Proof. We first bound the expected quadratic process that defines the smallest eigenvalue of \(S_A\). Concentration will give a margin at each fixed \(z\), and a finite mesh will extend it along the path.

We use two standard Gaussian facts. A real-valued \(L\)-Lipschitz function of a standard Gaussian vector satisfies \(\mathbb P\{|F-\mathbb EF|>u\}\le2\exp(-u^2/(2L^2))\). Also, if two centered Gaussian processes have ordered squared increments, the process with larger squared increments has the larger expected supremum (Sudakov–Fernique). The latter comparison remains valid after the same deterministic function is added to both processes; see the deterministic-shift comparison in Vitale [22]. Both facts apply by finite nets and continuity to the compact index sets below. The linear map from the independent standard Gaussian coordinates to \(W\), in Frobenius norm, has norm \(\sqrt{2j/n}\).

For the comparison, write \(w=A^{1/2}p\) for \(\lVert p\rVert=1\) and consider \[X_p=w^TWw,\qquad Y_p=2\sqrt{j/n}\,\lVert w\rVert\,g^Tw,\] where \(g\) is standard Gaussian in \(\mathbb R^n\). If \(w'=A^{1/2}p'\), direct calculation gives \[\begin{align*} &\mathbb E(Y_p-Y_{p'})^2-\mathbb E(X_p-X_{p'})^2\\ &\qquad=\frac{2j}{n} \left[(\lVert w\rVert^2-\lVert w'\rVert^2)^2 +2(\lVert w\rVert\lVert w'\rVert-w^Tw')^2\right]\ge0. \end{align*}\] Consequently, for each \(z\in[0,1]\), \[\mathbb E\sup_{\lVert p\rVert=1}\{zX_p-z^2j\chi_A\lVert w\rVert^2\} \le \mathbb E\sup_{\lVert p\rVert=1}\{zY_p-z^2j\chi_A\lVert w\rVert^2\}.\] Set \(T_g=\sqrt{j/n}\lVert A^{1/2}g\rVert\). Since \(g^Tw=p^TA^{1/2}g\le\lVert A^{1/2}g\rVert\), the last supremum is at most \[\sup_{0\le v\le\sqrt{a_+}} (2zT_gv-z^2j\chi_Av^2).\] The remaining supremum depends on the Gaussian vector only through \(T_g\). The variable \(T_g^2\) has mean \(j\chi_A\) and variance \(2j^2\mathop{\mathrm{Tr}}(A^2)/n^2\le C/n\). The inequality \(|\sqrt x-\sqrt y|\le\sqrt{|x-y|}\) therefore shows, uniformly in \(A\), that \(\mathbb E|T_g-\sqrt{j\chi_A}|=o(1)\). On replacing \(T_g\) by \(\sqrt{j\chi_A}\) in the preceding display, the expression becomes \(2d-d^2\), with \(0\le d=z\sqrt{j\chi_A}v\le\sqrt j\,a_+<1\). Thus, with \(\delta=(1-\sqrt j\,a_+)^2>0\), \[ \mathbb E\sup_{\lVert p\rVert=1} \{zp^TA^{1/2}WA^{1/2}p-z^2j\chi_Ap^TAp\} \le1-\delta+o(1), \tag{24}\] uniformly in \(A,z\). The supremum in (24) is \(a_+\)-Lipschitz in \(W\) for Frobenius norm. Gaussian concentration gives an exponentially small probability that it exceeds \(1-\delta/2\), uniformly for each fixed \(A,z\).

To pass from a fixed \(z\) to the entire interval, we also need a bound on \(\lVert W\rVert\). Taking \(A=I\) in the comparison without deterministic offsets gives \[\mathbb E\lambda_{\max}(W)\le2\sqrt{j/n}\mathbb E\lVert g\rVert\le2\sqrt j.\] The same holds for \(-W\). Concentration of these two eigenvalues proves the stated norm tail. On \(\lVert W\rVert\le R\), the function of \(z\) inside the supremum in (24) is uniformly Lipschitz, with constant at most \(a_+R+2ja_+^2\). A sufficiently fine constant mesh of \([0,1]\), followed by a union bound, therefore gives (23), for example with \(c_0=\delta/4\). ◻

Proof. The calculation has two parts. We first justify integration by parts for the truncated words and compute the inverse factor alone. We then obtain a recursion in the number of ordinary \(W\) factors; the same recursion determines the formal prediction exactly and the expectation up to \(O(n^{-1})\).

All constants in this proof depend only on the fixed length and parameter bounds. Lemma 3 shows that, for every fixed deterministic parameter tuple, the truncations agree with the actual matrices for all \(z\in[0,1]\), except on an event of probability \(Ce^{-c_2n}\). Gaussian integration by parts can be applied to the globally Lipschitz truncated factors, using weak derivatives. More explicitly, for a scalar Lipschitz function \(H\), \[ \mathbb E[W_{ab}H(W)] =\frac jn\mathbb E\bigl[\mathrm dH(W)[E_{ab}+E_{ba}]\bigr]. \tag{30}\] When \(a=b\), the elementary direction on the right is doubled, as required by the diagonal variance \(2j/n\).

For (30), remove the projection from the distinguished ordinary \(W\) factor. The resulting expectation error is exponentially small, since the other factors are bounded and Gaussian norm tails control the removed factor. On the event of agreement, the differential of the inverse is \[ \mathrm dK_A(z;W)[E]=zK_A(z;W)AEK_A(z;W). \tag{31}\] We may use the expressions on the right, with truncated factors, on the whole probability space. The error is exponentially small even after the index sums: all factors and their true truncated differentials are bounded, there are at most a fixed polynomial number of indices, and the disagreement event has exponentially small probability. More precisely, the elementary direction \(E_{ab}+E_{ba}\) has Frobenius norm at most \(2\). Both the true weak differential and its formal replacement are therefore bounded by \(C\) in each entry; their difference vanishes on the agreement event. Each such differential error has expected absolute value at most \(Ce^{-c_2n}\). Expanding the diagonal entry of a length-\(L\) word and applying the product rule uses at most \((L-1)n^{L-1}\) scalar contributions before any index identifications; the factor \(j/n\) in integration by parts only improves this bound. Thus their total error is at most a fixed polynomial in \(n\) times \(e^{-c_2n}\), hence at most \(C'e^{-c'n}\). Subsequent words have bounded lengths depending only on the original length. We therefore suppress hats during the loop calculation, with this convention understood.

We now compute the base case, in which the only nondiagonal factor is \(K=K_A(z;W)\). Write \(a_i=A_{ii}\), \(k_i(z)=\mathbb EK_{ii}\), and \(r(z)=n^{-1}\sum_i a_i k_i(z)\). Differentiating \(K_{bi}\) in (30) gives \[\begin{align*} \mathbb E(WK)_{ii} &=\frac{zj}{n}\mathbb E\left[(\mathop{\mathrm{Tr}}KA)K_{ii}\right] +\frac{zj}{n}\sum_b\mathbb E[(KA)_{bi}K_{bi}]+O(e^{-c'n})\\ &=zjr(z)k_i(z)+O(n^{-1}). \end{align*}\] For the second sum, the absolute value of the sum over \(b\) is at most \(\lVert KAe_i\rVert\lVert Ke_i\rVert\le C\). The factorization in the first term follows from (28); its covariance is at most \(C n^{-3/2}\), which is more than enough. The inverse identity \(K=I+zAWK-z^2B_AAK\) now yields \[ \mathbb E(WK)_{ii}=zjr k_i+O(n^{-1}),\qquad k_i=1+z^2ja_i(r-\chi_A)k_i+O(n^{-1}). \tag{32}\]

The loop equations must select the solution near \(k_i=1\). We establish this by continuation from \(z=0\), where the inverse is the identity. Put \(h(z)=r(z)-\chi_A\). The truncated \(k_i\) are uniformly bounded and continuous in \(z\). If \(|h(z)|\le\delta_1\), the second equation in (32) gives \(\max_i|k_i(z)-1|\le C\delta_1+C/n\). Choose \(\delta_1>0\) small, using (16), so that throughout this region and for large \(n\), \[\left|z^2j\frac1n\sum_i a_i^2 k_i(z)\right| \le1-\delta_2\] for some fixed \(\delta_2>0\). Averaging the second equation in (32) with weights \(a_i\) gives \[h(z)=z^2j\left(\frac1n\sum_i a_i^2k_i(z)\right)h(z)+O(n^{-1}),\] and hence \(|h(z)|\le C/n\) while \(|h(z)|\le\delta_1\). Since \(h(0)=0\), continuity rules out a first exit from that region for large \(n\). It follows uniformly in \(z\) that \[ \mathbb EK_{ii}=1+O(n^{-1}),\qquad \mathbb E(WK)_{ii}=zB_A+O(n^{-1}). \tag{33}\]

To complete the base case, we verify the formal identity \(\mathsf P(K_A(z;W))=I\) independently of the expectation calculation. The cancellation takes place at each coefficient of the series. In the expansion (21), a block \(-z^2B_AA\) may be regarded as the negative contraction of two adjacent blocks \(zAW,zAW\), since their contraction is \(z^2j\chi_AA=z^2B_AA\). Fix a completely noncrossing-paired chain \((AW)^{2m}\) with \(m>0\). Let \(\mathcal A\) be the nonempty set of its pairs which join two originally adjacent occurrences of \(W\). It is nonempty because every nonempty noncrossing pairing of a linear order has an adjacent pair. All pairs in \(\mathcal A\) are disjoint. The terms of (21) which give this paired chain are in bijection with subsets \(T\subseteq\mathcal A\): contract the pairs in \(T\) into the negative deterministic blocks, and pair the remaining occurrences normally. The magnitude of the final diagonal contribution is unchanged, and its sign is \((-1)^{|T|}\). Thus the total multiplier is \(\sum_{T\subseteq\mathcal A}(-1)^{|T|}=0\). This argument is finite at each power of \(z\) and proves the asserted formal identity. Multiplication by surrounding diagonal matrices proves both claims of the lemma for words with no ordinary \(W\) letters.

