p_{q_\beta}(1-)\le\alpha\mathbb Ef_{\beta,z}(1,Z^\beta_1)^2\), passes to \(1\le L^2\alpha\mathbb E(h^*)_-^2=2eL^2\). Finally the derivative martingale identifies \(g_{\beta,z}(0,0)\to\mu\), so (66) implies \(\alpha\mu^2L^2\le1\). The lower bound for \(\mu\) proves the upper bound for \(L\). Dividing the two estimates proves the final assertion. This conclusion concerns the thermal endpoint atom; it does not preclude additional layers concentrating at the endpoint in the subsequent density limit. ◻

Proof. The bounds \(L\ge(2e)^{-1/2}\), \(\alpha\mu\ge c\sqrt e\), and \(\alpha\mu^2L^2\le1\) give the two bounds for \(\mathbb ER=L\mu\). Moreover \(p=z_c/\alpha\) is bounded away from zero because \(z_c\ge1\) and \(\alpha\le B_0\). Thus the conditional mean \(\mathbb E[R\mid R>0]=L\mu/p\) is bounded above and below. Apply Lemma 39 to the corresponding mean-one force to obtain the displayed small-value bound. The final assertion is \(z_c\to1\). ◻

Proof. Remove a tested row. Its insertion denominator is at most one, whereas its penalty vanishes on the tested positive interval. Conditional on the cavity spin, the removed projection is standard Gaussian. Its density is bounded below on \([-1,0]\), which proves the lower bound. The same proof works for the small mixtures, conditional on their variances, with a uniform lower density bound.

For the upper bound, exact ground minimizers alone are insufficient: the observable first takes \(N\to\infty\) at positive temperature. We therefore prove the counting bound throughout a thin energy sublevel and only then concentrate the Gibbs measure on it. The needed replacement for the exact contact count \(k\ge N\) is a lower bound \(k\ge(1-2\eta')N\) on that whole sublevel. We derive it before counting the positive gaps.

Fix the UNSAT density and a tolerance \(\rho>0\). On the good operator-norm event the surface-cap argument implies that, for all sufficiently large \(\beta\), Gibbs mass outside \[\mathcal L_\rho=\{y:H(y)\le H_{\min}+\rho N\}\] is exponentially small in \(N\). Along any great circle the energy derivative has Lipschitz constant \(CN\). Descending in the negative tangent-gradient direction therefore gives \[ \|\nabla_S H(y)\|\le C\sqrt\rho\,N \qquad(y\in\mathcal L_\rho). \tag{101}\] Indeed a derivative of magnitude \(d\) permits an energy decrease at least \(c d^2/N\), contradicting near-minimality if \(d>C\sqrt\rho N\).

Set \(t=\sqrt e\) and require \(\rho\le e/4\). On the event \(H_{\min}/N\in[3e/4,5e/4]\), every point of \(\mathcal L_\rho\) then satisfies \(t\sqrt N\le\|r(y)\|_2\le2t\sqrt N\). Choose from the outset \[0<\eta'\le\min\left\{\frac14, \frac1{2(1+|\log t|)}\right\}.\] At a sufficiently small fixed cutoff \(v=v(e,\eta')>0\), and then sufficiently small \(\rho>0\), no point of \(\mathcal L_\rho\) has \(\eta'N\) absolute gaps at most \(v\). To see this, use the net and conditional-gradient argument with projection factor \((Cv)^{\lfloor\eta'N/2\rfloor}\) and residual lower bound \(t\sqrt N\). Its union probability is at most \(a^{-1}C^N t^{-(N-1)}v^{\lfloor\eta'N/2\rfloor}\). Choose \(v\) small, then \(a\ll\min\{t,v\sqrt{\eta'}\}\), then \(\sqrt\rho\ll a\). This is exponentially small by (101).

Every point of \(\mathcal L_\rho\) must consequently have \(k\ge(1-2\eta')N\) active rows, after decreasing \(\rho\) if necessary. Otherwise impose zero image on all active and all absolute-near-zero rows and intersect their kernel with \(y^\perp\). This subspace has dimension at least \(\eta'N-1\). Lemma 28 gives a unit vector \(v_0\) there with \(\|Av_0\|_\infty\le C/\sqrt{\eta'}\). For small fixed \(b>0\), depending on \(v\) and \(\eta'\), set \(y_b=\sqrt{1-b^2}y+bv_0\). With \(c_b=\sqrt{1-b^2}\), \[h_i(y_b)=c_b h_i(y)+(1-c_b)+bA_i v_0.\] Choose \(b\) so small that \(c_b\ge1/2\) and \(bC/\sqrt{\eta'}<v/4\). Positive gaps above \(v\) then remain positive. All rows with \(|h_i(y)|\le v\) have \(A_i v_0=0\), so their nonnegative gaps remain nonnegative. The same vanishing projection on every active row gives \(r_i(y_b)\le c_b r_i(y)\). Therefore \(H(y_b)\le(1-b^2)H(y)\), an improvement of at least \(b^2 eN/2\) with high probability. This contradicts membership in \(\mathcal L_\rho\) when \(\rho<b^2e/2\).

Now suppose that \(\eta N\) positive gaps are at most \(u\). Transfer to a sufficiently fine net, retaining at least \(j=\lfloor\eta N/2\rfloor\) of these rows with absolute gap at most \(2u\); these row indices are disjoint from the active set of the original point. The active coordinates at the net point lie in a Euclidean ball of radius \(Ct\sqrt N\), with the same \(t=\sqrt e\) fixed above. Since \(k\ge N/2\), the joint projection probability is at most \(C^N t^k u^j\). The net estimate (98) still applies, by choosing \(\rho\) after the mesh. Since \(t=\sqrt e\le\sqrt{B_0}\), \(k\ge(1-2\eta')N\), and \(2\eta'|\log t|\le1\), \[t^{k-N}\le \max\left\{\max(1,\sqrt{B_0})^{B_0N}, \exp\bigl(2\eta'N|\log t|\bigr)\right\} \le C^N,\] with a uniform \(C\), even as \(e\downarrow0\). The union probability is consequently bounded by \(a^{-1}C^N u^{\lfloor\eta N/2\rfloor}\), and tends to zero for \(u\le\exp(-C'/\eta)\), where \(C'\) is uniform in the density.

The sequence of choices is: density and \(\eta\); then \(\eta'\), \(v\), and the corresponding descent step; then the uniform exponential cutoff \(u\); then the net mesh and \(\rho\); finally \(\beta\) large enough for Gibbs concentration, and \(N\to\infty\). Thus no interchange of the temperature and size limits is used. The row count bound \(\eta N\), divided by \(M\), is at most a uniform multiple of \(\eta\), since the densities stay bounded away from zero. Taking \(\eta=C/|\log u|\) proves (100), since the counting event included all positive gaps in \((0,u]\). At fixed density, the finite-temperature row marginal and Proposition 35 converge along the chosen subsequence at both endpoints 0 and \(u\), which are atomless. Thus the bound passes directly to every subsequential law of \(h^*\), without removing a positive lower cutoff or identifying positive tests at exact ground minimizers. ◻

Proof. The denominator is positive. The diffusion representation writes \(g(0,0)\) as a negative terminal expectation minus \(\frac12\mathbb E\int\gamma g_z^2\); atomlessness with a positive density on negative intervals makes the terminal expectation strictly negative. Recovery trials in Proposition 34 remain valid when \(e=0\). In that regime all trial functionals are nonnegative, so \(\alpha\) is at most every ratio in (102). For \(e>0\), an attained negative functional gives a ratio strictly below \(\alpha\). Approaching the threshold from the two sides proves the formula.

To verify the normalization assertion, scale a fixed pair by \(c>0\). If \(g_c\) is the solution for \((c\gamma,c\Delta)\), then \(c g_c\) solves the equation with coefficient \(\gamma\) and terminal value \(-V/(\Delta+1/c)\). The ratio for the scaled pair is therefore the ratio for this rescaled equation. It decreases as \(c\uparrow\infty\) and converges to the ratio with terminal \(-V/\Delta\), by the PDE comparison and stability already proved. Every shape can be normalized to \(\bar\chi(0)=1\), and every normalized limiting ratio is approached by its finite scalings. The two infima are equal. ◻

Proof. Use gap coordinates \(h=1+z\) and write \(f(t,h)=g(t,h-1)\) and \(a=f_h\). The zero-temperature equation and forward diffusion are \[f_t+\tfrac12f_{hh}+\tfrac12\gamma f_h^2=0, \quad f(1,h)=-\frac{h_-^2}{2(1+\Delta)}, \qquad H_t=1+B_t+\int_0^t\gamma(v)a(v,H_v)\,\mathrm dv,\] where \(h_-=\max\{-h,0\}\). The coefficient is time integrable; \(a\) has linear growth and an integrable Lipschitz constant after multiplication by \(\gamma\). The diffusion therefore has finite second moments. Itô’s formula, first on \([0,T]\) and then with \(T\uparrow1\), gives \[f(0,1)=\mathbb E\left[-\frac{(H_1)_-^2}{2(1+\Delta)} -\frac12\int_0^1\gamma(t)a(t,H_t)^2\,\mathrm dt\right].\] For every nonnegative \(R\), the pointwise quadratic dual inequality is \[-\frac{h_-^2}{2(1+\Delta)} \le Rh+\frac{1+\Delta}{2}R^2.\] Insert \(h=H_1\), condition \(R\) against \(\mathcal F_t\) in the drift term, and use \(r_ta-a^2/2\le r_t^2/2\). This proves \[f(0,1)\le \mathbb E\left[R(1+B_1)+\frac{1+\Delta}{2}R^2\right] +\frac12\int_0^1\gamma(t)\mathbb Er_t^2\,\mathrm dt.\] For \(R=a(1,H_1)=(H_1)_-/(1+\Delta)\) the differentiated equation makes \(a(t,H_t)\) a square-integrable martingale with terminal value \(R\). Consequently \(r_t=a(t,H_t)\), and both inequalities are equalities. Equation (104) proves the claimed formula. Strict convexity of \(\mathbb ER^2/2\) proves uniqueness for fixed \(s\). All uses of Itô’s formula can be justified with smooth terminal data and bounded smooth coefficients; the curvature and growth bounds above dominate the terms when those approximations are removed. ◻

Proof. For \(0<q<1\), the reciprocal of \(qs_1+(1-q)s_2\) is the weighted harmonic mean \(\mathsf H(x,y)=(q/x+(1-q)/y)^{-1}\) evaluated at \((\chi_1,\chi_2)\). This function is increasing in each coordinate and concave on the positive quadrant: differentiation gives a negative semidefinite Hessian. Hence \(\mathsf H(\chi_1(t),\chi_2(t))\) is concave and nonincreasing in \(t\). It extends continuously to the endpoint, and its negative derivative is nonnegative, nondecreasing, and integrable. Integrability of the new \(s\) is immediate from its definition as a convex combination. The same argument preserves \(L^2\) integrability and the normalization.

For a real Hilbert space \(\mathcal H\), the perspective \((x,u)\mapsto\|x\|_{\mathcal H}^2/u\), \(u>0\), is convex. Its convexity defect vanishes exactly when \(x_1/u_1=x_2/u_2\). Apply this once with \(\mathcal H=\mathbb R\) to \(c\), and for almost every \(t\) with \(\mathcal H=L^2(\Omega)\) to \(b_t\). Integration preserves nonnegative defects, so equality in the sum implies the asserted almost-everywhere equalities. The remaining terms are linear, except for \(\alpha\lambda\mathbb ER^2/2\), whose strict convexity gives the last assertion. ◻

Proof. Two minimizing pairs have the same \(R\) and therefore the same \(c\) and \(b\). For a PDE optimizer, \(b_t=f_{hh}(t,H_t)<0\) almost surely for \(t<1\). To see the strict inequality, the curvature is nonpositive, and its differentiated diffusion representation propagates the strictly negative terminal curvature on \((-\infty,0)\). On every interval ending before time \(1\) the drift is Lipschitz, its flow derivative is positive, and the diffusion has positive probability of reaching that interval. Equivalently the strong maximum principle applied after smoothing gives strict negativity; the representation retains strictness on removing the smoothing. Equality in (106) therefore gives \(s_1=s_2\) almost everywhere, hence everywhere before the endpoint by continuity. Their continuous susceptibilities have the same endpoint value as well. The PDE and its forward diffusion are then identical. The subsequential description in (89) consequently has only one possible limit. ◻

Proof. Nonnegativity of the limiting functional. First suppose \(\Delta>0\). For \(A>0\), direct substitution gives the exact homogeneity identity \[ A J_{\alpha,1}(R/A,s/A) =J_{\alpha,0}(R,s)+\frac{\alpha}{2A}\mathbb ER^2. \tag{109}\] The reciprocal of \(s/A\) is \(A\chi\), so this is an admissible zero-temperature trial. At \(\alpha_c\), zero ground energy and Proposition 44 make its value nonnegative. Letting \(A\to\infty\) proves the assertion. If \(\Delta=0\), replace \(\chi\) by \(\chi+\eta\). The quadratic term converges by dominated convergence, while \(1/(\chi+\eta)\uparrow1/\chi\); let \(\eta\downarrow0\).

