{"id":"8959996e-4ceb-4858-874f-b16a10bc5fd7","arxiv_id":"2507.09073","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Variation-after-projection with few non-orthogonal Slater determinants reproduces exact shell-model energies in sd/pf shells and yields a 78Ni binding energy below the best Lanczos diagonalization.","lead":"A variational method using a small number of non-orthogonal Slater determinants, optimized after angular momentum projection, is shown to reproduce exact nuclear shell-model solutions in light nuclei and 48Cr, and to give a 78Ni ground state binding energy lower than the largest Lanczos diagonalization performed to date. If the method's exactness holds, it could extend shell-model calculations to nuclei beyond current limits.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper's own benchmarks show DNO-SM(VAP) misses exact SM energies by up to ~14 keV in 48Cr yrast states; the 78Ni gain over Lanczos is only 16 keV, so the 'fully recovered exact diagonalization' claim is not supported.","rationale":"The reader identified global convergence to the projected-energy minimum as the key untested assumption. My reading agrees that this is a load-bearing concern, but I find a more direct internal problem: the paper's own benchmark tables show that DNO-SM(VAP) does not actually reproduce exact SM energies at the precision implied by 'exact' and 'fully recovered'. The sd-shell ground-state discrepancies are a few tenths of a keV, and the 48Cr excited-state discrepancies are as large as 14 keV. These are variational upper bounds, so they are consistent with the method being a good approximation, but they are inconsistent with the literal claim of having recovered the exact eigensolution. The 78Ni result is only 16 keV below a truncated Lanczos calculation and 4 keV below an extrapolation, so the same convergence issues that cause 8-14 keV errors in 48Cr could plausibly affect the 78Ni number. No convergence thresholds, restarts, or numerical stability diagnostics are reported, and no independent diagonalization exists for 78Ni. Thus the central claim is not yet established, although the method is promising and the sd-shell and 48Cr results are strong evidence of near-exactness. The reader's CONDITIONAL verdict remains appropriate; the concern strengthens the condition but does not change the verdict.","tokens_in":12580,"tokens_out":5486,"duration_ms":70789,"concrete_test":"Recompute the 48Cr yrast states of Table 2 with the same DNO-SM(VAP) implementation and stopping criterion used for 78Ni, starting from 10 independent random reference Slater determinants and using at least twice the number of VAP states; compare each state to the exact SM energy. If any state fails to reproduce the exact energy within 1 keV, or if any state moves below the Table 2 value by more than 1 keV, then the 78Ni claim of exact convergence lacks an error bar and should be reported as an upper-bound variational estimate.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3's central numerical proof is undercut by the reported numbers. In Table 1, the DNO-SM(VAP) ground-state energies lie above the exact SM values: 24Mg -87.10405 vs -87.10445 (0.40 keV less bound), 28Si -135.86003 vs -135.86073 (0.70 keV), 26Al -105.74901 vs -105.74934 (0.33 keV). In Table 2 for 48Cr, the yrast 2+, 4+, 6+, 8+, 10+, and 12+ states are all above the exact SM values by 8-14 keV (e.g., 2+: -32.135 vs -32.148), despite the text calling the matching 'perfect' and claiming pairing correlations are 'fully captured'. These are variational upper bounds, so the optimization has not demonstrably reached the exact eigensolution even when the exact answer is known. The 78Ni headline result is -372.73275 MeV, 16.07 keV below the 10p10h Lanczos value and 4.25 keV below the exponential extrapolation. That margin is at the same scale as the benchmark discrepancies. Because no independent full diagonalization exists for 78Ni, and because the paper reports no convergence thresholds, no restart analysis, and no condition-number or precision checks for the non-orthogonal generalized eigenproblem, the claim that exact diagonalization is 'fully recovered' is an extrapolation from approximate benchmark agreement, not a demonstrated fact.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the Discrete Non-Orthogonal Shell Model with variation after projection (DNO-SM(VAP)), a variational method that represents nuclear wave functions as superpositions of angular-momentum-projected, non-orthogonal Slater determinants obtained by a hybrid quasi-Newton optimization of the projected energy. The authors benchmark the method in the sd shell (20Ne, 24Mg, 28Si, 26Al) with the USDB interaction, in 48Cr with KB3, and in 78Ni with PFSDG-U in the pf-sdg space. They report that the method reproduces exact shell-model ground-state energies in the sd shell to within 0.7 keV and the 48Cr yrast band to within 14 keV, and that for 78Ni the DNO-SM(VAP) energy (-372.73275 MeV) is lower than the 10p10h Lanczos result (-372.71668 MeV) and the exponential extrapolation (-372.72850 MeV). Based on these results, the paper