The Möbius–Klein–Laplace Transformation II: Beyond One Fixed Seam

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. Part I worked with one fixed seam, a causal signal of exponential order and a linear seam. This article removes these limits one by one and asks what each removal brings. The growth condition is not needed: in Mikusiński's operator field the seam closure is a locally finite operator series, and the unsealing inverts it for every continuous signal. The other side of the seam row carries the anticausal closure. It is the causal closure with the inverse seam parameter and a negative period, its response precedes the input, and the pre-echo has the time constant T/ln|γ|. The bilateral closure is a Zak transform with a complex quasi-momentum. The periodic seam has a slope theorem in place of a final value, a repeated seam has a Jordan block and secular growth, and a seam with diffusion between the returns has an algebraic tail. The seam series has a closed form in Apostol–Bernoulli polynomials, which turns the seam defects into a convergence accelerator. A varying period is straightened by an Abel function: in warped time the record is again an MKLT. Moore's function of a moving mirror is such an Abel function, and a Neumann–Dirichlet cavity is the Möbius seam in Moore time. Several seams give exponential polynomials whose roots lie in a band bounded by an explicit abscissa. A seam that contains a seam, a controller inside the loop, is exactly a closure with 2 seams after one unsealing. A seam parameter that depends on the returning amplitude makes every strand of the record iterate one map. Its fixed point lies where the seam parameter is 1, and the linearised seam parameter is one plus the elasticity of the seam function. Period doubling sets in at the elasticity −2, where the seam row moves onto the Möbius row; the distance from it is the doubling detuning, with an exact identity for the image. The logistic, Ricker and Hill seams follow this rule, the cascade runs with the Feigenbaum ratio, and the Lyapunov exponent is the mean logarithm of the effective seam parameter. A pointwise seam selects no wave shape. A Kerr ring with dispersion, whose seam carries the detuning, the loss and the nonlinear phase, holds a stable cavity soliton and creates it by a detuning sweep. Korteweg–de Vries solitons arise when the seam delay depends on the amplitude. The path-average soliton of an amplified fibre link is the soliton of a lossy seam with gain. A random seam is stable almost surely when the mean logarithm of its modulus is negative, and its moments follow closures with averaged seam parameters. The fugacity is a seam parameter, the Matsubara frequencies are its seam row, and the dynamic Casimir effect in a mixed cavity is shifted by the Möbius seam in Moore time. Estimators with instruments and forgetting track a changing seam. Every new statement is proved and checked by deterministic scripts with counter-checks.