Compactness on the common space. Normalization and concavity give \(1\ge\chi_n(t)\ge1-t\). On each \([0,T]\) their slopes are bounded by a constant depending only on \(T\): monotonicity of \(\gamma_n\) gives \(\gamma_n(t)\le(1-t)^{-1}\). Arzelà–Ascoli and a diagonal selection give the stated local uniform convergence. The limit is concave and nonincreasing; its continuous endpoint value is \(\Delta=\lim_{t\uparrow1}\chi(t)\), which is not initially identified with \(\lim_n\chi_n(1)\). Its derivative gives an admissible integrable \(\gamma=-\chi'\). At differentiability points of \(\chi\), the concavity inequalities imply \(\gamma_n\to\gamma\); the same local bound then gives \(L^1(0,T)\) convergence for every \(T<1\). Fatou’s lemma gives \(s\in L^2\).

The bound on \(R_n\) gives a weakly convergent subsequence on the fixed Wiener space. The cone of nonnegative \(L^2\) variables is weakly closed. Martingale representation, being a bounded linear isometry after the constant component is included, gives weak convergence of \(b_n\) and convergence of \(c_n\). In particular \(c\ge c_0\).

Passage to the minimum. For each \(T<1\), uniform convergence of the positive weights \(\chi_n\) on \([0,T]\) and weak convergence of \(b_n\) imply \[\int_0^T\chi(t)\mathbb Eb_t^2\,\mathrm dt \le\liminf_n\int_0^T\chi_n(t)\mathbb Eb_{n,t}^2\,\mathrm dt.\] Let \(T\uparrow1\) for the corresponding global lower semicontinuity. In contrast the entropy term actually converges: local convergence of \(s_n\) combines with \(\int_T^1s_n\le C\sqrt{1-T}\) from (108). The linear term converges weakly, and \(\lambda_n\mathbb ER_n^2\to0\). Equations (107) and (108) therefore give \[J_{\alpha_c,0}(R,s) \le\liminf_nJ_{\alpha_n,\lambda_n}(R_n,s_n)=0.\] Nonnegativity proves equality. ◻

Proof. Comparison estimates. The curvature bounds are parabolic comparison bounds. Specifically, if \(b=(f_n)_{hh}\), then \(b_t+\tfrac12b_{hh}+\gamma_n a_n b_h+\gamma_n b^2=0\). The spatially constant function \(-(\lambda_n+\chi_n(t))^{-1}\) solves this equation, and zero is the upper comparison solution. Differentiating once more gives the linear equation \(d_t+\tfrac12d_{hh}+\gamma_n a_n d_h+3\gamma_n b d=0\) for \(d=(f_n)_{hhh}\), which preserves nonnegativity backwards from the convex terminal derivative. These comparisons follow first for smooth terminal data and smooth coefficients; the bounds survive approximation. The first-order solution is \(q_n(t,h)=-h_-^2/(2(\lambda_n+\chi_n(t)))\). Its missing diffusion term is bounded in absolute value by \(s_n(t)/2\). Comparison with \(q_n\pm\tfrac12\int_t^1s_n\) proves the value bound. The derivative bound follows by taking secants of length \(\sqrt{(\int_t^1s_n)/s_n(t)}\) and using the curvature bound; the zero-length limiting case is immediate.

Convergence with a terminal cutoff. At a fixed cutoff \(T<1\) the value estimates compare \(f_n(T,\cdot)\) and \(f_m(T,\cdot)\) with quadratic functions whose coefficients converge. The error left after this quadratic comparison is bounded uniformly in space by \(C\sqrt{1-T}\), uniformly in \(n,m\). On \([0,T]\), the difference \(w=f_n-f_m\) solves a linear equation with drift \(\gamma_n(a_n+a_m)/2\) and source \((\gamma_n-\gamma_m)a_m^2/2\). The drift has uniformly bounded Lipschitz and linear-growth constants on that interval, and its diffusion has uniformly bounded second moments on bounded starting sets. Its Feynman–Kac representation propagates a constant terminal error without amplification. The quadratic terminal error tends to zero at fixed \(T\), as does the source error because \(\gamma_n\to\gamma\) in \(L^1(0,T)\). For each fixed \(T\), first let \(n,m\to\infty\): only the constant terminal error remains, bounded on every earlier compact set by \(C\sqrt{1-T}\). Then let \(T\uparrow1\). This proves the Cauchy property and shows that the limit depends only on \(\chi\). Curvature bounds and secants give convergence of the first derivatives.

Regularity for a measurable time coefficient. On a compact time interval ending strictly before \(1\), \(\gamma\) is bounded. We need spatial smoothness and local time-Lipschitz bounds, including at jumps of \(\gamma\), for the inverse-curvature map used below. We give the regularity argument in a form that does not require continuity of this coefficient in time. Reverse time and multiply \(a\) by smooth space and time cutoffs supported in a larger interior rectangle, with the time cutoff zero at its initial edge. The resulting inhomogeneous heat equation has bounded source: the nonlinear term is \(\gamma aa_h\), and the cutoff terms involve only \(a\) and \(a_h\), already bounded on that rectangle. For the heat semigroup \(P_u\), differentiation of the Gaussian kernel gives \[\|\partial_hP_uF\|_\infty\le C u^{-1/2}\|F\|_\infty, \qquad \|\partial_{hh}P_uF\|_\infty \le C_\eta u^{-1+\eta/2}[F]_{C^\eta_h}, \quad0<\eta<1.\] The second estimate uses the zero integral of the differentiated kernel to subtract \(F(h)\). The first estimate and its spatial increment version in Duhamel’s formula give a uniform spatial \(C^{1,\eta}\) bound for every \(\eta<1\): splitting the time integral at the square of the spatial increment bounds the gradient increment by a constant times \(|z|(1+|\log|z||)\). The source is therefore spatially \(C^\eta\). The second displayed estimate is integrable at \(u=0\) and yields \(C^{2,\eta'}\) on a smaller rectangle for \(0<\eta'<\eta\). More generally, if \(a\) is already bounded in \(C_h^{m,\eta}\), \(m\ge1\), then the localized source is bounded in \(C_h^{m-1,\eta}\): its derivatives of order \(m-1\) use derivatives of \(a\) of order at most \(m\). Apply the second heat-kernel estimate to \(\partial_h^{m-1}F\) to obtain \(C_h^{m+1,\eta'}\) on a smaller rectangle. This induction uses no derivative of \(a\) that it is trying to establish. It gives every spatial derivative on nested interior rectangles. Multiplication by the bounded function \(\gamma(t)\) preserves all spatial Hölder bounds, so this bootstrap is valid for measurable time coefficients. Smooth coefficient approximation justifies the intermediate differentiations with these same bounds. Finally the PDE bounds the almost-everywhere time derivative of every spatial derivative on smaller rectangles. This gives the claimed local time-Lipschitz property, including across jumps of \(\gamma\). The uniform spatial Hölder and time-Lipschitz bounds make these derivatives jointly continuous. At the start of this argument the source uses the bounded weak derivative \(a_h\) supplied by the curvature estimate; the distributional PDE passes from \(a_n\) by local uniform convergence and \(L^1\) convergence of \(\gamma_n\). Thus the initial Duhamel representation does not presume smoothness.

Strict curvature and identification of the martingale. The value bound on \(h>0\) and \(a\ge0\) imply \(a(t,h)\to0\) as \(h\to+\infty\). On the negative half-line, (110) makes \(a\) unbounded. Thus \(a\) is not affine, its nonnegative second derivative is nontrivial at every later interior time, and its nonpositive first derivative is nontrivial there. The strong maximum principle, on an interval between two strictly interior times, gives \(f_{hhh}>0\) and \(f_{hh}<0\) at the earlier time.

The coupled forward diffusions converge in \(L^2\) uniformly on each compact time interval, by the local gradient convergence, \(L^1\) convergence of \(\gamma_n\), and the common linear-growth bounds. For precision, those deterministic growth bounds and Gronwall’s inequality give uniformly bounded moments of every fixed finite order for \(\sup_{t\le T}|H_{n,t}|\) and \(\sup_{t\le T}|a_n(t,H_{n,t})|\), for each \(T<1\). The fourth-moment bound permits spatial localization to be removed in \(L^2\) after the coupled convergence is proved on bounded sets. Consequently \(a_n(t,H_{n,t})\to a(t,H_t)\) strongly in \(L^2\) for every \(t<1\). But \(a_n(t,H_{n,t})=\mathbb E[R_n\mid\mathcal F_t]\) converges weakly to \(\mathbb E[R\mid\mathcal F_t]\). This identifies \(r_t\); Itô’s formula identifies \(b_t=f_{hh}(t,H_t)\) and proves (111). Finally Doob’s inequality gives \(\mathbb E\sup_{t\le1}|r_t|^2\le4\mathbb ER^2\). Since \(\gamma\in L^1\), the drift integral in (111) converges both almost surely and in \(L^2\) at \(1\). The martingale itself has its continuous \(L^2\) extension with terminal value \(R\). ◻

Proof. Equality identifies the normalized curvatures along the states. Both pairs minimize the globally nonnegative critical functional. Their midpoint is admissible and also has value zero. Lemma 45 and \(s_i(0)=1\) yield \(c_1=c_2=:c>0\) and \(\chi_1b_1=\chi_2b_2\) almost everywhere. Put \(k_i(t,h)=-\chi_i(t)(f_i)_{hh}(t,h)\). The proof uses this matched normalized curvature to express one state process as a deterministic transform of the other. Their common Brownian noise forces the transform to have spatial derivative one; the common martingale mean then identifies the susceptibilities. Interior continuity upgrades the curvature identity to \[k_1(t,H_{1,t})=k_2(t,H_{2,t}),\qquad 0<t<1,\] on a single event of probability one. Each \(k_i(t,\cdot)\) is strictly decreasing with strictly negative derivative by Lemma 48. Each \(H_{i,t}\) has full support in \(\mathbb R\): on every interior compact time interval its additive-noise diffusion has locally bounded Lipschitz drift, and localized Girsanov equivalence, or the Brownian support property followed by continuity of the solution map, reaches every open interval. Thus the support of \(k_i(t,H_{i,t})\) is the closure of the range of \(k_i(t,\cdot)\). Equality of their laws identifies the two ranges, which are open intervals. We may therefore define \[\Psi(t,h)=k_2(t,\cdot)^{-1}(k_1(t,h)),\qquad H_{2,t}=\Psi(t,H_{1,t}).\] These inverse functions have uniform local bounds. At a fixed \((t_0,h_0)\) put \(y_0=\Psi(t_0,h_0)\). Strict decrease gives \[k_2(t_0,y_0-1)>k_1(t_0,h_0)>k_2(t_0,y_0+1).\] Joint continuity preserves these inequalities on a neighborhood of \((t_0,h_0)\), trapping \(\Psi\) between \(y_0-1\) and \(y_0+1\). Compactness and uniqueness of the inverse give continuity there. The negative derivative of \(k_2\) is bounded away from zero on a slightly smaller compact rectangle. The implicit derivative \(\Psi_h(t,h)=(k_1)_h(t,h)/(k_2)_h(t,\Psi(t,h))\) is therefore jointly continuous, all needed spatial derivatives are bounded locally, and the mean-value theorem with the time-Lipschitz bounds on \(k_1,k_2\) proves a uniform local time-Lipschitz bound on \(\Psi\). Finite covers give the same assertions on every compact subset of \((0,1)\times\mathbb R\).

The common noise forces a spatial translation. Taking quadratic covariations with the driving Brownian motion on an arbitrary interval \([u,t]\subset(0,1)\) gives \[t-u=[H_2,B]_t-[H_2,B]_u =\int_u^t\Psi_h(v,H_{1,v})\,\mathrm dv.\] This chain rule needs no continuous time derivative. After localization, the temporal increments in a partition contribute at most \(C\sum\Delta t\,|\Delta B|\), which tends to zero. The spatial Taylor remainders contribute at most \(C\max|\Delta B|\sum|\Delta H_1|^2\), which tends to zero in probability, since quadratic-variation sums of a continuous semimartingale are tight. The first-order sums give the displayed integral. Rational interval endpoints and continuity give all these identities on one event of probability one. It follows that \(\Psi_h(t,H_{1,t})=1\) for almost every time, almost surely. Full support gives \(\Psi_h(t,h)=1\) for every \(h\) at almost every \(t\); continuity extends it to every interior \(t\). Hence \(\Psi(t,h)=h+d(t)\).

