Song–Zhang v2: the improved inner iteration - The KLS theorem and its methods

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 dossier reconstructs the operator argument of Section 6 of Song & Zhang, 2026. A restricted inverse operator gives a low-energy gradient. Normalization pays for its skew linear moment. Averaging an orbit then reduces the startup losses to cubic order in the inverse radius. A joint dyadic frame controls all subsequent losses. The resulting comparison iterates a static coefficient radius; the additive conversion from that radius to $C_P$ is used only once at the end.

Dependencies. We use Lemma 7.1, Theorem 7.1, Theorem 7.2, the fixed-depth profiles of Theorem 7.3, and the three new foundations Lemma 10.1, Lemma 10.2, Proposition 10.1. The latter have their full proofs in the separate foundations dossier and require independent review. No BKL result or consequence of KLS is used.

Fix a centered regular measure of covariance $\Sigma\preceq I$ and Hessian $D^2W\succeq aI>0$ . Use the analytic operator $H$ , $B=H^{-1}$ on centered functions, $P_+f=f-\mathbb Ef$ , $D=P_+\nabla H^{-1/2}$ , $Lf=\mathbb E[Xf]$ , and $\mathcal T=P_+\nabla B=DB^{1/2}$ . Operators act componentwise on finite families and append ordered derivative slots. Put $P=C_P=\lambda^{-1}$ . All inverse powers, gradients and second derivatives below are justified by the form-domain and Bochner results in Lemma 7.1.

Restricted operator and normalized hierarchy¶

Lemma 137.1 (Restricted radius and extremizer)

For the restricted quotient

R=\sup_{\mathbb E\nabla f=0} \frac{\operatorname{Var}f}{\mathbb E|\nabla f|^2},

(137.1)

one has

\mathcal T^*\mathcal T=B-L^*L,\qquad \|\mathcal T\|^2=R,\qquad R\le P\le R+1.

(137.2)

There is a centered unit $f\in\operatorname{Dom}H$ with

\mathbb E\nabla f=0,\quad \mathbb E|\nabla f|^2=R^{-1}, \quad Bf=Rf+q\cdot X,\quad q=Lf.

(137.3)

The centered unit gradient $U=\sqrt R\nabla f$ has energy $\lambda\le\|H^{1/2}U\|^2\le R^{-1}$ .

Proof 137.1

Write $A=LH^{1/2}$ , initially on the form domain. Its adjoint is $A^*z=H^{1/2}(z\cdot X)$ and the Dirichlet identity gives $AA^*=I$ . Thus $A$ extends boundedly and $E=I-A^*A$ is an orthogonal projection. The mean removed from $\nabla H^{-1/2}h$ is $Ah$ , so $D^*D=E$ . The substitution $h=H^{1/2}f$ turns the restricted quotient into $\|EBE\|$ . Also $B^{1/2}EB^{1/2}=B-L^*L$ . Equality of the norms of $T^*T$ and $TT^*$ for $T=B^{1/2}E$ proves the radius identities. In the splitting $\operatorname{ran}A^*\oplus\operatorname{ran}E$ , the first diagonal block of $B$ is $ABA^*=\Sigma\preceq I$ and the second is at most $RI$ . Positivity controls the mixed term, hence

\langle(x,y),B(x,y)\rangle \le(\|x\|+\sqrt R\|y\|)^2 \le(1+R)(\|x\|^2+\|y\|^2).

(137.4)

This proves $P\le R+1$ ; the reverse $R\le P$ is immediate. The compact positive operator $B-L^*L$ has a positive top eigenvalue: $E$ has nonzero range since it removes only finitely many directions from an infinite-dimensional centered space. Its unit top eigenvector satisfies $Bf=Rf+(Lf)\cdot X$ . Both terms other than $Rf$ are in $\operatorname{Dom}H$ , so $f$ is too. Taking gradient means gives $Lf=R\mathbb E\nabla f+Lf$ , and pairing its $H$ -image with $f$ gives $1=R\mathbb E|\nabla f|^2$ . For $h=Bf$ , Bochner yields

1=R^2\mathbb E\|D^2f\|^2+ \mathbb E[(R\nabla f+q)^TD^2W(R\nabla f+q)].

(137.5)

Dropping the last nonnegative term gives the upper energy bound for $U$ ; the gap gives the lower bound.

Lemma 137.2 (Normalized hierarchy and delayed energy)

Start with any centered unit form-domain family $F_0$ of energy $\nu$ . Set $u^j=DF_j$ . If $u^j\ne0$ , put

\beta_{j+1}=\frac{\|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.

