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+A DIMENSION-FREE LOGARITHMIC SOBOLEV INEQUALITY
+FOR SUBGAUSSIAN LOG-CONCAVE MEASURES
+
+An expository rewrite: one theorem, every step, in plain language.
+
+
+0. HOW TO READ THIS
+
+This document proves a single theorem and explains every step of its
+proof. You need single-variable calculus, basic linear algebra, and a
+first course in probability. Anything beyond that is defined the moment
+it is used. Long arguments are split into labelled lemmas (Lemma,
+Proposition, Theorem) so you can jump around. The proof is a proof by
+contradiction that runs through six sections; Section 4 gives the whole
+roadmap before any heavy machinery starts, so read it early.
+
+Notation used throughout (all ASCII, so it copies cleanly):
+
+  R^n         n-dimensional Euclidean space (n can be any positive
+              integer; nothing below depends on how big n is).
+  |x|         length of the vector x;  x.y = dot product.
+  D           gradient (the vector of first partial derivatives).
+  D^2         Hessian (the matrix of second partial derivatives).
+  Delta       Laplacian (the trace of the Hessian).
+  int f dmu   the integral of f against the measure mu.
+  mu(A)       probability that a mu-random point lands in the set A.
+
+A measure mu on R^n is a probability measure when mu(R^n)=1. It has a
+density rho when mu(A) = int_A rho(x) dx for every set A; then rho is a
+nonnegative function integrating to 1. "Full-dimensional support" means
+the set where rho>0 is not flat (it has volume in every direction).
+
+
+1. THE CLAIM, IN PLAIN WORDS
+
+1.1 Entropy.
+
+For a nonnegative function g and a probability measure mu, the entropy
+of g with respect to mu is
+
+  Ent_mu(g) = int g log g dmu - (int g dmu) log( int g dmu ),
+
+with the convention 0 log 0 = 0. Entropy measures how uneven g is: it
+is zero exactly when g is constant, and it grows as g becomes spiky.
+The subtraction term is there so that adding a constant to log g, or
+rescaling g by a positive factor, changes nothing.
+
+1.2 Dirichlet energy.
+
+For a smooth function f, its Dirichlet energy with respect to mu is
+
+  int |Df|^2 dmu,
+
+the average squared steepness of f. It is zero exactly when f is
+constant. It measures how much f wiggles.
+
+1.3 The logarithmic Sobolev inequality (LSI).
+
+Saying mu satisfies an LSI with constant R means
+
+  Ent_mu(f^2) <= 2 R int |Df|^2 dmu          for all smooth f,
+
+again with constant functions giving equality at zero. In words: if f
+wiggles only a little (small energy), then f^2 cannot be very uneven
+(small entropy). The factor 2 is a convention; R is what carries the
+information. The smallest such R is written R(mu).
+
+1.4 Why "dimension-free" is the whole point.
+
+Many inequalities in n dimensions carry constants that blow up like n,
+sqrt(n), or worse as n grows. A constant that is completely independent
+of n is called dimension-free, or absolute if it is a fixed number like
+17. Bounds whose constants grow with n become useless for the
+high-dimensional objects (convex bodies, machine-learning models,
+statistical estimators) that motivate the question. We want one fixed
+number that works in every dimension.
+
+1.5 Log-concave measures.
+
+A density rho on R^n is log-concave when rho = exp(-V) for a convex
+function V, on the set where rho>0. Convexity of V means its graph
+bends upward; equivalently the superlevel sets {rho >= t} are convex.
+Examples include the uniform measure on any convex body (a ball, a
+cube, a simplex), the Gaussian, the exponential, and products and
+Gaussian convolutions of all of these. Log-concave measures are the
+standard model for "a probability sitting on a convex region and
+decaying outward without bumps."
+
+1.6 Subgaussian linear functionals.
+
+Fix a centered random vector X with law mu (centered means E X = 0).
+For a unit vector theta, the scalar theta.X is one linear shadow of X.
+We call X linearly a-subgaussian when
+
+  E exp( (theta.X)^2 / a^2 ) <= 2      for every unit vector theta.
+
+This says every one-dimensional shadow has tails no heavier than a
+Gaussian of variance about a^2: it controls all the linear functions of
+X at once, but only the linear ones. The parameter a is exactly the
+"width" of these shadows.
+
+The question the paper answers: do these Gaussian tails of the LINEAR
+shadows already force an LSI for ALL (including nonlinear) test
+functions, with a constant independent of dimension?
+
+
+
+2. THE THEOREM AND ITS TRANSPORT CONSEQUENCE
+
+2.1 The main theorem.
+
+Theorem 1 (Main). There is an absolute constant C < infinity with the
+following property. For every n >= 1, every centered log-concave
+probability measure mu on R^n that has a Lebesgue density, and every
+a > 0 such that a random vector X with law mu is linearly a-subgaussian,
+
+  Ent_mu(f^2) <= C a^2 int |Df|^2 dmu      for every smooth compactly
+                                            supported f on R^n.
+
+The same C works in every dimension and for every such mu.
+
+In the convention of Section 1.3 (Ent(f^2) <= 2 R(mu) int|Df|^2), the
+theorem says R(mu) <= (C/2) a^2: the logarithmic Sobolev constant is
+bounded by a universal multiple of the squared subgaussian width. Note
+what is NOT assumed: the density need not be smooth, and no positive
+curvature (no D^2 V >= kappa > 0) is required. So uniform measures on
+convex bodies, whose potentials are flat and only convex, are included.
+Before this result, only a dimension-dependent bound like R(mu) <= C n
+a^2 or C sqrt(n) a^2 was known in general.
+
+2.2 Why the converse is free (so the hypothesis is sharp).
+
+The LSI already forces Gaussian tails for 1-Lipschitz functions by the
+classical Herbst argument; in particular it forces exactly the
+subgaussian bound on linear functions that Theorem 1 assumes. So the
+hypothesis "linear functions are a-subgaussian" is, up to universal
+constants, also necessary for an LSI with constant of order a^2. The
+theorem is therefore not wasting any assumption: it upgrades a
+necessary condition to a sufficient one, at the same scale.
+
+2.3 A consequence for transport.
+
+Two standard objects. For two probability measures alpha, beta on R^n
+with finite second moment, the squared 2-Wasserstein distance is
+
+  W2(alpha,beta)^2 = inf over couplings pi of int |x-y|^2 dpi(x,y),
+
+where a coupling is a joint law with the two given marginals. It is the
+cheapest way to move the mass of alpha onto beta, paying squared
+distance. The relative entropy of alpha with respect to beta is
+
+  H(alpha | beta) = int p log p dbeta   when alpha = p beta,
+                  = +infinity            when alpha is not absolutely
+                                         continuous with respect to beta.
+
+Corollary 1 (Quadratic transport-entropy inequality). Under the
+hypotheses of Theorem 1, every probability nu with finite second moment
+satisfies
+
+  W2(nu,mu)^2 <= C a^2 H(nu | mu),
+
+with the same absolute constant C. This follows from the LSI through the
+Otto-Villani theorem, which converts any LSI into a
+transport-entropy inequality with matching constants; the proof at the
+level of this document is in Section 5.7 and only needs an
+approximation argument to remove the smoothness of the density.
+
+The corollary is the useful form in applications: it says it costs very
+little mass-movement energy (order a^2 times a bit of relative entropy)
+to reach mu from any competitor nu. This is what turns "control of
+linear functions" into concrete concentration statements.
+
+
+
+3. HISTORY AND WHY IT MATTERS
+
+3.1 The inequality and the Gaussian.
+
+The logarithmic Sobolev inequality was introduced by Leonard Gross in
+1975 in his study of hypercontractivity, the property that a semigroup
+pushes functions into higher integrability. For the standard Gaussian
+measure the LSI holds with R = 1, independent of dimension, and Gross
+showed the constant cannot be improved. This paper asks how far the
+"constant 1, no dimension" phenomenon survives beyond the Gaussian.
