Song–Zhang v2: fixed-cost repeated height reduction

Part of the second version of Song–Zhang, Chapter Song–Zhang, second version: repeated refinement with summable losses; the reading order is on the full proofs page.

Overview. This reconstructs the fixed-cost height reduction in Section 8 of Song & Zhang, 2026. Polynomial testing and a joint tensor frame give delayed loss estimates. Actual norm-attaining orbits then produce starting families at arbitrarily high odd orders. We use a fixed, enlarged radius floor in this section; approaching the seed radius with loss tending to zero is a separate issue in Section 9. A static radius is iterated through all inner depths before its conversion to the Poincaré constant.

Dependencies. We use Lemma 10.1, Lemma 10.3, Lemma 10.4, Lemma 10.5, Proposition 10.1, Theorem 9.1, and Proposition 9.1. Generic sequence and restart lemmas used below have no height-reduction or small-loss profile as a premise. There is no use of BKL, KLS, or a bound derived from either.

Throughout, the measure is centered and regular, with covariance at most $I$ and $aI\preceq D^2W\preceq bI$ for $0<a\le b<\infty$ . Use the operators and hierarchy of Lemma 10.3 and Lemma 10.4; put $R=\|\mathcal T\|^2$ and $\lambda=C_P^{-1}$ . Every tensor norm includes all ordered output indices. Write $C_F=10^4$ , $g(x)=\log(e+x)$ , $\ell_r=g^{\circ r}$ , $t(x)=1+\log^*(x+2)$ and $t_j=t^{\circ j}$ . Constants denoted $C$ may increase only by universal factors, unless their displayed arguments specify fixed seed-comparison constants.

Joint-frame interfaces¶

We use the independently reconstructed Lemma 10.6 and Lemma 10.7. Their raw estimate is denoted (B1) below and their delayed normalized loss estimate is denoted (B2).

Seeds and elementary bounds uniform in the height stage¶

Lemma 143.1 (Capped height seeds)

There is a universal $C_6\ge1$ for which $G_*(x)=C_6t(x)^{1/6}$ satisfies $c_d\le G_*(d)^{d-1}$ for every centered log-concave covariance contraction and every $d\ge1$ . For $j\ge1$ , $A\ge1$ , let $G(x)=\min\{G_*(x),At_j(x)^{1/6}\}$ and $g_0=G^2$ . The same conclusions below hold for $G=G_*$ , with constants independent of $j,A$ :

G(Cx^p)\le C_{C,p}G(x),\quad G(k)^6\le C_0k, \tag{B4}

(143.1)

g_0(R^Q)\le C g_0(8Q)+C G_*(R)^2\quad(R,Q\ge3), \tag{B5}

(143.2)

g_0(C(Qs)^2)\le C_Cg_0(Q)s^{1/Q}\quad(Q\ge3,s\ge1),\tag{B6}

(143.3)

2C_F2^Q\sum_{k\ge Q}k^2g_0(k)^{Q-1}4^{-(k-Q)} \le[C_1g_0(Q)]^Q. \tag{B7}

(143.4)

Proof 143.1

The inner curvature theorem gives $C_P\le C(r+1)^{1/3}\ell_r(a^{-1})^2$ . Repeated Poincaré on Appell derivatives, starting with $c_1\le1$ , gives $c_k\le C_P^{(k-1)/2}$ for each regular measure. Apply the static coefficient transfer at a depth $r\ge r_*$ chosen so that $r\le C't(d)$ and $\ell_r(d)\le5$ . Such a depth exists: while the ordinary logarithm iterate $y$ is at least four, $g(y+1)\le\log y+1$ , and $g$ preserves $[0,5]$ . Induction along the ordinary logarithms, followed by the fixed additional $r_*$ steps, proves both requirements. The transfer now gives $c_d\le[C''t(d)^{1/6}]^{d-1}$ uniformly over all log-concave covariance contractions, as claimed.

The log-star thresholds $E_0=1$ , $E_{k+1}=e^{E_k}$ imply $t(x+b)\le t(x)+b$ for integers $b\ge0$ ; consecutive thresholds on the domain $x\ge1$ have gaps greater than one. Induction preserves this inequality for every $t_j$ . Moreover $t(3)=3$ , $t(n)\le n$ for integers $n\ge3$ , and hence $3\le t_j(x)\le t(x)\le4x$ . For $x,y\ge1$ , $(x+2)(y+2)\le e^{\max(x+2,y+2)}$ : indeed $2\log u\le u$ for $u\ge3$ . Since both $xy+2$ and $x+y+2$ are at most this product, it follows that $t(xy),t(x+y)\le t(x)+t(y)$ . Apply this inductively to obtain the same inequalities for every $t_j$ .

For fixed $C,p\ge1$ , $Cx^p+2\le e^{x+2}$ beyond a fixed threshold; the bounded remaining interval supplies an integer $b_{C,p}$ such that $t(Cx^p)\le t(x)+b_{C,p}$ everywhere. Integer-shift stability propagates this bound through any number of compositions. Likewise $x^x+2\le\exp(\exp(x+2))$ gives $t_j(x^x)\le t_j(x)+2$ . Since all heights are at least three, both branches of $G$ satisfy (B4). The original cap gives $G(k)^6\le C_6^6t(k)\le4C_6^6k$ . For (B5) let $M=\max(R,Q)$ . Then $R^Q\le M^M$ , so $G(R^Q)\le C G(M)$ . If $M=Q$ , use $G(M)\le G(8Q)$ ; otherwise use $G(M)\le G_*(R)$ .

