Song–Zhang v2: block construction and propagation

Part of the second version of Song–Zhang, Chapter Song–Zhang, second version: technical estimates; the reading order is on the full proofs page.

Overview. This proves the generic mechanisms of Lemma 10.8, Lemma 10.9, and Lemma 10.10. The starting assumptions of each lemma are explicit; no claim is made here that a particular degree majorant satisfies the raw-frame hypotheses. These mechanisms underlie Sections 8–9 of Song & Zhang, 2026.

Dependencies. Use Lemma 7.1, Lemma 10.3, and Lemma 10.4. The skew compensation is derived below. No height-reduction theorem, small-loss profile, summable profile, BKL result, or KLS conclusion is used.

Write $H=-\Delta+\nabla W\cdot\nabla$ on centered $L^2(\mu)$ , $B=H^{-1}$ , $Lf=\mathbb E[Xf]$ , and $\mathcal T=P_+\nabla B$ , where $P_+$ centers every output component. All operators act componentwise on finite direct sums with all ordered output indices retained. The law is centered and regular, has covariance at most $I$ , and curvature at least $aI$ , $a>0$ . The generic sequence and norm lemmas also apply abstractly.

Lemma 145.1 (Two-block Green 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,

(145.1)

and

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

(145.2)

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).

(145.3)

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.

(145.4)

Proof 145.1

All sums below are finite; $C$ depends only on $K$ . Put $S=\sum_{j=0}^{m-1}b_j$ , $L_1=\sum_{j=1}^m l_j$ and $D_1=\sum_{j=1}^mD_j$ . The first telescope is

D_1\leq L_1+z(b_{m+1}-b_1)\leq L_1,

(145.5)

since $b_{m+1}\leq b_1$ . On this block the terminal coherent sum is at most $\gamma S$ , and all convolution source indices are less than $m$ . Thus

L_1\leq2+K/z+\gamma S+(K/z)D_1.

(145.6)

Choose $z\geq2K$ . Absorption gives $D_1,L_1\leq C(1+\gamma S)$ and $\sum_{j=2}^m l_j\leq C/z+C\gamma S$ . It also gives the boundary estimate $z(b_1-b_{m+1})\leq L_1$ .

For $\delta_j=b_j-b_{j-1}$ , the defect inequality implies $\delta_{j+1}\geq\delta_j-l_j/z$ . Since $\sum_{j=1}^m\delta_j=0$ , some $\delta_j$ is nonnegative; propagating from that index gives $\delta_{m+1}\geq-L_1/z$ . On the other hand, $b_{2m}\leq1$ and $\delta_{2m+1}\leq2b_{2m}/z\leq2/z$ . Telescoping the defect inequality over the second block and adding the first block yields

D_{\rm tot}:=\sum_{j=1}^{2m}D_j \leq\sum_{j=1}^{2m}l_j+2+L_1.

(145.7)

The coherent sum through $2m$ is at most $K/z+2\gamma S$ , because $\sum_{j=0}^{2m-1}b_j\leq2S$ . A second absorption proves

D_{\rm tot}\leq C+C\gamma S,\qquad \sum_{j=2}^{2m}l_j\leq C/z+C\gamma S. \tag{G1}

(145.8)

Fix $1\leq q<m$ , and extend $g_q(j)=\min\{j,q\}(m-\max\{j,q\})/m$ by zero outside $1\leq j<m$ . It is nonnegative and globally one-Lipschitz. The discrete second differences give exactly

\sum_jg_q(j)(b_{j+1}-2b_j+b_{j-1})=1-b_q.

(145.9)

For $E=\sum_jg_q(j)l_j$ and $F=\sum_jg_q(j)D_j$ this gives $F\leq E+z(1-b_q)\leq E+z$ . The term $j=1$ contributes at most two. The early coherent terms contribute at most $K/z$ , since $g_q(j)\leq j$ ; the later terms at most $\gamma mS$ . Reindex the convolution and use $g_q(j+s)\leq g_q(j)+s$ to obtain

E\leq2+K/z+\gamma mS+(K/z)F+(K/z)D_1.

