Thermodynamic Gaplessness of the Spin-1/2 Kagome Heisenberg Antiferromagnet: An Exact Computer-Assisted Proof

This record contains the full author preprint, its mathematical supplement, and the LaTeX source archive. AbstractWe prove that the nearest-neighbor spin-1/2 antiferromagnetic Heisenberg model on the infinite Kagome lattice is gapless in the bulk GNS sense. The argument constructs six explicit local preimages and their complete infinite-lattice Hamiltonian commutators. Two rigorous 17-spin positivity certificates, each decomposed into 31 nontrivial swap-character blocks and nine SU(2) multiplicity blocks, give a state-uniform off-kernel mass floor. Exact operator transport on the two genuine outward Kagome continuations propagates the same physical family through a finite stationary interface. A root/detail/current decomposition, a local current–boundary identity, and state-uniform norm estimates yield a horizon-independent Dirichlet-energy bound. With harmonic weights, for every infinite-volume ground state ω and every R ≥ 1 the same finite family satisfies M_R(ω) ≥ (15/32) ∑ₖ₌₁R 1/k and E_R(ω) ≤ 761429817528146602401/25088000. Any positive GNS gap would force the second quantity to diverge with the first. Hence every admissible ground-state representation has zero gap, and the declared bulk gap is zero. All finite operator identities and matrix inequalities are checked from exact algebraic data, with rigorous ball arithmetic used for the large positive-definiteness tests. Lean 4 verifies the arbitrary-horizon harmonic implication and the final variational contradiction. The proof uses neither finite-size gap extrapolation nor a spectral cutoff. Companion public verification code, Lean 4 formalization and exact certificates: https://doi.org/10.5281/zenodo.23638365 Distribution and reuse are governed by the Public Scientific Evaluation and Reproducibility Terms included with this record. The deposit excludes the author's private discovery engine and proprietary know-how.