claims that the Broeckhove-Deumens theorem is numerically realized and that exact diagonalization is 'fully recovered' in 78Ni.","tokens_in":12930,"tokens_out":5068,"duration_ms":57327,"significance":"If the claim of exact recovery were justified, the method would be a significant algorithmic advance for shell-model calculations in spaces beyond the reach of direct diagonalization. The small-space benchmarks are encouraging and give the method credibility as a variational tool: the sd-shell ground states are within about 0.7 keV of exact results and the 48Cr yrast spectrum is qualitatively reproduced with only tens of determinants. Independent external comparisons against exact shell-model diagonalization in small spaces are a strength of the paper. However, the central exactness claim is not supported by the data: in the small spaces where exact results are known, the variational energies are upper bounds and deviate from exact values by up to 14 keV, and in 78Ni no exact solution exists to validate the claim. The paper therefore presents a promising method with an overreaching interpretation.","major_comments":[{"comment":"The DNO-SM(VAP) ground-state energies in Table 1 do not reproduce exact shell-model energies: 24Mg gives -87.10405 MeV versus -87.10445 MeV, 28Si gives -135.86003 versus -135.86073 MeV, and 26Al gives -105.74901 versus -105.74934 MeV. Since these are variational upper bounds, the statement in Section 3 that the approach 'captures all correlations to recover exactly the SM solution' is contradicted by the reported numbers. Please quantify these residuals and specify a convergence criterion (e.g., energy plateaus with respect to the number of determinants and optimization tolerances) that would support the 'exact recovery' claim.","section":"Section 3, Table 1"},{"comment":"For the 48Cr yrast band, the DNO-SM(VAP) energies for the 2+, 4+, 6+, 8+, 10+, and 12+ states lie 8-14 keV above the exact shell-model values (e.g., 2+ is -32.135 MeV vs -32.148 MeV), while only the 0+, 14+, and 16+ states match within the quoted precision. The text states that 'the two curves becomes indistinguishable with a perfect matching' and that proton-neutron pairing correlations are 'fully incorporated'. These statements are not supported by the table. Please provide a quantitative comparison of discrepancies and show whether increasing the number of determinants reduces them systematically.","section":"Section 4, Table 2 and Fig. 2"},{"comment":"The headline claim that 'this is the first time using a variational method, the exact diagonalization is fully recovered' in 78Ni is not established. The DNO-SM(VAP) energy -372.73275 MeV is a variational upper bound on the exact ground-state energy; the 10p10h Lanczos value -372.71668 MeV is also an upper bound in a truncated space, and the exponential extrapolation -372.72850 MeV is an estimate with an assumed functional form. The difference of 4-16 keV is comparable to the residuals shown in Table 2 where exact results are known (up to 14 keV). Because no independent exact diagonalization exists for 78Ni, the word 'exact' is unsubstantiated. Please reframe the claim as 'the variational approximation improves on the largest Lanczos diagonalization' and provide a convergence analysis with error estimates.","section":"Section 5, Fig. 3"},{"comment":"The hybrid VAP optimization relies on a quasi-Newton local minimization of the projected energy functional, but the paper provides no proof or numerical evidence that the algorithm reaches a global minimum, no explicit convergence thresholds, no restarts from different initial conditions to check for local minima, and no condition-number or precision checks for the generalized eigenvalue problem (3) with potentially near-linear-dependent non-orthogonal states. Since the 'exact recovery' claim depends on global convergence of the optimization, these diagnostics are load-bearing. Please add them or explicitly qualify the results as variational approximations.","section":"Section 2, Eqs. (3)-(7) and algorithm description"}],"minor_comments":[{"comment":"The abstract states 'exact shell-model solutions are obtained', but the results in Table 1 show differences of a few tenths of a keV from exact energies. Please use more qualified language, such as 'accurate to within 0.7 keV' or 'variational approximations'.","section":"Abstract and Section 3"},{"comment":"The 26Al row appears to have a typographical issue: the entries in the DNO-SM(VAP) and Exact SM columns are presented as '105.74901' and '105.74934' without a leading minus sign. Please correct the formatting.","section":"Table 1"},{"comment":"The phrase 'the two curves becomes indistinguishable' has a subject-verb agreement error; it should be 'the two curves become indistinguishable'. Also, the caption uses 'DNO-SM(VAP)' and 'SM' without defining them at first use in the caption.","section":"Section 4, Fig. 2 caption and text"},{"comment":"The extrapolated value -372.72850 MeV assumes exponential convergence in the number of NpNh excitations. Please state the uncertainty of this extrapolation or show an alternative extrapolation (e.g., with different fitting ranges) to support the comparison.","section":"Section 5, Fig. 3"},{"comment":"Reference [33] contains a stray '3.' after the DOI (Phys. Rev. Lett. 25 (1970) 782. 