Mathematical and Observational Consistency in Constructing Quantum Gravity and Unifying General Relativity

The unification of Einstein’s General Relativity (GR) and Quantum Mechanics (QM) into a consistenttheory of Quantum Gravity confronts two foundational challenges at the Planck scale (ℓPl = √Gℏ/c3 ∼1.62 × 10−35 m, EPl ∼ 1.22 × 1019 GeV): the resolution of classical curvature singularities (Big Bang andblack hole interiors) and the preservation of quantum unitarity during black hole evaporation (Hawk-ing’s information paradox). In this paper, we present a comprehensive mathematical and numerical syn-thesis bridging the two leading paradigms of modern quantum gravity: background-independent LoopQuantum Gravity / Loop Quantum Cosmology (LQG/LQC) and holographic Superstring The-ory / AdS/CFT Quantum Information. We execute five self-contained computational verifications:(1) LQG Discrete Spatial Geometry & Isolated Horizon Microstates (§2): From the SU (2)holonomy–flux algebra, we compute the discrete spectrum of the area operator ˆAS , solve the exact transcen-dental state-counting equations for isolated horizons to determine the Barbero–Immirzi parameter γU (1) =0.2375329580 (γSU (2) = 0.2740668582) and the fundamental area gap ∆ = 4√3πγℓ2Pl = 5.170046 ℓ2Pl, andevaluate logarithmic quantum corrections −α ln(A/∆) (αU (1) = 1/2, αSU (2) = 3/2) to the Bekenstein–Hawking entropy. (2) LQC Big Bounce Dynamics & Singularity Resolution (§3): Integrating theholonomy-modified Friedmann–Raychaudhuri–Klein–Gordon system under a Starobinsky R2 inflationarypotential, we demonstrate that the classical Big Bang singularity (a → 0, ρ → ∞) is dynamically re-placed by a nonsingular Quantum Bounce at the universal critical density ρc = √332π2γ3G2ℏ ≈ 0.4092 ρPlwith numerical constraint conservation to 5.38 × 10−7. (3) LQC Primordial Perturbations & CMBLow-Multipole Suppression (§4): Solving the dressed-metric Mukhanov–Sasaki equation across thebounce, we show that super-horizon modes (k ≲ kB = 2.20 × 10−4 Mpc−1) undergo a 57.8% infraredpower suppression, naturally alleviating the Planck 2018 large-angle (ℓ ≤ 30) temperature anomaly with∆χ2 = −46.91 relative to standard ΛCDM while preserving high-ℓ acoustic peaks. (4) Holographic Is-land Formula, Replica Wormholes & the Unitary Page Curve (§5): Extremizing the generalizedentropy functional Sgen(R ∪ I) = Area(∂I)4GN ℏ + Ssemi-cl(R ∪ I), we demonstrate the phase transition at thePage time tP ≈ 0.545 tevap from the trivial No-Island saddle (I = ∅) to the nontrivial Quantum ExtremalSurface (QES) Island saddle (I̸ = ∅), restoring full quantum unitarity (S(R) → 0) and resolving the AMPSfirewall paradox via ER = EPR and computational complexity constraints. (5) Holographic QuantumError-Correcting Codes (QECC) & Emergent Spacetime (§6): Constructing the hyperbolic pen-tagonal HaPPY tensor network ([[5, 1, 3]] perfect stabilizer code), we verify the Ryu–Takayanagi geodesicscaling (r = 0.9993 correlation with continuum CFT2) and establish that radial bulk depth directly en-codes logical error-correction resilience up to the pc = 50.0% boundary erasure threshold. Finally, in §7,we establish the mathematical isomorphism between LQG spin networks and holographic tensor networks.