Song–Zhang, second version: technical estimates - The KLS theorem and its methods

The estimates in this chapter explain how the bounds of Chapter Song–Zhang, second version: repeated refinement with summable losses are realized by actual functions. Three issues recur: removing means loses norm, normalization changes energy, and the derivative tensors are only partly symmetric. These losses must be controlled for the same family of functions. Separate estimates achieved by unrelated functions would not supply the iteration.

Static transfer and tensor symmetries¶

Localization transfers a coefficient cap valid simultaneously in every degree to a better cap. The starting threshold below is independent of the size of that cap, which is essential when the cap is improved repeatedly.

Proposition 10.1 (Transfer of a static coefficient cap — Proved (from a preprint) · agent review (gpt-6-astra, 2026-10-06) · prop:sz-v2-static-coefficient-transfer · Correction)

There is a universal integer $r_*\ge2$ with the following property. Fix a dimension range consisting either of all positive integers or of $1,\ldots,n$ . Let $r\ge r_*$ and $\Gamma\ge1$ , and use $\ell_r$ from Theorem 9.1. Suppose that every measure in Definition 9.1 in this range satisfies, for every admissible lower curvature bound $a>0$ and every integer $k\ge1$ ,

c_k(\mu)\le[\Gamma\ell_r(a^{-1})]^{k-1}.

(10.1)

Then every centered log-concave probability measure in the same dimension range, with covariance at most $I$ , satisfies

c_d(\mu)\le[(1+r^{-2})\Gamma\ell_r(d)]^{d-1}\qquad(d\ge1).

(10.2)

The threshold $r_*$ is independent of $\Gamma$ , the dimension range and the law.

The factor $1+r^{-2}$ is summable over the inner depths. The transfer retains the exponent $d-1$ from the common-radius normalization and passes to general log-concave laws only after obtaining a uniform regular-law estimate. The next algebraic estimate controls a tensor by one symmetric component and its failures of symmetry on successively larger groups of slots.

Lemma 10.1 (Joint partial symmetrization — Proved (from a preprint) · agent review (gpt-6-astra, 2026-10-06) · lem:sz-v2-joint-frame · Correction)

Let $d\ge1$ be dyadic and let a finite-dimensional real Hilbert space carry an orthogonal representation of the symmetric group on $2d$ letters. For each dyadic $k\le d$ , let $\mathsf P_k$ average permutations of the first $k$ letters. For dyadic $k<d$ , let $\mathsf S_k$ average permutations of the next $3k$ letters, numbered $k+1,\ldots,4k$ . Then every vector $T$ satisfies

\|T\|^2\le10^4\left[d\|\mathsf P_dT\|^2+ \sum_{\substack{k<d\\k\text{ dyadic}}} k^2\|(I-\mathsf S_k)\mathsf P_kT\|^2\right].

(10.3)

This includes tensor-slot permutations with arbitrary finite output direct sums.

The dyadic groups let the proof charge an error to the scale where symmetry first fails. Finite output direct sums are included, so the same estimate applies to windows of several inverse-gradient iterates. Normalization supplies a second useful cancellation: an antisymmetric covariance term is paid by a deficit between two spectral energies.

Lemma 10.2 (Skew loss paid by normalization — Proved (from a preprint) · agent review (gpt-6-astra, 2026-10-06) · lem:sz-v2-skew-credit · Correction)

Let $\mu$ be a measure in Definition 9.1, with diffusion operator $H$ . Let $U\in H^1(\mu;\mathbb R^n)$ be centered, with $\|U\|_{L^2(\mu)}=1$ and symmetric $\mathbb E_\mu\nabla U$ . Define

e=\|H^{1/2}U\|_2^2,\qquad \beta=\|H^{-1/2}U\|_2^{-2},\qquad LU=\mathbb E_\mu[X\otimes U],

(10.4)

where inverse operators act componentwise and $\operatorname{Skew}(M)=(M-M^{\mathsf T})/2$ . Then

\beta\|\operatorname{Skew}(LU)\|_{\mathrm{HS}}^2\le1-\beta/e.

(10.5)

The nonnegative difference $1-\beta/e$ measures the slack in the comparison between direct and inverse spectral energies. Using it to pay the skew term avoids charging normalization and antisymmetry independently. This cancellation enters the compensated restart below.