(145.10)

By the preceding estimates, absorption gives $E\leq C+C\gamma mS$ . Since $F\geq0$ , the Green identity implies $b_q\leq1+C/z+C\gamma mS/z$ . Setting $M=\max_{0\leq q\leq m}b_q$ and using $S\leq mM$ yields $M\leq1+C/z+Cc_0M$ . Fix $z_0$ large and then $c_0$ small so $M\leq2$ . The shift inequality propagates this upper bound to all indices.

For the lower bound use the weight

n(j)=\#\{k\in\{2,\ldots,m+1\}:k\leq j\leq m+k-1\}.

(145.11)

It is one-Lipschitz, supported in $[2,2m]$ , and bounded by $\min\{j-1,m\}$ . With $E_n=\sum_jn(j)l_j$ , $F_n=\sum_jn(j)D_j$ , telescoping each window gives

F_n\leq E_n+z(b_{2m+1}-2b_{m+1}+b_1) \leq E_n+L_1.

(145.12)

The second inequality uses $b_{2m+1}\leq b_{m+1}$ and the first boundary estimate. The coherent contribution is at most $K/z+2\gamma mS$ . Convolution reindexing, now using (G1), gives

E_n\leq K/z+2\gamma mS+(K/z)F_n+(K/z)D_{\rm tot} \leq C/z+C\gamma mS+(K/z)E_n.

(145.13)

Therefore $E_n\leq C/z+C\gamma mS$ . Apply the Dirichlet Green identity on $[q,q+m]$ at $m$ . Both endpoint values are at most $b_q$ , since $b_{q+m}\leq b_q$ ; the lower bound $\Delta^2b_j\geq-l_j/z$ therefore implies

1=b_m\leq b_q+z^{-1}\sum_{j=q+1}^{q+m-1}G_{q,m}(j)l_j,

(145.14)

where $G_{q,m}(j)=(\min\{m,j\}-q)(q+m-\max\{m,j\})/m$ . For $j\leq m$ , this is at most $j-q\leq j-1=n(j)$ ; for $j>m$ it is at most $q+m-j\leq2m-j\leq n(j)$ . Thus $b_q\geq1-C/z^2-C\gamma mS/z\geq1/2$ by $S\leq2m$ and the same universal choices. Endpoints already equal one.

Finally replace the $m$ windows in $n$ by the $J$ windows starting at $2,\ldots,J+1$ . The new weight is still one-Lipschitz, at most $\min\{j-1,J\}$ , and supported inside $[2,2m]$ . Its boundary telescope is $z(b_{m+J+1}-b_{m+1}-b_{J+1}+b_1)\leq z(b_1-b_{m+1})\leq L_1$ , using $b_{m+J+1}\leq b_{J+1}$ . Its coherent sum is at most $K/z+2\gamma Jm$ , now that $b_j\leq2$ . The identical reindexing and absorption give the asserted window estimate. This proves the lemma.

Lemma 145.2 (A single restarted family with a matched budget)

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),

(145.15)

\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),

(145.16)

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.

(145.17)

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},

(145.18)

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

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

(145.19)

Proof 145.2

Set $\mathcal W=\bigoplus_{j=0}^{m-1}w_j$ , $S_k=\sum_{j=k}^{m+k-1}b_j$ and $C_k=\sum_{j=k}^{m+k-1}l_j$ . Then $S_0=S_1\geq m/2$ , and $S_{k+1}\leq S_k$ because $S_{k+1}-S_k=b_{m+k}-b_k\leq0$ . The family

U={\mathcal T\mathcal W\over\sqrt{zS_0}}

(145.20)

has norm one, since $\|\mathcal T\mathcal W\|^2=zS_1$ . Each component is the gradient of $B\mathcal W$ minus a constant vector, hence its derivative is symmetric in its two newest slots. Bochner gives $\|H^{1/2}\mathcal T\mathcal W\|^2\leq\|\mathcal W\|^2=S_0$ , so its energy $e$ is at most $u$ . The direct-sum norm and the assumed loss bounds imply

s:=\|\operatorname{Sym}(LU)\|^2 ={1\over S_0}\sum_{j=1}^m\|\operatorname{Sym}(Lw_j)\|^2 \leq C/(zm)+C\gamma.

