AOne noise law, two energies
Take noise that multiplies instead of adds: X = e^Y with Y Gaussian. Ask how much a function f(X) can wobble, measured against how steep f is. Measure steepness the multiplicative way, as x f'(x), and the answer is bounded: never more than s^2 times. Measure it the ordinary way, as f'(x), and there is no bound at all.
That is the Gaussian Poincare inequality read through x = e^y. It suggests sampling in log coordinates, and the sampler table below shows that this helps on some targets and hurts on others.
- weighted gap 1/s^2
- 4
- weighted Poincare constant s^2
- 0.25
- ordinary Langevin gap
- none (essential spectrum reaches 0)
- weighted ratio at k = 4
- 0.245421
- ordinary ratio at k = 4
- 2.03181
- equality case (weighted)
- f affine in log x
Log coordinates are a choice, not a free gain
A sampler that works in y = log x sees the potential W(y) = V(e^y) - y. Where W'' dips below zero the target is not log-concave in y. For a Gaussian truncated to x > 0 that happens for every y below ln(a/2). For the lognormal law W is an exact quadratic.
| target | gain | reading |
|---|---|---|
| truncGauss(a=0.5) | 0.36 | loses |
| truncGauss(a=2.0) | 0.14 | loses |
| truncGauss(a=5.0) | 0.64 | loses |
| lognormal(s=0.5) | 5.52 | gains |
| lognormal(s=1.5) | 72.60 | unreliable finite-run estimate: the x-coordinate chain failed the 4-MCSE bias check (not mixed) |
The precise statement
Let Y ~ N(0, s^2 I_d) and X = exp(Y) componentwise, f: (0, oo)^d -> R with f(e^y) in H^1. Then Var f(X) <= s^2 E[sum_i X_i^2 (d_i f(X))^2], with equality exactly when f is affine in log x. For f = x^k the weighted ratio is (1 - e^(-k^2 s^2)) / (k^2 s^2), at most 1.
The Dirichlet form E[X^2 f'^2] belongs to dX = X (1 - log X / s^2) dt + sqrt(2) X dW, the exponential of an Ornstein-Uhlenbeck process. Its eigenvalues are n / s^2 with Hermite eigenfunctions He_n(log x / s), so the gap is 1/s^2. The ordinary Langevin diffusion for the same law has essential spectrum [0, oo): no gap.
Where it breaks
The ordinary inequality Var f <= C E[f'^2] has no constant C. The counterexample is x^k: at s = 1 the ratio is 99.01 for k = 2 and 1.67e+11 for k = 8, while the weighted ratio at k = 8 is 0.0156.
The weighted constant needs the noise to be exactly lognormal; for other multiplicative noise it is the Poincare constant of the law of log X, which may not exist. And log coordinates are not a free improvement: on a Gaussian truncated to x > 0 the measured MALA gain was 0.36, 0.14, 0.64 at a = 0.5, 2 and 5, all below 1. The log map removes the boundary at 0 and wrecks the conditioning.
Prior art and sources
Classical results it reduces to
- Gaussian Poincare inequality and its Hermite equality cases
- Exponential OU / Black-Karasinski diffusion
- Change of variables for MCMC with the Jacobian term
- Reed-Simon IV for the half-line essential spectrum
NNC anchors
- The First Systems of Weighted Differential and Integral Calculus (grossman_1980_weighted)
- Bigeometric Calculus: A System with a Scale-Free Derivative (grossman_1983_bigeometric)
OpenAI families named (context only)
- Dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures (family 093)
- Subpolynomial query complexity for log-concave sampling (family 139)
Model-written and unreviewed. No result on this page depends on one.
Write-up and checks: docs/research/nnc-openai-crossanalysis-2026-10/X1/FINDINGS.md. Every number on this page except the MALA sampler table, which is measured and read from the committed X1/results.json is computed in your browser by lib/crossovers/, tested against independent closed forms in lib/crossovers.test.ts.