Exact operator identities and normalized losses¶

Lemma 10.3 (Restricted operator, orbit defects, and compensated normalization — Proved (from a preprint) · agent review (gpt-6-astra, 2026-10-06) · lem:sz-v2-operator-block-primitives · Correction)

Let $\mu$ be a measure in Definition 9.1, with $D^2W\succeq aI>0$ . Let $H$ be its gradient-form operator on centered $L^2(\mu)$ , $B=H^{-1}$ , $Lh=\mathbb E[Xh]$ , $P_+h=h-\mathbb Eh$ , $D=P_+\nabla H^{-1/2}$ , $\mathcal T=P_+\nabla B$ , and $\lambda=\CP(\mu)^{-1}$ . Operators extend componentwise to finite families and append ordered derivative slots. Define

R=\sup_{\substack{f\in H^1(\mu),\ f\text{ nonconstant}\\\mathbb E\nabla f=0}} \operatorname{Var}(f)/\mathbb E|\nabla f|^2.

(10.6)

Then $\mathcal T$ is compact, $\mathcal T^*\mathcal T=B-L^*L$ , $\|\mathcal T\|^2=R$ , $R\le \CP\le R+1$ , and $\|H^{1/2}\mathcal Th\|\le\|h\|$ .

For a centered unit $f$ , $z>0$ , $w_j=z^{-j/2}\mathcal T^jf$ , $b_j=\|w_j\|^2$ , $l_j=\|Lw_j\|^2$ , and $D_j=\|\nabla(B-zI)w_j\|^2$ ( $j\ge1$ ),

D_j\le z(b_{j+1}-2b_j+b_{j-1})+l_j,

(10.7)

w_{j+1}=\sqrt zP_+\nabla w_j+z^{-1/2}P_+\nabla(B-zI)w_j.

(10.8)

The swap of the newest two derivative slots in $w_{j+1}$ has norm at most $2\sqrt{D_j/z}$ . Subsequent maps propagate it by $\mathcal T/\sqrt z$ .

For a centered unit form-domain vector family $U$ with symmetric $\mathbb E\nabla U$ , of energy $e=\|H^{1/2}U\|^2$ , put $F=\sqrt\beta B^{1/2}U$ , $\beta=\|B^{1/2}U\|^{-2}$ . If $e^{-1}>\|\operatorname{Sym}(LU)\|^2$ , centering $DF$ , inverse-normalizing, and rescaling to unit norm produces a family of energy at most $[e^{-1}-\|\operatorname{Sym}(LU)\|^2]^{-1}$ .

The restricted operator identifies the energy scale after means are removed. Its orbit defect satisfies a discrete second-difference inequality, while swapping adjacent derivative slots costs the square root of that defect. The final normalization identity uses the skew credit to retain only the symmetric covariance loss. These identities connect the operator norm, tensor symmetry and energy on the same orbit.

Lemma 10.4 (Normalized hierarchy and exact orbit-window losses — Proved (from a preprint) · agent review (gpt-6-astra, 2026-10-06) · lem:sz-v2-normalized-hierarchy · Correction)

Use the operator notation of Lemma 10.3. Start with a centered unit form-domain family $F_0$ of energy $\nu$ . Let $u^j=DF_j$ , and when $u^j\ne0$ define

\beta_{j+1}=\|u^j\|^2/\|B^{1/2}u^j\|^2,\qquad F_{j+1}=\sqrt{\beta_{j+1}}B^{1/2}u^j,\qquad \chi_j=\|H^{1/2}u^j\|^2-\beta_{j+1}\|u^j\|^2.

(10.9)

After a zero successor set all later families to zero, $\beta=\lambda$ , $\chi=0$ . Write $v_j=\|F_j\|^2$ , $e_j=\|H^{1/2}F_j\|^2$ , $p_j=\|LH^{1/2}F_j\|^2$ , and $P_N=\sum_{j<N}p_j$ , $X_N=\sum_{j<N}\chi_j$ , $V_N=\sum_{j<N}v_j$ . Then

v_{j+1}=v_j-p_j,\quad e_{j+1}\le e_j-av_j-\chi_j, \quad 0\le p_j\le\min(v_j,e_j),\quad\chi_j\ge0,

(10.10)

\lambda\le\beta_j\le e_{j-1}/v_j\quad(v_j>0),\qquad v_N=1-P_N,

(10.11)

aV_N+X_N\le\nu-e_N,\quad v_{N+1}\le R e_N, \quad aV_N+X_N\le\nu-R^{-1}+R^{-1}P_{N+1}.

