HEXAGONbeta

Cancellation of all ultraviolet divergences of quantum gravity by a scale-invariant gas of branes

A mix of human-written and AI-generated textHuman understanding: some partsmath-ph — Mathematical Physicsmath.PR — Probability

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya ↗

Version 2 / Oct 18, 2026 / CC BY 4.0

Abstract

A frozen Poisson gas of branes with no preferred size, each brane a layer of independent cells that shift the proper time of a crossing graviton, multiplies the graviton kinetic operator by an entire function without zeros, $\exp[\tfrac n2\Ein(\Box^2/M^4)]$ times a random factor that tends to a constant at high momentum, in almost every realization. We prove that for n≥4n\ge4 the theory of gravity built on it has no ultraviolet divergence at any order of perturbation theory: every graph with two or more loops converges, the one-loop poles are those of a local theory and are cancelled by three operators quartic in curvature with finite coefficients, Newton's constant and the cosmological constant do not run, the zero-point energy of the graviton is 3nM4/(64π2)3nM^4/(64\pi^2) for every regulator, and the physical SS matrix is unitary at every loop order. The propagator keeps the graviton pole alone, and the potential of a point mass is finite at the origin. A one-loop graph in closed form confirms the cutting rule and grows faster than any exponential of the energy, so the loop expansion is an expansion in a small parameter only below an energy between 1.4M1.4M and 1.9M1.9M. The free Euclidean field is a Gaussian measure with finite curvature that is not reflection positive, which states what a construction beyond perturbation theory would have to supply.

Provenance statement

Artificial-intelligence tools (Claude Code, Anthropic) assisted with drafting and editing the text, with algebraic and numerical calculations, with the verification code and with the scripts that draw the figures. Deep Bhattacharjee carried out all computations and checks, in C, Python, Julia and Lean 4, with the assistance of Claude Code. Every result is derived analytically in the text; no statement rests on a numerical computation or on an AI output. The verification suite (195 Python, 84 C and 114 Julia checks, 17 Lean 4 theorems) and five independent audit scripts (74 checks) re-evaluate the formulas: release v3.2.0, https://github.com/creelie/brane-cluster-gases/releases/tag/v3.2.0. The result is perturbative: finite and unitary at every order for n >= 4; no construction beyond perturbation theory is claimed. Both authors approved the final version.
AI provenance filesAI-PROVENANCE-HEXAGON.md · 2 KB

Tools used

Anthropic
Claude CodeVersion Opus 5.5 Ultracode

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Version history

  1. v1Submitted by Deep BhattacharjeeInitial depositSupersededOct 18, 2026
  2. v2Submitted by Deep BhattacharjeeSubmitted as a new versionCurrentOct 21, 2026