For the induction step, select an ordinary occurrence of \(W\) in a general word and apply integration by parts at that occurrence. We describe the resulting finite recursion directly, without requiring convergence of the formal series. In each product-rule term, mark the partner of the selected occurrence: for a derivative falling on an ordinary factor, this is the \(W\) being differentiated. If the derivative falls on \(K_A\), replace that factor by \(zK_AA\,\boxed W\,K_A\) and take the boxed occurrence as the partner. In either case, list the two marked positions in their linear order, so the marked product reads \[U\,\boxed W\,V\,\boxed W\,Q.\] The two orientations in (30) contribute respectively \[ \frac jn\mathbb E\bigl[(UQ)_{ii}\mathop{\mathrm{Tr}}V\bigr],\qquad \frac jn\mathbb E(UV^TQ)_{ii}. \tag{34}\] There is an additional factor \(z\) when the derivative fell on \(K_A\). The second expression in (34) is \(O(n^{-1})\) by operator-norm bounds, including when it contains two copies of \(K_A\). The first expression is \[ j\left(\frac1n\mathbb E\mathop{\mathrm{Tr}}V\right)\mathbb E(UQ)_{ii}+O(n^{-1}), \tag{35}\] again by (28). The index sums explain the two terms. The opposite orientation identifies the two interior indices and gives \(\mathop{\mathrm{Tr}}V\); the equal orientation reverses the interior path and gives \(V^T\).

Consequently \(\mathbb E\widehat Q_{ii}\) is the finite sum of the leading products (35) over all ordinary partners, plus \(z\) times the corresponding leading product for the one \(K_A\) factor if present, with an \(O(n^{-1})\) remainder. Each word \(V\) and \(UQ\) in this recursion has fewer ordinary \(W\) letters than the original word. Each contains at most one \(K_A\): if the derivative fell on \(K_A\), the splitting places one of its two copies in each word. Extra diagonal factors \(A\) cause no problem, since at most one is added per decrease of the induction index.

The decrease in the number of ordinary letters holds whether the selected occurrence lies before or after the inverse. The two possible orders can be checked explicitly. If the original word is \(P K_A R W T\), the contraction with \(K_A\) contributes \[zj\,\mathbb E\left[(P K_A A T)_{ii}\frac{\mathop{\mathrm{Tr}}(K_A R)}n\right].\] If it is \(P W R K_A T\), that contribution is \[zj\,\mathbb E\left[(P K_A T)_{ii}\frac{\mathop{\mathrm{Tr}}(R K_A A)}n\right].\] Here \(P,R,T\) contain no inverse factor. The displayed outer and inner words each have one inverse factor, even though the original distinguished occurrence may lie anywhere among the ordinary letters.

It remains to compare this expectation recursion with the formal prediction. Sort the noncrossing pairings by the partner of the selected ordinary occurrence. The inside and outside may then be paired independently; their predictions are respectively \(\mathsf p(V)\) and \(\mathsf P(UQ)\). If the partner occurs inside the series for \(K_A(z;W)\), marking that occurrence in the formal series is exactly the formal differentiation identity \(\mathrm dK_A[E]=zK_AAEK_A\). Thus the entire sum over positions in that series is the single \(z\)-weighted product just described.

We can now induct on the number \(r\) of ordinary \(W\) letters. An ordinary partner leaves a total of \(r-2\) such letters in the two smaller words; a partner in \(K_A\) leaves \(r-1\). The base prediction is constant. The recursion therefore proves that the formal prediction is a polynomial of degree at most \(r\). It also proves (29): use the induction hypothesis for the two expectations in each leading product, and average the diagonal induction hypothesis to handle its normalized trace. There are only finitely many products and all factors are uniformly bounded. The same induction, with no inverse factor, proves the statement for ordinary words. ◻

Proof of Lemma 2. We first make the centered estimates uniform over the diagonal parameters, then add the expectation estimate from Lemma 4. Fix a letter pattern and let \(\theta\) range over its parameter box, which has at most \(C_Ln\) real coordinates. Work initially with the truncated words at \(z=1\). Conjugation by any diagonal sign matrix preserves the law of \(W\) and conjugates \(\widehat Q_\theta\) by that same sign matrix. The diagonal parameters commute with the conjugation, and both the projection and the functional calculus respect it. Thus \(\mathbb E(\widehat Q_\theta)_{ab}=0\) whenever \(a\ne b\).

Define the two matrix seminorms \[N_2(M)=\lVert \mathop{\mathrm{diag}}M\rVert_2,\qquad N_4(M)=\left(\sum_{a\ne b}|M_{ab}|^4\right)^{1/4}.\] Both are \(1\)-Lipschitz with respect to Frobenius norm. The entry moment bounds following (28) give \[ \mathbb EN_k(\widehat Q_\theta-\mathbb E\widehat Q_\theta)\le C \quad(k=2,4). \tag{36}\] Indeed, sum the \(n\) second moments for \(k=2\), and the at most \(n^2\) fourth moments for \(k=4\), then apply Jensen’s inequality. For two parameter tuples at distance at most \(\Delta\), the same argument, using (27), gives \[\mathbb EN_k\bigl((\widehat Q_\theta-\mathbb E\widehat Q_\theta) -(\widehat Q_{\theta'}-\mathbb E\widehat Q_{\theta'})\bigr) \le C\sqrt\Delta.\] The random norms in this last display concentrate around their means at Gaussian scale \(C\sqrt{\Delta/n}\), again by (27). The centered expectations are deterministic and do not change Lipschitz constants.

To control every parameter tuple, approximate it successively on finer meshes. For each \(\ell\ge0\), choose a sup-norm mesh \(\mathcal N_\ell\) of spacing \(2^{-\ell}\) in the parameter box. It may be chosen with \[|\mathcal N_\ell|\le\exp(C_L(\ell+1)n).\] For a tuple \(\theta\), choose nearest mesh points \(\theta_\ell\in\mathcal N_\ell\). Then \(\lVert \theta_\ell-\theta_{\ell-1}\rVert_\infty\le C2^{-\ell}\). For all pairs of mesh points at this distance, concentration and a union bound show that their centered increments have both \(N_2\) and \(N_4\) norm at most \[C'\sqrt{\ell+1}\,2^{-\ell/2},\] except with probability \(Ce^{-c(\ell+1)n}\), provided \(C'\) is chosen sufficiently large. Specifically, the squared ratio of this threshold to the Gaussian scale is a constant multiple of \((\ell+1)n\), and its constant can be chosen to dominate the logarithm of the number of mesh pairs. The initial mesh is bounded in the same way using (36). Summing the failure probabilities over \(\ell\), and summing the convergent increment bounds along each chain, proves, with probability \(1-Ce^{-c_1n}\), \[ \sup_\theta N_k(\widehat Q_\theta-\mathbb E\widehat Q_\theta)\le C \quad(k=2,4). \tag{37}\] At each fixed \(n\), continuity of the truncated word and its expectation in \(\theta\) justifies taking the limit of the mesh chain. There are only finitely many letter patterns of length at most \(L\), so their events may be intersected.

The centered words are now controlled on one event for every parameter tuple. Lemma 4 supplies the remaining bias estimate, \(N_2(\mathbb E\widehat Q_\theta-\mathsf P(Q_\theta))\le C/\sqrt n\), uniformly in \(\theta\). Sign symmetry makes the off-diagonal of \(\mathbb E\widehat Q_\theta\) identically zero. Combining these facts with (37) proves the desired diagonal and off-diagonal assertions for the truncated words. Their operator norms are already bounded by construction. On \(\lVert W\rVert\le R\) and at parameters with \(S_A(1;W)\succeq cI\), the truncations equal the exact words, proving the GOE version.

The GOE estimate transfers to the zero-diagonal model through \(J=W-D_{\mathop{\mathrm{diag}}W}\). This last step uses a Frobenius bound on the removed diagonal. Since \[\lVert D_{\mathop{\mathrm{diag}}W}\rVert_{\mathrm F}^{2} \ \stackrel{\mathrm{law}}=\ \frac{2j}{n}\chi_n^2,\] it is bounded by a fixed constant except with exponentially small probability. The norm tail for \(W\), and a sufficiently large choice of \(R\), ensure also that both \(W\) and \(J\) lie in the operator-norm ball except with exponentially small probability. For every parameter tuple, (26) then bounds the Frobenius norm of the difference of the truncated word evaluated at \(W\) and at \(J\) by a fixed constant. Both \(N_2\) and \(N_4\) of this difference are bounded by its Frobenius norm. The prediction is unchanged by this substitution. At parameters with \(S_A(1;J)\succeq cI\), the truncated factors evaluated at \(J\) equal the exact factors. This proves all of (22). The polynomial assertion was proved in Lemma 4. ◻

Proof. For each of the specified bilinear forms, we calculate its expectation and then apply concentration. Use the truncations above with a cutoff agreeing with the inverse on the path-stability event. Sign conjugation makes the off-diagonal expectations of each relevant matrix vanish. Equation (33) at \(z=1\) gives \[\mathbb E\widehat K_{ii}=1+O(n^{-1}),\qquad \mathbb E(\widehat W\widehat K)_{ii}=B_A+O(n^{-1}).\] The identities, valid on the event of agreement, \[G_AW=K_A(W)(I+B_AA)-I, \qquad WG_AW=WK_A(W)(I+B_AA)-W\] therefore give the three diagonal expectation predictions \(A\), \(B_AA\), and \(B_AI+B_A^2A\), with entrywise errors \(O(n^{-1})\). Their use with truncated factors changes expectations only exponentially little, by the same argument as in Lemma 4. In particular, their errors in any of the specified bilinear forms are \(O(n^{-1})\): for a diagonal error matrix \(E\), one has \(|v_r^TEv_s|\le\lVert E\rVert\lVert v_r\rVert\lVert v_s\rVert\).

The expectation calculation has identified the three required predictions. To control fluctuations, each truncated bilinear form is \(C M_1^2\)-Lipschitz in \(W\) for Frobenius norm, by (26). Gaussian concentration gives failure probability \(Ce^{-c_3n}\) at any fixed positive tolerance. Intersect these events for the \(3m^2\) forms and with Lemma 3. For large \(n\) the expectation errors are less than half the specified tolerance, proving the result. ◻

Proof. We first compute the mean log determinant with \(E=0\) and a fixed deterministic \(V\). Regularization will then provide a global Lipschitz bound, allowing a finite mesh in \(V\) and finally the finite-rank perturbation. Set \[L_z=I+z^2BV-zWV, \qquad K(z)=(I-zVW+z^2BV)^{-1}.\] Since \(V\le I\), we use the proof of Lemma 3 with the diagonal upper bound equal to \(1\); its stability and inverse-norm constants here depend only on \(j\). On the path-stability event from Lemma 3 with \(A=V\), the determinant of \(L_z\) equals the positive determinant of \(S_V(z;W)\) by \(\det(I-UV)=\det(I-VU)\), and both \(L_z^{-1}\) and \(K(z)\) have uniformly bounded operator norm. Indeed \(L_z=K(z)^{-T}\), and (18) applies. Differentiating the normalized log determinant on this event gives \[\begin{align*} \frac{\mathrm d}{\mathrm dz}\frac1n\log\det L_z &=\frac1n\mathop{\mathrm{Tr}}\bigl[L_z^{-1}(2zBV-WV)\bigr]\\ &=2zB\frac1n\mathop{\mathrm{Tr}}(K(z)V) -\frac1n\mathop{\mathrm{Tr}}(WK(z)V). \tag{42}\end{align*}\] Here \(VL_z^{-1}=K(z)V\), which follows by multiplying \((I-zVW+z^2BV)V=V L_z\), and cyclicity of trace was used.