The martingale mean identifies the susceptibility. The equality of \(k\)’s now holds for all \(h\). Integrate the matched curvatures from \(h\) to \(+\infty\), using gradient decay there, to obtain \[\chi_1(t)a_1(t,h) =\chi_2(t)a_2(t,h+d(t)).\] Along the states this gives \(\chi_1r_{1,t}=\chi_2r_{2,t}\). Taking expectations yields \(c\chi_1=c\chi_2\), so \(\chi_1=\chi_2\). The PDE is determined by \(\chi\), and the two forward diffusions have the same initial condition and Brownian motion. They agree, and their martingales, including their \(L^2\) terminal values, agree as well. ◻

Proof. The critical pair minimizes without normalization, since its value is the global lower bound zero. By (104), the Fréchet derivative in \(R\) is the \(L^2\) variable \(\alpha_c(H_1+\Delta R)\). Explicitly the quadratic operator is \[\mathcal A Q=\Delta Q+ \int_0^1\gamma(t)\mathbb E[Q\mid\mathcal F_t]\,\mathrm dt, \qquad \|\mathcal A\|_{L^2\to L^2}\le\chi(0).\] Conditional expectations are self-adjoint contractions, so \(\mathcal A\) is bounded and self-adjoint; the functional’s gradient is \(\alpha_c(1+B_1+\mathcal AR)\), as asserted. In particular \(\|H_1\|_2\le1+\|B_1\|_2+\|\gamma\|_1\|R\|_2\) also when \(\Delta=0\). Minimization on the nonnegative cone means that this variable is nonnegative and has zero pairing with \(R\); these are exactly (112).

Adding \(\varepsilon\mathbf 1_{[t,1)}\) to \(\gamma\) is an admissible positive variation. Its susceptibility increment at \(v\) is \(\varepsilon(1-\max\{t,v\})\), bounded by \(\varepsilon\chi(v)/\chi(0)\) by concavity. Differentiating the entropy term is therefore justified by domination: if \(q(v)=1-\max\{t,v\}\), its difference quotient is bounded by \(s(v)/\chi(0)\) for the positive variations used here. For scaling \(\gamma\), the susceptibility increment is \(q=\chi-\Delta\le\chi\), and the difference quotient is bounded by \(2s\) for \(|\varepsilon|<1/2\). Both bounds require only \(s\in L^1\). Tonelli’s theorem in the quadratic term gives the directional derivative \(K(t)/2\). This proves \(K\ge0\). Scaling \(\gamma\) by \(1+\varepsilon\) permits either sign for small \(\varepsilon\), and gives \(\int\gamma J=0\). Fubini gives \(\int\gamma J=\int K\,\mathrm d\nu\): its absolute version is bounded by \(\|J\|_\infty\int\gamma<\infty\). Equivalently the Stieltjes boundary term at \(1\) vanishes because \(K(t)=O(1-t)\) and \((1-t)\gamma(t)\to0\). Thus \(K\) vanishes on \(\mathop{\mathrm{supp}}\nu\).

Since \(J(0)=\alpha_c c^2>0\) and \(K'=-J\), nonnegativity of \(K\) forces \(K(0)>0\). Hence \(\nu\) vanishes on an initial interval, including its possible atom at zero. An interior support point is a local minimum of the nonnegative \(C^2\) function \(K\), so \(J=0\) and \(J'\le0\). At a non-isolated support point, differentiability of \(J\) and nearby zeros give \(J'=0\).

If \(\Delta>0\), two-sided variation of \(\Delta\) gives \(0=(P(1)-I(1))/2\). If \(\Delta=0\) and the support did not accumulate at \(1\), then \(\gamma\) would be constant on a final interval and \(\chi(t)\) would be a positive constant times \(1-t\). This contradicts \(s\in L^2\). Thus in this case interior support points approach \(1\), and continuity of \(J\) again gives \(J(1)=0\). ◻

Proof. Introduce \(W(t)=s(t)+\alpha_c\mathbb Eb_t\). The curvature equation along the forward diffusion gives \(\frac{\mathrm d}{\mathrm dt}\mathbb Eb_t=-\gamma(t)\mathbb Eb_t^2\) almost everywhere. Since \(s'=\gamma s^2\), we have \(W'=-\gamma J'\). The Stieltjes product rule and \(J\,\mathrm d\gamma=0\) now give \[ W(t)=W(0)-\gamma(t)J(t)\quad\text{almost everywhere}, \qquad \int_0^1W(t)\,\mathrm dt=W(0). \tag{115}\] The initial constant uses the initial zero interval of \(\gamma\).

Itô’s formula for \(H_tr_t\) and (112) yield \[-\Delta P(1)=\alpha_c c +\int_0^1\bigl(\gamma(t)P(t)+\alpha_c\mathbb Eb_t\bigr)\,\mathrm dt.\] Every term is integrable: \(P\) is bounded, \(\gamma\in L^1\), and \(|\mathbb Eb_t|\le s(t)\in L^1\). The stochastic products are justified by localization and the \(L^2\) maximal bounds for \(H\) and \(r\). Moreover (113) and integration by parts give \[\int_0^1\gamma P=\int_0^1\gamma I =\int_0^1s-\Delta I(1).\] Since \(P(1)=I(1)\), these identities show \[ W(0)=-\alpha_c c<0. \tag{116}\]

Suppose for contradiction that \(\Delta>0\). Then \(s\le1/\Delta\), and the value and derivative bounds in (110) give \(a(t,h)\to h_-/\Delta\) as \(t\uparrow1\), uniformly on bounded sets. Convex secant bounds give \(f_{hh}(t,h)\to-\mathbf 1_{\{h<0\}}/\Delta\) locally away from zero. The endpoint \(H_1\) has no atom at zero. Here is a direct justification: condition on the Brownian bridge and write \(B_t=tZ+\mathcal B_t\), where \(Z\) has a Gaussian density independent of the bridge. The map \(Z\mapsto H_1\) is increasing, and its derivative is \[\int_0^1\exp\left\{\int_u^1\gamma(v)a_h(v,H_v)\,\mathrm dv\right\}\mathrm du \ge\exp\left\{-\Delta^{-1}\int_0^1\gamma\right\}>0.\] The estimate follows first by smoothing and then by the Lipschitz flow comparison. Thus the conditional endpoint law, and hence the unconditional one, is atom-free. With \(p=\mathbb P(H_1<0)\), bounded convergence gives \[W(1-)=\frac{1-\alpha_c p}{\Delta},\qquad J'(1-)=\frac{\alpha_c p-1}{\Delta^2}.\] Because \(J(1)=0\) and \(J'\) is bounded when \(\Delta>0\), \(|J(t)|\le C(1-t)\). Integrability and monotonicity of \(\gamma\) imply \(\gamma(t)J(t)\to0\). Hence (115)–(116) give \(W(1-)=-\alpha_c c<0\), so \(\alpha_cp>1\) and \(J'(1-)>0\). It follows that \(J(t)<0\), and therefore \(K(t)<0\), for \(t\) sufficiently close to \(1\). This contradicts Lemma 50.

Thus \(\Delta=0\), and the support accumulation follows from that lemma. At a support point \(J=0\), (115) and (116) give the first identity in (114). Although (115) was initially obtained almost everywhere, it applies here by continuity of \(W\): approach the support point through times where that identity holds, and use local boundedness of \(\gamma\) and \(J(t)\to0\). Multiplying \(J'=\alpha_c\mathbb Eb_t^2-s^2\le0\) by \(\chi^2\) gives the second. The equality statement follows from \(J'=0\) at non-isolated support points. ◻

Proof. Use stars temporarily for the unique critical pair and put \(S(s)=\tfrac12\int_0^1s\), \(d_n=\alpha_n-\alpha_c\). The critical pair is a legitimate limiting normalized trial at any \(\lambda>0\): replace \(\chi_*\) by \((\chi_*+\eta)/(1+\eta)\), keep \(R_*\) fixed, and send \(\eta\downarrow0\). Its entropy and quadratic terms converge by domination and monotone convergence. Since its critical value is zero, its value at \((\alpha_n,\lambda)\) is \[-\frac{d_n}{\alpha_c}S_*+ \frac{\alpha_n\lambda}{2}\mathbb ER_*^2.\] The original unnormalized trial has an additional factor \(\lambda\). Choosing \(\lambda=d_nS_*/(\alpha_c\alpha_n\mathbb ER_*^2)\) gives \[e(\alpha_n)\ge \frac{d_n^2S_*^2}{2\alpha_c^2\alpha_n\mathbb ER_*^2}.\] Here \(S_*>0\) and \(\mathbb ER_*^2>0\), so (108) implies \(\lambda_n\ge c_1d_n\).

Compare now at the physical value \(\lambda_n\). Optimality and the same trial give \[ \left(1+\frac{d_n}{\alpha_c}\right) J_{\alpha_c,0}(R_n,s_n) +\frac{\alpha_n\lambda_n}{2} (\mathbb ER_n^2-\mathbb ER_*^2) \le\frac{d_n}{\alpha_c}(S_n-S_*). \tag{117}\] The first term is nonnegative, \(S_n\to S_*\) by uniform integrability, and \(d_n/\lambda_n\) is bounded. Therefore \(\limsup\mathbb ER_n^2\le\mathbb ER_*^2\). Weak convergence and lower semicontinuity give the reverse inequality. The Hilbert-space identity for \(\|R_n-R_*\|_2^2\) proves strong convergence. Uniqueness of all weak subsequential limits makes this conclusion valid for the original sequence.

Local \(L^1\) convergence of \(\gamma_n\) becomes global because \(\int_T^1\gamma_n\le\chi_n(T)\to\chi(T)\), and \(\chi(T)\downarrow0\) by Proposition 51. The same estimate gives \(\chi_n(1)\to0\). Doob’s inequality applied to \(\mathbb E[R_n-R\mid\mathcal F_t]\) gives \(L^2\) convergence uniformly in time of the martingales. In (111), split the drift difference into \(\gamma_n(r_n-r)+(\gamma_n-\gamma)r\). Conditional-expectation contraction and Minkowski’s inequality give the explicit bound \[\left\|\int_0^1(\gamma_nr_n-\gamma r)\,\mathrm dt\right\|_2 \le\|\gamma_n\|_1\|R_n-R\|_2 +\|\gamma_n-\gamma\|_1\|R\|_2\longrightarrow0.\] Hence \(H_{n,1}\to H_1\) in \(L^2\).

The terminal dual optimizer satisfies \(R_n=(H_{n,1})_-/(\lambda_n+\chi_n(1))\). On its positive half-line the physical gap equals \(H_{n,1}\); on its negative half-line the proximal formula (89) gives \(h_n^*=-\lambda_nR_n\). These two cases are exactly \(h_n^*=H_{n,1}+\chi_n(1)R_n\). The asserted limits follow. Finally (112) with \(\Delta=0\) gives \(H_1\ge0\) and \(RH_1=0\). ◻

Proof. Let \(p_n=\mathbb P(R_n>0)\). Corollary 41 gives \(\alpha_np_n\to1\) and the direct cutoff bound \[\lim_{u\downarrow0}\sup_n\mathbb P(0<R_n\le u)=0\] after discarding a finite initial part of the density sequence. Apply strong convergence at positive continuity cutoffs and then send the cutoff to zero. This gives \(\mathbb P(R>0)=\lim_np_n=1/\alpha_c\).

Similarly the direct bound in (100), passed to the atomless endpoint law at each fixed UNSAT density, gives \(\lim_{u\downarrow0}\sup_n\mathbb P(0<h_n^*\le u)=0\). Its constants and its upper cutoff are uniform in the density. This is a bound on the whole interval \((0,u]\) at fixed density; no interchange between a lower gap cutoff and the density limit is used. At every positive continuity point \(u\) of the law of \(H_1\), the \(L^2\) convergence \(h_n^*\to H_1\) gives convergence of the probabilities of \((-\infty,u]\). Subtracting the negative mass \(p_n\) and then sending \(u\downarrow0\) proves \(\mathbb P(H_1=0)=1/\alpha_c\); fixed-density endpoint laws have no atom at zero, as established in (89). Complementarity gives the inclusion \(\{R>0\}\subset\{H_1=0\}\). Equality of their probabilities proves the final assertion. ◻

Proof. The force variable. Monotonicity of \(s\) gives \[s(t)\int_t^1s(v)\,\mathrm dv\le\int_t^1s(v)^2\,\mathrm dv\longrightarrow0.\] Thus (110) approximates \(a(t,h)\) by \(s(t)h_-\) with an error \(\varepsilon_t\to0\) uniformly in \(h\). Fix \(x>0\). When \(x/2\le a(t,h)\le2x\) and \(t\) is sufficiently close to \(1\), one has \(h<0\) and \(-h\ge(x/2-\varepsilon_t)/s(t)\). A forward secant of length \(x/(8s(t))\) stays negative. Convexity of \(a\) gives \[a_h(t,h)\le-s(t)+16\varepsilon_t s(t)/x\le-s(t)/2.\] Consequently \(1/2\le k(t,h)\le1\) in that region, whereas \(0\le k\le1\) globally.