+
+3.2 The curvature approach and its blind spot.
+
+Bakry and Emery (1985) gave a powerful criterion. Write the density as
+rho = exp(-V). Suppose the potential is strictly convex in the strong
+sense D^2 V >= kappa Id with kappa > 0. Then mu satisfies the LSI with
+R <= 1/kappa. The proof runs a diffusion whose drift is D V and shows
+the flow contracts distances at rate kappa.
+
+The catch: log-concavity says only D^2 V >= 0, i.e. kappa = 0. The
+curvature criterion then gives nothing finite. A uniform measure on a
+convex body is the extreme case: its potential is infinite outside the
+body and flat inside, so it has zero curvature in the interior and
+infinite curvature on the boundary. The Bakry-Emery machine is blind to
+exactly the examples we care about. The whole difficulty is to replace a
+local curvature hypothesis by the global tail hypothesis that linear
+functions are subgaussian.
+
+3.3 Necessity and the earlier dimension-dependent bounds.
+
+The Herbst argument (see Ledoux's 1999 survey) shows that an LSI with
+constant R gives Gaussian concentration: for every 1-Lipschitz u,
+
+  log E exp( t (u - E u) ) <= R t^2 / 2.
+
+Applying this to linear functions, an LSI with constant of order a^2
+forces the linear subgaussian bound with width of order a. So some
+subgaussian control of linear functions is necessary. The open problem
+was sufficiency: does it give the LSI back, dimension-free?
+
+Sergey Bobkov (1999) gave a norm-based criterion: the entropy constant
+is bounded by a universal multiple of the squared subgaussian norm of
+the scalar |X|. Combining this with the linear subgaussian assumption,
+which controls |X| only through a Holder step E exp(|X|^2/(n a^2)) <= 2,
+yields an entropy constant of order n a^2, with a dimension factor.
+
+Pierre Bizeul (conjectured in his 2023 preprint, cited here as the 2026
+journal version) formulated precisely that linear subgaussian tails
+should suffice for a dimension-free LSI within the log-concave class.
+Bizeul proved an improvement with constant of order sqrt(n) a^2 in
+general, and the fully dimension-free statement in the special case of
+rotationally invariant log-concave measures. The survey of Klartag and
+Lehec (2024/2025) restates the conjecture together with these bounds.
+
+3.4 What is genuinely new.
+
+The linear subgaussian hypothesis controls only one-dimensional shadows,
+while the LSI conclusion controls the entropy of genuinely nonlinear
+test functions. The gap between "linear control" and "nonlinear
+conclusion" is where the entire argument lives. Emanuel Milman (2010)
+had shown that for log-concave measures, Gaussian concentration for all
+Lipschitz functions is equivalent to Gaussian isoperimetry, which in
+turn gives an LSI; but "all Lipschitz functions" is much stronger than
+"linear functions." The theorem closes the remaining gap: linear
+functions turn out to already force the concentration of all Lipschitz
+functions, hence the LSI. Theorem 1 is the first proof that the
+conjecture is true in full generality.
+
+
+
+4. THE PLAN OF THE PROOF (READ THIS FIRST)
+
+The proof is a proof by contradiction. We assume Theorem 1 fails and
+derive nonsense. Everything technical is in service of one idea: a
+counterexample that violates the LSI would have to behave like a
+Gaussian in a very rigid way, both from above (entropy contraction) and
+from below (an information test), and those two descriptions cannot
+both hold.
+
+4.1 Assume a counterexample, then clean it up.
+
+Failure of Theorem 1 gives a sequence of centered, compactly supported,
+full-dimensional log-concave laws mu such that the LSI constant R(mu)
+is normalized to 1 while the linear subgaussian width a tends to 0. The
+ratio R(mu)/a^2 then blows up; that is the failure. Two harmless
+operations keep the failure: restricting to a large ball and tilting by
+exp(-epsilon |x|^2 / 2). They keep the law log-concave, compactly
+supported, and smooth enough to work with.
+
+4.2 Gaussian regularization.
+
+Convolve mu with a tiny Gaussian of variance s. Write
+mu_s = law(X + sqrt(s) G), with G standard Gaussian independent of X,
+and fix s = 1/8 once and for all. Convolution makes the density smooth
+and strictly positive and keeps it log-concave (a Gaussian is
+log-concave and a convolution of log-concave densities is
+log-concave). Its subgaussian width a_s still tends to 0. Crucially,
+the regularized LSI constants R_s tend to 1, because s is fixed and
+small. So from now on we have a smooth, positive, log-concave law mu_s
+sitting at the edge: it is one unit of entropy-constant away from a
+counterexample.
+
+4.3 The critical law and a near-first-eigenfunction.
+
+From mu_s we build a second probability eta and a vector field w. There
+are two cases, and we must handle both because we cannot predict which
+occurs:
+
+  Strict branch. The entropy quotient is attained at a positive
+  optimizer f solves the logarithmic Euler equation. Put
+  eta = f^2 mu_s and w = D log f^2. Then eta satisfies its own LSI
+  with constant <= 1/lambda, and w is an approximate first
+  eigenfunction of the diffusion A_eta = -Delta - D log(eta).D:
+
+    (A_eta - 1) w is small in L^2,     E_eta|w|^2 = I > 0.
+
+  Gap branch. The LSI and Poincare constants coincide. Then w = D phi,
+  where phi is an eigenfunction of A_{mu_s} with eigenvalue lambda, and
+  eta = mu_s. The same near-eigenfunction relation holds.
+
+In both cases eta satisfies a logarithmic Sobolev inequality with a
+constant close to 1, and w is close to a first eigenfunction of its
+diffusion. That is the single object the rest of the proof works with.
+
+4.4 The prediction curve.
+
+Add fresh Gaussian noise. For r > 0 form U_r = Y + sqrt(r) G with
+Y ~ eta and G standard, and let m(r) be the fraction of w that the best
+predictor can recover from U_r:
+
+  m(r) = (1/I) int |E[w(Y) | U_r]|^2 deta.
+
+If w were an exact first eigenfunction, the semigroup would give
+m(r) = e^{-r} exactly. We prove this is essentially true anyway:
+
+  Theorem (prediction curve). For every fixed 0 < r < 1/4,
+  m(r) -> e^{-r} along the counterexample sequence.
+
+The proof pairs two facts. Upper bound: entropy contraction of the
+diffusion (the LSI of eta) caps how much information about w can
+survive the noise. Lower bound: an entropy-loss comparison lemma
+tracks the loss under repeated Gaussian observations; observing many
+independent copies lets us throw away at most one direction per copy,
+accumulating negligible cost. Together they pin m(r) to e^{-r}.
+
+4.5 From the curve to tensors.
+
+The exact identity m(r) = e^{-r}, integrated against Gaussian Hermite
+polynomials, says the prediction is Bessel-tight at every derivative
+order. This is the bridge from a statement about the vector w to a
+statement about all its derivatives. It produces symmetric tensor
+fields T_l of order l+1 with
+
+  ||T_l||^2 = (1+o(1)) I,     ||DT_l - T_{l+1}|| = o(sqrt I),
+
+  ||(A-1) T_l|| = o(sqrt I),  <T_l, K T_l> = o(I),
+
+where K = D^2(-log mu_s) is the curvature. So derivatives keep the
+same size, stay nearly eigenfunctions, and avoid the curved directions
+of mu_s. This holds simultaneously for any fixed finite list of levels.
+
+4.6 Matrices and the frozen law.