(137.6)

At a zero successor continue by zero, set $\beta=\lambda$ , $\chi=0$ . With $v_j=\|F_j\|^2$ , $e_j=\|H^{1/2}F_j\|^2$ , $p_j=\|LH^{1/2}F_j\|^2$ , and partial sums $P_N=\sum_{j<N}p_j$ , $X_N=\sum_{j<N}\chi_j$ , $V_N=\sum_{j<N}v_j$ ,

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

(137.7)

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

(137.8)

aV_N+X_N\le\nu-\lambda+\lambda P_N, \qquad aV_N+X_N\le\nu-\sigma+\sigma P_{N+1},\quad\sigma=R^{-1}.

(137.9)

Every finite nonzero prefix lies in the required form domains.

Proof 137.2

Cauchy–Schwarz in the spectral measure gives $\|u\|^4\le\|B^{1/2}u\|^2\|H^{1/2}u\|^2$ , proving the bounds on $\beta$ and $\chi\ge0$ . The squared $L^2$ norm of $\nabla H^{-1/2}F_j$ is $v_j$ and its mean is $LH^{1/2}F_j$ . Centering proves the mass identity. Bochner bounds the derivative energy of $u^j$ by $e_j-av_j$ , and inverse normalization subtracts $\chi_j$ . Thus each successor is again in the form domain and all inequalities hold inductively, including the zero convention. Since $L$ is a contraction, $p_j\le e_j$ . Telescoping, using $e_N\ge\lambda v_N$ , gives the first budget. For the second, write $v_{N+1}=\|DF_N\|^2=\|\mathcal T H^{1/2}F_N\|^2\le Re_N$ . The telescoped inequality $aV_N+X_N\le\nu-e_N$ then gives the claim.

The defect in inverse normalization also gives the identity

u^{j+1}=\beta_{j+1}^{-1/2}P_+\nabla u^j+P_+\nabla z^j, \qquad\|\nabla z^j\|^2=\chi_j/\beta_{j+1}. \tag{1}

(137.10)

Indeed take $z^j=B^{1/2}F_{j+1}-\beta_{j+1}^{-1/2}u^j$ and expand its Dirichlet norm. The first term has symmetric newest derivative slots. Thus a fresh adjacent swap has norm at most $2\sqrt{\chi_j/\lambda}$ . An older swap is acted on by $\sqrt\beta\mathcal T$ , with its scalar normalizer fixed to that of the original whole family, and never by an uncontrolled differentiation.

For Appell testing put $Q_kh=\mathbb E[\mathcal A_kh]$ . Its norm is at most $k!c_k$ . Integration by parts and the derivative rule for Appell polynomials give $Q_{k+1}/(k+1)=P_{k+1}Q_k\mathcal T$ and

P_k(\sqrt{\beta_j}Q_1u^{j-1}) =\frac{\prod_{i=0}^{k-1}\sqrt{\beta_{j-i}}}{k!}Q_ku^{j-k}. \tag{2}

(137.11)

These hold on full output direct sums. The removed mean gradient contributes zero since every positive-degree Appell polynomial is centered.

The initial restart and a mesoscopic orbit¶

A coarse coefficient seed needed below follows without a new localization argument. At the fixed depth $r_*$ of the static transfer, the certified v1 profile bounds $C_P$ by $\Gamma^2\ell_{r_*}(a^{-1})^2$ for one universal $\Gamma$ . Repeated Poincaré inequalities on Appell derivatives give $c_k\le C_P^{(k-1)/2}$ : start with covariance normalization at degree one, and use $K_k\le C_P k^2K_{k-1}$ . Apply Proposition 10.1. Since the fixed iterates of $g$ satisfy $\ell_{r_*}(d)\le Cg(d)$ , this proves

c_k\le[C_{\rm seed}\log(e+k)]^{k-1}\qquad(k\ge1). \tag{3}

(137.12)

The constant is uniform over all log-concave covariance contractions. In particular choose a universal $\bar c_3\ge c_3$ . Letwin gives $K_2\le8$ .

Proof 137.3

Normalize $U$ by $F_0=\sqrt\beta B^{1/2}U$ , $\beta=\|B^{1/2}U\|^{-2}$ . Its first loss is $p_0=\beta\|LU\|^2$ . The skew-credit theorem gives $1-p_0\ge\beta(e^{-1}-\|\operatorname{Sym}LU\|^2)$ . The next unit family $F_1/\sqrt{v_1}$ has energy at most $\beta/(1-p_0)$ , proving (4). Use now the restricted gradient $U=\sqrt R\nabla f$ and $\sigma=R^{-1}$ . Quadratic testing of $Hf=R^{-1}(f-H(q\cdot X))$ gives $2\operatorname{Sym}\mathbb E[X\otimes\nabla f]=R^{-1}Q_2f$ : the affine correction has zero pairing with the centered quadratic Appell polynomial, since its gradient has zero mean. Hence $\|\operatorname{Sym}LU\|^2\le K_2/(4R)$ , proving the energy bound. Also $\lambda\le\beta\le e\le\sigma$ , $\sigma-\lambda\le\sigma^2$ , and $\sigma/\lambda\le17/16$ . Since $K_2/4\le2$ ,

\nu-\lambda\le\frac{\sigma}{1-2\sigma^2} -\frac{\sigma}{1+\sigma}\le2\sigma^2.