To prove (B6), put $y=(\log s)/Q$ and $X=CQ^2e^{2Qy}$ . There is fixed $C'$ with $\log(X+2)\le C'Q(1+y)$ . The product and fixed-argument estimates therefore give

t_j(X)\le1+t_j(C'Q(1+y)) \le1+t_j(Q)+b_{C',1}+4(1+y) \le C''t_j(Q)(1+y).

(143.5)

Both branches of $g_0$ obey the resulting cube-root bound; their minimum does too. Divide by $s^{1/Q}=e^y$ and use boundedness of $(1+y)^{1/3}e^{-y}$ .

Finally for $k\ge Q$ , the product inequality gives $t_j(k)\le t_j(Q)+4k/Q\le5(k/Q)t_j(Q)$ , hence $g_0(k)\le Cg_0(Q)(k/Q)$ for either branch and their minimum. For $\beta=Q+1$ ,

\sum_{k\ge0}k^\beta4^{-k} \le \sup_{x\ge0}(x^\beta2^{-x})\sum_{k\ge0}2^{-k} =2[\beta/(e\log2)]^\beta.

(143.6)

Substitute this estimate into (B7). All remaining factors are bounded by a universal constant to the power $Q$ : in particular $Q^{2/Q}$ is bounded for $Q\ge3$ . This proves the lemma.

Proof 143.2

For $k\le F^3$ , $\rho_k\le C_0/F\le F^{-1/2}$ after increasing $L$ , so this part is bounded by $\sum_{k\ge h+1}k^MF^{-(k-h)/2}$ , which tends to zero as $F\ge4\sqrt L$ tends to infinity. For $k>F^3$ discard $\rho_k$ , remove $e^{-c\epsilon F^3/2}$ , and bound the remaining exponential moment by $C\epsilon^{-M-1}$ using an integral over unit intervals. Since $\epsilon^{-1}\le F^2/L$ and $\epsilon F^3\ge LF$ , the tail is at most $CL^{-M-1}F^{2M+2}e^{-cLF/2}$ . The exponential dominates its polynomial uniformly on $F\ge4\sqrt L$ , proving both assertions.

Realizing an arbitrary odd block order¶

Lemma 143.3 (Local orbit and actual initialization)

Assume $G\le G_*$ is nondecreasing and bounds all coefficients. There are universal $C,c>0$ with the following properties. Let $d\ge2$ be dyadic, $m\ge2d$ , $z=\|\mathcal T^m\|^{2/m}$ , and

0\le R-z\le2,\quad z\ge\max\{4G(d)^2,C\},\quad \gamma=C_Fd c_d^2z^{-(d-1)},\quad \gamma m^2/z\le c. \tag{B8}

(143.8)

A unit norm-attaining vector of $\mathcal T^m$ gives an orbit satisfying $1/2\le b_j\le2$ for $0\le j\le m$ and $b_j\le2$ for all $j\ge0$ . It also satisfies

\sum_{j=2}^ml_j\le C/z+C\gamma m,\qquad \sum_{k=2}^{J+1}\sum_{j=k}^{m+k-1}l_j\le C/z+C\gamma Jm \quad(1\le J\le m/4). \tag{B9}

(143.9)

Suppose $p\ge3$ is odd, $\kappa\ge1$ , $d$ is the least dyadic integer at least $4p$ , and additionally

\frac{z^{p-2}}{4\kappa}\le m\le\frac{4z^{p-2}}\kappa, \quad J_0\le m/4,\quad J_0\gamma zm\le1, \quad C\kappa z^{-p}\le1/8. \tag{B10}

(143.10)

There is one centered unit starting family with $u=z^{-1}$ ,

\nu\le\frac{u}{1-C\kappa u^p},\quad P_J\le C\kappa u^p\ (1\le J\le J_0),\quad aV_N+X_N\le\nu-u+uP_{N+1}\ (N\ge0). \tag{B11}

(143.11)

Proof 143.3

Compactness in the operator primitive supplies the singular vector. Submultiplicativity gives $b_0=b_m=1$ , $b_{j+m}\le b_j$ , $b_{j+1}\le t b_j$ with $t=R/z\le1+2/z$ , and $b_j\le t^j$ before any orbit estimate. Use the local form of (B1) at the largest available dyadic degree, capped at $d$ . At degree two and index two, the three-slot frame $\|Y\|^2\le8(\|\mathsf P_2Y\|^2+ \|(I-\mathsf S_1)Y\|^2)$ gives

l_2\le(2K_2b_1+8D_1)/z.

(143.12)

That frame is immediate on the trivial and alternating representations; on the two-dimensional standard representation its lower frame eigenvalue is $1-\sqrt3/2>1/8$ . Consequently $l_j\le h_j+\sum_s\alpha_sD_{j-s}$ for $j\ge2$ , where $h_2=2K_2b_1/z$ , the later $h_j$ is the coherent term at the available degree, and the finite nonnegative kernel has $8/z$ at delay one and, for each dyadic $k<d$ , the weight $108C_Fk^5c_k^2z^{-k}t^{3k-2}$ at each delay $k,\ldots,4k-2$ .

