The Quantum Holographic Form and Conformal Mellin Transform: A Bounded Optimization Framework for Quantum Gravity and Cosmological Scaling
(Carson) Kai Shun Lam
(Carson) Kai Shun Lam
1) Bachelor of Science (Hons) & M.Sc. (Hons) in Communication Engineering & M.Sc. (Merits) in Information Technology
Education &M.Phil. (Distinction) in Christian Education, Hong Kong.
2) Fellow of Scholar Academic Scientific Society & Dignitary (or Honorary) Fellow, International Organization for Academic &
Scientific Development, India.
We present a non-perturbative framework for quantum gravity that bypasses ultraviolet (UV) divergences
by reframing bulk graviton scattering as a bounded, convex optimization problem on the bounding
horizon. Utilizing the AdS/CFT correspondence as a computational dictionary, we replace the traditional
invariant Einstein-Hilbert coupling constant with a scale-dependent, non-local holographic prefactor
locked directly to boundary horizon microstate distributions. By mapping four-point stress-energy tensor
correlation functions into Mellin space, we construct the Grand Unified Quantum Holographic Field
Equation, anchoring low-energy Effective Field Theory (EFT) expansions to high-energy conformal
bootstrap crossing relations optimized via semidefinite programming (SDP). Evaluating this master
relation across spin sectors ℓ ∈{2,4,6} reveals that while the baseline duality gap width (∆gap) widens
monotonically at high scaling dimensions, the activation of non-local Quantum Extremal Surface (QES)
Island contributions triggers a total phase suppression, completely collapsing the unphysical excluded
space (∆gap = 0) for external dimensions ∆ ≤ 2.0. This establishes that non-local horizon entanglement
structures are strictly required to enforce crossing-symmetry unitarity. Extending the architecture
onto non-flat cosmological geometries (k ̸ = 0) establishes a critical mass-energy threshold at ρcrit =
32.18012 Planck units for closed configurations (k = +1). Below this cutoff, spatial curvature drag
forces early cosmic turnaround and freeze-out; above it, primordial momentum drives the system past
the deceleration barrier toward stable de Sitter convergence where cosmic entropy saturates (S →
Smax). Finally, we demonstrate that the multi-instanton anomalous dimension exponent (γanom) serves
as a cosmic phase operator. The renormalization group (RG) flow drives γanom → 0 in the super-critical
infrared expansion limit, eliminating infinite one-loop zero-mode fluctuations, ensuring exact scale
invariance, and revealing classical general relativity as an emergent, thermodynamic manifestation of
the bounding horizon.
10.33425/2690-8077.1245
Published: 01 Sep, 2026