The derivative is expressed in the two matrices whose diagonal expectations were computed in (33). Let \(\mathcal E_V\) be the path-stability event intersected with \(\lVert W\rVert\le R\), for a fixed \(R>2\sqrt j\). This one event applies to the entire interval of \(z\). Equations (33), and replacement by the truncated factors off \(\mathcal E_V\), show uniformly in \(z,V\) that the expectation of the right-hand side of (42), multiplied by \(\mathbf 1_{\mathcal E_V}\), is \[2zBb-zBb+O(n^{-1})=zjb^2+O(n^{-1}).\] All differentiated quantities are bounded on \(\mathcal E_V\), so integration and expectation may be interchanged. Since \(L_0=I\), it follows that \[ \mathbb E\left[\mathbf 1_{\mathcal E_V}\frac2n\log\det L_1\right] =jb^2+O(n^{-1}). \tag{43}\] On \(\mathcal E_V\), the singular values of \(L_1\) are bounded below by a positive constant. Therefore \[0\le\frac1n\log\det(L_1^TL_1+\gamma I) -\frac2n\log\det L_1 \le C\gamma.\] Off \(\mathcal E_V\), the absolute value of the regularized normalized log determinant is bounded by \[|\log\gamma|+C+2\log(1+\lVert W\rVert),\] for \(0<\gamma\le1\). The norm tails of \(W\) and (23) make its expected contribution exponentially small, for each fixed \(\gamma\). Consequently, uniformly in deterministic \(V\), \[ \mathbb E\left[\frac1n\log\det(L_1^TL_1+\gamma I)\right] \le jb^2+C\gamma+O(n^{-1})+O_\gamma(e^{-c n}). \tag{44}\]

The mean bound is uniform in deterministic \(V\). To obtain a single event for all \(V\), we use the concentration supplied by regularization. Define on all real matrices \[F_\gamma(M)=\frac1n\log\det(M^TM+\gamma I).\] Its Frobenius gradient is \(2n^{-1}M(M^TM+\gamma I)^{-1}\). Each singular value of this gradient is at most \(1/(n\sqrt\gamma)\), and hence \[ \mathop{\mathrm{Lip}}_{\mathrm F}(F_\gamma)\le\frac1{\sqrt{n\gamma}}. \tag{45}\] For fixed \(V\), the map \(W\mapsto L_1\) has Frobenius Lipschitz constant at most one. Gaussian concentration therefore gives \[ \mathbb P\{|F_\gamma(L_1)-\mathbb EF_\gamma(L_1)|>u\} \le2\exp\left(-\frac{\gamma n^2u^2}{4j}\right). \tag{46}\]

Choose \(\gamma_0\le1\) small enough that \(C\gamma_0\) in (44) is at most \(\varepsilon/8\). For fixed \(\gamma\le\gamma_0\) and sufficiently large \(n\), the expectation bound and (46) bound all points of any fixed sup-norm mesh of the cube \([0,1]^n\) by \(jb^2+\varepsilon/2\), with failure \(\exp(-c_\gamma n^2)\). Indeed a mesh of spacing \(\Delta>0\) has at most \((C/\Delta)^n\) points, whose logarithm is only of order \(n\). On \(\lVert W\rVert\le R\), if \(\lVert V-V'\rVert\le\Delta\), then \[|b-b'|\le\Delta,\qquad \lVert (I+(BI-W)V)-(I+(B'I-W)V')\rVert_{\mathrm F} \le\sqrt n(2j+R)\Delta.\] By (45), the corresponding \(F_\gamma\) values differ by at most \((2j+R)\Delta/\sqrt\gamma\), whereas \(|jb^2-j(b')^2|\le2j\Delta\). Choose \(\Delta\), depending only on \(j,R,\gamma,\varepsilon\), so that these errors sum to at most \(\varepsilon/4\). Including the exponentially likely norm event proves, simultaneously for all \(V\), \[F_\gamma(I+(BI-W)V)\le jb^2+3\varepsilon/4.\]

We have obtained the simultaneous bound for \(E=0\) with room for the perturbation. The same Lipschitz estimate (45) gives, for every allowed \(E\), \[|F_\gamma(M+E)-F_\gamma(M)| \le\frac{\lVert E\rVert_{\mathrm F}}{\sqrt{n\gamma}} \le M_2\sqrt{\frac{r_0}{n\gamma}}.\] This is at most \(\varepsilon/4\) for large \(n\), uniformly even when \(E\) is chosen from the observed matrix. It proves (41) and the lemma. ◻

Proof. We first evaluate the integral without its indicator. For fixed \(m\), the GOE covariance is \[\mathop{\mathrm{Cov}}(Wm)=jqI+\frac jnmm^{\mathsf T}.\] Convolution with the heat kernel of variance \(\sigma^2\) and with \(\sqrt t\,g\) therefore produces the Gaussian vector \(Wm+\sqrt{t+\sigma^2}\,g'\), of covariance \(sI+jmm^{\mathsf T}/n\). Its determinant is \(s^nS/s\), and its inverse is \[\frac1sI-\frac{j}{nsS}mm^{\mathsf T}.\] Moreover, \(F_{\widetilde J,h_0}(y)=z-Wm-\sqrt t\,g\). The Gaussian density at \(z\), multiplied by \(e^{njb^2/2}\), differs from \(\prod_i\varphi_{d,s}(y_i)\) by the prefactor \(\sqrt{s/S}\) and the exponential whose logarithm divided by \(n\) is \[-\frac{Ba_y}{s}-\frac{B^2q}{2s} +\frac{j(a_y+Bq)^2}{2sS}+\frac{jb^2}{2} =\frac{j(a_y-sb)^2}{2sS}.\] This calculation gives the identity for the unrestricted integral. To retain the indicator, condition the joint Gaussian pair formed by \(W\) and the convolved observation on the value \(z\) of the latter. The resulting conditional probability is exactly the factor in (65). All integrands are nonnegative, so Tonelli’s theorem justifies the interchange of integrations. ◻

Proof. We first obtain a Gaussian tail envelope, then prove a compact-set bound by a finite cover. Strict negativity away from the equality laws will give the second and third assertions; an exponential tilt will give the fourth.

Throughout, \(|d|\le T+j\) and \(s\le T+1+j\). Also \(a(P)^2\le q\int(y-d)^2\,\mathrm dP\) and \(jq/S\le1/2\). The elementary inequality \((r-u)^2\le\frac32r^2+3u^2\) consequently gives \[n\psi\le\frac{3jq}{4sS}\sum_i(y_i-d)^2+\frac{3nj}{2} \le\frac{3}{8s}\sum_i(y_i-d)^2+\frac{3nj}{2}.\] On \(s\ge s_{\min}\), it follows that \[ G_{t,\sigma,n}(y)\le \exp\left(Cn-c\sum_i y_i^2\right), \tag{67}\] where \(c>0\) and \(C<\infty\) depend only on \(j,T,s_{\min}\). On \(\sum_i y_i^2>nR^2\), use half of the remaining Gaussian square for integration and the other half for the required exponential decay. This proves the first assertion of (i).

The estimates on bounded second moments follow from one compactness argument, which we now give in full. Put \[\mathcal P_R=\left\{P:\int y^2\,\mathrm dP\le R^2\right\}.\] Tightness and preservation of the second-moment bound under weak limits make this set weakly compact. Integration of a continuous function of at most linear growth is continuous on it: the second-moment bound makes the contribution of \(|y|>K\) uniformly \(O(K^{-1})\), while bounded truncations are weakly continuous. In particular, \(d,s,a(P),\psi\) are continuous on \(\mathcal P_R\times[0,T]\times[0,1]\) where \(s\ge s_{\min}\).

For every compact subset \(\mathcal C\) of that region, \[ \limsup_{n\to\infty}\frac1n\log \sup_{t,\sigma}\int_{\{(P_y,t,\sigma)\in\mathcal C\}} G_{t,\sigma,n}(y)\,\mathrm dy \le\sup_{\mathcal C}\bigl\{\psi-D(P\Vert N(d,s))\bigr\}, \tag{68}\] where \(P_y=n^{-1}\sum_i\delta_{y_i}\); empty integrals are zero. To establish this bound, fix a strict finite upper bound \(r\) for its right-hand side. At each \(\xi=(P,t,\sigma)\in\mathcal C\), choose a bounded continuous test function \(\varphi_\xi\) from the variational formula for relative entropy so that \[\psi(\xi)-\int\varphi_\xi\,\mathrm dP +\log\int e^{\varphi_\xi}\,\mathrm dN(d,s)<r.\] Such a test also exists when the entropy is infinite. Retain positive slack in the inequality and then choose a neighborhood of \(\xi\) on which continuity controls \(\psi\), the empirical average of \(\varphi_\xi\), and \(d,s\). Moreover, on \(\sum_i y_i^2\le nR^2\), the logarithmic density ratio between \(N(d',s')^{\otimes n}\) and the fixed law \(N(d,s)^{\otimes n}\) is at most \(n\) times a quantity tending uniformly to zero as \((d',s')\to(d,s)\); this follows directly by expanding their Gaussian log densities. For \(y\) in that neighborhood, insert \(e^{\sum_i\varphi_\xi(y_i)}\) and bound its expectation under the fixed product Gaussian. The integral on the neighborhood is then at most \(e^{nr}\), after using less than the reserved slack for the three continuity errors. Cover \(\mathcal C\) by finitely many such neighborhoods. An arbitrarily small enlargement of \(r\) absorbs their number and proves (68). Combining this bound with (66) on the ball and (67) outside it proves the second assertion of (i).