(137.15)

We verify the three actual restart losses, rather than only their symmetric parts. On three slots let $P=(I+s_{12})/2$ , $S=(I+s_{23})/2$ . Then $\|T\|^2\le8(\|PT\|^2+\|(I-S)T\|^2)$ : on the trivial and sign representations $P+I-S=I$ , and on the standard representation its eigenvalues are $1\pm\sqrt3/2>1/8$ . This also follows directly from the two-dimensional reflection matrices. Let $\chi_{\rm pre}=e-\beta\le\sigma^2$ . For the original $p_1$ , use (1) with the initial normalization and (2) at degree two; the three-slot inequality gives

p_1\le2K_2\beta\beta_1+8(\beta_1/\beta)\chi_{\rm pre}.

(137.16)

For $j\ge2$ it gives

p_j\le2K_2\beta_j\beta_{j-1}v_{j-1} +8(\beta_j/\beta_{j-1})\chi_{j-2}. \tag{5}

(137.17)

Through $j=3$ , the crude loss bound $p_j\le\sigma$ implies $v_j\ge1-j\sigma\ge13/16$ , $\beta_j\le4\sigma/3$ , $\beta_j/\beta_{j-1}\le4/3$ , and $\beta_j\beta_{j-1}v_{j-1}\le4\sigma^2/3$ . The ratio estimate follows from decreasing energy and $p_{j-1}\le e_{j-1}=\beta_{j-1}v_{j-1}$ . The first energy budget gives $\chi_0\le2\sigma^2$ and $\chi_1\le3\sigma^2$ . Equation (5) therefore gives respectively $p_1\le32\sigma^2$ , $p_2\le(128/3)\sigma^2$ , $p_3\le(160/3)\sigma^2$ . Divide their sum by $v_1\ge1-\sigma$ for the unit restart. Its first three losses are at most $128\sigma^2/(1-\sigma)\le160\lambda^2$ . All families used are nonzero by the mass estimates.

Proof 137.4

First, if required normalizers satisfy $\beta_i\le3\sigma$ and $\beta_i/\lambda\le6$ , block recovery from the certified curvature dossier, with one coordinate slot and three derivative slots, gives

p_j\le A_0\sigma^3+B_0(\chi_{j-2}+\chi_{j-3})\quad(j\ge3), \quad A_0=486\bar c_3^2,\ B_0=50000. \tag{7}

(137.19)

Indeed its recovery constants are 3 and 4. The degree-three coherent term in (2) has squared norm at most $27\bar c_3^2\sigma^3$ . The two derivative swaps in (1), one propagated once by a map of norm at most $\sqrt3$ , give a symmetrization error at most $6\sqrt{\beta_j/\lambda}(\sqrt{\chi_{j-2}}+\sqrt{3\chi_{j-3}})$ . Its square is at most $1296(\chi_{j-2}+\chi_{j-3})$ . Squaring the recovery inequality with weights $18,32$ proves (7).

Put $K=512+4A_0$ , $M=\lfloor R\rfloor$ , and $p_*=K\sigma^2$ . Increase $R_0$ so that $R_0\ge\max(16,4B_0,\sqrt{8K})$ . Use the compensated restart, whose energy obeys $\nu\le\sigma+3\sigma^3$ and $P_3\le160\sigma^2$ . Before a first exit of $P_N$ above $p_*$ , its required normalizers are at most $\nu/(1-p_*)\le3\sigma$ with ratio to $\lambda$ at most six. The possible exit successor still has mass at least $1-p_*-\nu>0$ . Sum (7) through $N\le M$ , use the delayed budget at $N-2$ , and obtain

P_N\le160\sigma^2+A_0N\sigma^3 +2B_0(3\sigma^3+\sigma P_N).

(137.20)

Since $N\le\sigma^{-1}$ and $2B_0\sigma\le1/2$ , this gives $P_N\le(323+2A_0)\sigma^2<3K\sigma^2/4$ . The first three prefixes satisfy that bound directly, so no exit occurs.

