The residual dichotomy of the bootstrap - The KLS theorem and its methods

Part of the fixed-cut archive, Chapter The fixed cut: remaining problems; the reading order is on the full proofs page.

Overview. This dossier proves Corollary 29.1, conditional on the quantified completion in Assumption 29.1. The completion remains an antecedent throughout.

Match the stopping width and the near-worst parameter to the bootstrap, and choose a dimension threshold on which the published covariance window reaches the fixed completion time.
Use Corollary 33.1 to supply the completion whenever the dimension’s worst Cheeger constant is sufficiently small.
Apply the completion to an actual sequence of balanced near-minimizers. Its perimeter lower bound contradicts the assumed small Cheeger constant.

Refined statement.

Proof 109.1

Fix constants $T_0\in(0,1/8)$ and $\kappa,c_g,\eps_g,\delta_g>0$ witnessing Assumption 29.1. Throughout the proof the stopping width is exactly

\eta=T_0^{1/3}\in(0,1/2).

(109.2)

Choose once and for all

\eps=\frac12\min\{1,\eps_g,T_0^{1/3}\}>0.

(109.3)

The published discharge Corollary 26.2 supplies Assumption 26.1 with universal constants $c_0,C_1>0$ and $t_1(n)=c_0(\log n)^{-2}$ for $n\ge3$ . Since $t_1(n)\to0$ , choose an integer $N\ge3$ such that $t_1(n)\le T_0$ for every $n\ge N$ . Enlarge $N$ if necessary so that

L_n:=1+\log\log n\ge1\qquad(n\ge N).

(109.4)

Let $C_L>0$ be the universal constant in the supply inequality (33.11). Set

a=\min\left\{\frac{\kappa T_0}{4C_L},\frac{c_g}{2}\right\}>0.

(109.5)

Fix $n\ge N$ and suppose, for a contradiction, that $\hstar_n\le a/L_n$ . The established finite-dimensional bound Theorem 26.2 gives $\hstar_n>0$ . Also $\hstar_n<\infty$ : the standard Gaussian is an admissible isotropic law and a balanced coordinate halfspace has finite perimeter. The definition (0.5) of the infimum therefore supplies an isotropic log-concave probability $\mu$ on $\mathbb R^n$ with

h_\mu\le(1+\eps)\hstar_n\le2\hstar_n.

(109.6)

This is simultaneously within the near-worst class of the completion and the class of Theorem 33.1, with $\eps\le T_0^{1/3}$ .

By Lemma 33.1, $I_\mu(1/2)=h_\mu/2$ . By the definition of $I_\mu(1/2)$ as an infimum over balanced measurable cuts, for every integer $j\ge1$ there is such a cut $E_j$ with

0\le e_j:=\mu^+(E_j)-I_\mu(1/2)<\min\{\delta_g,j^{-1}\}.

(109.7)

In particular, every $E_j$ has finite initial lower outer Minkowski perimeter, is admissible for the completion, and satisfies $e_0(E_j)=\bar e_0(E_j)=e_j$ . No existence of an exactly minimizing cut is needed.

All time, dimension, and near-worst hypotheses of Corollary 33.1 now hold. Apply its supply estimate separately to each fixed $E_j$ , using its own mass process and stopping time $\tau_\eta(E_j)$ . Since $T_0^{4/3}\le1\le L_n$ , it gives

\begin{aligned} \int_0^{T_0}\mathbb E\bigl[\bar e_t(E_j)\mathbf1_{\{t<\tau_\eta(E_j)\}}\bigr]\,\mathrm dt &\le T_0e_j+C_Lh_\mu(T_0^{4/3}+L_n)\\ &\le T_0e_j+4C_L\hstar_nL_n\\ &\le T_0e_j+\kappa T_0. \end{aligned}

(109.8)

Thus Assumption 29.1 implies $\mu^+(E_j)\ge c_g$ for every $j$ . Passing to the limit in the scalar perimeter values, which converge to $I_\mu(1/2)$ by construction, gives

h_\mu=2I_\mu(1/2)\ge2c_g.

(109.9)

On the other hand, the hypothesized smallness and the choice of $a$ give

h_\mu\le2\hstar_n\le\frac{2a}{L_n}\le c_g,

(109.10)

a contradiction. We have proved $\hstar_n>a/L_n$ , which implies the claimed weak inequality. Finally, once $\log\log n\ge1$ , one has $1+\log\log n\le2\log\log n$ , proving the last assertion.

Remark 109.1 (Matched-time implication, separate from the corollary)

Here is the additional quantitative condition needed for the full KLS conclusion mentioned in the surrounding manuscript discussion. Let $C_B>0$ be a universal constant valid in the clean bootstrap inequality (33.7). Suppose, in addition to the completion, that there is a universal $\eps_t>0$ such that, for every $n\ge2$ and every isotropic log-concave $\mu$ with $h_\mu\le(1+\eps_t)\hstar_n$ ,

h_\mu\bigl(T_0^{4/3}+\Xi_{T_0}(\mu)\bigr) \le\frac{\kappa}{C_B}T_0,

(109.11)

at the very same $T_0$ fixed by the completion. Then $\hstar_n\ge2c_g$ for every $n\ge2$ .

Indeed, fix $n\ge2$ and choose positive $\eps_k\downarrow0$ with $\eps_k\le\min\{1,\eps_g,\eps_t,T_0^{1/3}\}$ . Positivity of $\hstar_n$ as above supplies near-worst measures $\mu_k$ with $h_{\mu_k}\le(1+\eps_k)\hstar_n$ . The clean bootstrap at $\eta=T_0^{1/3}$ and the displayed additional hypothesis give the completion’s supply for every balanced cut of $\mu_k$ with initial excess at most $\delta_g$ . Applying the same balanced near-minimizing sequence argument gives $h_{\mu_k}\ge2c_g$ . Therefore $(1+\eps_k)\hstar_n\ge2c_g$ ; letting $k\to\infty$ proves the assertion. Monotonicity from Lemma 33.2 gives $\hstar_1\ge\hstar_2$ , and the Cheeger formulation in Conjecture 0.1 yields KLS.

The premise of Conjecture 29.4 supplies a time depending on its requested error level. Even choosing that level at most $\min\{1,\kappa/C_B\}$ does not identify its time with the completion’s time. This remark does not assert the additional matched-time estimate or a time-uniform completion, and neither is used in the proof of Corollary 29.1.

Fences respected. The node has no registered bounded_by edges. The contextual constraints are respected as follows. The circularity warning Remark 32.6 is met by using the externally defined $\hstar_n$ and certified near-worst bootstrap, without inserting an unknown posterior profile bound. The crude-input warning Remark 33.5 is met by using the published polylogarithmic covariance window and retaining its $\log\log n$ loss. The relative-scale ceiling Remark 33.6 is respected: no all-measure dimension-free covariance bound is claimed. The spectator obstructions Proposition 29.1 and Proposition 29.2 are not bypassed: the completion is an explicit near-worst antecedent, and no uniform superlinear excess remainder is proved. The completion stays open, and the corollary’s dimension-dependent conclusion does not settle KLS.