Song–Zhang v2: near-unit refinement - 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. The aim is Proposition 9.3, from Section 9 of Song & Zhang, 2026. The proof combines scalar height estimates, the independently reconstructed finite-chain block theorem, a near-unit terminal-depth induction, and exact coefficient return. It does not rely on the claimed conclusion in the source.

Dependencies. The coefficient radius is Definition 9.1, with comparison Proposition 9.1. The polynomial normalization and regular measure class are those of Theorem 7.1 and Lemma 7.1. No BKL bound, consequence of BKL, or assertion of KLS is an input.

Scalar functions and their exact properties¶

Write $E_0=1$ , $E_{k+1}=\exp(E_k)$ and $\log^*x=\min\{k\geq0:x\leq E_k\}$ for $x\geq1$ . For $x\geq1$ set

t(x)=1+\log^*(x+2),\quad \kappa(x)=\min\{j\geq0:t^{\circ j}(x)\leq3\},\quad \chi(x)=\max\{3,1+\kappa(x)\}.

(140.1)

For $f=t,\chi$ , define

\bar f(x)=\max\left\{4,{1\over4}\int_0^4f(x+s)\,ds\right\}, \qquad W_m(x)=\bar t(\bar\chi^{\circ m}(x)).

(140.2)

We use $g(x)=\log(e+x)$ for $x\geq0$ , $\ell_r=g^{\circ r}$ , and $\mathcal L_r=\ell_r/\varrho$ , where $g(\varrho)=\varrho$ .

Proof 140.1

The first jump of $t$ on this domain is at $E_2-2=e^e-2>13$ ; all successive jump locations are $E_k-2$ and their gaps increase. In particular $t=3$ on $[1,13]$ . On integer arguments $n\geq4$ , $t(n)\leq n-1$ . Indeed this is direct below the first jump; above it, $\log^*(n+2)\leq n-2$ , since $E_{n-2}\geq n+2$ by induction starting at $n=4$ . On real arguments $3<x<4$ , $t(x)=3<x$ ; for $x\geq4$ , monotonicity and the integer estimate give $t(x)\leq\lceil x\rceil-1\leq x$ , with strict inequality at integers as well. The first image is an integer, and subsequent images decrease strictly until they reach three. This proves finiteness and monotonicity of $\kappa$ .

For integers $n\geq3$ , induction gives $0\leq\kappa(n+1)-\kappa(n)\leq1$ . The initial pair $(3,4)$ has values $(0,1)$ . At later pairs use $\kappa(n)=1+\kappa(t(n))$ , the fact that $t(n+1)-t(n)$ is zero or one, and that the arguments $t(n),t(n+1)$ precede the pair under consideration. For $x>3$ the same recurrence shows that jumps of $\kappa$ can occur only at jumps of $t$ , with size at most one. The possible jump at three disappears upon taking $\chi=\max\{3,1+\kappa\}$ . Thus $0\leq f(x+4)-f(x)\leq1$ for either function.

The forward average is locally absolutely continuous, with almost-everywhere derivative $(f(x+4)-f(x))/4$ . Its maximum with a constant has the same Lipschitz bound. Monotonicity sandwiches the average between $f(x)$ and $f(x)+1$ ; the additional floor obeys $4\leq f(x)+1$ . Since $t=3$ through eight and $\chi=3$ through ten, the asserted plateaus follow. Composition multiplies Lipschitz constants. For $x\geq4$ ,

0\leq\bar\chi^{\circ m}(x)-4\leq4^{-m}(x-4).

(140.4)

Choose finite $m$ so the right side is at most two; one additional iterate then equals four. If $x<4$ , the first iterate already equals four.

The contraction and fixed point also give $\bar\chi(x)\le x$ for $x\ge4$ : $\bar\chi(x)\le4+(x-4)/4\le x$ . Hence monotonicity implies $W_m(x)\le\bar t(x)$ for $x\ge4$ . On $1\le x\le4$ both sides are four, so this holds throughout the domain.

Proof 140.2

For sufficiently large $x$ , depending only on $C,p$ , $Cx^p+2\leq\exp(x+2)$ ; therefore $t(Cx^p)\leq t(x)+1$ there. On the remaining compact interval $t(Cx^p)-t(x)$ is bounded above. Consequently $t(Cx^p)\leq t(x)+b$ everywhere, for a fixed integer $b$ . The integer nonexpansiveness of $\kappa$ proved above gives, when $x>3$ ,

\kappa(Cx^p)=1+\kappa(t(Cx^p)) \leq1+\kappa(t(x))+b=\kappa(x)+b.

(140.7)

The bounded interval $x\leq3$ is handled by enlarging $b$ . Taking the maximum with three proves the fixed-power bound for $\chi$ too.

