Conditional fibers: inverse-variance frames of line resamplings

Overview of the mechanism¶

The idea. Replace Euclidean directions by one isotropic frame of conditional line resamplings, chosen from the measure but not from the test function, and normalized by the conditional variances; then ask for a dimension-free spectral gap of the resulting form.

What it would add. An elementary mechanism for the Poincaré bound, with constant $4C$ , resting only on one-dimensional log-concave inequalities and a choice of directions — none of the localization or polynomial machinery of the three proofs. Its link to KLS is a sufficient condition, and the diagram must be read as such:

\text{conditional-fiber frame gap} \ \xRightarrow[\ \text{sufficient, not equivalent}\ ]{}\ \text{KLS},

(22.1)

where the left-hand side is Conjecture 22.1 and the arrow is (22.5). The normalization is designed so that linear functions carry exactly the Euclidean energy, while the sharp one-dimensional log-concave Poincaré inequality bounds the form above by the usual gradient form. A negative answer to the frame question would rule out this mechanism, whose sharpest result so far is itself negative (Proposition 22.1).

What it builds on. The sharp one-dimensional log-concave Poincaré inequality, from the literature. Set up here: Lemma 22.1, that the inverse-conditional-variance line-resampling form is densely defined and closable, which is what makes the question well posed at all.

What it gives. A precise question, a failing frame, and a limitation of polynomial tests. The frame formalism is set up with its isotropy constraint (22.3) and its form (22.4), and the most natural choice — the $A_{m-1}$ root frame on the isotropic uniform simplex — fails: by Proposition 22.1 its form gap is $O(m^{-2})$ . The fixed-degree floor of Lemma 22.3 explains why polynomial tests of bounded degree cannot detect this failure.

What blocks it. Conjecture 22.1: exhibit, for every isotropic log-concave $\mu$ , one even test-independent admissible frame with a universal lower form gap on the maximal closed form domain — or decide the all-frame simplex dual against it.

What fails, and why. Proposition 22.1 rules out the root frame outright: a vertex-cap indicator drives the normalized Rayleigh quotient down. The stronger obstruction Lemma 22.3 excludes asymptotically vanishing all-frame upper certificates at every fixed polynomial degree. Its reverse inequality on each chord explains why fixed-degree tests miss the small cap. Proposition 22.2 is the same vertex-cap mechanism for a related negative-exponent exchange model, from the literature.

What would settle it. A universal form gap for a constructed frame would complete the argument. In the other direction, an asymptotically vanishing all-frame upper certificate on the simplex must use polynomial degrees growing with dimension or nonpolynomial tests, by Lemma 22.3.

How to read it. The form construction and root-frame obstruction use the sharp one-dimensional Poincaré inequality and no localization apparatus. The fixed-degree floor also uses the simplex Poincaré bound from Corollary 17.3; its moment-map proof need not be read to follow the chord argument here. Read Section Why the simplex root frame fails for the root-frame obstruction, which is the shortest complete argument among the alternative mechanisms.

The construction is as follows.

Let $\mu$ be a full-dimensional probability on $\R^d$ with finite second moment. For $\theta\in S^{d-1}$ , disintegrate along the affine lines parallel to $\theta$ :

\int h(x)\dd\mu(x) =\int_{\theta^\perp}\int_\R h(z+t\theta) \dd\mu_{\theta,z}(t)\dd\bar\mu_\theta(z).

(22.2)

Write $\sigma_{\theta,z}^2=\Var_{\mu_{\theta,z}}(T)$ . An admissible frame is an even Borel probability $\rho$ on $S^{d-1}$ satisfying

d\int_{S^{d-1}}\theta\theta^T\dd\rho(\theta)=I_d.

(22.3)

For such a frame define

\mathcal D_{\mu,\rho}(f) :=d\int_{S^{d-1}}\int_{\theta^\perp} \frac{\Var_{\mu_{\theta,z}}(f(z+T\theta))} {\Var_{\mu_{\theta,z}}(T)} \dd\bar\mu_\theta(z)\dd\rho(\theta).

(22.4)

Null fibers and fibers with zero denominator contribute zero. The maximal form domain is $\{f\in L^2(\mu):\mathcal D_{\mu,\rho}(f)<\infty\}$ , using jointly measurable conditional versions.

Lemma 22.1 (Conditional-fiber form and factor-4 bridge — Proved · agent review (2026-08-27) · lem:conditional-fiber-form · Correction)

For a full-dimensional probability with density and finite second moment, and an admissible frame, the quadratic form (22.4) is densely defined, closed, symmetric, Markovian, and reversible. Its nonlocal self-adjoint generator is understood in form sense, and agrees with the conditional pair-jump formula on the natural sufficient Bochner domain even when pointwise total jump rates are infinite.