3.); please correct this citation.","section":"References"},{"comment":"The symbols 'PA V' and 'V AP' appear with inconsistent spacing throughout the text (e.g., 'PA V' vs 'PAV' and 'V AP' vs 'VAP'). Please standardize the notation, e.g., PAV and VAP.","section":"Notation"}],"recommendation":"major_revision","confidential_remarks":"The paper's marketing language ('exact recovery', 'first time') is stronger than the evidence justifies, and the benchmark residuals in Tables 1 and 2 should be prominently discussed. The method itself is promising and likely of interest to the nuclear structure community, but the authors should be asked to revise the claims and provide convergence diagnostics before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something genuine: a small set of variation-after-projection non-orthogonal Slater determinants reproduces shell-model ground-state energies in the sd shell to within a keV of exact diagonalization, and the 48Cr yrast band comes within 8–14 keV. That is a real step beyond the authors' earlier DNO-SM and is worth taking seriously. The 78Ni result, a binding energy about 16 keV below the largest Lanczos run and 4 keV below the exponential extrapolation, is suggestive and potentially important if it holds.\n\nThe soft spots are in the claims, not the method's core. The text calls the 48Cr matching 'perfect' and says pairing is 'fully captured,' but Table 2 shows every yrast state above the exact value by 8–14 keV. Those are variational upper bounds, so the optimization has not demonstrably reached the exact eigensolution even where the exact answer is known. The 78Ni gain over Lanczos is at the same energy scale as those benchmark discrepancies, so calling it 'the exact diagonalization is fully recovered' is an extrapolation, not a demonstrated fact. The Broeckhove–Deumens theorem guarantees existence of a discrete non-orthogonal spanning set in a dense continuum, but it says nothing about whether this particular VAP construction reaches it; the paper's own numbers show it gets very close but not exactly there in every tested case.\n\nThe lack of reproducibility is a minor but real issue: no code, no convergence thresholds, no condition-number or precision checks for the non-orthogonal generalized eigenproblem. For a method paper whose main selling point is exactness, that matters.\n\nWho is this for? Practitioners doing large-scale shell-model calculations, especially near 78Ni and heavy nuclei where Lanczos is stretched. It deserves a serious referee. My recommendation: send to peer review, but require the authors to temper the 'exact' and 'fully recovered' language to match the numbers, report convergence criteria quantitatively, and ideally release code or at least a detailed convergence log for one benchmark. If they can do that, this becomes a solid contribution.","headline":"Real variational progress on the shell-model secular problem, but the 'exact recovery' claim overstates what the reported numbers show.","tokens_in":587,"tokens_out":604,"would_cite":true,"duration_ms":24365,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that a small set of non-orthogonal Slater determinants, optimized by variation after angular-momentum projection, reproduces exact shell-model eigensolutions; in $^{78}$Ni it converges to $-372.73275$ MeV, below the…","keywords":["Nuclear Structure","Non-orthogonal Shell Model","Variation after Projection","Symmetry breaking and restoration","Slater determinants","Pairing correlations","78Ni","Backbending"],"falsifier":"An independent calculation of the $^{78}$Ni ground state with the same PFSDG-U interaction that reaches an energy below $-372.73275$ MeV, for example a Lanczos diagonalization carried beyond the $10p10h$ truncation or a different exact method in the full $pf$--$sdg$ space, would refute the claim that the VAP result is the exact eigensolution. In the small spaces where exact diagonalization is available, restarting the VAP optimization from many random initial Slater determinants and finding multiple distinct converged energies would reveal local-minimum trapping and undermine the global-convergence assumption.","tokens_in":12362,"feed_emoji":"⚛️","tokens_out":14514,"duration_ms":156271,"temperature":0.7,"pith_summary":"This paper claims that a small set of non-orthogonal Slater determinants, chosen by minimizing the angular-momentum-projected energy, reproduces the exact solutions of the nuclear shell-model secular problem. In sd-shell nuclei and in the yrast band of $^{48}$Cr, the variation-after-projection energies match full shell-model diagonalization to within a few keV, including the backbending caused by pairing. In $^{78}$Ni, whose full $pf$--$sdg$ valence space contains about $2\\times 10^{11}$ Slater determinants, the method converges to $-372.73275$ MeV, lower than the $10p10h$ Lanczos value ($-372.71668$ MeV) and lower than the exponentially