Song–Zhang v2: finite chains with retained bounds

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 analytic finite-chain construction behind Proposition 9.23 of Song & Zhang, 2026, including its one-inherited-floor specialization. It proves Proposition 10.2 from the explicit raw-frame, normalization, Green, restart, and finite-power inputs below. No small-loss or summable profile is an input.

Dependencies. Use Definition 9.1, Proposition 9.1, Theorem 7.1, Lemma 10.3, Lemma 10.4, Lemma 10.5, Lemma 10.6, Lemma 10.7, Lemma 10.8, Lemma 10.9, and Lemma 10.10. Every argument is finite. No BKL result or KLS assertion is used.

A degree sum uniform in the number of retained bounds¶

Lemma 148.1 (Disjoint degree ranges)

Fix $M,h\geq0$ integral and $c>0$ . For a finite sequence $0<\alpha_N\leq\cdots\leq\alpha_0\leq1/16$ , set $K_i=\lceil L\alpha_i^{-2}\rceil$ . Let nondecreasing integers $\Xi_i$ satisfy $\Xi_i\geq K_{i+1}$ for $i<N$ . Suppose $0\leq\rho_k\leq1$ , $\rho_k\leq C_0^{-1}$ for $k\leq16K_0$ , and $\rho_k\leq(1+\alpha_i)^{-1}$ for $k\leq\Xi_i$ . Then

\sum_{k\geq h+1}k^M\rho_k^{k-h}e^{-c\alpha_N(k-h)}\leq C_{M,h,c},

(148.1)

uniformly in $N$ , all margins and all such sequences. The bound can be made arbitrarily small by fixed increases of $L,C_0$ .

Proof 148.1

Choose $L$ so $K_0\geq2h+2$ . The range $k\leq K_0$ is bounded by $\sum_{j\geq1}(j+h)^MC_0^{-j}$ . Partition the remaining integers into $(K_0,\Xi_0]$ , $(\Xi_{i-1},\Xi_i]$ for $1\leq i<N$ , and $(\Xi_{N-1},\infty)$ . Repeated endpoints give empty intervals and contribute nothing. In the $i$ th finite interval $k>K_i$ and hence $\alpha_i k\geq\sqrt{Lk}$ . Moreover

\rho_k^{k-h}\leq\exp[-\alpha_i(k-h)/2] \leq\exp[-\sqrt{Lk}/4].

(148.2)

In the last interval the same conclusion, with constant $c/2$ , follows from $k>K_N$ and the explicit $\alpha_N$ exponential factor. Because these intervals are disjoint, their total is bounded by the single series

\sum_{k>K_0}k^M\exp[-b\sqrt{Lk}],\qquad b=\min\{1/4,c/2\}>0.

(148.3)

For integers $l\geq1$ , the range $l^2\leq k<(l+1)^2$ contains at most $2l+1$ terms, each bounded by $(l+1)^{2M}e^{-b\sqrt L\,l}$ . The resulting polynomial times geometric series converges. With $L\geq1$ its summands are dominated by the summable series at $L=1$ and decrease to zero as $L\to\infty$ . The finite initial range is made small by $C_0$ . There is no factor proportional to the number of intervals.

Actual block realization¶

Before the height bookkeeping, the degree sum can be used to retain a finite chain of coefficient floors in an actual block construction. The following formulation records separately the orbit and normalized frame inputs, so the dependency is visible.

Lemma 148.2 (Finite chains of coefficient floors)

Let $G_*$ 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. Let $G,H_0,\ldots,H_{N-1}$ be nondecreasing valid coefficient majorants with $1\leq G\leq H_l\leq G_*$ and $c_k\leq G(k)^{k-1}$ . 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,

(148.4)

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.

(148.5)

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,

(148.6)

and constants $a_Q,s_Q\geq0$ obeying

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

(148.7)

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),

(148.8)

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.

(148.9)

For each odd $p\leq Q$ , 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.

Proof 148.2

Write $D=1+\epsilon$ and $\kappa_q=(F^2/4)^{q-1}$ for odd $q\geq1$ . The constants for the two joint frames are fixed before any cutoffs. First fix $C_{\rm cut}$ and $C_{\rm deg}$ , then the Green, restart and extension constants, and finally increase $C_0,C_X$ to exceed their finitely many universal thresholds. Enlarging $C_0,C_X$ only raises the floor or available degree range.

For each tested degree put

\rho_k=\min\{1,G_*(k)^2/F^2,H_l(k)^2/F^2:0\leq l<N\}.

(148.10)

The preceding disjoint-range lemma gives uniformly bounded, and when needed uniformly small, moments with any of the finitely many powers $k^M$ used below. If $G(d)\leq F$ and $z\geq DF^2$ , then for $k\leq d$

G(k)^2/z\leq\rho_k e^{-\epsilon/2}.