(10.12)

If $Y$ is a finite centered family and $F_0=B^{1/2}Y/\|B^{1/2}Y\|$ , then at every nonzero generation $F_k$ is a positive scalar multiple of $B^{1/2}\mathcal T^kY$ and

\frac{p_k}{v_k}= \frac{\|L\mathcal T^kY\|^2} {\langle\mathcal T^kY,B\mathcal T^kY\rangle}.

(10.13)

In particular if $Y=\bigoplus_{j=0}^{m-1}w_j$ is a window in a $z$ -normalized orbit, $S_k=\sum_{j=k}^{m+k-1}\|w_j\|^2$ and $C_k=\sum_{j=k}^{m+k-1}\|Lw_j\|^2$ , then

p_k/v_k=C_k/(zS_{k+1}+C_k).

(10.14)

Here $P_N$ records lost mass and $X_N$ records the normalization deficit. The identity $v_N=1-P_N$ makes their role explicit: a hierarchy survives as long as its accumulated centering loss stays small. For an orbit window the ratio $p_k/v_k$ is exact, not a separate upper estimate. It permits an averaged window with small covariance loss to restart an actual normalized hierarchy.

On the initial range of powers, the squared operator scale stays within $C_*/R$ of $R$ . This starts the construction without a bound derived from KLS. Longer blocks require the joint symmetry and loss estimates, which we state next.

From symmetry to delayed centering losses¶

Lemma 10.6 (Raw joint frame — Proved (from a preprint) · agent review (gpt-6-astra, 2026-10-06) · lem:sz-v2-raw-joint-frame · Correction)

Use the regular class and operators of Lemma 10.3, and set $C_F=10^4$ . Let $f$ be a finite centered $L^2$ family and $E\ge1$ . Let $z>0$ , $w_j=z^{-j/2}\mathcal T^jf$ , $b_j=\|w_j\|^2$ , $l_j=\|Lw_j\|^2$ , and $D_j=\|\nabla(B-zI)w_j\|^2$ for $j\ge1$ . If $\|\mathcal T^h\|\le E z^{h/2}$ for $h\ge0$ , then for dyadic $d\ge2$ and $j\ge2d-1$ ,

l_j\le C_Fd c_d^2z^{-(d-1)}b_{j-d+1} +108C_FE^2\sum_{k<d\ {\rm dyadic}}k^5c_k^2z^{-k} \sum_{h=1}^{3k-1}D_{j-k+1-h}. \tag{B1}

(10.16)

Without a global power bound, if $0<z\le R$ , the same formula holds with $E^2$ in the $k$ summand replaced by $(R/z)^{3k-2}$ . If instead $\|\mathcal T^h\|\le E(h+1)^\alpha z^{h/2}$ for $0\le\alpha\le1$ , (B1) holds with $E^2$ replaced by $E^2(3k)^{2\alpha}$ in its $k$ summand.

The symmetric term is controlled by the degree- $d$ coefficient; the remaining terms are orbit defects at smaller dyadic degrees. A bound on operator powers propagates each defect to the required position. The versions with exponential or polynomial propagation make explicit which power estimate is available at each stage of the construction.

Lemma 10.7 (Delayed losses under polynomial propagation — Proved (from a preprint) · agent review (gpt-6-astra, 2026-10-06) · lem:sz-v2-propagated-joint-loss · Correction)

Use the operators and hierarchy of Lemma 10.4, and let $C_F=10^4$ , $z>0$ , $E\ge1$ , and $B_*>0$ . For an actual normalized hierarchy with a centered unit starting family, suppose instead $\|\mathcal T^h\|\le E(h+1)^\alpha z^{h/2}$ , where $0\le\alpha\le1$ . Let $2\le d_0\le d$ be dyadic and $J=2d_0-1$ . On any prefix whose required normalizers are at most $B_*$ put $t_* =\max\{1,B_*z\}$ and

\theta=5184C_FE^2\sum_{k<d\ {\rm dyadic}} k^8(B_*t_*^3)^k c_k^2,

(10.17)

\Delta=P_J+2C_F\sum_{d_0\le k<d\ {\rm dyadic}}k^2B_*^kc_k^2, \qquad \tau=C_FdB_*^dc_d^2.