+
+Pack the tensors of a band of levels into a big column Psi, separate
+one slot, and let
+
+  rho = Psi Psi^*,      R = rho^{1/2}
+
+be a positive matrix field. Many symmetric slots let us average the
+action over slots with only O(1/k) error, and shifting the band costs
+only its endpoints; this yields sharp energy: ||DR||^2 -> 1. From the
+derivative-action estimate one builds a bounded symmetric matrix column
+H_i with a Sylvester-type relation D_i R ~ (H_i R + R H_i)/2 and almost
+commuting H_i. Then, running the stationary diffusion for time t and
+freezing H_i at the initial point, R(Y_t) is described by a matrix
+lognormal:
+
+  sigma = direct-sum over spectral blocks beta^2 exp( sqrt(8t) H[N] - 4t ),
+
+with N standard Gaussian. This is the frozen diffusion law. It is the
+engine of the final contradiction, and it is where the many symmetric
+tensor slots earn their keep: they make H[N] behave like a scalar
+Gaussian even though the H_i need not commute.
+
+4.7 The information contradiction.
+
+We now observe a Gaussian vector whose covariance is a bounded
+increasing function of R(Y_t)^2, with a scale T:
+
+  Z_T = ( Id + f_L(T R(Y_t)^2) )^{1/2} G,   f_L(q) = q/(1+q/L).
+
+Upper bound (from the LSI): the mutual information between the state
+and the observation, integrated against dT/T^2, is at most a universal
+constant. The factor dT/T^2 is the load-bearing trick. In a spectral
+block of size beta^2 the substitution x = T beta^2 gives
+
+  dT/T^2 = beta^2 dx/x^2,
+
+assigning to each block exactly the weight beta^2 that the matrix
+(frozen) trace uses, even when the spectrum is arbitrarily diffuse.
+
+Lower bound (from the frozen law): on the same event, the observation
+reproduces a scalar Gaussian variance-mixture experiment. The entropy
+of that experiment, integrated over the same scales, grows without bound
+once the diffusion time t and the covariance cap L are taken large
+enough.
+
+Upper bound is a constant, lower bound is unbounded. Contradiction.
+
+4.8 The order of limits.
+
+Throughout, every auxiliary parameter (time t, cap L, truncation m,
+band size k) is fixed BEFORE the counterexample index is sent to
+infinity. Only after that sequence limit do we let t and L grow. No
+uniformity in the growing parameters across the sequence is ever
+required. This order is repeated where it matters so it is never in
+doubt.
+
+4.9 Reading order for the technical part.
+
+  Section 5  analytic toolkit: entropy, information, curvature,
+             diffusion, entropy dissipation, Gaussian channels.
+  Section 6  normalization and construction of the critical law
+             (both branches).
+  Section 7  the prediction curve.
+  Section 8  the tensor hierarchy.
+  Section 9  the matrix square root and the frozen law.
+  Section 10 the covariance observations and the final information
+             contradiction, then the assembly of Theorem 1.
+
+Figure 1 (description): two arrows from "strict branch" and "gap
+branch" merge into "common critical law and field." From there the
+prediction curve feeds the tensor hierarchy, the hierarchy feeds the
+matrix frozen law, and the frozen law plus the LSI both feed the
+information contradiction. Only Theorem 1 needs the two branches to
+rejoin at the end.
+
+
+
+5. TOOLKIT: ENTROPY, DIFFUSION, AND GAUSSIAN CHANNELS
+
+This section sets up the analytic language used in every proof that
+follows. Nothing here is specific to the counterexample; it is the common
+vocabulary.
+
+5.1 Two time conventions that set all factors.
+
+We will run two different "time" evolutions and must not confuse them.
+
+  (a) Gaussian convolution. Convolving a law with a centered Gaussian of
+      covariance r Id has generator (1/2) Delta. Its time parameter is r,
+      a variance.
+
+  (b) Stationary diffusion. A reversible diffusion with invariant law pi
+      and generator -A_pi, with noise coefficient sqrt(2). Its time
+      parameter is a genuine clock time.
+
+The two conventions differ by factors of 2 throughout. We fix them once
+and state every dissipation identity in the convention used, so no
+factor can drift.
+
+5.2 Relative entropy, Fisher information, and the two constants.
+
+For probabilities p, pi with p absolutely continuous relative to pi,
+
+  H(p | pi) = int log(dp/dpi) dp,     H(p | pi) = +infinity otherwise.
+
+The Fisher information of p relative to pi is
+
+  Ical(p | pi) = 4 int |D sqrt(dp/dpi)|^2 dpi.
+
+When the density is positive and smooth this equals
+int |D log(dp/dpi)|^2 dp: the average squared score. The value is
++infinity unless the square root of the density lies in the weighted
+Sobolev space.
+
+The logarithmic Sobolev constant R(pi) and the Poincare constant
+R^P(pi) are defined by the best constants in
+
+  Ent_pi(f^2) <= 2 R(pi) int |Df|^2 dpi,
+
+  Var_pi(f) <= R^P(pi) int |Df|^2 dpi,   Var_pi(f)=int f^2 dpi - (int f dpi)^2.
+
+Equivalently, in "relative" form, the LSI constant R = R(pi) is the
+smallest number with
+
+  2 H(p | pi) <= R Ical(p | pi)      for all p.
+
+The Dirichlet form is int |Df|^2 dpi. Constant functions have zero
+entropy and zero energy, which is why the entropy is measured relative
+to the mean.
+
+Linearization: expanding Ent about f = 1 + epsilon g to second order in
+epsilon gives the variance bound, so R^P(pi) <= R(pi). The Poincare
+constant is never larger than the log-Sobolev constant, as one expects:
+a variance bound is weaker than an entropy bound.
+
+5.3 Sobolev space and why approximations are allowed.
+
+For a smooth positive density the weighted Sobolev space is the
+completion of the smooth compactly supported functions in the norm with
+squared norm int (f^2 + |Df|^2) dpi. Cutoffs, truncation of values, and
+smoothing on compact sets show this is also the space of functions whose
+weak gradient is square-integrable. A compact-test LSI extends to the
+whole space by lower semicontinuity of the entropy and the Fisher
+information, and by approximation. Every inequality below stated for
+smooth compact tests therefore holds on this space; we use this silently.
+
+5.4 The entropy-duality formula.
+
+For a probability pi, one exact identity underpins most of the analysis:
+
+  H(p | pi) = sup over bounded g of { int g dp - log int e^g dpi }.
+
+Jensen's inequality gives the upper bound (log int e^g dpi is an
+exponential-moment), and bounded truncations of log(dp/dpi) give
+equality. Three consequences we will use.
+
+  (i) Data processing. Pushing a probability through a measurable map,
+      or averaging through a Markov kernel, never increases relative
+      entropy. This is conditional Jensen.
+
+  (ii) Chain rule. Disintegrating a joint density and splitting its
+      logarithm gives
+      H(P_{U,Z} | Q_U x nu) = H(P_U | Q_U) + E_{P_U} H(P_{Z|U} | nu),
+      with conditional entropies averaged over U. Mutual information is
+      I(U;Z) = H(P_{U,Z} | P_U x P_Z).
+
+  (iii) Mean lower bound. Taking g(z)=v.z in the duality formula and
+      optimizing over v gives H(p | gamma) >= |E_p Z|^2/2 for a Gaussian
+      gamma of unit covariance: relative to a Gaussian, an entropy
+      budget bounds how far the mean can be pushed.
+
+
+
+5.5 Positive curvature: the Bakry-Emery lemma.
+
+Lemma 1 (Positive curvature). Let S be a positive definite matrix on R^n,
+and let U be a proper lower semicontinuous convex function, allowed to be
++infinity (so the support may be a convex set with boundary). Suppose
+
+  pi(x) = Z^{-1} exp( -(1/2) x.Sx - U(x) ) dx
+
+is a probability with full-dimensional support. Then
+
+  Ent_pi(f^2) <= 2 E_pi < Df, S^{-1} Df >,
+  Var_pi(f)  <=   E_pi < Df, S^{-1} Df >,
+
+first for compact smooth f, then for all f in the form domain.
