Recharacterises spacetime as an emergent projection from a Type III₁ von Neumann factor algebra, coarse-grained via GKSL into an effective non-Hermitian Hamiltonian. Derives dual bounds on the minimum geometric sampling frequency f_min in Planck units. The modular-to-geometric dictionary is grounded in the JLMS relation. Establishes non-Markovian robustness analytically via the Fast Scrambling Bound. Results: dual f_min bounds bracket the physical threshold; at R_crit~ℓ_P, eigenvalues coalesce at a higher-order Exceptional Point; the secular ringdown signature h(t)∝t·exp(−γt) is derived from the Jordan block matrix exponential. CMB isotropy follows from primordial modular non-locality without a fine-tuned inflaton. Prediction: t·e^{−γt} QNM envelope falsifiable at SNR≥25 with LISA. Structured Abstract Background General Relativity treats spacetime as a continuous smooth manifold that inevitably diverges at gravitational singularities (r → 0) and the Big Bang (a → 0). All existing quantum gravity frameworks apply quantum corrections to preexisting geometric structures. This ontological assumption is the source of the singularity problem: geometry cannot simultaneously be the regulator and the quantity being regulated. No framework has derived the informational threshold at which continuous geometry ceases to be physically computable, nor the mechanism by which its dissolution is topological rather than statistical. Gap Missing: (a) a first-principles informational threshold for geometric dissolution derived from Unruh thermal noise, Bekenstein-Hawking entropy, and the Margolus-Levitin speed limit; (b) an explicit algebraic mechanism governing the transition; (c) robustness of the mechanism under non-Markovian corrections to GKSL; (d) a non-circular modular-to-geometric dictionary anchored to geometric area operators; (e) a falsifiable observational signature accessible to next-generation detectors. Approach We recharacterize spacetime as an emergent projection from a Type III₁ von Neumann factor algebra, coarse-grained via GKSL into an effective non-Hermitian Hamiltonian H_eff. We derive dual bounds on the minimum geometric sampling frequency f_min in Planck units. The modular-to-geometric dictionary is grounded in the JLMS relation (Ĥ_ω = Â_QES/(4G) + K̂_bulk), resolving the Riemann-tensor circularity of prior derivations. Non-Markovian robustness is established analytically via the Fast Scrambling Bound, yielding a calculable memory kernel and R_crit correction. The 2×2 Teukolsky truncation is reframed as the leading contribution to infinite-dimensional QNM pseudospectral instability near extremality. Results Dual f_min bounds: f_min^{ML} = 2/(3πR) [O(R⁻¹)] and f_min^{BV} = 2/(3πR²) [O(R⁻²)] bracket the physical threshold. At R_crit ∼ ℓ_P, eigenvalues of H_eff coalesce at a higher-order Exceptional Point (EPᵈ); the QFI metric develops a structural pole from JLMS-derived area fluctuations. The Fast Scrambling Bound gives M₁^{NZ} ∼ Ο(R²) at the Planck scale, analytically bounding the R_crit shift to < 2ζR² for dimensionless ζ ≪ 1. The secular ringdown signature h(t) ∝ t·exp(−γt) derives exactly from the Jordan block matrix exponential. Near-extremal Kerr Bekenstein area corrections strengthen the EP prediction for high-spin black holes. Cosmological EPᵈ is mathematically formalised as a Riemann-sheet transition where the modular time parameter acquires a complex component. Implications Gravitational and cosmological singularities are resolved by topological dissolution without modifying equations of motion. The Big Bang is an atemporal algebraic phase on a distinct Riemann sheet of the time parameter. CMB isotropy follows from primordial modular non-locality without a fine-tuned inflaton. The t·e^{−γt} QNM envelope is pre-registerable and falsifiable at SNR ≥ 25 with LISA, with a secondary condition-number observable. All key claims have explicit non-Markovian robustness bounds derived from first principles.