Theorem U′-Local: The Local Exponent Law of the Syntonic Window

Theorem U′ gives the optimal window of a first-order tracker the exponent \(\alpha=1/(2H+1-\gamma)\). The statement is exact for two-power losses and holds at leading order on the regular-variation class, which is how the Corpus Status Register records it. Real low-frequency noise is rarely one power law: the Allan curve of a gyroscope is a sum of regimes, and the window an engine chooses can sit in a crossover. This note gives the local law. For any separable loss \(J(\tau)=A(\tau)+\sigma^2\beta(\tau)\) whose balance ratio \(R=A'/(-\beta')\) increases from \(0\) to \(\infty\), the optimum is unique and\[\alpha_{\mathrm{loc}}(\sigma^2)=\frac{d\log\tau^\star}{d\log\sigma^2}=\frac{1}{\mathsf{s}[A'](\tau^\star)-\mathsf{s}[-\beta'](\tau^\star)},\qquad \mathsf{s}[f](\tau)=\frac{\tau f'(\tau)}{f(\tau)},\]an identity with no asymptotics. For the exponential window the tracking term is half the Laplace average of the structure function, \(A(\tau)=\tfrac12\mathbb{E}[D(\tau E)]\) with \(E\) standard exponential. Under white noise, with the same monotone balance, the optimal window is the root of \(\tau\,\mathbb{E}[(E-1)D(\tau E)]=\sigma^2\), a nonparametric form of the syntonic law that needs no memory index, and the index that governs the exponent is a Laguerre-weighted functional of \(D\) around \(\tau\), not the log-slope of \(D\) at \(\tau\). The monotone balance is automatic for sums of power laws and not beyond: Brownian motion plus a sinusoid makes the optimum jump. For finite sums of independent fractional components this effective index is a mean of the component indices tilted towards the smoother ones, so it is never below the log-slope: reading \(H\) off the slope of the structure function overestimates the exponent. If \(D\) is regularly varying of index \(2H\), the optimal window is regularly varying of index \(1/(2H+1)\) and the local exponent converges to it, with no smoothness condition on the slowly varying factor, which in general still moves the exponent at finite scales. On a two-component example (\(H=1/2\) and \(0.9\)) the local law agrees with a numerical derivative to \(1.3\times10^{-10}\), while the slope reading is off by up to \(0.038\) at the crossover; a logarithmic correction keeps the local exponent \(0.008\) to \(0.044\) below its limit over eight decades of the window. In a pre-registered numerical bench with predictions frozen from a calibration, the local law matched the exponents of the optimal window within three standard errors in all 37 secants of four drifts, the slope reading overestimated them at every secant of the two crossover drifts, and where the balance stops being monotone the optimum jumped, by a factor \(4.23\) between noise levels \(2\,\%\) below and above the predicted jump level; the drifts were drawn from the laws assumed, so this verifies the mathematics and its numerical chain, not a sensor. Authorship and collaboration statement All prose in this corpus was drafted by large language models under the author's direction and revised in author-directed loops. The author set the questions, chose the hypotheses, designed and ran every experiment, decided every retraction, and designed the corpus as a system (immutable hub-and-spoke record, concept and version identifiers, errata register, hash-frozen firmware, pre-registration gates). Verification is independent of both: the algebraic core is machine-checked in Lean 4/Mathlib, numerical results follow pre-registered protocols, and flight logs are public. Where model-generated reasoning text survives in a published document, it is an artifact of this method and is recorded in the Corpus Status Register. The author alone approves every statement and is accountable for it; no language model is an author. RETURN TO HUB : https://doi.org/10.5281/zenodo.17254395