Take the restricted extremizer $f$ and let $W_k=\mathcal T^kf$ , $a_k=\|W_k\|^2$ , $b_k=\|LW_k\|^2$ . Then $a_0=1$ , $a_1=R$ , and $W_1/\sqrt R=U$ . The compensated normalized hierarchy is exactly $F_j=s_j B^{1/2}W_{j+2}$ for positive scalars $s_j$ ; this follows by substituting the definition of each successor. For $\eta_k=a_k/\langle W_k,BW_k\rangle$ this gives

e_j/v_j=\eta_{j+2},\qquad b_{j+2}/a_{j+2}=p_j/(v_j\eta_{j+2}),\qquad\eta_k\ge\lambda.

(137.21)

Thus the prefix estimate implies $\sum_{k=2}^{M+1}b_k/a_k\le K\sigma^2/[\lambda(1-p_*)]\le2K\sigma$ . The skew-credit argument already used in (4) gives $a_2/a_1=\|\mathcal TU\|^2=\eta_1^{-1}-\|LU\|^2 \ge e_U^{-1}-K_2/(4R)\ge R-2\sigma$ . Bochner gives the form inequality $\mathcal T\mathcal T^*\preceq B$ : $\|H^{1/2}\mathcal Th\|\le\|h\|$ and duality prove it. Consequently

a_{k+1}+b_k=\langle W_k,BW_k\rangle \ge\|\mathcal T^*W_k\|^2\ge a_k^2/a_{k-1}.

(137.22)

The ratios $a_k/a_{k-1}$ are therefore at least $R-(2+2K)/R$ through $M+2$ , using the finite, positive hierarchy above. Multiply these ratios to obtain the lower bound in (6), with $C_*=2+2K$ . The upper bound follows from $\|\mathcal T\|^2=R$ .

Averaged orbit and its static radius¶

Proof 137.5

Bochner gives $\|H^{1/2}w_j\|^2\le b_{j-1}/z$ . Also $\langle w_j,Bw_j\rangle=z b_{j+1}+l_j$ from $\mathcal T^*\mathcal T=B-L^*L$ . Expansion of the Dirichlet square $D_j=\langle w_j,Bw_j\rangle-2z b_j+z^2\|H^{1/2}w_j\|^2$ proves (8). Identity (9) follows by adding and subtracting $zI$ . Each $w_j$ , $j\ge1$ , is a centered gradient in its newest slot; subtracting a constant vector preserves this property. Its weak second derivatives commute, so the first term in (9) is symmetric in the two newest slots. The other term has norm at most $\sqrt{D_j/z}$ , proving the swap bound. All maps commute with permutations of older slots.

Lemma 137.6 (Averaged startup)

For all sufficiently large universal $R$ , set $m=\lfloor R\rfloor$ and $z=\|\mathcal T^m\|^{2/m}$ . There are universal $C_S,C_7,C_0$ and a centered unit finite family with energy $\nu$ and first seven losses satisfying

\nu\le\frac{z^{-1}}{1-C_Sz^{-3}},\quad S_7:=\sum_{j=0}^6p_j\le C_7z^{-3},\quad D_0:=\max(\nu-R^{-1},0)\le C_0z^{-3}. \tag{10}

(137.25)

Furthermore

\|\mathcal T^l\|\le C_{\rm blk}z^{l/2}\quad(l\ge0), \qquad C_{\rm blk}=(R/z)^{(m-1)/2}\le2. \tag{11}

(137.26)

Proof 137.6

The map $\mathcal T=DB^{1/2}$ is compact and (6) makes $\mathcal T^m\ne0$ . Choose a unit top right singular vector $f$ of $\mathcal T^m$ and use the orbit just defined. Then $b_0=b_m=1$ , $b_{j+1}\le(R/z)b_j$ , and $b_j\ge(z/R)^{m-j}$ for $0\le j\le m$ . Equation (6) gives $\log(R/z)\le2C_*/R^2$ beyond a fixed threshold. Consequently

b_j\le2\ (0\le j\le m+9),\qquad b_j\ge1/2\ (0\le j\le m). \tag{12}

(137.27)

For $N=m+8$ , summing (8) gives

\sum_{j=1}^ND_j\le\sum_{j=1}^Nl_j +z(b_{N+1}-b_N+1-b_1)\le\sum_{j=1}^Nl_j+2C_*.

(137.28)

The last bound uses $z(b_{N+1}-b_N)\le(R-z)b_N\le2C_*/R$ and $z(1-b_1)\le z[1-(z/R)^{m-1}]\le C_*$ after a fixed enlargement.