To obtain a strictly negative rate, observe that the same variational formula expresses relative entropy as a supremum of continuous functions. Thus it is jointly lower semicontinuous in \(P,d,s\) when \(s\) is bounded away from zero, and \(\psi-D\) is upper semicontinuous. At \(\sigma=0\), its zero set is precisely the equality curve identified above. On a compact set disjoint from any open neighborhood of that curve its supremum is strictly negative. The same remains true for all sufficiently small \(\sigma\): otherwise a convergent sequence with \(\sigma\downarrow0\) and rates approaching zero would give an equality point outside that neighborhood. Apply (68), with \(s_{\min}=t_0\), and then choose the second-moment tail rate in (i) strictly negative. This proves (ii).

Near time zero, the restriction on \(q\) supplies the positive variance needed for the same argument. For (iii), choose \(t_0\) so small that \(t_0/(1-j)<q_0/2\). The set \(q\ge q_0\) contains no equality point at \(\sigma=0\) and \(0\le t\le t_0\). On this set \(s\ge jq_0>0\), so the preceding compactness argument and the tail envelope apply without change, including at \(t=0\).

It remains to control the contribution of large coordinates in (iv). Fix \(t_0,\nu>0\), and take \(0<\lambda<1/[8(T+1+j)]\). Let \(h_{R_1}(y)=y^2\mathbf 1_{\{|y|>R_1\}}\). On a fixed second-moment ball, repeat the proof of (68), this time inserting \(e^{\lambda\sum_i h_{R_1}(y_i)}\). Begin with any small positive rate bound \(\varepsilon\) in that proof and its finite collection of tests \(\varphi_\xi\). Uniformly for \(|d|\le T+j\) and \(t_0\le s\le T+1+j\), \[\int\bigl(e^{\lambda h_{R_1}(y)}-1\bigr)\,\mathrm dN(d,s)(y) \longrightarrow0\qquad(R_1\to\infty),\] by Gaussian tails; the choice of \(\lambda\) leaves a uniformly negative quadratic exponent. The same holds with the integrand multiplied by any of the finitely many bounded \(e^{\varphi_\xi}\). The additional logarithmic moment can therefore be made arbitrarily small in every member of the finite cover. For sufficiently large \(R_1\), the tilted integral on the ball is at most \(e^{2\varepsilon n}\). On \(\langle h_{R_1}(y)\rangle>\nu\), remove the tilt at cost \(e^{-\lambda\nu n}\). Choosing \(2\varepsilon<\lambda\nu/2\) gives a strictly negative rate there. The integral off the ball is estimated without the tilt, using (i) with a larger negative rate. This proves (iv). ◻

Proof. We first compute the Gaussian regression and the finite-rank correction at an equality law. We then show that the empirical-law and diagonal hypotheses preserve its positive margin, uniformly over all admissible diagonals.

Proof. We integrate over a Gaussian ellipsoid centered at the approximate zero. Its covariance is chosen to control the linearized residual. Let \(g''\) be standard Gaussian and put \(\Delta=\sigma Q^{-1/2}g''\), and restrict to \(\mathcal B=\{\lVert g''\rVert\le2\sqrt n\}\). For \(2\sigma/\sqrt\gamma\le r\), all resulting points \(y+\Delta\) belong to \(E\). Changing variables in the integral gives the lower bound \[ (\det Q)^{-1/2}\mathbb P(\mathcal B) \mathbb E\left[\exp\left\{ \frac{\lVert g''\rVert^2}{2} -\frac{\lVert F(y+\Delta)\rVert^2}{2\sigma^2} +\frac{njb(y+\Delta)^2}{2}\right\}\,\middle|\,\mathcal B\right]. \tag{78}\] Here and for the remainder of this proof \(F=F_{\widetilde J,h_0}\). The conditioning has probability tending to one, and all estimates below remain valid under it with a uniform factor, since \(\mathbb P(\mathcal B)\ge1/2\) for large \(n\).

We must control the nonlinear residual on this ellipsoid uniformly in its center. Coordinatewise Taylor expansion gives \[m(y+\Delta)=m+V_y\Delta+r_m,\qquad |(r_m)_i|\le C\Delta_i^2.\] Writing \(b(y+\Delta)=b+\ell_b+r_b\), its linear part is \(\ell_b=-2\langle m(1-m^2)\Delta\rangle\), and \(|r_b|\le C\lVert \Delta\rVert^2/n\). Thus, when the product \(jb,m\) in \(F\) is expanded, its remainder is bounded in norm by \[C\left(\lVert r_m\rVert+\frac{\lVert \Delta\rVert^2}{\sqrt n}\right).\] For the product of first differences use \(|b(y+\Delta)-b(y)|\le C\lVert \Delta\rVert/\sqrt n\) and \(\lVert m(y+\Delta)-m(y)\rVert\le\lVert \Delta\rVert\). The remaining term \(-\widetilde Jm\) has remainder bounded by \(K\lVert r_m\rVert\). Since the covariance of \(\Delta\) is bounded above by \(\sigma^2I/\gamma\), Gaussian fourth moments give \[\mathbb E\sum_i\Delta_i^4\le Cn\sigma^4/\gamma^2,\qquad \mathbb E\lVert \Delta\rVert^4\le Cn^2\sigma^4/\gamma^2.\] Combining these Taylor estimates with the Gaussian moments, and writing \(R_F=F(y+\Delta)-F(y)-L\Delta\), yields \[ \mathbb E[\lVert R_F\rVert^2\mid\mathcal B] \le Cn\sigma^4/\gamma^2,\qquad \mathbb E[|b(y+\Delta)^2-b(y)^2|\mid\mathcal B] \le C\sigma/\sqrt\gamma. \tag{79}\] All constants depend only on \(j,K\).

The choice of ellipsoid now controls the linear term: the singular values of \(LQ^{-1/2}\) are at most one. Hence \(\lVert L\Delta\rVert\le\sigma\lVert g''\rVert\) pointwise, including on \(\mathcal B\). Expanding \(F(y+\Delta)=F(y)+L\Delta+R_F\) and applying Cauchy–Schwarz to its cross terms yields \[\begin{align*} &\frac1n\mathbb E\left[ \frac{\lVert g''\rVert^2}{2} -\frac{\lVert F(y+\Delta)\rVert^2}{2\sigma^2} \,\middle|\,\mathcal B\right]\\ &\hspace{1cm}\ge -C\left\{ \frac\rho\sigma+\frac\sigma\gamma +\left(\frac\rho\sigma\right)^2 +\left(\frac\sigma\gamma\right)^2 +\frac\rho\gamma\right\}. \end{align*}\] The last mixed term may also be absorbed into the two squares. Apply Jensen’s inequality to the conditional expectation in (78). Its logarithm divided by \(n\) is at least \[-\frac{1}{2n}\log\det Q+\frac{jb(y)^2}{2} -C\left\{\frac\rho\sigma+\frac\sigma\gamma +\left(\frac\rho\sigma\right)^2 +\left(\frac\sigma\gamma\right)^2 +\frac\sigma{\sqrt\gamma}\right\}+o(1).\] The determinant hypothesis cancels \(jb(y)^2/2\), with a loss of at most \(\varepsilon_1/2\). Choose the constants in the order in the statement; the other errors then have total size smaller than \(\varepsilon\). This proves the lower bound. ◻

Proof of Proposition 9. We assemble the scalar, matrix, and local-volume estimates under the planted law (61). After excluding unstable approximate zeros at a fixed time, we remove the auxiliary diagonal, control the whole path by a fixed mesh, and undo the planting.

Let \(J\) be \(\widetilde J\) with its diagonal zeroed. Choose \(\eta,\theta,q_0>0\) sufficiently small that \(j(1+\eta)^2<1\) and, whenever \(q\le2q_0\), \[ I+(j-3jq-2\sqrt j-2\theta)A\succeq c_{\mathrm{sm}}I \qquad(0\le A\le(1+\eta)I) \tag{80}\] for some \(c_{\mathrm{sm}}>0\). These choices are possible because at \(\eta=\theta=q=0\) the lower scalar bound is \(1+j-2\sqrt j=(1-\sqrt j)^2>0\). On \[ \lVert J\rVert\le2\sqrt j+\theta,\qquad \max_i|\widetilde J_{ii}|\le\theta, \tag{81}\] we have \(\lVert \widetilde J\rVert\le2\sqrt j+2\theta\). Since \(mm^{\mathsf T}/n\preceq qI\), (80) implies \(\mathsf H_{\widetilde J}(y,A)\succeq c_{\mathrm{sm}}I\) for all \(q\le2q_0\). Choose \(t_0\in(0,T]\) small enough for Lemma 12(iii).

For the remaining times \(t\in[t_0,T]\), the conditional Hessian lemma provides the margin near the equality curve. Choose a second-moment ball whose complement has a fixed negative rate, using Lemma 12(i) with \(s_{\min}=t_0\). Apply Lemma 13 on this ball. Take its equality neighborhood and tail tolerance, choose its tail cutoff \(R_1\) sufficiently large for Lemma 12(iv), and then choose its diagonal-distance radius. Decrease \(\sigma_*>0\) so that Lemma 12(ii) excludes the complement of that equality neighborhood, and so that the small-time scalar bound also holds. The scalar regions excluded by these choices all have fixed strictly negative rates, uniformly for \(0<\sigma\le\sigma_*\). On the region left over, the conditional matrix margin fails with probability at most \(e^{-a_*n}\).

Let \(\epsilon_*>0\) denote the full diagonal-distance radius just chosen. Choose \(c_E>0\) smaller than both \(c_*/2\) and \(c_{\mathrm{sm}}/2\). For \(t\ge t_0\), define \[E_t(W)=\left\{y:\exists A,\quad 0\le A\le(1+\eta)I,\quad \lVert A-V_y\rVert_{\mathrm F}\le\epsilon_*\sqrt n,\quad \lambda_{\min}(\mathsf H_{\widetilde J}(y,A))\le c_E\right\}.\] For \(0\le t<t_0\), instead put \(E_t(W)=\{y:q(y)\ge q_0\}\). There is \(a_0>0\) such that \[ \mathbb EI_{t,\sigma}(E_t)\le e^{-a_0n} \quad(0\le t\le T, 0<\sigma\le\sigma_*) \tag{82}\] for all sufficiently large \(n\), uniformly in \(t,\sigma\). For small times, this follows from the scalar bound and Lemma 11. For larger times, split the conditional integral into the excluded scalar regions and their complement. The excluded regions have negative rates. On the complement multiply the conditional probability \(e^{-a_*n}\) by the unrestricted scalar integral, whose rate is at most any prescribed positive number, by Lemma 12(i). Taking that number below \(a_*/2\) proves (82). Finitely many pieces can be absorbed by decreasing \(a_0\).