+
+Proof. A linear change of variables reduces to S = Id. First suppose
+V(x) = |x|^2/2 + U(x) is smooth with bounded Hessian. Run the diffusion
+with generator L = Delta - DV.D, drifting with -DV; it is reversible
+with invariant law pi because the weight exp(-V) makes DV the score.
+Let J_t be the derivative of the flow with respect to the starting
+point. Differentiating the flow equation gives J'_t = -D^2 V(X_t) J_t,
+so with D^2 V >= Id we get ||J_t|| <= e^{-t}: the flow contracts
+distances exponentially. Weighted Cauchy-Schwarz then gives, for a
+smooth positive q bounded away from zero,
+
+  |D P_t q|^2 / (P_t q) <= e^{-2t} P_t ( |Dq|^2 / q ),
+
+where P_t = e^{tL} is the semigroup. Integration by parts yields
+
+  d/dt int (P_t q) log(P_t q) dpi = - int |D P_t q|^2 / (P_t q) dpi.
+
+Couple a trajectory started at x to a stationary trajectory driven by
+the same Brownian motion; the contraction bound says the distance
+shrinks like e^{-t}. Since pi has Gaussian tails, P_t q -> E_pi q and
+its entropy -> 0. Integrating the two displayed identities from 0 to
+infinity gives Ent_pi(q) <= (1/2) int |Dq|^2/q dpi. Put q = f^2+eps and
+send eps -> 0.
+
+For general U, replace U by its quadratic infimal convolutions
+U_j(x) = inf_z { U(z) + j|x-z|^2 }. These are convex, increase to U,
+and have Lipschitz gradients, so U_j is smooth enough after mollifying
+at radii -> 0. The affine minorant of U gives a uniform Gaussian-times-
+linear domination, the LSI passes to the limit, and the boundary of a
+full-dimensional convex support has zero Lebesgue measure so boundary
+values do not matter. Inserting f = 1 + eps g and comparing eps^2
+coefficients yields the variance inequality. QED.
+
+The lemma is the entire content of the curvature method: curvature
+D^2 V >= Id buys an LSI with constant 1. A "minimum curvature kappa"
+version rescales to constant 1/kappa. And it is exactly this hypothesis
+that log-concavity (D^2 V >= 0) fails to provide.
+
+5.6 Products and Gaussian convolution add the constants.
+
+Suppose independent pieces pi_1, pi_2 satisfy LSIs with energy forms
+Q_1, Q_2. Their product satisfies the LSI with energy Q_1 + Q_2. Indeed
+decompose the entropy of f on the product into conditional entropy on
+the fiber plus entropy of the fiber integral, and use
+
+  |D_x ( int f(x,y)^2 dpi_2(y) )^{1/2}|^2 <= int |D_x f(x,y)|^2 dpi_2(y).
+
+The variance version uses conditional variance and differentiation of
+the fiber mean. In particular, for independent X and Z,
+
+  R( law(X+Z) ) <= R( law(X) ) + R( law(Z) ).
+
+The Gaussian of variance s Id has R = s; adding Gaussian noise adds s to
+the constant. This is why the regularized constants R_s in Section 4.2
+obey R_s <= 1 + s.
+
+5.7 Herbst's argument: entropy control forces concentration.
+
+Lemma 2 (Herbst). If pi satisfies the LSI with constant R, then every
+integrable 1-Lipschitz function u satisfies
+
+  log E_pi exp( t (u - E_pi u) ) <= R t^2 / 2      for all real t.
+
+Proof. Apply the LSI to e^{tu/2}, which has D(e^{tu/2}) = (t/2) e^{tu/2}
+Du. Writing psi(t) = log E_pi e^{tu}, the LSI reads
+t psi'(t) - psi(t) <= R t^2 / 2. That is the derivative of psi(t)/t:
+integrate from 0 to t using psi(0)=0 and continuity. Bounded Lipschitz
+truncations and Fatou extend to all integrable 1-Lipschitz u. QED.
+
+Distance to the mean under pi therefore has a subgaussian tail of
+variance R. Conversely, if every 1-Lipschitz function has a subgaussian
+tail of variance R, this is "concentration," and it is one route back to
+the LSI (Milman's theorem, cited in Section 3). Applied to linear
+functions u(x) = theta.x, Herbst says an LSI of constant R forces the
+linear subgaussian bound of Section 1.6 with a of order sqrt(R). This is
+the sharpness statement of Section 2.2.
+
+
+
+5.8 The Dirichlet operator and the stationary diffusion.
+
+Lemma 3 (Dirichlet operator). Let pi be a smooth strictly positive
+probability density on R^n. On smooth compactly supported functions
+define
+
+  A_pi = - Delta - D log(pi) . D.
+
+This nonnegative symmetric operator is essentially self-adjoint in
+L^2(pi), and smooth compact functions are a core for its closure. In
+particular, if f in L^2(pi) has A_pi f (in the distributional sense)
+in L^2(pi), then f is in the domain of the closed operator. The
+semigroup P_u = e^{-u A_pi} preserves constants and is realized by the
+stationary diffusion
+
+  dY_u = sqrt(2) dB_u + D log pi(Y_u) du,     Y_0 ~ pi.
+
+For f in the domain, the stationary Ito formula holds in L^2, with drift
+- A_pi f and martingale integrand sqrt(2) Df.
+
+Proof sketch, complete in its logical steps. Integration by parts gives
+<f, A_pi g> = int Df . Dg dpi, so A_pi is symmetric and nonnegative. If
+v in L^2(pi) is orthogonal to (1 + A_pi) of all smooth compact
+functions, then (1 + A_pi) v = 0 distributionally. Conjugating by
+sqrt(pi) reduces this to a Poisson equation with a smooth local
+potential, and local elliptic bootstrap makes v smooth. Testing against
+chi_R^2 v with a cutoff chi_R equal to 1 on a ball of radius R and
+|D chi_R| <= C/R, and using 2ab <= a^2/2 + 2b^2, gives
+
+  int chi_R^2 v^2 dpi + (1/2) int chi_R^2 |Dv|^2 dpi <= 2 int |D chi_R|^2 v^2 dpi,
+
+which forces v = 0. Hence the range of 1 + A_pi is dense. The bound
+||(1+A_pi)f||_2 >= ||f||_2 says the closure's range is also closed, so
+it is all of L^2; the closure is self-adjoint (solve the adjoint
+equation and use the orthogonal complement argument). This proves the
+domain and self-adjointness claims. The cutoffs converge to 1 while
+their energy tends to 0, so 1 is in the kernel and P_u 1 = 1. Solving
+the stochastic equation with killing on balls, whose kernels are
+symmetric sub-Markov by the Dirichlet heat equation, and increasing the
+balls, identifies the minimal semigroup with P_u; essential
+self-adjointness shows no mass is lost, so the diffusion is stationary
+and non-exploding. QED.
+
+The point of the lemma: it gives a genuine reversible Markov process
+with known generator and a well-defined energy form, on which "time"
+entropy gradients are computable. Everything downstream is a statement
+about this process.
+
+5.9 Entropy dissipation (de Bruijn identities).
+
+Lemma 4 (Entropy dissipation). For the semigroup of Lemma 3, if
+p_0 = q_0 pi has finite relative entropy and p_u = (P_u q_0) pi, then
+for 0 <= a <= b,
+
+  H(p_a | pi) - H(p_b | pi) >= int_a^b Ical(p_u | pi) du.
+
+If instead two probabilities both evolve by Gaussian convolution,
+p_r = p_0 * gamma_r and q_r = q_0 * gamma_r, then
+
+  H(p_a | q_a) - H(p_b | q_b) >= (1/2) int_a^b Ical(p_r | q_r) dr
+
+whenever the left-hand entropy at time a is finite; here gamma_r is the
+centered Gaussian of covariance r Id, and a = 0 is allowed. For the
+standard Gaussian Ornstein-Uhlenbeck semigroup the first inequality is
+an equality, including at time 0 when the initial entropy is finite.