(148.11)

The raw frame under $\|\mathcal T^h\|\leq4(h+1)z^{h/2}$ has the form

l_j\leq C_Fe c_e^2z^{-(e-1)}b_{j-e+1} +C\sum_{k<e,\ k\ {\rm dyadic}}k^7c_k^2z^{-k} \sum_{s=k}^{4k-2}D_{j-s},\qquad j\geq2e-1. \tag{F1}

(148.12)

Use at each $j$ the largest available dyadic $e\leq d$ . Together with the degree-two identity this gives a convolution kernel whose zeroth and first moments are at most $C/z$ : summing over the $O(k)$ delays gives powers $k^8$ and $k^9$ , respectively. Before the orbit has been bounded, propagation gives $b_j\leq16(j+1)^2$ . A degree $k<d$ is used only for $2k-1\leq j\leq4k-2$ . Its contribution to the early weighted coherent sum, after multiplication by $z$ , is at most $C k^6\rho_k^{k-2}e^{-\epsilon(k-2)/2}$ , because $G(k)^2\leq C_*k$ . Degree two is bounded directly by the fixed quadratic estimate. The moment lemma therefore gives weighted early sum $C/z$ . For later indices the coherent term is $\gamma b_{j-d+1}$ , with $\gamma=C_Fd c_d^2z^{-(d-1)}$ . These are exactly the hypotheses of the Green sequence lemma, with a universal $K$ independent of $N$ .

Consider extension at odd $q$ from an already initialized actual family. Let $\omega_q=1$ in the first band and equal the margin of its band otherwise; put $\beta=1+\omega_q/128$ . Before the exit level of Lemma 10.9, the normalizers are at most $\beta u$ when its two smallness bounds are at most $\omega_q/1024$ . The normalized joint frame has kernel bounded by

Cu\sum_{k<e_q,\ k\ {\rm dyadic}}k^8 (\beta^4G(k)^2/z)^{k-1}. \tag{F2}

(148.13)

In the small band all degrees are below $16K_0$ , so the geometric floor absorbs $\beta^4$ . In a later band the tested degrees lie below that band’s $\Xi_l$ . Earlier margins are no smaller than $\omega_q$ ; $4\log(1+\omega_q/128)\leq\omega_q/32$ consumes only a fixed fraction of $\log(1+\alpha_l)\geq\alpha_l/2$ . The last range uses $z/F^2\geq1+\epsilon$ . Thus (F2) is at most $C_\theta u$ , independently of the number of bands.

For $d=e_q$ , $g=G(d)^2$ , and $\rho=g/F^2$ , the terminal term $\tau=C_Fd(\beta u)^dc_d^2$ satisfies

{\tau\over\kappa_q\kappa_{q-2}u^{2q}} \leq C_Fd g^3 4^{2q-4}\rho^{d-4}\beta^dD^{2q-d}. \tag{F3}

(148.14)

This follows by substituting $c_d^2\leq g^{d-1}$ and $z\geq DF^2$ ; the identity $(F^2)^3\rho^{d-1}=g^3\rho^{d-4}$ removes the floor. In the small band $8q\leq d<16q$ , $\rho\leq C_0^{-1}$ , and (F3) is at most $Cq^4C_0^4(16\beta^{16}/C_0^8)^q$ . In another finite band $d\geq C_{\rm deg}q/\alpha_l$ , $\rho\leq(1+\alpha_l)^{-1}$ and $\beta=1+\alpha_l/128$ . The exponential damping dominates $4^{2q}$ ; since $q>K_l$ gives $\alpha_l^{-1}<\sqrt q$ , the residual polynomial is at most $Cq^6e^{-cC_{\rm deg}q}$ . In the final band use $D^{2q-d}$ instead of the $\rho$ factor and obtain the same bound. Fixed choices of constants make (F3) at most $C_F$ . Thus the delayed normalized frame and the exact matched budget satisfy all loss-estimate inputs of Lemma 10.9.

For initialization at order $p\geq5$ , suppose $m$ is within a factor four of $z^{p-2}/\kappa_{p-2}$ . The two coherent smallness expressions are bounded by

\gamma m^2/z\leq Cd g\rho^{d-2}4^{2p-6}D^{2p-d-4},\quad J_p\gamma zm\leq CJ_pd g^2\rho^{d-3}4^{p-3}D^{p-d},\quad d=d_p.

