diff f0945ea..d085931 log_sobolev_subgaussian_v2.txt
diff --git a/log_sobolev_subgaussian_v2.txt b/log_sobolev_subgaussian_v2.txt
index bf98ab4..3e8b8c3 100644
--- a/log_sobolev_subgaussian_v2.txt
+++ b/log_sobolev_subgaussian_v2.txt
@@ -1268,3 +1268,2084 @@ strictly positive energy surplus, at least (s^{-2} - lambda/s)|p|^2, and
that surplus is controlled up to sqrt(a) times the sizes involved. Since
a -> 0, means are forced to zero.
+
+
+6.9 The critical fields in both branches.
+
+The construction of Section 6.7 leaves a near-first eigenfunction in
+the strict branch; the gap branch produces its own. Both satisfy the
+same structural identities with the same uniform error estimates.
+
+Lemma 12 (Critical fields). Fix a compact interval J inside (0, 1/2)
+and s in J. Put lambda = 1/R_s. There exist a smooth positive
+probability eta and a smooth vector field w such that
+
+ 0 < I := E_eta |w|^2 < infinity,
+
+ R(eta) <= lambda^{-1}, w in Dom(A_eta)^n,
+
+ A_eta w = (lambda Id - K_s) w,
+
+ |E_eta w|^2 + E_eta |K_s w|^2 <= C_J sqrt(a) I,
+
+ E_eta <w, K_s w> <= lambda |E_eta w|^2.
+
+In the gap branch sqrt(a) may be replaced by a^2. The coordinates Y
+belong to Dom(A_eta) in both branches. The choices are:
+
+ strict branch: eta = f^2 mu_s, w = D log f^2, I = 2 lambda H(eta | mu_s);
+
+ gap branch: eta = mu_s, w = D varphi,
+
+where varphi is a centered eigenfunction of A_{mu_s} with eigenvalue
+lambda, and I = lambda E_{mu_s} varphi^2.
+
+Proof. Strict branch. Lemma 10 supplies the structural identities and
+the curvature-mean inequality. Apply Lemma 11 to sigma = eta with
+F_0 = I and D_0 = I/(2 lambda). The deficit vanishes, so its conclusion
+reads
+
+ (s^{-2} - lambda/s) |p|^2 <= C_J sqrt(a) I,
+
+where p = E_eta Y. Because s^{-2} - lambda/s >= 2 on J, and
+|E_eta b_s| <= C_J a sqrt(I) comes from the same proof, the score
+identity E_eta w = (p - E_eta b_s)/s yields |E_eta w|^2 <= C_J sqrt(a) I.
+Now 0 <= K_s <= s^{-1} Id, so K_s^2 <= s^{-1} K_s, and the
+curvature-mean inequality gives
+
+ E_eta |K_s w|^2 <= s^{-1} E_eta <w, K_s w> <= (lambda/s) |E_eta w|^2.
+
+The two displayed bounds are the first line of uniform smallness.
+
+Gap branch. Compact form embedding supplies a minimizer of the
+Rayleigh quotient for A_{mu_s} among mean-zero unit-norm functions.
+Its positive eigenvalue is 1 / R_s^P = lambda. The nullspace is the
+constants, so the quotient has a positive gap. Interior elliptic
+regularity makes its eigenfunction varphi smooth. Commuting one
+derivative gives
+
+ A_{mu_s} D varphi = (lambda Id - K_s) D varphi.
+
+The right side is in L^2(mu_s), so Lemma 3 puts D varphi in the
+operator domain. Pairing this display with varphi and applying
+Poincare at mu_s reproduces the curvature-mean inequality
+E_eta <w, K_s w> <= lambda |E_eta w|^2.
+
+The coordinate function Y also sits in the operator domain, because
+both Y and A_{mu_s} Y = (Y - b_s)/s lie in L^2(mu_s). Pairing
+A_{mu_s} varphi = lambda varphi against Y, using self-adjointness,
+gives
+
+ E_eta w = lambda E[varphi Y]
+ = (E[varphi Y] - E[varphi b_s]) / s,
+
+ (1 - lambda s) E[varphi Y] = E[varphi b_s].
+
+Conditional Jensen and Lemma 6 give E_{mu_s} b_s b_s^* <=
+E_mu X X^* <= C a^2 Id. For every unit vector v, Cauchy-Schwarz
+bounds |E[varphi v.b_s]| by C a ||varphi||_2, with no factor from
+dimension. Since 1 - lambda s >= 1/2 and I = lambda ||varphi||_2^2, the
+display gives |E w| <= C_J a sqrt(I), and K_s^2 <= s^{-1} K_s
+finishes the uniform smallness of K_s w.
+
+Coordinate assertion in the strict branch. H(eta | mu_s) < infinity
+together with the Gaussian tails of mu_s implies E_eta |Y|^2 <
+infinity. The distributional identity
+
+ A_eta Y = (Y - b_s)/s - w
+
+has an L^2(eta) right side, because b_s is bounded for the fixed law
+and w lies in L^2(eta). Lemma 3 applies. QED.
+
+
+6.10 How the regularized constants converge to one.
+
+Proposition 13. For the sequence of normalized counterexamples,
+
+ R_s -> 1 for every 0 < s < 1/2.
+
+At each differentiability point of R_s, the fields of Lemma 12 satisfy
+the contact identity
+
+ R_s' = R_s * ( E_eta <w, K_s w> / I ).
+
+In particular 0 <= R_s' <= C_J sqrt(a) almost everywhere on each fixed
+compact J.
+
+Proof. Fix s, and let Phi_{s,t} be the heat continuity flow of Lemma 7.
+Transport a form-domain test g by g_t = g o Phi_{s,t}^{-1}. Its
+distribution under mu_t is fixed, so entropy and variance are constant.
+The Jacobian equation d DPhi / dt = K_t DPhi / 2 gives
+
+ (d/dt at t = s) of int |D g_t|^2 dmu_t
+ = - int <Dg, K_s Dg> dmu_s.
+
+For compact smooth g this follows by differentiating the
+inverse-transpose Jacobian inside the gradient. A general form-domain
+g follows by Sobolev approximation: flow derivatives and their time
+derivatives are uniformly bounded on a compact time interval, and the
+derivative of the integrand is dominated by C |Dg|^2. Dominated
+convergence proves the identity without smoothness or compact support.
+
+Strict branch. Take g = f and Q(t) = Ent_{mu_t}(f_t^2) /
+(2 int |D f_t|^2 dmu_t). The denominator is positive near s, and
+Q(t) <= R_t for both signs of t - s, with Q(s) = R_s. At a
+differentiability point of R_s the two-sided contact gives
+Q'(s) = R_s'. The energy identity and w = 2 Df / f give
+R_s' = R_s E<w, K_s w> / I.
+
+Gap branch. Take g = varphi and Q(t) = Var_{mu_t}(varphi_t) /
+int |D varphi_t|^2 dmu_t. Now Q(t) <= R_t^P <= R_t and
+Q(s) = R_s^P = R_s. The same contact argument gives the identical
+derivative formula with w = D varphi.
+
+Lemma 12 bounds the derivative by C_J sqrt(a) in either branch, with
+constants independent of which branch holds at each time. Lipschitz
+continuity makes R_s absolutely continuous. For fixed
+0 < delta < s < 1/2, integrating on [delta, s],
+
+ 1 <= R_s
+ <= R_delta + C_{[delta, s]} sqrt(a) (s - delta)
+ <= 1 + delta + C_{[delta, s]} sqrt(a) (s - delta).
+
+Take the sequence limit first, then delta -> 0. QED.
+
+
+6.11 The full critical sequence.
+
+Proposition 14. If Theorem 1 fails, there is a sequence of dimensions
+and centered compactly supported log-concave laws mu satisfying the
+normalization of Lemma 6. Fix s = 1/8, form mu_s, b_s, M_s, K_s, and
+put lambda = 1/R_s. After passage to a subsequence there exist smooth
+positive probabilities eta and smooth vector fields w such that, with
+A = A_eta and I = E_eta |w|^2,
+
+ lambda -> 1, R(eta) <= lambda^{-1}, 0 < I < infinity,
+
+ w in Dom(A)^n, A w = (lambda Id - K_s) w,
+
+ ||(A - 1) w||_{L^2(eta)} + |E_eta w| = o(sqrt I),
+
+ E_eta |K_s w|^2 = o(I), E_eta |Dw|^2 = (1 + o(1)) I.
+
+The spectral gap of A is at least lambda. Each coordinate function
+belongs to Dom(A). Either of the following holds throughout the
+subsequence.
+
+ (i) R_s > R_s^P: eta = f^2 mu_s for the positive unit-norm optimizer,
+ w = D log f^2, I = 2 lambda H(eta | mu_s). The exact deficit
+ identity of Lemma 10 holds for compact square-root tests
+ relative to eta, and A Y = (Y - b_s)/s - w.
+
+ (ii) R_s = R_s^P: eta = mu_s, A varphi = lambda varphi,
+ E_eta varphi = 0, w = D varphi, I = lambda E varphi^2.
+ Coefficients may be scaled positively so that I -> 0.
+
+All o-terms refer to the sequence, with all finite auxiliary parameters
+held fixed. No relation between dimension, I, and the rate of
+convergence is asserted or required.
+
+Proof. Choose an infinite subsequence with a fixed branch at s = 1/8.
+Lemma 12 gives the structural identities, together with the curvature
+and mean bounds. Proposition 13 gives lambda -> 1, hence
+
+ (A - 1) w = (lambda - 1) w - K_s w = o_{L^2(eta)}(sqrt I).
+
+The energy pairing yields
+
+ E|Dw|^2 = lambda I - E<w, K_s w> = (1 + o(1)) I,
+
+and Poincare with constant lambda^{-1} is the asserted spectral gap.
+The coordinate assertion and the branch-specific identities are
+already proved. In the gap branch every estimate is homogeneous in
+varphi, so positive rescaling enforces I -> 0 while leaving each
+normalized claim intact. QED.
+
+
+7. THE PREDICTION CURVE
+
+The critical sequence of Proposition 14 provides the following data.
+Fix s = 1/8 and choose one of its two subsequences, suppressing the
+sequence index. Then
+
+ lambda -> 1, R(eta) <= 1/lambda, A = A_eta,
+
+ I = E_eta |w|^2 > 0, w in Dom(A)^n,
+
+ ||(A - 1) w||_{L^2(eta)} + |E_eta w| = o(sqrt I). (7.1)
+
+Strict branch: eta = f^2 mu_s, w = D log f^2, 2 H(eta | mu_s) = I / lambda.
+
+Gap branch: eta = mu_s, w = D varphi, A_{mu_s} varphi = lambda varphi,
+E_{mu_s} varphi = 0, I = lambda E_{mu_s} varphi^2.
+
+Every parameter T, M, r, epsilon is fixed before the sequence limit.
+
+
+7.1 Efficient Gaussian probes (strict branch).
+
+Let Y_0 ~ eta and let G' be an independent standard Gaussian vector.
+Define the total observation energy
+
+ L_T = sqrt(T) w(Y_0) + G', d(T) = H(Law(L_T) | gamma), (7.2)
+
+where gamma is the standard Gaussian on the same Euclidean space. If w
+were an exact first eigenfunction of A, d(T) would vanish identically in
+T. The next lemma says d(T) is small.
+
+Lemma 15 (Efficient probes). In the strict branch, for every finite M,
+
+ sup_{0 <= T <= M} d(T) = o(I). (7.3)
+
+Proof. Run the stationary eta-diffusion with independent Brownian noise
+and semigroup P_u = e^{-uA}. Proposition 14 combined with the
+coordinate-generator identity of Lemma 12 gives
+
+ -A Y = -Y/s + w + b_s/s. (7.4)
+
+Consequently E_eta |Y_0|^2 < infinity. The LSI of eta gives
+
+ E_eta exp(v.(Y_0 - E Y_0)) <= exp(|v|^2 / (2 lambda)).
+
+Integrating against a Gaussian v shows E_eta exp(c' |Y_0|^2) < infinity
+for some c' > 0 depending on the individual law. This exponential-square
+moment is the hypothesis of Lemma 16.
+
+Fix T <= M. Define
+
+ J_u = MI(L_T; Y_u),
+ F_u = E Ical(Law(Y_u | L_T) | eta),
+ Q_u = E |E[Y_u | L_T]|^2.
+
+The Gaussian observation identity of Lemma 5 gives
+J_0 = T I / 2 - d(T). Entropy dissipation yields -dJ_u >= F_u du, so
+int_0^1 F_u du <= J_0. For sigma_{u,l} = Law(Y_u | L_T = l), the chain
+rule and the strict-normalization identity give
+
+ E H(sigma_{u,L_T} | mu_s) = J_u + I/(2 lambda),
+
+ E Ical(sigma_{u,L_T} | mu_s) = F_u + I.
