Song–Zhang v2: joint loss estimates - The KLS theorem and its methods

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 reconstructs the joint raw-orbit estimate and its normalized hierarchy version from source Sections 8.3–8.4 of Song & Zhang, 2026. The two claims are independent of any height profile or block realization.

Dependencies. Use Definition 9.1, Lemma 10.1, Lemma 10.3, Lemma 10.4, and the Appell testing identity in Proposition 9.1. All measures are centered regular covariance contractions with positive lower curvature. All output indices are ordered and retained in the Hilbert–Schmidt norms. Put $C_F=10^4$ . The operator notation and hierarchy sums are those of the two operator lemmas. No BKL or KLS bound, or consequence of either, is used.

Two joint-frame estimates¶

Lemma 146.1 (Raw moments and propagated normalization losses)

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}

(146.1)

If $0<z\le R$ , without a global power bound 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.

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 $d_0\le d$ 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,

(146.2)

\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.

(146.3)

Then, on each such justified prefix,

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

(146.4)

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

Proof 146.1

Put $Q_kh=\mathbb E[\mathcal A_kh]$ . Its norm is $k!c_k$ . The testing identity proved in Proposition 9.1 gives

\mathsf P_kQ_1w_j= \frac{Q_kw_{j-k+1}}{k!z^{(k-1)/2}}. \tag{B3}

(146.5)

The first $k$ slots are the coordinate slot and the $k-1$ newest derivative slots. Each fresh adjacent swap in the remaining family has norm at most $2\sqrt{D_i/z}$ by the operator primitive. After $h-1$ further steps its norm is multiplied by at most $E$ under the assumed global bound. In the local version it is multiplied by $(R/z)^{(h-1)/2}$ . These componentwise operators commute with all permutations of older slots.

Sorting a permutation of $N$ slots by adjacent swaps uses each adjacent position at most $N$ times. To see this, insert the letters successively into their required positions; a given boundary is crossed by at most the number of letters. Telescoping the orthogonal products and averaging therefore bounds the distance to the symmetric subspace by $N$ times the sum of the adjacent errors. Apply this with $N=3k$ to the next $3k$ slots after the first symmetric block. It yields

\|(I-\mathsf S_k)\mathsf P_kQ_1w_j\| \le6kE c_kz^{-k/2} \sum_{h=1}^{3k-1}\sqrt{D_{j-k+1-h}}.

(146.6)

For $k<d$ and $j\ge2d-1$ , the smallest index is $j-4k+2\ge1$ , so every fresh swap used is a genuine derivative swap. The terminal projection in (B3) has norm at most $c_dz^{-(d-1)/2}\sqrt{b_{j-d+1}}$ . Cauchy–Schwarz over fewer than $3k$ defects and Lemma 10.1 prove (B1). In the local version the longest propagation has at most $3k-2$ steps and $R/z\ge1$ ; replacing its squared norm by $(R/z)^{3k-2}$ gives the stated variant. The polynomial-envelope variant charges at most $E^2(3k)^{2\alpha}$ for that entire propagation, by the same argument.

For (B2), define $T_j=\sqrt{\beta_j}Q_1u^{j-1}$ for $j\ge1$ ; its squared norm is $p_j$ . Repeated testing gives

\mathsf P_kT_j= \frac{\prod_{i=0}^{k-1}\sqrt{\beta_{j-i}}}{k!}Q_ku^{j-k}.

(146.7)

Since $\|u^{j-k}\|\le1$ , the terminal squared norm is at most $B_*^kc_k^2$ . For completeness the inverse-normalization defect is

z^i=B^{1/2}F_{i+1}-\beta_{i+1}^{-1/2}u^i, \qquad \|\nabla z^i\|^2=\chi_i/\beta_{i+1}.

(146.8)

This follows by expanding its Dirichlet square and using $F_{i+1}=\sqrt{\beta_{i+1}}B^{1/2}u^i$ . Thus $u^{i+1}=\beta_{i+1}^{-1/2}P_+\nabla u^i+P_+\nabla z^i$ . The first term has symmetric newest slots and the second contributes a swap of norm at most $2\sqrt{\chi_i/\lambda}$ . Propagating an old swap uses the scalars of the original entire family, not freshly chosen normalizers: the next $h-1$ maps have norm at most $Eh^\alpha t_*^{(h-1)/2}$ . The same sorting argument gives

\|(I-\mathsf S_k)\mathsf P_kT_j\| \le6kE B_*^{k/2}c_k\lambda^{-1/2} \sum_{h=1}^{3k-1}h^\alpha t_*^{(h-1)/2} \sqrt{\chi_{j-k-h}}.

(146.9)

The sum’s convolution kernel has total mass at most $3k(3k+1)^\alpha t_*^{3k/2}$ . Weighted Cauchy–Schwarz and summation over $j$ bound its squared convolution by the square of that mass times the sum of the source $\chi$ ’s. Multiplication by the joint-frame weight $C_Fk^2$ and $(3k+1)^{2\alpha}\le16k^2$ gives the stated $\theta$ . Every source index is at most $N-3$ in a prefix $j<N$ . Keep $p_0,\ldots,p_{J-1}$ exactly. Subsequently use the largest available dyadic degree, capped at $d$ ; degree $k$ is available when $j\ge2k-1$ . Each nonterminal degree is used at most $2k$ times. Their coherent terms sum to the second part of $\Delta$ , while the terminal coherent terms sum to at most $N\tau$ . This proves (B2).

Fences respected. The estimates are conditional on their explicitly stated power envelopes and normalizer bounds. Neither estimate constructs those bounds or asserts a uniform coefficient radius. No bounded_by edge is proposed.

References¶

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