{"id":"d667de04-bcb2-457d-85f7-9afd1051cded","arxiv_id":"2411.14490","paper_version":5,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For a particle-in-a-box toy model, Rayleigh-Ritz eigenvalues for a projected Hamiltonian converge from below (mostly), unlike the usual upper-bound behavior, with exact zero eigenvalues for the null space.","lead":"A short paper applies the standard Rayleigh-Ritz method to a projected Hamiltonian from a recent ensemble-state variational proposal, finding that variational eigenvalues often converge from below rather than above in a toy model. The result is a caution that the proposed variational framework does not inherit the usual upper-bound guarantee.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The conclusion that RRM eigenvalues of a projected Hamiltonian approach from below 'in most cases' is evidenced only for one polynomial basis and one Hamiltonian; no proof or second basis rules out basis-dependent counterexamples.","rationale":"The reader's weakest_assumption identifies exactly this point: the polynomial basis and the particle-in-a-box Hamiltonian may not be representative, and no theorem extends the numerical finding to general bases or Hamiltonians. I agree. The paper is an honest, narrowly framed toy-model study: the numerical tables and the simple rank-one bound in Eqs. (7)-(8) are credible, and the counterexample to Ding et al. is valid. The weak spot is the scope of the concluding generalization, not the internal computation. Because the paper explicitly presents itself as a toy model, this overgeneralization does not invalidate the central correction; it only means the broad 'most cases' claim should be understood as an observation about the tested example, not a proven property. The proposed monomial-basis recomputation is a feasible, concrete check that would determine whether the observed sign pattern is robust. Since the reader's acceptance already treats the toy-model limitation as acceptable, no verdict change is needed.","tokens_in":5393,"tokens_out":7646,"duration_ms":91369,"concrete_test":"Recompute Tables 2-4 and 7 for D=3 with the same projected Hamiltonian H_D but a different complete L^2 basis, for example the monomial basis u_i(x)=x^{i-1} on [0,1], for N=1,3,5,7,9. If W11, or W23, then the claim that RRM eigenvalues approach from below 'in most cases' is basis-specific; if the sign pattern is unchanged, the generalization is strengthened.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the abstract's statement that RRM eigenvalues of the projected Hamiltonian approach the exact ones from below 'in most cases but a few ones,' and the stronger Section 3.3 conclusion that 'the RRM yields upper and lower bounds for this kind of projected Hamiltonian operators.' The evidence consists of Tables 2-4 and 7, all computed with the single basis u_i(x)=x^i(1-x) and the particle-in-a-box Hamiltonian H=-1/2 d^2/dx^2 on [0,1]. No theorem is given. This matters because for a finite-rank positive semidefinite operator H_D, the eigenvalues of its compression to an N-dimensional trial subspace can lie either above or below the low-lying eigenvalues of H_D depending on the subspace; the min-max/interlacing inequalities do not force one side. The observed lower-bound pattern for N>1 is therefore a property of this specific basis, not a general consequence of the construction. The paper itself notes exceptions (W1>E1 for N=1, W2>E2 for N=3), so the effect is known to be non-universal, but the scope of 'most cases' is never quantified or delimited. The honest limitation is that the broad claim is an extrapolation from a single numerical experiment.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Rayleigh-Ritz method (RRM) applied to a projected Hamiltonian H_D = sum_{k=1}^D E_k |psi_k> are eigenstates of an infinite-dimensional Hamiltonian H. Using a particle-in-a-box toy model and the polynomial basis u_i(x)=x^i(1-x), the author numerically shows that the RRM eigenvalues W_k of H_D often converge from below to the exact eigenvalues E_k, contrary to the usual upper-bound property of the RRM for the full Hamiltonian H. Exceptions are reported for small trial-space dimensions N. The paper also discusses the effect of an energy shift and the behavior of the projected identity operator.","tokens_in":5647,"tokens_out":15799,"duration_ms":167878,"significance":"The observation is useful: it shows that the Rayleigh-Ritz upper-bound theorem does not automatically apply to finite-rank projections of a Hamiltonian, which is directly relevant to the ensemble variational principle proposed by Ding et al. The numerical tables are consistent with the rank argument that H_D has at most D nonzero RRM eigenvalues and the remaining ones are exactly zero; the zero-eigenvalue multiplicity is correctly explained by orthogonality of the corresponding Ritz vectors to the eigenvectors spanning H_D. The paper makes no parametric fits and uses textbook RRM theory, so the core computation is sound. The main weakness is that the 'most cases' claim is an extrapolation from a single basis set and a single Hamiltonian, with no theorem or second numerical experiment to delimit its scope.","major_comments":[{"comment":"The assertion that 'the