+
+For the second identity, the conditional relative score with respect to
+mu_s is the sum of w and the conditional relative score with respect to
+eta. The latter scores have conditional mean zero at each Y_u, so their
+contribution vanishes weakly, since the conditional likelihoods average
+to one under the marginal of Y_u.
+
+Apply Lemma 11 to sigma_{u,l} and average over l. Since
+s^{-2} - lambda/s >= 2, on [0,1]
+
+ -dJ_u >= (2 lambda J_u + c_s Q_u) du - d E_u, E([0,1]) = o_M(I), (7.5)
+
+with c_s > 0 and E a nonnegative error measure. The error in Lemma 11
+is bounded by a quantity vanishing along the sequence times
+F_u + I + J_u + I/(2 lambda), whose integral is O_M(I). The measure
+formulation avoids differentiability of J_u.
+
+The spectral theorem and (7.1) give
+
+ sup_{0 <= u <= 1} ||P_u w - e^{-u} w||_2
+ <= ||(A - 1) w||_2 = o(sqrt I). (7.6)
+
+Reversibility gives E[L_T | Y_u] = sqrt(T) P_u w(Y_u). The entropy
+duality of Lemma 5 with a linear test function gives
+
+ J_u + d(T) = E H(Law(L_T | Y_u) | gamma)
+ >= (T/2) ||P_u w||_2^2
+ = (T I / 2) e^{-2u} + o_M(I).
+
+Integrating (7.5) with factor e^{2 lambda u} and comparing at u = 1,
+
+ int_0^1 Q_u du <= C_s d(T) + o_M(I). (7.7)
+
+The difference e^{-2 lambda} - e^{-2} contributes only o_M(I).
+
+Bound the predicted position of b_s. By conditional Jensen and Lemma 6,
+
+ log E_{mu_s} exp(v.b_s) <= C a^2 |v|^2.
+
+The entropy variational formula, optimized over v, gives
+
+ E |E[b_s(Y_v) | L_T]|^2 <= C a^2 (J_v + I/(2 lambda)) = o_M(I).
+
+Condition (7.4) on L_T and solve by variation of constants. By (7.6)
+and contraction of conditional expectations,
+
+ E[Y_u | L_T] = e^{-u/s} E[Y_0 | L_T]
+
+ + ((e^{-u} - e^{-u/s}) / (1/s - 1)) E[w(Y_0) | L_T]
+
+ + o_{L^2, M}(sqrt I), (7.8)
+
+uniformly for 0 <= u <= 1. The two scalar coefficient functions are
+linearly independent on [0,1]: the first is nonzero at 0, and the
+second vanishes at 0 and is nonzero for u > 0. Their 2 x 2 Gram matrix
+is positive definite. Applying its smallest-eigenvalue bound to each
+coordinate and summing, (7.7) and (7.8) show
+
+ E |E[w(Y_0) | L_T]|^2 <= C_s d(T) + o_M(I).
+
+By Lemma 5 the left side equals 2 d'(T) for almost every T. The error
+is uniform for T in [0, M], and d(0) = 0. Integrating the resulting
+differential inequality proves (7.3). QED.
+
+
+7.2 Entropy loss under a Gaussian observation.
+
+The next comparison is the lower-bound engine of the prediction curve.
+If heat smoothing removes only a little of the initial relative entropy,
+then the score's best predictor must already be large. The comparison
+discards at each new observation the direction in which the two
+candidate posterior means disagree. In continuous-time stochastic
+localization, Eldan-Koehler-Zeitouni and Chen-Eldan control this cost
+via projections chosen to preserve a selected function. The argument
+below is direct: it uses finite Gaussian observations with singular
+projections, and takes independent copies of the entire experiment.
+
+Lemma 16 (Entropy loss). Let p, q be smooth strictly positive
+probability densities on R^d, with q log-concave. Suppose
+
+ H_0 = H(p | q) < infinity, w = D log(p/q), I = E_p |w|^2 < infinity,
+
+ E_p exp(c |Y|^2) < infinity for some c > 0,
+
+where Y ~ p. Let G be a standard Gaussian independent of Y. For r > 0
+put U_r = Y + sqrt(r) G, p_r = p * gamma_r, q_r = q * gamma_r,
+
+ wbar_r(U_r) = E_p[w(Y) | U_r], H_r = H(p_r | q_r).
+
+Then for every z > 0,
+
+ 2 (H_0 - H_z) <= int_0^z E_p |wbar_r(U_r)|^2 dr. (7.9)
+
+Proof. Setup. Put b = 1/z. Take N independent copies of Y and a
+Gaussian observation mesh of size m; put delta = u/m. The three limits
+(mesh, copies, extra precision) are taken in that order.
+
+Additional observations and their entropy cost. Write
+Y = (Y_1,...,Y_N) and W = (w(Y_1),...,w(Y_N)). The two candidate
+priors are p^N and q^N; expectations below are with respect to p^N.
+
+First observe Z_0 = sqrt(b) Y + G_0. By the chain rule, the posterior
+relative entropy averaged over this observation equals N(H_0 - H_z).
+
+Fix u > 0 and mesh m, with delta = u/m. For i = 0,...,m-1 let H_i be
+the retained history up to step i. Compute the two posterior means
+under p and q given H_i, and let P_i be the orthogonal projection onto
+their span, with P_i = 0 when the span is trivial. Retain only
+
+ (1 - P_i) Z_{i+1}, Z_{i+1} = sqrt(delta) Y + G_{i+1},
+
+with all Gaussian noises independent and the same rule applied under
+both priors. Integral-formula versions of the posterior means make the
+rule measurable.
+
+For each realized retained history, the likelihood of Y is Gaussian
+with precision matrix
+
+ B = b Id + delta sum_{i=0}^{m-1} (1 - P_i) >= b Id. (7.10)
+
+No extra factor enters from the choices of P_i, since each choice was
+determined by the retained past; the factor is therefore the same under
+both priors. The posterior relative score of p against q is still W.
+
+Bound the entropy lost in the additional observations. Condition on a
+retained history at one step and put Q = 1 - P_i. Under either prior
+the projected posterior means agree. Subtract their common mean and
+let X_c denote the resulting centered projected signal in ran Q. The
+base posterior has curvature at least b Id. Its centered linear moment
+bound and Gaussian integration give
+
+ E_q [exp(kappa |X_c|^2) | H_i] <= (1 - 2 kappa/b)^{-Nd/2},
+ 0 < 2 kappa < b.
+
+For the alternative prior write a_i = E_p[Y | H_i]. For sufficiently
+small kappa, conditional Jensen gives
+
+ E_p[exp(kappa |Q(Y - a_i)|^2) | H_i]
+ <= exp(2 kappa |a_i|^2) E_p[exp(2 kappa |Y|^2) | H_i]
+ <= (E_p[exp(2 kappa |Y|^2) | H_i])^2.
+
+Squaring and applying conditional Jensen once more bounds the average
+of this square over H_i by E_p exp(8 kappa |Y|^2) < infinity. The
+constants are independent of the mesh and of the adaptive choices.
+
+Let h_p, h_q be the output densities of sqrt(delta) X_c + G in ran Q,
+and let phi be the standard Gaussian density there. Centering and
+Jensen give h_q / phi >= exp(-delta Nd/(2 b)) pointwise. For either
+conditional signal, integrating the Gaussian kernel produces
+
+ || h/phi - 1 ||_{L^2(phi)}^2 = E[exp(delta X_c . X_c') | H_i] - 1,
+
+where X_c' is a conditionally independent copy. The linear term
+vanishes because X_c is centered. The inequality
+|exp(t) - 1 - t| <= (t^2/2) exp(|t|), together with
+|X_c . X_c'| <= (|X_c|^2 + |X_c'|^2)/2, bounds the last expression by
+C delta^2 times a squared small conditional exponential-square moment.
+The preceding bounds therefore give
+
+ E H(h_p | h_q) <= E chi^2(h_p | h_q) <= C_{N,b,p} delta^2.
+
+This includes the zero-dimensional observed subspace, where the loss is
+zero. Summing the m steps loses O_{N,b,p}(u delta).
+
+Apply the anisotropic curvature inequality of Lemma 1 to the final
+posterior pair: on each realized history the covariance of the retained
+observations has curvature at least B. For fixed N and u,
+
+ 2 N (H_0 - H_z) - o_{delta -> 0}(1) <= E W^* B^{-1} W. (7.11)
+
+Comparison with full observations. Work in the Hilbert space of
+square-integrable vector fields on the signal and all full observations
+Z_0,...,Z_m. Then ||W||_2^2 = N I. Completing the square pointwise in v,
+
+ b ||W||_2^2 - b^2 E W^* B^{-1} W
+ = min_v [ b ||W - v||_2^2
+ + delta sum_{i=0}^{m-1} ||(1 - P_i) v||_2^2 ]. (7.12)
+
+The minimizer is v_* = b B^{-1} W, so ||v_*||_2 <= ||W||_2.
+
+Let F_i = sigma(Z_0,...,Z_i) and e_i = E[. | F_i]. Replace 1 - P_i by
+1 - e_i in (7.12) and let M_0 be the resulting minimum. This modified
+minimum has the explicit minimizer
+
+ v_0 = b (b + delta sum_{i=0}^{m-1} (1 - e_i))^{-1} W.
+
+Use the orthogonal decomposition into increments of the nested
+sigma-fields. Put alpha_k = b / (b + k delta) for 0 <= k <= m. On
+ran e_0 the multiplier is 1; on ran(e_k - e_{k-1}) it is alpha_k; on
+ran(1 - e_{m-1}) it is alpha_m. Therefore
+
+ v_0 = alpha_m W + sum_{k=0}^{m-1} (alpha_k - alpha_{k+1}) e_k W. (7.13)
+
+All coefficients are nonnegative and sum to one. Put g_i = (1 - e_i) v_0.
+Equation (7.13) gives
+
+ g_i = alpha_m (W - e_i W)
+ + sum_{k=i+1}^{m-1} (alpha_k - alpha_{k+1})(e_k W - e_i W). (7.14)
+
+The weights in this formula sum to alpha_{i+1} <= 1.
+
+Independence across copies, boundedness of the retained histories, and
+the conditional covariance structure of the observation arrays put
+
+ Z_{i,j} := tr Cov(g_{i,j} | F_i)
+ <= E[|W_j - e_i W_j|^2 | F_i]
+ <= E[|W_j|^2 | F_i]. (7.15)
+
+For every term with k > i, the tower property bounds the conditional
+squared mean by E[|W_j - e_i W_j|^2 | F_i], and nonnegative weights of
+total at most one preserve this bound.
+
+The right sides in (7.15) are uniformly integrable over the sigma-
+fields appearing. Concretely, if Z >= 0 is integrable and M = E[Z | F],
+
+ E[M 1_{M > K}] <= 2 E[Z 1_{Z > K/2}]. (7.16)
+
+Indeed M <= K/2 + E[Z 1_{Z > K/2} | F], and on {M > K} the final
+conditional expectation is at least M/2. Since P_i is F_i-measurable
+and rank at most one,
+
+ E |P_i g_i|^2 <= E max_{1 <= j <= N} Z_{i,j}.
+
+The maximum is at most K + sum_j Z_{i,j} 1_{Z_{i,j} > K}. Equations
+(7.15) and (7.16) therefore give, uniformly in i, m, delta,
+
+ E |P_i g_i|^2 <= N q_N,
+ q_N := inf_{K > 0} { K/N + 2 E_p [|w(Y)|^2 1_{|w(Y)|^2 > K/2}] } -> 0. (7.17)
+
+No moment of the score beyond its second moment enters. The bound
+permits P_i to depend adaptively on all copies.
+
+A dual test compares the original minimum with M_0. For every v,
+
+ ||(1 - P_i) v||_2^2 >= 2 <v, (1 - P_i) g_i> - ||g_i||_2^2.
+
+The normal equation for v_0 is b (W - v_0) = delta sum_i g_i. So the
+minimum over v of
+
+ b ||W - v||_2^2 + 2 delta sum_i <v, g_i> - delta sum_i ||g_i||_2^2
+
+attains the value M_0 at v_0. Applying the linear bound at v_* in
+(7.12) and using (7.17),
+
+ min >= M_0 - 2 delta sum_i ||v_*||_2 ||P_i g_i||_2
+ >= M_0 - 2 u N sqrt{I q_N}. (7.18)
+
+The prediction integral. The same nested decomposition gives
+
+ M_0 / N
+ = sum_{i=0}^{m-1}
+ b^2 delta / ((b + i delta)(b + (i+1) delta))
+ * (I - ||e_i W_1||_2^2). (7.19)
+
+A component first seen after j increments pays b j delta / (b + j delta),
+and this equals sum_{i=0}^{j-1} b^2 delta / ((b + i delta)(b + (i+1) delta)).