(10.18)

Then, on each such justified prefix,

P_N\le\Delta+N\tau+(\theta/\lambda)X_{\max\{N-2,0\}}. \tag{B2}

(10.19)

For a prefix shorter than $J$ , its retained actual losses suffice.

Summing the raw estimate along the normalized hierarchy separates three contributions: actual initial losses $\Delta$ , a terminal-degree cost $N\tau$ , and delayed normalization deficits. The delay matters because a first-crossing argument may use only estimates justified before the crossing. Short prefixes keep their actual losses rather than being silently discarded.

Restarting and extending finite blocks¶

Lemma 10.8 (Green control and the actual averaged restart — Proved (from a preprint) · agent review (gpt-6-astra, 2026-10-06) · lem:sz-v2-orbit-green-restart · Correction)

The following two assertions hold.

Sequence estimate. Fix $K\geq1$ . There are $z_0(K)$ , $c_0(K)>0$ and $C(K)$ with the following property. Let integers $m\geq2d\geq4$ and numbers $z\geq z_0$ , $\gamma\geq0$ satisfy $\gamma m^2/z\leq c_0$ . Suppose nonnegative sequences $b_j$ ( $j\geq0$ ), $l_j,D_j$ ( $j\geq1$ ) obey

b_0=b_m=1,\quad b_{j+m}\leq b_j,\quad b_{j+1}\leq(1+2/z)b_j,\quad l_1\leq2,

(10.20)

and

D_j\leq z(b_{j+1}-2b_j+b_{j-1})+l_j.

(10.21)

Let $a_s\geq0$ have finite support and satisfy $\sum_sa_s\leq K/z$ , $\sum_ssa_s\leq K/z$ . Let $h_j\geq0$ . With $D_j=0$ for $j\leq0$ , assume, for $j\geq2$ ,

l_j\leq h_j+\sum_{s\geq1}a_sD_{j-s},\qquad \sum_{j=2}^{2d-2}j h_j\leq K/z,\qquad h_j=\gamma b_{j-d+1}\quad(j\geq2d-1).

(10.22)

Then $1/2\leq b_j\leq2$ for $0\leq j\leq m$ , $b_j\leq2$ for every $j\geq0$ , and for $1\leq J\leq m/4$ ,

\sum_{k=2}^{J+1}\sum_{j=k}^{m+k-1}l_j\leq C/z+C\gamma Jm.

(10.23)

Actual restart. Use $H$ , $B=H^{-1}$ , $Lf=\mathbb E[Xf]$ , $\mathcal T=P_+\nabla B$ and the finite-family normalized hierarchy of Lemma 10.4, for a centered regular log-concave law of covariance at most $I$ and curvature at least $aI$ , $a>0$ . Suppose $z=\|\mathcal T^m\|^{2/m}>0$ and a unit norm-attaining vector $f$ gives $w_j=z^{-j/2}\mathcal T^jf$ , $b_j=\|w_j\|_2^2$ , and $l_j=\|Lw_j\|^2$ . Assume

b_{j+m}\leq b_j,\quad b_0=b_m=1,\quad 1/2\leq b_j\leq2\ (0\leq j\leq m),\quad b_j\leq2\ (j\geq0),

(10.24)

\sum_{j=2}^m l_j\leq K/z+K\gamma m,\qquad \sum_{k=2}^{J+1}\sum_{j=k}^{m+k-1}l_j\leq K/z+K\gamma Jm \quad(1\leq J\leq J_0),

(10.25)

and $\|\operatorname{Sym}(Lw_1)\|^2\leq K/z$ . Here $1\leq J_0\leq m/4$ and $m\geq4$ . Let $p\geq3$ be odd, $\kappa\geq1$ , and assume

{z^{p-2}\over4\kappa}\leq m\leq {4z^{p-2}\over\kappa}, \qquad J_0\gamma zm\leq1.