If $\mu$ is log-concave, then for every locally Lipschitz $f$ in the form domain,

\mathcal D_{\mu,\rho}(f) \le4\int\abs{\nabla f}^2\dd\mu.

(22.5)

Consequently, a form gap $\Var_\mu(f)\le C\mathcal D_{\mu,\rho}(f)$ implies $\CP(\mu)\le4C$ . For every linear test $f_a(x)=a\cdot x$ , one has $\mathcal D_{\mu,\rho}(f_a)=\abs{a}^2$ ; hence linear tests have quotient one when $\mu$ is isotropic. Every admissible frame has form gap one for the standard Gaussian, and the coordinate frame has form gap one for a standardized product law.

Conjecture 22.1 implies KLS, with $\CP\le4C$ , by Lemma 22.1 ((22.5)); what it would add to the existing proofs is that the only analytic input is one-dimensional. The order of quantifiers is essential: $\rho_\mu$ may depend on $\mu$ , but it must be fixed before the test $f$ is chosen.

Why the simplex root frame fails¶

Let $P$ be uniform on the simplex $\Delta_{m-1}$ , put $H_0=\mathbf1^\perp$ and

X=\sqrt{m(m+1)}\left(P-\frac1m\mathbf1\right)\in H_0.

(22.7)

Then $X$ is isotropic in dimension $d=m-1$ . The even $A_{m-1}$ root frame is

\theta_{ij}=\frac{e_i-e_j}{\sqrt2}, \qquad \rho_{\rm root}=\frac1{m(m-1)}\sum_{i\ne j}\delta_{\theta_{ij}}.

(22.8)

It satisfies $d\int\theta\theta^T\dd\rho_{\rm root}=I_{H_0}$ . If $\Var_{ij}$ denotes conditional variance under redistribution of $(P_i,P_j)$ with their sum fixed, then

\mathcal D_{\rm root}(f) =\frac{12}{m^2(m+1)}\sum_{i<j} \E\frac{\Var_{ij}(f)}{(P_i+P_j)^2}.

(22.9)

Proposition 22.1 (Vertex-cap obstruction for the root frame — Proved · agent review (2026-08-27) · prop:conditional-fiber-root-obstruction · Correction)

For $m\ge2$ and $\varepsilon\in(0,1)$ , let $\eta_i=mP_i$ and $A_{m,\varepsilon}=\{\eta_1>m-\varepsilon\}$ . Then

p_{m,\varepsilon}:=\mu(A_{m,\varepsilon}) =\left(\frac\varepsilon m\right)^{m-1},

(22.10)

and the centered cap indicator belongs to the maximal root-form domain and satisfies

\frac{\mathcal D_{\rm root}(\one_{A_{m,\varepsilon}})} {\Var(\one_{A_{m,\varepsilon}})} \le \frac{12(m-1)}{(m+1)(m-\varepsilon)^2 \left(1-(\varepsilon/m)^{m-1}\right)}.

(22.11)

Consequently $\operatorname{gap}(\mathcal D_{\rm root})=O(m^{-2})$ .

The vertex-cap failure is an $L^2$ phenomenon and does not descend to fixed polynomial degree. At polynomial degree two the root-frame pencil can be computed exactly.

Lemma 22.2 (Root-frame degree-two pencil identity — Proved · agent review (2026-09-06) · lem:fiber-root-degree-two · Correction)

For every $m\ge3$ , on the degree- $\le2$ quotient $V_{m,2}$ of $L^2$ of the isotropic uniform simplex modulo constants, the $A_{m-1}$ root-frame pair form $K(f,f)=\tfrac{12}{m^2(m+1)}\sum_{i<j}\E\bigl[\Var_{ij}(f)/(P_i+P_j)^2\bigr]$ and $G=\Var$ satisfy

\lambda_{\min}(K,G)=\frac{(m+2)(m+3)}{5m^2},

(22.12)

attained exactly on the radial line $\R\,[\abs X^2-(m-1)]$ . Consequently, granting the identification of $K$ with the conditional-fiber root pencil (22.9), every degree-two dual certificate in the all-frame min–max framework has objective at least $(m+2)(m+3)/(5m^2)>1/5$ : no sequence of degree-two dual certificates has objective tending to zero, any such fixed-degree polynomial sequence requires degree at least three, and $\Lambda_{m,2}\ge(m+2)(m+3)/(5m^2)$ . No upper bound on $\Lambda_{m,2}$ , no degree-three statement, and no claim about the full $L^2$ gap (where the vertex-cap obstruction stands) is asserted.