For either $f$ , averaging and the bound $f(x)\leq\bar f(x)\leq f(x)+1$ now give $\bar f(Cx^p)\leq\bar f(x)+b+1$ . For $m\geq1$ apply this to the first $\bar\chi$ and use the $4^{-m}$ Lipschitz constant of the remaining composition $\bar t\circ\bar\chi^{\circ(m-1)}$ . For $m=0$ use the bound for $\bar t$ . For sums and products put $z=\max\{x,y\}$ and use $xy\leq z^2$ , $x+y\leq2z$ . Finally $\max\{u+c,h\}\leq\max\{u,h\}+c$ for $c\geq0$ proves the last assertion.

Proof 140.3

The function $g(x)-x$ is strictly decreasing, positive at one and negative at two; hence $1<\varrho<2$ is unique. Since $0<g'(x)\leq e^{-1}$ ,

|\ell_s(y)-\varrho|\leq e^{-s}|y-\varrho|\quad(y\geq0).

(140.9)

For $y\geq5$ one has $e+\log(e+y)\leq y$ : it holds at five ( $e<11/4$ and $\log(e+5)<9/4$ suffice), and the derivative of the left side is less than one. Thus $g^{\circ2}(y)\leq\log y$ . Apply this comparison repeatedly while the current value is at least five. After at most $2\log^*x$ iterations the current value is at most five; otherwise ordinary logarithm iteration would reach at most one while the current value remained at least five. The interval $[0,5]$ is invariant under $g$ . A further $s$ iterations give distance at most $5e^{-s}$ from $\varrho$ . Taking $s\geq D/\eta$ with fixed sufficiently large $D$ makes this distance at most $\eta/8$ ; the elementary inequalities $\log(1+u)\leq u$ and $\log(1-u)\geq-2u$ for $0\leq u\leq1/2$ give the two bounds for $\mathcal L_r$ . The same contraction started at zero proves the lower bound independently of $x$ . Finally $2\log(1+r^{-2})\leq2r^{-2}\leq\eta$ after increasing $D$ .

Moment estimates below and above degree cutoffs¶

Lemma 140.4 (Uniform three-range sum)

Fix integers $M,h\geq0$ and $c>0$ . Let $0<\epsilon\leq\delta\leq1/16$ , $K_\alpha=\lceil L\alpha^{-2}\rceil$ , and $\Xi\geq K_\epsilon$ . Suppose $0\leq\rho_k\leq1$ , $\rho_k\leq C_0^{-1}$ for $k\leq16K_\delta$ , and $\rho_k\leq(1+\delta)^{-1}$ for $k\leq\Xi$ . For sufficiently large fixed $L,C_0$ , the quantity

\sum_{k\geq h+1} k^M\rho_k^{k-h}e^{-c\epsilon(k-h)}

(140.10)

is uniformly bounded in $\delta,\epsilon,\rho$ . Its supremum tends to zero as $L$ and $C_0$ tend to infinity.

Proof 140.4

Choose $L$ so $K_\delta\geq2h+2$ . Up to $K_\delta$ , bound the sum by $\sum_{j\geq1}(j+h)^MC_0^{-j}$ , which tends to zero as $C_0\to\infty$ . For $K_\delta<k\leq\Xi$ , use $\log(1+\delta)\geq\delta/2$ and $k-h\geq k/2$ to bound each term by $k^Me^{-\delta k/4}$ . Beyond $\Xi$ , use $k^Me^{-c\epsilon k/2}$ . For $0<s\leq1$ , $K\geq1$ and fixed $b>0$ ,

\sum_{k>K}k^Me^{-bsk} \leq e^{-bsK/2}\sum_{k\geq1}k^Me^{-bsk/2} \leq C_{M,b}s^{-M-1}e^{-bsK/2}.

(140.11)

The last estimate follows by integration on each unit interval, or from $k^M\leq C_{M,b}s^{-M}e^{bsk/4}$ followed by a geometric sum. The two tails are therefore at most $C\delta^{-M-1}e^{-c_1L/\delta}$ and $C\epsilon^{-M-1}e^{-c_2L/\epsilon}$ . For $u\geq16$ , $u^{M+1}e^{-c_iLu}$ tends uniformly to zero as $L\to\infty$ : split the exponential into two equal factors, use one to dominate the polynomial, and bound the other by $e^{-8c_iL}$ . This proves both claims.

The claimed small-loss profile and its unresolved input¶

The two-block Green estimate and the actual restart are proved in Lemma 10.8. They apply once the raw frame and Bochner identities have supplied their explicit sequence hypotheses.

Set

\widehat W_{m,\delta}(x)=\max\{W_m(x),\bar t(\delta^{-1})\},\quad R_\delta=\lceil C_R\delta^{-12}\rceil,\quad S_m=1+2D\sum_{l=0}^{m-1}4^{-l}.