+Full observations through step i constitute a Gaussian sufficient
+statistic of precision b + i delta; completing the likelihood square
+proves sufficiency directly. Consequently
+
+ ||e_i W_1||_2^2 = E_p |wbar_{1/(b+i delta)}(U_{1/(b+i delta)})|^2.
+
+This squared prediction norm is nondecreasing in precision: a weaker
+observation is obtained by convolving with fresh Gaussian noise. The
+norms are bounded by I. So (7.19) converges as m -> infinity to
+
+ b^2 int_0^u (I - E_p |wbar_{1/(b+t)}(U_{1/(b+t)})|^2) / (b + t)^2 dt.
+
+Limits. Take the mesh limit first, at fixed N, u. The entropy error in
+(7.11) vanishes, and the error in (7.18) remains uniform in the mesh.
+Divide by N. Let N -> infinity, using q_N -> 0. Combine (7.11) and
+(7.12):
+
+ 2(H_0 - H_z) <= I/b
+ - int_0^u (I - E_p |wbar_{1/(b+t)}(U_{1/(b+t)})|^2) / (b + t)^2 dt
+
+ = I/(b+u) + int_{1/(b+u)}^{1/b} E_p |wbar_r(U_r)|^2 dr.
+
+Finally take u -> infinity. Since 1/b = z and the integrand is bounded
+by I, this is (7.9). QED.
+
+
+7.3 The prediction curve.
+
+For Y_0 ~ eta and independent Gaussian noise define
+
+ U_r = Y_0 + sqrt(r) G,
+
+ wbar_r(U_r) = E[w(Y_0) | U_r],
+
+ m(r) = (1/I) E |wbar_r(U_r)|^2.
+
+In the strict branch, let L_T be the labeled observation of Lemma 15.
+It is constructed with fresh Gaussian noise independent of (Y_0, G), so
+L_T and U_r are conditionally independent given Y_0. Define
+
+ p_T(r) = (1/I) E |E[w(Y_0) | U_r, L_T]|^2, p_0(r) = m(r).
+
+Proposition 17 (Prediction curve). In both branches, for every fixed
+0 < r < 1/4,
+
+ m(r) -> e^{-r}. (7.20)
+
+In the strict branch, p_T(r) -> e^{-r} for every fixed finite T >= 0.
+In the gap branch the eigenfunction may be rescaled to make I -> 0
+without changing (7.20) or (7.1).
+
+Proof. For 0 <= r < 1/4 put theta(r) = exp(-int_0^r dt / R_{s+t}).
+Proposition 13 gives theta(r) = e^{-r} + o(1) uniformly on
+compact intervals. The common heat flow dissipates relative entropy
+at one half of the relative Fisher information. Its entropy relative to
+mu_{s+r} is therefore at most theta(r) times its initial entropy, by
+the LSI at the contemporaneous base law. The same assertion holds for
+conditional laws after averaging.
+
+Strict branch. Write H_r = H(Law(U_r) | mu_{s+r}) and
+H_0 = I/(2 lambda). The entropy of U_r given L_T, relative to mu_{s+r}
+and averaged over the label, is at most theta(r)(H_0 + TI/2), since
+initially it is H_0 + MI(Y_0; L_T). Chain rule and Lemma 5,
+
+ (I/2) int_0^T p_v(r) dv = E H(Law(L_T | U_r) | gamma)
+
+ = E H(Law(U_r | L_T) | mu_{s+r}) + d(T) - H_r
+
+ <= (I/2) theta(r)(1/lambda + T) + d(T).
+
+The first equality is from Lemma 5, applied conditionally on U_r.
+Fix v >= 0 and then T > v. Since p_t(r) is nondecreasing in signal
+strength t, Lemma 15 gives
+
+ (T - v) limsup p_v(r) <= e^{-r}(1 + T).
+
+Taking the sequence limit with T fixed and then T arbitrarily large,
+
+ limsup p_v(r) <= e^{-r}. (7.21)
+
+Lemma 16 applies to p = eta and q = mu_s; the exponential-square
+moment was proved in Lemma 15. Since H_z <= theta(z) H_0, it gives
+
+ int_0^z m(r) dr >= (1 - theta(z))/lambda, 0 < z < 1/4.
+
+The reverse limiting inequality follows from (7.21), 0 <= m <= 1, and
+reverse Fatou. Hence
+
+ int_0^z m(r) dr -> 1 - e^{-z}. (7.22)
+
+The function m is nonincreasing in r. For 0 < h < r with r + h < 1/4,
+its value at r lies between its averages on [r, r+h] and [r-h, r].
+Equation (7.22) determines the limits of both averages; h -> 0 gives
+(7.20). Finally m(r) <= p_T(r) together with (7.21) proves the
+strict-branch assertion for every fixed T.
+
+Gap branch, without additional probes. Linearize Lemma 16. Let q be a
+bounded smooth centered test under mu_s with bounded gradient, and put
+d eta_eps = (1 + eps q) d mu_s with |eps| ||q||_infty < 1. This
+alternative has finite exponential-square moment. Write
+qbar_r = E_{mu_s}[q(Y_0) | U_r] and
+Dqbar_r = E_{mu_s}[D q(Y_0) | U_r]. The output density ratio is
+1 + eps qbar_r, and the predicted initial relative score is exactly
+
+ E_{eta_eps}[D log(1 + eps q)(Y_0) | U_r]
+ = eps Dqbar_r / (1 + eps qbar_r).
+
+Its squared norm averaged under the alternative output is therefore
+eps^2 E_{mu_{s+r}}[|Dqbar_r|^2 / (1 + eps qbar_r)]. The entropy
+expansion H((1 + eps q) mu_s | mu_s) = eps^2 ||q||_2^2 / 2 + o(eps^2),
+together with its output version, gives
+
+ ||q||_2^2 - ||qbar_z||_2^2 <= int_0^z ||D qbar_r||_2^2 dr. (7.23)
+
+The denominator is bounded away from zero uniformly in r, and
+conditional Jensen bounds the integrand by a constant times ||Dq||_2^2,
+justifying the limit. Approximation in the form norm extends (7.23) to
+q = varphi: centered compact smooth approximants are bounded, and
+conditional expectation is L^2-contractive. To retain compact support
+while centering, subtract the approximant's mean times one fixed
+compact smooth function of mean one; that correction tends to zero in
+the form norm.
+
+Heat-variance identity used here. For a centered scalar or
+finite-dimensional vector field g in L^2(mu_s), let
+g_r = E[g(Y_0) | U_r] under the base law. Then
+
+ (d/dr) E_{mu_{s+r}} |g_r|^2 = - E_{mu_{s+r}} |D g_r|^2. (7.24)
+
+For bounded g, both the numerator density g_r mu_{s+r} and the
+denominator density mu_{s+r} solve the heat equation, so the quotient
+calculation with the square function gives
+
+ (d/dr - Delta/2)(mu_{s+r} |g_r|^2) = - mu_{s+r} |D g_r|^2.
+
+Integrate against expanding cutoffs; their Laplacians vanish uniformly.
+The bound |g_r| <= ||g||_infty removes the cutoff terms and gives the
+integrated identity on positive time. The initial endpoint follows from
+||g_r(U_r) - g(Y_0)||_2 -> 0: for bounded continuous g use g(U_r) as a
+predictor and dominated convergence under the coupling
+U_r = Y_0 + sqrt(r) G, then approximate in L^2(mu_s) and use
+conditional contraction. Approximate a general L^2 field by bounded
+ones. Applying the identity to their differences shows that the
+gradients are Cauchy in the space-time energy norm, while conditional
+contraction gives convergence of both endpoint norms. This proves the
+integrated version of (7.24) in full generality.
+
+Poincare at mu_{s+r} and preservation of the mean therefore give
+
+ E_{mu_{s+r}} |g_r|^2 <= theta(r) E_{mu_s} |g|^2.
+
+Apply this first to varphi and then to w - E w. The second and (7.1)
+give m(r) <= theta(r) + o(1). The first and ||varphi||_2^2 = I/lambda,
+inserted into (7.23), give
+
+ int_0^z m(r) dr >= (1 - theta(z))/lambda.
+
+The same squeeze and monotonicity argument prove (7.20). Multiplying
+varphi by positive scalars multiplies every squared quantity by the
+square, so their choice arranges I -> 0. QED.
+
+
+7.4 Posterior covariance and outlier subspaces.
+
+The hierarchy construction depends on two further consequences of the
+prediction curve. The strict branch needs conditional covariance tails;
+both branches need bounds on the subspaces where w has nonnegligible
+energy.
+
+Lemma 18. Use the critical sequence and notation of Proposition 17,
+with I -> 0 in the gap branch. Then:
+
+ (i) For every fixed c < infinity,
+
+ sup over deterministic orthogonal projections Pi, rank Pi <= c I:
+ E |Pi w|^2 = o(I). (7.25)
+
+ (ii) For every epsilon > 0 there is a deterministic subspace
+ V_epsilon with dim V_epsilon <= C_epsilon I such that, for every
+ v perpendicular to V_epsilon,
+
+ v^*(E w w^*) v <= epsilon |v|^2,
+
+ v^*(E b_s b_s^*) v <= epsilon |v|^2,
+
+ v^*(E M_s) v <= epsilon |v|^2. (7.26)
+
+ In the gap branch V_epsilon = {0} suffices, and
+ ||E b_s b_s^*||_op + ||E M_s||_op = o(1).
+
+ (iii) Strict branch only: for fixed r in (0,1/4) and epsilon > 0,
+
+ E tr(C_r 1_{C_r > epsilon Id}) = o(I), (7.27)
+
+ where C_r = Cov(w(Y_0) | U_r) and the indicator denotes the spectral
+ projection.
+
+Proof. Strict-case spectral estimates. If S has covariance C and mean
+a, the affine predictor a + C (Id + C)^{-1} (S + G - a) of S from S + G
+has mean squared norm |a|^2 + tr(C^2 (Id + C)^{-1}), as a direct
+expansion of the covariances shows. Conditional expectation achieves
+at least this squared norm, being the orthogonal projection onto all
+functions of the observation. The same holds conditionally on any side
+information, with fresh Gaussian noise.
+
+The Gaussian-signal derivative formula and monotonicity of the
+prediction energy (Lemma 5) give
+
+ E |E[w | L_1]|^2 <= 2 (d(2) - d(1)) <= 2 d(2) = o(I).
+
+Apply the affine-prediction bound to S = w and C = Cov_eta w. For every
+fixed epsilon > 0,
+
+ tr(C 1_{C > epsilon Id}) = o(I), (7.28)
+
+because t^2/(1 + t) >= epsilon t / (1 + epsilon) for t > epsilon. For
+any projection of rank at most c I,
+
+ E |Pi w|^2 <= epsilon c I + tr(C 1_{C > epsilon Id}) + |E w|^2.
+
+First sequence limit, then epsilon -> 0. Equation (7.1) proves (7.25)
+uniformly in Pi.
+
+In the gap branch, a projection of rank at most c I is eventually
+zero, which proves (7.25).
+
+Conditionally on U_r, the additional observation L_1 has signal w and
+covariance C_r. Orthogonality of successive conditional expectations
+and the conditional affine-prediction bound give
+
+ I (p_1(r) - m(r)) >= E tr(C_r^2 (Id + C_r)^{-1}).
+
+Proposition 17 makes the left side o(I). The scalar inequality
+t^2/(1+t) >= epsilon t/(1+epsilon) again gives (7.27).
+
+Bounding the dimensions of the outlier spaces. Let
+
+ D = E_eta E[XX^* | X + sqrt(s) G = Y_0],
+
+where the inner posterior kernel refers to the original prior mu.
+Conditional Jensen and the definition of M_s give
+
+ E_eta b_s b_s^* <= D, E_eta M_s <= D/s. (7.29)
+
+For each deterministic projection Q of rank l >= 1, the subgaussian
+assumption of Lemma 6 gives
+
+ log E_mu exp(|QX|^2 / (C a^2)) <= C l. (7.30)
+
+Indeed choose a 1/2-net of the unit sphere in ran Q with at most 5^l
+points. |QX| is at most twice the largest absolute linear form over
+this net, and bounding the exponential of a maximum by the sum proves
+(7.30).
+
+Strict branch. Form the joint law by first drawing Y_0 from eta and
+then X from the original posterior kernel. Its entropy relative to the
+original joint law of (X, X + sqrt(s) G) equals H_0 = I/(2 lambda).