(10.26)

There is $C=C(K)$ such that, if $C\kappa z^{-p}\leq1/8$ , one centered unit starting family for the normalized inverse-gradient hierarchy has energy $\nu$ and actual centering losses satisfying

\nu\leq{u\over1-C\kappa u^p},\qquad P_J\leq C\kappa u^p\quad(1\leq J\leq J_0),\qquad u=z^{-1},

(10.27)

and the same family satisfies for every $N\geq0$

aV_N+X_N\leq\nu-u+uP_{N+1}.

(10.28)

The discrete Green estimate turns small forcing and controlled orbit defects into bounds on every shifted window. Averaging these windows provides one family with small centering losses. The compensated normalization then gives that family’s energy bound. The matched inequality for $aV_N+X_N$ is retained at every later length, so the restart can be used in the next block without changing the family.

Lemma 10.9 (Finite block extension — Proved (from a preprint) · agent review (gpt-6-astra, 2026-10-06) · lem:sz-v2-block-extension · Correction)

Use the same operators and hierarchy as in Lemma 10.8. Let $q\geq3$ be odd, $a_q,s_q\geq0$ , $C_F,C_\theta\geq1$ , $z>0$ , $u=z^{-1}$ , $\lambda\geq u/2$ , and suppose an actual centered unit hierarchy starts in direction $B^{1/2}Y$ , with energy $\nu\leq u/(1-a_qu^q)$ and matched budget $X_M\leq\nu-u+uP_{M+1}$ for every $M$ . Fix $\kappa_q,\kappa_{q-2}\geq1$ and define

K_q=4(s_q+C_F\kappa_{q-2}+4C_\theta a_q+1),\quad p_*=K_qu^q,\quad T_*=\lfloor z^q/\kappa_q\rfloor.

(10.29)

Assume $p_*\leq1/1024$ , $a_qu^q\leq1/1024$ , $z\geq8C_\theta$ , and the following delayed estimate is valid before, and at, a first crossing of $p_*$ , including all short prefixes:

P_N\leq s_qu^q+N\tau+{\theta\over\lambda}X_{\max\{N-2,0\}}, \qquad \tau\leq C_F\kappa_q\kappa_{q-2}u^{2q},\quad \theta\leq C_\theta u.

(10.30)

Then $P_N<p_*/2$ for every $N\leq T_*$ , and

\|\mathcal T^m\|^{2/m}\geq z-(a_q+4K_q)z^{1-q} \qquad(1\leq m\leq T_*).

(10.31)

Assume that the centering loss first reaches $p_*$ . Before that time, the delayed estimate is valid and the matched energy budget controls its normalization term. The resulting bound is below $p_*/2$ , a contradiction. Thus the hierarchy retains mass to length $T_*$ , and its survival gives a lower bound on every operator power up to that length.

Lemma 10.10 (Propagation from exact finite block norms — Proved (from a preprint) · agent review (gpt-6-astra, 2026-10-06) · lem:sz-v2-block-propagation · Correction)

Let $T$ be a bounded operator on a Hilbert space (or a compatible graded family), $R=\|T\|^2$ , and integers $1<m_0<m_1<\cdots<m_s$ satisfy $m_{l+1}\geq4m_l$ . Put $z_l=\|T^{m_l}\|^{2/m_l}>0$ . Let $\Delta_l,\eta,\eta_0\geq0$ . Suppose

z_l\leq R,\quad z_l\geq R/2,\quad z_{l+1}\geq z_l-\Delta_l,\quad \Delta_{l+1}\leq\Delta_l/2,

(10.32)

and $m_{l+1}\Delta_l/z_l\leq\eta$ for every $l<s$ . If $m_0(R-z_s)/z_s\leq\eta_0$ , then for all $h\geq0$ ,

\|T^h\|\leq z_s^{h/2} \exp\left({\eta_0\over2}+2\eta\min\{s,\log_4(h+1)+1\}\right).

(10.33)

Exact norms at a sequence of increasing block lengths control arbitrary powers by decomposing the length among those blocks. Geometric growth of the lengths and geometric decay of the scale losses prevent a cost proportional to the full length. This gives the polynomial propagation needed when the construction returns to the joint-frame estimate.

