BEYOND EINSTEIN: HOW THE BIG BANG AND BLACK HOLE SINGULARITIES DISSOLVE IN A NON-HERMITIAN ALGEBRAIC UNIVERSE A First-Principles Derivation of the Geometric Dissolution Threshold via Type III₁ von Neumann Algebras, Non-Hermitian Exceptional Points, and Falsifiable LISA Signatures

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 diverges at gravitational singularities (r → 0) and the Big Bang (a → 0). The quantum gravity frameworks we have examined apply quantum corrections to preexisting geometric structures; this ontological assumption is the source of the singularity problem, since geometry cannot simultaneously be the regulator and the quantity being regulated. We are not aware of a prior framework that derives the informational threshold at which continuous geometry ceases to be physically computable and supplies the physical driver, the explicit dissipative construction, and a pre-registered observational signature — the nearest neighbour, the exceptional-point cosmology of Znojil (2025), places an EP at t = 0 without these elements (Table 1.2). Gap Missing in the tradition: (a) a first-principles informational threshold for geometric dissolution from Unruh thermal noise, Bekenstein–Hawking entropy, and the Margolus–Levitin speed limit; (b) an explicit algebraic mechanism governing the transition; (c) robustness under non-Markovian corrections to GKSL; (d) a non-circular modular-to-geometric dictionary anchored to geometric area operators; (e) a falsifiable signature for next-generation detectors. Approach Spacetime is recharacterised as an emergent projection from a Type III₁ von Neumann factor, coarse-grained via GKSL into an effective non-Hermitian Hamiltonian H_eff, with the norm interpretation specified biorthogonally (Appendix B). Dual bounds on the minimum geometric sampling frequency f_min are derived in Planck units. The modular-to-geometric dictionary is grounded in the JLMS relation, resolving the Riemann-tensor circularity of prior derivations [D1*]. Non-Markovian robustness is established analytically via the Fast Scrambling Bound [D3]. The 2×2 Teukolsky truncation yields an exact Jordan-block secular signature [D2]; its extension to the infinite-dimensional QNM spectrum is classified as a conjecture pending pseudospectrum computations [C3]. Results (1) Dual bounds f_minML = 2/(3πR) and f_minBV = 2/(3πR²) bracket the dissolution threshold; all three rates converge at the Planck scale. (2) The critical radius is mass-locked and free of adjustable parameters, R_crit = (288π)¹ᐟ⁴ M¹ᐟ², by the companion derivation [Mattos, 2026b], which supersedes the α-dependent estimate of Revision 1.0; α now affects only signal amplitude and required SNR (Table 2.5). (3) The MSS-bounded NZ kernel gives M₁NZ ∼ O(R²) and an R_crit shift below 2ζC [D3]. (4) The secular ringdown signature h(t) ∝ t·e−γt follows exactly from the 2×2 Jordan block [D2]. (5) Near-extremal Kerr corrections strengthen the EP prediction for high-spin systems. (6) The cosmological EPᵈ is formalised as a Riemann-sheet transition of the modular time parameter; quantitative CMB consequences are carried by the companion cosmology paper [Mattos, 2026c]. Implications If the Class-A programme confirms (§6.4), gravitational and cosmological singularities are resolved by topological dissolution without modifying equations of motion, and the Big Bang is an atemporal algebraic phase on a distinct Riemann sheet [D1*–D3, C2]. The flagship’s own empirical scope is a single evidence class (defective-operator signatures: LISA + laboratory traps); near-term Class-B/C windows live in the companions and are not double-counted here. Falsification is symmetric and time-bounded: the LISA window closes after a pre-registered number of qualifying events, not ‘after exhausting all events’ (§9.2). Keywords: GR-CONT; NH-ALG; Type III₁ von Neumann algebra; exceptional points; non-Hermitian quantum mechanics; biorthogonal metric; emergent spacetime; GKSL; Nakajima–Zwanzig; fast scrambling; JLMS relation; holographic islands; QNM pseudospectrum; Tomita–Takesaki; quantum Fisher information; LISA; ALGUILAS-AI. Method ALGUILAS–AI Dialectical Engine (Mattos, J. C. de, 2026) · v3.2