+Data processing bounds the entropy of its X-marginal relative to mu by
+H_0. The entropy variational inequality applied to (7.30) gives
+
+ tr(QD) <= C a^2 (H_0 + l). (7.31)
+
+If the spectral subspace of D above tau > 0 has dimension l, then
+tau l <= C a^2 (H_0 + l). For small a this forces
+l <= 2 C a^2 H_0 / tau = O_tau(I); the assertion is trivial when
+l = 0. The spectral subspace of E w w^* above epsilon has dimension at
+most I/epsilon, since its trace is I. Take V_epsilon to be the span of
+these two spectral subspaces. Its dimension is at most C_epsilon I, and
+(7.29) gives the three bounds in (7.26).
+
+Gap branch. Averaging the original posterior under eta = mu_s gives
+D = E_mu XX^* <= C a^2 Id. Equation (7.29) therefore gives the two
+operator bounds directly. Also ||E w w^*||_op <= I -> 0, so
+V_epsilon = {0} suffices. QED.
+
+
+8. A SYMMETRIC HIERARCHY OF NEARLY FIRST EIGENFUNCTIONS
+
+The prediction curve measures much more than the size of a conditional
+mean: its Bessel inequality is asymptotically saturated at every fixed
+derivative order. That saturation yields symmetric tensor fields T_l
+whose derivatives remain nearly first eigenfunctions, and every tensor
+has negligible energy in the curvature directions of mu_s. Throughout,
+tensor norms are unnormalised Euclidean coefficient norms, including
+for high order.
+
+8.1 The hierarchy.
+
+Fix the critical sequence of Proposition 14, retaining lambda, A, eta,
+w, b_s, M_s, K_s, I. Let U_r = Y_0 + sqrt(r) G for Y_0 ~ eta and G
+standard Gaussian independent of Y_0. Write
+
+ h_r(y) = E[w(Y_0) | U_r = y], h_0 = w,
+ m(r) = ||h_r||_2^2 / I.
+
+Then m: [0, infty) -> [0, 1] is nonincreasing, and by Proposition 17
+
+ lim m(r) = e^{-r} for every fixed r in (0, 1/4). (8.1)
+
+Let g_r(y) = E_G h_r(y + sqrt(r) G) be the doubly smoothed prediction.
+
+Proposition 19 (Symmetric hierarchy). There exist symmetric tensor
+fields T_l, l = 0, 1, 2, ..., of order l+1, with T_0 = w and
+
+ ||T_l||_2^2 = (1 + o(1)) I, (8.2)
+
+ ||D^j T_l||_2^2 = (1 + o(1)) I for every fixed j, (8.3)
+
+ T_{l+1} = D T_l + o_{L^2}(sqrt I), (8.4)
+
+ (A - 1) T_l = o_{L^2}(sqrt I), (8.5)
+
+ <T_l, K_s T_l>_2 = o(I). (8.6)
+
+T_l is symmetric in all l+1 slots, and (8.5)-(8.6) hold also with every
+one-slot flattening. Moreover, for every fixed c < infinity and every
+deterministic orthogonal projection Pi acting on an existing slot,
+with rank Pi <= c I,
+
+ ||Pi T_l||_2^2 = o(I). (8.7)
+
+The o(.) errors are uniform over fixed l and over the finite lists used
+below.
+
+8.2 Bessel inequality for the prediction.
+
+The proof begins with a Gaussian-chaos decomposition of the doubly
+smoothed prediction.
+
+Lemma 20. With the data of Proposition 19,
+
+ <w, g_r>_2 = m(r) I,
+ sum_{j >= 0} (r^j / j!) ||D^j g_r||_2^2 <= m(r) I. (8.8)
+
+In particular every fixed derivative in (8.8) is in the form domain
+of A.
+
+Proof. The first identity is conditional expectation:
+E <w(Y), h_r(U_r)> = E |h_r(U_r)|^2.
+
+Fix y. Gaussian integration by parts identifies the order-j Hermite
+coefficient of h_r(y + sqrt(r) G) as r^{j/2} D^j g_r(y). Its projection
+onto that chaos is
+
+ (r^{j/2} / j!) D^j g_r(y) : H_j(G),
+
+where the vector output slot is not contracted. By the Hermite isometry,
+the squared norm of this projection equals r^j |D^j g_r(y)|^2 / j!.
+Bessel's inequality, integrated in y ~ eta, gives (8.8). QED.
+
+
+8.3 Cross-covariance and spectral smoothing.
+
+Two elementary Hilbert-space facts prevent dimension factors and
+unjustified differentiations in the induction.
+
+Lemma 21. (a) Let F be a square-integrable random variable with values
+in a finite-dimensional Hilbert space H, and let Z be a centered
+Euclidean random vector with Cov Z <= c Id. Then
+
+ |E(F ⊗ Z)|_HS^2 <= c E|F|^2. (8.9)
+
+(b) Let (A^{(k)}) be a sequence of nonnegative Dirichlet operators on
+fields with kernel the constants, whose spectral gap is at least
+lambda -> 1. If fields S lie in the form domain of A and satisfy
+
+ |E S|^2 = o(I), ||S||_2^2 = (1 + o(1)) I,
+ ||DS||_2^2 <= (1 + o(1)) I, (8.10)
+
+then S~ = e^{1 - A} S satisfies
+
+ ||S~ - S||_2 + ||D(S~ - S)||_2 = o(sqrt I),
+
+ || (A - 1) S~ ||_2 = o(sqrt I). (8.11)
+
+Proof. (a) The map u -> u.Z from H to scalar L^2(Prob) has operator
+norm at most sqrt(c). Its adjoint sends a scalar coefficient to
+E[Z u], and summing across coefficients gives
+
+ |E(F ⊗ Z)|_HS^2 = sum_k |E(F_k Z)|^2
+ <= c sum_k E|F_k|^2 = c E|F|^2.
+
+(b) Let E_S(mu) denote the spectral resolution of A applied to S. The
+identities
+
+ ||S||_2^2 = int_0^infty d||E_S(mu) S||_2^2,
+
+ ||DS||_2^2 = int_0^infty mu d||E_S(mu) S||_2^2,
+
+together with (8.10), give
+
+ int_{mu >= lambda} (mu - lambda) d||E_S S||_2^2
+ <= ||DS||_2^2 - lambda ||S||_2^2 + lambda |E S|^2
+ = o(I). (8.12)
+
+Spectral mass at mu = 0 carries weight |E S|^2 = o(I), and (8.12)
+bounds the mass of the remaining contribution at mu >= 1 + delta by
+o(I)/delta for every fixed delta > 0. Splitting the multiplier
+e^{1-mu} - 1 into the ranges {mu = 0}, [lambda, 1 + delta],
+(1 + delta, K], (K, infty) for large K: the first and fourth give
+o(I), the second is O(delta) by |e^{1-mu} - 1| <= C |mu - 1| near
+mu = 1, and the third is at most sqrt(o(I)/delta). Sending delta -> 0
+after the sequence limit, with K free, proves ||S~ - S||_2 = o(sqrt I).
+
+For the derivative norm,
+
+ ||D(S~ - S)||_2^2 = int mu |e^{1-mu} - 1|^2 d||E_S S||_2^2.
+
+On [lambda, 1 + delta] the factor mu |e^{1-mu} - 1|^2 is at most
+C delta^2. On (1 + delta, K] it is at most K, and the mass there is
+bounded by o(I)/delta. On (K, infty) it is at most mu, and
+int_{mu > K} mu d||E_S S||_2^2 = o(I) by (8.12) with lambda -> 1.
+This proves ||D(S~ - S)||_2 = o(sqrt I).
+
+The spectral multiplier of (A - 1) S~ is (mu - 1) e^{1 - mu}, vanishing
+at mu = 1 and bounded on [0, infty). The same piecewise splitting
+gives
+
+ int |(mu - 1) e^{1-mu}|^2 d||E_S S||_2^2
+ <= C delta^2 ||S||_2^2 + o(I)/delta = O(delta^2) I + o(I)/delta.
+
+Sending delta -> 0 after the sequence limit gives (8.11). QED.
+
+
+8.4 Hierarchy construction.
+
+The construction starts at T_0 = w and produces T_{l+1} from T_l. Two
+ingredients combine: Bessel tightness (Lemma 20) supplies an
+L^2-close symmetrized candidate, and the smoothing estimate of Lemma 21
+converts that candidate into a nearly first eigenfunction. The
+curvature estimate (8.6) requires separate machinery, deferred to
+Section 8.7.
+
+Lemma 22 (Construction induction). Suppose T_0,...,T_l satisfy
+(8.2)-(8.5) and (8.7) at orders 0,...,l. Then there exists a symmetric
+field T_{l+1} of order l+2 satisfying (8.2)-(8.5) and (8.7) at order
+l+1, with E T_{l+1} = 0. The starting vector obeys |E w|^2 <= o(I)
+by (7.1).
+
+Proof. Set V = DT_l and L = l+1.
+
+Step 1. Self-adjointness and (8.4) give
+
+ ||V||_2^2 = <T_l, A T_l> = ||T_l||_2^2 + o(I) = (1 + o(1)) I.
+
+For any admissible projection Pi on the derivative slot,
+
+ ||Pi_1 V||_2^2 = <Pi_1 T_l, A Pi_1 T_l>
+
+ <= ||Pi_1 T_l||_2^2 + ||Pi_1 T_l||_2 ||(A-1) T_l||_2 = o(I),
+
+uniformly in Pi, by (8.7) and (8.5).
+
+Step 2. Fix 0 < r < 1/4. Put V_j = T_j for 0 <= j <= l, and V_L = V.
+Successive form integration by parts gives
+
+ |I^{-1} <D^j g_r, V_j>_2 - e^{-r}| = o(1), 0 <= j <= L. (8.22)
+
+At j = 0 this is (8.8) with m(r) -> e^{-r}. At 1 <= j <= L,
+
+ <D^j g_r, DT_{j-1}>_2 = <D^{j-1} g_r, A T_{j-1}>_2
+ = <D^{j-1} g_r, T_{j-1}>_2 + o(I),
+
+by ||(A-1)T_{j-1}||_2 = o(sqrt I) and (8.8). At j = L the same
+identity applies with T_{l} = T_{L-1} and DT_l = V.
+
+Step 3. Multiply (8.8) by e^{2r}/I and use step 2 with
+||V_j||_2^2 = (1+o)I. Put
+
+ kappa_r(L) = e^r - sum_{j=0}^L r^j/j! > 0. (8.23)
+
+The exponential-series tail absorbs everything beyond the j <= L
+contributions matched in step 2:
+
+ limsup ||e^r D^L g_r - V||_2^2 / I
+ <= (L! / r^L) kappa_r(L) = O_L(r), (8.24)
+
+ limsup e^{2r} ||D^{L+1} g_r||_2^2 / I
+ <= ((L+1)! / r^{L+1}) kappa_r(L) = 1 + O_L(r). (8.25)
+
+Step 4. Choose r -> 0 slowly along the sequence so that
+S_0 := e^r D^L g_r satisfies
+
+ ||S_0 - V||_2 = o(sqrt I), ||DS_0||_2^2 <= (1 + o(1)) I. (8.26)
+
+S_0 is symmetric in its last L slots, V is symmetric in the first L
+since V = DT_l and T_l is symmetric. For L >= 2 the adjacent
+transpositions within each block generate the full symmetric group
+S_{L+1}; for L = 1, V = Dw is already symmetric because w = Dh. Let
+S be the full symmetrization of S_0. Then
+
+ ||S - V||_2 = o(sqrt I), ||S||_2^2 = (1 + o(1)) I,
+
+ ||DS||_2^2 <= (1 + o(1)) I. (8.27)
+
+Averaging coefficient permutations is an orthogonal contraction
+commuting with differentiation, and step 1 transfers to every
+existing slot of S.
+
+Step 5. Write C_y = Cov_eta(T_l, Y), C_b = Cov_eta(T_l, b_s),
+C_w = Cov_eta(T_l, w). Testing A T_l = T_l + (A-1)T_l against
+Y - E Y and using self-adjointness with the coordinate-generator
+A_eta Y = (Y - b_s)/s - w in the form domain gives
+
+ E V = C_y + E_l, |E_l|_HS <= lambda^{-1/2} ||(A-1)T_l||_2, (8.28)
+
+ E V = s^{-1} (C_y - C_b) - C_w. (8.29)
+
+The error bound uses cross-covariance (8.9) with Cov_eta Y <=
+lambda^{-1} Id (Poincare). The bounds E_eta b_s b_s^* <= C a^2 Id and
+|E_eta w|^2 = o(I) of Lemma 12 combine with (8.28)-(8.29):
+
+ |E V|_HS = o(sqrt I), |C_y|_HS = o(sqrt I). (8.30)
+
+Pass to the centered field S_c = S - E S. Then
+
+ ||S_c - V||_2 = o(sqrt I), |E S_c| = 0,
+
+ ||DS_c||_2 = ||DS||_2 <= (1 + o(1)) I,
+
+ ||S_c||_2^2 = (1 + o(1)) I. (8.31)
+
+Step 6. Apply Lemma 21(b) to S_c; the spectral gap of A_eta is at
+least lambda -> 1. The field T_{l+1} := e^{1-A} S_c satisfies
+
+ ||T_{l+1} - S_c||_2 + ||D(T_{l+1} - S_c)||_2 = o(sqrt I),
+
+ ||(A - 1) T_{l+1}||_2 = o(sqrt I). (8.32)
+
+Combining (8.31) and (8.32):
+
+ ||T_{l+1} - V||_2 = o(sqrt I),
+
+ ||T_{l+1}||_2^2 = (1 + o(1)) I,
+
+ ||D T_{l+1}||_2^2 <= (1 + o(1)) I,
+
+ ||(A - 1) T_{l+1}||_2 = o(sqrt I).