Put $F^2=z/4$ . For $k\le d$ , $\rho_k=G(k)^2/F^2\le1$ and $\rho_k\le C k^{1/3}/F^2$ . At large universal $z$ , $t^3G(k)^2/z\le\rho_k/2$ . Thus both kernel moments are at most $C/z$ : the factors $k^6$ and $k^7$ are summable against $\rho_k^{k-1}2^{-(k-1)}$ , by the preceding moment lemma with a fixed $\epsilon$ . The case $k=1$ is bounded separately by $C/z$ . The same estimates control the early coherent terms while only $b_i\le t^i$ is available. Degree $k<d$ is used at indices $2k-1,\ldots,4k-2$ . After multiplication by $z$ , their weighted contribution $\sum jh_j$ is at most

Ck^3t^{3k-1}G(k)^{2k-2}z^{-(k-2)} =Ct^5k^3G(k)^2(t^3G(k)^2/z)^{k-2} \le Ck^4\rho_k^{k-2}2^{-(k-2)}

(143.13)

for $k\ge4$ . The degree-two terms are bounded using the fixed universal $K_2$ . Hence $\sum_{j=2}^{2d-2}jh_j\le C/z$ . At later indices $h_j=\gamma b_{j-d+1}$ .

These are exactly the hypotheses of the two-block Green estimate Lemma 10.8, using the raw defect inequality from the operator primitive and $l_1\le b_1\le2$ . Its conclusion gives the orbit bounds and the second part of (B9). The first-block telescope in its proof also gives $\sum_{j=2}^ml_j\le C/z+C\gamma\sum_{j<m}b_j\le C/z+2C\gamma m$ .

Quadratic testing gives $\mathsf P_2Lw_1=Q_2f/(2\sqrt z)$ , so $\|\operatorname{Sym}(Lw_1)\|^2\le K_2/(4z)$ . Apply the actual-restart part of Lemma 10.8, with (B9) and (B10). It uses $Y=\bigoplus_{j=0}^{m-1}w_j$ and the unit restart at its second normalized successor. Its matched budget follows from the exact window monotonicity $S_{k+1}\le S_k$ for this same orbit, not by replacing $R^{-1}$ in the generic hierarchy budget. This proves (B11).

Lemma 143.4 (Extension at a fixed seed margin)

Let $G\le G_*$ be a coefficient majorant, $q\ge3$ odd, $F\ge1$ , $d$ the least dyadic integer at least $8q$ , and $\kappa_q=F^{2q-2}$ , $\kappa_{q-2}=F^{2q-6}$ . Assume $G(d)\le F$ , $0\le R-z\le2$ , $z\ge\max\{4F^2,C\}$ , and $\|\mathcal T^h\|\le4(h+1)z^{h/2}$ for all $h\ge0$ . Suppose one centered unit hierarchy has the matched budget (B11), $\nu\le u/(1-a_qu^q)$ and $P_J\le s_qu^q$ through $J=32q$ , where $u=z^{-1}$ and $a_q+s_q+1\le C_0\kappa_{q-2}$ . If $F$ exceeds a sufficiently large constant depending only on $C_0$ , then

\|\mathcal T^m\|^{2/m}\ge z-C_b\kappa_{q-2}z^{1-q} \quad(1\le m\le\lfloor z^q/\kappa_q\rfloor), \tag{B12}

(143.14)

where $C_b$ is independent of $q,F,G$ .

Proof 143.4

Use (B2) with its retained degree $d_0=d$ . Since $d<16q$ , its retained range $2d-1<32q$ is available and there are no intermediate coherent terms in $\Delta$ . Put $K_q=4(s_q+C_F'\kappa_{q-2}+4C_\theta a_q+1)$ , $p_*=K_qu^q$ , where $C_F'$ and $C_\theta$ are fixed below. Before, and at, a first exit of $p_*$ , required normalizers obey

B_*\le\frac{u}{(1-a_qu^q)(1-p_*)}\le(1+1/64)u.

(143.15)

Indeed $a_qu^q,p_*\le C F^{-6}4^{-q}$ ; choose the fixed lower bound on $F$ to make both at most $1/1024$ . Thus $t_*\le1+1/64$ , and $B_*t_*^3G(k)^2\le G(k)^2/(2F^2)$ for $k\le d$ . The same capped moment estimate bounds $\theta\le C_\theta u$ uniformly. Also $C_P\le R+1\le z+3$ gives $\lambda\ge u/2$ at $z\ge3$ .

For the terminal coefficient, write $\rho=G(d)^2/F^2\le1$ . Then

\frac{\tau}{\kappa_q\kappa_{q-2}u^{2q}} \le C_Fd F^6\rho^{d-1}(1+1/64)^d(z/F^2)^{2q-d} \le C d^2\rho^{d-4}(1+1/64)^d4^{2q-d}.

(143.16)

We used $F^6\rho^{d-1}=G(d)^6\rho^{d-4}$ and $G(d)^6\le Cd$ . Since $d\ge8q$ , the last expression is bounded by a universal constant; polynomial growth in $d$ is dominated by its geometric decay. Fix $C_F'$ to dominate it. Now Lemma 10.9 applies with these $\theta,\tau$ , the retained losses and the actual matched budget. Its other thresholds are $z\ge8C_\theta$ and the two already imposed smallness conditions. It proves (B12), since $a_q+4K_q\le C_b\kappa_{q-2}$ .

