Holographic Complexity in Cosmology: Positive Results, No-Go Theorems, and the Memory Integral Obstruction

Holographic Complexity in Cosmology: Positive Results, No-Go Theorems, and the Memory Integral Obstruction John Shun Leong Cheng — Independent Researcher — Revised April 2026 This preprint systematically investigates whether the growth of holographic computational complexity can drive cosmic acceleration through modifications of the Bekenstein–Hawking entropy-area law within the Jacobson thermodynamic programme. Four Positive Results 1. Complexity Filling Fraction: The ratio of accumulated Complexity=Action complexity to holographic capacity takes the parameter-free form fC ≈ 0. 057 for ACDM (based on an exact numerical integral of the Lloyd bound over cosmic history). This refines the 0. 10 "algebraic shortcut" used in earlier drafts, which is now shown to overestimate the exact integral by a factor of 1. 7. 2. Computability-Theoretic Separation: The Harlow–Hayden (HH) computational complexity barrier and Chaitin's halting probability Ω are separated by the Church-Turing limit. While both share high Kolmogorov complexity and high logical depth, HH lies in Class C (computable inversion, exponentially slow), while Ω lies in Class U (no computable inversion exists). 3. QEC Screening Mechanism: The Almheiri–Dong–Harlow quantum error correction framework suggests a heuristic screening mechanism for local gravitational constant variations. In this proposal, local variations of Gₑff are suppressed by exp (-Sₗocal) ≈ exp (-10⁹6), rendering them undetectable to all conceivable experimental precision. 4. de Sitter Exclusion Theorem: This theorem links the late-time fate of the universe to the computational complexity class of quantum gravity. If horizon dynamics involve a quantum lattice system with an undecidable spectral gap, the spacetime cannot be asymptotically de Sitter; this requires w > -1 at late times. Three No-Go Theorems Theorem 1 (Sign Problem): For any monotonic entropy-area modification, positive dark energy and a weakening gravitational constant are mutually exclusive. Theorem 2 (Memory Integral Catastrophe): The corrected entropy law produces a non-local memory integral that forces cosmic recollapse. Numerical integration confirms this "cybernetic" effect guarantees a zero-crossing of H² at redshifts z ≈ 1. 45 to 2. 40 depending on coupling strength. Theorem 3 (QEC Dark Energy Exhaustion): If the universe is modeled as a Maximum Distance Separable holographic QEC code, the resulting equation of state is a constant w ≈ -0. 966. This worsens the Hubble tension to 6. 3 sigma and is incompatible with DESI DR2 observations at >3 sigma. Furthermore, time-dependent corrections from entropy production are suppressed by ~17 orders of magnitude, rendering the dynamic correction physically empty. Conclusion These results collectively close the programme of complexity-driven dark energy within the Jacobson framework and its QEC extension. While the no-go theorems rule out the sourcing of cosmic acceleration via entropy-area modifications, the positive results (filling fraction, C/U separation, and the Exclusion Theorem) survive as independent building blocks for future approaches. Code Availability: A verification suite (SymPy + SciPy) reproducing every mathematical claim is available at: https: //github. com/cjlcv6061-svg/adm-verification. Keywords: holographic complexity, dark energy, Jacobson thermodynamic programme, Complexity=Action, no-go theorems, memory integral, quantum error correction, de Sitter holography, Harlow–Hayden, Chaitin Ω. To cite this work: Cheng, J. S. L. (2026), 'Holographic Complexity in Cosmology', 10. 5281/zenodo. 19242134.