+
+Since T_{l+1} - V = o(sqrt I) and V = DT_l, (8.4) holds at order
+l+1. (8.2) and (8.3) at every fixed derivative order hold by
+construction. Symmetry passes into T_{l+1} because e^{1-A} commutes
+with coefficient permutations. (8.7) transfers from S to T_{l+1} by
+step 4 and (8.32).
+
+Step 7. The field T_{l+1} is centered: E S_c = 0 and semigroups
+preserve the mean (1 is in the kernel of A), so
+E T_{l+1} = e E S_c = 0. This completes the induction for (8.2)-(8.5)
+and (8.7). The curvature assertion (8.6) is deferred to Section 8.7.
+QED.
+
+
+8.5 Vanishing operator tails.
+
+The curvature estimate of Section 8.7 tests the operator identity
+D(A F) = A(D F) + (K_s - Q) D F against a field P whose operator norm
+is small, so that Q P is genuinely negligible in L^2. That is what the
+next lemma supplies.
+
+Lemma 23 (Vanishing operator tails). In the strict branch put Q = Dw;
+in the gap branch Q = 0. For every fixed delta > 0 and every l >= 1,
+each one-slot flattening F of T_l, and also F = Q in the strict branch,
+obeys
+
+ E tr(F F^* 1_{F F^* > delta^2 I}) = o(I). (8.33)
+
+Equivalently, clipping every singular value at any fixed positive
+cutoff changes the field by o(sqrt I) in L^2.
+
+Proof. Base case. Take F = Q or F = T_1. Lemma 22 at L = 1 gives
+
+ ||e^r D g_r - Q||_2^2 <= (r/2 + O(r^2)) I, (8.34)
+
+so Q = Dw is L^2-close to a Gaussian-averaged derivative of a smooth
+field. A diagonal-smoothing approximation replaces T_1 by the same
+field up to o(sqrt I). The conditional covariance bound of Lemma 18(iii)
+puts the singular values of the local posterior covariances of w below
+any fixed threshold outside a subspace of vanishing trace;
+Gaussian-convolution regularity propagates this into pointwise
+operator-norm bounds on D g_r over compact y-sets, and the
+exponential-square moment of eta makes the outside mass vanish.
+Splitting ||Q 1_{||Q||_op > delta}||_HS^2 into {|y| <= R} and {|y| > R}:
+
+ ||Q 1_{||Q||_op > delta}||_HS^2
+ <= 2 ||D g_r 1_{||D g_r||_op > delta/2}||_2^2 + o(I) = o(I), (8.35)
+
+after sending r -> 0 slowly along the sequence, then R -> infty, then
+the sequence limit. The bound transfers to T_1 through
+||T_1 - Q||_2 = o(sqrt I).
+
+Propagation to l >= 2. Fix l >= 1 and t > 0. Apply the Hermite
+expansion to the stationary diffusion at time t, with starting level 1.
+Conditional on Y_0, the Gaussian-Hermite chaoses of P_t T_1 are
+orthogonal even after acting on the first tensor slot by an arbitrary
+deterministic orthogonal projection P(Y_0) (a function of Y_0 alone).
+In the expansion of P_t T_1 against the diffusion, the chaos containing
+T_l(Y_0) carries squared coefficient
+
+ c_{l,t} = e^{-2t} (2t)^{l-1} / (l-1)! > 0. (8.36)
+
+The eigenvalue exp(-t) of A contributes the exp(-2t) factor, the
+l-1 derivatives from T_l = (D^{l-1} T_1) contribute (2t)^{l-1},
+and projection is norm one on each chaos. Take the expansion to any
+fixed order m >= l-1, take the sequence limit at fixed m, then let
+m -> infty in the uniform remainder bound. The result is
+
+ liminf((E|P(Y_0)_1 T_1(Y_t)|^2
+ - c_{l,t} E|P(Y_0)_1 T_l(Y_0)|^2) / I) >= 0. (8.37)
+
+The remainder estimates are uniform in P, whose operator norm is at
+most one.
+
+Conditional contraction of P_t against P(Y_0) gives
+E|P(Y_0)_1 P_t T_1(Y_0)|^2 <= E|P(Y_0)_1 T_1(Y_t)|^2.
+Combining with (8.37),
+
+ c_{l,t} E|P(Y_0)_1 T_l(Y_0)|^2
+ <= E|P(Y_0)_1 T_1(Y_t)|^2 + o(I). (8.38)
+
+Choose P(Y_0) as the spectral projection of the first-slot flattening
+of T_l(Y_0) above threshold epsilon. Then
+
+ E rank P <= epsilon^{-2} ||T_l||_2^2
+ = (1 + o(1)) epsilon^{-2} I. (8.39)
+
+Let S_delta be the singular-value clipping of T_1 at delta. Pointwise
+
+ |P(Y_0) S_delta(Y_t)|_HS^2 <= delta^2 rank P(Y_0). (8.40)
+
+Stationarity and the base case give
+||S_delta(Y_t) - T_1(Y_t)||_2 = o(sqrt I) uniformly in t. Hence
+
+ E|P(Y_0)_1 T_1(Y_t)|^2
+ <= delta^2 (1+o) epsilon^{-2} I + o(I). (8.41)
+
+Combine (8.38) and (8.41):
+
+ c_{l,t} E|P(Y_0)_1 T_l(Y_0)|^2
+ <= delta^2 epsilon^{-2} I + o(I). (8.42)
+
+Send delta -> 0. The left side equals
+E tr(T_l T_l^* 1_{T_l T_l^* > epsilon^2 I}) on the first-slot
+flattening. This proves (8.33) at threshold epsilon for the
+first-slot flattening of T_l. Tensor symmetry supplies every other
+one-slot flattening, and exchanging the roles of P(Y_0) and T_1(Y_t)
+preserves the bound, so no independence between P(Y_0) and T_1(Y_t)
+was needed. QED.
+
+
+8.6 Weighted divergence and the differentiated operator.
+
+The next identity is the only place where differentiating the drift is
+legitimate without uniform control of third derivatives of the weighted
+density. It pairs the weighted divergence of a matrix field against a
+function in the operator domain, using only the second-order potential.
+
+Lemma 24. Let eta = e^{-V} be a smooth strictly positive probability
+density with A_eta = -Delta - DV.D and H = D^2 V. Let P = (P_{i alpha})
+have one vector slot i and a finite-dimensional coefficient index alpha.
+Suppose every coefficient lies in Dom(A_eta) and H P lies in L^2(eta).
+Then the weighted adjoint divergence
+
+ (D^* P)_alpha = - sum_i d_i P_{i alpha}
+ - sum_i (d_i V) P_{i alpha} (8.47)
+
+belongs to L^2(eta) and satisfies
+
+ ||D^* P||_2^2 <= ||DP||_2^2 + <P, H P>_2. (8.48)
+
+For every coefficient field F in Dom(A_eta),
+
+ <AF, D^* P>_2 = <DF, AP>_2 + <DF, H P>_2. (8.49)
+
+In particular (8.49) requires H P in L^2 and does not require H DF
+in L^2.
+
+Proof. For compactly supported smooth P, two first derivatives commute
+in weighted integration by parts, so
+
+ ||D^* P||_2^2
+ = sum_{i, j, alpha} <d_j P_{i alpha}, d_i P_{j alpha}>_2
+ + <P, H P>_2. (8.50)
+
+Cauchy-Schwarz applied to the arrays (i, j, alpha) and (j, i, alpha)
+bounds the double sum by ||DP||_2^2 without any factor from the numbers
+of coordinate or coefficient indices. This proves (8.48) for smooth
+compactly supported P.
+
+For the stated P, membership in Dom(A_eta) puts P in H^2_loc by local
+elliptic regularity, and D P in L^2(eta). Choose smooth compactly
+supported cutoffs 0 <= chi_R <= 1 with chi_R -> 1 monotonically and
+||D chi_R||_infty -> 0. Equation (8.50) applies to chi_R P by local
+Sobolev approximation, and its right side is controlled by
+
+ ||D(chi_R P) - DP||_2 -> 0,
+ <chi_R P, H chi_R P>_2 -> <P, HP>_2.
+
+The second convergence uses |<P, H P>| in L^1, a consequence of P, HP in
+L^2. Weak compactness in L^2 and distributional convergence of
+D^*(chi_R P) identify the limit as D^* P. Lower semicontinuity yields
+(8.48). Only first derivatives of the cutoffs enter the estimate.
+
+For compactly supported smooth F, differentiating the scalar operator
+
+ D(A F) = A(D F) + H D F. (8.51)
+
+Pairing this identity with P and using symmetry of H gives (8.49) for
+such F. For general F in Dom(A_eta), take compactly supported smooth
+F_j with F_j -> F and A F_j -> AF in L^2, as Lemma 3 supplies. The same
+convergence holds in the form norm, since
+
+ ||D(F_j - F)||_2^2 = <F_j - F, A(F_j - F)>_2 -> 0.
+
+All three factors D^* P, AP, HP in (8.49) lie in L^2, so every term
+converges. This proves the global identity. QED.
+
+
+8.7 The curvature estimate: completion of Proposition 19.
+
+Lemma 22 supplies the hierarchy fields and every assertion of
+Proposition 19 except (8.6). What remains is to show
+<T_l, K_s T_l>_2 = o(I) at each level, using Lemma 24 and the operator
+tails of Lemma 23.
+
+Step 1: The test field P. Fix l >= 0 and write F = T_l. View DF as a
+matrix with its derivative slot as row index and all old tensor slots
+as column index.
+
+Strict branch. Apply Lemma 23 with a fixed delta > 0 to the
+corresponding flattening of T_{l+1}. This gives a clipped field C with
+
+ ||C - T_{l+1}||_2 = o(sqrt I),
+ ||C(y)||_op <= delta. (8.66)
+
+Set P = e^{1-A} C = e P_1 C. Markov averaging and convexity of the
+operator-norm ball give ||P(y)||_op,infty <= e delta. The spectral
+multipliers (x-1) e^{1-x}, e^{1-x}, x^{1/2} e^{1-x} are bounded on
+[0, infty); the function e^{1-x} - 1 vanishes at x = 1 and satisfies
+|e^{1-x} - 1| <= C |x - 1| there. Combining these bounds with (8.5)
+and the derivative-closeness T_{l+1} = DF + o(sqrt I):
+
+ ||P - DF||_2 + ||(A-1) P||_2 = o(sqrt I), (8.67)
+
+ ||DP||_2 = O(sqrt I), (8.68)
+
+ ||QP||_2 <= ||P||_op,infty ||Q||_2 <= e delta (1+o(1)) sqrt I. (8.69)
+
+The clipping is used only to construct P, so it need not preserve
+tensor symmetry.
+
+Gap branch. Take P = T_{l+1} directly. Then Q = 0 and (8.67)-(8.69)
+follow from (8.3), (8.4), and (8.5).
+
+In both branches the field P obeys (8.67)-(8.69), P lies in Dom(A), and
+H P lies in L^2(eta) for H = K_s - Q.
+
+Step 2: The curvature identity. Lemma 22 gives the drift decomposition
+
+ H = D^2(-log eta) = K_s - Q, (8.70)
+
+with Q = 0 in the gap branch. Both K_s P and Q P lie in L^2(eta), since
+||K_s P||_2 <= s^{-1}||P||_2 and ||QP||_2 = o(sqrt I). Lemma 24 applies
+to F = T_l and P:
+
+ <A F, D^* P>_2 = <DF, AP>_2 + <DF, (K_s - Q)P>_2. (8.71)
+
+Insert A F = F + (A-1)F and A P = P + (A-1)P. The divergence identity
+for weighted integration by parts,
+
+ <F, D^* P>_2 = <DF, P>_2, (8.72)
+
+cancels the two zeroth-order terms. Therefore
+
+ <DF, K_s P>_2
+ = <(A-1)F, D^* P>_2 - <DF, (A-1)P>_2
+ + <DF, QP>_2. (8.73)
+
+Bounds:
+- ||D^* P||_2^2 <= ||DP||_2^2 + <P, H P>_2 <= O(I) + s^{-1}||P||_2^2
+ + ||P||_2 ||QP||_2 = O(I). So |<(A-1)F, D^* P>| <= ||(A-1)F||_2
+ ||D^* P||_2 = o(sqrt I) O(sqrt I) = o(I).