(145.21)

Put $\beta=\|B^{1/2}U\|^{-2}$ and $F=\sqrt\beta B^{1/2}U$ . It is a centered unit family of energy $\beta\leq e$ . The normalized hierarchy first applies $D=P_+\nabla H^{-1/2}$ and then applies $H^{-1/2}$ to the result, rescaling to preserve that result’s norm. Its removed mean at $F$ has squared norm $p_* =\beta\|LU\|^2$ .

The skew contribution satisfies $\beta\|\operatorname{Skew}(LU)\|^2\leq1-\beta/e$ . Indeed every unit skew tensor $C$ gives a vector family $CX$ of Dirichlet norm one orthogonal in Dirichlet inner product to $U$ . Project $H^{-1/2}U$ orthogonally to $H^{1/2}U$ ; its squared norm is $\beta^{-1}-e^{-1}$ because the inner product is $\|U\|^2=1$ . Duality over $C$ proves the claimed skew inequality, also for finite direct sums. Consequently

1-p_*\geq\beta/e-\beta s=\beta(e^{-1}-s)>0.

(145.22)

Bochner and Cauchy–Schwarz show that the mean energy of the normalized successor, after making it a unit family, is at most

{\beta\over1-p_*}\leq {e\over1-es} \leq {u\over1-us} \leq {u\over1-C/(z^2m)-C\gamma/z}.

(145.23)

Call this unit successor $G_0$ . Its direction is exactly $B^{1/2}\mathcal T^2\mathcal W$ , so it is the actual two-step restart of the orbit, rather than a separately selected low-energy family.

For any centered finite family $Y$ the identity $\|B^{1/2}Y\|^2=\|\mathcal TY\|^2+\|LY\|^2$ follows by subtracting the mean from $\nabla BY$ and using the Dirichlet identity. Thus the successive relative loss fractions of $G_0$ are exactly

{C_k\over zS_{k+1}+C_k},\qquad k=2,3,\ldots.

(145.24)

For $k\leq J_0+1$ , the interval defining $S_{k+1}$ contains at least $m-k\geq m-J_0-1\geq m/2$ indices in $[0,m]$ , and hence $S_{k+1}\geq m/4$ . Successive unnormalized masses are at most one. It follows that

P_J\leq {4\over zm}\sum_{k=2}^{J+1}C_k \leq C/(z^2m)+C\gamma J/z.

(145.25)

All required successors are nonzero on this prefix by the same lower window bounds. The length comparison gives $1/(z^2m)\leq4\kappa z^{-p}$ , and $J_0\gamma zm\leq1$ gives $\gamma J/z\leq1/(z^2m)$ . These observations prove the energy and prefix estimates with one $C$ .

Finally every later nonzero hierarchy member has the form $G_j=q_jB^{1/2}\mathcal T^{j+2}\mathcal W$ for a positive scalar $q_j$ . If $v_j,e_j$ denote its squared norm and energy, then

{v_{j+1}\over e_j} ={\|\mathcal T^{j+3}\mathcal W\|^2\over \|\mathcal T^{j+2}\mathcal W\|^2} =z{S_{j+3}\over S_{j+2}}\leq z.

(145.26)

The hierarchy’s Bochner/normalization recurrence gives $aV_N+X_N\leq\nu-e_N$ . Combining it with $e_N\geq u v_{N+1}=u(1-P_{N+1})$ proves the matched budget. If a later successor is zero, set all subsequent families to zero; then $P_{N+1}=1$ and the same inequalities persist. All operations are finite spectral-calculus and weak form operations covered by the analytic foundations; no differentiability of $HU$ is asserted.

Lemma 145.3 (Finite extension from a delayed loss estimate)

Let $q\geq3$ be odd, $C_F,C_\theta\geq1$ , $u=z^{-1}$ , $\lambda\geq u/2$ , 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.

(145.27)

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 $p_*$ , with its short prefixes justified by retained actual losses:

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.

(145.28)

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_*).

