Model geometries: the Gaussian and product brackets

A localization argument should be tested where the answer is known. This chapter records the exact behavior of the fixed-cut quantities on the two canonical models and identifies the genuine stress test.

Gaussian initial data: deterministic covariance, active damping, zero excess¶

Proposition 27.1 (Gaussian model — Proved · agent review (2026-08-27) · prop:gaussian-model · Correction)

Let $\mu=\gamma_n$ be the standard Gaussian. Then:

(i) the covariance is deterministic: $A_t=(1+t)^{-1}I_n$ for all $t$ ;

(ii) the centroid mechanism closes unconditionally: $\E\int_0^T\lmax(A_t)\dd t\le T$ , hence by (30.3) and Doob, $\Prob(\tau\le T)\le9T\le\tfrac12$ for $T\le\tfrac1{18}$ , and the conclusion of Theorem 30.1 holds — recovering the Gaussian isoperimetric bound up to universal constants, without any Carleson input;

(iii) for every halfspace cut $E=\{x_1\le x_0\}$ , the excess vanishes identically along the flow: $e_t(E)\equiv0$ ;

(iv) for every halfspace cut, the damping is strictly active: writing $\sigma_t^2=(1+t)^{-1}$ and $r_t=\sigma_t^2\varphi(\alpha_t)^2/s_t$ with $\alpha_t$ the normalized cut level,

r_t\le\frac2\pi\,\sigma_t^2, \qquad D_t=r_t\bigl(2\sigma_t^2-r_t\bigr)\ \ge\ \Bigl(2-\frac2\pi\Bigr)\sigma_t^2\,r_t\ >\ 0 .

(27.1)

Proof 27.1

(i) For Gaussian $\mu_t$ all third moments vanish, so the martingale part of the covariance SDE (23.7) vanishes and $\dot A_t=-A_t^2$ , $A_0=I$ , giving $A_t=(1+t)^{-1}I$ . (ii) is (30.10) with the deterministic bound $\lmax\le1$ . (iii) $\mu_t=N(a_t,\sigma_t^2I)$ , and after the similarity $x\mapsto(x-a_t)/\sigma_t$ the measure is standard Gaussian while Minkowski content scales by $\sigma_t^{-1}$ ; halfspaces are exact Gaussian isoperimetric minimizers at every volume, so $\mu_t^+(E)=\varphi(\beta_t)/\sigma_t=I_{\mu_t}(p_t)$ with $p_t=\Phi(\beta_t)$ . (iv) For the one-dimensional Gaussian marginal cut at normalized level $\alpha$ , $\abs{\delta_t}=\sigma_t\varphi(\alpha)/s$ , so $r=s\abs\delta^2=\sigma^2\varphi(\alpha)^2/s$ and, since $A_t$ is scalar, $D=2s\,\delta^TA\delta-r^2=2\sigma^2r-r^2=r(2\sigma^2-r)$ . The even function $\alpha\mapsto\varphi(\alpha)^2/(\Phi(\alpha)\Phi(-\alpha))$ has maximum value $4\varphi(0)^2=\tfrac2\pi$ at $\alpha=0$ : since $\varphi(\alpha)^2=\tfrac1{2\pi}e^{-\alpha^2}$ and $4\Phi(\alpha)\Phi(-\alpha)=1-(2\Phi(\alpha)-1)^2$ , the bound $\varphi(\alpha)^2\le\tfrac2\pi\Phi(\alpha)\Phi(-\alpha)$ is equivalent to $(2\Phi(\alpha)-1)^2\le1-e^{-\alpha^2}$ , i.e. to the elementary error-function inequality $\operatorname{erf}(x)^2\le1-e^{-2x^2}$ at $x=\alpha/\sqrt2$ . Hence $r\le\tfrac2\pi\sigma^2<2\sigma^2$ .

