Song–Zhang v1: the iteration of curvature profiles

Part of the first version of Song–Zhang, Chapter Song–Zhang, first version: polynomial estimates and curvature; the reading order is on the full proofs page.

Overview. This reconstructs Section 6 of Song & Zhang, 2026. First a curvature bound is extended to affine normalizations of possibly nonsmooth localization posteriors. Next the derivative hierarchy gives new polynomial coefficients. Retaining its zero initial data makes the additional coefficient loss tend to one at large depth. Finally the comparison Theorem 7.2 closes a finite-depth induction, with every initialization and admissibility threshold paid explicitly.

Extending a regular curvature profile¶

We record elementary estimates used uniformly in the depth. Concavity and $g(0)=1$ imply $g(cx)\le c g(x)$ for $c\ge1$; induction gives

For the last assertion, $e+2308d^2\le2309(e+d)^2$ and $\log2309<8$ give $g(2308d^2)\le10g(d)$; apply the scaling estimate to the remaining compositions. Furthermore

Indeed, setting $t=e+x$, the required inequality is $t\log t-2t+2e\ge0$; its derivative is $\log t-1\ge0$ for $t\ge e$, and its value at $e$ is $e$. Integration of this logarithmic derivative and composition yield

A coarse coefficient improvement¶

Suppose for some $r\ge1$ and $\Gamma\ge1$ we have the curvature profile $F(a)=\Gamma^2\ell_r(a^{-1})^2$ for every regular covariance contraction. Write $\ell=\ell_r$, $b_0=1$, and $b_s=\ell(s)^s/(s+1)^2$ for $s\ge1$. Monotonicity gives

The ratio statement includes $d=1$ by the definition of $b_0$. For the convolution sum, replace all logarithmic factors by $\ell(s)$ and split the index set at $s/2$. The reciprocal square from the larger index contributes at most $4/(s+1)^2$; summing the other reciprocal squares on each half gives a bound smaller than 16, including the endpoint $s-k=0$.

We prove, simultaneously for all isotropic log-concave measures,

Here $c_d=\sqrt{K_d}/d!$ has the Appell normalization of Theorem 7.1. Degree one follows from $c_1=1$ and $R\ge4$. For the induction step take a compactly supported isotropic initial law and $f=P_d[T]$, $T\ne0$. Use the global covariance-adapted localization Lemma 120.1 and its derivative hierarchy Lemma 120.2, both proved in the polynomial dossier. Explicitly, $A_t$ is the posterior covariance, $\Lambda_t=\int_0^tA_s^{-1}ds$ is its curvature matrix, $h_j(t)=\mathbb E_tD^jf$, and

The hierarchy gives $N_j'\le4j^2N_j+9j^2L_j$ almost everywhere. The lower-degree inductive bounds apply to every whitened posterior. The Appell expansion, followed by the $L^2$ triangle inequality over its full output tensor direct sum and over localization randomness, gives

Each additional derivative index carries the same covariance factor as the original derivative indices; this is an affine change of variables, not a scalar covariance bound.

Put $Q=(d!)^2\|T\|_{\rm HS}^2$, $w_s=R^{2s}b_s^2$, and $M(t)=\max_{1\le j\le d}N_j(t)/(Qw_{d-j})$. Appell centering implies $N_d(0)=Q$, $N_j(0)=0$ for $j<d$, and $M(0)=1$. Equations (3),(5) give $L_j\le256Qw_{d-j}M$, with $L_d=0$. Integrating the hierarchy gives $M(t)\le1+2308d^2\int_0^tM(s)ds$, hence $M(t)\le e^{2308d^2t}$. At $\tau=1/(2308d^2)$, for $0\le s\le\tau$,

Here are the variance and terminal-transfer identities needed in both coefficient inductions. The localization martingales for $f,f^2$ give $V'(t)=-\mathbb E_{\rm loc}|\operatorname{Cov}_t(f,\xi_t)|^2\ge-V(t)$, where $V(t)=\mathbb E_{\rm loc}\operatorname{Var}_t f$ and $\xi_t=A_t^{-1/2}(X-m_t)$ is isotropic. Bessel’s inequality supplies the last bound. Therefore $V(\tau)\ge e^{-\tau}\operatorname{Var}_0f$. Also

To verify (7) test on $v$ and expand $\int_0^\tau|A_s^{1/2}v-\tau A_s^{-1/2}\Lambda_\tau^{-1}v|^2ds$. The result is $\int_0^\tau v^TA_svds-\tau^2v^T\Lambda_\tau^{-1}v$. For $s\le\tau$, the posterior martingale of each fixed product $\partial_if\partial_jf$ and the $\mathcal F_s$-measurability of $A_s$ give

All quantities are integrable because the initial support is compact. The profile-inflation lemma at time $\tau$, together with (7),(8), consequently gives for any $\delta>0$

This identity retains the correlation between the earlier covariance and the terminal derivative products.

Choose $\delta=\tau$. Equations (1),(6),(9) bound the variance by $2e^2\cdot257\cdot100\Gamma^2\ell_r(d)^2 Q R^{2d-2}b_{d-1}^2$. By (3) this is at most $Q R^{2d}b_d^2$ because $16\cdot2e^2\cdot257\cdot100<2^{24}=R^2/\Gamma^2$ (use $e^2<8$ for the strict comparison). This proves (4) at degree $d$ for compact initial laws. Condition any isotropic law on increasing centered balls, then center and whiten. All fixed moments converge, as do the recursively specified Appell coefficients, so the degree-$d$ inequality passes to the limit. This completes induction over all measures. Finally, a centered covariance contraction inherits the estimate from its isotropic whitening, since the map $T\mapsto(\operatorname{Cov}\nu)^{\otimes d/2}T$ contracts Hilbert--Schmidt norm. Singular covariances are treated on their affine support. Thus (4) holds for every centered covariance contraction as well.

The coefficient improvement with summable additional loss¶

Closing the depth induction¶

Dependencies and fences. The argument uses Lemma 7.1, Theorem 7.1 (including its proved localization and hierarchy lemmas), and Theorem 7.2. Brascamp--Lieb is an established literature input Brascamp & Lieb, 1976Bakry et al., 2014, Theorem 4.9.1. There are no bounded_by edges on this node. In the brief’s threshold test, (14) is the explicit discharge for an exponentially growing sequence; it is not a construction of an admissible bounded sequence. The argument neither proves CMH nor provides universal-time occupation estimates. No infinite-depth limit or dimension-free KLS conclusion is taken.