Change deterministic time to \(v=I(t)=\int_0^ts^2\) and put \(T=I(1)<\infty\). The martingale \(r\) then has Brownian coefficient \(-k\): indeed, with \(t(v)=I^{-1}(v)\), \[\widetilde B_v=\int_0^{t(v)}s(q)\,\mathrm dB_q, \qquad \mathrm dr_{t(v)} =\frac{b_{t(v)}}{s(t(v))}\,\mathrm d\widetilde B_v =-k(t(v),H_{t(v)})\,\mathrm d\widetilde B_v.\] The process \(\widetilde B\) is Brownian in the filtration \(\mathcal F_{t(v)}\), since its integrand and time change are deterministic and its quadratic variation is \(v\). The coefficient has absolute value at most one and at least \(1/2\) whenever \(|r-x|\le x/2\) in a sufficiently short final time interval. For \(\eta>0\) write \(\ell=T-v+\eta^2\) and consider \[\Phi(v,r)=\ell^{-\zeta/2} \exp\{-a_0(r-x)^2/\ell\}, \qquad a_0=\tfrac14,\quad\zeta=\tfrac1{16}.\] Its drift divided by \(\Phi\) is \[\frac{\zeta/2-a_0k^2}{\ell} +\frac{(-a_0+2a_0^2k^2)(r-x)^2}{\ell^2}.\] On \(|r-x|\le x/2\) both terms are nonpositive. Off this interval the second is at most \(-x^2/(32\ell^2)\) and the first at most \(1/(32\ell)\), so their sum is nonpositive when \(\ell\le x^2\). Choose \(\delta>0\) so that \(v_0=T-\delta\) lies in that final region and \(2\delta\le x^2\). For \(\eta^2\le\delta\) one has \(\ell\le2\delta\). At this fixed \(\eta\), \(\Phi\) and its stochastic derivative are bounded, so Itô’s formula gives \[\mathbb E\Phi(T,R)\le\mathbb E\Phi(v_0,r_{t(v_0)}) \le\delta^{-\zeta/2},\] uniformly in \(\eta\). Since \(\Phi(T,R)\ge e^{-a_0}\eta^{-\zeta}\) on \(|R-x|\le\eta\), \[\mathbb P(|R-x|\le\eta)\le C_x\eta^{\zeta}.\] This proves the first assertion.

The gap variable. We show that the state drift is bounded on spatial intervals away from the contact point. Positivity of the drift and monotonicity of \(a\) imply, for \(t<t'<1\), \[a(t,h)=\mathbb Ea(t',H_{t'}^{t,h}) \le\mathbb Ea(t',h+B_{t'-t}).\] Fix \(x>0\), write \(\tau=1-t\), and set \(t_j=1-2^{-j}\tau\) and \(x_j=x/2+(x/2)2^{-j/4}\). Split the last expectation according to whether the Brownian increment is below \(-(x_j-x_{j+1})\). The complement costs at most \(a(t_{j+1},x_{j+1})\). On the exceptional event, the global linear-growth bound in (110) and the Gaussian tail estimate bound the cost by \[C_x(1+s(t_{j+1})) \exp\{-c_x2^{j/2}/\tau\}.\] Concavity and \(\chi(1)=0\) give \(s(t_{j+1})\le2^{j+1}s(t)\). The remaining term \(a(t_j,x_j)\) tends to zero by the uniform error \(\varepsilon_{t_j}\). Summing the rapidly decreasing tail series proves \[\sup_{h\ge x}a(t,h) \le C_xs(t)\exp\{-c_x/(1-t)\}\] for \(t\) close to \(1\). Since \(\chi(t)=\int_t^1\gamma(v)\,\mathrm dv\ge(1-t)\gamma(t)\), the drift \(\gamma(t)a(t,h)\) is bounded uniformly near time \(1\) on every compact interval contained in \((0,\infty)\); explicitly it is at most \(C_x(1-t)^{-1}\exp\{-c_x/(1-t)\}\) for \(h\ge x\).

Start at a fixed late time, stop on leaving such an interval, and set the drift to zero after exit. The resulting diffusion has bounded adapted drift up to time \(1\). The exponential martingale in Girsanov’s theorem is integrable because that drift is bounded; its law is equivalent to Brownian motion with its initial value. Its terminal law therefore has no atoms. The original diffusion agrees with it on the event of no exit. Continuity of \(H\) implies that every event \(\{H_1=x\}\) with \(x>0\) is covered by a countable union of these no-exit events, using rational interval endpoints and rational starting times. Thus \(\mathbb P(H_1=x)=0\). ◻

Proof. Thermodynamic and zero-temperature limits at fixed density. First take \(N\to\infty\) at fixed \((\beta,\alpha)\), using the inner empirical-marginal result proved earlier. At fixed UNSAT density, Corollary 46 and (89) give the zero-temperature endpoint law, with no atoms. For completeness, let \(h_{\beta,\alpha}\) denote the inner limiting gap variable and put \(p_{\beta,\alpha}=\mathbb P(h_{\beta,\alpha}<0)\) and \(\mu_{\beta,\alpha}=\mathbb E(h_{\beta,\alpha})_-\). Proposition 35 supplies uniform integrability of the negative first moment, and the absence of an atom at zero gives \[p_{\beta,\alpha}\longrightarrow p_\alpha>0,\qquad \mu_{\beta,\alpha}\longrightarrow\mu_\alpha>0.\] Positivity follows from \(\alpha\mathbb E(h_\alpha^*)_-^2=2e(\alpha)>0\). The inner limiting force CDF at \(s>0\) is \[\frac1{p_{\beta,\alpha}} \mathbb P\left(-s\frac{\mu_{\beta,\alpha}}{p_{\beta,\alpha}} \le h_{\beta,\alpha}<0\right).\] Both limiting interval endpoints are atomless under the law of \(h_\alpha^*\), so this expression converges at every \(s>0\). Thus the full \(\beta\to\infty\) limit exists for all the required gap and force arguments, and its conditioning mass is preserved before the density is changed. Uniqueness of every subsequential endpoint law yields convergence along the full temperature limit.

The critical-density limit and the final gap cutoff. Now approach \(\alpha_c\) from above. Proposition 52 gives convergence of the physical endpoint gaps and the scaled residuals, and Lemma 54 permits every positive cutoff, rather than just a selected continuity sequence.

For gaps keep \(0<\delta<u\) fixed throughout these three limits; the answer is \(\mathbb P(\delta<H_1\le u)\). Only then let \(\delta\downarrow0\), which removes the contact atom and gives the first formula in (118). For forces the conditional mean at density \(\alpha_n\) is \(c_n/p_n\), which converges to \(\alpha_cc\). Lemma 53 preserves the conditioning mass, and its cutoff argument loses no positive residual mass to zero. Subtracting the noncontact mass and applying the positive-atom assertion gives the second formula at every \(s>0\). Since \(\mathbb ER=c\) and \(\mathbb P(R>0)=1/\alpha_c\), its mean is one. Its conditional probability of \(\{0\}\) is zero; the uniform cutoff bound also explicitly prevents an atom from appearing there in the limit of the physical conditional laws.

Finally the Gaussian and mixture inner-limit observables agree at each fixed temperature and density in a neighborhood above the common threshold, by Section 5. Equality of these functions passes through exactly the same ordered limits. The limits \(u\downarrow0\) and \(s\downarrow0\), which determine their exponents, are taken only in the subsequent sections. ◻

Proof. First \(m\ge h_-\). Indeed \(X_h=l\ge0\), and the bounded difference between \(f(t,h)\) and \(-h_-^2/(2\chi)\) in (110) forces \(X(t,h)\to0\) as \(h\to-\infty\): a nonzero limit of the monotone function \(X\) would make the integral of \(f_h+h/\chi=X/\chi\) unbounded on that half-line. Thus \(X\ge0\); on the other half-line use \(m\ge0\).

The heat evolution of the hinge is the lower function in (120). It is a subsolution of (119), since \(m(1+m_h)\ge0\). For the upper function \(U\), direct differentiation of \(\rho(z)=\phi(z)/\Phi(z)\), using \(\rho'=-z\rho-\rho^2\), gives \(U_\tau=U_{hh}/2+\tau^{-1}U(1+U_h)\) and \(-1\le U_h\le0\). It is therefore a supersolution for \(d\le\tau^{-1}\).

Here comparison must start before the singular endpoint. At \(\tau'>0\), (110) gives \[0\le m(\tau',h)-h_-\le C\chi(\tau')\sqrt{w(\tau')} \quad\hbox{uniformly in }h.\] For the difference between two solutions, the zeroth-order coefficient in the linear comparison equation is at most \(d\). Thus the positive error at \(\tau'\) is amplified by at most \(\exp(\int_{\tau'}^\tau d)=\chi(\tau)/\chi(\tau')\). The resulting error is at most \(C\chi(\tau)\sqrt{w(\tau')}\), which tends to zero as \(\tau'\downarrow0\). Comparison may first be made with smooth coefficients and on bounded spatial intervals; the uniform linear growth and curvature bounds permit the limits.

Finally, for a convex decreasing function \(m\), its negative derivative at \(h\) is bounded above by the backward secant over a distance \(\sqrt\tau\). Applying the upper value barrier gives the positive Gaussian tail for \(k\). Apply the forward secant to the convex increasing function \(X\) on the negative half-line, using \(X(t,h)\le C\tau/|h|\) there; this gives the stated bound for \(l\). The integral estimate follows by using \(kl\le1\) on \(|h|\le\sqrt\tau\) and integrating the two tails. ◻

Proof. At a support point, stability says \(\mathbb E(k^2-\pi)\le0\). Suppose instead that along support points \(D\ge\tau^{-\varepsilon}\) for some \(\varepsilon>0\). Since \(h=X-\chi r\ge-\chi r\), (125) gives \(q_g\ge\Phi(-v)\). Thus (126) implies \(k^2-\pi\ge q_g/2\) when \(2\le v\le a\sqrt D\), for a fixed sufficiently small \(a>0\). When \(v>a\sqrt D\) its negative part is bounded by \(kl\le C\exp(-c_0D)\). When \(v<2\) and \(h\le-K\sqrt\tau\), take \(K\) sufficiently large in (121); again \(k^2-\pi\ge q_g/2\). In particular, on \[I_-=\left[-\tfrac12\chi\sqrt w,-\tfrac14\chi\sqrt w\right]\] there is a fixed positive lower bound for \(k^2-\pi\) once \(D\) is large. This follows from \(q_g\ge\Phi(-1/2)\), the negative-tail slope bound, and the upper value barrier, which gives \(v\le1\) on this interval. Writing \(p_-:=\inf_{I_-}p_t\), the positive contribution is at least \(c_0\chi\sqrt w\,p_-\). Equation (123) gives \(p_-\ge c_0\) for late times.

For clarity, the required density comparison has the following quantitative form. If \(h_2>h_1\), \(\delta=(h_2-h_1)/\chi\), and \(P\) is fixed, then \[ p_t(h_2)\le \exp\!\{C_P[\delta\sqrt{|\log\tau|}+\delta^2+1]\}\,p_t(h_1) +\tau^P. \tag{128}\] To prove it, use the decreasing inverse Jacobian, the bound \(|\mathcal I_t(h_2)-\mathcal I_t(h_1)|\le\delta\), and the Gaussian density ratio on \(|\mathcal I_t(h_2)|\le C_P\sqrt{|\log\tau|}\). On the complement use \(\mathcal J_t\le\chi^{-1}\le C\tau^{-1/2}\); the last inequality follows from \(\tau/\chi^2\le w\le w(t_0)\). Increasing \(C_P\) makes the discarded Gaussian density at most \(\tau^P\).

Apply this with \(h_1\in I_-\) and \(-K\sqrt\tau\le h_2\le C_P\sqrt{\tau|\log\tau|}\). Here \(\delta\le C_P\sqrt{w(1+|\log\tau|)}\), so the exponential factor is \(\tau^{-o(1)}\). The negative contribution in this layer is at most \(C\sqrt\tau\,\tau^{-o(1)}p_-\) by \((k^2-\pi)_-\le kl\) and (121). Above the layer the positive Gaussian tail of \(k\) makes the contribution an arbitrarily small power of \(\tau\). Below it the only remaining negative contribution is \(C\exp(-c_0D)\). Both are negligible compared with \(\chi\sqrt w\,p_-=D\sqrt\tau\,p_-\), a contradiction.