We need full linear moments, not just symmetric moments. At degree two, Appell testing gives $P_2Q_1w_2=Q_2w_1/(2\sqrt z)$ . The three-slot frame used in (5) and the fresh defect (9) yield

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

(137.29)

For $j\ge3$ , block recovery on one coordinate and three derivative slots has constants $3,4$ . The degree-three coherent term is $P_3Q_1w_j=Q_3w_{j-2}/(3!z)$ , while the symmetrization error is at most $6z^{-1/2}(\sqrt{D_{j-1}}+\sqrt{R/z}\sqrt{D_{j-2}})$ . Since $R/z\le2$ , squaring gives

l_j\le18\bar c_3^2z^{-2}b_{j-2} +10^4z^{-1}(D_{j-1}+D_{j-2}). \tag{13}

(137.30)

Since $l_1\le b_1\le2$ , (12),(13), and $N\le4z$ imply

\sum_{j=1}^Nl_j\le2+4K_2/z+36\bar c_3^2N/z^2 +(8+2\cdot10^4)z^{-1}\sum_{j=1}^ND_j.

(137.31)

Absorb the last term in the preceding defect-sum inequality by increasing one universal radius threshold. It follows that

\sum_{j=1}^{m+8}D_j\le C_D, \qquad\sum_{j=2}^{m+8}l_j\le C_L/z \tag{14}

(137.32)

for universal constants.

Use the finite direct sum $Y=\bigoplus_{j=0}^{m-1}w_j$ and put $S_k=\sum_{j=k}^{m+k-1}b_j$ , $C_k=\sum_{j=k}^{m+k-1}l_j$ . The family $U=\mathcal TY/\sqrt{zS_0}$ has norm one because $b_m=b_0$ . It is centered and a genuine gradient in each output column; Bochner gives $e_U\le z^{-1}$ . Quadratic testing also gives $\|\operatorname{Sym}Lw_j\|^2\le K_2b_{j-1}/(4z)$ for $j\ge1$ . Use this for $j=1,2$ , and (14) for the remaining indices. Since $S_0\ge m/2$ and $m$ is comparable to $z$ , it follows that $\|\operatorname{Sym}LU\|^2\le C_S z^{-2}$ .

Start the hierarchy at $F_0=B^{1/2}Y/\|B^{1/2}Y\|$ . Its first successor, renormalized to unit mass, is the inverse normalization of $U$ . Apply (4); the unit restart at $F_2$ then has energy at most $(z-C_Sz^{-2})^{-1}$ , as claimed. To verify its losses exactly, each $F_k$ is a positive scalar multiple of $B^{1/2}\mathcal T^kY$ , and

\|\mathcal T^kY\|^2=z^k S_k,\qquad \langle\mathcal T^kY,B\mathcal T^kY\rangle =z^k(zS_{k+1}+C_k).

(137.33)

Consequently its loss fraction is

p_k/v_k=C_k/(zS_{k+1}+C_k). \tag{15}

(137.34)

For $2\le k\le8$ , (14) gives $C_k\le C_L/z$ , while the interval in $S_{k+1}$ contains $k+1,\ldots,m$ . Equation (12) gives $S_{k+1}\ge m/4\ge z/8$ at $m\ge16$ . Thus these seven loss fractions are at most $8C_Lz^{-3}$ . They are exactly the first seven losses of the unit restart, whose masses are at most one; take $C_7=56C_L$ . Every family through $F_9$ is nonzero because its window contains positive terms in the interior of (12). Equation (6) gives $z^{-1}-R^{-1}=O(z^{-3})$ , and the remaining energy error is $O(z^{-4})$ . This proves the bound on $D_0$ . Finally, for $l=qm+r$ with $0\le r<m$ , $\|\mathcal T^l\|\le R^{r/2}z^{qm/2}$ , proving (11). The estimate on $\log(R/z)$ already proves $C_{\rm blk}\le2$ .

Proof 137.7

Choose a universal $R_*$ above the preceding thresholds with $C_*/R_*\le1$ , and such that for $R\ge R_*$ , $C_{\rm seed}^2\log(e+R)^2\le R/2\le z$ . Put $H_0=R_*+1$ . For $R<R_*$ define $Z=H_0$ ; otherwise define $Z=\max(H_0,z)$ . Iteration of $Q_{k+1}/(k+1)=P_{k+1}Q_k\mathcal T$ across $m$ degrees, keeping older output slots and using contraction of every symmetrization, gives

c_d\le\|\mathcal T^m\|c_{d-m}=z^{m/2}c_{d-m} \quad(d\ge m+1).

(137.36)

For $1\le d\le m$ , the seed (3) bounds $c_d$ by $z^{(d-1)/2}$ . Repeated subtraction of $m$ proves this for every degree, including the remainder degree one. If $R<R_*$ , repeated Poincaré on Appell derivatives gives $c_d\le C_P^{(d-1)/2}\le H_0^{(d-1)/2}$ . For large $R$ , $C_P\le R+1\le z+2$ by (6), and $z\le R\le C_P$ . These statements and the small- $R$ case give (16). Only finite-degree polynomial inequalities are used in future transfers; no continuity of $Z$ or $m$ is asserted or needed.

