Proof. The identity \(\partial_i(P^*)=\operatorname{flip}((\partial_iP)^*)\) and the symmetry of \(\tau\otimes\tau\) show that \(\xi_i^*\) is a conjugate variable whenever \(\xi_i\) is. Uniqueness follows from polynomial \(L^2\) density, so \(\xi_i\) is self-adjoint. The expression under the supremum in (1) equals \[\sum_i\big(2\tau(\xi_iP_i(Z))-\|P_i(Z)\|_2^2\big) =\sum_i\big(\|\xi_i\|_2^2-\|\xi_i-P_i(Z)\|_2^2\big).\] For completeness, the closed span of polynomial vectors is invariant under each bounded self-adjoint left multiplication operator \(Z_j\). Its orthogonal projection therefore commutes with these operators and with \(W^*(Z)\); since it fixes \(1\), it fixes every bounded algebra vector. Those vectors are dense in \(L^2\), proving polynomial density. Symmetrizing the approximants proves the claimed supremum. Approximating their finitely many coefficients proves the countable version. Each functional in that countable supremum is continuous in the moments, which proves measurability. ◻

Proof. We first record bounds independent of the deterministic input. A \(1/4\)-net of the unit sphere of \(\mathbb C^d\), viewed in real dimension \(2d\), has cardinality at most \(9^{2d}\). For a Hermitian matrix \(G\), \(\|G\|\leq2\max_v|v^*Gv|\) over this net. For standard GUE, each \(v^*Gv\) is a real centered Gaussian of variance \(1/d\), since \(\operatorname{Tr}((vv^*)^2)=1\). Hence \[ \mathbb P(\|G\|>L) \leq 2\exp\big(d(2\log9-L^2/8)\big). \tag{3}\] In particular, the coordinate norms of any fixed finite GUE tuple are at most \(8\) with probability tending to one. For sufficiently large \(L\) the right side of (3) is at most \(2\exp(-dL^2/16)\). Integrating that tail shows that every fixed moment of the operator norm is bounded uniformly in \(d\).

Now take a deterministic input \(a\) with coordinate norms at most \(R\), and put \(F=\operatorname{tr}_d P(a,G)\) for a fixed polynomial \(P\). Let \((E_\alpha)\) be a Hermitian basis orthonormal for the unnormalized Hilbert–Schmidt inner product, and write \(G_j=\sum_\alpha g_{j\alpha}E_\alpha\). The real coordinates \(g_{j\alpha}\) are independent Gaussians with variance \(1/d\). Cyclicity of trace gives \[\partial_{g_{j\alpha}}F=d^{-1}\operatorname{Tr}(E_\alpha B_j),\qquad \sum_{j,\alpha}|\partial_{g_{j\alpha}}F|^2 =d^{-1}\sum_j\operatorname{tr}_d(B_j^*B_j),\] where \(B_j\) is the sum of the cyclically ordered products obtained by deleting each occurrence of the \(j\)th GUE letter in \(P\). The basis also complexifies to an orthonormal basis of all matrices, so the second identity applies to complex polynomials.

For a polynomial in standard real Gaussian coordinates, expansion in orthonormal Hermite polynomials shows that its variance is at most the expected squared gradient: differentiation weights each nonconstant coefficient’s squared modulus by its positive total degree. Rescaling the coordinates to variance \(1/d\), and applying this to real and imaginary parts, gives \[ \mathbb E|F-\mathbb EF|^2 \leq d^{-2}\sum_j\mathbb E\operatorname{tr}_d(B_j^*B_j) =O_{P,R}(d^{-2}). \tag{4}\] The final bound follows from the norm-moment estimate, since each \(B_j\) has fixed degree and the deterministic coordinates are uniformly bounded. Thus the cited expected-trace theorem yields convergence in probability for each deterministic moment-convergent input sequence.

This convergence is uniform over the sets \(K_d\). Otherwise, for some polynomial and positive error threshold, one could choose a subsequence and points \(a_d\in K_d\) whose error probabilities stay bounded away from zero. These points form a bounded deterministic moment-convergent sequence, contradicting the preceding conclusion. Conditioning on \(A^{(d)}\) proves convergence in probability for the random input.

