The moment map: construction - The KLS theorem and its methods

This chapter is the construction layer of the moment-map approach. Chapters The moment map: CMH and the linear test and The moment map: exact cases fixed the target $\mathrm{CMH}(4)$ , showed that it implies the affine Poincaré inequality, computed it in the tractable cases and isolated the two falsification tests; what remains is to prove it, and that is what this chapter attempts. Its aim is to extend Theorem 4.2 from constant matrices to multiplier fields selected by an arbitrary test function. Its conclusion is not established here: the identities displayed in this chapter outside labelled statements are formal calculations, not proofs, and the labelled problems below say what closing each gap would deliver. The steps are those of Remark 15.1: Haar compression, Schur–Piola transport, and the square-root commutator that the argument must control.

To avoid a collision with the stochastic quantities $H_t,S_t,K_t$ and $A_t$ used earlier, all objects in this chapter are stationary moment-map objects: $H=D^2\phi$ is the Hessian metric, $N$ is a weighted elliptic operator, and $K_M$ is a compressed multiplier.

Haar compression and the commutator error¶

On the mean-zero subspace, formally set

N=D_\nu^*H^{-1}D, \qquad Q_M=D_\nu^*H^{-1/2}MH^{-1/2}D,

(18.1)

and

R=H^{-1/2}DN^{-1/2}, \qquad K_M=R^*MR.

(18.2)

On a common core, $R^*R=I$ and

Q_M=N^{1/2}K_MN^{1/2}.

(18.3)

For normalized Haar contrast multipliers $M_S$ , the formal error is therefore

e_S(u)=Q_{M_S}u-K_{M_S}Nu =[N^{1/2},K_{M_S}]N^{1/2}u.

(18.4)

Formally, the complete-tree Bessel deficit is

\mathfrak B(h) =\left(1-\frac1n\right)\norm{h}^2-\sum_S\norm{K_{M_S}h}^2 =\sum_S\norm{(I-RR^*)M_SRh}^2.

(18.5)

Numerical experiments produce rotated examples with negative individual sibling deficits, and Laguerre near-extremizers that use nearly all descendant slack. These are evidence, not proofs, and no such example is recorded here as an exact counterexample; what they suggest is that neither nodewise positivity nor a fixed fractional allocation of slack along the tree can work.

Schur–Piola transport and local algebra¶

For a $1+d$ split write

H=\begin{pmatrix}h&b^\top\\ b&C\end{pmatrix},\qquad v=C^{-1}b,\qquad s=h-b^\top C^{-1}b,\qquad \Delta=\det C,

(18.6)

and $X=\partial_r-v\cdot\nabla_q$ . Block algebra, Piola’s identity, and Hessian compatibility formally give

\operatorname{cof}H =\Delta\begin{pmatrix}1&-v^\top\\-v&vv^\top+sC^{-1}\end{pmatrix}, \qquad \operatorname{div}(\Delta X)=0,

(18.7)

Xv=C^{-1}\nabla_qs, \qquad XC=J^\top C=CJ, \qquad \Tr(C^{-1}XC)=\operatorname{div}_qv.

(18.8)

If $z=\nabla_q\phi$ is the child target coordinate, then $Xz=0$ ; for $x=\nabla\phi$ , $Xx_\perp=0$ and $Xx_0=s$ . Hence $s^{-1}X$ is the pullback of $\partial_{x_0}$ . In particular, the trace/conformal derivative is constrained by the Schur connection and is not a free scalar mode.

At a Schur-normal $1+2$ point, let $T\in\operatorname{Sym}_2$ , $a,g\in\R^2$ , let $R_\perp e_1=e_2$ and $R_\perp e_2=-e_1$ , and use $\operatorname{sym}(u\otimes v)=\tfrac12(u\otimes v+v\otimes u)$ . Put

B_i=\operatorname{sym}(g\otimes R_\perp^\top e_i).

