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

We investigate whether the Discrete Symplectic Cosmology (DSC) framework---in which cosmic evolution is modeled as a non-autonomous symplectic map on a Planck-scale lattice with a 1/² (t) adiabatic cooling law---can reproduce key statistical features of the cosmic microwave background (CMB) temperature anisotropies. Using a St\"ormer--Verlet integrator on two- and three-dimensional lattices, we demonstrate that: (i) the symplectic evolution produces near-Gaussian one-point statistics (skewness = +0. 06 0. 31, excess kurtosis = -0. 10 0. 26 over 20 independent realizations), while a dissipative coupled-map-lattice baseline fails with kurtosis = -1. 60; (ii) acoustic-like oscillation peaks emerge naturally in the power spectrum D (k) = k² P (k), with positions alignable to a mock CDM reference via three lattice parameters; (iii) the 1/² (t) cooling produces a finite acoustic horizon through asymptotic freeze-out of wave propagation; and (iv) Bayesian optimization recovers all three lattice parameters from power spectra in a twin experiment (MSE = 8. 9 10^-5). Full-sky maps generated by projecting a 128³ lattice onto a Mollweide sphere exhibit multi-scale temperature fluctuations visually resembling Planck observations, with pixel distributions close to those of the Planck SMICA map. These results establish the DSC lattice as a minimal computational framework capable of reproducing several CMB phenomenological features from symplectic dynamics alone, though the kurtosis sign differs from Planck and the acoustic peak spacing requires parameter fitting.