(148.15)

In the small band these are at most $Cp^2C_0^2(16/C_0^4)^p$ and $Cp^4C_0^3(4/C_0^4)^p$ . In each other band its own damping bounds them by $Cp^3e^{-cC_{\rm deg}p}$ and $Cp^6e^{-cC_{\rm deg}p}$ . These estimates are uniform in the band index, without summing them. Also $m\geq cF^2 4^{p-3}D^{p-2}$ , whereas $J_p,d_p\leq Cp^{3/2}$ , so $m\geq2d_p$ and $J_p\leq m/4$ . The Green and actual-restart lemmas now give $a_p+s_p+1\leq C\kappa_{p-2}$ once propagation has been verified. The extension smallness bounds follow from $C\kappa_{q-2}z^{-q}\leq CF^{-6}4^{3-q}D^{-q}$ ; dividing by any nonsmall-band margin costs at most $\sqrt q$ .

It remains to construct the blocks in an order that supplies propagation before invoking Green. Work first in the branch $\mathcal A>H_Q$ . Start with $m_3=\lfloor R\rfloor$ and $z_3=\|\mathcal T^{m_3}\|^{2/m_3}$ . The mesoscopic estimate gives $R-z_3\leq C_*/R$ . Greedy division by $m_3$ with the remainder estimated by $R$ gives a base propagation factor $\exp[m_3(R-z_3)/(2z_3)]\leq2$ for large $R$ . Here $d_3=16,e_3=32,J_3=128$ ; the fixed-degree original seed makes the two coherent errors $O(z_3^{-14})$ and $O(J_3z_3^{-13})$ . The lower-radius argument below applies also at this base.

Given the constructed order $q$ , set

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

(148.16)

Extension gives $z_{q+2}\geq z_q-\Delta_q$ . Every proposed length is at most $R^Q$ . If $R\leq X$ , its lower coefficients are bounded by $G(X^X)^2\leq F^2<\mathcal A$ . If $X<R<2\mathcal A$ , use $Q<R$ and $G_*(R^R)^2\leq R/2<\mathcal A$ . In either case the common-radius block maximum forces $z_{q+2}\geq\mathcal A$ . If $R\geq2\mathcal A$ , use the mesoscopic gap and the previously proved decrement ratios: their sum, including the current bridge, is at most $2\Delta_3=O(R^{-2})$ , so $z_{q+2}\geq R-C_*/R-O(R^{-2})\geq\mathcal A$ . There is no use of the new ratio in proving this lower bound.

Now the new ratio can be estimated:

{\Delta_{q+2}\over\Delta_q} ={F^4\over16z_{q+2}^2}(z_q/z_{q+2})^{q-1}\leq1/8.

(148.17)

For an upward radius step the last factor is at most one. For a downward step use $q\Delta_q/z_q\leq CF^{-6}$ and enlarge $C_0$ so the last factor is at most two. Thus all constructed radii satisfy $R-z_q\leq C/R\leq1$ . The length ratio for $q\geq5$ , before integer rounding, is $16(z_q/F^2)^2(z_q/z_{q-2})^{q-2}$ ; the downward correction is $e^{-O(F^{-6})}$ . Rounding loses an arbitrarily small fixed fraction because all real lengths exceed $F^24^{q-1}$ . The first ratio is $16(z_3/F^2)^2(z_3/R)$ up to the same rounding error. Consequently successive lengths grow by at least four.

Apply Lemma 10.10 to this constructed prefix. Its remainder cost is $O(1/R)$ and $m_{q+2}\Delta_q/z_q\leq C/F^4$ . With a fixed large floor this yields $\|\mathcal T^h\|\leq4(h+1)z_q^{h/2}$ for every $h$ . Counting only the $O(q)$ constructed levels also gives a global factor $E_q$ with $\log E_q\leq C/R+Cq/F^4$ . At the proposed next length this implies

q(\log(z_{q+2}/z_q))_+ \leq Cq4^{-q}/(F^2R)+Cq^24^{-q}/F^6.

(148.18)

Together with the downward bound it places $m_{q+2}$ within a factor four of $z_{q+2}^q/\kappa_q$ . At this point the new radius, envelope, and length hypotheses are established. Apply Green and actual restart to the norm-attaining orbit at this length. This completes the finite order induction, without using its new Green estimate to establish its own propagation hypothesis.

Take $Z_Q=z_Q$ in this branch; otherwise take $Z_Q=H_Q$ and make no assertion about a starting family. The coefficient inequalities follow from $\mathcal A\leq Z_Q$ . All final amplitude estimates follow from $a_Q+s_Q+1\leq C\kappa_{Q-2}$ , and all choices of constants preceded the finite chain length. The raw and normalized frame inputs (F1)–(F2), including the delayed-prefix applicability, are exactly the two joint-frame dependencies named in the statement. No profile conclusion is used.

Fences respected. The block radius is bounded using the exact common-radius maximum before the new decrement ratio is estimated. Propagation is established before Green is applied, and the retained family supplies its own matched budget. No small-degree or depth threshold is removed by assertion.

References¶

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