(140.12)

Analytic mechanisms available for the iteration¶

The actual restart, delayed finite extension, and exact-block propagation are proved separately in Lemma 10.8, Lemma 10.9, and Lemma 10.10. Their proofs do not depend on the profile targeted here.

For the source’s block scales $m_{q+2}\simeq z_q^q/\kappa_q$ , $\kappa_q=(F^2/4)^{q-1}$ and $\Delta_q=C_b\kappa_{q-2}z_q^{1-q}$ , the per-level quantity is at most $C/F^4$ . The lemma explains how a polynomial propagation bound can be established from already constructed exact block norms before applying the Green lemma at the next block. It does not by itself verify the lower radius, length comparability, or degree-frame hypotheses needed to construct that next block.

Proof 140.5

For $Q\leq k\leq N$ , monotonicity bounds $g(k)$ by $g(N)$ . Writing $k=Q+l$ , and using $\sum_{l\geq0}l^je^{-\epsilon l} \leq C_j\epsilon^{-j-1}$ for $j=0,1,2$ , the corresponding sum is at most $C\epsilon^{-3}(Q+1)^2g(N)^{Q-1}$ .

For $k>N$ , the logarithm of $2C_Fk^2g(k)^{Q-1}$ is at most $C_1Q\log(e+k)$ , for a constant depending only on $C_g,C_F$ . Choose $D$ so $C_1Q\log(e+k)\leq\epsilon k/4$ for $k\geq N$ . To justify a uniform choice, $\log(e+k)/k$ decreases on $[1,\infty)$ , so it suffices to check $N$ . Up to universal factors the needed ratio is at most

{C_1\epsilon\over DQ} [\log(D+2)+2\log(\epsilon^{-1})+2\log(Q+1)].

(140.14)

The terms involving $\epsilon\log(\epsilon^{-1})$ are bounded, and $\log(Q+1)/Q$ is bounded. Increasing $D$ dominates $\log(D+2)$ and makes this ratio at most $1/4$ . The tail is therefore at most $C\epsilon^{-1}\exp(\epsilon Q-3\epsilon N/4)\leq1$ after increasing $D$ . Since $g(N)\geq1$ , the whole sum is at most $[C\epsilon^{-3}(Q+1)^2+1]g(N)^Q$ . For $Q\geq C_Q\epsilon^{-4}$ , $\log[C\epsilon^{-3}(Q+1)^2+1]\leq\epsilon Q$ with fixed sufficiently large $C_Q$ : the ratio of the left side to $Q$ decreases beyond a fixed threshold, and at that threshold the remaining quantity is bounded by a constant times $\epsilon^3\log(\epsilon^{-1})$ plus $\epsilon^3\log(C_Q+2)$ , divided by $C_Q$ . Taking $Q$ th roots proves the assertion.

The terminal-depth induction with an arbitrarily small cost¶

Lemma 140.7 (Near-unit depth step)

Fix $0<\epsilon\le1/16$ and odd $Q\ge C_Q\epsilon^{-4}$ . Suppose $G,H_l,\alpha_l$ are coefficient bounds and margins as in Proposition 10.2. Write $g_0=G^2$ and suppose, for some $M\ge M_0^*$ , where $M_0^*$ is a fixed universal constant,

F^2\le e^{6\epsilon}M,\quad \sup_{s\ge1}g_0(C_k(Qs)^2)s^{-1/Q}\le e^{5\epsilon}M, \quad g_0(X^X)\le e^{5\epsilon}M. \tag{D1}

(140.17)

Assume also $g_0(k)\le Ck^{1/3}$ with a fixed universal constant. Let $r_0\ge\max\{r_*,\lceil C_bt(Q)\rceil,\lceil C_b'/\epsilon\rceil\}$ . If the common radius satisfies the initial profile

\mathcal A\le e^{20\epsilon}M(r_0+1)^{1/Q} \mathcal L_{r_0}(a^{-1})^2, \tag{D2}

(140.18)

uniformly over the regular law class, then for every $r\ge r_0$ ,

\mathcal A\le e^{21\epsilon}M(r+1)^{1/Q} \mathcal L_r(a^{-1})^2. \tag{D3}

(140.19)

The constants are independent of the number of inherited bounds.

Proof 140.7

Set $L=e^{20\epsilon}M$ and

J_{r_0}=1,\quad J_{r+1}=J_r(1+r^{-2})^2\zeta_r^2, \quad\Gamma_r^2={L\over\varrho^2}(r+1)^{1/Q}J_r, \quad\zeta_r=1+2(e+1)^{-r/2}.