Retaining coefficient caps and controlling terminal degrees¶

Proposition 10.2 (Finite chains of coefficient floors — Proved (from a preprint) · agent review (gpt-6-astra, 2026-10-06) · prop:sz-v2-finite-chain-blocks · Correction)

Let $G_*:[1,\infty)\to[1,\infty)$ be a fixed universal coefficient seed. Assume it is nondecreasing, $G_*\geq1$ , $G_*(k)^2\leq C_*k$ , and $G_*(R^R)^2\leq R/2$ for all sufficiently large $R$ , with fixed universal constants and a fixed universal threshold. The constants below can be chosen depending only on this fixed seed. Let $N\ge1$ be an integer and let $G,H_0,\ldots,H_{N-1}:[1,\infty)\to[1,\infty)$ be nondecreasing valid coefficient majorants with $1\leq G\leq H_l\leq G_*$ and $c_k\leq G(k)^{k-1}$ . There are positive universal constants $C_{\rm cut},C_{\rm deg},C_X,C_0,C$ with the following property. Fix $0<\epsilon=\alpha_N\leq\alpha_{N-1}\leq\cdots\leq\alpha_0\leq1/16$ . Put

K_l=\lceil C_{\rm cut}\alpha_l^{-2}\rceil,\quad \Xi_l=\lceil4C_{\rm deg}\alpha_l^{-1}K_{l+1}\rceil, \quad X=C_XQ^2,

(10.34)

F^2=\max\{G(X^X)^2,C_0G_*(16K_0)^2,C_0, (1+\alpha_l)H_l(\Xi_l)^2:0\leq l<N\}, \quad H_Q=(1+2\epsilon)F^2.

(10.35)

Here $Q\geq3$ is odd. Use the regular measure class and operators of Definition 9.1. The joint-frame inputs are Lemma 10.6 and Lemma 10.7, and the initial power estimate is Lemma 10.5. There is an assignment $Z_Q$ with

H_Q\leq Z_Q\leq\max\{H_Q,R\},\qquad \mathcal A\leq Z_Q,

(10.36)

and constants $a_Q,s_Q\geq0$ obeying

a_Q+s_Q+1\leq C(F^2/4)^{Q-3}\leq F^{2Q}.

(10.37)

If $Z_Q>H_Q$ , write $z=Z_Q$ , $u=z^{-1}$ . One actual centered unit starting family has energy and losses satisfying

\nu\leq{u\over1-a_Qu^Q},\qquad P_J\leq s_Qu^Q\quad(1\leq J\leq J_Q),\qquad aV_M+X_M\leq\nu-u+uP_{M+1}\quad(M\geq0),

(10.38)

and

\|\mathcal T^h\|\leq4(h+1)z^{h/2}\quad(h\geq0),\qquad 0\leq R-z\leq\min\{1,C/R\},\qquad a_Qu^Q\leq1/8.

(10.39)

For each odd $3\leq p\leq Q$ , choose $d_p,e_p$ to be the least dyadic integers at least $4p,8p$ when $p\leq K_0$ . When $K_l<p\leq K_{l+1}$ choose both to be the least dyadic integer at least $C_{\rm deg}p/\alpha_l$ ; when $p>K_N$ use $\epsilon$ instead. Set $J_p=4e_p$ . Constants are independent of $N$ , the margins, majorants and order. The assertion about an actual family is made only above the floor.

The coefficient caps cover separate degree ranges, chosen by the margins $\alpha_l$ . Their contributions enter the common floor $F^2$ through a maximum, rather than a sum over all retained caps. Above that floor, repeated restarts and block extensions produce one actual family with the stated energy, centering and propagation bounds. Below the floor, the coefficient-radius estimate is already sufficient; no such family is asserted there. Uniformity in the number of caps is the feature needed by the summable-cost induction.