Finally, any fixed product \(Q\) of normalized polynomial traces satisfies \[|Q(A^{(d)},G^{(d)})| \leq C(1+\max_j\|G_j^{(d)}\|)^D\] for constants \(C,D\) depending only on the polynomials and the common cutoff. The same estimate squared has uniformly bounded expectation. The products are therefore uniformly integrable, so their convergence in probability implies convergence of their expectations. ◻

Proof. Define \(U_{\mathbf j}\) from the word \(S_{j_1}\cdots S_{j_r}\) by summing over sets of disjoint adjacent pairs in the original index list: delete each chosen pair with factor \(-\delta_{j_a,j_{a+1}}\). The empty set gives the original word. This definition also gives \(U_{\mathbf j}^*=U_{(j_r,\ldots,j_1)}\).

Expand \(\tau(U_{\mathbf j}^*P(Z))\) by noncrossing cumulants. After multilinearly expanding \(Z=C+\sqrt u S\), freeness and semicircularity show that every block meeting an initial \(S\) position is a pair. Its covariance is \(\delta_{ij}\) against another initial \(S\) and \(\sqrt u\,\delta_{ij}\) against a selected \(S\) term from \(Z\).

To account for the deletion terms, fix a choice of \(C\) or \(S\) at every \(P\)-position in the multilinear expansion, and a noncrossing partition \(\pi\) on the original positions. Let \(A(\pi)\) be its initial-initial pair blocks whose endpoints are originally adjacent. For these fixed choices, contributions correspond exactly to subsets \(F\subseteq A(\pi)\): delete the pairs in \(F\) and restore them after expanding the remaining word. Restoring adjacent pairs preserves noncrossingness and the covariance factors, so their total coefficient is \[\sum_{F\subseteq A(\pi)}(-1)^{|F|}=(1-1)^{|A(\pi)|}.\] If \(\pi\) has any initial-initial pair, an innermost one is originally adjacent. Indeed, its intervening positions are all initial, and each must pair inside that interval to avoid crossing. Thus \(A(\pi)\) is nonempty and the partition cancels. This argument includes nested internal pairs and does not require deleting newly adjacent positions.

In each surviving partition all initial positions pair externally with positions of \(P\), in reverse order. The initial indices in \(U_{\mathbf j}^*\) are \(j_r,\ldots,j_1\), so the selected indices in \(P\) occur as \(j_1,\ldots,j_r\). The external pair arches prevent any remaining block from connecting distinct gaps. Summing the unrestricted partitions and the choices of \(C\) or \(S\) within each gap therefore gives its trace, even though the entries of \(C\) need not be free. Each external pair contributes \(\sqrt u\), proving (5).

Apply this calculation with zero signal and \(u=1\). A Wick word tests to zero against a shorter word and gives the corresponding Kronecker delta against a word of its own degree. Since every other term of a Wick polynomial has smaller degree, this proves orthonormality against Wick words of equal or smaller degree; conjugate symmetry covers the remaining cases. For \(r=1\), \(D_i(P)=(\tau\otimes\tau)(\partial_iP(Z))\), so (5) and conditional expectation give the stated conjugate variables. ◻

Proof. Let \(Z=V_u\) and let \(\xi_i\) be supplied by Lemma 4. Contraction of conditional expectation gives \(\Phi^*(Z)\leq n/u\). Also \(\tau(Z_i\xi_i)=1\) by testing the conjugate identity with \(Z_i\). Since \(\|Z_i\|_2^2=v_i(u)\), Cauchy–Schwarz gives \(\|\xi_i\|_2^2\geq v_i(u)^{-1}\) for every \(u>0\).

For the upper deficit bound, expand the block containing the first position in \(\tau(Z_iP(Z))\). Its other positions select the letters in \(D_{\mathbf j}(P)\), while its gaps contribute their moments. Thus \[\tau(Z_iP(Z))=\sum_{r\geq1,\mathbf j} \kappa_{r+1}(Z_i,Z_{j_1},\ldots,Z_{j_r})D_{\mathbf j}(P).\] Only \(r\leq\deg P\) contributes. Centering removes the order-one cumulant, and diagonal covariance makes the order-two term \(v_i(u)D_i(P)\). Adding free semicircular noise leaves all higher cumulants equal to those of \(C\).