It remains to cross a support gap \(\tau_1<\tau<\tau_2\). Affineness gives the exact identity \[ w(\tau)=w_1+\frac{\tau-\tau_1}{\chi_1\chi(\tau)}. \tag{129}\] If \(\tau_2/\tau_1\le2\), concavity gives \(\chi/\chi_1\le2\), so the assertion transfers immediately. Otherwise \(\chi_2/\chi_1\le2D_2^2\), by applying (129) at \(\tau_2\). Hence \[D(\tau)^2\le D_1^2\frac{\tau_1}{\tau}(2D_2^2)^2+2D_2^2.\] For every fixed \(\eta>0\) the endpoint estimates bound \(D_i\) by \(C_\eta\tau_i^{-\eta}\). The factor \(\tau_1/\tau\) absorbs the possible disparity between \(\tau_1\) and \(\tau\); since \(\tau_2\ge\tau\), for \(0<\eta<1/2\) we obtain explicitly \[D(\tau)^2\le C_\eta\frac{\tau_1^{1-2\eta}}{\tau}\tau_2^{-4\eta} +C_\eta\tau_2^{-2\eta} \le C_\eta(\tau^{-6\eta}+\tau^{-2\eta}).\] As \(\eta\) is arbitrary, this proves the same subpower conclusion in the gap. ◻

Proof. In the inverse formula, \(\log \mathcal J_t=\int d(u)k(u,\widetilde H_u)\,\mathrm du\). Starting at \(h\in[-a\sqrt\tau,0]\), the inverse drift is nonpositive, so at every reverse time its probability of a negative position is at least \(1/2\). The barriers and secants give a uniform lower bound \(k\ge\kappa>0\) for negative positions. Moreover the inverse moment bound proved above gives \(\mathbb E\log p_0(\mathcal I_t(h))\ge-C\) uniformly: its initial normalized negative position has square at most \(a^2\tau/\chi^2\le a^2w\). Jensen’s inequality therefore yields \(p_t(h)\ge C^{-1}\chi^{-\kappa/2}\). Here both \(k\) and \(l\) have fixed positive lower bounds on \([-a\sqrt\tau,0]\) if \(a>0\) is sufficiently small. To see this uniformly in time, the heat barrier gives \(m(0)\ge\phi(0)\sqrt\tau\), while the upper barrier makes \(m(K\sqrt\tau)/\sqrt\tau\) arbitrarily small for fixed large \(K\). The secant from \(0\) to \(K\sqrt\tau\) bounds \(k(0)\) below, and \(k(h)\ge k(0)\) for \(h\le0\). Likewise \(X(h)\ge X(0)-|h|\ge(\phi(0)-a)\sqrt\tau\) on that interval, whereas \(X(-K\sqrt\tau)\le C\sqrt\tau/K\). The backward secant from \(-K\sqrt\tau\) to \(h\) therefore gives \[l(h)\ge\frac{X(h)-X(-K\sqrt\tau)}{h+K\sqrt\tau} \ge\frac{\phi(0)-a-C/K}{K}>0\] after fixing \(a<\phi(0)/2\) and then a sufficiently large \(K\). Together with (124) this gives \[\mathbb Ekl\ge C^{-1}\sqrt\tau\,\chi^{-\kappa/2} \ge C^{-1}\tau^{b_0},\qquad b_0=\tfrac12-\tfrac\kappa8<\tfrac12.\] The drift identity for \(k\), including its endpoint limit, gives \(\mathbb Ek-1/\alpha=\int_t^1d\,\mathbb Ekl\). Thus, if \(\tau_2\) is a late support point and \(0<\tau_1<\tau_2\), \[ \chi(\tau_2)\ge C^{-1}\tau_1^{b_0} \log\frac{\chi(\tau_2)}{\chi(\tau_1)}. \tag{130}\]

Put \(b=\limsup_{\tau\downarrow0}\log\chi(\tau)/\log\tau\). The bounds already proved imply \(1/4\le b\le1/2\). Fix \(0<\varepsilon<1\) and choose \(\tau_n\downarrow0\) realizing this limsup. For sufficiently small \(\delta>0\), eventually \(\chi(u)\ge u^{b+\delta}\) at all small \(u\), whereas along a subsequence \(\chi(\tau_n)\le\tau_n^{b-\delta}\). At \(u_n=\tau_n^{1-\varepsilon}\), choosing \((2-\varepsilon)\delta<b\varepsilon/4\) gives \[\frac{\chi(u_n)}{\chi(\tau_n)} \ge \tau_n^{-b\varepsilon+(2-\varepsilon)\delta} \ge \tau_n^{-3b\varepsilon/4}.\] If \(u_n\) lies in a gap, move to its smaller endpoint \(\tau_{2,n}\); otherwise put \(\tau_{2,n}=u_n\). The loss in susceptibility is subpower in \(1/u_n\). Indeed it is at most two on a gap of time ratio at most two; on any other gap it is at most \(2D(\tau_2)^2\), by (129), and \(\tau_2\ge u_n\). Lemma 57 therefore gives, for all large \(n\), \[\frac{\chi(\tau_{2,n})}{\chi(\tau_n)} \ge \tau_n^{-b\varepsilon/2}.\] In particular \(\tau_n<\tau_{2,n}\le u_n\). This is the ordering needed to apply (130) with lower endpoint \(\tau_n\). Concavity then gives \[\chi(\tau_n)\ge\frac{\tau_n}{\tau_{2,n}}\chi(\tau_{2,n}) \ge C^{-1}\tau_n^{b_0+\varepsilon}|\log\tau_n|.\] Therefore \(b\le b_0+\varepsilon\). Letting \(\varepsilon\downarrow0\) gives \(b\le b_0<1/2\). Choose any exponent strictly between \(b_0\) and \(1/2\) and use the defining limsup to obtain the asserted bound on \(s\); integration gives the bound on \(w\). ◻

Proof. We first prove a conditional steering estimate; the martingale need not itself have the Markov property. After any stopping time \(S\), apply Brownian time change to the shifted continuous martingale \(Z_{S+v}-Z_S\) in the filtration \(\mathcal F_{S+v}\). Extending it with independent Brownian increments after the terminal time if necessary, this gives \(Z_{S+v}=Z_S+W_{Q_v}\), where \(Q_v=\int_S^{S+v}\sigma^2\,\mathrm dt\) and \(W\) is Brownian with respect to a filtration containing \(\mathcal F_S\) at time zero. Its Brownian law thus holds conditionally on \(\mathcal F_S\). All estimates below are uniform in that initial information. Conditional spatial and time scaling permits us to set the remaining time at \(S\) equal to one.

Consider first a single step with time ratio \(\rho\in[\sqrt2,2]\), elapsed time \(L=1-\rho^{-1}\), and starting position \(x\in[1,2]\). Put \[L_{\min}=1-1/\sqrt2,\qquad \kappa=c_{1/2},\qquad q_* =\kappa^2L_{\min}/2,\qquad y_\rho=\frac{3}{2\sqrt\rho}.\] The terminal first bin is \([\rho^{-1/2},2\rho^{-1/2}]\). Take the fixed spatial corridor \((1/2,3)\). Up to its first exit, (131) gives \(\kappa^2\le\dot Q\le1\), since the remaining time is at most one. Let \(g_{x,\rho}(q)\) move linearly from \(x\) to \(y_\rho\) on \([0,q_*]\), and stay at \(y_\rho\) on \([q_*,1/2]\). It lies in \([1,2]\). On the Brownian event \[\sup_{0\le q\le1/2}|x+W_q-g_{x,\rho}(q)|<1/16,\] no first corridor exit can occur before elapsed time \(L\): at such an exit, \(Q_v\le v\le L\le1/2\), whereas the displayed event places the exit position strictly inside the corridor. This first-exit argument establishes the lower clock bound without assuming it on the event being proved. In particular \(Q_L\ge\kappa^2L\ge2q_*\), so \(Z_{S+L}\) is within \(1/16\) of \(y_\rho\) and lies in the terminal first bin. The paths \(g_{x,\rho}-x\) form a compact family of piecewise linear paths starting at zero. Brownian motion has positive probability to stay in a fixed uniform neighborhood of each such path; a finite cover by neighborhoods of radius \(1/32\) gives one positive lower bound for the displayed event, independent of \(x\) and \(\rho\). This elementary tube fact follows, for example, by requiring finitely many Gaussian increments to lie in prescribed open intervals and the intervening Brownian bridges to stay in prescribed open strips.

For any total ratio \(u'/u\ge\sqrt2\), partition it into factors in \([\sqrt2,2]\) and iterate the conditional estimate. The number of steps is at most \(1+\log_2(u'/u)\), so the resulting probability is at least \(c(u/u')^{P_1/2}\) for a fixed \(P_1\). On the first step the starting interval may instead be \([1.5,J]\): use corridor \((1/2,J+1)\) and the same piecewise linear path, obtaining a first-step constant depending only on fixed \(J\), with the same subsequent power. There is also a uniform bound when starting at exactly \(1.5\) and \(u'/u\in[1,\sqrt2]\). In normalized units the event \(\sup_{q\le L_{\min}}|W_q|<1/16\) suffices, since \([1.5-1/16,1.5+1/16]\) lies inside every corresponding terminal first bin. Here only \(Q_L\le L\le L_{\min}\) is used; no lower clock bound is necessary.

The steering estimates now give a polynomial lower bound for reaching the first bin across the scale ratios used below. We use that bound in an induction over decreasing dyadic slabs of remaining time, comparing the other bins to the first and controlling distant starting bins by Gaussian tails. On \([u_0/2,u_0]\), \(M_1\) has a positive minimum, so the assertion follows from \(M_j\le1\). Suppose the assertion has been established on earlier slabs and let \(u\) lie in the next slab. For \(j\ge2\), set \(u_a=\min(j^2u,u_0)\ge2u\). If \(u_a=u_0\), steering from its first bin gives \(M_1(u)\ge c(u/u_0)^{P_1/2}M_1(u_0)\). Since \(j^2u\ge u_0\), the trivial bound \(M_j(u)\le1\) proves the desired inequality with power \(P_1\) and a fixed constant.

In the uncapped case \(u_a=j^2u\), condition at \(u_a\). Any path that starts below \(1.5\sqrt{u_a}\) and ends in bin \(j\) at time \(u\) must hit the moving boundary \(Z=1.5\sqrt{\text{remaining time}}\). Let \(A\) be the event of starting below this boundary and hitting it before the terminal observation, and let \(S\) be the first hitting time. On \(A\) its remaining time \(u_h\) belongs to \([u,u_a]\). The conditional steering estimate just proved, restarted at \(S\), gives a probability at least \(cj^{-P_1}\) of ending in the first bin, since \(u_h/u\le u_a/u=j^2\); its short-ratio case covers \(u_h/u\in[1,\sqrt2]\). Therefore \[M_1(u)\ge \mathbb E\!\left[\mathbf 1_A\mathbb P(Z\text{ ends in the first bin}\mid\mathcal F_S) \right]\ge cj^{-P_1}\mathbb P(A).\] The target contribution from starting positions below the boundary is at most \(\mathbb P(A)\), because every such target path must hit it. This proves the desired bound \(Cj^{P_1}M_1(u)\) without conditioning on later target success. Starting positions between \(1.5\sqrt{u_a}\) and \(J\sqrt{u_a}\) obey the same bound, with \(C_J\), by direct steering.

For an ancestor in bin \(i\ge J\) at \(u_a\), ending in the target bin requires a martingale displacement at least a constant times \(i\sqrt{u_a}\), once \(J\) is a fixed sufficiently large integer: the upper target endpoint divided by \(\sqrt{u_a}\) is \((j+1)/j\le3/2\). The exponential martingale bound therefore gives probability at most \(C e^{-c_0i^2}\). The induction hypothesis at \(u_a\) bounds the ancestor mass by \(C_*i^P M_1(u_a)\), and steering gives \(M_1(u_a)\le Cj^{P_1}M_1(u)\). Thus the remaining contribution is at most \[C_*Cj^{P_1}M_1(u)\sum_{i\ge J}i^P e^{-c_0i^2}.\] Choose \(P\ge P_1\), then \(J\) making the coefficient of \(C_*\) at most \(1/2\), and finally \(C_*\) large enough to absorb the nonrecursive terms and the initial slab. These choices are independent of the slab and close the induction. ◻

Proof. Repeat the stability argument in Lemma 57, now along an arbitrary support sequence on which \(D\to\infty\). Use the force bin \(1\le r/\sqrt w\le2\) for the positive contribution. The first estimate in (126), valid also for \(v\ge1\), places its state interval below \(-K\sqrt\tau\) when \(D\) is large. There \(q_g\ge\Phi(-2)\) and \(l\to0\) uniformly, so its contribution is at least \(c_0M_1(w)\). Its interval has length at least \(\chi\sqrt w\), because \(|r_h|\le1/\chi\).