Lemma 143.5 (Finite blocks at every odd order)

Let a nondecreasing $1\le G\le G_*$ bound the coefficients of all regular covariance contractions and satisfy (B4),(B5) with fixed comparison constants. For every odd $Q\ge3$ there is a static assignment $Z_Q$ and constants $H_Q,a_Q,s_Q>0$ such that

H_Q\le K G(8Q)^2,\quad a_Q+s_Q+1\le[KG(8Q)^2]^Q, \tag{B13}

(143.17)

c_d\le Z_Q^{(d-1)/2},\quad H_Q\le Z_Q\le\max\{H_Q,C_P\},\quad C_P\le Z_Q+3.\tag{B14}

(143.18)

Whenever $Z_Q>H_Q$ , it is an actual finite block radius $z$ , and one centered unit family satisfies, with $u=z^{-1}$ and $J_Q=\lceil128Q\log(e+G(8Q))\rceil$ ,

\nu\le u/(1-a_Qu^Q),\quad P_J\le s_Qu^Q\ (J\le J_Q), \quad aV_N+X_N\le\nu-u+uP_{N+1}, \tag{B15}

(143.19)

\|\mathcal T^h\|\le4(h+1)z^{h/2},\quad 0\le R-z\le1,\quad a_Qu^Q\le1/8. \tag{B16}

(143.20)

All constants depend only on the fixed comparison constants and universal analytic constants, not on $Q$ , the measure, or the smaller seed $G$ .

Proof 143.5

Choose a universal multiplier $C_2$ sufficiently large in the order specified below, and put $F=C_2G(8Q)$ , $R_0=16F^2$ , $H_Q=R_0+1$ , $\kappa_q=F^{2q-2}$ for odd $q\ge1$ . The fixed-argument comparison in (B4) permits increasing $C_2$ once so that every initialization degree below $8Q$ and extension degree below $16Q$ has seed at most $F/C_3$ , where $C_3$ is any prescribed large universal constant. Take $\widetilde J_p=\lceil128p\log(e+F)\rceil$ .

If $R<R_0$ , set $Z_Q=H_Q$ . The common-radius comparison gives $c_d\le\max(1,R)^{(d-1)/2}\le H_Q^{(d-1)/2}$ , and $C_P\le R+1$ gives (B14). Only the large- $R$ case requires a family. Suppose therefore $R\ge R_0$ .

Base block. Let $m_3=\lfloor R\rfloor$ , $z_3=\|\mathcal T^{m_3}\|^{2/m_3}$ . The mesoscopic powers lemma gives $R-C/R\le z_3\le R$ , and dividing arbitrary powers into blocks of length $m_3$ gives

\|\mathcal T^h\|\le(R/z_3)^{(m_3-1)/2}z_3^{h/2} \le2z_3^{h/2}

(143.21)

at a universal large threshold. Also $m_3$ lies between $z_3/4$ and $4z_3$ , so its actual divisor is $\kappa_1=1$ . Its initialization degree is sixteen, with a fixed coefficient bound from $G_*$ . Thus $\gamma m_3^2/z_3\le Cz_3^{-14}$ and $\widetilde J_3\gamma z_3m_3\le C\widetilde J_3z_3^{-13}$ . Since $\widetilde J_3=O(\log(e+F))$ and $z_3\ge15F^2$ , all conditions (B8),(B10), including $\widetilde J_3\le m_3/4$ , hold at a fixed large $C_2$ . The local initialization gives one family with $a_3+s_3+1\le C\kappa_1$ .

Initialization conditions at any subsequent order. Suppose $p\ge5$ , $z\ge15F^2$ , $0\le R-z\le1$ , and $z^{p-2}/(4\kappa_{p-2})\le m\le4z^{p-2}/\kappa_{p-2}$ . Let $d$ be the least dyadic integer at least $4p$ , $G_p=G(d)$ , and $\rho=G_p^2/F^2\le C_3^{-2}$ . The coefficient cap $G_p^6\le Cd$ and the preceding length bounds give

\gamma m^2/z \le CdG_p^2\rho^{d-2}15^{-(d-2p+4)} \le Cp^{4/3}C_3^{-8p+4}15^{-2p}, \tag{B17}

(143.22)

\widetilde J_p\gamma zm \le C\widetilde J_p dG_p^6F^{-2}\rho^{d-4}15^{p-d} \le Cp^3\frac{\log(e+F)}{F^2}C_3^{-8p+8}15^{-3p}. \tag{B18}

(143.23)

The constants harmlessly cover $d<8p$ and the ceilings. The two right sides are uniformly as small as required by choosing $C_3$ , then $C_2$ , universally large. Also $\widetilde J_p/m\le C p\log(e+F)/(F^2 15^{p-2})$ is uniformly small, so $\widetilde J_p\le m/4$ and $m\ge2d$ . Finally $C\kappa_{p-2}z^{-p}\le CF^{-6}15^{-p}$ is small. Thus local initialization gives $a_p+s_p+1\le C\kappa_{p-2}$ through the entire longer prefix. The same constant works for every order.

Extension and radius control. Given the block at odd order $q<Q$ , define

m_{q+2}=\lfloor z_q^q/\kappa_q\rfloor,\qquad z_{q+2}=\|\mathcal T^{m_{q+2}}\|^{2/m_{q+2}},\qquad \Delta_q=C_b\kappa_{q-2}z_q^{1-q}. \tag{B19}

(143.24)

The extension lemma uses only the already constructed family’s losses through $32q$ , contained in $\widetilde J_q$ , and its already proved power envelope. It gives $z_{q+2}\ge z_q-\Delta_q$ . At $z_q\ge15F^2$ ,

\Delta_q/z_q\le C_bF^{-6}15^{-q},\qquad q\Delta_q/z_q\le CF^{-6}. \tag{B20}

(143.25)

The decrement sequence decreases by at least a factor two: indeed, once the new radius is above $15F^2$ ,

\frac{\Delta_{q+2}}{\Delta_q} =\frac{F^4}{z_{q+2}^2}(z_q/z_{q+2})^{q-1} \le15^{-2}\exp(CF^{-6})\le1/2.

