A uniform floor at every fixed polynomial degree - The KLS theorem and its methods

Part of the conditional-fiber mechanism, Chapter Conditional fibers: inverse-variance frames of line resamplings; the reading order is on the full proofs page.

Overview. A polynomial on a uniform interval satisfies a reverse Poincaré inequality with a constant depending only on its degree. Applying it on every chord and averaging with the frame identity bounds the fiber form below by the ordinary gradient energy. The established affine Poincaré bound for the uniform simplex then gives a dimension-independent floor at every fixed degree. No optimization over frames and no computation enter this argument.

Theorem 128.1 (Fixed-degree floor for every simplex frame)

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).

(128.1)

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)},

(128.2)

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 $m\to\infty$ .

Proof 128.1

Let $U$ be uniform on $[-1,1]$ and let $P_n$ denote the Legendre polynomial normalized by $P_n(1)=1$ . Its standard elementary identities are

\mathbb E[P_n(U)P_j(U)]=\frac{\delta_{nj}}{2n+1}, \qquad P_n(-1)=(-1)^n.

(128.4)

These identities follow from Rodrigues’ formula $P_n(u)=(2^n n!)^{-1}(d/du)^n(u^2-1)^n$ : integration by parts $n$ times proves orthogonality, since the boundary terms vanish; its leading coefficient is $(2n)!/(2^n(n!)^2)$ . Integrating Rodrigues’ formula against $P_n$ gives $\int_{-1}^1P_n^2= (2n)!/[2^{2n}(n!)^2]\int_{-1}^1(1-u^2)^n\,du =2/(2n+1)$ , where the last integral follows by the beta integral (or its integration-by-parts recurrence). Evaluating Rodrigues’ formula at the endpoints gives the displayed endpoint values.

For $0\le j<n$ , integration by parts and orthogonality yield

\int_{-1}^1P_n'(u)P_j(u)\,du =[P_nP_j]_{-1}^1-\int_{-1}^1P_nP_j'\,du =1-(-1)^{n+j}.

(128.5)

Expanding the polynomial $P_n'$ in $P_0,\ldots,P_{n-1}$ therefore gives

P_n'=\sum_{\substack{0\le j<n\\n-j\ \mathrm{odd}}}(2j+1)P_j, \qquad \mathbb E|P_n'(U)|^2 =\sum_{\substack{0\le j<n\\n-j\ \mathrm{odd}}}(2j+1) =\frac{n(n+1)}2.

(128.6)

The last equality is the sum of an arithmetic progression, for either parity of $n$ . Thus $e_n=\sqrt{2n+1}P_n$ form an orthonormal basis, and a centered polynomial $h=\sum_{n=1}^k a_ne_n$ satisfies, by pointwise Cauchy–Schwarz,

\mathbb E|h'(U)|^2 \le\left(\sum_{n=1}^ka_n^2\right) \sum_{n=1}^k\mathbb E|e_n'(U)|^2 =\operatorname{Var}(h(U))\sum_{n=1}^k\frac{n(n+1)(2n+1)}2 =\frac{A_k}{4}\operatorname{Var}(h(U)).

(128.7)

The final sum telescopes because the difference of $n(n+1)^2(n+2)/4$ at consecutive $n$ is $n(n+1)(2n+1)/2$ . For $I=[a,a+L]$ substitute $T=a+L(U+1)/2$ and $h(u)=q(a+L(u+1)/2)$ . The derivative transformation supplies $4/L^2$ , proving the first inequality. Since $\operatorname{Var}(T)=L^2/12$ , it also proves the second.

Proof 128.2 (Proof of the fixed-degree floor)

The support of $\mu_m$ is a compact convex body with nonempty interior relative to $H_0$ . For every fixed direction $\theta$ , orthogonal Fubini shows that its conditional law on almost every chord is uniform on a nondegenerate bounded interval. Parameterize that interval by the Euclidean arc-length coordinate $t$ in $z+t\theta$ ; its variance is exactly $L^2/12$ . Restriction of $f$ to it has degree at most $k$ , and derivative $\partial_\theta f$ . The interval inequality therefore gives

\int_{\theta^\perp} \frac{\operatorname{Var}(f(z+T\theta)\mid z)} {\operatorname{Var}(T\mid z)}\,d\bar\mu_\theta(z) \ge\frac{12}{A_k}\int|\partial_\theta f|^2\,d\mu_m.

(128.8)

Exceptional null fibers are assigned zero, as in the form definition. All polynomials have bounded gradients on the support, hence are Lipschitz on that convex support. If $M$ bounds this gradient, the independent-copy variance formula on each chord gives $\operatorname{Var}(f(z+T\theta)\mid z)\le M^2\operatorname{Var}(T\mid z)$ . Consequently $\mathcal D(f)\le dM^2<\infty$ , proving maximal-domain membership directly.

Integrate the preceding lower bound against $d\rho(\theta)$ . Tonelli and frame isotropy give

d\int\!\int|\partial_\theta f|^2\,d\mu_m\,d\rho(\theta) =\int\nabla f^T\left(d\int\theta\theta^T\,d\rho\right)\nabla f\,d\mu_m =\int|\nabla_{H_0}f|^2\,d\mu_m.

(128.9)

This proves the first theorem inequality. The law $\mu_m$ is an isotropic linear image of a log-concave Dirichlet law. By Corollary 17.3, its ordinary Poincaré constant on $H_0$ is at most 4, giving the second inequality. This use has no unresolved CMH premise: it invokes the established Dirichlet corollary, not universal CMH.

Taking the infimum over nonconstant polynomials, then the supremum over admissible frames, preserves the lower bound. Every nonzero linear function has quotient one by Lemma 22.1, so each infimum is at most one. Admissible frames exist, for example the even uniform measure on an orthonormal basis of $H_0$ . Finally $A_3=240$ .

A valid all-frame upper certificate has objective at least $\Lambda_{m,k}$ by its defining upper-bound property. At fixed $k$ , the lower bound $3/A_k$ is independent of $m$ and strictly positive, so these objectives cannot tend to zero.

Hypotheses and fences. Uniform density and convex support supply uniform interval conditional laws; bounded support ensures domain membership; polynomial degree supplies the inverse inequality; the test-independent frame identity recovers the full gradient energy; isotropy and the certified Dirichlet Poincaré bound supply the dimension-independent variance comparison. No additional hypothesis is used. Neither the proposed node nor Conjecture 22.1 has a recorded bounded_by edge. The root cap obstruction Proposition 22.1 and negative exchange obstruction Proposition 22.2 concern unrestricted $L^2$ tests and are compatible with a degree-dependent floor. The exact quadratic floor Lemma 22.2 is stronger at that degree and is not used. This does not reverse the sufficient-only bridge from fiber gaps to KLS.