+- |<DF, (A-1)P>| <= ||DF||_2 ||(A-1)P||_2 = O(sqrt I) o(sqrt I) = o(I).
+- |<DF, QP>| <= ||DF||_2 ||QP||_2 = O(sqrt I) e delta O(sqrt I) = o(I)
+ as delta -> 0.
+
+Together,
+
+ <DF, K_s P>_2 = o(I). (8.74)
+
+Replace P by DF. Since P - DF = o(sqrt I) and ||K_s||_op,infty <= s^{-1},
+
+ |<DF, K_s (P - DF)>| <= ||DF||_2 s^{-1} ||P - DF||_2 = o(I). (8.75)
+
+So <DF, K_s DF>_2 = o(I). Replacing DF by T_{l+1} costs o(I) by the
+derivative estimate: <T_{l+1}, K_s T_{l+1}>_2 = o(I).
+
+Step 3: Every slot. The identity <T_{l+1}, K_s T_{l+1}>_2 = o(I) now
+holds on the derivative slot. By tensor symmetry (8.6) holds on every
+slot. Level zero is the initial vector: E_eta <w, K_s w> <= lambda
+|E_eta w|_2^2 = o(I) by Lemma 12 and |E w|^2 = o(I) from (7.1). Every
+assertion of Proposition 19 is in place. QED.
+
+
+9. A MATRIX FIELD AND A FROZEN DIFFUSION LAW
+
+The hierarchy has fixed finite levels, but no single well-defined matrix.
+Packing a band of levels produces a positive matrix field with sharp
+energy, from which a Gaussian matrix fluctuation law emerges. The
+diagonal choice made at the start of this section fixes every finite
+auxiliary constant used below.
+
+9.1 The band reduction.
+
+Diagonal choice. For each finite integer K, impose the conclusions of
+Proposition 19 through level 3K, the fixed-projection inequalities for
+rank coefficients at most K, and the Hermite expansion bounds for every
+starting level 0 <= l < 2K and every truncation order 0 <= m <= K, on
+time intervals 0 <= t <= K. Only levels through 3K enter these
+expansions, including their remainders. For every finite list, the
+hierarchy defects and the excesses over the normalized Hermite
+remainder bounds can be made arbitrarily small. Choose a positive
+integer k that increases along the sequence slowly enough that all these
+defects vanish uniformly. Further parameters below are chosen by the
+same procedure. Normalized endpoint energies remain bounded when a band
+of k levels is shifted by a fixed number of levels.
+
+Regard
+
+ Psi = direct_sum_{k <= l < 2k} T_l / sqrt(k I) (9.1)
+
+as a matrix by separating its first tensor index from the remaining
+indices and the direct-sum label. Define
+
+ rho = Psi Psi^*, R = rho^{1/2}. (9.2)
+
+The matrix field R is positive semidefinite; it is distinct from the
+scalar log-Sobolev constant R(eta), and context distinguishes them. The
+normalization reads
+
+ E tr R^2 = ||Psi||_2^2 -> 1,
+ ||D Psi||_2^2 -> 1. (9.3)
+
+9.2 Curvature and centering.
+
+Two consequences of the hierarchy pass to the matrix field.
+
+First, the curvature conclusion (8.6), at every level 0 <= l < 2k,
+
+ <T_l, K_s T_l>_2 = o(I).
+
+Summing over the band with weights 1/(k I) identifies the left side
+with tr(K_s rho) up to a uniform o(1). Since K_s = (Id - M_s)/s,
+
+ E tr((1 - M_s) R^2) = o(1). (9.4)
+
+Second, an independence statement. Fix delta > 0 and let P_delta be the
+spectral projection of E M_s above delta. Lemma 18(ii) supplies
+rank P_delta = O_delta(I), and the deterministic-projection bound (8.7)
+gives
+
+ tr(P_delta E R^2) = o(1). (9.5)
+
+Since 0 <= M_s <= Id, this bounds the left side of
+tr((E M_s)(E R^2)) by o(1) + delta E tr R^2 = o(1) + delta. Send
+delta -> 0 after the sequence limit:
+
+ tr((E M_s)(E R^2)) = o(1). (9.6)
+
+In the gap branch of Lemma 18, ||E M_s||_op = o(1) makes (9.6) immediate.
+
+9.3 Energy lower bound for R.
+
+Let R' denote R evaluated at an independent copy of Y_0. Insert
+(1 - M_s) + M_s between R and R' in ||E R||_HS^2 = E tr(R R').
+
+Hilbert-Schmidt Cauchy-Schwarz bounds the term containing 1 - M_s by
+
+ (E tr((1 - M_s) R^2))^{1/2} (E tr R^2)^{1/2} = o(1), (9.7)
+
+using (9.4) and (9.3).
+
+The remaining term is bounded by
+
+ (E tr(M_s R^2))^{1/2} (tr((E M_s)(E R^2)))^{1/2} = o(1), (9.8)
+
+because E tr(M_s R^2) <= E tr(R^2) is at most 1 + o(1) and
+tr((E M_s)(E R^2)) = o(1) by (9.6).
+
+Together,
+
+ ||E R||_HS = o(1). (9.9)
+
+Poincare for eta and (9.3) give
+
+ Var_eta(R) = E tr R^2 - ||E R||_HS^2 = 1 + o(1),
+
+ Var_eta(R) <= R_eta^P ||D R||_2^2 <= (1 + o(1)) ||D R||_2^2,
+
+so
+
+ ||D R||_2^2 >= 1 + o(1). (9.10)
+
+The upper bound is the elementary inequality
+||D R||_2^2 <= ||D Psi||_2^2 = 1 + o(1), following from the chain rule
+D rho = D R · R + R · D R applied to the positive square root and the
+normalized energy of Psi in (9.3). Consequently
+
+ ||D R||_2^2 = 1 + o(1). (9.11)
+
+
+9.4 Averaging over symmetric tensor slots.
+
+The next estimate is the reason for using tensors with many symmetric
+slots. It compares averaging an action over those slots with the
+geometric, rather than arithmetic, weighting by the marginal matrix.
+
+Lemma 25. Let psi be a finite direct sum of symmetric tensors, each
+with at least k >= 2 slots. Let rho be its one-slot reduction and
+R = rho^{1/2}. For a self-adjoint matrix H let H_av act on each
+component by averaging the action of H over its slots. Then
+
+ ||H_av psi||_2^2 <= tr(R H R H) + (2/k) tr(rho H^2). (9.12)
+
+Proof. For a component psi_d with d slots put
+
+ a_d = <psi_d, H_(1)^2 psi_d>_2,
+ c_d = <psi_d, H_(1) H_(2) psi_d>_2. (9.13)
+
+Symmetry and Cauchy-Schwarz give |c_d| <= a_d, and direct expansion
+gives
+
+ ||H_av psi||_2^2 = sum_d (a_d/d + (1 - 1/d) c_d)
+ <= sum_d c_d + (2/k) sum_d a_d. (9.14)
+
+It remains to prove |sum_d c_d| <= tr(R H R H). This correlation bound
+is a case of Lieb's Wigner-Yanase trace-concavity theorem. A direct
+proof by weighted interpolation runs as follows.
+
+Define the linear map C by
+
+ tr(X C(Y)) = sum_d <psi_d, X_(1) Y_(2) psi_d>_2. (9.15)
+
+Flattening all remaining indices exhibits C(Y) as a sum of terms
+A_alpha Y^T A_alpha^*. Thus C is positive and C(1) = C^*(1) = rho.
+On the support of rho, on which inverse powers are defined, put
+
+ U(Y) = rho^{-1/2} C(Y) rho^{-1/2}. (9.16)
+
+The map U is unital and operator-norm contractive: its preceding
+representation is an isometric compression of copies of Y^T. It also
+satisfies tr(rho U(Y)) = tr(rho Y). For 0 <= Re z <= 1 define
+
+ U_z(Y) = rho^{z/2} U(rho^{-z/2} Y rho^{-z/2}) rho^{z/2}. (9.17)
+
+On Re z = 0, multiplication on either side by imaginary powers of
+rho is isometric, so U_z is operator-norm contractive. At z = 1,
+
+ U_1(Y) = C(rho^{-1/2} Y rho^{-1/2}), U_1^*(1) = 1. (9.18)
+
+Its adjoint is a unital positive map with the same compression and
+transpose representation, hence is operator-norm contractive. Duality
+makes U_1 trace-norm contractive; imaginary powers give the same at
+Re z = 1.
+
+The Wigner-Yanase / Wigner interpolation theorem, applied on the strip
+0 <= Re z <= 1 to the family U_z evaluated at Y = rho^{1/2} H rho^{1/2},
+bounds |tr(U(H) rho^{1/2} H rho^{1/2})| by the endpoint norms. Since
+the boundary z = 0 is operator-norm contractive and z = 1 is
+trace-norm contractive, the strip argument yields
+
+ |sum_d c_d| = |tr(C(H) H)|
+ = |tr(U(H) rho^{1/2} H rho^{1/2})|
+ <= tr(R H R H). (9.19)
+
+Substitute this in (9.14). QED.
+
+
+9.5 Saturation via the Sylvester equation.
+
+Lemma 26 (Saturation of the square-root energy). There are measurable
+symmetric matrix fields H_i, 1 <= i <= n, solving the Sylvester
+equation
+
+ D_i R = (H_i R + R H_i) / 2 (9.20)
+
+with zero kernel-kernel block, such that
+
+ ||D R||_2^2 -> 1, sum_i E ||[H_i, R]||_HS^2 = o(1),
+
+ sum_i E tr(R^2 H_i^2) -> 1. (9.21)
+
+Moreover
+
+ sum_i (||D_i R - H_i R||_2^2
+ + ||D_i Psi - (H_i)_(1) Psi||_2^2) = o(1). (9.22)
+
+The first slot can be replaced by any existing slot in each component.
+
+Proof. In an eigenbasis of R with eigenvalues r_a, define
+
+ (H_i)_{ab} = 2 (D_i R)_{ab} / (r_a + r_b) (9.23)
+
+when r_a + r_b > 0, and zero otherwise. The kernel-kernel block of
+D_i R vanishes almost everywhere. Indeed, for constant vectors v, w in
+the kernel of R, the quadratic form v^* R w vanishes identically in
+position; differentiating gives v^* (D_i R) w = 0. Since R is self-
+adjoint and the Sylvester formula is symmetric in (a, b), the matrix H_i
+is symmetric. The formula is measurable.
+
+Energy identities. Set
+
+ e = ||D R||_2^2, a = sum_i E tr(R^2 H_i^2),
+ c = sum_i E ||[H_i, R]||_HS^2, d = ||D Psi||_2^2. (9.24)
+
+From the Sylvester equation, |(H_i R)_{ab}|^2
+= 4 |(D_i R)_{ab}|^2 r_b^2/(r_a + r_b)^2 <= 4 |(D_i R)_{ab}|^2,
+because 0 <= r_b <= r_a + r_b. Summing over (a, b) and i,
+
+ sum_i ||H_i R||_HS^2 <= 4 ||D R||_2^2. (9.25)
+
+Writing A = H_i R and B = R H_i, the parallel and perpendicualr parts
+satisfy ||A + B||^2 - ||A - B||^2 = 4 <A, B>_HS = 4 tr(R H_i R H_i),
+and ||A||_HS^2 = ||B||_HS^2 = a_i by symmetry of H_i. Therefore
+
+ e = a - c/4,
+ sum_i E tr(R H_i R H_i) = e - c/4. (9.26)
+
+Differentiating R^2 = Psi Psi^* in the weak form sense and using
+symmetry of D_i Psi on its slots:
+
+ e = sum_i E <D_i Psi, (H_i)_(1) Psi>_2
+ = sum_i E <D_i Psi, (H_i)_av Psi>_2. (9.27)
+
+Lemma 25 gives ||(H_i)_av Psi||_2^2 <= tr(R H_i R H_i) + (2/k) tr(rho H_i^2).
+Summing over i and applying Cauchy-Schwarz to (9.27):
+
+ e^2 <= d (e - c/4 + 2a/k). (9.28)
+
+Since a <= 4e by (9.25) and d -> 1 by (9.3), equation (9.28) forces
+limsup e <= 1. Combined with liminf e >= 1 from (9.10), this gives
+e -> 1; then c -> 0 and a -> 1.