Joint losses and finite comparison¶

Lemma 137.8 (Loss bound for the averaged startup)

For the averaged starting family, suppose a prefix’s required normalizers are at most $B_*$ , and set $t=\max(1,B_*z)$ . For dyadic $d\ge4$ , define

\theta=324C_FC_{\rm blk}^2 \sum_{k<d\text{ dyadic}}k^6(B_*t^3)^kc_k^2,

(137.37)

\Delta=S_7+2C_F\sum_{4\le k<d\text{ dyadic}}k^2B_*^kc_k^2, \qquad\tau_d=C_FdB_*^dc_d^2,\qquad C_F=10^4.

(137.38)

Then

P_N\le\Delta+N\tau_d+(\theta/\lambda)X_{\max(N-2,0)}. \tag{17}

(137.39)

Proof 137.8

Apply the joint frame to $T_j=\sqrt{\beta_j}Q_1u^{j-1}$ . Equation (2) bounds its degree- $k$ coherent projection by $B_*^{k/2}c_k$ . A fresh derivative swap is bounded by $2\sqrt{\chi/\lambda}$ . After $h-1$ additional steps, (11) bounds it by $2C_{\rm blk}t^{(h-1)/2}\sqrt{\chi/\lambda}$ : the block constant appears once for the entire propagation. These maps commute with old slot permutations. Sorting the $3k$ derivative slots uses each adjacent swap at most $3k$ times; telescoping products of isometries and averaging gives

\|(I-S_k)P_kT_j\| \le6kC_{\rm blk}B_*^{k/2}c_k\lambda^{-1/2} \sum_{h=1}^{3k-1}t^{(h-1)/2}\sqrt{\chi_{j-k-h}}.

(137.40)

The inner convolution kernel has $\ell^1$ norm at most $3kt^{3k/2}$ . Weighted Cauchy–Schwarz, summation in $j$ , and the frame weight $C_Fk^2$ produce the asserted coefficient $324C_FC_{\rm blk}^2k^6$ . Every defect index in a prefix $j<N$ is at most $N-3$ , explaining the delay in (17). At generation $j\ge7$ , use the largest available dyadic degree, capped at $d$ ; availability is $2k-1\le j$ . Each degree $k<d$ is used for $2k$ generations and contributes coherent loss at most $C_FkB_*^kc_k^2$ per generation. The first seven losses are kept separately. The terminal contribution is bounded by $N\tau_d$ . Shorter prefixes satisfy the same bound by nonnegativity. Every required slot and defect index is present before invoking the frame.

Lemma 137.9 (Delayed finite comparison)

Let a centered unit hierarchy start at energy $\nu\le1/4$ and write $D_0=\max(\nu-\sigma,0)\le\sigma$ . Fix $0<p\le1/8$ and $B_*=\nu/(1-p)$ . Suppose the bound

P_N\le\Delta+N\tau+(\theta/\lambda)X_{\max(N-2,0)}, \qquad p_0+p_1\le\Delta

(137.41)

holds for every prefix whose required normalizers are at most $B_*$ , including its first exiting prefix. If

\theta\sigma/\lambda\le1/8,\qquad \Delta+\theta D_0/\lambda\le p/64,

(137.42)

then $a<12(\sigma+D_0/p)\tau$ .

Proof 137.9

For $N\ge3$ , the delayed budget at $N-2$ gives $X_{N-2}\le D_0+\sigma P_{N-1}\le D_0+\sigma P_N$ . For $N\le2$ the startup estimate suffices. Thus every justified prefix obeys

(1-\theta\sigma/\lambda)P_N\le\Delta+N\tau+\theta D_0/\lambda.

(137.43)

The first two successors exist since each loss is at most $\nu\le1/4$ . The delayed budget at one yields $a\le D_0+\sigma(p_0+p_1) \le D_0+\sigma p$ . Suppose $a\ge12(\sigma+D_0/p)\tau$ . Put $m=\lceil(D_0+\sigma p)/a\rceil$ and $M=m+1$ . Then $M\le3(D_0+\sigma p)/a$ and $M\tau\le p/4$ . Before a first $N\le M$ with $P_N>p$ , required normalizers use preceding masses at least $1-p$ , hence are at most $B_*$ . The possible successor has mass at least $1-p-\nu>0$ . The preceding inequality gives $P_N\le p/64+p/4+P_N/8$ , hence $P_N\le cp$ with $c=17/56<1$ . There is no exit and this holds throughout the finite horizon. At $m=M-1$ , the delayed budget and surviving mass then give respectively

aV_m\le D_0+c\sigma p,\qquad aV_m\ge(1-cp)(D_0+\sigma p).