(140.20)

The product estimate proved above gives $J_r\le e^\epsilon$ . Induct on the statement $\mathcal A\le\Gamma_r^2\ell_r(a^{-1})^2$ . The static transfer at this fixed depth gives

c_k^2\le[\widehat\Gamma^2\ell_r(k)^2]^{k-1}, \qquad\widehat\Gamma=(1+r^{-2})\Gamma_r. \tag{D4}

(140.21)

There is no assumption on continuity of the finite block assignment in this use of static transfer. Its premise follows from the common-radius bound, and $\Gamma_r\ge1$ follows by fixing the initial universal amplitude.

Put $s=r+1$ , $\xi=(64Qs)^{-1}$ , $p=\xi/128$ , $b=(1-2p)^{-1}$ , and $k_0=\lceil100\xi^{-1}\log(M_0/\xi)\rceil$ , with fixed large $M_0$ . Then $k_0\le C_k(Qs)^2$ . For a dyadic $d\ge d_0$ , where $d_0$ is the least dyadic integer at least $Q$ , suppose

\mathcal A>b^4e^\xi\widehat\Gamma^2\ell_r(d)^2. \tag{D5}

(140.22)

Because $\ell_r(d)\ge\varrho$ and $L\ge e^{20\epsilon}M$ , this exceeds the floor $H_Q=(1+2\epsilon)F^2\le e^{8\epsilon}M$ . Thus the finite-chain construction supplies an actual radius $z\ge\mathcal A$ with its actual family and matched budget. Write $u=z^{-1}$ . Then

b^4u\widehat\Gamma^2\ell_r(d)^2<e^{-\xi},\quad u\le L^{-1}s^{-1/Q},\quad u^Q\le L^{-Q}/s. \tag{D6}

(140.23)

In the notation of that construction, its initial energy $\nu$ and loss prefix satisfy $\nu\le u/(1-a_Qu^Q)$ and $P_{2d_0-1}\le s_Qu^Q$ . Until a first exit of mass $p$ , all normalizers are at most $bu$ , and the power envelope has constant 4 and polynomial exponent 1. The propagated joint-loss lemma therefore gives

\theta\le C_\theta'\sum_{k<d}k^8(b^4u)^kc_k^2,\quad \Delta\le s_Qu^Q+2C_F\sum_{Q\le k<d}k^2(bu)^kc_k^2, \quad\tau=C_Fd(bu)^dc_d^2, \tag{D7}

(140.24)

where enlarging the sums from dyadic degrees to all integers only increases these upper bounds.

Here the low-degree kernel has a uniform bound even when the number of inherited bounds grows. For $k\le k_0$ , (D1),(D6) give $b^4ug_0(k)\le e^{-14\epsilon}b^4\le e^{-13\epsilon}$ . For the initial range $k\le16K_0$ , the floor gives the stronger bound $b^4ug_0(k)\le2/C_0$ . In each subsequent inherited range it gives $b^4ug_0(k)\le e^{-\alpha_l/2}$ : the factor $b^4$ is absorbed since $\log b^4\le16p\ll\epsilon\le\alpha_l$ , and $z\ge H_Q$ . Here is the degree sum explicitly. Below $16K_0$ the ratio is at most $2/C_0$ , so its polynomially weighted powers have a uniformly convergent geometric sum. Partition the remaining degrees into the successive inherited intervals ending at $\Xi_l$ , and the last interval above $\Xi_{N-1}$ . In the $l$ th inherited interval $k>K_l$ ; therefore $\alpha_l k\ge\sqrt{C_{\rm cut}k}$ . In the last interval $k>K_N$ and the same inequality holds with $\epsilon=\alpha_N$ . After removing the factor $b^4u$ , every term is bounded by $Ck^8\exp[-c\sqrt{C_{\rm cut}k}]$ . The intervals are disjoint, so their sum is bounded by this single convergent series, not by a constant times the number of intervals. To check convergence, group $k$ between $v^2$ and $(v+1)^2$ and bound the resulting sum by $\sum_{v\ge1}(2v+1)(v+1)^{16}e^{-c\sqrt{C_{\rm cut}}v}$ . Thus the low kernel is at most $Cu$ , uniformly in the cap chain. For the low coherent sum use the coherent-root estimate above:

2C_F\sum_{Q\le k\le k_0}k^2(bu)^kc_k^2 \le b^Q u^Q E_{Q,\epsilon}=D_Qu^Q, \qquad D_Q^{1/Q}\le e^{7\epsilon}M. \tag{D8}

(140.25)

Indeed $bu g_0(k)\le e^{-\epsilon}$ on this range, and $E_{Q,\epsilon}^{1/Q}\le e^\epsilon g_0(N)$ with $N=\lceil D\epsilon^{-2}Q^2\rceil\le X^X$ ; also $b\le e^\epsilon$ .

