The Thermodynamic Limit, Scale-Invariant Gravity, and Topological Slow-Roll in Discrete 1D Spacetime

This manuscript presents the computational culmination of the Information-Topological Register Model (Mass-Gap Cosmology), validating the emergence of macroscopic three-dimensional spacetime and cosmological inflation from a strictly discrete, one-dimensional binary network. By scaling the network to the thermodynamic limit (N = 300, 000 nodes), we demonstrate a fundamental spatial dichotomy: a pure vacuum network exhibits fractal roughness with a spectral dimension of Ds ≈ 2. 92, whereas the introduction of a 10% matter fraction intrinsically curves the local space, raising the macroscopic dimension to Ds ≈ 3. 06 via scale-invariant topological gravity. Furthermore, by translating topological node degrees into local density fluctuations, we extract the primordial power spectrum P (k) using Fast Fourier Transforms (FFT). While the pure topological base model inherently generates a perfectly scale-invariant Harrison-Zel'dovich spectrum (ns ≈ 1. 0), we introduce a deterministic mechanism for inflation: Topological Slow-Roll. By implementing an exponential topological friction term — analogous to the cosmological e-fold expansion — the scale invariance is organically broken. The highly non-linear response of the discrete manifold naturally generates the red-tilted primordial spectrum (ns 1. 0: survivalₚrob = 1. 0 if np. random. rand () Genesis beendet in infₜime: . 2f Sekunden. Kanten: numₑdges: , ") print (f"Aktiviere Kinematik. . . ") startₜime = time. time () finalₙodes = kinematicₗoop (TICKS, numₙodes, bit, left, right) thermₜime = time. time () - startₜime print (f"-> Kinematik beendet in thermₜime: . 2f Sekunden. ") print ("Starte Observatorium (200. 000 Wanderer). . . ") Pₜ = measureₛpectraldimension (finalₙodes, numₑdges, left, right, edgesᵤ, edgesᵥ, bit) valid = Pₜ > 0 t = np. arange (1, len (Pₜ) + 1) tᵥalid = tvalid Pᵥalid = Pₜvalid logₜ = np. log (tᵥalid) logP = np. log (Pᵥalid) fitₘask = (tᵥalid > 50) & (logP > -11. 5) if np. sum (fitₘask) > 5: m, c = np. polyfit (logₜfitₘask, logPfitₘask, 1) Dₛ = -2 * m else: Dₛ = 0. 0 m, c = 0, 0 print (f"=======================================") print (f" GEMESSENE SPEKTRALDIMENSION: Dₛ = Dₛ: . 4f") print (f"=======================================") plt. figure (figsize= (10, 6) ) plt. plot (logₜ, logP, 'b. ', label=r'Return Probability P (t) ', alpha=0. 6) if np. sum (fitₘask) > 5: plt. plot (logₜfitₘask, m * logₜfitₘask + c, 'r-', linewidth=2, label=f'Fit (Steigung \ -Dₛ/2: . 3f \ Dₛ \ Dₛ: . 2f) ') plt. title (f'Emergenz der Dimension (Mass-Gap 10% Materie, N=NINITIAL) ') plt. xlabel ('Log (Schritte t) ') plt. ylabel ('Log (Return Probability P (t) ) ') plt. legend () plt. grid (True, linestyle='--') plt. show () if _ₙame__ == "_ₘain__": runₜopologicalᵤniverse ()