Lean Verification for "Weak Typicality of von Neumann Entanglement Entropy in Gaussian Boson Sampling"

This software archive contains the Lean 4 formalization accompanying the paper “Weak Typicality of von Neumann Entanglement Entropy in Gaussian Boson Sampling.”

It provides typed Lean counterparts and machine-readable verification statuses for all 57 labeled equations in the paper. Conditional on three explicitly stated external mathematical inputs—full unitary Haar concentration, Mirsky’s singular value inequality, and Gaussian normal form evidence—the kernel verified development proves the finite covariance and symplectic matrix identities, the singular value entropy representation, logarithmic endpoint regularization, exact cutoff estimates, COE second moments, the extensive entropy mean lower bound, the relative concentration bound of order exp(−c ε² n²/log²(en)), weak typicality, almost sure convergence under arbitrary couplings with Haar marginals, the logarithmic variance bound, deviations below every polynomial scale, a typical volume law, and logarithmic tightness. The Gaussian entropy formula is derived internally from the supplied thermal product spectrum rather than assumed as an additional theorem.

The Page curve transfer separately uses the previously published mean Page curve limit for the chosen subsystem sequence. Definitions, manuscript assumptions, the published Page mean input, external boundary statements, and derived kernel theorems are distinguished in the equation registry.

The archive pins the official stable Lean release leanprover/lean4:v4.33.0 and Mathlib revision 641fbd329d4ffb62bef83c51f54088469056bd36. It contains the complete source code, the 57-equation registry and crosswalk, reproducible build instructions, permanent axiom audits, citation metadata, copyright and license notices, and a stable toolchain build report. The source contains no unfinished proofs or project-specific global axioms. The three external mathematical inputs are ordinary explicit theorem parameters. The axiom reports for the proved endpoints contain only Lean and Mathlib’s standard logical foundations: propext, Classical.choice, and Quot.sound.

The formal verification concerns the ideal lossless and noiseless, fixed nonzero equal squeezing, Haar random Gaussian boson sampling model at proportional subsystem size. It uses an explicit spectral Gaussian state interface but does not construct bosonic Fock space, trace class density operators, partial traces, or operator logarithms, and it does not internally identify the abstract trace-log transcription with the spectral entropy model. It also does not formalize unequal squeezing, loss, finite depth interferometers, sampling hardness, anticoncentration of output probabilities, or the manuscript’s later human proof of failure of strong typicality.

Version 1.0.1 is a reproducibility and archival metadata correction release. The 129 Lean source files are mathematically unchanged from version 1.0.0.

The archive is licensed under the GNU General Public License v3.0 only (GPL-3.0-only).