Product measures: the genuine stress test¶

Proposition 27.2 (Product structure under localization; KLS for products — Proved · agent review (2026-08-27) · prop:products · Correction)

Let $\mu=\bigotimes_{i=1}^n\mu^{(i)}$ be a product of isotropic one-dimensional log-concave measures. Then:

(i) $\mu_t$ is a product measure for every $t$ , pathwise: the localization tilt $\exp(c_t\cdot x-\tfrac t2\abs x^2)$ factorizes across coordinates; consequently $A_t$ is diagonal, with entries $A_t^{(i)}=\Var(\mu_t^{(i)})$ evolving as one-dimensional variance processes;

(ii) each $A^{(i)}$ is a nonnegative supermartingale (drift $-(A^{(i)})^2$ ), so $\Prob(\sup_{s\le t}A_s^{(i)}\ge\lambda)\le1/\lambda$ for every $\lambda\ge1$ ;

(iii) $h_{\mu_t}\ \ge\ c\,\min_i\bigl(A_t^{(i)}\bigr)^{-1/2} =c\,\lmax(A_t)^{-1/2}$ pathwise, with $c$ universal; in particular KLS holds for products, and the excess of any cut propagates through the direct profile bound, with no bootstrap needed.

Proof 27.2

(i) The density of $\mu_t$ is proportional to $\prod_ie^{-V_i(x_i)}\cdot\exp(c_{t,i}x_i-\tfrac t2x_i^2)$ , a product; the coordinates remain independent and the variance dynamics decouple. (ii) is the scalar case of the covariance SDE. (iii) The Poincaré constant tensorizes, $\CP(\mu_t)=\max_i\CP(\mu_t^{(i)})$ , each one-dimensional log-concave factor satisfies $\CP(\mu_t^{(i)})\le C\Var(\mu_t^{(i)})$ , and for log-concave measures the Cheeger and Poincaré constants are equivalent up to universal factors (Buser–Ledoux; see Bakry et al., 2014Lee & Vempala, 2018). Hence $h_{\mu_t}\ge c\min_i\Var(\mu_t^{(i)})^{-1/2}$ , which is (iii) since $\lmax(A_t)=\max_iA_t^{(i)}$ .

Remark 27.3 (Product stress test of the all-cut estimate)

Decide whether Assumption 28.1 holds for product measures. A proof cannot go through $\lmax(A_t)$ control — the operator norm genuinely reaches $\log n$ — and is therefore forced to exploit the cut-specific quantities $G_t,K_t,D_t$ , which is exactly the discipline the general case demands; the available structure is independence, one-dimensional log-concave estimates, and the explicit variance dynamics of Proposition 27.2. A counterexample cannot use a fixed cut depending on a single coordinate, even one whose variance later inflates (Corollary 31.1); it needs a fixed cut of unbounded coordinate complexity whose source is aligned with the inflation excursions of many coordinates at once (Conjecture 31.1).

The two outcomes do not weigh the same. A proof for products would exhibit, where the dangerous covariance regime is visited, the cut-aware mechanism that the prefix form Assumption 28.2 needs in general. A failure would refute the every-interval estimate Assumption 28.1 on products, where the Poincaré bound holds by tensorization, but not the weaker prefix form unless the same cut violates it too. Nor would a failure favor the near-Cheeger variant, whose package already fails on products (Proposition 29.1).

The non-model obstruction¶

The two-tail family of Proposition 25.2 marks the boundary of what model verification can establish: it is a family of perfectly explicit Gaussian configurations on which the slice-wise form of the Stein-trace estimate fails, while every genuinely dynamical question about it (the expected occupation of the inflated states from isotropic initial data) is of Carleson type. Model geometries can validate machinery and falsify formulations — both have happened in this manuscript, the second through the product witnesses of Proposition 29.1 — but the Carleson hypotheses Assumption 28.1 and Assumption 28.2, and the trace estimate Conjecture 29.3, are irreducibly dynamical.

References¶

Bakry, D., Gentil, I., & Ledoux, M. (2014). Analysis and Geometry of Markov Diffusion Operators (Vol. 348). Springer. 10.1007/978-3-319-00227-9
Lee, Y. T., & Vempala, S. S. (2018). The Kannan–Lovász–Simonovits Conjecture. https://arxiv.org/abs/1807.03465
Chen, Y. (2021). An Almost Constant Lower Bound of the Isoperimetric Coefficient in the KLS Conjecture. Geometric and Functional Analysis, 31(1), 34–61. 10.1007/s00039-021-00558-4
Klartag, B., & Lehec, J. (2022). Bourgain’s Slicing Problem and KLS Isoperimetry up to Polylog. Geometric and Functional Analysis, 32(5), 1134–1159. 10.1007/s00039-022-00612-9