Differentiable Discrete Symplectic Cosmology: Forward Simulation and Inverse Reconstruction of CMB-like Fluctuations

An inverse-logarithmic-square evolution law, motivated by numerical studies of non-autonomous maps and Riemann zeros, has been proposed as a generator of CMB-like fluctuations. We isolate its linear lattice sector and ask which features of a temperature sky can identify the evolution law. An exact modal transfer gives $P_N=|T_N|^2P_0$: Gaussianity is inherited from Gaussian initial conditions, and a freely adjustable initial mode spectrum can absorb a change of cooling law wherever the transfer is nonzero. We verify this degeneracy numerically and compare $\ln^{-p}$ schedules with $\(p=0,1,2,3\)$, matched in accumulated propagation, using 2048 simulated skies and the observed Planck PR3 SMICA temperature map. With fixed geometry and either of two initial power spectra, all four schedules show substantial mismatch in reserved low-multipole bands. Their ordering depends on the initial spectrum; a Planck-spectrum Gaussian reference performs better under the same shape diagnostic. These results constrain the tested generators, without excluding every logarithmic model. A separate initial-field reconstruction attains $\(r=0.997\)$ on fitted pixels, illustrating the freedom rather than selecting the dynamics. A distinct thin-shell spectral extension reaches acoustic multipoles, but a three-parameter fit still misses their observed sequence. Reusing the exact transfer also accelerates repeated linear final-field calculations, as quantified against the direct implementation. The contribution is an exact, reproducible separation of dynamical effects from initial-condition freedom, together with a route to tests using newly released polarization and lensing products.