(143.26)

There is no circular assumption here. At the next step the decrements already bounded give first $R-z_{q+2}\le C/R+\sum_{p\le q}\Delta_p \le C/R+2\Delta_3<1$ after increasing $C_2$ . Since $R\ge16F^2$ , this establishes $z_{q+2}\ge15F^2$ before its new decrement is compared. Increasing actual radii can only improve these upper bounds. In particular every finite constructed prefix has $0\le R-z_q<1$ and $(z_p-z_q)_+\le2\Delta_p$ for $p<q$ .

The power envelope before the next initialization. The lower length comparison already available at order $q$ gives $m_q\le4z_q^{q-2}/\kappa_{q-2}$ . The new defining length is at least half its unfloored value, since $z_q^q/\kappa_q\ge15^qF^2$ . Therefore

m_{q+2}/m_q\ge z_q^2/(8F^4)\ge15^2/8>4.

(143.27)

Moreover $m_{q+2}\Delta_q/z_q\le C_b/F^4$ . For any prefix ending at $p$ , its base remainder satisfies

m_3(R-z_p)/z_p\le C/R+C/F^4.

(143.28)

The greedy propagation lemma Lemma 10.10 therefore gives

\|\mathcal T^h\|\le z_p^{h/2} \exp\{C/R+C/F^4+(C/F^4)\log_4(h+1)\} \le4(h+1)z_p^{h/2}. \tag{B21}

(143.29)

Fixing $C_2$ large makes both the prefactor and exponent no greater than the displayed values. This uses only exact norms of constructed blocks; it precedes initialization of the new block.

New length compared with its own radius. Apply the previous envelope at $h=m_{q+2}$ to get

q\log(z_{q+2}/z_q) \le\frac q{m_{q+2}}[\log16+2\log(m_{q+2}+1)].

(143.30)

The right side is uniformly small for all $q\ge3$ at large $F$ : $m_{q+2}\ge15^qF^2/2$ , and $\log(x+1)/x$ is decreasing for $x>0$ . The resulting bound is $Cq(q+\log(e+F))/(15^qF^2)$ . The negative part is bounded by (B20). With $C_2$ fixed large, $z_{q+2}^q/z_q^q$ is between $1/2$ and two. The floor loses at most a further factor two. Hence

\frac{z_{q+2}^q}{4\kappa_q}\le m_{q+2} \le\frac{4z_{q+2}^q}{\kappa_q}.

(143.31)

All initialization conditions at order $q+2$ were verified in (B17),(B18), so its new actual family exists with the required longer prefix and matched budget. This completes a finite induction up to $Q$ .

All static degrees. The constructed $m_Q$ satisfies $m_Q\le R^Q$ . By (B5),

G(m_Q)^2\le C G(8Q)^2+C G_*(R)^2\le R/2\le z_Q.

(143.32)

The first term is at most $R/4$ because $F=C_2G(8Q)$ and $R\ge16F^2$ ; the second is at most $R/4$ above a fixed universal threshold since $G_*(R)^2=O(t(R)^{1/3})=o(R)$ . Increase $C_2$ to include that threshold. The common-radius finite-block comparison now gives $c_d\le z_Q^{(d-1)/2}$ simultaneously for all $d\ge1$ . Put $Z_Q=\max\{H_Q,z_Q\}$ . We have $z_Q\le R\le C_P\le R+1\le z_Q+2$ , which proves (B14). Whenever $Z_Q>H_Q$ , the constructed radius and family are the required ones. The uniform initialization constants may be enlarged to fix $a_Q,s_Q$ independently of the particular law; they satisfy $a_Q+s_Q+1\le CF^{2Q-6}\le(KG(8Q)^2)^Q$ . The retained prefix contains $J_Q$ , and (B16) follows from the preceding induction. All choices of constants are made before any inner-depth or outer-height iteration.

Logarithmic distortion and the full inner-depth profile¶

Lemma 143.6 (Uniform burn-in for terminal-degree distortion)

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}

(143.33)

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 $0<a\le1$ , $r\ge r_Q$ and $d<2C_TQ^2(r+1)^2\log(e+a^{-1})$ ,

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

(143.34)

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 $R$ .

Proof 143.6

First $g(Mx)-g(x)\le\log M$ . If $y\ge1$ and $v\ge0$ , $g(y+v)-g(y)\le\log(1+v/(e+1))$ . Iteration proves (B22). The map $F(v)=\log(1+\eta v)$ obeys $F(v)\le\eta v$ everywhere and $F(v)\le\log v$ for $v\ge2$ , because $1+\eta v\le v$ there. Starting from $\log M$ , its orbit reaches $[0,2]$ in at most $\log^*M$ steps; once in that interval it contracts by $\eta$ . This gives the displayed bound, also directly if $1\le M\le e$ .