Lemma 58 makes the exponential factor in (128) uniformly bounded when comparing this bin to \([-K\sqrt\tau,C_P\sqrt{\tau|\log\tau|}]\). Using the infimum density on the bin, (121) bounds the negative layer contribution by \(CM_1(w)/D\). Errors \(\tau^P\) are negligible compared with \(M_1(w)\): the density lower bound and the bin length give \(M_1(w)\ge c_0\chi\sqrt w\ge c_0\sqrt\tau\). For \(v>a\sqrt D\), sum (126) over the force bins and use Lemma 59. This contribution is at most \(CM_1(w)\sum_{j\ge a\sqrt D-1}j^{P+2}e^{-c_0j^2}=o(M_1(w))\). The other regions have nonnegative contribution as before. Stability is contradicted. Formula (129) transfers the result across gaps, with the same two cases as in the subpower proof. The estimate for \(\int s\) follows from Cauchy–Schwarz for the upper bound and monotonicity for the lower bound. ◻

Proof. For a lower bound, start the relevant martingale in the scaled interval \([0.4,0.6]\). The barriers, bounded \(D\), and (134)–(137) put at least \(c_0\chi\) mass in this interval. Its remaining quadratic variation is at most the square of the corresponding scale. Brownian time change shows that its displacement stays within \(0.2\) of that scale with a fixed positive probability: it suffices that standard Brownian motion stay in \([-0.2,0.2]\) up to time one. Its endpoint is then strictly positive and below the desired cutoff.

For the force upper bound, paths starting in force bin \(j\ge2\) must make a displacement of at least \((j-1)\sqrt w\) to finish below \(\sqrt w\), and hence pay a Gaussian tail. Paths starting in gap bin \(j\ge2\) must make a displacement at least \(j\sqrt\tau\) to have \(R>0\), since that event forces \(Y=0\). Sum the polynomial bin estimates against these Gaussian tails. The remaining region \(\{r\le2\sqrt w,\ X\le2\sqrt\tau\}\) is in a fixed scaled spatial interval and has mass \(O(\chi)\). For the gap upper bound interchange the two martingales, using \(Y>0\Rightarrow R=0\). This proves both statements. ◻

Proof. We first establish the claim for a regularization that makes all endpoint comparisons classical. In the approximants of (110), replace \(m\) by \((\lambda_n+\chi_n)a_n\). Its terminal value is the hinge, and it solves (119) with \(d=\gamma_n/(\lambda_n+\chi_n)\), \(0\le\tau d\le1\). At fixed \(n\) this coefficient is integrable. Approximate it in \(L^1\) by bounded coefficients satisfying the same inequality. Next replace the initial hinge by its heat smoothing of variance \(e'>0\) and replace the bounded coefficient \(d(\tau)\) by \(d(\tau)\tau/(\tau+e')\). Put \(T=\tau+e'\), and denote the replaced coefficient again by \(d\). Then \(D_0=Td\in[0,1]\). These operations converge on interior compact sets, first as \(e'\downarrow0\), then in the coefficient approximation, then as \(n\to\infty\): the linear difference equation and the uniform curvature and value bounds give local uniform convergence, and the semigroup gradient estimates give convergence of spatial derivatives. It is therefore enough to prove the bounds uniformly for this regularization, with \(\sqrt T\) in the definition of \(V\).

The slopes remain in \([-1,0]\) and are strictly monotone. Their endpoint values are \(-1\) and \(0\). Write \(v=-k_h\), and let \(h=h(\tau,k)\) be the inverse position. From the equation for \(k\), \[h_\tau=\tfrac12v_k-dm+\frac{d k(1-k)}v,\qquad h_k=-1/v.\] Differentiate the first identity in \(k\), use \(m_k=k/v\), and compare with the time derivative of \(h_k\). For \(V=\sqrt T\,v\) the result is \[ \partial_{\log T}V=\tfrac12V^2V_{kk} +[\tfrac12+D_0(1-3k)]V-D_0k(1-k)V_k. \tag{140}\] The endpoint values are \(V(0)=V(1)=0\), uniformly on compact time intervals in this regularization. To justify this at spatial infinity, \(v\) solves \(v_\tau=v_{hh}/2+d m v_h+d(1-3k)v\), with Gaussian initial datum, Lipschitz drift of at most linear growth, and bounded potential. Its reverse diffusion representation shows that \(v\) vanishes uniformly at spatial infinity on a compact time interval: starting positions tending to infinity remain there in probability, by the bounded integrated Lipschitz norm. The corresponding equations for \(k\) and \(1-k\) show that their fixed intermediate levels stay in bounded spatial intervals. These facts give the asserted endpoint behavior in inverse coordinates.

Let \(t_k=k(1-k)\) and \(b_+=(6/5)t_k^{3/4}\). Direct differentiation gives \[-b_+b_+''=\frac{36}{25} \left(\frac34\sqrt{t_k}+\frac{3}{16\sqrt{t_k}}\right),\qquad 1-3k-t_k b_+' /b_+=\tfrac14-\tfrac32k.\] For \(k\ge1/6\), \(-b_+b_+''\ge27/25>1\) suffices to make the right-hand side of (140) nonpositive at \(b_+\). For \(k<1/6\), the required inequality is \[\frac{27}{100}\frac{1+4t_k}{\sqrt{t_k}} \ge\frac32\sqrt{1-4t_k}.\] After squaring, it follows from \(x(1-x)/(1+x)^2\le1/8\) for \(0\le x\le1\), with \(x=4t_k\). Thus \(b_+\) is a stationary supersolution for every \(D_0\in[0,1]\).

For \(b_-=Ck(1-k)^{3/2}\), the right-hand side divided by \(b_-\) is bounded below by \[\frac{1-k}{2} \left[1-C^2\left(3k(1-k)-\frac34k^2\right)\right].\] The quadratic in parentheses has maximum \(3/5\); with \(C=9/7\), its product with \(C^2\) is \(243/245<1\). This proves the subsolution inequality.

At the heat-smoothed initial time, the inverse profile is \(V_0(k)=\phi(\Phi^{-1}(k))\) and satisfies \(V_0''=-1/V_0\). Both barriers have zero endpoint values. Their stationary inequalities at \(D_0=0\) give \(b_-b_-''\ge-1\) and \(b_+b_+''\le-1\). If \(b_--V_0\) had a positive interior maximum, then \(b_-''\le V_0''=-1/V_0<-1/b_-\), a contradiction. The same argument at a positive maximum of \(V_0-b_+\) proves \(V_0\le b_+\). Parabolic comparison in (140) now yields the bounds at all later times. One may apply the maximum principle on \([\delta,1-\delta]\) and then let \(\delta\downarrow0\). The zeroth-order coefficient in the linearized difference is bounded above: it involves \((V+b)b''\), which is bounded above for the supersolution and, for the subsolution, on the only relevant region \(V<b_-\). Multiplying the difference by a sufficiently decreasing exponential in time deals with this bound. The uniform endpoint vanishing deals with the spatial boundary. Smooth approximation handles measurable time coefficients. Passing through the regularizations proves (139). ◻

Proof. We first prove an unweighted inequality for every limiting profile satisfying (139). Put \[F_A(z)=V_z^2+6A(2k-1)V^2+6A^2k^2(1-k)^2, \qquad 0\le A\le1.\] Its cumulative integral from \(-\infty\) is nonnegative until \(k=1/2\), since every term is then pointwise nonnegative. For an endpoint with \(k=k_0\le1/2\), change variables from \(z\) to \(k\) to obtain \[\int_{-\infty}^{z(k_0)}F_A(z)\,\mathrm dz=H+6AC_1+6A^2B_1,\] where all following integrals are over \([k_0,1]\) and \[\begin{aligned} H&=\int V V_k^2,& C_1&=\int(2k-1)V,& B_1&=\int t_k^2/V,\\ T_1&=\int t_k,& U_b&=\frac65\int t_k^{3/4},& B_b&=\frac79\int k\sqrt{1-k}. \end{aligned}\] Cauchy–Schwarz applied to \((V^{3/2})'\) on \([1/2,1]\) gives \[H\ge\frac89 V(1/2)^3\ge\frac{81}{5488\sqrt2}>\frac1{100}.\] Put \(b=T_1/U_b\). Integration by parts and weighted Cauchy–Schwarz give \[\begin{align*} C_1&=V(k_0)\{t_{k_0}-bV(k_0)/2\} +\int V(t_k/V-b)V_k\\ &\ge V(k_0)\{t_{k_0}-bV(k_0)/2\} -\sqrt{HB_1\left(1-\frac{T_1^2}{U_bB_b}\right)}. \end{align*}\] Indeed the square integral in this application is \(B_1-2bT_1+b^2\int V\le B_1-T_1^2/U_b\); (139) gives \(\int V\le U_b\) and \(B_1\le B_b\).

Here are explicit bounds for the ratio appearing in this estimate. Under probability weight \(t_k/T_1\), the mean of \(t_k^{-1/4}\) on \([k_0,1]\), \(0\le k_0\le1/2\), is at most its full-interval mean. This follows by symmetry and monotonic decrease on \([0,1/2]\): the integral of \(t_k\) times its deviation from the full mean over \([0,k_0]\) is nonnegative. Cauchy–Schwarz bounds the full mean by \(\sqrt{3\pi/4}\), since \(\int_0^1\sqrt{k(1-k)}=\pi/8\) and \(\int_0^1k(1-k)=1/6\). The mean of \((1-k)^{-1/2}\) increases with the lower cutoff; at \(k_0=1/2\) it is \(7\sqrt2/5<2\). If \(k_0<1/50\), its value is less than \(17/10\): integrate \(k\sqrt{1-k}\) explicitly and evaluate at \(k_0=1/50\). Consequently, using \(\pi<22/7\), \[\frac{U_bB_b}{T_1^2}<\frac{28}{15}\sqrt{\frac{33}{14}} <\frac{287}{100}, \qquad \frac{U_bB_b}{T_1^2}<\frac{119}{75}\sqrt{\frac{33}{14}} <\frac{61}{25}\quad(k_0<1/50).\]

The endpoint term is nonnegative unless \(t_{k_0}<1/64\). Indeed \(b\le5/(6\sqrt2)\) and \(V(k_0)\le(6/5)t_{k_0}^{3/4}\). When \(t_{k_0}<1/64\), necessarily \(k_0<1/50\), and the endpoint term is still greater than \(-1/20000\). To verify the latter, its dependence on \(V(k_0)\) is concave, so its minimum on the permitted interval is at an endpoint. With \(x=t_{k_0}^{1/4}\), the nonzero endpoint is bounded below by \((6/5)x^6(x-1/(2\sqrt2))\). The absolute value of this polynomial’s minimum on \(x\ge0\) is \(6^7/(5\cdot7^7\cdot1024\sqrt2)<1/20000\).

Complete the square in \(A\) in \(H+6AC_1+6A^2B_1\) and use \(A\le1\) for the endpoint loss. The result is at least \[H\left[1-\frac32\left(1-\frac{T_1^2}{U_bB_b}\right)\right] -\frac6{20000}\mathbf 1_{\{t_{k_0}<1/64\}}.\] The preceding rational bounds make this greater than \[\min\left\{\frac{13}{57400},\frac7{6100}-\frac3{10000}\right\} >\frac1{5000}.\] Thus, if \(C_A(z)=\int_{-\infty}^zF_A\), then \(C_A\ge0\) everywhere and \(C_A\ge1/5000\) after the unique point \(z_*\) at which \(k=1/2\).

Let \(P\) be any positive nonincreasing limiting density in (142). Integration by parts against its Stieltjes measure gives \[\int_\mathbb RF_A P\,\mathrm dz =C_A(+\infty)P(+\infty)+\int_\mathbb RC_A\,\mathrm d(-P) \ge \frac1{5000}P(z_*)>0.\] The negative boundary product is zero by the Gaussian tail of \(C_A\) and the exponential bound for \(P\); one may first truncate both tails. The inequality includes the case \(P(+\infty)>0\), which contributes the displayed boundary term.

Finally suppose (143) failed along a sequence to the endpoint. Extract a limit of \(A_0\in[0,1]\), of the spatial profiles, and of the density ratios. Equations (134) and (141) give an integrable Gaussian majorant for the normalized integrands, so dominated convergence applies. The level \(z_*\) stays in a fixed compact by the value and slope barriers, and the limiting density is uniformly bounded below there. The preceding strictly positive lower bound contradicts failure. This proves (143). ◻

Proof. On an open gap, \(\gamma\) is constant. Write \(b=f_{hh}<0\) and \(H_b=\mathbb Eb(t,H_t)^2\). The diffusion equations and Itô’s formula give \[H_b'=\mathbb Eb_h^2-2\gamma\mathbb Eb^3,\qquad H_b''=\mathbb Eb_{hh}^2-12\gamma\mathbb E(b b_h^2)+6\gamma^2\mathbb Eb^4.\] For example \(\mathrm db=b_h\,\mathrm dB-\gamma b^2\,\mathrm dt\) and \(\mathrm db_h=b_{hh}\,\mathrm dB-3\gamma b b_h\,\mathrm dt\); these yield both identities by the product rule. Localization and the spatial tails justify taking expectations.