For $k_0<k<d$ , (D4),(D6) give respectively $(b^4u)^kc_k^2\le b^4u e^{-\xi(k-1)}$ and $(bu)^kc_k^2\le e^{-\xi k}$ . For fixed integer $h\ge0$ ,

\sum_{k>k_0}k^he^{-\xi k} \le C_h\xi^{-h-1}e^{-\xi k_0/2} \le C_h\xi^{-h-1}(\xi/M_0)^{50}.

(140.26)

This follows by integrating the remaining half-exponential over unit intervals. Choose $M_0$ once to obtain

\theta\le C_\theta u,\qquad \Delta\le(s_Q+D_Q)u^Q+p/512. \tag{D9}

(140.27)

The block theorem gives $(a_Q+s_Q+1)^{1/Q}\le C^{1/Q}F^2$ . For all $Q\ge C_Q\epsilon^{-4}$ , increasing $C_Q$ once ensures $\log(CQ)\le\epsilon Q$ for every fixed constant $C$ appearing here. Consequently (D1),(D8) and the gap between $20\epsilon$ and $7\epsilon$ ensure simultaneously

L\ge\max\{16,16C_\theta\},\qquad L^Q\ge2^{24}Q(a_Q+s_Q+D_Q+4C_\theta a_Q+1). \tag{D10}

(140.28)

The first assertion uses the fixed lower bound on the universal amplitude, not a loss depending on $\epsilon$ .

For clarity the finite curvature comparison is included here. Set $D_0=\max\{\nu-u,0\}\le2a_Qu^{Q+1}$ and let $\lambda=C_P^{-1}$ . The actual-radius comparison gives $\lambda\ge u/2$ . Equations (D6),(D9),(D10) give

D_0\le u,\quad\nu\le1/4,\quad \theta u/\lambda\le1/8,\quad \Delta+\theta D_0/\lambda\le p/64,\quad D_0/p\le u.

(140.29)

If $a\ge12(u+D_0/p)\tau$ , put $m=\lceil(D_0+up)/a\rceil$ , $M_1=m+1$ . The matched budget at the initial time implies $a\le D_0+up$ , so $M_1\tau\le p/4$ . Before or at the first exit $N\le M_1$ , the budget gives $X_{N-2}\le D_0+uP_N$ and the delayed loss bound gives $P_N\le p/64+p/4+P_N/8$ , hence $P_N\le cp$ with $c=17/56$ . The retained prefix supplies the first two losses. All subsequent normalizers use earlier masses at least $1-p$ ; the next mass is positive because $1-p-\nu>0$ . No first exit can occur. At time $m$ ,

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

(140.30)

which is impossible since the left minus right is at least $up(1-2c-cp)>0$ . Thus $a<12(u+D_0/p)\tau\le24u\tau$ .

For $0<a\le1$ take dyadic $d$ between $C_TQ^2s^2\log(e+a^{-1})$ and twice this quantity. A fixed $C_T$ ensures $d\ge d_0$ and $\log(24C_Fd/a)\le\xi d$ : the logarithm is bounded by $C+2\log Q+2\log s+2\log(e+a^{-1})$ , whereas $\xi d\ge(C_T/64)Qs\log(e+a^{-1})$ . Writing $B=\widehat\Gamma^2\ell_r(d)^2\ge1$ , (D4),(D6) contradict

a<24C_Fd(bu)^dc_d^2 \le24C_Fd B^{-1}(buB)^d\le24C_Fd e^{-\xi d}\le a.

(140.31)

Hence (D5) is false. For $a\ge1$ , Brascamp--Lieb gives $\mathcal A\le1$ , already within the desired bound.

Apply Lemma 10.11 to the chosen $d$ . Since $\log(b^4e^\xi)\le16p+\xi\le Q^{-1}\log(1+1/s)$ , the resulting upper bound on $\mathcal A$ is exactly $\Gamma_{r+1}^2\ell_{r+1}(a^{-1})^2$ . This completes the finite induction on $r$ . Using $J_r\le e^\epsilon$ gives (D3).

Returning coefficients to the next height¶

Lemma 140.8 (Outer round with a retained profile)

Suppose $V=W_m$ or $V=\widehat W_{m,\delta}$ , $1\le S\le S_*$ , $A\ge A_0$ , $R_\delta=\lceil C_R\delta^{-12}\rceil$ , and

\mathcal A\le A[V(r)+S]^{1/3}\mathcal L_r(a^{-1})^2 \qquad(r\ge R_\delta). \tag{E1}

(140.32)

Retain its valid coefficient cap

H(x)^2=\min\{G_*(x)^2,e^{3\delta}A[V(r_\delta(x))+S]^{1/3}\}, \quad r_\delta(x)=\max\{R_\delta,\lceil C_rt(x)\rceil+ \lceil D_r/\delta\rceil\}.

