The Topological Inversion Model (TIM): A Unified Non-Singular Framework for Cosmology, Quantum Topology, and Gravitational-Wave Phenomenology

We present Version 5 of the Topological Inversion Model (TIM), a unified non-singular cosmological framework based on a Planck-scale reciprocal inversion R = L2p/r. A complete definition of 'Absolute Nothing' requires the negation of itself, implying the existence of its complement. In its simplest form: +1 demands −1, and the universe is the remainder thatwill not cancel. This logical self-negation of the null state is what we call the Law of Nothing (designated the //Gaunab, the Khoekhoe deity associated with annihilation, chaos and destruction). Six results are now established. (1) A topological snap at r = Lp generates an elastic recoil field Ψ whose self-consistent scattering amplitude gives echo reflectivity Γinner ≈ 0.42 at ξΨ = 0.002; the observable Fabry-Pérot echo is Γecho ≈ 0.125 (Section 11b). (2) A calibrated BBN that yields ψBBN = 1.059±0.012, partially mitigating the 7Li problem. (3) An overshoot texture mechanism raises the predicted H0 from 67.4 to 71.1kms−1 Mpc−1, reducing the Hubble tension from 5.0σ to 1.7σ and forward predicting the NANOGrav 15-year PTA signal. (4) Polymer one-loop analysis counter terms UV Finiteness without counter terms and vacuum stability (m2 eff = 1.016m2Ψ > 0). (5) Dark matter is identified with the gradient energy of the thinning field δΨ(r) = Ψeq j0(mΨr): no new particle is required, the density profile is naturally cored (ρDM ∝ r0 at r → 0, resolving the core-cusp problem), rotation curves are at, and the required field mass mΨ ≈ 3.8 × 10−23 eVc−2 falls within the observational window for ultralight dark matter. (6) The Standard Model is shown to emerge as the low-energy effective theory on the post-Snap manifold: topological defects of the Skyrme field carry quantum numbers matching the observed fermion spectrum, and integrating out high-momentum modes recovers LSM exactly. All results are reproducible from the provided Python code.