non-ergodic sampling, VMC (GCNN), -0.4461111
Finding. Flag upheld from the primary text, not only from the Comment. (1) Update rule and sector are stated by the paper itself: Sec. III, last paragraph before Sec. IV (arXiv:2401.02866v2, sources/2401.02866.txt lines 1013-1020; identical wording in v1, sources/2401.02866v1.txt lines 3040-3047): "Whilst for the 48 sites cluster the best variational energy is obtained by the free energy minimization within the total Sz = 0 sector where the samples are generated by exchanging spins at two randomly chosen lattice sites, for the 108 sites cluster the best results are found by sampling within the whole Hilbert space with samples generated by flipping a spin at random lattice site." Sec. II defines the Z2 spin-parity symmetry P_Z2 = prod_i sigma^x_i only "for the eigenstates with Mz = sum_i sigma(r_i) = 0", yet the 108-site run is not restricted to that sector; the Hamiltonian (Eq. 7, H = J sum_<ij> S_i.S_j, no constant shift) conserves S^z_tot, so single spin flips out of the S^z_tot = 0 sector have vanishing acceptance once the ansatz concentrates there. N_s = 2^13 samples per step (Sec. II). Neither arXiv version prints an acceptance rate, autocorrelation time or number of chains (grep "acceptance": 0 hits in v1 and v2). (2) The energy is printed as E0 = -48.18(0) (v2 Sec. III text and Fig. 5 caption; v1 Fig. 12 caption "E_chi0 = -48.18(0)") while the five other 108-site states in the same figure carry (3), (4), (5), (7), (8): a statistical error rounding to zero at the 0.01 level from 2^13 samples is what identical, frozen samples produce, so the printed (0) is evidence for the defect rather than a usable sigma. The 48-site run with the exchange sampler carries a normal error, -21.00 +/- 0.01 (Sec. III). (3) Nothing has changed since the Comment: arXiv:2401.02866 has v1 (5 Jan 2024) and v2 (9 Mar 2025, the PRX text, dated March 11, 2025), both with -48.18(0) and the same sampler sentence; the Comment arXiv:2605.28861 has only v1 (21 May 2026, sources/2605.28861.txt: acceptance exactly zero beyond 5000 iterations, Fig. 1(a); exchange-update optimisation of the same 4x4 GCNN converges to E ~ -45.6 vs DMRG -47.3, spin-flip-trained parameters re-evaluated ergodically give E ~ -42.6, Fig. 1(b); split-R-hat 1.39 frozen vs 1.03 ergodic; numerics in the (Gamma, chi0) sector, argued to carry over to (M, chi0)). No Reply exists: arXiv API au:Duric AND au:Sengupta lists only 2401.02866v2, 2512.11670 and 2606.14101 (sources/arxiv_api_duric_sengupta.xml); Crossref record of 10.1103/PhysRevX.15.011047 has empty updated-by (no erratum) and a bibliographic search for the Comment title returns no PRX Comment or Reply (sources/crossref_prx15_011047.json, crossref_comment_search.json); none of the 16 OpenAlex citing works is a Reply (sources/openalex_citing_prx15_011047.json). No code repository: GitHub repository search on seven title/author/method queries returns 0 hits and no author handle with public repositories; the paper has no code or data availability statement. The authors later preprint arXiv:2512.11670 (1/9 plateau, same 108-site cluster) keeps the unconstrained single-spin-flip scheme (sources/2512.11670.txt lines 416-433) and does not revisit -48.18. (4) Flag stands as written: the row keeps bound_type variational (it is what the method claims) and is skipped for the record under RULES.md 6.1 and 10 (non-ergodic sampling clause).
Diagnosis. The reported energy is an artifact of non-ergodic Monte Carlo sampling, not a property of the ansatz. The kagome Heisenberg Hamiltonian is SU(2) symmetric and its ground state lies in a fixed total-magnetisation sector, so a single-spin-flip update - which changes S^z_tot - is incompatible with the symmetry. As the network concentrates on the physical S^z_tot = 0 sector the acceptance rate collapses to exactly zero beyond 5000 iterations and the chains freeze, so the reported average is taken over a non-representative set of configurations. With the magnetisation-preserving exchange update the same architecture and optimizer converge stably to E ~ -45.6 against a DMRG value of E ~ -47.3; re-evaluating the spin-flip-optimised PARAMETERS under ergodic sampling raises the energy further, to E ~ -42.6. The published -48.18 lies below all of them.
Ruled out. NOT a variational-principle violation, and that is the point: DMRG at finite bond dimension is itself an upper bound, so an energy below it is not on its own evidence of error - it would ordinarily just be a better state. Nothing about the number alone identifies it as wrong. The refutation had to come from the sampler.
Evidence. arXiv:2605.28861 (Kamal, Kufel, Vu, Laumann & Yao), Fig. 1(a) acceptance-rate collapse and Fig. 1(b) energy convergence; Ðurić et al., Phys. Rev. X 15, 011047 (2025), arXiv:2401.02866, Sec. IV.