(137.44)

Subtract the upper bound from the lower. The difference is $\sigma p(1-c-cp-cD_0/\sigma)\ge\sigma p(1-2c-cp)>0$ , a contradiction. No infinite hierarchy limit is used.

Closing the depth recurrence¶

Proof 137.10

We first iterate the static radius $Z$ in (16). Let $r\ge r_*$ and suppose its already established profile is $Z\le\widetilde\Gamma_r^2\ell_r(a^{-1})^2$ , where $\widetilde\Gamma_r\ge G(r+1)^{1/6}$ ; $G$ will be one large universal constant. Require $G^2>2H_0$ . Static transfer then gives

c_k\le[Q\ell_r(k)]^{k-1},\qquad Q=(1+r^{-2})\widetilde\Gamma_r.

(137.46)

Together with (3), there are two simultaneous coefficient bounds, the second independent of $r$ . Put

s=r+1,\quad\delta=(32s)^{-1},\quad p=(4096s)^{-1}, \quad b=(1-2p)^{-1},\quad q=e^{-\delta},

(137.47)

M_0=2^{44}(1+C_F),\qquad k_0=\lceil64\delta^{-1}\log(M_0/\delta)\rceil.

(137.48)

For a dyadic $d\ge4$ , suppose for contradiction

Z>b^4e^\delta Q^2\ell_r(d)^2. \tag{18}

(137.49)

This exceeds the floor, so $Z=z$ is the actual block radius. Put $u=z^{-1}$ . Then $b^4uQ^2\ell_r(d)^2<q$ , $u\le G^{-2}s^{-1/3}$ , and $u^3\le G^{-6}s^{-1}$ . The averaged startup gives (10); choose $G$ so that $C_Su^3\le p$ and all universal radius thresholds hold. Prior to an exit of cumulative mass $p$ , the needed normalizers satisfy

B_*\le u/(1-p)^2\le bu,\qquad \max(1,B_*z)\le b.

(137.50)

Since $C_{\rm blk}\le2$ , the preceding joint loss estimate is bounded by

\theta_d=1296C_F\sum_{k<d\text{ dyadic}}k^6(b^4u)^kc_k^2,

(137.51)

\Delta_d=C_7u^3+2C_F\sum_{4\le k<d\text{ dyadic}}k^2(bu)^kc_k^2, \qquad\tau_d=C_Fd(bu)^dc_d^2. \tag{19}

(137.52)

We check these sums uniformly in $r,d$ . There is a universal $C_M$ such that $k_0\le C_Ms^2$ for every $s\ge1$ , since $\log s\le s$ . Therefore $s^{-1/6}\log(e+k_0)$ is bounded uniformly: bound it by $\log(e+C_M)+2s^{-1/6}\log s$ , and use $\sup_{s\ge1}s^{-1/6}\log s=6/e$ . One fixed enlargement of $G$ thus ensures $\widetilde\Gamma_r\ge2C_{\rm seed}\log(e+k_0)$ . For $k\le k_0$ , the seed and (18) give

b^4u[C_{\rm seed}\log(e+k)]^2\le1/4,

(137.53)

(b^4u)^kc_k^2\le b^4u\,4^{-(k-1)},

(137.54)

(bu)^kc_k^2\le b^3u^3C_{\rm seed}^4\log(e+k)^4\,4^{-(k-3)} \quad(k\ge4).

(137.55)

The resulting integer series, with weights $k^6$ and $k^2$ , converge. They contribute respectively at most a universal constant times $u$ and one times $u^3$ . For $k_0<k<d$ , the iterated-logarithm coefficient bound and monotonicity give

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

(137.56)

For $0<\delta\le1$ , comparison with the integral of $(x+1)^je^{-\delta x/2}$ gives, with the generous fixed constants shown,

\sum_{k>k_0}k^6e^{-\delta k} \le2^{30}\delta^{-7}e^{-\delta k_0/2},\qquad \sum_{k>k_0}k^2e^{-\delta k} \le2^{10}\delta^{-3}e^{-\delta k_0/2}.