(140.33)

Optionally retain earlier caps, provided their floors in the finite-chain construction are at most $A(4+S)^{1/3}$ ; its fixed original-seed floor must also be at most that quantity. With a fixed sufficiently large $b_*$ with $b_*\le C\log(e+S_*)$ for one universal $C$ , put

D_*=V(\epsilon^{-1})+S+b_*4^{-m},\qquad0<\epsilon\le\delta.

(140.34)

Then, for every $j\ge1$ , odd $Q\ge Q_\epsilon$ , and $r\ge r_0(Q):=\max\{R_\delta,\lceil C_bt(Q)\rceil+ \lceil C_b'/\epsilon\rceil\}$ ,

\mathcal A\le e^{16\delta+24\epsilon j}A \max\{t_j(Q),D_*\}^{1/3}(r+1)^{1/Q}\mathcal L_r(a^{-1})^2. \tag{E2}

(140.35)

For the one-cap case, the original-seed floor can instead be bounded by $e^{4\delta}A D_*^{1/3}$ . The same conclusion holds.

Proof 140.8

The cap $H$ is valid by static transfer at the fixed depth $r_\delta(d)$ : the squared transfer and burn-in factors cost at most $e^{2\delta}$ . The third $\delta$ leaves room for integer rounding and increasing the fixed burn-in constants. The coefficient bound is simultaneous on the whole class, hence the cap may be retained at later stages. Throughout the proof take minima also with the fixed bound $C_2t_2(x)^{1/3}$ on squared coefficients supplied by the first height-reduction stage. This ensures the growth hypothesis $g_0(x)\le Cx^{1/3}$ .

Here are the elementary bounds needed for both initialization and coefficient return. For fixed constants $C,p$ ,

V(C\epsilon^{-p})\le V(\epsilon^{-1})+b(C,p)4^{-m}. \tag{E3}

(140.36)

For the maximum defining $\widehat W$ , this follows by taking the maximum with its constant branch. Also $V$ is $4^{-m-1}$ -Lipschitz. For $Y=C_k(Qs)^2$ , $h=t(Q)$ and $y=\log(s)/Q$ , the tower definition gives

t(X^X)\le h+B_h,\qquad t(Y)\le h+B_h+y. \tag{E4}

(140.37)

Indeed $\log(Y+2)\le C Q(1+y)$ ; the product bound for the remaining iterated logarithms gives $t(Y)\le t(Q)+C+t(1+y)-3$ , and $t(1+y)\le y+4$ . The self-power bound follows by applying the same argument to $\log(X^X)=X\log X$ . Choose $C_h=C(1+S_*)^2$ with a sufficiently large universal $C$ , and call a height high if $h\ge C_hR_\delta\epsilon^{-5}$ . At low heights (E4) implies

r_\delta(Y)\le C(\epsilon^{-17}+y),\qquad r_\delta(X^X)\le C\epsilon^{-17}. \tag{E5}

(140.38)

Combining (E3), Lipschitz continuity, and $D_*\ge5$ gives

e^{-y}[V(r_\delta(Y))+S]^{1/3}\le D_*^{1/3}, \tag{E6}

(140.39)

provided $b_*$ absorbs the fixed constants in (E3). More explicitly, split (E5) into $C\epsilon^{-17}+C_r'y$ , use (E3) on the first term, and Lipschitz continuity on the second. This gives $V(r_\delta(Y))+S\le D_*+C_r'4^{-m-1}y$ . Choose $b_*$ above the constant required by (E3), and also so that $D_*\ge C_r'4^{-m}/12$ for every $m$ . Then $\log(1+C_r'4^{-m}y/(4D_*))/3-y\le0$ . The constant in (E5) is polynomial in $C_h$ , while the fixed-power allowance (E3) is at most $C\log(e+C_h)$ : this follows by counting the additional ordinary logarithms needed to remove a fixed factor and applying the $4^{-m}$ contraction. Thus all these choices permit $b_*\le C\log(e+S_*)$ . Thus (E6) has coefficient exactly one; no fixed multiplicative loss is introduced. The analogous estimate at $X^X$ follows with $y=0$ .

We first initialize $j=1$ . Use the original cap $H$ , and all earlier caps if present, as the working coefficient majorant. At high heights, $r_\delta(Y)\le C(h+\epsilon^{-1}+y)$ and $V(r_\delta(Y))+S\le h+y$ after increasing $C_h$ . To justify this also for $\widehat W$ , note its constant branch $\bar t(\delta^{-1})$ is smaller than $h/4$ here. For the nonconstant branch, $W_m\le\bar t\le t+1$ and $t(C(h+\epsilon^{-1}+y))\le t(C'(h+\epsilon^{-1}))+t(1+y)+C$ ; its first term plus the fixed $S_*$ and constants is at most $h/2$ . For $0\le y\le1$ use the same slack to get a bound by $h$ ; for $y\ge1$ use $t(1+y)\le y+4$ . Then $e^{-y}(h+y)^{1/3}\le h^{1/3}$ . The same argument applies to $X^X$ . At low heights use (E6). Consequently the two dynamic bounds (D1) hold with $M=e^{16\delta}A\max\{t(Q),D_*\}^{1/3}$ . The starting depth $r_0(Q)$ is treated in exactly these two ranges: high heights give $V(r_0)+S\le h$ , and low heights give $V(r_0)+S\le D_*$ . Thus (E1) supplies (D2).