Define the ambient \(L^2\) series \[R_i=\sum_{r\geq2,\mathbf j}u^{-r/2} \kappa_{r+1}(C_i,C_{j_1},\ldots,C_{j_r})U_{\mathbf j}(S)^*.\] The adjoint Wick family is orthonormal, and the assumed summability gives convergence in \(L^2\) with \[\sum_i\|R_i\|_2^2=\sum_{r\geq2}u^{-r}\|\kappa_{r+1}(C)\|_2^2.\] For a fixed polynomial \(P\), all terms with \(r>\deg P\) have zero trace pairing by (5). Moreover, \(A\mapsto\tau(AP(Z))\) is continuous on \(L^2\). The finite cumulant expansion above therefore gives \[\tau\big((Z_i-v_i(u)\xi_i)P(Z)\big)=\tau(R_iP(Z)).\] The coefficients in \(R_i\) need no complex conjugation: this identity uses the bilinear trace pairing, rather than a Hilbert inner product. Testing with adjoint polynomials converts it into Hilbert-space orthogonality. Polynomial \(L^2\) density then identifies \[Z_i-v_i(u)\xi_i=E[R_i\mid W^*(Z)].\]

The conjugate variables are self-adjoint by their conditional-expectation formula. Expanding the residual norm using \(\tau(Z_i\xi_i)=1\) yields \[\|Z_i-v_i(u)\xi_i\|_2^2 =v_i(u)^2\big(\|\xi_i\|_2^2-v_i(u)^{-1}\big).\] Summing after division by \(v_i(u)^2\), using projection contraction and \(v_i(u)\geq u\), proves (6). ◻

Proof. There are at most \(4^m\) noncrossing partitions and their Möbius coefficients have magnitude at most \(4^m\), by the Catalan product formula and the interval factorization of the noncrossing lattice [7]. Each product of block moments has absolute value at most \(2^m\), since \(\|B_i\|=2\) and \(\|Y\|\leq1\). Thus each cumulant is at most \(32^m\) in absolute value.

Suppose \(b\) of the \(m\) positions have \(B\) indices. A nonzero product of block moments in Möbius inversion requires, after deleting the \(Y\) positions in each block, a semicircular word with a noncrossing pairing matching equal indices. Indeed, tensor independence factors each block moment into its \(B\)-word moment and its \(Y\)-moment. The union of these pairings is still noncrossing: crossing pairs from distinct blocks would cross the containing partition. For fixed \(b\) positions, at most \(4^b p^{b/2}\) labelings can therefore contribute. Summing over positions bounds the number of potentially nonzero entries by \((1+4\sqrt p)^m\). Consequently \[\|\kappa_m(C)\|_2\leq32^m(1+4\sqrt p)^{m/2} \leq(32\sqrt5)^m p^{m/4}\leq K^m p^{m/4}.\] ◻

Proof. Use the deterministic models for \(C\) above, in dimensions \(d_k=pq_k\), and add independent GUE noise of variance \(h\). If \(M\) bounds the coordinate norms of those models, take \(R_0=M+8\sqrt h\). Lemma 3 and (3) show that these noised models lie in every fixed moment neighborhood of \(X\), within that cutoff, with probability tending to one. Their density is bounded above by \((d/(2\pi h))^{nd^2/2}\). If \(q_d\) is the indicated probability, then \[\operatorname{Vol}\Gamma_{R_0}(X;m,d,\varepsilon) \geq q_d(2\pi h/d)^{nd^2/2}.\] Since \(q_d\to1\) along the model dimensions, the normalized limsup logarithm is at least \(\frac n2\log(2\pi h)\) for each \(m,\varepsilon\). Take the infimum and then the supremum over cutoffs. The limsup definition makes this subsequence of matrix sizes sufficient. ◻