+All assertions hold after averaging over conditioning labels whose
+noises are independent given those labels.
+
+Proof. For a bounded density q bounded away from zero, the form chain
+rule, first for smooth functions and then by form approximation, gives
+
+  (d/du) int (P_u q) log(P_u q) dpi = - int |D P_u q|^2 / (P_u q) dpi.
+
+Spectral smoothing puts P_u q in the operator domain for u > 0.
+Approximating an arbitrary finite-entropy density from above and below
+and normalizing, the initial entropies converge; L^1 contraction keeps
+later densities close; lower semicontinuity of entropy and of the
+Fisher information, plus Fatou in time, gives the first display from
+time 0. Restarting at a gives arbitrary a <= b.
+
+For simultaneous heat evolution put z = p_r / q_r. Direct
+differentiation with (d/dr) p_r = Delta p_r / 2 gives, for convex
+Phi_k increasing to z log z - z + 1,
+
+  (d/dr - (1/2) Delta) { q_r Phi_k(z) } = -(1/2) q_r Phi_k''(z) |Dz|^2.
+
+Integrate against spatial cutoffs, let the cutoff go to infinity, then
+k -> infinity, and use monotone convergence. Data processing gives
+H(p_eps | q_eps) <= H(p_0|q_0), so the initial time is included. For
+Ornstein-Uhlenbeck evolution relative to a standard Gaussian,
+generator Delta - x.D, the same convex truncation stacked with expanding
+radial cutoffs gives equality on positive time; relative entropy is
+continuous at time 0 by data processing plus lower semicontinuity.
+Tonelli handles conditional versions. QED.
+
+In words: relative entropy is nonincreasing under both evolutions, and
+its rate of decrease equals the Fisher information. This is the "engine"
+that converts an entropy budget into an integral information budget.
+
+
+
+5.10 The Gaussian channel identity (Guo-Shamai-Verdu).
+
+This is the standard "I-MMSE" identity: the derivative of entropy
+through a Gaussian channel is half the power of the minimum-mean-square
+predictor. We need its conditional, time-integrated form in our
+conventions.
+
+Lemma 5 (Gaussian signals). Let S be an R^d-valued random vector with
+finite second moment, let U be any side information, and let G be a
+standard Gaussian independent of (S,U). Write Z_t = sqrt(t) S + G and
+
+  d(t) = E_U H( Law(Z_t | U) | gamma ),
+
+the conditional relative entropy against the standard Gaussian gamma,
+averaged over the side information. Then d(0) = 0, d(t) <= t E|S|^2/2,
+and
+
+  d(t) = (1/2) int_0^t E |E[S | Z_v, U]|^2 dv.
+
+The integrand is nondecreasing in v. Taking U constant gives the
+unconditional identity.
+
+Proof. Conditional convexity of relative entropy and the shift formula
+H( gamma(. - sqrt(t) s) | gamma ) = t|s|^2/2 give the upper bound.
+Gaussian kernel differentiation gives the score of the output,
+
+  D_z log( dLaw(Z_t|U) / dgamma )(z) = sqrt(t) E[S | Z_t = z, U].
+
+This is justified at every t > 0 by the Gaussian kernel and the finite
+second moment, or by truncating S. Running Ornstein-Uhlenbeck evolution
+for time u replaces t by t e^{-2u}. Its entropy dissipation identity
+together with dt/du = -2t gives d'(t) = (1/2) E |E[S|Z_t,U]|^2 almost
+everywhere on (0,infinity). The upper bound gives d(t) -> 0 at zero,
+so integrating from 0 proves the identity. Monotonicity of the
+prediction energy: the channel at v < t is obtained from the channel at
+t by multiplying the output by sqrt(v/t) and adding fresh Gaussian noise
+of covariance (1 - v/t) Id, and conditional Jensen gives
+nondecreasingness. QED.
+
+The lemma turns a Gaussian observation into a computing device: the
+information it carries is exactly the integral of the squared best
+prediction of the signal. We will use it both directions - to bound
+information from a predictor, and to read off a predictor from
+information.
+
+5.11 The transport consequence of the LSI.
+
+Proof of Corollary 1. Jensen in each coordinate gives
+int |x|^2 dmu <= n a^2 log 2, so mu has finite second moment. Only the
+case H(nu|mu) < infinity needs proof. For s > 0 put gamma_s =
+N(0, s Id), mu_s = mu * gamma_s, and nu_s = nu * gamma_s. The density of
+mu_s is smooth and strictly positive.
+
+Claim: mu_s is log-concave with convex potential V_s. For X ~ mu and an
+independent standard Gaussian G, the posterior potential of X given
+X + sqrt(s) G = y is the original convex potential plus
+|x - y|^2/(2s), so Lemma 1 bounds the posterior covariance by s Id.
+Hence the Hessian of V_s satisfies
+
+  D^2 V_s(y) = s^{-1} Id - s^{-2} Cov( X | X + sqrt(s) G = y ) >= 0,
+
+and Gaussian differentiation is valid without compact support. So mu_s
+is centered and log-concave, smooth.
+
+Next, mu_s is a_s-subgaussian with a_s = a + beta_s and
+beta_s = sqrt(8s/3). For a scalar standard Gaussian Z,
+E exp( z^2 / beta_s^2 ) = 2. The elementary inequality
+
+  (u+t)^2/(a+beta_s)^2 <= (a/(a+beta_s)) u^2/a^2 + (beta_s/(a+beta_s)) t^2/beta_s^2
+
+and Holder show that if X is a-subgaussian and sqrt(s)G has width
+beta_s, then X + sqrt(s)G is a_s-subgaussian. Theorem 1 with the Sobolev
+extension gives the LSI with constant C a_s^2.
+
+In the conventions of Otto-Villani, Ent(f^2) <= L int|Df|^2 is LSI(2/L),
+whose transport conclusion is W2^2 <= L H. Their theorem applies to the
+smooth convex potential of mu_s, hence
+
+  W2(nu_s, mu_s)^2 <= C a_s^2 H(nu_s | mu_s) <= C a_s^2 H(nu | mu).
+
+The last step is data processing for the common Gaussian kernel, as in
+Section 5.4. Coupling a vector with itself plus independent Gaussian
+noise gives W2(mu_s, mu) <= sqrt(n s) and the same for nu. Let
+s -> 0 at fixed n, with a_s -> a. QED.
+
+The whole proof of the corollary is therefore three moves: regularize
+by Gaussian convolution (which buys smoothness and curvature), invoke
+the main inequality plus Otto-Villani on the regularized law, and let
+the regularization vanish. The only inputs are Lemmas 1 and the data
+processing bound.
+
+
+
+6. THE CRITICAL LAW AND A NEAR-FIRST-EIGENFUNCTION
+
+Suppose Theorem 1 is false. This section converts "a counterexample
+exists" into a single smooth object: a probability eta satisfying an LSI
+with constant close to 1, plus a vector field w that is almost a first
+eigenfunction of eta's diffusion. There are two constructions of (eta,w);
+we keep both alive because we cannot predict which occurs. Every limit
+in this section is along the counterexample sequence; we suppress its
+index.
+
+6.1 Normalized counterexamples.
+
+Lemma 6 (Normalized counterexamples). Failure of Theorem 1 produces a
+sequence of centered, compactly supported, full-dimensional log-concave
+probabilities mu such that
+
+  R(mu) = 1,   sup_{|theta|=1} E_mu exp( (X.theta)^2 / a^2 ) <= 2,   a -> 0.
+
+For an absolute constant C they also satisfy
+
+  log E_mu exp( v.X ) <= C a^2 |v|^2,     Cov_mu(X) <= C a^2 Id.