(137.57)

For example $k^je^{-\delta k/2}$ is integrated over $[k-1,k]$ after replacing $k$ by $x+1$ and weakening the exponential by $e^{1/2}$ ; expanding $(x+1)^j$ and integrating each monomial gives these bounds. Also $e^{-\delta k_0/2}\le(\delta/M_0)^{32}$ . The chosen $M_0$ therefore makes the tail contributions at most $u$ and $p/512$ , respectively. Replacing dyadic sums by integer sums only increases them. We have universal constants $C_\theta,C_\Delta$ such that

\theta_d\le C_\theta u,\qquad \Delta_d\le C_\Delta u^3+p/512. \tag{20}

(137.58)

Because $C_P\le z+2$ , we have $\lambda\ge1/(z+2)$ . Also $\sigma=R^{-1}\le u$ and $\sigma\ge u/2$ above a fixed threshold. Choose $G$ once so large that, in addition to all previous requirements,

2C_\theta/G^2\le1/8,\qquad (C_\Delta+2C_\theta C_0)/G^6\le2^{-20},

(137.59)

C_S/G^6\le1/4096,\qquad G^{-2}+4096C_0G^{-6}\le1,\qquad G^4\ge2C_0.

(137.60)

All these are compatible lower bounds on a single universal constant. They imply

\theta_d\sigma/\lambda\le1/8,\qquad \Delta_d+\theta_dD_0/\lambda\le p/64,\qquad \sigma+D_0/p\le1.

(137.61)

Indeed $\theta_dD_0/\lambda\le2C_\theta C_0u^3$ for $z\ge2$ , and $u+4096sC_0u^3\le G^{-2}+4096C_0G^{-6}$ . The same thresholds ensure $\nu\le1/4$ and $D_0\le\sigma$ . The delayed comparison consequently gives

a<12C_Fd(bu)^dc_d^2. \tag{21}

(137.62)

If $0<a\le1$ , put $h=\log(e+a^{-1})$ and choose dyadic $d$ with

C_Ts^2h\le d<2C_Ts^2h

(137.63)

for a sufficiently large universal $C_T$ . It can be chosen so that $\log(12C_Fd/a)\le\delta d$ for all $s,h\ge1$ : the left side is at most $\log(24C_FC_T)+2\log s+2h$ , while the right is at least $(C_T/32)sh$ . It suffices that $C_T/32\ge\log(24C_FC_T)+4$ and $d\ge4$ . Writing $L_*=Q^2\ell_r(d)^2\ge1$ , the assumed (18) and (21) imply

a<12C_FdL_*^{-1}(buL_*)^d \le12C_Fd e^{-\delta d}\le a,

(137.64)

a contradiction. Thus

Z\le b^4e^\delta(1+r^{-2})^2\widetilde\Gamma_r^2\ell_r(d)^2. \tag{22}

(137.65)

For $a\ge1$ , Brascamp–Lieb gives $C_P\le1$ ; by (16), $Z\le H_0$ , so the large universal profile already covers this case.

The logarithmic elasticity estimate from the static-transfer proof gives

\ell_r(d)\le\zeta_r\ell_{r+1}(a^{-1}),\qquad \zeta_r=\exp(2^{-r}\log(2C_T(r+1)^2)).

(137.66)

Furthermore

\log(b^4e^\delta)\le16p+\delta =9/(256s)\le\tfrac13\log(1+1/s),

(137.67)

where $\log(1+1/s)\ge1/(2s)$ suffices. Define

B_{r_*}=1,\quad B_{r+1}=B_r(1+r^{-2})^2\zeta_r^2,\qquad \widetilde\Gamma_r^2=G^2(r+1)^{1/3}B_r.

(137.68)

The product is bounded above and below by positive universal constants, since its logarithm is at most a constant multiple of $\sum r^{-2}+\sum2^{-r}(1+\log(r+1))<\infty$ . At the one fixed depth $r_*$ , the v1 curvature theorem and (16) initialize the $Z$ profile by one further enlargement of $G$ . Equation (22) now proves the next profile at each finite depth, with the chosen amplitude, because the factor $b^4e^\delta$ is paid by $((r+2)/(r+1))^{1/3}$ . Every coefficient premise used at a given step has therefore already been proved at its previous finite depth. Cover the finitely many smaller depths by the certified v1 profiles and increase one universal constant. Finally (16) and $\ell_r\ge1$ give $C_P\le(\widetilde\Gamma_r^2+2)\ell_r(a^{-1})^2$ . The additive two is absorbed once here, after the radius induction. This proves the asserted universal polynomial envelope in $r$ .

Fences respected. No bounded_by edge is proposed for this node. The brief’s small-degree initialization requirement is met by the static transfer starting at degree two and the fixed seed (3). All large-radius requirements are universal and are imposed before starting the depth induction; their common constant is $G$ . This profile is still curvature dependent, and its dimension transfer still depends on dimension. No CMH, occupation or trace antecedent is discharged. There is no infinite-depth limit and no use of a KLS-equivalent coefficient assertion without its explicit profile premise.

References¶

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