RRM eigenvalues approach those of the projected Hamiltonian from below in most cases' is supported only by numerical experiments for one Hamiltonian (particle in a box) and one basis set u_i(x)=x^i(1-x). No theorem or independent numerical test is provided, and the paper itself records exceptions (e.g., W1>E1 for N=1 in Table 4). Since this claim appears in the abstract and in the conclusions, the authors should either add a proof or a second basis/Hamiltonian experiment, or explicitly restrict the claim to 'in the cases studied here.'","section":"Abstract; Section 3.2, Tables 2-4; Section 3.3, Table 7"},{"comment":"The treatment of the projected identity operator is unclear and appears to conflict with the spectral properties of compressions of a projection. For I_D = sum_{k=1}^D |psi_k> = E_n ||^2, or the same expression in quotient form.","section":"Equation (7)"},{"comment":"The name 'Cauchy-Schwartz' should be 'Cauchy-Schwarz'.","section":"Throughout"},{"comment":"In Eq. (13), the summation on the right-hand side runs over D, but the left-hand side is a sum over N Ritz vectors; since W_i=0 for i>D, the equality is correct only if the summation indices are understood consistently, and this should be stated explicitly.","section":"Equation (13)"},{"comment":"The statement that 'the remaining roots vanish W_k = 0, D < k <= N as expected' would be more precise if it noted that the matrix (H_D) has rank at most D, so the generalized eigenproblem has at least N-D zero eigenvalues.","section":"Section 3.2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a short numerical note whose main value is the counterexample showing that the RRM upper-bound property fails for projected Hamiltonians. The editor may wish to weigh whether the unsupported 'most cases' generalization can be satisfactorily tempered in revision; if the authors restrict it to the cases actually computed, the paper would be sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper does what it says—applies RRM to the projected Hamiltonian of Ding et al. on a toy model and shows the eigenvalues converge from below in many cases, with a few exceptions. The math in Eqs. (7)–(8) is elementary Cauchy–Schwarz but it is applied correctly, and the numerical tables are consistent with the rank argument for the zero eigenvalues. The identity-operator discussion in Sec. 3.4 is a nice extra: RRM gives upper bounds for N1 they come from below. That is a real counterexample to the naive expectation that the usual RRM upper-bound theorem transfers to the projected operator. The reader's acceptance is well founded; the paper is a legitimate correction.\n\nWhere the paper is softer: the abstract and Conclusions say 'in most cases' and Sec. 3.3 goes further, claiming 'the RRM yields upper and lower bounds for this kind of projected Hamiltonian operators.' That generality is not established. All numerical evidence uses the single basis u_i(x)=x^i(1-x) and H=-1/2 d^2/dx^2. The min-max principle for finite-rank operators does not force one side; compression eigenvalues can sit on either side of the target eigenvalues depending on the subspace. The observed pattern is a property of this polynomial basis and the sine eigenfunctions, not a general theorem. The paper itself notes exceptions (W1>E1 for N=1, W2>E2 for N=3), so the effect is non-universal, but the 'most cases' scope is never quantified or delimited. A second basis or a simple argument would have helped. This is a moderate weakness, not a fatal one, because the title and core result are explicitly about applying RRM to this projected Hamiltonian, and the correction to Ding et al. stands.\n\nThe citation pattern is fine; the self-citations are to the author's prior RRM analyses, but the upper-bound property for ordinary Hamiltonians is textbook MacDonald (1933), and the new result is not circular.\n\nWho is this for? Practitioners using the Ding et al. ensemble variational framework and anyone teaching the limits of the RRM. It deserves a serious referee; I would accept it as a technical comment or short paper after the author tones down the 'most cases' language to 'for the examples studied.'","headline":"A sound, narrowly scoped correction to Ding et al.'s projected-Hamiltonian variational claim, overstating the generality of the lower-bound pattern.","tokens_in":6089,"tokens_out":1797,"would_cite":false,"duration_ms":18347,"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":"For a projected Hamiltonian, the Rayleigh-Ritz method gives lower bounds in most cases and exact zero eigenvalues—the reverse of the usual upper-bound rule.","keywords":["Rayleigh-Ritz method","projected Hamiltonian","variational bounds","lower bounds","eigenvalue convergence","ensemble states","particle in a box","secular determinant"],"falsifier":"For the same particle-in-a-box projected Hamiltonian with $D=3$, repeat the Rayleigh-Ritz calculation with a different complete basis that respects the boundary conditions, for example $u_i(x)=x^i(1-x)^2$; if the lowest eigenvalue $W_1$ for $N=2$ or $N=4$ lies above $E_1=\\pi^2/2$, the paper's claim that lower bounds occur in most cases would be shown to be basis-dependent rather than generic.","tokens_in":5217,"feed_emoji":"⚛️","tokens_out":14411,"duration_ms":123421,"temperature":0.7,"pith_summary":"The