Substitute \(b=-sk\), \(b_h=sV/\sqrt\tau\), \(b_{hh}=sV_z/\tau\), and \(\gamma s\tau=A_0\). Then \[\begin{align*} \frac{2H_bH_b''-3(H_b')^2}{2H_bs^2/\tau^2} ={}&\mathbb EV_z^2+12A_0\mathbb E(kV^2)+6A_0^2\mathbb Ek^4\\ &-\frac{3}{2\mathbb Ek^2}(\mathbb EV^2+2A_0\mathbb Ek^3)^2. \end{align*}\] Use \(\mathbb Ek^3/\mathbb Ek^2\le1\) and the identity \[\mathbb Ek^4-\frac{(\mathbb Ek^3)^2}{\mathbb Ek^2} =\mathbb E[k^2(1-k)^2]-\frac{(\mathbb E[k^2(1-k)])^2}{\mathbb Ek^2}.\] The last display is therefore at least the left-hand side of (143) minus \[\frac{6A_0^2(\mathbb E[k^2(1-k)])^2}{\mathbb Ek^2} +\frac{3(\mathbb EV^2)^2}{2\mathbb Ek^2}.\] Both errors are \(O(\chi^2)\): (135), (137), and the Gaussian derivative tails bound each numerator’s unsquared expectation by \(C\chi\). Moreover \(\mathbb Ek^2\to\mathbb P(R>0)=1/\alpha>0\) by bounded convergence. The main term in (143) is at least \(c_0\chi\). Hence \(2H_bH_b''-3(H_b')^2>0\) on every sufficiently late open gap. Equivalently \(\psi=(\alpha H_b)^{-1/2}\) is strictly concave there.

Choose a support point \(t_*\) beyond the time at which this determinant inequality holds uniformly. Every support gap to the right of \(t_*\) has both endpoints in the increase support: the left endpoint lies at or after \(t_*\), and accumulation at \(1\) excludes a final gap. For any such gap, stability gives \(\psi\ge\chi\) at those endpoints, whereas \(\chi\) is affine on the gap. Strict concavity consequently gives \(\psi>\chi\) at every interior point. But then \(J'=\alpha H_b-s^2<0\) throughout the gap, contradicting \(J=0\) at both endpoints. Thus there are no complementary intervals to the right of \(t_*\). Closedness of the support proves that it contains \((t_*,1)\).

On this interval the variational conditions give \(J=J'=0\), \(\mathbb Ek^2=1/\alpha\), and \(\mathbb Ekl=c\chi\). Apply Itô’s formula to \(k(t,H_t)^2\): \[0=\frac{\mathrm d}{\mathrm dt}\mathbb Ek^2 =\frac1\tau\mathbb EV^2-\frac{2\gamma}{\chi}\mathbb E[k^2(1-k)] \quad\hbox{for almost every }t.\] The denominator is strictly positive. Interior spatial regularity, the diffusion representation, and dominated convergence make the quotient on the right of (144) continuous in \(t\). It supplies the continuous version of \(A_0\), and hence of \(\gamma\), and proves the asserted identity everywhere on the terminal interval. ◻

Proof. The density estimate (134), together with \(p_{1-\tau}(0)\sqrt\tau/\chi\asymp1\), bounds \(P\) above and below on spatial compact sets. The same estimate bounds it globally by \(C\exp(C|z|+o(1)z^2)\), which is integrable against the Gaussian tail bounds already proved. The value and curvature barriers bound the coefficients of (145) locally, uniformly after sufficiently late translation. Interior estimates for the localized heat equation, applied forward in \(v\) to \(M\) and forward in \(u\) to \(P\), give local bounds and equicontinuity for their spatial derivatives. Applying the equations on a slightly smaller cylinder gives time equicontinuity. For precision, one may first differentiate the localized heat-kernel formula in space: the initial \(C^1\) bound controls the drift term, and iteration gives the derivatives needed below. Bounded, initially merely measurable, time coefficients suffice in this argument.

On the full-support endpoint interval, (144) expresses \(A\) as a quotient of spatial integrals. Its denominator, after division by \(\chi\), is bounded below uniformly. Indeed the value barriers place an interior interval of slope values in a bounded spatial interval; inversion and (139) give positive spatial width, and central positivity of \(P\) gives a positive integral of \(k^2lP\). Gaussian tails and the density bound permit uniform truncation of both integrals. Consequently the same compactness gives compactness of \(A\) itself. Pass first in the integral form of the equations; the quotient then shows that the limiting \(A\) is continuous.

The critical identities in physical units are \(\mathbb E[r_tX_t]=c\tau\) and \(\mathbb E[kl]=c\chi\). They become the first two integrals in (146). Their passage to the limit, and passage in the quotient, use the value barriers, the negative-tail estimate (126), and the Gaussian domination just described. The constants in all these arguments are uniform under translation, so the conclusions hold on the whole time axis. ◻

Proof. We first show that the right side of (148) is strictly positive whenever \(0.41\le A\le1\). Denote \(t_k=k(1-k)\). Its unweighted integrand is positive while \(k\ge5/8\). For a cumulative integral ending at a slope \(k_0\le5/8\), change variables from \(z\) to \(k\), so the cumulative runs from \(k_0\) to \(1\). Put \[b=2(0.6-k),\qquad E=-0.4t_k,\qquad K=12Ak-5A-1-b^2+0.8(1-2k).\] Integrating the inequality \(V(V_k-b-E/V)^2\ge0\) gives the following lower bound for that cumulative: \[ -(bV^2+2EV)(k_0) +\int_{k_0}^1 \left[2V^2+KV-2bE+(6A^2-0.16)\frac{t_k^2}{V}\right]\mathrm dk. \tag{149}\] The boundary term at \(k=1\) vanishes by (139).

The finite check checkA proves strict positivity of (149), uniformly for \(A\in[0.41,1]\) and \(k_0\in[0,5/8]\). Its input is only the two explicit bounds in (139). It partitions \(A\) and \(k\), subdivides the interval between the lower and upper allowed values of \(V\), and integrates cellwise lower bounds. For the boundary quadratic its maximum is at an endpoint when \(b\ge0\); when \(b<0\) it is decreasing for \(V\ge0\). The two omitted endpoint cells each cost at most \(9\) times their width, and on the first cell the boundary expression before its negation is less than \(0.001\). A possible positive partial cell is subtracted rather than included in full. These are exactly the corrections in checkA; Section 12 verifies the arithmetic.

For completeness, positive cumulatives are preserved under the weight \(P\). If \(f\) is the unweighted integrand and \(F(z)=\int_{-\infty}^z f\), then \(F>0\) at every finite point and its terminal value is positive. Since \(P\) is positive and nonincreasing, Stieltjes integration by parts gives \(\int fP=F(\infty)P(\infty)+\int F\,\mathrm d(-P)>0\); limits of truncated expressions give the same formula when a tail value is infinite. The Gaussian/exponential bounds make the left endpoint product zero and justify the limits. Strictness follows either from \(P(\infty)>0\) or from the nonzero positive measure \(\mathrm d(-P)\).

Finally let \(L\) be the endpoint limsup of the original rescaled \(A\). Choose translates realizing \(L\) at time zero and take an entire limit. In that limit \(A(0)=L\) and \(A(u)\le L\) for all \(u\). If \(L\ge0.41\), (148) makes \(A_u(0)>0\), contradicting this maximum. Thus \(L<0.41\), which implies the claimed, slightly weaker bound for every entire limit. ◻

Proof. The central reference check and the tail estimates proved below give \(-1.12\le T_0\le0.13\). Let \(\alpha=q+0.41\cdot0.13\) and solve \(C'=C(\alpha-qC^2)\) with \(C(v)\to\infty\) as \(v\downarrow v_0\). Equation (152) says that \(CW_0\) is a supersolution for \(V_i\) in the \(k\) equation. This comparison does not initially assume a bounded ratio. Indeed in the equation for \(V_i-CW_0\) the extra zeroth-order coefficient is \(\tfrac12(V_i+CW_0)CW_{0,kk}\le0\), because \(W_{0,kk}=-1/W_0\). The remaining positive coefficient is bounded independently of \(C\). At time \(v_0+h\), the positive part of \(V_i-CW_0\) has supremum tending to zero as \(h\downarrow0\): on interior \(k\)-intervals \(C\to\infty\), and near both endpoints the uniform upper bound (139) tends to zero. Comparison, followed by \(h\downarrow0\) and \(v_0\to-\infty\), gives \(R\le\sqrt{\alpha/q}\).

To obtain a positive lower ratio without assuming one, set \(U=1/R\) in heat-reference position. Differentiating (152) and dropping its nonpositive gradient-square term gives \[U_v\le\tfrac12R^2U_{zz} +\left[-qR^2z+A\frac{t_k}{W_0}\right]U_z -\delta_0U+qR, \qquad \delta_0=q-0.41\cdot1.12>0.\] Here \(0\le At_k/W_0\le C\) and the already proved upper bound controls \(R\). The lower bound in (139) and elementary normal-tail estimates give, uniformly in entire time, \[U(z)\le C\bigl(1+z_+^2+e^{z_-^2/3}\bigr).\] The diffusion representation for this subsolution has no surviving boundary term at infinity. Here is a direct justification. For \(r<q\) and an even integer \(m\ge2\), use a \(C^1\), piecewise \(C^2\) Lyapunov function comparable to \(1+z_+^m+e^{rz_-^2}\). On the negative half-line its generator is bounded above by a constant: the quadratic exponential contribution has coefficient \(2r(r-q)R^2z^2\le0\), and the additional drift is nonnegative, hence contributes nonpositively there. On the positive half-line its generator is at most \(C(1+z_+^{m-1})\). Stopped Itô and Jensen therefore bound its expectation over duration \(T\) by \(C_z(1+T)^m\). Taking \(m=4\), \(r=0.45\) supplies uniform integrability for the smaller growth bound on \(U\), permitting removal of stops at each finite duration. The representation thus yields \[U(v,z)\le e^{-\delta_0T}\mathbb E[U(v-T,Z_T)] +q\sup R\int_0^T e^{-\delta_0s}\,\mathrm ds.\] The first term tends to zero polynomially times an exponential. We obtain \(U\le q\sup R/\delta_0\), a uniform positive lower bound for \(R\).

Now compare \(R\) to the spatially constant solution of \(r'=r(\delta_0-qr^2)\) starting below that positive lower bound. The coefficients in (152) are bounded in diffusion and at most linear in drift, so the whole-line comparison stated above applies. Let its starting time tend to \(-\infty\); this gives \(R\ge\sqrt{\delta_0/q}\). The quotient of the upper and lower constants in (153) is less than \(3.8\). ◻

Proof. Let \(d=\sup|A-a_*|<0.01\). If \(d=0\) there is nothing to prove; assume \(d>0\). Interpolate from the root to the input as above, with \(c_+=0.3\) and \(\xi=a_*\). Normalize the two densities at the same spatial position \(z_c=z_*(1/2)\) to have value one. This normalization preserves both identities (159). Unlike the previous normalization, it does not require the input density at its own median slope to be one.

Our target is \(|A(v)-a_*|\le c_{\rm loc}d\) at every entire time, with \(c_{\rm loc}<1\). We obtain it by bounding the change in the signed functional from its zero value at the root and dividing by the positive denominator in the identity for \(N_{a_*}\). The estimates below control that one quotient, including its endpoint tails, by a constant times \(d\).

Write \(W=W_*\). Equation (161) implies \[\frac{|\epsilon|}{d}\le E_m:=\frac{L}{1-0.01L},\qquad |(V_i/W)^{-\sigma}-1|\le\sigma dE_m, \qquad E_m<3.1.\] For the second inequality, if \(V_i/W<1\), convexity of \(t\mapsto(V_i/W)^{-t}\) puts its value below the chord on \([0,1]\); if \(V_i/W\ge1\), use \(1-e^{-x}\le x\) and the logarithmic bound. Integrating \(E_m/W\) with weights \(k\) and \(l\) in (158) gives bounds \(L_M,L_X\) for primitive changes divided by \(\sigma d\). Their sum \(L_z\) bounds the position change: \[|z_i(k)-z_*(k)|\le dL_z(k).\] The code arrays lm, lx, lz enclose these functions; the global bounds \(3.11M_*\) and \(3.11X_*\) give convenient caps and tail allowances.