We now use local volume to exclude every unstable approximate zero. An individual witness will force a ball of the excluded set to contribute too much to the integral. Suppose \[ \lVert F_{\widetilde J,h_0}(y)\rVert\le\rho\sqrt n,\qquad \lVert A-V_y\rVert_{\mathrm F}\le\tfrac12\epsilon_*\sqrt n,\qquad \lambda_{\min}(\mathsf H_{\widetilde J}(y,A))\le\tfrac12c_E \tag{83}\] for some \(0\le A\le(1+\eta)I\), and assume (81). There is a constant \(r>0\), independent of this witness and of \(n,t\), such that \(\lVert y'-y\rVert\le r\sqrt n\) implies \(y'\in E_t(W)\). For \(t\ge t_0\), the same \(A\) witnesses instability throughout a sufficiently small ball. Indeed, \[\lVert V_{y'}-V_y\rVert_{\mathrm F}\le2\lVert y'-y\rVert,\qquad |b(y')-b(y)|\le2\lVert y'-y\rVert/\sqrt n,\] and \[\lVert m(y')m(y')^{\mathsf T}/n-m(y)m(y)^{\mathsf T}/n\rVert \le2\lVert y'-y\rVert/\sqrt n\] preserve the full diagonal-distance constraint and the bound \(\lambda_{\min}\le c_E\) if \(r\) is small enough. For \(t<t_0\), (80) and the chosen \(c_E\) imply \(q(y)>2q_0\); the same Lipschitz bound for \(q\) then gives \(q(y')\ge q_0\) for a smaller \(r\).

Set \(L=F'_{\widetilde J,h_0}(y)\). It is a bounded-rank, bounded-norm perturbation of \[I+(BI-W)V_y: \quad L=I+(BI-W)V_y -\frac jn\mathbf 1\mathbf 1^{\mathsf T}V_y -\frac{2j}{n}mm^{\mathsf T}V_y.\] Here \(n^{-1}\mathop{\mathrm{Tr}}V_y=b(y)\), so the determinant rate in Lemma 6 matches the weight in (62). That lemma says that, for any chosen \(\varepsilon_1>0\), one may choose a fixed \(\gamma>0\) sufficiently small so that, except on an event of probability \(Ce^{-a_1n}\), simultaneously at every \(y\), \[\frac1n\log\det(L^{\mathsf T}L+\gamma I) \le jb(y)^2+\varepsilon_1.\] The lemma’s rank-perturbation allowance applies here with two rank-one terms of norms at most \(j\) and \(2j\). Choose \(\varepsilon<a_0/2\), choose \(\varepsilon_1\) for Lemma 14, then choose \(\gamma\) by the log-determinant lemma. Next fix \(\sigma\in(0,\sigma_*]\) sufficiently small for the local-volume lemma, and finally fix \(\rho>0\) sufficiently small for it. A witness (83) on the determinant event forces \(I_{t,\sigma}(E_t)\ge e^{-\varepsilon n}\). Markov’s inequality and (82) show that the probability of such a witness, under (81), is at most \(Ce^{-a_2n}\), uniformly in \(t\in[0,T]\). Here the uniformity of the negative rates permits the later choice of \(\sigma\) required by the local-volume estimate.

The integral argument used a full GOE matrix. We now pass to its zero-diagonal version, retaining slack in both residual and Hessian bounds. Choose a fixed \(\delta_D>0\) so small that \[\delta_D\le\theta,\qquad \delta_D\le\rho/2,\qquad (1+\eta)\delta_D\le c_E/4.\] The diagonal entries of \(\widetilde J\) are independent \(N(j/n,2j/n)\), so \(\mathbb P(\max_i|\widetilde J_{ii}|>\delta_D)\le Ce^{-a_Dn}\) for large \(n\). On its complement, \[\frac{\lVert F_{\widetilde J,h_0}(y)-F_{J,h_0}(y)\rVert}{\sqrt n} \le\delta_D,\qquad \lVert \mathsf H_{\widetilde J}(y,A)-\mathsf H_J(y,A)\rVert \le(1+\eta)\delta_D.\] Thus a zero-diagonal witness with residual at most \(\rho\sqrt n/2\), radius \(\epsilon_*\sqrt n/2\), and Hessian minimum at most \(c_E/4\) would give a witness (83). We have proved an exponentially small fixed-time failure probability for the zero-diagonal matrix, restricted only to \(\lVert J\rVert\le2\sqrt j+\theta\).

To pass from a fixed time to the whole interval, we control observation increments on a deterministic mesh. Take a mesh of size \(\Delta>0\) that includes both endpoints. For a scalar Brownian motion, the reflection bound implies that \(\sup_{0\le u\le\Delta}|B(u)|^2/\Delta\) has an exponential moment at a fixed positive argument. The coordinate Brownian motions are independent. Exponential Markov applied to the sum of their squared suprema consequently gives, for each desired \(\kappa_0>0\), a mesh \(\Delta\) small enough that \[\mathbb P\left(\max_k\sup_{t_k\le t\le t_{k+1}} \frac{\lVert B_0(t)-B_0(t_k)\rVert}{\sqrt n}>\kappa_0\right) \le Ce^{-a_3n}.\] The deterministic drift contributes at most \(\Delta\sqrt n\) to each increment. Choose the mesh so the full normalized increment is at most \(\rho/4\). Set \(\rho_0=\rho/8\), \(\epsilon_A=\epsilon_*/2\), and \(c=c_E/4\). A pathwise violation of (60) then gives a fixed-time violation at a mesh point: the Hessian and its witness do not use the field, and the residual increases by at most \(\rho\sqrt n/4\). A union bound over the fixed finite mesh proves an annealed path failure bound \(Ce^{-a_4n}\) under the planted law, restricted to \(\lVert J\rVert\le2\sqrt j+\theta\).

We have obtained a path estimate under planting. To recover the original disorder law, introduce the partition-function density \[Z(J)=\sum_x e^{x^{\mathsf T}Jx/2},\qquad L(J)=Z(J)/\mathbb EZ(J).\] Size-bias the original joint law of \(J,X\), with \(X\sim\mu_0\), by \(L(J)\). Its density in \(J,X\) is proportional to \(e^{x^{\mathsf T}Jx/2}\) times the original Gaussian density. Gaussian tilting makes \(X\) uniform and, conditionally on \(X\), makes the off-diagonal entries of \(J\) independent with means \(jX_iX_j/n\) and variance \(j/n\). Add independent diagonal entries with law \(N(j/n,2j/n)\)—a centered GOE diagonal plus \(jI/n\)—and conjugate by \(D_X\); the resulting matrix and observation have exactly the law (61). All witness conditions are equivariant under this conjugation, because \(m(D_Xy)=D_Xm(y)\), \(V_{D_Xy}=V_y\), and diagonal \(A\) commutes with \(D_X\). Conditional observation paths given \(J\) are unchanged by size bias.

Undoing this change of law requires a lower bound for \(L(J)\) on typical original matrices. For independent uniform signs \(\varepsilon_i\), Gaussian averaging gives \[\frac{\mathbb EZ^2}{(\mathbb EZ)^2} =e^{-j/2}\mathbb E\exp\left\{\frac j{2n} \left(\sum_i\varepsilon_i\right)^2\right\} \le(1-j)^{-1/2}.\] To obtain the inequality, introduce an independent standard Gaussian \(g\) and use \(\mathbb E_\varepsilon e^{\sqrt{j/n}g\sum_i\varepsilon_i} =\cosh(\sqrt{j/n}g)^n\le e^{jg^2/2}\). Paley–Zygmund then yields a constant \(p_j>0\) with \(\mathbb P(Z\ge\mathbb EZ/2)\ge p_j\). As a function of the independent standard off-diagonal Gaussians, \(\log Z\) has gradient squared at most \((j/n)\binom n2\le jn/2\). Gaussian concentration about a median therefore has scale \(O(\sqrt n)\). The preceding positive-probability lower bound implies that this median is at least \(\log\mathbb EZ-O_j(\sqrt n)\): otherwise its upper concentration tail would be smaller than \(p_j\). It follows that, for every fixed \(\delta>0\), \[ \mathbb P\{L(J)<e^{-\delta n}\}\longrightarrow0. \tag{84}\]

We can now turn the annealed planted estimate into the required conditional observation bound. Let \(p_n(J)\) denote the conditional observation-path failure probability for (60). The planted estimate is \[\mathbb E\bigl[L(J)p_n(J) \mathbf 1_{\{\lVert J\rVert\le2\sqrt j+\theta\}}\bigr] \le Ce^{-a_4n}.\] Choose \(\delta<a_4/2\). On \(L(J)\ge e^{-\delta n}\), this gives an original expectation bound \(Ce^{-(a_4-\delta)n}\) for the restricted \(p_n(J)\). Markov’s inequality makes \(p_n(J)\le e^{-an}\) outside an event of probability tending to zero, for a fixed sufficiently small \(a>0\). Equation (84) and the usual GOE norm bound, including diagonal removal, show that the restrictions on \(L(J)\) and \(\lVert J\rVert\) hold with probability tending to one. Intersect these events to obtain \(\mathcal J_n\) and finish the proof.

We finish by checking measurability of the witness events used in the argument. At fixed \(y\), the matrix-witness sets are projections over the compact diagonal range. For existence of \(y,A,t\) along a continuous path, additionally restrict \(\lVert y\rVert\), \(\lVert J\rVert\), and \(\sup_{t\le T}\lVert Y(t)\rVert\) by integers. The witness inequalities are closed and the witness parameter space is compact, so these restricted projections are closed. Taking a countable union removes the bounds. This also justifies the conditional probabilities and the restricted integrals above. ◻

Proof. We first bound the derivative inverse on the region in the hypothesis. We then prove existence and uniqueness of a zero, and integrate the inverse bound along a path to that zero.

Put \(L_1=BI-J-2jmm^{\mathsf T}/n\). Since \(F'(y)=I+L_1V_y\), the elementary inverse identity gives \[F'(y)^{-1} =I-L_1V_y^{1/2} (I+V_y^{1/2}L_1V_y^{1/2})^{-1}V_y^{1/2}.\] Under (85), this inverse exists and has norm at most \(1+(K+3j)/c\), since \(0<V_y\le I\) and \(\lVert L_1\rVert\le K+3j\).

