Twenty-Three Certified Decimals: Rigorous Ground-State Energies, Correlations and a Coupling Curve for the Half-Filled 4×4 Hubbard Model: certificate bundle v1.0.0

We prove the ground-state energy of the periodic, half-filled 4×4 repulsive Hubbard model with t = 1 to at least 23 decimal places at U = 1, 2, 3, 4, 5, 6, 7, 8, 10, 12 and 16: every enclosure has width below 10⁻²⁴. The double occupancy, the spin correlations and the staggered structure factor have certified interval widths below 10⁻¹⁰, and the kinetic energy below 10⁻⁹. Hypercube, spin and pseudospin symmetries reduce the problem to a 19,186-dimensional block that contains the ground state by Lieb's theorem and a character argument. A binary64 Cholesky factorization, verified using a rigorous rounding-error bound evaluated in exact rational arithmetic, certifies spectral separation within this block; exact Rayleigh quotients and residuals then give Temple enclosures of the energy and bounds on the angle to the ground state, which certify the observables. Separation certificates at both ends and a perturbative trial family enclose E₀(U) for every U in [3.9, 4.1], with uniform width at most 1.6 × 10⁻⁹. Two independent implementations replay the certificates, the interlacing and Temple steps are proved in Lean 4, and all computations run on one laptop. At U = 4 and 8, exact-arithmetic brackets that use neither Lieb's theorem nor floating point are also given, and the enclosures bound their deficits. At U = 4 the certified energy rounds to −13.62185 and excludes the published value −13.62192 [Physical Review B 94, 085103].

Contents of the bundle

• paper/ — the manuscript (Markdown and PDF), its figures, and supplementary/ with five videos generated from the certificate files.
• rayleigh/symmetry/ — the enclosure pipeline (block dimensions, exact basis, certificates, observables, the interval certificate) and recheck/, the independent Rust reference checker; certificates/ holds every certificate, trial vector, run log, checker log and the clean-replay report.
• sos/, rayleigh/demo/, cluster/ — the exact-arithmetic bracket certificates and the expected outputs of their checkers (golden_ files); the bracket checkers belong to the authors' certification software and are not included.
• formal/ — the Lean 4 statements (rank-one interlacing, Temple's inequality, the sum-of-squares remainder bound).
• tests/ — the generators of every table, figure and video of the manuscript.
• replay.sh, REPLAY.md — the automated replay and the step-by-step commands.
• AUDIT_AXIOMS.txt — #print axioms of the formal statements. SHA256SUMS — checksums of every file.

Verification

PYTHON=<python with numpy and scipy> ./replay.sh rebuilds the symmetry block from the Hamiltonian, reproduces every certificate from the stored integer vectors, runs the Rust reference checker, recomputes the observables and checks the manuscript's tables, and writes REPLAY_REPORT.md. It needs about 12 GB of memory and 16 GB of free disk, and takes about half an hour on the hardware above; its work directory is removed at the end. REPLAY.md lists the commands step by step; rayleigh/symmetry/certificates/REPLAY_REPORT_2026-10-14.md is a clean replay of this material. Not included: the two 662,547,600-byte full-sector vectors of the upper bracket certificates (their SHA-256 digests are in rayleigh/demo/). Integrity: shasum -a 256 -c SHA256SUMS inside the bundle; the SHA-256 of the tarball is in the .sha256 file beside it.

Environment

• Python: 3.13.7 with numpy 2.5.4, scipy 1.18.1, matplotlib 3.11.2 (producer-requirements.txt)
• Rust: 1.90.0 (reference checker; cargo build --release in rayleigh/symmetry/recheck)
• Lean: leanprover/lean4:v4.33.1 with Mathlib v4.33.1 (formal statements)
• Hardware: Apple M3 Pro, 18 GiB (all production and replay)

License: CC BY-NC 4.0. Commercial use requires written permission from Origin Neural AI (labs@smartledger.solutions).