Proof. Every uniform law on a positive-volume microstate set at cutoff \(R\) has \(\sum_i\mathbb E\operatorname{tr}_d(A_i^2)\leq nR^2\). Comparison with independent GUE matrices of variance \(R^2\) bounds its normalized entropy above by \(\frac n2\log(2\pi eR^2)\). Thus \(\chi_R(X)\) is a finite real number. Choose a new increasing sequence of dimensions \(d_k\), \(k\geq1\), such that the normalized log-volume at parameters \((m,\varepsilon)=(k,1/k)\) is at least \(\chi_R(X)-1/k\). Such arbitrarily large dimensions exist by the defining limsup, and the selected sets have positive volume. The pairs \((k,1/k)\) are cofinal among all moment accuracies. Their uniform laws satisfy \[ \liminf_d H_d(A^{(d)})\geq\chi_R(X). \tag{14}\] Here and below \(d\) ranges over the selected dimensions \(d_k\). Every tuple in the support of \(A^{(d)}\) has the same fixed cutoff and increasingly accurate moments of \(X\). The selection does not depend on the subsequent noise time or polynomial tests.

Put \(Z^{(d)}(s)=A^{(d)}+\sqrt s G^{(d)}\), and let \(f_s\) be its density for \(s>0\). In the \(nd^2\) real Hilbert–Schmidt coordinates, the added Gaussian has covariance \((s/d)I\). Thus \(\partial_s f_s=(1/(2d))\Delta f_s\). Differentiating the entropy and integrating by parts gives the normalized de Bruijn identity \[ \frac{d}{ds}H_d(Z^{(d)}(s))=\tfrac12 I_d(s),\qquad I_d(s)=\frac1{d^3}\int\|\nabla\log f_s\|^2 f_s =\sum_i\mathbb E\operatorname{tr}_d(F_i^2), \tag{15}\] where \(F_i=-(1/d)\nabla_i\log f_s\) evaluated at \(Z^{(d)}(s)\). Gaussian convolution of a compactly supported distribution justifies differentiation and integration by parts on compact positive-time intervals. Indeed, derivatives have Gaussian decay, a Gaussian lower bound on \(f_s\) bounds \(|\log f_s(x)|\) by \(C(1+\|x\|^2)\), and \[\nabla\log f_s(x) =-\frac d s\big(x-\mathbb E[A^{(d)}\mid Z^{(d)}(s)=x]\big)\] has at most linear growth. Here the norm and gradient are Euclidean, and the conditional expectation is bounded at each fixed dimension. In particular the entropy and Fisher information are finite at positive times. Entropy concavity under mixtures of translations gives \(H_d(Z^{(d)}(\varepsilon))\geq H_d(A^{(d)})\). Integrating first from \(\varepsilon\) to \(t\) and then letting \(\varepsilon\downarrow0\) proves \[ H_d(Z^{(d)}(t))\geq H_d(A^{(d)})+\tfrac12\int_0^t I_d(s)\,ds. \tag{16}\] No continuity of entropy at zero is needed.

For fixed \(s>0\), integration by parts against any complex noncommutative polynomial \(P\) gives \[ \mathbb E\operatorname{tr}_d(F_i P(Z^{(d)}(s))) =\mathbb E(\operatorname{tr}_d\otimes\operatorname{tr}_d)((\partial_iP)(Z^{(d)}(s))). \tag{17}\] To verify this identity including its normalization, take an orthonormal Hermitian basis \((E_\alpha)\) for unnormalized Hilbert–Schmidt inner product. It satisfies \(\sum_\alpha E_\alpha D E_\alpha=\operatorname{Tr}(D)1\) for every complex matrix \(D\). Scalar integration by parts, separately on real and imaginary parts, turns the left side of (17) into \[d^{-2}\sum_\alpha\mathbb E\operatorname{Tr}(E_\alpha D_iP(Z^{(d)}(s))[E_\alpha]).\] Here \(D_iP(Z)[E_\alpha]\) is the directional derivative in the \(i\)th matrix coordinate along \(E_\alpha\). At an occurrence \(P=LZ_iR\) in a monomial, its contribution is \[d^{-2}\sum_\alpha\operatorname{Tr}(E_\alpha L E_\alpha R) =d^{-2}\operatorname{Tr}(L)\operatorname{Tr}(R)=\operatorname{tr}_d(L)\operatorname{tr}_d(R).\] Summing over occurrences proves (17) with the prefix and suffix in the same order as the free difference quotient.