The newest floor is also covered: with $\Xi=\lceil4C_{\rm deg}\delta^{-1}\lceil C_{\rm cut}\epsilon^{-2}\rceil\rceil$ , we have $r_\delta(\Xi)\le C\epsilon^{-12}$ and hence

(1+\delta)H(\Xi)^2\le e^{4\delta}A D_*^{1/3}. \tag{E7}

(140.40)

Older floors are covered by hypothesis. So is the original-seed floor. The constant $C_0$ itself is covered by $A_0$ . This proves the floor part of (D1). The near-unit depth lemma gives (E2) for $j=1$ .

Now suppose (E2) proved for a given $j$ . Write $A'=e^{16\delta+24\epsilon j}A$ and $P_j(x)=\max\{t_j(x),D_*\}^{1/3}$ . Define

r_\epsilon(x)=\max\{R_\delta,\lceil C_rt(x)\rceil+ \lceil D_r/\epsilon\rceil\}, \quad k_\epsilon(x)=\text{least odd integer at least } \max\{Q_\epsilon,\epsilon^{-1}\log(r_\epsilon(x)+1)\}.

(140.41)

With universal large $D_r,C_r$ , one has $r_\epsilon(x)\ge r_0(k_\epsilon(x))$ . Indeed $R_\delta$ is included, the logarithmic part of $k_\epsilon$ has height at most $C\log(r_\epsilon+2)$ , and the $Q_\epsilon$ part has height $O(\log(\epsilon^{-1}+2))$ ; both fit below $r_\epsilon/2C_b$ after the fixed burn-in increase. Apply the known profile at this order and depth and then static transfer. The order factor, squared burn-in factor and squared transfer factor cost at most $e^{3\epsilon}$ in total, and therefore

g_0(x):=\min\{G_*(x)^2,C_2t_2(x)^{1/3},H_l(x)^2, e^{5\epsilon}A'P_j(k_\epsilon(x))\} \tag{E8}

(140.42)

is a valid nondecreasing squared coefficient majorant. The minimum includes every retained $H_l$ and the newest $H$ .

Set $M=A'P_{j+1}(Q)$ . At high heights, $k_\epsilon(X^X)\le h$ , and for the cutoff $k_\epsilon(Y)\le h$ when $y\le1$ ; when $y\ge1$ ,

k_\epsilon(Y)\le h/2+2+y/(\epsilon h)\le h+y\le h(1+y).

(140.43)

These follow from $k_\epsilon(x)\le Q_\epsilon+3+ \epsilon^{-1}\log[C(R_\delta+t(x)+\epsilon^{-1})]$ and the choice $h\ge C_hR_\delta\epsilon^{-5}$ . The same logarithmic bound at $y=0$ is at most $h/2$ ; its increment is at most $y/(\epsilon h)$ , with at most two for odd rounding. The iterated-height product estimate used here follows directly from the tower thresholds. For $x_1,x_2\ge1$ set $n=\max\{t(x_1),t(x_2)\}\ge3$ and $E=E_{n-1}$ . Then $x_l\le E-2$ , and both $x_1+x_2$ and $x_1x_2$ are at most $e^E-2=E_n-2$ ; indeed $2(E-2)$ and $(E-2)^2$ are below that quantity for $E\ge e^e$ . Thus $t(x_1+x_2),t(x_1x_2)\le n+1\le t(x_1)+t(x_2)$ . Apply the sum bound successively to the product bound to obtain $t_j(x_1x_2)\le t_j(x_1)+t_j(x_2)$ for every $j\ge1$ . For $y\ge1$ , the already proved $t(n)\le n$ on integers $n\ge3$ gives $t_j(1+y)\le t(1+y)\le y+4\le5y$ . Consequently $t_j(k_\epsilon(Y))\le t_j(h)+5y$ , as required. Since $\max\{t_j(h),D_*\}\ge5$ , multiplication by $e^{-y}$ after taking cube roots removes this additive $5y$ exactly. Thus (E8) proves both dynamic inequalities (D1) at high heights. At low heights use the retained newest cap and (E6), whose cost $e^{3\delta}A$ is smaller than $A'$ . The floor proof (E7) is unchanged.