extrapolated estimate ($-372.72850$ MeV). If correct, this means exact-quality nuclear-structure calculations no longer require diagonalizing the full shell-model space: a few dozen optimized determinants can carry the same physics, and a previously formal theorem on discrete non-orthogonal spanning sets becomes a practical numerical tool.","feed_headline":"Tiny variational basis beats 200-billion-state shell-model limit","feed_subtitle":"Non-orthogonal Slater determinants match exact energies in benchmarks and go lower in 78Ni than the largest Lanczos run.","key_machinery":"The machinery is the Discrete Non-Orthogonal Shell Model in its variation-after-projection variant (DNO-SM(VAP)). Trial states are linear combinations $|\\psi^{J\\pi}_n\\rangle = \\sum_{q,K} C^{J\\pi}_{n,qK}\\hat P^J_{MK}\\hat P^\\pi|\\phi_q\\rangle$ of non-orthogonal Slater determinants projected onto good angular momentum $J$ and parity $\\pi$. Each determinant is written in Thouless form $|\\phi_q\\rangle = \\mathcal N_0 e^{\\sum_{ij} Z^{(q)}_{ij} a_i^\\dagger a_j}|\\phi^{(q)}_0\\rangle$, with $Z^{(q)}$ a skew-symmetric complex matrix, and the projected energy $E = \\langle\\psi|H|\\psi\\rangle/\\langle\\psi|\\psi\\rangle$ is minimized with respect to both the mixing coefficients $C$ and the matrices $Z$. The stationarity condition is a generalized Brillouin condition, meaning the energy is stable against one-particle--one-hole excitations of all determinants in the set, and a quasi-Newton algorithm carries out the minimization. A greedy basis-selection procedure grows the non-orthogonal set from the Hamiltonian itself, so parity and rotational symmetry are built in from the start rather than imposed afterward; this is the mechanism that turns the formal spanning theorem into a converging numerical solver.","core_discovery":"The central discovery is that symmetry-restored non-orthogonal Slater determinants exactly span the full shell-model valence space. Applying the Ritz variational principle to angular-momentum- and parity-projected linear combinations of Slater determinants, and optimizing the determinants themselves at the same time, yields energies indistinguishable from exact shell-model diagonalization wherever that comparison can be made. In $^{48}$Cr the entire yrast band is reproduced with 12--57 determinants, capturing proton--neutron pairing without breaking particle number. In $^{78}$Ni the converged ground-state energy $-372.73275$ MeV lies below the value obtained by the largest Lanczos diagonalization and below the extrapolated Lanczos limit; the paper reads this as the first variational recovery of the exact diagonalization at this scale. This is presented as numerical proof that a completeness theorem for discrete non-orthogonal bases is realized in realistic shell-model calculations, and that variation after projection is numerically equivalent to including very high particle--hole excitations on top of a deformed or spherical reference state.","pith_inferences":["The authors do not state it, but if the $^{78}$Ni VAP energy is indeed the exact eigensolution, the extrapolated Lanczos value ($-372.72850$ MeV) underestimates the binding energy by about 4 keV, implying the truncation error of the $10p10h$ diagonalization is slightly larger than the exponential extrapolation assumed.","A natural and testable extension is to compute excited states and electromagnetic transitions in $^{78}$Ni with the same VAP determinant set; with no full diagonalization available, agreement with the measured low-lying spectrum would independently test the variational completeness claim.","The results suggest a more general compression principle: the low-energy sector of a strongly correlated fermionic Hamiltonian can be spanned by far fewer non-orthogonal determinants than the dimension of the full Fock space, a property that could transfer to other fermionic many-body problems beyond the nuclear shell model, although the paper itself does not pursue this.","Running the VAP optimization from many random initial reference determinants and checking that all trajectories land on the same energy plateau would quantify the local-minimum risk; the paper does not report such a stability analysis, so this remains an open check."],"forward_implications":["Exact shell-model ground and yrast states can be represented by a dozen to a few dozen non-orthogonal Slater determinants rather than by the $\\sim 2\\times 10^{11}$ orthonormal basis states of a conventional M-scheme diagonalization.","Variation after projection with non-orthogonal Slater determinants is numerically equivalent to including very high particle--hole excitations on top of deformed or spherical reference states, which explains why it recovers the correlation energy missing in simpler projected mean-field treatments.","Strong proton--neutron pairing, including the $^{48}$Cr backbending, is fully captured without breaking particle number, so the method provides a particle-number-conserving alternative to symmetry-restored quasiparticle approaches.","Because well-deformed heavy nuclei show nearly perfect $J(J+1)$ rotational