The Holographic Path Integral: Volume-Dependent Cutoffs, Axiomatic Rigor, and the Topological Regulation of Quantum Field Theory Anomalies

Abstract

This paper re-evaluates the mathematical architecture of Quantum Field Theory (QFT) by replacing the assumption of an infinite spatial integration domain with the physical, thermodynamic bounds of the Tri-Scale Informational Topology. By explicitly truncating the Feynman Path Integral measure ($\mathcal{D}\phi$) at the finite Bekenstein-Hawking capacity limit of the macroscopic Kodama horizon ($R_K$), this work natively regulates the continuous divergences of the Standard Model and restores the foundational mathematical axioms of perturbation theory without altering the local Lagrangian. Key Breakthroughs: Resolution of UV Divergences: Derives a strictly volume-dependent UV momentum cutoff that dynamically regulates global vacuum fluctuations at $\sim 214\text{ MeV}$. This naturally suppresses the $10^{120}$ cosmological constant error and mitigates the Higgs hierarchy divergence, while permitting local collider cutoffs of $\sim 10^{14}\text{ GeV}$ to preserve $13\text{ TeV}$ scattering cross-sections. Resolution of IR Divergences & RG Flow: Demonstrates that geometrically bounding the spatial wavelength limit by $R_K$ organically eradicates the soft photon anomaly in Bremsstrahlung scattering. Concurrently, anchoring downward Renormalization Group (RG) flow to a zero-initialization at the Planck scale guarantees absolute Higgs vacuum stability. Axiomatic Rigor (Haag's Theorem & The Measure Problem): Restores the mathematical legality of the Interaction Picture by bypassing Haag's Theorem, utilizing the finite boundary ledger to uphold the Stone-von Neumann theorem globally. Furthermore, the framework defines a rigorous, finite-dimensional Lebesgue measure for the Path Integral based on strictly $10^{122}$ discrete boundary variables. Dyson Series Convergence: Proves that because the physical universe possesses a finite informational capacity ($N_{\max} \sim 10^{122}$), the number of independent interaction vertices in a Feynman diagram cannot extend to infinity. The perturbative Dyson Series is mathematically truncated into a finite polynomial, rendering it strictly convergent and exact. Gravity Renormalizability: Demonstrates that the infinite cascade of counterterms required for Einstein-Hilbert spin-2 tensor perturbations is organically truncated by the geometric Bekenstein bound, suggesting that macroscopic gravity functions consistently as an effective field theory without requiring continuous bulk renormalization. By bounding the spatial integration domain, this work reframes the infinities and axiomatic failures of theoretical physics as mathematical artifacts of an unbounded continuum, offering a unified, empirically consistent resolution to QFT's most persistent anomalies.