For existence and uniqueness, pass to the coordinates \(m\in(-1,1)^n\) and define the potential \[\Phi(m)=\sum_i\int_0^{m_i}\mathop{\mathrm{atanh}}z\,\mathrm dz-h_0^{\mathsf T}m -\frac12m^{\mathsf T}Jm+\frac j2\lVert m\rVert^2 -\frac j{4n}\lVert m\rVert^4.\] Its critical-point equation is \(F_{J,h_0}(\mathop{\mathrm{atanh}}m)=0\). At any critical point, \[V_y^{1/2}\Phi''(m)V_y^{1/2} =\mathsf H_J(y,V_y)\succeq cI,\qquad y=\mathop{\mathrm{atanh}}m,\] so the critical point has positive definite Hessian and is isolated. Choose \(a\in(0,1)\) close enough to one that \[\mathop{\mathrm{atanh}}a>1+\max_i\left(|(h_0)_i|+\sum_j|J_{ij}|+j\right).\] Every critical point lies in the interior of \([-a,a]^n\), and on each face of this box the negative gradient \(-\nabla\Phi\) points strictly inward. The minimum of \(\Phi\) on the box is therefore attained at an interior critical point. This proves existence.

To prove uniqueness, consider the negative gradient flow from any point of the box. The inward-pointing condition keeps the trajectory in the box for all positive times. Its potential decreases and is bounded below, so \(\int_0^\infty\lVert \nabla\Phi(m(t))\rVert^2\,\mathrm dt<\infty\). The squared gradient is uniformly continuous in time, because the gradient and Hessian are bounded on the box. It follows that the gradient tends to zero: a nonzero limsup would, by uniform continuity, force disjoint intervals of fixed positive integral. Every limit point of a trajectory is therefore critical. The critical set is finite, since it is a compact set of isolated zeros. The limit set of a trajectory is connected: it is the intersection of the nested compact connected closures of its tails. Each trajectory therefore converges to one critical point.

It remains to show that all trajectories converge to the same point. The positive Hessian makes every critical point a strict local attractor. For clarity, in a sufficiently small ball about it one has \((m-m_*)^{\mathsf T}\nabla\Phi(m)\ge c_*\lVert m-m_*\rVert^2\), which gives exponential contraction of the distance along the flow and invariance of the ball. Continuous dependence of the flow on its starting point, up to the time it enters that ball, shows that each attraction basin is relatively open in \([-a,a]^n\). These disjoint nonempty basins cover the connected box. Hence there is only one basin and one critical point, proving uniqueness of \(r\).

We now derive the distance estimate. Suppose \(\lVert F(y)\rVert<r_0\sqrt n\). Starting at \(y(0)=y\), solve on the open region \(\lVert F\rVert<r_0\sqrt n\) the ordinary differential equation \[\dot y(s)=-F'(y(s))^{-1}F(y).\] The chain rule gives \(F(y(s))=(1-s)F(y)\) while the solution exists. Its velocity has norm at most \(C\lVert F(y)\rVert\). If its maximal existence interval ended at or before time one, this velocity bound would give a finite limiting point, whose residual is still strictly less than \(r_0\sqrt n\) and whose derivative is invertible. Local existence would extend the solution, a contradiction. The solution consequently reaches time one, at which its value is the unique zero \(r\). Integrating the velocity bound along this path gives (86). ◻

Proof. We first bound the formal matrices in (89), then compare them with the actual discrete derivatives. Expanding \(H_{l+1}\) produces the chain \[J D_{1-(m^l)^2}J\cdots D_{1-(m^1)^2}J\] and the terms obtained by contracting disjoint pairs of originally adjacent \(J\)’s. Contracting \(J D_{1-(m^s)^2}J\) contributes \(-j b_s I\). This description follows inductively from the two terms defining \(H_{l+1}\): either keep its first link, or contract that link with the next one. Multiplicities are retained when different contractions happen to give the same matrix.

The contraction terms cancel the diagonal predictions. To see this, fix a complete noncrossing pairing of the original chain. Reinserting each contracted adjacent pair identifies its contribution with the prediction of this full pairing, multiplied by the signs of the contractions. The originally adjacent pairs belonging to a pairing are mutually disjoint, and every nonempty complete noncrossing pairing contains at least one such pair. Summing over their subsets gives \((1-1)^a=0\), where \(a\geq1\) is their number. An odd chain has no complete pairing. Lemma 2 therefore bounds the diagonal \(\ell^2\) norm of \(H_l\), and bounds its operator and off-diagonal \(\ell^4\) norms term by term. Its full \(\ell^4\) norm is bounded as well, since \(\lVert \mathop{\mathrm{diag}}H_l\rVert_4\leq\lVert \mathop{\mathrm{diag}}H_l\rVert_2\).

We now compare these formal bounds with the actual derivatives. For a scalar function \(\psi\) with bounded second derivative, Taylor’s theorem gives \[\lvert d_j\psi(h_i^l)-\psi'(h_i^l)d_jh_i^l\rvert \leq C|d_jh_i^l|^2.\] The Frobenius norm of the remainder is at most \(C\lVert \nabla h^l\rVert_{\ell^4}^2\). This proves the magnetization part of (90) once the field part is known. The same argument for \(\psi(z)=\mathop{\mathrm{sech}}^2z\) gives an operator bound for its site derivative matrix \(Q_l\). Averaging its rows yields \[db_l=n^{-1}Q_l^{\mathsf T}\mathbf 1,\qquad \lVert db_l\rVert\leq n^{-1/2}\lVert Q_l\rVert.\] In differentiating \(b_lm^{l-1}\), the term from \(db_l\) is a rank-one matrix of Frobenius norm at most \(\lVert m^{l-1}\rVert\lVert db_l\rVert\leq C\). The discrete product error is \(\nabla m^{l-1}\) times a diagonal whose diagonal has norm at most \(2\lVert db_l\rVert\). Its Frobenius norm is bounded, since \(\lVert \nabla m^{l-1}\rVert\leq C\). Starting at \(\nabla h^1=J\), these estimates close the induction. The resulting bounded Frobenius perturbation preserves the operator, diagonal \(\ell^2\), and entrywise \(\ell^4\) bounds of the formal matrices.

For the final assertion, apply the same comparison to a smooth function of several arguments. Expand along the segment between the arguments at \(x\) and \(x^{(j)}\). The preceding bounds control the primary linear terms. A linear term involving \(db_s\) is a rank-one matrix with a bounded-entry left vector, hence has bounded Frobenius norm. The squared-average remainder is controlled by \[n\sum_j |d_jb_s|^4\leq n\lVert db_s\rVert^4\leq C.\] Mixed remainders are bounded by sums of pure squares, and primary quadratic remainders are controlled by the entrywise \(\ell^4\) bounds. This proves the assertion. ◻

Proof. We first control the coefficients and solve the implicit linear system. Only then do we differentiate it, using the larger region to cover spin flips from the smaller one. The uniform kernel and quotient bounds proved above will control the coefficients even when some \(A_i\) are small.

Put \(\ell_i=(\partial_z\Phi/\Phi)(h_i^k;r_i,a)\). At \(z=r_0\) this ratio equals \(-\tanh r_0\), for every \(a\). Comparison with \(z=r_0,a=0\), followed by (97), gives \[ \lVert A-D_{1-(m^k)^2}\rVert_{\mathrm F}\leq C\rho\sqrt n, \qquad \lVert \ell+t_*\rVert\leq C\rho\sqrt n, \qquad |\chi-b_k|\leq C\rho. \tag{103}\] In particular, the comparisons put \(A\) among the diagonals allowed by the stable-field implication, once \(\rho\) is decreased.

Let \(u_0=w_0/A\) coordinatewise and \(p_0=A\partial_k u_0\). The quotient bounds just proved give \(\lVert u_0\rVert+\lVert p_0\rVert\leq C\); their local derivatives have coordinatewise bounds proportional to \(|e_i|\) in the first case and to \(n^{-1/2}\) in the second. Direct differentiation with the specified partial convention gives \[\partial_k w=D_\ell w+p_0,\qquad c_1=\langle \ell w+p_0\rangle.\] We can now eliminate \(y^{\rm imp}\) and \(c_1\) and solve for the source alone. The resulting linear equation is \[ \left[I-A(J-j\chi I) +\frac jn A(m^{k-1}+t_*)\ell^{\mathsf T}\right]w =A\left[u_0-j(m^{k-1}+t_*)\langle p_0\rangle\right]. \tag{104}\] Write the matrix on the left as \(I+AL_2\). Its symmetrically normalized version \(I+A^{1/2}L_2A^{1/2}\) differs in operator norm by \(O(\rho)\) from \[I+jb_k A-A^{1/2}JA^{1/2} -\frac{2j}{n}A^{1/2}m^k(m^k)^{\mathsf T}A^{1/2}.\] For the rank term, use (97) and (103) in the elementary inequality \(\lVert uv^{\mathsf T}-u'v'^{\mathsf T}\rVert \leq\lVert u-u'\rVert\lVert v\rVert+\lVert u'\rVert\lVert v-v'\rVert\). The displayed symmetric matrix has the fixed positive margin supplied by (60). Decrease \(\rho\) until the perturbation is less than half that margin. The normalized matrix then has bounded inverse, even though it need not itself be symmetric. The identity \[(I+AL_2)^{-1} =I-A^{1/2}(I+A^{1/2}L_2A^{1/2})^{-1}A^{1/2}L_2\] gives the same bound for the unnormalized inverse, since \(\lVert L_2\rVert\) is bounded. The right side of (104) is bounded in norm, because \(|\langle p_0\rangle|\leq\lVert p_0\rVert/\sqrt n\). This establishes the size bounds. To obtain the last assertion of (101), remove the negative rank term from the stable-field matrix and change \(b_k\) to \(\chi\). The constants used to solve the system and obtain these bounds depend only on the fixed stability and operator margins.