spectra, where pairing is weaker than deformation, the present variational strategy is expected to be even more efficient there, opening exact-quality shell-model calculations in the heavy-mass region.","In the $^{78}$Ni region, where shape coexistence is pervasive, the demonstrated convergence allows low-lying states and effective interactions to be studied without waiting for larger Lanczos diagonalizations."],"supporting_citations":[{"why":"States the theorem that a dense continuous set of non-orthogonal states contains a countable subset spanning the same space; this is the existence result the paper numerically realizes.","marker":"[8]"},{"why":"Defines the M-scheme shell-model framework and Lanczos diagonalization used as the exact benchmark and as the comparison standard for 78Ni.","marker":"[10]"},{"why":"Earlier implementation of the discrete non-orthogonal shell model with (β,γ) generator coordinates and the basis-selection technique that the present work extends to particle-hole excitations and VAP.","marker":"[30]"},{"why":"Specifies the pf-sdg valence space and PFSDG-U interaction for 78Ni and supplies the doubly-magic experimental context.","marker":"[39]"},{"why":"Supplies the Thouless parametrization of Slater determinants, angular-momentum and parity projection operators, and the generator-coordinate formalism used throughout.","marker":"[40]"},{"why":"Provides the theorem used to parametrize non-orthogonal Slater determinants around a reference state for the variational optimization.","marker":"[41]"},{"why":"Introduces the resonating Hartree-Fock idea of simultaneously varying several non-orthogonal determinants, which the paper's hybrid VAP strategy follows.","marker":"[42]"},{"why":"Supplies the USDB sd-shell interaction used in the exact benchmarks for 20Ne, 24Mg, 28Si and 26Al.","marker":"[46]"}],"fun_headline_variants":["Exact shell-model energies from a few non-orthogonal determinants","Non-orthogonal Slater determinants beat massive diagonalization","Tiny basis surpasses 200-billion-state shell-model limit","Variation after projection outmatches Lanczos in nickel-78"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the numerical minimization of the projected energy reaches the true ground state rather than stopping in a local minimum; for $^{78}$Ni, where no independent full diagonalization exists, this is inferred from the plateau of the convergence curve rather than proved.","fun_headline_variants_meta":{"raw":{"variants":["Exact shell-model energies from a few non-orthogonal determinants","Non-orthogonal Slater determinants beat massive diagonalization","Tiny basis surpasses 200-billion-state shell-model limit","Variation after projection outmatches Lanczos in nickel-78"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000699,"raw_usage":{"total_tokens":3174,"prompt_tokens":978,"completion_tokens":2196,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":594,"completion_tokens_details":{"reasoning_tokens":2124}},"tokens_in":594,"tokens_out":2196,"duration_ms":19891,"temperature":1.0,"reasoning_tokens":2124,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:06:02.206079+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"An independent calculation of the $^{78}$Ni ground state with the same PFSDG-U interaction that reaches an energy below $-372.73275$ MeV, for example a Lanczos diagonalization carried beyond the $10p10h$ truncation or a different exact method in the full $pf$--$sdg$ space, would refute the claim that the VAP result is the exact eigensolution. In the small spaces where exact diagonalization is available, restarting the VAP optimization from many random initial Slater determinants and finding multiple distinct converged energies would reveal local-minimum trapping and undermine the global-convergence assumption.","supporting_citations":[{"cited_title":"Broeckhove, E","cited_arxiv_id":null,"evidence_quote":"States the theorem that a dense continuous set of non-orthogonal states contains a countable subset spanning the same space; this is the existence result the paper numerically realizes."},{"cited_title":"Taniuchi, C","cited_arxiv_id":null,"evidence_quote":"Specifies the pf-sdg valence space and PFSDG-U interaction for 78Ni and supplies the doubly-magic experimental context."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Thouless parametrization of Slater determinants, angular-momentum and parity projection operators, and the generator-coordinate formalism used throughout."},{"cited_title":"Thouless, Stability conditions and nuclear rotations in the hartree-fock theory, Nucl","cited_arxiv_id":null,"evidence_quote":"Provides the theorem used to parametrize non-orthogonal Slater determinants around a reference state for the variational optimization."},{"cited_title":"Fukutome, Theory of resonating quantum fluctuations in a fermion system: Resonating hartree-fock approximation, Prog","cited_arxiv_id":null,"evidence_quote":"Introduces the resonating Hartree-Fock idea of simultaneously varying several non-orthogonal determinants, which the paper's hybrid VAP strategy follows."}],"review_version":1}