paper examines what happens when the Rayleigh-Ritz method (RRM) is applied to a projected Hamiltonian, an operator built from a finite set of eigenstates of a physical Hamiltonian. Normally RRM is prized because its approximate eigenvalues converge from above to the exact ones; the author shows that for a projected Hamiltonian the convergence is typically from below, giving lower bounds instead. Using a particle-in-a-box model with polynomial trial functions, the paper demonstrates this behavior numerically, identifies exceptions for small basis sizes, and shows that the $N-D$ zero eigenvalues of the projected operator are reproduced exactly. It also studies how an energy shift changes the direction of the bound and how projecting the identity operator restores the usual upper-bound behavior. The motivation is to clarify the variational properties of a recently proposed ensemble-state variational principle built on such a projected Hamiltonian.","feed_headline":"Projected Hamiltonians: Rayleigh-Ritz bounds approach from below","feed_subtitle":"The usual upper-bound rule fails: for a projected Hamiltonian, Rayleigh-Ritz gives lower bounds in most cases.","key_machinery":"The central object is the projected Hamiltonian $H_D=\\sum_{k=1}^{D} E_k |\\psi_k\\rangle\\langle\\psi_k|$, a finite-rank operator on an infinite-dimensional Hilbert space, treated by the RRM through the secular determinant $|H_D - W S|=0$ with overlap matrix $S_{ij}=\\langle u_i|u_j\\rangle$. Two mechanisms carry the argument. First, because $H_D$ has rank $D$, its $N\\times N$ matrix representation has at least $N-D$ zero eigenvalues for any $N>D$, and these zeros are obtained exactly. Second, for a trial state orthogonal to every eigenstate except one, the Cauchy-Schwarz inequality shows the expectation value lies below or above the eigenvalue depending on the sign of that eigenvalue; this is what allows lower bounds to appear. The non-orthogonal polynomial basis $u_i(x)=x^i(1-x)$ supplies the explicit numerical demonstration.","core_discovery":"The central discovery is that the Rayleigh-Ritz method, applied to a projected Hamiltonian $H_D=\\sum_{k=1}^{D} E_k |\\psi_k\\rangle\\langle\\psi_k|$ formed from the $D$ lowest eigenstates of a Hermitian operator, does not obey the standard variational upper-bound theorem. For the toy model $H=-\\frac12 \\frac{d^2}{dx^2}$ on $[0,1]$ with basis $u_i(x)=x^i(1-x)$, the RRM eigenvalues $W_k$ approach the nonzero eigenvalues $E_k$ from below for most $N$, while the remaining $N-D$ eigenvalues are exactly zero. Exceptions do occur: for $D=3$ and $N=1$ the lowest RRM value is an upper bound, and for $D=4$ and $N=3$ the second eigenvalue is an upper bound; for $N>3$ the convergence is from below. The paper proves that for a trial state orthogonal to all but one eigenstate, the expectation value lies below or above the eigenvalue according to the sign of that eigenvalue (via the Cauchy-Schwarz inequality), and it shows numerically that the pattern survives an energy shift and arbitrary real coefficients $\\alpha_k$ in the projection. The author's conclusion is that for this kind of projected Hamiltonian the RRM yields both upper and lower bounds, with the lower-bound direction predominant.","pith_inferences":["The exact reproduction of the $N-D$ zero eigenvalues suggests a general rank-imposed constraint: for any finite-rank operator, a variational subspace of dimension $N>D$ will always contain a null space of dimension at least $N-D$, independent of basis quality; this could serve as a diagnostic in truncated configuration-interaction or density-matrix embeddings.","Because the bound direction depends on shifting the zero of energy, the apparent 'lower-bound rule' is not an intrinsic property of the projected operator but an artifact of its spectrum's position; adding a large positive constant would restore the familiar upper-bound regime, so any test of this behavior must specify the energy reference.","If the ensemble-state variational principle built on projected Hamiltonians inherits this lower-bound direction, variational optimization over ensemble weights could converge from below, which would change how convergence is monitored in such calculations.","A natural next step is to test the pattern on degenerate or quasi-degenerate eigenstates; the rank argument and Cauchy-Schwarz bound still apply, but the ordering of the eigenvalues may become ambiguous."],"forward_implications":["For any $N>D$, the projected Hamiltonian's $N-D$ zero eigenvalues are obtained exactly by the RRM, even though the corresponding wavefunctions only become exact in the $N\\to\\infty$ limit.","The direction of the variational bound is tied to the sign of the target eigenvalue: via the Cauchy-Schwarz argument, a trial state overlapping only one eigenstate gives a lower bound for positive eigenvalues and an upper bound for negative ones.","Adding a constant energy shift can flip the bound direction; with the shift $-5$, the lowest RRM eigenvalue becomes an upper bound while the higher ones remain lower bounds.","If the projected identity $I_D$ replaces the full identity in the overlap matrix, the RRM gives upper bounds for $1\\le N