It remains to bound the discrete derivatives. Compare the equations at \(x\) and \(x^{(i)}\). On the stated inner region both configurations belong to \(\Omega_{2\rho}\) for large \(n\), by Lemma 16. In every coefficient-difference term, retain the solution vector at the common point \(x\). In particular, \[\begin{align*} d_i c_1 &=\langle (d_i\ell)w(x)+d_i p_0\rangle +n^{-1}\ell(x^{(i)})^{\mathsf T}d_iw,\\ d_i\{c_1(m^{k-1}+t_*)\} &=c_1(x)d_i m^{k-1} +(m^{k-1}(x^{(i)})+t_*)d_ic_1. \end{align*}\] Use the same product arrangement for \(A\) and \(\chi\) in \[w=A\{u_0+(J-j\chi I)w-jc_1(m^{k-1}+t_*)\}.\] For clarity, put \(v=m^{k-1}+t_*\) and \(\mathcal B=u_0+(J-j\chi I)w-jc_1v\), and use primes for evaluation at \(x^{(i)}\) in the next display. With \(\zeta_i=n^{-1}\{(d_i\ell)^{\mathsf T}w+\mathbf 1^{\mathsf T}d_ip_0\}\), the exact equation is \[\begin{align*} &\left[I-A'(J-j\chi'I)+\frac jn A'v'(\ell')^{\mathsf T}\right]d_iw \\ &\qquad=(d_iA)\mathcal B+A'd_iu_0 -jA'w\,d_i\chi-jc_1A'd_im^{k-1}-jA'v'\zeta_i. \tag{105}\end{align*}\] Thus the left matrix is precisely the matrix in (104) evaluated at \(x^{(i)}\). The bracket \(\mathcal B\) on the right is consequently one common-point vector for all columns. This allows the following estimates to control the Frobenius norm of the entire right side.

We bound its terms in turn. A site coefficient difference times a common vector \(s\) of bounded norm has bounded Frobenius norm, since \[\sum_{i,j}s_i^2|d_j\psi_i|^2 \leq \lVert s\rVert^2\max_i\sum_j|d_j\psi_i|^2\leq C.\] This applies to \(A,\ell,u_0,p_0\), using their coefficient bounds and Lemma 16; the seed factors are retained for \(u_0,p_0\). The bracket multiplying \(d_iA\) is bounded in norm by the already established size estimates, so no bound on \(1/A_i\) is needed. Also \(\lVert d\chi\rVert\leq C/\sqrt n\), because \(A_i\) is a primary site function with an average parameter of that derivative scale. If \(\zeta_i=\langle (d_i\ell)w+d_i p_0\rangle\), then \[\lVert \zeta\rVert\leq n^{-1/2} \bigl(\lVert D_w\nabla\ell\rVert_{\mathrm F} +\lVert \nabla p_0\rVert_{\mathrm F}\bigr) \leq C/\sqrt n.\] Its multipliers \(m^{k-1}(x^{(i)})+t_*\) have norm at most \(2\sqrt n\), which still gives a bounded Frobenius norm after summing the squared column norms. Finally \(|c_1|\lVert \nabla m^{k-1}\rVert_{\mathrm F}\leq C\), by its operator bound and \(|c_1|\leq C/\sqrt n\). Applying the uniformly bounded flipped inverses column by column proves \(\lVert \nabla w\rVert_{\mathrm F}\leq C_k\). Returning to the two displayed difference identities gives \(\lVert dc_1\rVert\leq C_k/\sqrt n\) and the required bound for \(\nabla y^{\rm imp}\). The operator in (104), the regions \(\Omega_\rho\), and the required buffer inclusions do not depend on the directional seed \(e\). All bounds for its right side and its derivatives use only \(\lVert e\rVert\leq1\); indeed \(u_0=e\) and \(p_0=0\) for that source. The inverse margins, the threshold on \(n\), and all these estimates are therefore uniform over the full Euclidean unit ball of seeds. ◻

Proof. The proof has two parts. We bound the errors introduced by discrete differentiation, then cancel the trace predictions of the retained formal terms. Write \(\lVert E\rVert_{\rm nuc}\) for the sum of the singular values of \(E\). The error estimates repeatedly use \[ |\mathop{\mathrm{Tr}}(PE)|\leq\lVert P\rVert\lVert E\rVert_{\rm nuc},\qquad \lVert BC\rVert_{\rm nuc}\leq\lVert B\rVert_{\mathrm F}\lVert C\rVert_{\mathrm F},\qquad \lVert E\rVert_{\rm nuc}\leq\sum_j\lVert E_{\cdot j}\rVert. \tag{108}\] Every bounded-length diagnostic word \(P\) has bounded operator norm and bounded off-diagonal entrywise \(\ell^4\) norm. For the errors arising before we solve the differentiated implicit equations, \(P\) may contain one inverse \(K\) from (17). Its required margin is supplied by the last assertion of (101).

Proof. We induct strongly on \(p\) over all finite admissible recipes, proving the two assertions together. Each step may enlarge the original recipe, but it adds only finitely many auxiliary operations and invokes only smaller residual indices. Consequently every particular instance of the induction uses finitely many operations and a finite word-length bound.

At the base level \(p=1\), condition on all spins except the one being differentiated. Its conditional mean is \(m_i^1\) and its conditional variance is \(v_i^1\). A function of this single spin is affine, with linear coefficient \(d_i\) of the function. Taking its conditional covariance with the spin gives the exact identity \[ \mathbb E\sum_i(x_i-m_i^1)GU_i =\mathbb E\sum_i v_i^1d_i(GU_i). \tag{117}\] By (2), its right side is \[\mathbb E\sum_i v_i^1Gd_iU_i +\mathbb E\sum_i v_i^1(d_iG)\{U_i-2x_id_iU_i\}.\] The diagonal derivative bound makes the first term at most \(C\mathbb E|G|\). For the second term, the coefficient vector has pointwise norm at most \(\lVert U\rVert+2\lVert \mathop{\mathrm{diag}}\nabla U\rVert_2\leq C\). Weighted Cauchy–Schwarz and \(0<v_i^1\leq1\) therefore bound that term by \(C\mathcal D_{h_0}(G)^{1/2}\). This proves (115). For a supported local test, its derivatives vanish outside the one-flip neighborhood of its support; all bounds used here are available there.

For (116), absorb the bounded multiplier by replacing \(U\) in the base identity with \(\widetilde U=H_0U\). This vector still has bounded norm and bounded diagonal derivative \(\ell^1\) norm: the additional term obeys \(\sum_i|U_i d_iH_0|\leq\lVert U\rVert\lVert dH_0\rVert\), and the product-error term is bounded by the same argument using \(\lVert \mathop{\mathrm{diag}}\nabla U\rVert_2\). Apply (117) with \(G=f^2\) and use \[d_i(f^2)=2f\,d_i f-2x_i(d_i f)^2.\] The diagonal-derivative term is bounded by \(C\mathbb Ef^2\); the remaining terms are bounded by \(C\sqrt{\mathbb Ef^2\,\mathcal D_{h_0}(f)}+C\mathcal D_{h_0}(f)\). The base case is now proved in both forms. Before proceeding to higher residuals, we record how bounded scalar multipliers affect the tests. If \(|\vartheta|+\lVert d\vartheta\rVert\leq C\), then \[ \lVert \vartheta G\rVert_*\leq C'\lVert G\rVert_*. \tag{118}\] Indeed arrange the product rule as \(d_i(\vartheta G)=\vartheta(x^{(i)})d_iG+G(x)d_i\vartheta\), square, and sum with weights \(v_i^1\leq1\). In the square-density case such a multiplier can be absorbed into \(H_0\); the product retains its bounded size and difference norm. For local tests only the buffer bounds are needed, since multiplication does not enlarge their support.

For the induction step, suppose both assertions hold through level \(p\) and take \(U\) admissible from level \(p+1\). The scalar integral identity suggests the following new source. Set \[a_p=j(b_p-b_{p-1}),\qquad U'_i=\Phi(h_i^p;h_i^{p+1},a_p)U_i.\] This is a source admissible from level \(p\). Its coefficient and all needed derivatives are bounded because \(|a_p|\leq j\) and the bounds for \(\Phi\) proved with (95) hold uniformly in its two real field arguments. The level condition on \(U\) makes its site partial at level \(p\) vanish: \(\partial_pU\) differentiates only explicit primary arguments. Parameters and auxiliary arguments remain fixed, including when their values depend on earlier primary fields. Applying (95) coordinatewise therefore yields \[R_{p+1}^{\mathsf T}U =(h^p-h^{p+1}-a_pm^p)^{\mathsf T}U' -a_p n\langle \partial_pU'\rangle.\] The primary recursion gives \[h^p-h^{p+1}-a_pm^p =JR_p+a_pR_p-jb_{p-1}R_{p-1},\] where the last term is absent for \(p=1\). Add a last auxiliary field \(y'\) by (93) with source \(U'\), and write \(\alpha_l=\langle \partial_lU'\rangle\) and \(\gamma_b=\langle \partial_{y^b}U'\rangle\). Symmetry of \(J\) now gives \[\begin{align*} R_{p+1}^{\mathsf T}U ={}&R_p^{\mathsf T}y'+a_pR_p^{\mathsf T}U' -jb_{p-1}R_{p-1}^{\mathsf T}U' +j\sum_b\gamma_bR_p^{\mathsf T}w^b\\ &+j\sum_{l\geq p}\alpha_lR_p^{\mathsf T}m^{l-1} -a_pn\alpha_p. \tag{119}\end{align*}\] The first line now consists of earlier residuals with admissible sources and bounded multipliers. After testing, (118), (107), and the level constraints allow us to apply the induction hypothesis to each term. In particular the enlarged recipe is admissible from level \(p\), which is also sufficient for its \(R_{p-1}\) term.