Lemma 10.11 (Terminal-degree logarithmic distortion — Proved (from a preprint) · agent review (gpt-6-astra, 2026-10-06) · lem:sz-v2-terminal-distortion · Correction)

Use $t$ and $\ell_r$ from Definition 9.2. Let $\eta=(e+1)^{-1}$ . For $M,x\ge1$ and integer $r\ge1$ ,

0\le\ell_r(Mx)-\ell_r(x) \le [v\mapsto\log(1+\eta v)]^{\circ(r-1)}(\log M). \tag{B22}

(10.40)

For $r\ge1+\log^*M$ the right side is at most $2\eta^{r-1-\log^*M}$ . Consequently, for fixed $C_T\ge1$ , sufficiently large universal $C_b$ and $r_*\ge2$ , set $r_Q=\max\{r_*,\lceil C_bt(Q)\rceil\}$ . For odd integers $Q\ge3$ , $0<a\le1$ , integers $r\ge r_Q$ , and $1\le d<2C_TQ^2(r+1)^2\log(e+a^{-1})$ ,

\ell_r(d)\le\zeta_r\ell_{r+1}(a^{-1}),\qquad \zeta_r=1+2\eta^{r/2}. \tag{B23}

(10.41)

The products $\prod_{r=r_Q}^R(1+r^{-2})^2\zeta_r^2$ have a universal upper bound, independent of $Q$ and the finite terminal depth $R$ .

Each additional logarithm contracts a multiplicative perturbation of the argument. After a number of logarithms comparable to $\log^*M$ , the remaining distortion decays geometrically. Applied to the terminal polynomial degree, this compares its logarithms with those of inverse curvature. The products of these distortions and the static-transfer factors stay bounded independently of the terminal depth.

The outer round with retained caps¶

The finite-chain estimate above, with the terminal-degree distortion, gives one outer round of the refinement used in Chapter Song–Zhang, second version: repeated refinement with summable losses: a radius bound yields a coefficient cap, and the cap, together with earlier ones kept on their own degree ranges, yields an improved radius bound.

Lemma 10.12 (Outer round with a retained profile — Proved (from a preprint) · agent review (gpt-6-astra, 2026-10-06) · lem:sz-v2-profile-refinement · Correction)

Fix a universal valid coefficient seed $G_*$ satisfying the seed conditions of Proposition 10.2, with its fixed cutoff constants. Use the functions in Definition 9.2, and set $\widehat W_{m,\delta}(x)=\max\{W_m(x),\bar t(\delta^{-1})\}$ . All radius bounds below are uniform over every dimension and every measure in Definition 9.1, for every admissible lower curvature $a>0$ . There are universal $A_0,C_R,C_Q,C_b,C_b',C_r,D_r\ge1$ such that the following holds for every finite $S_*\ge1$ (the additional allowance $b_*$ may depend on $S_*$ ). For each integer $m\ge0$ and $0<\delta\le1/16$ , suppose $V=W_m$ or $V=\widehat W_{m,\delta}$ , $1\le S\le S_*$ , $A\ge A_0$ , $R_\delta=\lceil C_R\delta^{-12}\rceil$ , and

\mathcal A\le A[V(r)+S]^{1/3}\mathcal L_r(a^{-1})^2 \qquad(r\ge R_\delta).

(10.42)

Then the following function is a valid coefficient cap, meaning $c_k\le H(k)^{k-1}$ for every centered log-concave covariance contraction:

H(x)^2=\min\{G_*(x)^2,e^{3\delta}A[V(r_\delta(x))+S]^{1/3}\}, \quad r_\delta(x)=\max\{R_\delta,\lceil C_rt(x)\rceil+ \lceil D_r/\delta\rceil\}.

(10.43)

Optionally retain earlier caps, with margins $\delta=\alpha_{N-1}\le\cdots\le\alpha_0\le1/16$ , where $N\ge1$ is finite and the newest cap is $H_{N-1}=H$ . For every $0<\epsilon\le\delta$ , put $\alpha_N=\epsilon$ , $K_l=\lceil C_{\rm cut}\alpha_l^{-2}\rceil$ and $\Xi_l=\lceil4C_{\rm deg}\alpha_l^{-1}K_{l+1}\rceil$ . Require $(1+\alpha_l)H_l(\Xi_l)^2\le A(4+S)^{1/3}$ for $l<N-1$ , and $\max\{C_0G_*(16K_0)^2,C_0\}\le A(4+S)^{1/3}$ . All $H_l$ are nondecreasing valid coefficient majorants between 1 and $G_*$ for every centered log-concave covariance contraction. With a fixed sufficiently large $b_*$ with $b_*\le C\log(e+S_*)$ for one universal $C$ , put

D_*=V(\epsilon^{-1})+S+b_*4^{-m},\qquad0<\epsilon\le\delta.

