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Ghosts in Poisson Brane Gases

A mix of human-written and AI-generated textHuman understanding: some partsmath-ph — Mathematical Physics

Contributed by Deep Bhattacharjee ↗, Ushashi Bhattacharya ↗

Version 1 / Oct 18, 2026 / CC BY 4.0

Abstract

A Poisson gas of branes with mass intensity ν\nu dresses the graviton propagator by the random entire function ∏k(1+p2/tk)\prod_k(1+p^2/t_k). We prove that the gas average of the propagator equals $p^{-2}\exp[-\int p^2\nu(\dd t)/(p^2+t)]$, so that the massive states carry total weight e−ν(0,∞)−1<0e^{-\nu(0,\infty)}-1<0. For $\nu=\lambda(t)\,\dd t/t$ the averaged spectrum consists of ghosts alone exactly when λ≤1\lambda\le1, and every average with a finite four-dimensional tadpole has states of both signs. Frozen gases keep alternating residues.

Provenance statement

The proofs, the text, the figures, the Lean 4 formalization and the Python, C and Julia checks were carried out by Deep Bhattacharjee with the help of AI assistance (Claude, Anthropic) in the Claude Code environment. The paper proves by hand that the gas average of the dressed propagator is exp(-phi(z))/z with phi a complete Bernstein function (not 1/(z E F(z))), that this average has no positive Kallen-Lehmann representation, that for nu = lambda(t) dt/t its massive spectrum is purely ghost-like exactly when lambda <= 1 (Plemelj-Privalov boundary values), that every average with a finite four-dimensional tadpole carries states of both signs, and that the residues of a frozen gas alternate in sign. The finite and elementary steps (sign of the residues, the sum rule, the inequalities of Appendix A, Im phi <= pi, the sign of the spectral density and a partial fraction) are proved in Lean 4 with Mathlib (13 theorems in ghosts/lean/BraneGhosts.lean), depending only on propext, Classical.choice and Quot.sound, with no sorry or added axiom; the analytic steps are proved in the paper and are not formalized. Python (19 checks), C (5) and Julia (9) programs recheck the formulas numerically and by simulation. The paper does not prove ultraviolet finiteness or completeness of quantum gravity; its Theorem D shows that this averaged model cannot have both a finite tadpole and a positive spectrum. Both authors approved the final version. Tools used: Anthropic - Claude (Claude Code); Lean community - Lean 4 (v4.33.0) with Mathlib; Python (NumPy, SciPy, mpmath); C; Julia; shell. arXiv identifier: none (not posted on arXiv). Formalization: https://github.com/creelie/brane-cluster-gases/blob/main/ghosts/lean/BraneGhosts.lean (archived at Zenodo, release v3.0.0: https://doi.org/10.5281/zenodo.23449534; all versions 10.5281/zenodo.22866290)
AI provenance filesAI_PROVENANCE_GhostsBraneGases.md · 3 KB

Tools used

Anthropic
Claude CodeVersion Opus 5.5 Ultracode

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Version history

  1. v1Submitted by Deep BhattacharjeeInitial depositCurrentOct 18, 2026