On the second line, the terms with \(l>p\) use the terminal source \(m^{l-1}/\sqrt n\), admissible from level \(p\), and multiplier \(\sqrt n\alpha_l\). The multiplier bounds follow from Lemma 19. The remaining term is handled by the identity \[ R_p^{\mathsf T}m^{p-1} =n(b_p-b_{p-1})-R_p^{\mathsf T}m^p. \tag{120}\] This follows from \((m^{p-1}-m^p)^{\mathsf T}(m^{p-1}+m^p) =\lVert m^{p-1}\rVert^2-\lVert m^p\rVert^2\). The first term on the right of (120) cancels \(a_pn\alpha_p\) in (119); the remaining term is handled with the admissible source \(m^p/\sqrt n\) and multiplier \(\sqrt n\alpha_p\). This completes the induction for both (115) and (116). In the local version it stops at level \(k\); each added source respects the support and buffer requirements of the initial implicit construction. ◻

Proof. The vector in braces is \(JR_k-jb_{k-1}(R_{k-1}+R_k)\). To express its pairing with the implicit source in terms of earlier residuals, use symmetry of \(J\), the equation \(Jw=y^{\rm imp}+j\chi w+jc_1m^{k-1}\), and (120). The bracketed expression becomes \[R_k^{\mathsf T}y^{\rm imp} +j(\chi-b_{k-1})R_k^{\mathsf T}w -jb_{k-1}R_{k-1}^{\mathsf T}w -jc_1R_k^{\mathsf T}m^k.\] Apply Lemma 20 to the first three terms, absorbing their bounded parameters into the test. For the last term, use source \(m^k/\sqrt n\) and multiplier \(\sqrt n c_1\). The bounds needed for this multiplier are precisely (101) and (102). ◻

Proof. We use the scalar residual pairing \(S_l=R_l^{\mathsf T}m^l/\sqrt n\) to control the total squared increment. First, the primary derivative bounds give \(\lVert dS_l\rVert\le C_l\) pointwise. In the product rule, the linear terms are controlled by the operator norms of \(\nabla R_l\) and \(\nabla m^l\), multiplied by \(\lVert m^l\rVert/\sqrt n\) and \(\lVert R_l\rVert/\sqrt n\). The cross term for coordinate \(i\) has size at most \(2\lVert d_i R_l\rVert\lVert d_i m^l\rVert/\sqrt n\); summing its square uses the bounded column norms and gives the same scale.

Apply Lemma 20 with target \(m^l/\sqrt n\) and test \(G=S_l\). It gives \[\mathbb ES_l^2\le C_l\bigl(\mathbb ES_l^2+C_l\bigr)^{1/2}, \qquad\text{hence}\qquad \mathbb ES_l^2\le C_l'.\] For the square-density version use \(H_0=S_l/\sqrt n\), which is bounded with bounded difference norm. Dividing the resulting estimate by \(N\sqrt n\) gives \[ \mathbb E_\nu\left(\frac{S_l}{\sqrt n}\right)^2 \le\frac{C_l}{\sqrt n}\left(1+\frac{\mathcal D_{h_0}(f)}{N}\right). \tag{122}\] We now use the squared-increment identity (88). Choose a fixed \(d\) with \(\lfloor d/2\rfloor\rho^2/4>2\), followed by \(\varepsilon>0\) with \(2d\varepsilon<1\). Whenever \(\max_{l\le d}|S_l|/\sqrt n\le\varepsilon\), telescoping gives \(\sum_{l=1}^d\lVert R_l\rVert^2/n<2\). If every consecutive pair contained a residual larger than \(\rho\sqrt n/2\), the disjoint pairs \((1,2),(3,4),\ldots\) would force the sum to exceed this bound. At least one pair therefore meets the required threshold.

To implement the selection smoothly, apply a scalar cutoff equal to one on \([0,1/4]\) and zero on \([1,\infty)\) to each of \(\lVert R_k\rVert^2/(\rho^2n)\) and \(\lVert R_{k-1}\rVert^2/(\rho^2n)\), and multiply the results to define \(C_k\). The primary operator and column bounds, with the product rule, give \(\lVert d(\lVert R_l\rVert^2/n)\rVert\le C_l/\sqrt n\), hence the claimed derivative bound. Finally \(D=\prod_{k=2}^d(1-C_k)\): positive deficit implies \(\max_{l\le d}|S_l|/\sqrt n>\varepsilon\). The preceding two moment bounds and a finite union bound give the ordinary and square-reweighted estimates. ◻

Proof. Fix an index \(k\ge2\) and work on the small-residual region (96), with the buffered implicit construction (100). Abbreviate \(m=m^k\), \(p=m-t_*\), and put \[a=j(1-b_{k-1}-q_*),\qquad L_* =\sqrt n\bigl(\chi-(1-q_*)\bigr).\] Here \(\chi=\langle A_i\rangle\) and \(A_i=\Phi(h_i^k;r_i,a)\) are defined globally by (98), even though the implicit vector is only needed locally. Thus \(|L_*|\le C\sqrt n\) and \(\lVert dL_*\rVert\le C_k\) pointwise, by the primary derivative bounds and \(\lVert db_{k-1}\rVert=O(n^{-1/2})\).

For either of the two allowed sources \(w_0\), the following pointwise identity holds on the buffered region: \[\begin{align*} &\bigl(h^k-h_0-(J-jb_{k-1}I)m\bigr)^{\mathsf T}w -j(b_k-b_{k-1})nc_1 \\ &\quad=\sum_i\bigl[(h_i^k-r_i-am_i)-a\partial_k\bigr]w_{0,i} +j(1-q_*-\chi)p^{\mathsf T}w-jc_1R_k^{\mathsf T}p. \tag{124}\end{align*}\] Here the zeroth-order term in the first sum acts by multiplication on \(w_{0,i}\). For the verification, subtract the root equation to write the vector on the left as \(h^k-r-am-Jp+j(1-q_*)p\), then substitute \(Jw=y^{\rm imp}+j\chi w+jc_1m^{k-1}\). Apply (95) to the contribution \(A(y^{\rm imp}-jc_1t_*)\), holding its auxiliary arguments and parameters fixed, and use \(nc_1=\sum_i\partial_k w_i\). The remaining scalar terms cancel by \[p^{\mathsf T}(m^{k-1}+t_*)+n(b_k-b_{k-1}) =n(1-b_{k-1}-q_*)+R_k^{\mathsf T}p,\] which follows by expanding \(p=m-t_*\) and \(R_k=m^{k-1}-m\).

Lemma 21 controls the tested left-hand side of [eq:implicit-comparison]. The last term on the right also has tested expectation at most \(C\lVert G\rVert_*\): apply Lemma 20 to target \(p/\sqrt n\) and multiplier \(\sqrt n c_1\). The latter is bounded with bounded difference norm on the buffer. Meanwhile the size estimates for the implicit solution and Lemma 15 give \[ \lvert j(1-q_*-\chi)p^{\mathsf T}w\rvert\le C_0\rho|L_*|. \tag{125}\] The source size and stability margins determine \(C_0\) independently of \(k\) and the selection depth. This lets us choose \(\rho\) first so that \(C_0\rho<1/2\) and all stability and flip-buffer conditions hold. The selection depth in Lemma 22 is chosen afterward. This order will permit absorption of the scalar error without a self-dependent choice of depth.

First control the averaged coefficient by taking \(w_{0,i}=\partial_{r_0}\Phi(h_i^k;r_i,a)/\sqrt n\). The second identity in (99) makes the first sum in [eq:implicit-comparison] equal to \(L_*\). Test with \(G=C_k^2L_*\). The cutoff and difference bounds give \[\lVert C_k^2L_*\rVert_*\le C+\lVert C_kL_*\rVert_{L^2(\mu_{h_0})}.\] Indeed its difference vector is bounded pointwise: \(|L_*|=O(\sqrt n)\) multiplies \(\lVert d(C_k^2)\rVert=O(n^{-1/2})\), and \(\lVert dL_*\rVert=O(1)\). Absorbing (125) now yields \[ \lVert C_kL_*\rVert_{L^2(\mu_{h_0})}\le C_k'. \tag{126}\]

Now test a direction by taking \(w_0=Ae\), where \(e\) is any deterministic vector with \(\lVert e\rVert\le1\). By the first identity in (99), the first sum in [eq:implicit-comparison] is \(p^{\mathsf T}e\). Test with \(\widehat C_kG\). The residual estimates control the left side and the last term by \(C\lVert G\rVert_*\); the middle term is bounded by (126) and Cauchy–Schwarz, since \(0\le\widehat C_k\le C_k\). Thus \[\lvert \mathbb E[\widehat C_kG(m^k-t_*)^{\mathsf T}e]\rvert\le C\lVert G\rVert_*.\] Summing Lemma 20 with the constant target \(e\) over \(l=1,\ldots,k\) gives the same bound for \(\mathbb E[\widehat C_kG(x-m^k)^{\mathsf T}e]\). Sum over \(k\). The deficit has contribution at most \[2\sqrt n\,\lVert G\rVert_{L^2(\mu_{h_0})} \mathbb P_{\mu_{h_0}}(D>0)^{1/2}\le C\lVert G\rVert_*.\] Thus \(|\mathbb E[G(x-t_*)^{\mathsf T}e]|\le C\lVert G\rVert_*\) uniformly over \(\lVert e\rVert\le1\). Centering \(G\) and taking the supremum over this seed class gives (123). The uncentered estimate also records the ordinary-mean consequence: take \(G=1\) to obtain \(\lVert \mathbb E_{\mu_{h_0}}x-t_*\rVert\le C\). This norm is the unnormalized Euclidean norm. It compares the mean itself with \(t_*\); no approximation of the derivative of the mean map is used here. The local construction required \(n\) above a fixed threshold. In the remaining dimensions, the elementary bounds \(\lVert \mathop{\mathrm{Cov}}(G,x)\rVert^2\le n\mathop{\mathrm{Var}}(G)\) and \(\lVert \mathbb E_{\mu_{h_0}}x-t_*\rVert\le2\sqrt n\) extend both conclusions after enlarging \(C\). ◻

Proof. Select the cutoffs with a fixed threshold \(\rho\le\delta\) small enough for Lemma 15. On each cutoff support, the residual-to-root comparison gives \[\lVert m^k-t_*\rVert\le C\delta\sqrt n.\] Here the constant comes from the recurrence and the inverse bound in Lemma 15, and is independent of \(k\). For \(\lVert e\rVert\le1\), sum the square-density version of Lemma 20 with target \(e\) through \(k\), using multiplier \(\widehat C_k\). After division by \(N\), this bounds \[\lvert \mathbb E_\nu[\widehat C_k(x-m^k)^{\mathsf T}e]\rvert \le C_\delta\left(1+\frac{\mathcal D_{h_0}(f)}{N}\right).\] The cutoff weights sum to at most one, so all root-comparison terms together cost at most \(C\delta\sqrt n\). The leftover mass contributes at most \(2\sqrt n\mathbb P_\nu(D>0)\), which is controlled by the second part of Lemma 22. Add the terms, divide by \(\sqrt n\), and take the supremum over \(e\) to obtain (127). The leading \(C\) comes from root stability; only \(C_\delta\) and the finite diagnostic length depend on the selected tolerance. Any dimensions below a fixed threshold required by this construction are covered by the elementary bound \(\lVert \mathbb E_\nu x-t_*\rVert/\sqrt n\le2\) after increasing \(C_\delta\); this leaves the leading constant unchanged. The argument has used no implicit test vector. ◻