+
+Final identities. From the Sylvester equation,
+
+ ||D_i R - H_i R||_2^2 = (1/4) ||[H_i, R]||_HS^2, (9.29)
+
+ ||(H_i)_(1) Psi||_2^2 = a_i = tr(R^2 H_i^2). (9.30)
+
+Therefore
+
+ sum_i ||D_i R - H_i R||_2^2 = c/4 -> 0,
+
+ sum_i ||D_i Psi - (H_i)_(1) Psi||_2^2 = d + a - 2e -> 0. (9.31)
+
+Tensor symmetry of Psi in each component lets the first slot stand
+for any existing slot. QED.
+
+
+9.6 Bounded columns and balance.
+
+Lemma 27 (Balance and bounded column). The matrices H_i of Lemma 26
+satisfy
+
+ E ||sum_i H_i rho H_i - rho||_1 = o(1),
+
+ E ||R (sum_i H_i^2) R - rho||_1 = o(1). (9.45)
+
+They can be replaced by symmetric matrix fields, still denoted H_i, so
+that (9.20), the commutator conclusion of (9.21), and (9.45) hold, and
+
+ S := sum_i H_i^2 <= 4 Id,
+ E ||(S - 1) R||_HS^2 = o(1). (9.46)
+
+The total change satisfies
+sum_i E ||(H_i^new - H_i^old) R||_HS^2 = o(1). (9.47)
+
+Proof.
+
+Perturbation bound. From the identity X X^* - Y Y^* = (X - Y) X^*
++ Y (X - Y)^*,
+
+ ||X X^* - Y Y^*||_1
+ <= ( ||X||_HS + ||Y||_HS ) ||X - Y||_HS. (9.48)
+
+Balance identities. By (9.22), the sum sum_i E ||D_i Psi - (H_i)_(1) Psi||_2^2
+is o(1), so the one-slot reduction of D Psi with the derivative index
+included among the column indices is close in expected trace norm to
+sum_i H_i rho H_i. The hierarchy coupling identifies this reduction
+with
+
+ (1 / (k I)) sum_{k <= l < 2k} T_{l+1} T_{l+1}^*. (9.49)
+
+The trace-norm error from the coupling is o(1) by (9.48) and (9.3).
+Shifting the band by a fixed number of levels changes its endpoints by
+O(1/k), so the first line of (9.45) holds.
+
+Comparing the arrays (H_i R)_i and (R H_i)_i proves the second line of
+(9.45) by the same inequality, using the summed commutator estimate in
+(9.21).
+
+Bounded columns. Write H v = (H_i v)_i, S = H^* H, J = (1 + S)^{-1},
+and
+
+ K = H R - R^{direct sum n} H. (9.50)
+
+By (9.22), E ||K||_HS^2 = o(1). Algebra gives
+
+ [S, R] = H^* K - K^* H,
+ [R, J] = J [S, R] J. (9.51)
+
+The bounds ||H J H^*||_op <= 1, ||H J||_op <= 1/2, ||J||_op <= 1 yield
+
+ ||H [R, J]||_HS <= 2 ||K||_HS. (9.52)
+
+For every fixed function f(x) = a + b/(1 + x) with bounded real a, b,
+
+ E ||H [f(S), R]||_HS^2 = o(1). (9.53)
+
+The squared norms of H f(S) R and H R f(S) differ by o(1): the latter
+has bounded mean-square norm, and (9.53) controls their difference.
+Testing the second balance equation in (9.45) against the bounded matrix
+f(S)^2 consequently gives
+
+ E tr(R^2 (S - 1) f(S)^2) = o(1). (9.54)
+
+Inserting f = 1, f = 1 + J, f = 1 - J and the identity
+(S - 1)^2/(1 + S) = (S - 1) - 2 (S - 1) J yields
+
+ E tr(R^2 ((S - 1)^2 / (1 + S))) = o(1). (9.55)
+
+All expressions are integrable: their absolute values are bounded by a
+constant multiple of tr(R^2 (1 + S)).
+
+Tail truncation. Let Q = 1_{(4, infty)}(S), P = 1 - Q. On this spectral
+region (S - 1)^2/(1 + S) dominates both a positive constant and a
+positive multiple of S. Thus
+
+ E tr(R^2 Q) + E tr(R^2 S Q) = o(1). (9.56)
+
+The input truncation costs
+
+ sum_i E ||H_i Q R||_HS^2 = E tr(R^2 S Q) = o(1).
+
+The output truncation costs
+
+ sum_i E ||Q H_i R||_HS^2
+ = E tr(Q sum_i H_i rho H_i)
+ <= E tr(Q rho) + E ||sum_i H_i rho H_i - rho||_1 = o(1).
+
+So replacing H_i by P H_i P has total right-R squared cost o(1). The
+new column obeys sum_i (P H_i P)^2 <= P S P <= 4 P.
+
+Perturbation accounting. Suppose symmetric \tilde H_i have
+
+ Delta := sum_i E ||(\tilde H_i - H_i) R||_HS^2 = o(1). (9.57)
+
+Then symmetry gives the same bound with R on the left. Each derivative
+error in (9.22) changes in norm by at most sqrt(Delta), and the summed
+commutator norm changes by at most 2 sqrt(Delta). The two balance
+errors change in expected trace norm by at most
+
+ ( (sum_i E ||H_i R||_HS^2)^{1/2}
+ + (sum_i E ||\tilde H_i R||_HS^2)^{1/2} ) sqrt(Delta),
+
+by (9.48), which is o(1). The new summed energy is bounded by the old
+plus the perturbation in norm. These statements apply to the
+compression just made. QED.
+
+
+9.7 Spectral blocks and a bounded input map.
+
+Ordering the operator column by the spectral range of R decouples its
+action in the later Gaussian test. The matrix field R varies with
+position, so its eigenbasis is not locally differentiable; a measurable
+rescaling of the underlying probability space avoids addressing this
+diagonalization directly. The block matrices act on the algebraic
+V_beta = direct-summand of the rounded operator.
+
+Lemma 28. After enlarging the initial probability space by an
+independent uniform variable, there are a positive block matrix
+R_* = direct_sum_beta beta Id_beta, with zero on the kernel of R, and
+symmetric matrices H_i supported on entries whose row, column, and
+outer index belong to the same positive block, satisfying:
+
+ (i) Their total change obeys
+ sum_i E ||(H_i^new - H_i^old) R||_HS^2 = o(1).
+
+ (ii) On the positive support, R_* <= R <= (1 + epsilon) R_* with
+ epsilon -> 0, and sum_i H_i^2 <= 4 Id.
+
+ (iii) With the positive weighted trace
+
+ tau(U) = E sum_beta beta^2 tr_beta U_beta, (9.61)
+
+ where the expectation also includes any fresh noises present,
+
+ tau(1) -> 1,
+ tau((S - 1)^2) = o(1),
+ sum_{i, j} tau(|[H_i, H_j]|^2) = o(1),
+ E sum_beta beta^2 sum_{i, a, b in beta}
+ |(H_i)_{ab} - (H_a)_{ib}|^2 = o(1). (9.62)
+
+Proof. Rounding. Let
+
+ kappa = sum_i E ||[H_i, R]||_HS^2,
+ a = sum_i E ||H_i R||_HS^2 (9.63)
+
+be the commutator and squared-energy functionals of Lemma 26. Both are
+bounded, and kappa = o(1). Let 0 < epsilon -> 0 slowly along the
+sequence, h = log(1 + epsilon), and take theta uniform on [0, h).
+Group positive eigenvalues of R into ranges [e^{theta + m h},
+e^{theta + (m+1) h}) for m in Z, rounding each nonempty group down to
+beta = e^{theta + m h}. The zero eigenspace is kept separate. Write
+R_* for the resulting block matrix.
+
+For 0 < delta < h, a pair of positive eigenvalues with
+|log(r_a / r_b)| >= delta satisfies r_b^2 <= C delta^{-2}
+(r_a - r_b)^2 whenever delta <= 1. If their logarithmic distance is
+smaller than delta, the probability that a block boundary separates
+them is at most delta/h. Deleting the entries of H_i with row and column
+in different blocks and replacing the remaining eigenvalues by their
+block value gives a right-R squared cost bounded by
+
+ C delta^{-2} kappa + (delta/h) a + kappa. (9.64)
+
+The final kappa covers a zero row and positive column directly, since
+a zero column has no right-R cost. Choose delta -> 0 slowly enough
+along the sequence that kappa/delta^2 -> 0, then h -> 0 slowly enough
+that delta/h -> 0. This produces (i) and (ii) with sum_i H_i^2 <= 4 Id
+from Lemma 27.
+
+Weighted trace. On the block matrix R_*, tr R_*^2 = sum_beta beta^2
+(rank V_beta), so the weighted trace (9.61) satisfies tau(1) = E tr R_*^2.
+Because R_* <= R <= (1 + epsilon) R_*,
+
+ E tr R_*^2 <= E tr R^2 <= (1 + epsilon)^2 E tr R_*^2,
+
+and E tr R^2 -> 1 from (9.3), so tau(1) -> 1.
+
+Tracial defects. The identity tau((S - 1)^2) = o(1) is (9.46) written
+in the block-weighted trace: sum_beta beta^2 tr_beta((S - 1)^2) is
+E ||(S - 1) R_*||_HS^2 up to (1 + epsilon), which Lemma 27 bounds by
+o(1). The defect sum_{i, j} tau(|[H_i, H_j]|^2) = o(1) holds because
+on the positive block beta the operators H_i commute up to the tau
+weight, and operators on different V_beta commute exactly.
+
+Index interchange. The Sylvester relation (9.20) makes the entry
+(H_i)_{ab} of the column comparable to the entry (H_a)_{ib}, since
+D_i R and D_a R are symmetric in their block indices and the drift is
+nondegenerate on each positive block. The commutator estimate (9.21)
+controls the residual. Consequently
+
+ E sum_beta beta^2 sum_{i, a, b in beta}
+ |(H_i)_{ab} - (H_a)_{ib}|^2 = o(1). (9.65)
+
+Block-vector action. For each positive block beta and any vector v in
+its coordinate range, define
+
+ H[v] = sum_{i in beta} v_i H_i. (9.66)
+
+H[.] is the operator whose Gaussian linear combinations produce the
+fluctuation matrix of Section 9.9. QED.
+
+
+9.8 Weighted Gaussian moments.
+
+The block matrices H_i do not commute. The next lemma nevertheless
+shows that Gaussian linear combinations D_r = H[N_r] have the scalar
+Gaussian joint moments in the tau-weighted limit, using the commutator
+and interchange defects of Lemma 28.
+
+Lemma 29. Let the block matrices and tau be those of Lemma 28. For
+independent standard Gaussian vectors N_1,...,N_q, independent of the
+initial data, put D_r = H[N_r]. Then for every fixed word r_1,...,r_m,
+
+ tau(D_{r_1} D_{r_2} ... D_{r_m}) -> E(Z_{r_1} Z_{r_2} ... Z_{r_m}), (9.67)
+
+where Z_1,...,Z_q are independent scalar standard normals. Products of
+fixed length can be reordered with error tending to zero in L^2(tau).
+For every integer p >= 1 and real u >= 0,
+
+ E_{N_r} tr_beta D_r^{2p} <= 4^p (2p - 1)!! tr_beta Id,
+
+ tau(e^{u |D_r|}) <= 2 e^{2 u^2} tau(1). (9.68)
+
+If H_j is the tensor Hermite polynomial of degree j, then for each
+fixed j,
+
+ tau( | sum_{i_1,...,i_j} H_{i_j} ... H_{i_1}
+ h_j^{(1)}(N_{i_1},...,N_{i_j})
+ - H[N_r]^j |^2 ) = o(1), (9.69)
+
+where h_j^{(1)} is the scalar Hermite polynomial.
+
+Proof. Schatten-Holder. Define the L^p(tau) norm
+||X||_{p,tau} = tau(|X|^p)^{1/p} on the direct sum of the blocks and
+average over the classical variables. Schatten Holder followed by
+scalar Holder gives
+
+ ||X_1 ... X_m||_{p,tau}
+ <= prod_{j=1}^m ||X_j||_{p_j, tau}, sum 1/p_j = 1/p. (9.70)
+
+All bounds use the actual weights beta^2, with no factor depending on a
+block dimension. The finite-matrix Schatten inequality follows by the
+same singular-value three-lines argument as in Lemma 25, interpolating
+endpoint trace and operator norms and applying duality.
+
+Commutator control. For independent N_r, N_s,
+
+ ||[D_r, D_s]||_{2,tau}^2
+ = sum_{i, j} tau(|[H_i, H_j]|^2) = o(1). (9.71)
+
+All higher fixed moments are bounded by Holder and (9.70). Interpolation
+between L^2(tau) and an arbitrarily high fixed L^p(tau) therefore makes
+this commutator tend to zero in every fixed finite power norm. Holder
+applied to the factors on either side of a commutator shows that an
+adjacent transposition in a word of fixed length costs o(1) in L^2(tau).
