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Riemann Zeros from the Edge of Spacetime

  We do not attempt to prove the Riemann Hypothesis. However we do show how it emerges as the unitarity condition for holographic RG flow   1. Physical Setup Consider a quantum gravitational system in a de Sitter static patch of radius R with UV cutoff $$y_{\min} = 2\ell_P$$ imposed by the intersection of Schwarzschild radius and Compton wavelength at the Planck scale. The system saturates the MSS bound, so the Lyapunov exponent achieves its maximum $\lambda_L = c/R$, establishing a maximally chaotic fixed point . 2. Universal Effective Theory By the Almheiri-Polchinski universality theorem for near-AdS₂ throats, the effective theory of any system saturating the MSS bound is JT gravity with Schwarzian boundary action . The symmetry breaking $$\text{Diff}(S^1) \to \text{SL}(2,\mathbb{R})$$ leaves the one-dimensional conformal algebra ${H, D, K}$. The dilation operator $$D = \tfrac{1}{2}(xp + px) = -i\left(y\partial_y + \tfrac{1}{2}\right)$$ generates holographic RG flo...
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Anomalous Running Gravity

  How running gravity, anomaly-driven vacuum energy, and quantum error correction combine to explain cosmic tensions, while preserving ΛCDM    ΛCDM fits the CMB, large-scale structure, and nucleosynthesis exceptionally well. And yet: The locally measured Hubble constant ($H_0$) is higher than the Planck CMB prediction Weak lensing surveys find lower clustering amplitude ($S_8$) than ΛCDM predicts These are small but persistent discrepancies. Rather than discarding ΛCDM, what if these tensions are subtle signals about how vacuum energy and gravity behave dynamically? 1. Core Idea: Mildly Running Gravity Standard ΛCDM Vacuum energy is constant: $$\rho_\Lambda = \text{const.}$$ ARG 3.3: Running Gravity with Anomaly Source In Anomalous Running Gravity (ARG 3.3) , we promote vacuum energy and Newton's constant to dynamic quantities sourced by the trace anomaly of quantum fields and a Gauss–Bonnet term: $$S \supset \int d^4x \sqrt{-g} \frac{b}{(4\pi)^2}\ln\left(\frac...
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Dark Energy as the Computational Cost of Spacetime

  Or, dark energy is the thermodynamic cost of maintaining quantum error correction (QEC) in an expanding computational substrate—spacetime itself is a quantum code . In a QEC spacetime, you don’t need a literal CPU (this isn't The  Matrix ). The computation is encoded in the dynamics of the underlying microscopic degrees of freedom, and topological protection ensures coherence. The “code” and the “hardware” are unified.     1. Introduction Dark energy is commonly modelled as a cosmological constant, a fluid, or a scalar field. We explore another idea kicking around: dark energy is the computational cost of maintaining quantum coherence in spacetime. This framework can unify three previously distinct approaches: Viscoelastic / stochastic spacetime : local elastic and viscous responses, stochastic stress from coarse-grained quantum fluctuations Topological Berry phase : global invariants of the vacuum manifold, protecting $\Lambda$ QEC / computational : microscop...
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When Dark Energy Has a Shear: From Noise to ΛCDM

Dark Energy as a Viscoelastic Stochastic Medium The cosmological dark sector can be understood as a relativistic viscoelastic medium coupled to gravity via a stochastic Einstein–Langevin equation. ΛCDM emerges as a special limit of this more general framework.   1. Emergent Gravity, Elasticity, and Dissipation 1.1 Elasticity of Spacetime Gravity may be emergent rather than fundamental: Jacobson (1995) : Einstein equations as a thermodynamic equation of state, $\delta Q = T dS$. Padmanabhan : Spacetime behaves like an elastic solid; diffeomorphisms are deformations; horizons are defects carrying area entropy. Sakharov : Einstein–Hilbert action arises from induced metrical elasticity of quantum vacuum fluctuations. In this view, as we previously  discussed, spacetime is elastic at long wavelengths, with an effective modulus set by vacuum energy: $$Y_\Lambda = \frac{\Lambda c^4}{8\pi G} = \rho_\Lambda c^2.$$ This modulus corresponds to a "cosmic Young's modulus," g...
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Bulk Viscosity from Geometry, Not Only Kinetic Theory

The idea that bulk viscosity  could be an alternative to dark energy for a cosmological effective theory   has been around for a while. For example, Gagnon , 2011 Dark goo : bulk viscosity as an alternative to dark energy,  or Hu, 2024   Viscous universe with cosmological constant ,  or Khan, 2025  Spatial Phonons : A Phenomenological Viscous Dark Energy Model for DESI .  Now, Paul , 2025  Origin of bulk viscosity in cosmology and its thermodynamic implications , takes a kinetic/thermodynamic slant.    Paul found: $$p_{\text{vis}}=-3\zeta H,\qquad \dot S_h>0,\qquad \dot S_m<0,\qquad T_m\neq T_h$$ lead to $$\dot S_{\text{tot}} = \dot S_h + \dot S_m >0$$ for an expanding FLRW universe with the apparent horizon treated as a thermodynamic boundary. What entropy?  $\dot S_{BH}$ is the   Bekenstein–Hawking entropy. It does not increase .  $\dot S_m$ is the coarse-grained, hydrodynamic entrop y of the vacuum fluid,...
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The Horizon's Poisson Ratio: Extremal by Necessity

  The traditional view is that spacetime is not a thing , it is a mathematical object and doesn't have material properties. However,  when a gravitating mass recedes from a region of space-time the curvature diminishes. The field equations of General Relativity don’t have an explicit term for this elastic property , but the framework as a whole does   have that property.  So, if we apply  the principles of continuum mechanics to the scaling of the cosmological horizon, we uncover a startling possibility: the vacuum of our universe may be an auxetic medium , characterised by a negative Poisson ratio that "flips" its fundamental rigidity at the holographic boundary. The Scaling Strain: Measuring the Unmeasurable In traditional engineering, the Poisson ratio ($\nu$) measures how a material deforms. If you stretch a rubber band, it gets thinner (positive $\nu$). If you stretch an auxetic foam, it actually gets thicker (negative $\nu$). To apply this to cosmolog...
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Spacetime Has a Clock: Horizon Thermodynamics and the de Sitter Seesaw

  De Sitter space as a global/semi-classical  thermodynamic equilibrium During both inflation and late-time cosmic acceleration, the Universe is well-approximated by a de Sitter (dS) spacetime with nearly constant curvature radius ($\ell_\Lambda $). Our present Universe may therefore be regarded as a quasi–de Sitter state , possessing a cosmic event horizon (CEH) associated with its vacuum energy density (cosmological constant). A defining feature of de Sitter space is that the cosmological horizon is not merely a causal boundary but a thermodynamic object, endowed with temperature, entropy, and energy. In this context, the total bare (rest) energy associated with the horizon, defined via the Brown–York quasilocal energy, can be written as: \begin{equation} E_0 = 2\, k_B T_{dS} S_{dS} = 2\, m_{CEH} c^2 = 2E_H  \end{equation} This relation is a horizon version of the entanglement first law , from which the Einstein equations themselves can be derived. Here: $S_{dS}$...
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