It remains to initialize the new depth induction; a coefficient bound alone would not do this. At high heights apply the preceding stage at order $k=k_\epsilon(x_0)$ with $x_0$ any argument of height $h$ and choose its logarithmic order using $r_0(Q)$ directly: $k$ is the least odd integer at least $\max\{Q_\epsilon,\epsilon^{-1}\log(r_0(Q)+1)\}$ . Then $k\le h$ , $r_0(k)\le r_0(Q)$ , and its order factor is at most $e^\epsilon$ . This supplies (D2) with the new $M$ . At low heights the original profile (E1) and (E3) supply it instead. The near-unit depth lemma costs $e^{21\epsilon}$ , covered by the next $e^{24\epsilon}$ . Finite induction on $j$ proves (E2).

Proof 140.9 (Initialization and analytic outer round)

For $m=0$ , the original inner profile together with the fixed-cost height theorem gives $\mathcal A\le C[t(r)+1]^{1/3}\mathcal L_r^2$ : choose the least odd $Q\ge\log(r+1)$ in its $j=1$ profile. Then $(r+1)^{1/Q}\le e$ , $t(Q)\le t(r)$ , and the starting depth condition holds for all sufficiently large $r$ . Increase $C_R,A_0$ once to cover the fixed initial threshold. This proves the initial profile since $\widehat W_{0,\delta}\ge t$ .

Assume the stage- $m$ profile. The outer-round lemma applies with $V=\widehat W_{m,\delta}$ and $S=S_m$ , which is uniformly bounded by $1+8D/3$ . Retain just its one cap $H$ , so $N=1$ in the block theorem. The original seed satisfies $G_*(x)^2=C_6^2t(x)^{1/3}$ ; therefore its initial floor at $16K_\delta$ is at most $C[t(\delta^{-1})+1]^{1/3}\le A_0D_*^{1/3}$ after fixing $A_0$ . The cap’s newest floor is covered by (E7). Choose $D$ to dominate the constant $b_*$ of the outer-round lemma. There is no circular choice involving $S_*$ : only the high-height threshold $C_h$ depends on $S_*$ ; the additive allowances in (E3),(E5) then grow at most logarithmically in $C_h$ , so $D\ge C\log(e+D)$ suffices and has a universal solution. Alternatively fix the bound $S_*\le1+8D/3$ and take this last scalar inequality as the explicit final choice of $D$ . The result is, for all indicated $j,Q,r$ ,

\mathcal A\le e^{16\delta+24\epsilon j}A \max\{t_j(Q),\widehat W_{m,\delta}(\epsilon^{-1})+S_m+D4^{-m}\}^{1/3} (r+1)^{1/Q}\mathcal L_r(a^{-1})^2.

(140.45)

Thus the outer-round estimate is supplied by the analytic proof, not assumed as an additional premise.

Proof 140.10 (Completion of the outer induction)

Fix a permitted $r$ and let $j=\kappa(r)$ , $\epsilon=\delta/(j+1)$ . Choose the least odd $Q$ at least $\max\{Q_\epsilon,\delta^{-1}\log(r+1)\}$ . Since $\kappa(r)\leq t(r)$ and $t(r)\leq C r^{1/8}$ , $Q_\epsilon\leq C\delta^{-4}r^{1/2}$ ; hence $Q\leq r$ for $r\geq C_R\delta^{-12}$ after a universal enlargement of $C_R$ . The elementary growth bound follows from $t(r)\leq C\log(r+2)$ and $\log(r+2)\leq C r^{1/8}$ . The second term in $Q$ fits below $r/2$ by the same growth estimate. Furthermore $C_bt(Q)+C_b'/\epsilon\leq C\delta^{-1}t(r)\leq r$ . Thus the analytic estimate just proved applies at this same $r$ .

Monotonicity gives $t^{\circ j}(Q)\leq3<D_*$ , while $(r+1)^{1/Q}\leq e^\delta$ and $\epsilon j\leq\delta$ . Since $j+1\leq\chi(r)$ , the product estimate above yields

\widehat W_{m,\delta}(\epsilon^{-1}) \leq\widehat W_{m+1,\delta}(r)+D4^{-m},

(140.46)

after fixing $D$ sufficiently large. Here $W_m\circ\bar\chi=W_{m+1}$ , $\bar\chi\geq\chi$ , and $W_m(\delta^{-1})\leq\bar t(\delta^{-1})$ . The two additive allowances total $2D4^{-m}=S_{m+1}-S_m$ . Consequently the next amplitude is at most $e^{41\delta}A$ , and $C=42$ suffices. Together with the initialization and analytic outer round, finite induction on $m$ proves the stated small-loss profile.

Fences respected. No new bounded_by edges are proposed. The argument does not discharge a CMH, occupation, or cut-relative premise. Uniform admissibility is retained explicitly, and exponential coefficients are not obtained by invoking the already proved target.

References¶

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