For self-adjoint polynomial tests \(P_i\), completion of squares gives \[I_d(s)\geq\sum_i\mathbb E\operatorname{tr}_d\big(2F_iP_i(Z^{(d)}(s)) -P_i(Z^{(d)}(s))^2\big).\] First use (17) to replace every term involving the score by a product of normalized polynomial traces. Lemma 3 then gives the limit of the right side as \[\sum_i\left[2(\tau\otimes\tau)((\partial_iP_i)(V_{h+s})) -\tau(P_i(V_{h+s})^2)\right].\] The limiting law is \(V_{h+s}\) because the new semicircular noise is free from \(X=V_h\). Conjugate variables exist by Lemma 4. Taking the supremum over these tests in Lemma 2 gives \[ \liminf_d I_d(s)\geq\Phi^*(V_{h+s}). \tag{18}\] This holds for every fixed \(s>0\) along the same selected dimensions. The moments of \(V_{h+s}=C+\sqrt{h+s}S\) are continuous in \(s\), so Lemma 2 also gives measurability of the free Fisher information. Applying Fatou’s lemma to the nonnegative measurable functions \(I_d\), and then using (14) and (16), yields \[\liminf_d H_d(Z^{(d)}(t)) \geq\chi_R(X)+\frac12\int_0^t\Phi^*(V_{h+s})\,ds.\] This proves (13), even with liminf in place of limsup. The free Fisher integral is finite since \(\Phi^*(V_{h+s})\leq n/(h+s)\) and \(h>0\). Throughout the proof the terminal law is that of the same \(A^{(d)}+\sqrt tG^{(d)}\) selected above; no entropy-maximizing law at the terminal distribution has been chosen. ◻

Proof. For the first bound use the witness in (19), before adding the noise. Conditionally on \(A^{(d)}\), its projections are fixed and independent of \(G^{(d)}\). Put \(r_l=\operatorname{rank}(P_l)\). The block-diagonal Hermitian subspace has real dimension \(\sum_l r_l^2\), so the GUE covariance \(1/d\) and the normalized trace give \[\mathbb E\big[\|\sqrt t\mathcal D_P G^{(d)}_{[p]}\|_{2,d}^2\mid A^{(d)}\big] =tp\sum_l(\operatorname{rank}(P_l)/d)^2\leq2t.\] Markov’s inequality at squared threshold \(p/16\) bounds failure by \(32t/p<1/8\). On success the projected total norm exceeds \(\sqrt{p/2}-\sqrt p/4>\sqrt p/4\). Independence and centering of the noise also give \[\mathbb E\|A^{(d)}_{[p]}+\sqrt t G^{(d)}_{[p]}\|_{2,d}^2 =\mathbb E\|A^{(d)}_{[p]}\|_{2,d}^2+tp\longrightarrow vp.\] Markov’s inequality at the squared norm \(16vp\) bounds the failure of the norm condition in (21) by \(1/16+o(1)<1/8\). The union bound shows that both conditions hold with probability at least \(3/4\), proving the weaker claimed bound.