Lemma 22.3 (Fixed-degree floor for every simplex frame — Proved · agent review (gpt-6-astra, 2026-10-04) · lem:fiber-polynomial-floor · Correction)

For integers $m\ge2$ and $k\ge1$ , let $\mu_m$ be the law on $H_0=\mathbf1^\perp\subset\mathbb R^m$ of $X=\sqrt{m(m+1)}(P-\mathbf1/m)$ , where $P$ is uniform on $\{p_i\ge0:\sum_i p_i=1\}$ . Set $d=m-1$ and $A_k=k(k+1)^2(k+2)$ . For every even Borel probability $\rho$ on the unit sphere of $H_0$ with $d\int\theta\theta^T\,d\rho=I_{H_0}$ , and every real polynomial $f$ on $H_0$ of total degree at most $k$ , $f$ belongs to the maximal form domain of (22.4) and

\mathcal D_{\mu_m,\rho}(f) \ge \frac{12}{A_k}\int_{H_0}|\nabla_{H_0}f|^2\,d\mu_m \ge \frac{3}{A_k}\operatorname{Var}_{\mu_m}(f).

(22.13)

In particular, defining

\Lambda_{m,k} =\sup_{\rho\ \mathrm{admissible}} \inf_{\substack{\deg f\le k\\\operatorname{Var}_{\mu_m}(f)>0}} \frac{\mathcal D_{\mu_m,\rho}(f)}{\operatorname{Var}_{\mu_m}(f)},

(22.14)

one has $3/A_k\le\Lambda_{m,k}\le1$ . For every $m\ge2$ , $\Lambda_{m,3}\ge1/80$ . Thus no sequence of valid all-frame upper certificates supported on polynomials of one fixed finite degree can have objectives tending to zero as $m\to\infty$ .

The proof of Lemma 22.3 starts on a single chord. For a polynomial $q$ of degree at most $k$ on a uniform interval of length $L$ , expansion in normalized Legendre polynomials bounds differentiation by $\mathbb E|q'|^2\le A_k\operatorname{Var}(q)/L^2$ . The coordinate variance is $L^2/12$ , so the conditional variance ratio controls derivative energy with constant $12/A_k$ , independently of the chord length. Frame isotropy turns the average directional energy into the full gradient energy. The simplex Poincaré bound $\CP\le4$ from Corollary 17.3 then gives the stated floor. Compactness bounds the gradient of each polynomial and ensures that it belongs to the maximal form domain.

For example, $A_3=240$ gives $\Lambda_{m,3}\ge1/80$ in every dimension. The same reasoning excludes an asymptotically vanishing upper certificate at any fixed degree. It does not determine the exact values of $\Lambda_{m,3}$ , even for $m=3,\ldots,8$ , and its constant tends to zero as the degree grows. There is therefore no conflict with the small full $L^2$ gap of the root frame.

The vertex-cap failure mechanism has published prior art in negative-rate energy-exchange models.

This is a published consequence of Sasada’s vertex-cap argument Sasada, 2015. Caputo proves a dimension-free gap for the unweighted flat simplex heat bath, while Carlen–Posta–Tóth prove uniform gaps for nonnegative exponents $s\in[0,1]$ Caputo, 2008Carlen et al., 2025. The inverse-variance root form lies at $s=-2$ , outside those positive results.

The root calculation does not refute Conjecture 22.1. A general permutation-invariant admissible frame may mix continuously many direction orbits, and no proof shows that the root orbit is optimal. The decisive simplex alternative is an exact all-frame dual certificate with objective tending to zero, or a uniform full-domain lower bound after optimizing over all admissible frames. By Lemma 22.3, the former must use degrees growing with dimension or nonpolynomial tests. Exact optimization at fixed finite dimension remains undetermined; it would require an upper inequality valid over the continuous sphere of directions and a matching admissible frame. Floating-point computations over finitely many frames, and tests restricted to the root frame, decide neither alternative.

References¶

Sasada, M. (2015). Spectral Gap for Stochastic Energy Exchange Model with Nonuniformly Positive Rate Function. Annals of Probability, 43(4), 1663–1711. 10.1214/14-AOP916
Caputo, P. (2008). On the Spectral Gap of the Kac Walk and Other Binary Collision Processes. ALEA. Latin American Journal of Probability and Mathematical Statistics, 4, 205–222. https://alea.math.cnrs.fr/articles/v4/04-10.pdf
Carlen, E. A., Posta, G., & Tőth, I. P. (2025). Spectral Gap for the Stochastic Exchange Model. Stochastic Processes and Their Applications, 190, 104769. 10.1016/j.spa.2025.104769