(145.29)

Proof 145.3

Let $D_0=\max\{\nu-u,0\}\leq2a_qu^{q+1}$ . For $N\geq3$ the matched budget bounds $X_{N-2}\leq D_0+uP_{N-1}\leq D_0+uP_N$ . The assumed retained-prefix bound handles $N\leq2$ . At every justified $N\leq T_*$ , including a putative first exit,

(1-2C_\theta u)P_N \leq s_qu^q+C_F\kappa_{q-2}u^q+4C_\theta a_qu^{q+1} \leq K_qu^q/4.

(145.30)

The left coefficient is at least $3/4$ , so $P_N\leq p_*/3$ . A first exit cannot occur. The lower bound $1-P_N>0$ and the retained short prefixes ensure all families used in this argument exist. This is a finite induction over $N$ , not an estimate obtained by assuming that normalizers remain bounded forever.

Write $Y_j=\mathcal T^jY$ , $A_j=\|Y_j\|^2$ , $B_j=\|LY_j\|^2$ . Each hierarchy member is a positive multiple of $B^{1/2}Y_j$ . Its mean energy $\eta_j=A_j/\langle Y_j,BY_j\rangle$ is at least $\lambda$ , and its relative loss is

{p_j\over v_j}={B_j\over\langle Y_j,BY_j\rangle} =\eta_j{B_j\over A_j}.

(145.31)

Consequently $\sum_{j<m}B_j/A_j\leq P_m/[\lambda(1-p_*)] \leq4K_qu^{q-1}$ . The identity $\mathcal T^*\mathcal T=B-L^*L$ gives $A_1/A_0=\nu^{-1}-B_0/A_0$ . For $j\geq1$ , Bochner gives $\|H^{1/2}Y_j\|^2\leq A_{j-1}$ , and Cauchy–Schwarz gives

A_j^2\leq\|H^{1/2}Y_j\|^2\langle Y_j,BY_j\rangle \leq A_{j-1}(A_{j+1}+B_j).

(145.32)

Thus $A_{j+1}/A_j\geq A_j/A_{j-1}-B_j/A_j$ . Every ratio through $m$ is therefore at least $\nu^{-1}-4K_qu^{q-1}\geq z-(a_q+4K_q)u^{q-1}$ . Multiplication bounds $\|\mathcal T^mY\|^2/\|Y\|^2$ below by the $m$ th power of this quantity. It is positive under the smallness assumptions. Finite direct-sum amplification preserves the operator norm, proving the assertion.

Lemma 145.4 (Propagation from exact finite block norms)

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 $z_l=\|T^{m_l}\|^{2/m_l}>0$ . 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,

(145.33)

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).

(145.34)

Proof 145.4

Greedily divide $h$ by $m_s,m_{s-1},\ldots,m_0$ to write $h=\sum_{l=0}^sn_lm_l+r$ , $0\leq r<m_0$ , with $n_lm_l<m_{l+1}$ for $l<s$ . Submultiplicativity and the exact norm at each block give

{\|T^h\|\over z_s^{h/2}} \leq (R/z_s)^{r/2}\prod_{l<s}(z_l/z_s)^{n_lm_l/2}.

(145.35)

The remainder contributes at most $m_0(R-z_s)/(2z_s)\leq\eta_0/2$ to its logarithm. The decrement hypotheses give $(z_l-z_s)_+\leq\sum_{j=l}^{s-1}\Delta_j\leq2\Delta_l$ . Since $z_l/z_s\leq2$ , each nonzero lower-level contribution is at most $2m_{l+1}\Delta_l/z_l\leq2\eta$ . There are at most $s$ such levels. A level contributing a nonzero digit has $m_l\leq h$ , and $m_l\geq4^lm_0$ ; hence there are at most $\log_4(h+1)+1$ of them. This proves the claim.

Fences respected. The operator input uses the actual starting family and its own matched budget. The scale $z$ is obtained from the exact orbit window identity, not by replacing the global norm $R$ with $z$ in a generic hierarchy estimate. No source-specific polynomial or height bound is claimed without its hypotheses. No new bounded_by edges are proposed.

References¶

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