The logarithmic density difference splits as \[\begin{split} \pi(k)={}&\log P_i(z_*(k))-\log P_*(z_*(k))\\ &+\log P_i(z_i(k))-\log P_i(z_*(k)). \end{split}\] The first term is zero at \(k=1/2\) and has \(k\)-derivative bounded by \(2d/W\), by (162). The absolute value of the second is bounded by \(d\) times shifts: use the position interval \(z_*\pm dL_z\) and (162) to bound \(Q_i\) there by a range query for \(Q_*\) plus \(2d\). Hence \[\frac{|\pi(k)|}{d}\le2|z_*(k)-z_c|+\texttt{shifts}(k).\] These estimates enclose the interpolated \(v,p,M,X\) on each of the four parameter subintervals used by localcheck.

Apply (160) to the middle contribution, divided by \(d\). The curvature term has the enclosure cv; the density term has the enclosure cp. For the latter, integrate by parts only the first, anchored, part of \(\pi\) just displayed, and bound the shift part directly. The primitive-tail error now uses \(4\) instead of \(6\): indeed \[\frac{|\epsilon|}{d}\frac{W}{v} \le\frac{3/0.97}{0.97}<4.\] This holds on the full open interval \((0,1)\), including the omitted tails, because (161) was proved there analytically. For example the omitted lower primitive derivative obeys \[\frac1d\int_0^\kappa\frac{j|\epsilon(j)|}{v(j)}\,\mathrm dj \le4\int_0^\kappa\frac j{W(j)}\,\mathrm dj=4M_*(\kappa),\] and the upper primitive has the corresponding bound \(4X_*(1-\kappa)\). These are precisely the two factors in the local terror bound. The certificate again checks this error below \(3\cdot10^{-7}\). It also gives a positive lower bound, called den, for \[D_i=2\int_0^1\frac{P_i}{V_i}kt_k\,\mathrm dk,\] by integrating its lower enclosure on the middle grid only. The last parameter box includes \(\sigma=1\), so the denominator enclosure applies to the actual input.

The full reference functional is exactly zero by the stationary identities at \(a_*\). Therefore its tail contribution must be subtracted from the input tail contribution before applying the local bound. We bound this difference by differentiating along the interpolation, not by an error independent of \(d\). Here are explicit bounds for that step. Equations (157) and (162), with the certified tight reference interval, give \(|Q_i|\le1.5\). The value barriers give \(M_*+X_*\le x+1.6\) and the certificate gives \(|z_c|<2\). Thus \[\frac{|\pi|}{d}\le2(x+2)+1.5\cdot3.11(x+1.6)<7x+12.\] With \(s=x-x_c\ge0\), the interpolated density satisfies \[p\le5e^5e^{0.47s}\quad\text{on the negative spatial tail}, \qquad p\le5e^{0.07s}\quad\text{on the positive spatial tail}.\] At the edge the normalization estimate above, \(d<0.01\), and \(x_c<20\) permit the factor \(5\). Along a tail one may use the reference density bound together with \(p=P_*\exp(\sigma\pi)\) and the \(7s\) part of the last display. On the negative side the reference has logarithmic growth bounded by \(0.4\); the positive-side reference density is at most its edge value. This gives exactly the displayed exponents.

Every primitive derivative has absolute value at most \(3.1d\) times the primitive itself: the full-interval bound \(|\epsilon(j)|<3.1d\) and positivity imply \[|\partial_\sigma M_\sigma(k)| \le3.1d\int_0^k\frac j{v_\sigma(j)}\,\mathrm dj =3.1dM_\sigma(k),\] and likewise for \(X_\sigma\). The reference tail bounds and \(v_\sigma\ge0.97W\) dominate these differentiated integrals, so differentiation under their endpoint integrals is justified. Therefore the pointwise absolute functional derivative, divided by \(d\), is at most \[\frac pv\left[ \left(\frac{|\pi|}{d}+3.1\right) \bigl(v^2+0.3(MX+t_k)+2a_*kt_k\bigr) +0.3\cdot6.2MX\right].\] Use \(\mathrm dk/v\le\mathrm dx/0.97\). On the negative side, \(MX\le2.6l\) and \(v^2\le1.061(x+1)^2l^2\); on the positive side, \(MX\le1.2k\) and \(v^2\le1.061(x+2)^2k^2\). For the product bound on the negative side, \(p_*:=q-a_*\ge0.2063465\) and \(x\ge10\) give explicitly \[\frac{MX}{l}\le \frac{1/(2p_*)+\kappa/(400p_*^2)}{0.97^2}<2.576<2.6;\] on the positive side, \(x\ge7\) gives \(MX/k\le(1+\kappa/49)/0.97^2<1.063<1.2\). The reference tail bounds give \(l\) or \(k\) at most \(\kappa e^{-2s}\), and \(x_c<20\). Integrating proves a total tail error below \(10^{-6}\) after division by \(d\). More explicitly, the non-\(v^2\) part on the negative side is bounded by \[\frac{5e^5}{0.97}\,\kappa \int_0^\infty\bigl[(156+7s)1.68+4.84\bigr]e^{-1.53s}\,\mathrm ds.\] The positive-side non-\(v^2\) counterpart is \[\frac5{0.97}\,\kappa \int_0^\infty[(156+7s)1.26+2.232]e^{-1.93s}\,\mathrm ds.\] The two \(v^2\) contributions are bounded respectively by \[\begin{split} &\frac{5e^5}{0.97}\,1.061\kappa^2 \int_0^\infty(156+7s)(21+s)^2e^{-3.53s}\,\mathrm ds,\\ &\frac5{0.97}\,1.061\kappa^2 \int_0^\infty(156+7s)(22+s)^2e^{-3.93s}\,\mathrm ds. \end{split}\] Using \(e^5<149\) and \(\int_0^\infty s^n e^{-bs}\,\mathrm ds=n!/b^{n+1}\) reduces their sum to a rational expression below \(5.035\cdot10^{-7}\). This proves the stated \(10^{-6}\) allowance. Including the primitive-tail error, the \(2\cdot10^{-6}\) used by localcheck is sufficient.

The certified middle derivative bound plus these errors, divided by the positive denominator enclosure, gives a number \(c_{\rm loc}<0.95\). Hence at each entire time \[|A(v)-a_*|=\frac{|N_{a_*}(V_i,P_i)|}{D_i} \le c_{\rm loc}\,d.\] Taking the supremum yields \(d\le c_{\rm loc}d\), contradicting \(d>0\). Thus every such density input has \(A\equiv a_*\). In particular this holds for every critical entire limit. Compactness now implies convergence of the original \(A\) to \(a_*\): otherwise a sequence with a fixed nonzero deviation would produce an entire limit with that deviation at time zero. Finally \(\mathrm d\log\chi/\mathrm d\log\tau=A\); integrate between a fixed positive \(\tau_0\) and \(\tau\), divide by \(\log\tau\), and use the convergence of \(A\) to obtain the logarithmic susceptibility limit.

A stationary reference at any other root in the certified interval is a density input: its value and curvature bounds were proved above, and its positive eigenfunction has bounded nonpositive logarithmic slope satisfying (147). The stationary integral identities are exactly (159). The implication just proved therefore forces its constant coefficient to equal \(a_*\). This proves uniqueness in the interval without assuming that the other reference arises as a critical translation limit. ◻

Proof. Let \(E=M_a-\widetilde M\) and let \(r\) denote the residual in (165). Subtraction gives \[ E''+2B E'-(2p-2a\widetilde M')E=-r. \tag{167}\] The zero-order damping satisfies \(2p-2a\widetilde M'\ge .18-.0082>.17\). The true deviations from the hinge at \(\pm24\) are bounded, respectively, by \(e^{-q24^2}/24\) and \(e^{-p24^2}/(2p\,24)\), by (150). Combining these with (166), the maximum principle applied to the constant barriers \(\pm10^{-22}\) yields the first claim. At an interior join a strict positive maximum of the difference is excluded by the same one-sided argument, since its first derivative is continuous.

On either half-line write \(x=|z|\), \(\beta(x)=\varepsilon B(\varepsilon x)\), and \(F=\partial_xE(\varepsilon x)\). Equation (167) reads \[F'+2\beta F=f,\qquad |f|<2.1\cdot10^{-22}.\] On the positive half-line \(\beta\ge qx\); on the negative half-line \(\beta=px-aX\ge px-.34\), using \(X\le .8\). Consequently the total magnitude of the negative part of \(2\beta\) is at most \(.34^2/p<1.29\). On \([0,.1]\) we also have \(\beta\le .38\). The integrating-factor formula for \(F\), integrated once more between zero and \(.1\), and \(|E(.1)-E(0)|<2\cdot10^{-22}\) now imply \(|F(0)|<2.4\cdot10^{-21}\). Explicitly, the coefficient of \(F(0)\) in this second integration is at least \(.1e^{-.076}\), whereas the source contribution has absolute value at most \(.1^2e^{1.29}(2.1\cdot10^{-22})/2\). Finally, on \([0,24]\) the same formula gives \[|F(x)|\le e^{1.29}\bigl(2.4\cdot10^{-21} +24\cdot2.1\cdot10^{-22}\bigr) <4\cdot10^{-20}.\] ◻

Proof. First replace the coefficients in (168) by their limiting values \(\beta=p_\varepsilon x\) and \(q-ak=p_\varepsilon\). For the residual \(\mathcal R(S)=S'-2\beta S+S^2-2(q-ak-\lambda)\), direct expansion gives \[\mathcal R(T)=\frac{(2c-5)e+d^2}{x^6} +\frac{2de}{x^8}+\frac{e^2}{x^{10}}.\] Thus \(x^6|\mathcal R(T)|\le6.62|e|+d^2+1\) for \(x\ge12\). Replacing \(T\) by \(T\pm Kx^{-7}\) adds the leading term \(\mp2p_\varepsilon Kx^{-6}\) and a remainder whose absolute value, after multiplication by \(x^6\), is at most \[\frac{|-7+2c+2d/x^2+2e/x^4|K}{x^2}+\frac{K^2}{x^8} \le .7p_\varepsilon K.\] The true coefficients differ from their limiting values by at most \[|\beta-p_\varepsilon x| \le\frac{.41e^{-p_\varepsilon x^2}}{2p_\varepsilon x}, \qquad |q-ak-p_\varepsilon|\le .41e^{-p_\varepsilon x^2}.\] Their additional residual cost, multiplied by \(x^6\), is less than \(8\). For example, the largest exponential envelope is obtained with \(p_\varepsilon=.09\) at \(x=12\), since \(x^6e^{-p_\varepsilon x^2}\) is decreasing for these values; inserting \(|T\pm Kx^{-7}|<.071\) proves the stated bound. The definition of \(K\) leaves strict room: \[1.3p_\varepsilon K-(6.62|e|+d^2+1+8) =6.38|e|+2.9d^2+30>0.\] Accordingly the lower barrier has positive residual and the upper barrier negative residual. These are the required signs for integration from larger to smaller \(x\).

Start at \(T(R)\) at a finite \(R>12\) and integrate inward. Comparison keeps the solution within the barriers. Compactness on finite intervals supplies a limit as \(R\to\infty\). If two such limits are compared, their difference satisfies a homogeneous first-order equation with coefficient \(2\beta-S_1-S_2\ge p_\varepsilon x\) for \(x\ge12\). Inward integration therefore forces their difference to vanish as the comparison point tends to infinity. The same formula with the additional source \(-2(\lambda_1-\lambda_2)\) proves continuity in \(\lambda\) on compact subintervals.

The first estimate in (171) follows immediately from (170). For the second, substitute the enclosure into (168). The resulting upper bound is \[\begin{align*} \frac{2|d|}{x^3}+\frac1{x^5}\bigg(&4|e|+6.62|e|+d^2+1+8 +2p_\varepsilon K\\ &+\frac{(1+2\cdot.82)K}{x^2}+\frac{K^2}{x^8}\bigg). \end{align*}\] To justify the constant \(.82\) in this expression, observe that \(d\) and \(e\) have opposite signs and \(|e|/(|d|x^2)=|2c-3|/(2p_\varepsilon x^2)<1\) when \(d\ne0\). Thus \(xT\) has the sign of \(c\) and magnitude at most \(|c|<.81\); the case \(d=0\) is immediate. The displayed bound, using (170), is less than \(.00514\). Here one must retain the identity \(p_\varepsilon K=10|e|+3d^2+30\); replacing the two factors by separate worst-case bounds would lose the required estimate.

Finally, integrating \(S=c/x+O(x^{-3})\) gives \(P(\varepsilon x)=C x^c(1+O(x^{-2}))\) with \(C>0\). After the conjugation \(P=e^{\int B}\psi\), this is the square-integrable branch of the Schrödinger equation. Reduction of order gives a second independent solution growing by the reciprocal Gaussian factor; hence no other square-integrable branch is available at this endpoint. ◻