For the Gaussian bound enumerate at most \((d+1)^p\) rank lists and choose an operator-norm \(\delta\)-net of the unitary group, with \(\delta=1/(64\sqrt v)\). A maximal \(\delta\)-separated subset of \(U(d)\) has unitary centers and is a \(\delta\)-net. Its ambient operator-norm balls of radius \(\delta/2\) are disjoint and lie in the ball of radius \(1+\delta/2\) in the real \(2d^2\)-dimensional space \(M_d(\mathbb C)\). Volume comparison gives cardinality at most \((1+2/\delta)^{2d^2}\). Every decomposition of a fixed rank list is a unitary conjugate of its standard coordinate blocks \(P^0\), including lists with zero ranks. For unitaries \(U,V\) with \(\|U-V\|\leq\delta\), \[\|U^*D_{[p]}U-V^*D_{[p]}V\|_{2,d} \leq2\delta\|D_{[p]}\|_{2,d}.\] Fixed-block compression is contractive. Thus replacing a witnessing unitary by a net point loses at most \(2\delta\cdot4\sqrt{vp}=\sqrt p/8\) on (21). Hence some rank list satisfying the bounds and some net unitary \(U\) must satisfy \[\|\mathcal D_{P^0}(U^*D_{[p]}U)\|_{2,d}\geq\sqrt p/8.\] For a fixed list and unitary, unitary invariance preserves the independent variance-\(v\) GUE law of the first \(p\) coordinates. Their projected Hilbert–Schmidt vector has \(q=p\sum_l\operatorname{rank}(P_l^0)^2\leq2d^2\) real coordinates of variance \(v/d\). The further division by \(d\) in the normalized squared norm gives the law \((v/d^2)Q\), where \(Q\) is chi-squared of degree \(q\). Since \(\mathbb Ee^{Q/4}=2^{q/2}\), its probability of reaching the squared-norm threshold \(p/64\), equivalently \(Q\geq pd^2/(64v)\), is at most \[\exp\left[d^2\left(\log2-\frac{p}{256v}\right)\right].\] The full union bound consequently has logarithm at most \[p\log(d+1)+d^2\left[2\log(1+128\sqrt v)+\log2-\frac{p}{256v}\right].\] Since \(v<2+t\), Equation (11) makes the bracket less than \(-8\). The first term is \(o(d^2)\) because \(p\) is fixed. This proves the bound \(e^{-4d^2}\) for large \(d\). ◻

Proof of Theorem 1. Retain the same cutoff, dimension subsequence, and ensembles throughout. Put \(\alpha=\mu_d(\mathcal E_d)\) and \(\beta=\gamma_d(\mathcal E_d)\). Binary data processing for relative entropy and the bound on binary entropy give \[\operatorname{KL}(\mu_d\Vert\gamma_d) \geq\alpha\log\frac\alpha\beta +(1-\alpha)\log\frac{1-\alpha}{1-\beta} \geq\alpha\log\frac1\beta-\log2.\] Thus Lemma 9 implies \(\liminf_d d^{-2}\operatorname{KL}(\mu_d\Vert\gamma_d)\geq2\). We retain only one unit of this normalized deficit in the comparison below. At positive time the bounded-input Gaussian convolution has finite entropy and finite second moments, and its relative entropy with respect to \(\gamma_d\) is finite. The Gaussian reference has density \[g_d(D)=\prod_{i=1}^n \left[ \left(\frac{d}{2\pi v_i(h+t)}\right)^{d^2/2} \exp\left(-\frac{d^2\operatorname{tr}_d(D_i^2)}{2v_i(h+t)}\right) \right].\] Using \(-\int f\log f=-\int f\log g_d-\operatorname{KL}(f\Vert g_d)\) and the normalization of \(H_d\) gives the exact identity \[\begin{align*} H_d(Z^{(d)}(t)) &=\frac12\sum_i\left[ \log(2\pi v_i(h+t))+ \frac{\mathbb E\operatorname{tr}_d(Z_i^{(d)}(t)^2)}{v_i(h+t)}\right] -d^{-2}\operatorname{KL}(\mu_d\Vert\gamma_d). \end{align*}\] The expected second moments equal those of \(A_i^{(d)}\) plus \(t\), and converge to \(v_i(h+t)\). In particular, \[\limsup_d H_d(Z^{(d)}(t))\leq L(h+t)-1.\] This upper bound concerns exactly the terminal law \(Z^{(d)}(t)=A^{(d)}+\sqrt t G^{(d)}\) in Lemma 8. Combining the two bounds on its entropy and then (9)–(10) gives \[\begin{align*} \chi_R(X) &\leq L(h+t)-1-\frac12\int_0^t\Phi^*(V_{h+s})\,ds\\ &\leq\chi^*(V_{h+t})-\tfrac12 -\frac12\int_0^t\Phi^*(V_{h+s})\,ds\\ &=\chi^*(V_h)-\tfrac12. \end{align*}\] This holds for every fixed cutoff with \(\chi_R(X)>-\infty\); for the remaining cutoffs it is automatic. Taking the supremum over cutoffs preserves the bound. Lemma 7 gives \(\chi(X)>-\infty\), and (8) shows that \(\chi^*(X)\) is a finite real number. The bounded matrix realization in Section 4 supplies the asserted faithful normal tracial state and bounded self-adjoint tuple. ◻