For $M=2C_TQ^2(r+1)^2$ , there is fixed $c_T$ such that $1+\log^*M\le c_T+\max\{t(Q),\log^*(r+1)\}$ . Indeed with $N=\max\{16,2C_T,Q,r+1\}$ , $M\le N^5\le e^N$ ; the bounded small-argument range is absorbed in $c_T$ . For sufficiently large $r$ , $\log^*(r+1)\le1+\log(r+1)\le r/4$ . Choose $C_b,r_*$ so $r\ge r_Q$ also implies $r\ge4t(Q)$ and $r\ge4c_T$ . Then $1+\log^*M\le r/2$ . Apply (B22) at $x=g(a^{-1})\ge1$ ; its unperturbed iterate equals $\ell_{r+1}(a^{-1})\ge1$ , proving (B23). Finally the logarithm of the product is bounded by

2\sum_{r\ge r_Q}r^{-2}+4\sum_{r\ge r_Q}\eta^{r/2},

(143.35)

which is universally finite for $r_Q\ge2$ .

Lemma 143.7 (A full profile from a seed and one starting depth)

Suppose $G$ satisfies the preceding coefficient-seed and block-realization hypotheses, including (B6),(B7). For each odd $Q\ge3$ , suppose $M_Q\ge G(Q)^2$ and every regular covariance contraction satisfies

C_P\le M_Q\ell_{r_Q}(a^{-1})^2\quad(a>0). \tag{B24}

(143.36)

Then uniformly for $r\ge r_Q$ ,

Z_Q\le C M_Q(r+1)^{1/Q}\ell_r(a^{-1})^2,\qquad C_P\le C M_Q(r+1)^{1/Q}\ell_r(a^{-1})^2. \tag{B25}

(143.37)

Here $r_Q$ is the preceding universal starting depth; the constant depends only on the fixed seed-comparison constants, not on $M_Q,Q,r$ or the law.

Proof 143.7

By (B4),(B13) and $M_Q\ge G(Q)^2$ , $H_Q\le CM_Q$ and $a_Q+s_Q+1\le(CM_Q)^Q$ . Set $A_Q=A M_Q$ , where one sufficiently large universal $A$ will be chosen. At an already established depth $r\ge r_Q$ suppose $Z_Q\le\Gamma_r^2\ell_r(a^{-1})^2$ , where $\Gamma_r^2\ge A_Q(r+1)^{1/Q}$ . The block radius’s static coefficient bound and the static coefficient transfer give simultaneously

c_k^2\le[\widehat\Gamma^2\ell_r(k)^2]^{k-1}, \quad c_k^2\le g_0(k)^{k-1},\quad \widehat\Gamma=(1+r^{-2})\Gamma_r. \tag{B26}

(143.38)

The transfer’s only premise is the previously established scalar profile; no continuity of $Z_Q$ or of its finite defining block length is used.

Put $s=r+1$ , $\delta=(64Qs)^{-1}$ , $p=\delta/128$ , $b=(1-2p)^{-1}$ , and $k_0=\lceil100\delta^{-1}\log(M/\delta)\rceil$ , where a universal $M$ larger than all fixed frame constants is fixed once. Let $d_0$ be the least dyadic integer at least $Q$ , and for dyadic $d\ge d_0$ suppose for a contradiction that

Z_Q>b^4e^\delta\widehat\Gamma^2\ell_r(d)^2. \tag{B27}

(143.39)

Take $A$ so large that $A_Q>2H_Q$ . Then (B27) excludes the floor: $z=Z_Q$ is an actual radius with the single family of (B15). For $u=z^{-1}$ ,

b^4u\widehat\Gamma^2\ell_r(d)^2<e^{-\delta}, \quad u\le A_Q^{-1}s^{-1/Q},\quad u^Q\le A_Q^{-Q}s^{-1}. \tag{B28}

(143.40)

Its retained range covers $2d_0-1<4Q$ , and, provided $a_Qu^Q\le p$ ,

\nu\le u/(1-a_Qu^Q),\quad P_{2d_0-1}\le s_Qu^Q, \quad D_0:=\max(\nu-u,0)\le2a_Qu^{Q+1}.

(143.41)

Before a first exit of mass $p$ , required normalizers are at most $B_*\le u/(1-p)^2\le bu$ ; hence $t_*\le b$ in (B2). Its loss estimate is bounded by

\theta_d=C_\theta'\sum_{k<d\ {\rm dyadic}}k^8(b^4u)^kc_k^2, \quad \Delta_d=s_Qu^Q+2C_F\sum_{d_0\le k<d\ {\rm dyadic}} k^2(bu)^kc_k^2, \quad\tau_d=C_Fd(bu)^dc_d^2, \tag{B29}

(143.42)

where $C_\theta'=5184C_F\cdot16$ because the block envelope has $E=4$ .

Uniform kernel estimates. Since $k_0\le C(Qs)^2$ , (B6) implies $g_0(k_0)\le Cg_0(Q)s^{1/Q}\le\Gamma_r^2/4$ after increasing $A$ . For $k\le k_0$ , the seed and (B28) give

(b^4u)^kc_k^2\le b^4u\,4^{-(k-1)}, \quad (bu)^kc_k^2\le b^Qu^Qg_0(k)^{Q-1}4^{-(k-Q)}\ (k\ge Q).

(143.43)

Therefore the low-degree kernel is at most $Cu$ , and its low-degree coherent sum is at most $D_Qu^Q$ , with $D_Q^{1/Q}\le Cg_0(Q)$ by (B7); use $b\le2$ . For $k_0<k<d$ , the first bound in (B26) gives

(b^4u)^kc_k^2\le b^4u e^{-\delta(k-1)},\qquad (bu)^kc_k^2\le e^{-\delta k}.

