Gravity and Information Geometry

An eight-paper research program on a precise question: when can gravity be understood as the infrared consistency law of a geometry reconstructed from quantum information?

Read the complete series on the publication site.

The project treats a spacetime metric as a coarse-grained description of distinguishability, localization, and causal structure. It then asks what additional hypotheses are needed for the leading consistency equation to take Einstein form. The answer is deliberately scoped. Information geometry by itself does not produce Lorentzian spacetime, diffeomorphism invariance, or general relativity.

The series

1. Microscopic Distinguishability and the Data of Geometric Reconstruction
2. Infrared Axioms for Reconstructed Lorentzian Geometry
3. The Infrared Consistency Equation for Reconstructed Metrics
4. Relative Entropy, Entanglement Equilibrium, and Linearized Gravity
5. Reconstruction Defects, Diffeomorphism Ward Identities, and Conservation
6. Renormalization of Reconstructed Geometry and Metric Beta Functionals
7. Newton's Constant as the Stiffness of the Infrared Metric
8. An Infrared Reconstruction Consistency Theorem

Together, the papers separate several arguments that are often blended together: the geometry of state distinguishability, the assumptions behind a local Lorentzian reconstruction, effective-action stationarity, entanglement first laws, Ward identities, metric beta functionals, and the interpretation of Newton's constant as a response coefficient.

Formal and executable models

The Lean 4 library records the logical dependency structure behind the series. It checks the relative-entropy Hessian sign, tangent-completeness implication, first-law bookkeeping, defect identities, beta-gradient relation, stiffness scaling, and the final implication from explicit assumptions to a leading defect equation. It does not claim to derive those physical assumptions from an arbitrary microscopic system.

Each paper also has a small Haskell model under src/. These programs test finite analogues, coefficient identities, dependency removal, and counterexamples under strict compiler warnings. They are executable diagnostics, not substitutes for continuum proofs.

Repository map

papers/latex/       arXiv-style LaTeX manuscripts
papers/pdf/         publication PDFs
reviews/            separate AGY peer-review reports
research/           prior-art map and claim register
lean/GravityInfo/   dependency-level Lean formalization
src/                per-paper Haskell models
images/             2550 x 3300 paper covers
website/            Next.js publication site and MathML readers
scripts/            build and verification entry points

The scientific source is reviewed before presentation treatment. Once a review round accepts the body, the accepted source is frozen and hashed; only then is the restrained GrokRxiv identifier added to page one. Review vocabulary and internal workflow labels do not appear in the manuscripts.

Verify the project

The verification scripts are independent gates:

bash scripts/verify_papers.sh
bash scripts/verify_haskell.sh
bash scripts/verify_lean.sh
bash scripts/build_readers.sh
bash scripts/verify_readers.sh

The web readers are generated by LaTeXML with MathML, then checked for HTML structure, duplicate IDs, broken internal references, raw TeX leakage, and unsafe links.

To build the publication site:

cd website
npm install
npm run lint
npm run typecheck
npm run build

The local toolchain uses TeX Live, GHC with QuickCheck, Lean 4.32.1, LaTeXML, and Node.js 22.22 or newer.

Research boundary

The strongest result defended by the series is a universality statement within a defined infrared reconstruction class. Locality, a smooth Lorentzian metric, diffeomorphism redundancy, conservation without an obstructing anomaly, tangent completeness, scale separation, and the absence of additional unsuppressed long-range tensor modes are inputs. Under those inputs, the leading local two-derivative metric equation has Einstein form, while higher-curvature, additional-field, and nonlocal corrections remain visible.

That is narrower than saying every quantum system generates gravity. Establishing which microscopic systems satisfy the reconstruction hypotheses is the open problem the series is designed to expose.