+
+Proof. Start from laws and compact smooth tests for which the ratio
+(entropy)/(energy)/a^2 tends to infinity. Choose the tests with positive
+energy: a smooth test of zero energy is constant on the connected
+interior of the convex support, and its entropy then vanishes. Restrict
+the law to a large ball; this preserves each test quotient arbitrarily
+well. Multiply the restricted density by exp(-eps |x|^2/2) and
+normalize, with eps > 0 small enough to preserve the quotient again.
+Both operations keep log-concavity, the combined density relative to the
+original law is bounded by 2, and Lemma 1 gives a finite LSI constant
+for the resulting compact law.
+
+A bounded density increase changes the exponential-square expectation by
+at most that factor. If E exp(Z^2/A^2) <= 4, Jensen gives
+E exp(Z^2/(2A^2)) <= 2. Centering costs only another absolute factor:
+E Z^2 <= A^2 log 4 and (Z - E Z)^2 <= 2Z^2 + 2(E Z)^2. Apply this to
+each unit linear form with the same constants. Translation preserves
+the test quotient. Thus the centered compact laws still have finite
+R with R/a^2 -> infinity. Dilation by R^{-1/2} makes R(mu) = 1.
+
+For the moment bound: if a centered scalar Z obeys
+E exp(Z^2/a^2) <= 2, then E|Z|^k <= 2 a^k Gamma(k/2+1) by integrating
+its tail. Expansion of exp(tZ), with the linear term absent, gives
+log E exp(tZ) <= C a^2 t^2 for a|t| <= c; for a|t| >= c use
+tZ <= Z^2/a^2 + a^2 t^2/4. Apply to Z = X.theta. Differentiating the
+centered moment generating function at zero gives the covariance bound.
+QED.
+
+6.2 Gaussian regularization and posterior identities.
+
+Fix s = 1/8. Let G be a fresh standard Gaussian and set
+
+  mu_s = law( X + sqrt(s) G ),     X ~ mu,
+  R_s = R(mu_s),
+  b_s(y) = E[ X | X + sqrt(s) G = y ]      (posterior mean),
+  V_s(y) = - log mu_s(y),                    (potential of mu_s).
+
+Gaussian differentiation and the posterior covariance give the three
+basic identities
+
+  D V_s(y) = ( y - b_s(y) ) / s,
+
+  M_s(y) = D b_s(y) = Cov( X | X + sqrt(s) G = y ) / s,
+
+  K_s(y) = D^2 V_s(y) = ( Id - M_s(y) ) / s.
+
+The posterior of X given y is the original convex potential plus
+|x - y|^2/(2s); Lemma 1 (applied to its linear forms) bounds its
+covariance by s Id. Hence
+
+  0 <= M_s <= Id,     0 <= K_s <= s^{-1} Id.
+
+All these functions are smooth. Compact support of X also gives
+|b_s| <= sup over the support of mu of |x|, so b_s is bounded.
+
+The matrix field K_s is the curvature of mu_s. Much of the proof is
+about showing that the near-eigenfield w avoids K_s; K_s is only
+nonnegative and can be huge near the boundary of the support, which is
+exactly why a uniform radius eigenvalue argument is unavailable and the
+whole argument must be structured around "w sees almost no curvature."
+
+6.3 The regularized constants barely change.
+
+Lemma 7 (Bounds for regularized constants). The function s -> R_s is
+nondecreasing and 1-Lipschitz on (0, infinity), and
+
+  1 <= R_s <= 1 + s.
+
+Proof. The heat equation for mu_s is a continuity equation with
+velocity D V_s / 2. On a compact interval of positive times this
+velocity is globally Lipschitz and grows at most linearly, so its smooth
+flow is a diffeomorphism carrying mu_s to mu_t. Its Jacobian J satisfies
+J' = K_t J / 2. Since K_t >= 0 the flow expands distances, so its
+inverse is 1-Lipschitz for t >= s. Transport a test function by that
+inverse: its entropy is unchanged and its Dirichlet energy cannot
+increase. Hence R_t >= R_s: monotone.
+
+The convolution bound R_{s+h} <= R_s + h from Section 5.6 and
+R_s <= R_0 + s = 1 + s give the upper bounds. Finally, for every compact
+smooth test the entropy and the energy converge to those under mu as
+s -> 0, so taking the supremum of the quotient gives
+liminf_{s->0} R_s >= R(mu) = 1; monotonicity then forces the whole
+range to start at >= 1. QED.
+
+So for the fixed tiny regularization s = 1/8 we have R_s very close to
+1, while each mu_s is smooth, positive, and log-concave with a
+well-behaved diffusion. That is our working law.
+
+
+
+6.4 Compactness and the Gaussian entropy defect.
+
+To extract an optimizer we need two facts about the regularized law
+nu = mu * gamma_s: its form domain is compactly embedded in L^2(nu),
+and the LSI holds with a small additive defect that lets maximizing
+sequences be compactified.
+
+Lemma 8 (Compactness and Gaussian entropy defect). Fix s > 0 and a
+compactly supported probability mu on R^n, and put nu = mu * gamma_s.
+Its form domain embeds compactly into L^2(nu). For every eps > 0 there
+is a finite C_eps, depending on this fixed law, such that
+
+  Ent_nu(u^2) <= 2(s+eps) ||Du||_2^2 + C_eps ||u||_2^2.
+
+If u_j -> u in L^2(nu), ||u_j||_2 = 1, and ||D u_j||_2^2 -> E_* <
+infinity, then
+
+  limsup_j Ent_nu(u_j^2) <= Ent_nu(u^2) + 2s ( E_* - ||Du||_2^2 ).
+
+The sharp coefficient 2s governs escape into the Gaussian tails; the
+additive constant governs maximizing sequences that would collapse to a
+constant.
+
+Proof. Write r = dnu/dgamma_s. The Gaussian kernel formula gives
+D log r = b_s / s, which is bounded, and |log r(y)| <= C(1+|y|). For
+v = u sqrt(r),
+
+  Ent_nu(u^2) = Ent_{gamma_s}(v^2) - int v^2 log r dgamma_s,
+
+  Dv = sqrt(r) ( Du + (1/2) u D log r ).
+
+Gaussian integration by parts followed by Cauchy-Schwarz gives
+
+  int |y|^2 v^2 dgamma_s = s n ||v||_2^2 + 2s int v (y.Dv) dgamma_s
+                         <= 2 s n ||v||_2^2 + 4 s^2 ||Dv||_2^2.
+
+Hence the linear-growth entropy correction is bounded by
+delta ||Dv||_2^2 + C_delta ||v||_2^2 for every delta > 0. Gaussian LSI
+and the derivative formula, with another small Cauchy-Schwarz
+parameter, give the defective inequality.
+
+The same moment estimate shows tightness of the L^2(nu) mass of any
+bounded form sequence. On a fixed ball the density is bounded above and
+away from zero, so the ordinary local Rellich theorem gives L^2
+precompactness; a diagonal extraction plus tightness give global
+compactness.
+
+For the sharp coefficient 2s in the entropy-defect display, set
+v_{j,L} = (|u_j| - L)_+ and m_{j,L} = ||v_{j,L}||_2^2. Strong L^2
+convergence gives lim_{L} sup_j m_{j,L} = 0. The energies of v_{j,L}
+and |u_j| wedge L sum to ||D u_j||_2^2, so lower semicontinuity for the
+clipped functions gives
+
+  lim_{L} limsup_j ||D v_{j,L}||_2^2 <= E_* - ||Du||_2^2.
+
+For a nonnegative v with squared norm m <= 1, the negative part of
+v^2 log v^2 has integral at most C sqrt(m) (use -z log z <= C sqrt(z) on
+0 <= z <= 1), and m log m <= 0, so the defective inequality gives
+
+  int v^2 log_+ v^2 dnu <= 2(s+eps)||Dv||_2^2 + C_eps m + C sqrt(m).