(10.44)

Then, for every $j\ge1$ , odd $Q\ge \min\{q\ge C_Q\epsilon^{-4}:q\text{ odd}\}$ , and $r\ge r_0(Q):=\max\{R_\delta,\lceil C_bt(Q)\rceil+ \lceil C_b'/\epsilon\rceil\}$ , $$ \mathcal A\le e^{16\delta+24\epsilon j}A \max{t_j(Q),D_*}^{1/3}(r+1)^{1/Q}\mathcal L_r(a^{-1})^2.

$ $For the one-cap case, the original-seed floor can instead be bounded by$ e^{4\delta}A D_*^{1/3}$. The same conclusion holds.

The radius bound first gives a coefficient cap through static transfer. One then divides degrees among retained caps and applies the finite-block construction with the corresponding margins. Terminal-degree distortion controls the logarithms introduced by localization. The factors $e^{16\delta}$ and $e^{24\epsilon j}$ record distinct costs: forming the new cap and repeating the inner improvement. Their explicit dependence on both margins is what permits a later summable choice.

Scalar control of changing starting depths¶

Lemma 10.13 (Scalar calculus of the contracted height profiles — Proved (from a preprint) · agent review (gpt-6-astra, 2026-10-06) · lem:sz-v2-profile-calculus · Correction)

Use the functions of Definition 9.2. The maps $\bar t,\bar\chi$ are nondecreasing and $1/4$ -Lipschitz on $[1,\infty)$ . For every integer $m\geq0$ , $W_m\geq4$ , $W_m(4)=4$ , and $W_m$ is nondecreasing and $4^{-m-1}$ -Lipschitz. For every $x\geq1$ some finite $m$ satisfies $W_m(x)=4$ . For fixed real $C,p\geq1$ there is $b_{C,p}<\infty$ such that, uniformly in $m\geq0$ and $x\geq1$ ,

W_m(Cx^p)\leq W_m(x)+b_{C,p}4^{-m}.

(10.45)

There is a universal $b$ such that for all $x,y\geq1$ and $m\geq0$ ,

\max\{W_m(xy),W_m(x+y)\} \leq\max\{W_m(x),W_m(y)\}+b4^{-m}.

(10.46)

There are universal $C_0,D_0,M<\infty$ such that, for $x\geq1$ , $0<\eta\leq1/16$ and every integer $r\geq C_0t(x)+D_0/\eta$ ,

\mathcal L_r(x)^2\leq e^\eta,\qquad \mathcal L_r(0)^2\geq e^{-\eta},\qquad (1+r^{-2})^2\leq e^\eta.

(10.47)

For every integer $r\geq C_0t(x)$ , $\mathcal L_r(x)\leq M$ .

The averaging in the definition makes each height map Lipschitz. Iterating its contraction reduces perturbations by $4^{-m}$ , while fixed powers, products and sums of arguments change the height by only this order. Separately, iterated logarithms approach their positive fixed point, allowing both upper and lower normalization estimates with arbitrarily small multiplicative loss.

Lemma 10.14 (Absorption of growing starting depths — Proved (from a preprint) · agent review (gpt-6-astra, 2026-10-06) · lem:sz-v2-profile-threshold · Correction)

Use Definition 9.2 and put $\alpha_i=2^{-i}/16$ for integers $i\geq0$ . For every fixed $C,p\geq1$ there is $b_{C,p}<\infty$ such that

W_i(C\alpha_i^{-p})\leq4+b_{C,p}2^{-i}\qquad(i\geq0).

(10.48)

For every fixed $C_R,C\geq1$ , set $R_i=\lceil C_R\alpha_i^{-12}\rceil$ . There is $b<\infty$ , depending only on $C_R,C$ , such that for all $i\geq0$ and $x\geq1$ ,

W_i(R_i+\lceil Ct(x)\rceil)\leq W_i(x)+b2^{-i}.

(10.49)

First apply the fixed-power estimate to $C(16\cdot2^i)^p$ , then the contraction estimate at $2^i$ . This reduces the growing threshold to a summable error of order $2^{-i}$ . The sum estimate then absorbs the additional logarithmic starting depth. This is the scalar estimate used in the final choice of parameters for each fixed measure.