Gravitational Waves in Geometric–Information Cosmological Evolution: Transition Sources, Continuous Frequency Bands, and Inter-Epoch Transfer — Conditional Mechanisms, Energy Transfer, and Formal Proof Sources

We investigate a conditional gravitational-wave mechanism during scalar instabilities in an SL(6,C)-motivated geometric-information framework. The four-dimensional Lorentzian background arises via a regular reduction with two positive internal directions; statistical localization alone does not establish it. Within a derivative-coupled scalar theory, dilution of the homogeneous kinetic background drives the neutral order parameter through a mass instability. For negative effective mass squared, noncollinear scalar modes give a nonzero transverse-traceless stress at every wavenumber in 0<K<2. In a frozen-background approximation, a finite-duration exponential source leaves strictly positive residual tensor energy, extended to finite wave packets by continuous spectral weights. These local existence results are distinguished from a stochastic background through unequal-time stress correlations and retarded propagation. Continuous-time finite-rank positivity, Gaussian benchmarks, and covariance-error bounds characterize spatial and temporal coherence. An off-shell energy identity connects parent-field equations to gravitational-wave, dark-matter, and visible-sector transfer rates; combined with receiver continuity and Raychaudhuri evolution it yields a conditional conservation chain consistent with the Friedmann constraint. Redshift relations and uncertainty bounds show that observed frequencies depend on source scales and thermal history. We establish a conditional route from seeded scalar instability to tensor radiation and identify the source statistics, transfer rates, and evolution required for quantitative predictions.