+
+If |u_j| > M L with M > 1 fixed then |u_j| <= c_M v_{j,L} with
+c_M = M/(M-1). Passing bounded truncations of z^2 log z^2 to the
+limit by strong L^2 convergence, and the negative part likewise,
+assembles the lost positive entropy as at most 2s(E_* - ||Du||_2^2).
+The normalization terms vanish because ||u||_2 = ||u_j||_2 = 1. QED.
+
+The coefficient 2s is where "escape into the Gaussian tails costs
+entropy at rate at most 2s" is made precise. The next subsection uses
+this defect to prove that a maximizer of the entropy quotient exists.
+
+
+
+6.5 Existence of an optimizer (strict branch).
+
+The strict branch is the case where the entropy constant strictly beats
+both the Poincare constant and the Gaussian-regularization scale. There
+the entropy quotient is attained.
+
+Lemma 9 (Existence of an optimizer). Let nu = mu * gamma_s, with s > 0
+and mu compactly supported. Suppose R = R(nu) and P = R^P(nu) satisfy
+infinity > R > max(P, s). Then there is a smooth strictly positive
+nonconstant f with
+
+  int f^2 dnu = 1,     Ent_nu(f^2) = 2 R int |Df|^2 dnu,
+
+and f satisfies the logarithmic Euler equation
+
+  A_nu f = lambda f log f,     lambda = R^{-1},
+
+first distributionally and then classically.
+
+Proof of existence and the Euler equation. Take nonnegative unit-norm
+maximizing tests f_j. Replacing a test by its absolute value preserves
+entropy and energy, so this loses nothing. The defective inequality of
+Lemma 8, applied with s + eps < R, bounds the energies ||D f_j||_2.
+
+First exclude energies tending to zero; this is the only place the
+strict inequality R > P is used. Suppose eps_j^2 = ||D f_j||_2^2 -> 0,
+and write c_j = E_nu f_j, h_j = (f_j - c_j)/eps_j. Poincare gives
+||h_j||_2^2 <= P, while ||D h_j||_2^2 = 1 and 1 - c_j^2 = eps_j^2
+||h_j||_2^2, so c_j -> 1. By compactness, after extracting, h_j -> h in
+L^2 with E h = 0, ||Dh||_2^2 <= 1, and ||h||_2^2 <= P ||Dh||_2^2.
+
+For c > 0, eps > 0 and z >= -c/eps define
+
+  Q_{c,eps}(z) = eps^{-2} [ (c+eps z)^2 log((c+eps z)^2/c^2)
+                            - (c+eps z)^2 + c^2 ].
+
+Normalization and -log c_j^2 - 1 + c_j^2 >= 0 give
+
+  eps_j^{-2} Ent(f_j^2) <= int Q_{c_j,eps_j}(h_j) dnu.
+
+The scalar estimate we need: for 1/2 <= c <= 1 and 0 < eps <= 1/2,
+
+  0 <= Q_{c,eps}(z) <= (2+2log2) z^2 + z^2 log_+ z^2,
+  Q_{c,eps}(z) -> 2 z^2  uniformly on bounded z-intervals.
+
+To verify, put t = eps z / c, so Q = z^2 B(t) with
+B(t) = 2 + 4 int_0^1 (1-r) log(1 + r t) dr, -1 <= t < infinity. For
+t <= 0, B(t) <= 2; for t >= 0, t <= z gives
+B(t) <= 2 + 2 log(1+z) <= 2 + 2log2 + log_+ z^2.
+
+Now use the general tail bound: the tail argument in Lemma 8 never
+used unit norm until its last line, so for any bounded form sequence
+u_j -> u in L^2 with energies -> E_*,
+
+  lim_{K} limsup_j int_{|u_j|>K} u_j^2 log_+ u_j^2 dnu
+    <= 2s (E_* - ||Du||_2^2).
+
+Apply the compact convergence in the scalar estimate on |h_j| <= K and
+the upper bound on the complement. Strong L^2 convergence removes the
+quadratic tail as K -> infinity; the general tail bound controls the
+entropy tail. Therefore
+
+  limsup_j eps_j^{-2} Ent(f_j^2)
+    <= 2||h||_2^2 + 2s(1 - ||Dh||_2^2) <= 2 max(P, s) < 2R,
+
+contradicting the maximizing property. So energies do not vanish.
+
+Hence we may extract f_j -> f strongly in L^2 with
+||D f_j||_2^2 -> E_* > 0. Put E = ||Df||_2^2. The entropy defect and
+the sharp inequality give
+
+  2R E_* <= Ent(f^2) + 2s(E_* - E) <= 2R E + 2s(E_* - E).
+
+Since R > s and E <= E_*, equality forces E = E_* > 0. Thus f is a
+nonnegative, nonconstant optimizer.
+
+The homogeneous functional Ent(g^2) - 2R ||Dg||_2^2 is nonpositive and
+is zero at f. Its derivative along every compact smooth variation zeta
+is
+
+  4 int f log f zeta dnu - 4R int Df.Dzeta dnu = 0.
+
+The entropy normalization term cancels the additional linear term, and
+differentiation at zeros is legitimate because z^2 log z^2 has zero
+derivative at zero. This is the Euler equation A_nu f = lambda f log f
+in distributional form. Smoothness and strict positivity are handled
+next. QED.
+
+So in the strict branch we have a bona fide optimizer f, and w = D log
+f^2 must be a nearly first eigenfunction. Turning the Euler equation
+into that statement, and proving f is smooth and positive, is the rest
+of the section.
+
+
+
+6.6 Regularity and strict positivity of the optimizer.
+
+We have a nonnegative optimizer satisfying the weak Euler equation. To
+use its logarithmic derivative we need local regularity and strict
+positivity.
+
+Regularity. Write rho = exp(-V) for the smooth density of nu, set
+v = exp(-V/2) f >= 0. Differentiating the weak equation directly gives
+
+  -Delta v = lambda v log v
+             + ( lambda V/2 - |DV|^2/4 + Delta V/2 ) v.
+
+The conjugating factor and its reciprocal are bounded on each fixed
+ball. Since |z log z| <= C_delta (1 + z^{1+delta}) for z >= 0, local
+L^p integrability of v puts the right side in L^q_loc with
+q = p/(1+delta) > 1. Localizing the right side on a larger ball and
+taking its Newtonian potential u solving -Delta u = F, the interior L^q
+estimate gives u in W^{2,q}_loc; the difference v - u is
+distributionally harmonic, hence smooth by Weyl lemma. Then Sobolev
+embedding improves the integrability of v step by step: before
+boundedness is reached, the reciprocal integrability exponent 1/p
+improves to (1+delta)/p - 2/n, a gain. After finitely many steps q >
+n/2, so v, hence f, is locally bounded and Holder continuous; the
+local Schauder estimate then gives f in C^{2,alpha}_loc. In dimension
+one the same conclusion follows by twice integrating. Every constant
+here concerns one fixed law and dimension and never enters a uniform
+estimate along the sequence.
+
+Positivity. Suppose f(x_0) = 0. Choose a ball B about x_0 on which
+0 <= f <= b < 1 on its closure. Start the nu-diffusion of Lemma 3 at
+x_0 and stop it at the exit time tau from B. Put Z_t = f(Y_{t and tau}),
+m(t) = E Z_t, g(z) = -lambda z log z. The stopped Ito formula gives,
+for almost every t,
+
+  m'(t) = E[ 1_{t<tau} g(Z_t) ] <= E g(Z_t) <= g(m(t)),   m(0) = 0,
+
+because g is nonnegative and concave on [0,b] and g(0)=0. Also m' >= 0.
