The Möbius–Klein–Laplace Transformation III: Random, Uncertain and Measured Seams

The Möbius–Klein–Laplace transformation (MKLT) is the Laplace transform of a seam-closed signal: every part of a causal signal returns after each period T, multiplied by a seam parameter γ. Read as a loop, γ is the gain of a delayed feedback and T its delay, and Parts I and II fixed both or let them follow a known rule. This article lets them be random, uncertain or measured. A delay that jitters from one return to the next has a mean response that is again a seam closure, with the Laplace transform of the delay law in place of e^(−sT). On a sinusoid the power has a closed form in which the delay law and the gain enter separately, and the spectrum is a coherent line on a background modulated with the loop period. Gaussian jitter bends the low rows of the seam to the left, quadratically in the row index, and the high rows as the square root of it, and adds a second branch of roots far to the right. It comes from the negative delays of the Gaussian tail and disappears under truncation. White jitter keeps the closure; correlated jitter breaks it with weights that are geometric only asymptotically, and persistent long-memory jitter with weights that decay faster than geometrically. A fractional seam driven by white noise has a power-law spectrum at every seam line. A random seam parameter with a jittering phase has a reduced mean and an unchanged mean square. A phase drawn afresh at every return gives a closure, a phase frozen for a record a mixture of closures. A random seam parameter and a random delay together enter through one joint transform, which differs from the product of the 2 marginal transforms when they are correlated. Random operator seams of the reflection family decouple into 2 scalar seams; non-commuting random operators mix the sectors, and they grow almost surely at the top Lyapunov exponent, which is bounded above by the mean-square rate. The unsealing is robust: a wrong seam parameter and a wrong delay change the recovered input by at most the sum of the 2 errors, the delay error weighted by the bandwidth, plus the out-of-band part of a causal record. The least-squares seam estimate reaches the Cramér–Rao bound, the lag and a delay kernel are identified from one record, and the cell start is not identifiable. Numerical inversion with a deformed contour converges only algebraically on the infinite vertical rows of poles of a scalar or an operator seam, and slowly on a distributed seam; Fourier-series methods converge in all 3 cases, and a hybrid that inverts the input and closes in time is as accurate as the inversion of the input. The discrete closure is inverted by a fast Fourier transform on a circle outside the seam poles. A window resolves 2 seams with real seam parameters when it is longer than T_1T_2/|T_1−T_2|, provided the lines are narrower than their separation; a delay drifting at the rate β has a best window of the order T/√(kβ) for the k-th line. Noise in a nonlinear seam switches a bistable seam map between its 2 states at a rate exponential in the inverse noise power, makes the Lyapunov exponent positive below the onset of chaos and in a periodic window, and, with an amplitude-dependent delay, makes the timing jitter grow as the cube of the number of returns unless the amplitude is restored, in which case it grows linearly. All 4 forms are covered, and every number comes from a deterministic script deposited with this record.