(143.44)

For each fixed integer $h\ge0$ ,

\sum_{k>k_0}k^he^{-\delta k} \le C_h\delta^{-h-1}e^{-\delta k_0/2},\qquad e^{-\delta k_0/2}\le(\delta/M)^{50}.

(143.45)

For the first inequality remove half of the exponential and integrate $(x+1)^he^{-\delta x/2}$ over unit intervals; expansion into monomials gives $C_h\delta^{-h-1}$ . Choosing $M$ fixed large makes the two tail contributions at most $u$ and $p/512$ respectively. Thus

\theta_d\le C_\theta u,\qquad \Delta_d\le(s_Q+D_Q)u^Q+p/512. \tag{B30}

(143.46)

The matched comparison. Choose $A$ once so that

A_Q\ge\max\{2H_Q,16,16C_\theta,Cg_0(Q)\},

(143.47)

A_Q^Q\ge2^{24}Q(a_Q+s_Q+D_Q+4C_\theta a_Q+1). \tag{B31}

(143.48)

These simultaneous inequalities are possible with one universal $A$ : every term’s $Q$ th root is bounded by a constant times $M_Q$ , and $Q^{1/Q}$ is uniformly bounded. The block comparison gives $\lambda\ge1/(z+3)\ge u/2$ . Equations (B28)–(B31) imply

\nu\le1/4,\quad D_0\le u,\quad a_Qu^Q\le p, \quad\theta_du/\lambda\le1/8, \quad\Delta_d+\theta_dD_0/\lambda\le p/64, \quad u+D_0/p\le2u. \tag{B32}

(143.49)

For example $\theta_dD_0/\lambda\le4C_\theta a_Qu^{Q+1}$ and $D_0/p\le16384Qa_QA_Q^{-Q}u\le u$ ; all remaining inequalities follow by inserting $u^Q\le A_Q^{-Q}/s$ into (B31).

Here is the finite comparison with this matched scale, to make explicit that no generic $R$ -budget is substituted for it. If a delayed bound $P_N\le\Delta+N\tau+(\theta/\lambda)X_{\max(N-2,0)}$ has $p_0+p_1\le\Delta$ and the conditions in (B32), then

a<12(u+D_0/p)\tau. \tag{B33}

(143.50)

Indeed the actual matched budget gives $X_{N-2}\le D_0+uP_N$ and $a\le D_0+u(p_0+p_1)\le D_0+up$ . If the reverse of (B33) held, let $m=\lceil(D_0+up)/a\rceil$ , $M_1=m+1$ ; then $M_1\tau\le p/4$ . Before or at a first exiting prefix $N\le M_1$ all required normalizers use earlier masses at least $1-p$ and so are at most $\nu/(1-p)$ ; the possible successor remains positive since $1-p-\nu>0$ . The delayed estimate yields $P_N\le p/64+p/4+P_N/8$ , hence $P_N\le cp$ with $c=17/56<1$ . No exit occurs. At time $m$ the budget and surviving masses imply

(1-cp)(D_0+up)\le aV_m\le D_0+cup.

(143.51)

Their difference is at least $up(1-2c-cp)>0$ , a contradiction. This proves (B33). The retained prefix supplies its two initial losses. Apply it to (B29) to get $a<24u\tau_d\le24C_Fd(bu)^dc_d^2$ .

For $0<a\le1$ , select a dyadic integer

C_TQ^2s^2\log(e+a^{-1})\le d <2C_TQ^2s^2\log(e+a^{-1}). \tag{B34}

(143.52)

A universal large $C_T$ ensures $d\ge d_0$ and $\log(24C_Fd/a)\le\delta d$ : the former logarithm is at most $\log(48C_FC_T)+2\log Q+2\log s+2\log(e+a^{-1})$ , whereas $\delta d\ge(C_T/64)Qs\log(e+a^{-1})$ . Writing $L=\widehat\Gamma^2\ell_r(d)^2\ge1$ , (B26),(B28) would give

a<24C_FdL^{-1}(buL)^d\le24C_Fd e^{-\delta d}\le a.

(143.53)

Thus (B27) is impossible. For $a\ge1$ , Brascamp–Lieb gives $C_P\le1$ and $Z_Q\le H_Q$ , already covered by the same large amplitude.

Closing all depths. The distortion lemma now implies

Z_Q\le b^4e^\delta(1+r^{-2})^2\Gamma_r^2 \zeta_r^2\ell_{r+1}(a^{-1})^2.

(143.54)

Since $\log(b^4e^\delta)\le16p+\delta=9/(512Qs) \le Q^{-1}\log(1+1/s)$ , define

J_{r_Q}=1,\quad J_{r+1}=J_r(1+r^{-2})^2\zeta_r^2, \quad\Gamma_r^2=A_Q(r+1)^{1/Q}J_r.

(143.55)

All $J_r$ are universally bounded. The initial bound follows from $Z_Q\le\max(H_Q,C_P)$ and (B24) by increasing the same universal $A$ . Finite induction gives the first inequality in (B25), with its coefficient transfer at each step justified by the previous profile. Finally $C_P\le Z_Q+3$ and $M_Q,\ell_r\ge1$ absorb the additive constant once, proving the second inequality.

Repeating the height reduction¶

Theorem 143.1 (Fixed-cost repeated height reduction)

There are universal $A_1,C_*,C_e,C_b\ge1$ and $r_*\ge2$ such that, with $A_j=A_1C_*^{j-1}$ and $r_Q=\max\{r_*,\lceil C_bt(Q)\rceil\}$ , every regular covariance contraction satisfies

C_P\le A_jt_j(Q)^{1/3}(r+1)^{1/Q}\ell_r(a^{-1})^2 \quad(j\ge1,\ Q\ge3\text{ odd},\ r\ge r_Q).