+If m(t_1) > 0 with t_1 > 0, and t_a is the first time m = a, then for
+0 < a < m(t_1),
+
+  int_a^{m(t_1)} dz/( -lambda z log z ) <= t_1 - t_a <= t_1,
+
+contradicting the divergence of the integral as a -> 0. Hence m = 0.
+The killed nondegenerate diffusion reaches every open subset of B with
+positive probability (follow a smooth interior path and require the
+Brownian motion to stay in a thin tube), so f vanishes on all of B, and
+its zero set is both open and closed. Since R^n is connected and f is
+nonzero, f must be strictly positive. Smooth elliptical bootstrap then
+makes f smooth everywhere. QED.
+
+6.7 Change of reference at an optimizer.
+
+An optimizer supplies a second probability with its own LSI, and no
+convexity of that second probability is needed.
+
+Lemma 10 (Change of reference). Under Lemma 9, suppose in addition that
+nu = exp(-V) is log-concave with bounded curvature K = D^2 V. Define
+
+  eta = f^2 nu,   h = log f^2,   w = D h,   I = E_eta |w|^2.
+
+Then I = 2 lambda H(eta | nu) > 0, R(eta) <= 1/lambda, each component
+of w lies in the domain of A_eta, and
+
+  A_nu h = lambda h + (1/2)|w|^2,        (i)
+  A_eta w = (lambda Id - K) w.           (ii)
+
+Moreover E_eta <w, K w> <= lambda |E_eta w|^2.               (iii)
+
+Proof. Since w = 2 Df/f, I = 4 ||Df||_{L^2(nu)}^2; optimizer equality
+gives I = 2 lambda H(eta|nu) > 0. The chain rule applied to the Euler
+equation gives (i). Commuting one derivative with A_nu gives
+
+  A_nu w = D(A_nu h) - K w = lambda w + (Dw) w - K w.
+
+Since A_eta = A_nu - w.D and Dw is symmetric, (ii) follows
+distributionally. Its right side is in L^2(eta) because K is bounded and
+w in L^2(eta), so Lemma 3 puts w in the operator domain; no unproved
+higher integrability has been used.
+
+For the inequality, let p = g^2 eta have unit mass with g smooth compact.
+Direct expansion and integration by parts give
+
+  Ical(p|nu) = Ical(p|eta) + int g^2 |w|^2 deta + 2 int D(g^2).w deta
+             = Ical(p|eta) + int g^2 (|w|^2 + 2 A_eta h) deta
+             = Ical(p|eta) + 2 lambda int h dp.
+
+On the other hand H(p|nu) = H(p|eta) + int h dp. Subtracting
+2 lambda times this gives the exact deficit identity
+
+  Ical(p|nu) - 2 lambda H(p|nu) = Ical(p|eta) - 2 lambda H(p|eta).
+
+Apply the nu-inequality to the compact smooth function g f: its deficit
+is nonnegative, so the right side is nonnegative. Homogeneity and
+Sobolev approximation give R(eta) <= 1/lambda. Finally, the domain
+identity gives E_eta|Dw|^2 = lambda I - E_eta<w,Kw>, while Poincare for
+eta gives the lower bound lambda(I - |E_eta w|^2). Subtract to obtain
+(iii). QED.
+
+The lemma does two things at once. It transfers a logarithmic Sobolev
+inequality from nu to the tilted law eta = f^2 nu (so eta inherits the
+good entropy constant), and it writes the score w as an approximate
+first eigenfunction of eta's diffusion. Everything after this point
+happens under eta.
+
+
+
+6.8 Translation gain: uniform control of means.
+
+The critical law eta has a near-first-eigenfield w, but we also need to
+know that its mean and its curvature-action are small. The tool is a
+stability estimate for the LSI under translating a probability.
+
+Lemma 11 (Translation gain). Let mu satisfy the normalized
+counterexample conditions of Lemma 6 with a <= 1. Fix a compact
+interval J inside (0, 1/2). There is a constant C_J, independent of the
+dimension and of the support radius of mu, with the following property.
+For s in J, put lambda = R_s^{-1}. If sigma is any probability with
+
+  D_0 = H(sigma | mu_s) < infinity,    F_0 = Ical(sigma | mu_s) < infinity,
+
+and p = E_sigma Y is its mean, then
+
+  F_0 - 2 lambda D_0 >= ( s^{-2} - lambda/s ) |p|^2
+                        - C_J sqrt(a) ( F_0 + D_0 ).
+
+Proof. The centered linear moment generating functions under mu_s obey
+
+  log E_{mu_s} exp(v.Y) <= C_J |v|^2,
+  log E_{mu_s} exp(v.b_s(Y)) <= C a^2 |v|^2,
+
+the first by independent Gaussian addition and the moment bound of
+Lemma 6, the second by conditional Jensen. Entropy duality, optimized
+over the scalar multiple of each unit vector, gives
+
+  |p| <= C_J sqrt(D_0),     |E_sigma b_s(Y)| <= C a sqrt(D_0).
+
+Compact support of X makes E_{mu_s} exp(c|Y|^2) finite for some c > 0,
+so all these expectations exist; by duality E_sigma |Y|^2 < infinity.
+The latter is used only for integrability, not for a dimension bound.
+
+Let sigma_u be the translate of sigma by -u p, 0 <= u <= 1, and let
+D_u = H(sigma_u | mu_s). Change of variables in entropy and the
+Hessian bound give
+
+  D_u = D_0 + E_sigma[ V_s(Y - up) - V_s(Y) ]
+      <= D_0 - u|p|^2/s + u p.E_sigma b_s(Y)/s + u^2 |p|^2/(2s)
+      <= C_J D_0.
+
+Thus the mean estimate applied to sigma_u gives
+|E_sigma b_s(Y - up)| <= C_J a sqrt(D_0) uniformly in u. Differentiating
+the exact entropy expression and integrating gives
+
+  D_1 = D_0 - |p|^2/(2s) + O_J(a D_0).
+
+Its derivative is -(1-u)|p|^2/s + p.E_sigma b_s(Y - up)/s.
+
+Write r(Y) = D log( dsigma / dmu_s )(Y) in the weak square-root sense.
+Its mean is
+
+  E_sigma r = E_sigma D V_s = ( p - E_sigma b_s ) / s.
+
+To justify this for finite Fisher information, the Lebesgue density q of
+sigma has weak derivative Dq = q(r - D V_s) in L^1: r is square
+integrable, and D V_s has linear growth. Integrating against expanding
+cutoffs gives int Dq = 0.
+
+In moving coordinates Y, the score of sigma_1 relative to mu_s is
+r(Y) - p/s + d(Y)/s with d(Y) = b_s(Y) - b_s(Y - p). Since
+0 <= D b_s <= Id, integration along the segment gives
+|d(Y)|^2 <= int_0^1 |M_s(Y - up) p|^2 du <= p.d(Y). The mean estimates
+at the two endpoints therefore imply
+
+  E_sigma |d|^2 <= C_J a D_0.
+
+Expanding the score square and using the mean identity,
+
+  E_sigma |r - p/s|^2 = F_0 - |p|^2/s^2 + 2 p.E_sigma b_s/s^2.
+
+The remaining cross term is bounded by C_J sqrt(a D_0)(sqrt(F_0) +
+sqrt(D_0)), and the d-square term by C_J a D_0. Therefore the translated
+information is finite and
+
+  F_1 = F_0 - |p|^2/s^2 + O_J( sqrt(a) (F_0 + D_0) ).
+
+These calculations use only the weak derivatives already justified, so
+they apply when the density of sigma vanishes. Apply the LSI to sigma_1,
+F_1 >= 2 lambda D_1, and combine with the two translated displays. Since
+2/3 <= lambda <= 1 on J, the result is the claim. QED.
+
+The lemma says: any probability close to mu_s in entropy and Fisher
+information cannot have a large mean. A nonzero mean would force a
+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.
+