(18.9)

The fixed-target coordinate form expands exactly as

\begin{aligned} Q(a,T,g) &=8|a|^2+\frac12\sum_i\norm{\{B_i,T\}}_{\HS}^2-4a\cdot R_\perp Tg\\ &=8\left|a-\frac14R_\perp Tg\right|^2 +\frac32|Tg|^2+\frac14(\Tr T)^2|g|^2\ge0. \end{aligned}

(18.10)

An independent coordinate expansion checked this identity. What is missing is the invariant identification of the tensorial lift actually generated by the global operator, and a reduction covering all higher-dimensional splits. Positivity in $1+2$ , $1+3$ , and $2+2$ alone would not be a dimension-free theorem; the missing step is Conjecture 18.1.

Two solenoidal channels¶

Let $S(z)=\E[C\mid z]$ be the inherited child block of the parent Stein kernel and let $K(z)$ be the canonical child moment-map Stein kernel. Both solve the same Stein equation, so

\mathsf D=S-K, \qquad \partial_\beta(\rho\mathsf D_{\alpha\beta})=0.

(18.11)

In a two-dimensional child, subject to topology and boundary conditions,

\rho\mathsf D=R_\perp^\top D^2\lambda R_\perp

(18.12)

is the Airy representation. For a parent datum $h$ , the canonical flux residual has, formally, the form

r_h=r_{\mathrm{cond}}+(I-\Pi_K)\mathsf D\nabla v.

(18.13)

Thus a conditional-flux channel and a canonical-versus-inherited Stein-kernel channel must be controlled separately: a single local conserved vector would identify the two projections in (18.13), which differ in general.

The main global obstacle¶

The cubic symbol does not cancel in general. With $A_M=H^{-1/2}MH^{-1/2}$ , a direct calculation gives

C_3(\xi)=2\left[(H^{-1}\xi)^r(\partial_rA_M)(\xi,\xi) -(A_M\xi)^r(\partial_rH^{-1})(\xi,\xi)\right].

(18.14)

It is expected, but not shown here, to equal

2(H^{-1}\xi)^k\xi^\top H^{-1/2}[\Omega_k,M]H^{-1/2}\xi, \qquad \Omega_k=\frac12\left(H^{-1/2}\partial_kH^{1/2} -(\partial_kH^{1/2})H^{-1/2}\right),

(18.15)

exhibiting eigenframe rotation rather than conformal variation. The square-root bridge is the standard resolvent formula

[N^{1/2},K_M] =\frac1\pi\int_0^\infty t^{1/2}(N+t)^{-1} [N,K_M](N+t)^{-1}\dd t,

(18.16)

provided all forms and domains are justified.

Conjecture 18.2 (Global square-root commutator — Not settled here · conj:mm-square-root-commutator · Idea · Counterexample)

Let $\mathfrak R_{\mathrm{Letwin}}(u)$ be the retained nonnegative remainder made up of the available variable-multiplier, Codazzi, Monge–Ampère, corrector, and descendant squares, and of nothing else. Over the complete Haar tree

2\sum_S\operatorname{Re}\langle K_{M_S}Nu,e_S(u)\rangle +\sum_S\norm{e_S(u)}^2 \le \mathfrak B(Nu)+\mathfrak R_{\mathrm{Letwin}}(u)

(18.17)

with a dimension-free constant and stable operator domains; and the complete-tree estimate supplies the global analytic input required by the fully defined CMH target of Remark 15.2.

This is the main obstacle, but not the only gap. The invariant lift, all-split reduction, Hodge boundary conditions, and one-edge flux identity must be settled alongside it; the endpoint duality is not among them, since it is Theorem 16.1. Note also that Proposition 16.3 places a floor under the difficulty: even the constant-multiplier, static specialization of the estimate above is not available from (4.15) by algebra, so no argument here may treat the commutator as a lower-order correction. Numerical experiments on model cases suggest that no scalar, conformal-only, or nodewise shortcut works. This is evidence rather than proof, and it points to an argument that is global and couples the matrix structure.