(143.56)

Every centered log-concave covariance contraction also satisfies

c_d\le[C_e\sqrt{A_j}\,t_{j+1}(d)^{1/6}]^{d-1} \quad(j,d\ge1).

(143.57)

This is Proposition 9.2.

Proof 143.8

For $G=G_*$ , the initial inner theorem at $r_Q\le Ct(Q)$ gives $C_P\le Ct(Q)^{1/3}\ell_{r_Q}(a^{-1})^2$ . Its amplitude dominates $G_*(Q)^2$ after one enlargement. Apply (B25) with $M_Q=Ct(Q)^{1/3}$ to obtain the asserted curvature profile at $j=1$ , choosing one universal $A_1\ge1$ .

We next extract coefficients from any established stage $j$ , without changing its constant. Fix $d\ge2$ , write $h=t(d)$ and choose $r=\lceil C_rh\rceil$ for a fixed sufficiently large $C_r$ . The logarithm comparison used for the original seed gives $\ell_r(d)\le5$ . Let $k$ be the least odd integer at least $\max\{3,\log(r+1)\}$ . For all $h$ above a fixed threshold,

k\le h,\qquad r_k\le r,\qquad(r+1)^{1/k}\le e.

(143.58)

Indeed $k\le\log(C_rh+2)+3$ grows slower than $h$ , and $r_k\le C t(k)\le C'k$ grows slower than $r$ ; the finite remaining range is handled below. The stage- $j$ profile at order $k$ yields, uniformly in the regular law,

C_P\le e A_jt_j(k)^{1/3}\ell_r(a^{-1})^2 \le e A_jt_{j+1}(d)^{1/3}\ell_r(a^{-1})^2.

(143.59)

Repeated Poincaré on Appell derivatives then supplies the coefficient premise of the static transfer with $\Gamma^2=e A_jt_{j+1}(d)^{1/3}\ge1$ . The transfer at this fixed $r$ gives $c_d\le[5(1+r^{-2})\sqrt{e A_j}\,t_{j+1}(d)^{1/6}]^{d-1}$ . If $h$ is below the fixed threshold, $G_*(d)$ is bounded by a universal constant. Since $A_j\ge1$ and $t_{j+1}(d)\ge3$ , increasing one $C_e$ covers this entire exceptional range uniformly in $j$ . Degree one is the covariance bound. The transfer already covers singular covariances and nonsmooth laws, so the coefficient conclusion has exactly the stated class.

Suppose the curvature profile is established through stage $j-1$ . Its coefficient conclusion permits the capped seed

G_j(x)=\min\{G_*(x),C_e\sqrt{A_{j-1}}t_j(x)^{1/6}\}.

(143.60)

The seed lemma shows that every block and kernel constant required for (B25) is uniform in $j$ and $A_{j-1}$ . It remains to initialize at the same $r_Q$ , rather than introduce a stage-dependent depth. Let $k$ be the least odd integer at least $\max\{3,\log(r_Q+1)\}$ . If $t(Q)$ exceeds a fixed threshold, then $k\le t(Q)$ , $r_k\le r_Q$ , and $(r_Q+1)^{1/k}\le e$ , by the same growth comparisons as above. Applying the previous profile at order $k$ and depth $r_Q$ gives

C_P\le e A_{j-1}t_{j-1}(k)^{1/3}\ell_{r_Q}(a^{-1})^2 \le e A_{j-1}t_j(Q)^{1/3}\ell_{r_Q}(a^{-1})^2.

(143.61)

If $t(Q)$ is below that threshold, $r_Q$ is bounded by a fixed constant. Use order three instead: $r_3\le r_Q$ by monotonicity, $t_{j-1}(3)=3$ , and $(r_Q+1)^{1/3}$ is uniformly bounded. This gives the same initial estimate with a larger universal constant. Therefore we may take $M_Q=C_M A_{j-1}t_j(Q)^{1/3}$ in (B24), choosing $C_M$ also to dominate $C_e^2$ , so $M_Q\ge G_j(Q)^2$ . The full-profile lemma proves the next curvature stage with $A_j=C_*A_{j-1}$ for a single universal $C_*$ . The coefficient extraction just proved gives its accompanying coefficient estimate. Finite induction completes both statements, with the same depths $r_Q$ at every stage.

Source mapping and fences. The two independent joint-frame interfaces reconstruct Lemmas 8.11 and 8.13. The local orbit and restart reconstruct the mechanism of Lemmas 8.19–8.20. The fixed-margin extension and finite odd-order induction give the interface of Proposition 8.22 with enlarged universal constants; no near-unit floor is claimed. The final three arguments reconstruct Lemma 8.23 and Propositions 8.24, 8.25 and 8.1. Small degrees use the original cap at every repetition; block floors and depth thresholds are fixed before their respective inductions. This respects the uniform-admissibility warning in the brief. The cost $C_*^{j-1}$ still grows, so this proof does not itself establish KLS or uniform exponential coefficients. No additional bounded_by edge is proposed, and no CMH, occupation, or trace premise is discharged.

References¶

Song, Z., & Zhang, X. (2026). An O(1) Bound for the KLS Constant. https://arxiv.org/abs/2610.01447v2