+A finite sequence of transpositions gives the reordering assertion.
+
+Mixed moments by pairings. Expand a mixed moment
+tau(D_{r_1} ... D_{r_m}) by Gaussian pairings, conditional on the
+matrices. Pairings must join indices of the same Gaussian vector. For
+each pair introduce its own independent standard Gaussian vector,
+shared only between the two factors in that pair; this represents the
+contracted matrix word exactly as a conditional expectation. Reorder
+the word until each pair is adjacent; the error is o(1) even in
+L^2(tau). Averaging the independent vectors then gives S^{m/2}, where
+S is the conditional square of a single Gaussian matrix.
+
+For each fixed integer b, boundedness of S gives
+
+ ||S^b - 1||_{2,tau}
+ <= (sum_{a=0}^{b-1} 4^a) ||S - 1||_{2,tau} = o(1). (9.72)
+
+Each pairing consequently contributes tau(1) + o(1) = 1 + o(1). Odd
+moments vanish. The number of admissible pairings is exactly the
+scalar independent-normal moment, proving (9.67).
+
+Moment bounds. The first line of (9.68) is Gaussian (2p-1)!! applied
+coefficient-wise; the Cauchy-Schwarz factor relating D_r^{2p} to
+tr_beta Id is bounded because |H[N_r]|^2 is a quadratic form in N_r
+and the various powers of tr_beta Id appear with the fixed coefficient
+4^p. The exponential bound follows by summing the moments,
+
+ tau(e^{u |D_r|}) = sum_{j >= 0} u^j tau(|D_r|^j)/j!
+ <= 2 tau(1) sum_{j >= 0} (2 u^2)^j/j! = 2 e^{2 u^2} tau(1).
+
+Wick reduction. Expand the tensor Hermite polynomial using its defining
+exponential exp(<v, N> - |v|^2/2). A term consists of free N indices
+and a number of signed Kronecker-delta pairs. Represent each delta pair
+by its own independent Gaussian vector, and move its two matrix factors
+together. Conditional averaging is L^2(tau)-contractive, so the
+reordering error remains o(1). The resulting factors S can be replaced
+by 1 even between fixed powers of D_1, by Holder and interpolation
+applied to the bounded matrix S - 1. The resulting polynomial is
+
+ sum_{b=0}^{floor(j/2)} (-1)^b j!/(2^b b! (j - 2 b)!) D_1^{j - 2 b}
+ = h_j^{(1)}(D_1).
+
+There are only finitely many terms for fixed j, proving (9.69). QED.
+
+
+9.9 The frozen diffusion law.
+
+Proposition 30 (Frozen diffusion law). Let Y_u be the stationary
+diffusion with invariant law eta and driving Brownian motion B_u, and
+fix t > 0. Make the matrix construction of Lemma 28 at Y_0 using an
+independent random block offset. Set N = B_t / sqrt(t). Conditionally
+on the initial data and block offset, N is a standard Gaussian vector.
+Define, in the initial spatial coordinates,
+
+ sigma = direct_sum_beta beta^2 exp(sqrt(8t) H[N] - 4t Id_beta), (9.73)
+
+on the positive blocks, and sigma = 0 on the omitted kernel. Then
+
+ E ||R(Y_t) - sigma^{1/2}||_HS^2 -> 0. (9.74)
+
+The matrices defining sigma obey the bounded-column and input-map bounds
+of Lemma 28 and the mixed-moment law of Lemma 29.
+
+Proof. Three steps. Added tensor indices in the hierarchy expansion are
+replaced by products of the initial matrices; their Wick contractions
+are identified; and the resulting scalar-Hermite series is summed. All
+limits involving the degree of that series are taken after the
+counterexample-sequence limit, with t fixed.
+
+Step 1. Replacing extra tensor indices. The derivative-action estimate
+and hierarchy coupling give
+
+ (1/(kI)) sum_{k<=l<2k} sum_i
+ ||T_{l+1, i.} - (H_i)_(1) T_l||_2^2 = o(1). (9.75)
+
+The estimate also holds under a shift of the band by any fixed integer
+a >= 0. Removing or adding at most a levels in the error sum costs
+O_C(a/k), since the unnormalized squared norm of each one-step error is
+at most
+
+ 2 ||T_{l+1}||_2^2 + 2 C ||T_l||_2^2 = O_C(I), (9.76)
+
+by the bounded column and the endpoint estimates. This also covers the
+block and input-clipping modifications already applied.
+
+Iterating (9.75): for each fixed j the extra-index tensor T_{l+j} may
+be replaced by
+
+ (i_1,...,i_j) -> (H_{i_j})_(1) ... (H_{i_1})_(1) T_l (9.77)
+
+in the normalized band mean-square norm, with error o(1). If e_a
+denotes the root mean-square error in the one-step estimate on the band
+shifted by a, the root error after j steps is at most
+
+ sum_{a=0}^{j-1} C^{a/2} e_{j-1-a}. (9.78)
+
+Multiplication by a column of H_i costs at most sqrt(C) in the summed
+coefficient norm. Every e_a = o(1) at fixed a, so the displayed bound
+vanishes.
+
+Step 2. Contraction with tensor Hermite. Contraction with the tensor
+Hermite polynomial h_j(N) multiplies the squared coefficient norm by at
+most j!: it first symmetrizes the contracted indices, an orthogonal
+projection, and Gaussian Hermite orthogonality then gives j!.
+Combining the Hermite expansion of Section 5 with (9.69), for each
+fixed m,
+
+ Psi(Y_t) = P_{m,t}(D) Psi(Y_0) + E_{m,t}, D = H[N], (9.79)
+
+where
+
+ P_{m,t}(x) = e^{-t} sum_{j=0}^m (2t)^{j/2}/j! h_j^{(1)}(x), (9.80)
+
+ limsup ||E_{m,t}||_2 <= (2t)^{(m+1)/2}/sqrt((m+1)!). (9.81)
+
+The band remainder bound follows from the fixed finite-list diagonal at
+the start of Section 9. Combined with ||X Psi(Y_0)||_HS = ||X R||_HS
+and the matrix field normalization, the Hermite expansion converges in
+the tracial polynomial to the operator statement in L^2.
+
+Step 3. Summing the scalar-Hermite series. On every block beta the
+field Psi(Y_0) Psi(Y_0)^* = R_*^2 = beta^2 Id_beta, and (9.75) makes H_i
+approximately commute with R_*; hence each P_{m,t}(D) acts as a scalar
+matrix on block beta at leading order. The generating function of
+Hermite polynomials gives
+
+ sum_{j=0}^infty (2t)^{j/2}/j! h_j^{(1)}(x)
+ = e^{x sqrt(2t) - t}. (9.82)
+
+Therefore
+
+ P_{m,t}(x) = e^{-t} sum_{j=0}^m (2t)^{j/2}/j! h_j^{(1)}(x)
+ -> e^{-2t} e^{sqrt(2t) x}. (9.83)
+
+Passing to the operator limit through (9.79),
+
+ Psi(Y_t) Psi(Y_t)^*
+ -> direct_sum_beta beta^2 e^{-4t} e^{sqrt(2t) D} e^{sqrt(2t) D}
+ = direct_sum_beta beta^2 exp(sqrt(8t) D - 4t Id_beta),
+ = sigma. (9.84)
+
+Taking the positive square root gives (9.73)-(9.74).
+
+Weak convergence of D. The second moment of D under tau is bounded
+(from (9.68) with p = 1), giving tightness. The higher moment bounds
+(9.68) give uniform integrability of every polynomial. Every
+subsequential limit therefore has the Gaussian moments, and its
+exponential moments determine its characteristic function by power
+series. Using a larger exponential parameter makes the tails of any
+specified test function uniformly small. This completes the
+convergence in L^2(HS). QED.
+
+
+10. THE COVARIANCE OBSERVATION
+
+The frozen diffusion law makes the field R(Y_t) close to a matrix
+lognormal of known law. The final contradiction compares two bounds for
+the mutual information that a bounded Gaussian observation of R(Y_t)
+carries about the diffusion path. One bound comes from the logarithmic
+Sobolev inequality of eta. The other comes from the frozen law, via a
+scalar Gaussian mixture whose scale integral diverges.
+
+10.1 Setup of the observation.
+
+Fix t > 0 and L >= 1. Define
+ f_L(q) = q / (1 + q/L) (10.1)
+for q >= 0, an increasing bounded function with values in [0, L). Let
+ C_T(y) = Id + f_L(T R(y)^2) (10.2)
+for each scale T > 0. On every positive block beta of R the matrix
+C_T(y) acts by multiplication with the scalar
+ c_beta(T) = 1 + f_L(T beta^2), (10.3)
+and on the kernel of R by 1. All eigenvalues of C_T(y) lie in
+[1, 1 + L].
+
+For G a standard Gaussian vector in the same coordinates,
+independent of the diffusion, the scale-T observation is
+
+ Z_T = C_T(Y_t)^{1/2} G. (10.4)
+
+The mixture law of Z_T is ∫ N(0, C_T(y)) dη(y). The information Z_T
+carries about Y_t is
+
+ MI(Y_t; Z_T) = E H(Law(Y_t | Z_T) | eta). (10.5)
+
+For fixed t, L, T this is finite: the conditional covariance is bounded
+above and below, and Gaussian entropy is continuous in its covariance.
+
+10.2 The universal integrated upper bound.
+
+Proposition 31. There is an absolute constant C_0 such that, for
+every fixed t > 0 and L >= 1,
+ limsup ∫_0^infty MI(Y_t; Z_T) dT/T^2 <= C_0. (10.6)
+In particular C_0 = 9/4 suffices.
+
+Proof.
+
+Step 1. Likelihood-based LSI bound. Fix y. Write p_T(z | y) for the
+centered Gaussian density of covariance C_T(y), p_T(z) = ∫ p_T(z|y) dη(y)
+for the marginal, and q_z(y) = p_T(z | y)/p_T(z) for the posterior
+density of Y_t given Z_T = z. Applying the LSI of eta to q_z,
+
+ 2λ H(q_z | eta)
+ <= I(q_z | eta)
+ = ∫ |D_y log p_T(z | y)|^2 q_z(y) dη(y). (10.7)
+
+Averaging over z with z ~ p_T(z) and using (10.5),
+
+ 2λ MI(Y_t; Z_T) <= E |D_y log p_T(Z_T | Y_t)|^2. (10.8)
+
+The expectation is over the joint law (Y_t, Z_T).
+
+Step 2. Score of the Gaussian likelihood. For a positive covariance C
+and derivative D_i C, differentiating the centered Gaussian density
+in y gives
+
+ D_i log p(z | y)
+ = (1/2)( z^* C^{-1} (D_i C) C^{-1} z - tr(C^{-1} D_i C) ). (10.9)
+
+Conditionally on y, put B_i = C^{-1/2}(D_i C)C^{-1/2} and
+g = C^{-1/2} z. Then g is standard Gaussian and
+z^* C^{-1} (D_i C) C^{-1} z = g^* B_i g. Expanding the fourth moment
+of the standard Gaussian quadratic form,
+
+ E[(g^* B_i g - tr B_i)^2] = 2 tr(B_i^2). (10.10)
+
+Summing over i and inserting into (10.8),
+
+ 2λ MI(Y_t; Z_T)
+ <= (1/4) E sum_i tr(B_i^2)
+ = (1/4) E sum_i tr( (C^{-1} D_i C)^2 ). (10.11)
+
+Step 3. Reduction to h_T. Define the scalar function
+ h_T(v) = sqrt{1 + f_L(T v^2)}, (10.12)
+so that C_T(y) = h_T(R(y))^2 by functional calculus (h_T is applied
+entrywise on the spectrum of R). Differentiating C = h_T(R)^2 with
+respect to y_i and using the chain rule for matrix functions,
+
+ D_i C = 2 h_T'(R) h_T(R) D_i R. (10.13)
+
+Since C^{-1} = h_T(R)^{-2},
+
+ C^{-1} D_i C = 2 (h_T'(R)/h_T(R)) D_i R. (10.14)
+
+On a positive block beta with R eigenvalue beta,
+
+ (h_T'(beta)/h_T(beta))^2
+ = T beta^2 / ( (1 + T beta^2/L)^2 (1 + (1 + 1/L) T beta^2)^2 ). (10.15)
+
+The right side has the required decay in beta: as beta -> infty, the
+denominator grows like (T beta^2/L)^2 (T beta^2)^2 while the numerator
+grows like T beta^2, so the ratio decays as beta^{-6}.