REVIEW 5 minor 34 references
Stationarity is not enough: tightness of the quantum mechanical bootstrap and the copositive cone
T0 review · 0 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash
Pith's one-line read The paper proves that the level-three stationary quantum bootstrap, in two dimensions, can certify a rational moment vector that satisfies every constraint yet is the moment sequence of no quantum state.
desk verdict A rare exact-arithmetic counterexample showing the stationary bootstrap is not tight in 2D, with an honest separation of proved, numerical, and conjectural content. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The argument turns on the gap between nonnegative polynomials and sums of squares in two variables, realized inside the feasible set of the stationary bootstrap. The machinery has three parts: the level-$K$ moment matrix $M_K$ whose positive semidefiniteness encodes positivity against sums of squares; the two constraint families (S) (stationarity, $\langle[H,O]\rangle=0$) and (E) (eigenstate, $\langle OH\rangle=E\langle O\rangle$) with their differing degree budgets; and the separating polynomial $p_c$, whose nonnegativity is certified not by being a sum of squares but by the multiplier identity $(x^2+y^2+w^2)P_c$ being SOS. The copositive cone enters when reflection symmetry reduces quartic witnesses to matrices $M$ with $x^\top M x\ge 0$ on the nonnegative orthant; the classical 5x5 matrix outside $P+N$ then places the first possible failure of the symmetric eigenstate bootstrap at $d=5$, with the degree-indexed hierarchy of copositive cones explaining why level three closes it.
What would settle it
Run the standalone verification script supplied with the paper in a different computer algebra system and check three things: every level-three (S) constraint evaluates to zero in exact rational arithmetic; all four blocks of $M_3$ have positive exact $LDL^\top$ pivots; and the Gram matrix factorization of $(x^2+y^2+w^2)P_c$ is exact. Additionally, for the conjecture, find any moment vector in $F^{SE}_3(E)$ for the double well with $\langle p_1^4\rangle > 10^3$, which would refute the claimed momentum bound.
Extended reading notes
Core claim
The central result is Theorem 1: there exists a moment vector $m$ with rational entries such that every level-three constraint of type (S) for the potential $V = 20[(q_1^2-1)^2+(q_2^2-1)^2] - q_1^2 q_2^2$ holds exactly, together with $\langle 1\rangle=1$ and $\langle H\rangle=E=12$; all four blocks of the moment matrix $M_3$ are positive definite; and $L_m(p_c) = -10649385831371/122880000000000 < 0$ for a polynomial $p_c\ge 0$ on $\mathbb{R}^2$. Consequently $m$ is feasible for the level-three stationary bootstrap yet is the moment sequence of no quantum state, pure or mixed. Every step is verified in exact rational arithmetic: the equalities by substitution, positive definiteness by exact $LDL^\top$ with smallest pivot $1.721\times 10^{-3}$, and nonnegativity of $p_c$ by the explicit sum-of-squares certificate $(x^2+y^2+w^2)P_c$, where $P_c$ is the homogenization of $p_c$. The paper further reports that the eigenstate bootstrap, which adds $\langle OH\rangle = E\langle O\rangle$, removed the violation in every setting tested, with the only exception occurring at a truncation so low that almost no eigenstate constraints survive.
Load-bearing premise
The whole result rests on the exact-arithmetic checks being bug-free: if the constraint list of 218 rows, the $LDL^\top$ pivot test, or the rational sum-of-squares certificate has a flaw, the counterexample collapses.
Editorial extensions
If this is right
- Any low-order stationary bootstrap result in more than one spatial dimension should be screened for the kind of witness constructed here; the paper proposes a linear-programming diagnostic that minimizes $L_m(f)$ over a fixed family of nonnegative non-SOS polynomials and flags a negative value.
- The eigenstate bootstrap, by contrast, emerged without a certified violation in five settings across two, three, and five degrees of freedom at truncation levels three and four; if the pattern holds, energy-eigenvalue bootstrap results in these regimes are on safer ground.
- The momentum-moment bound $\langle p^{2k}\rangle \le C(E,k)$ for $k\le K-1$, if the conjecture is correct, would give the Archimedean-type boundedness that convergence proofs for noncommutative moment problems require, and would put the eigenstate bootstrap within reach of existing Positivstellensatz theory.
- The copositivity analysis says that under sign symmetry, a violation of the eigenstate bootstrap at $d=5$ and $K=2$ is possible (and was found), while $K=3$ is closed to symmetric witnesses for structural reasons; an asymmetric potential would be needed to break the $K_1$ barrier.
Reading between the lines
- One testable extension: apply the paper's diagnostic to published one- and two-dimensional bootstrap spectra; if any reported low-order stationary result in $d\ge 2$ returns a negative witness, the corresponding spectrum claim would need reinterpretation.
- The thermal condition imposed in thermal bootstrap formulations is a further nonlinear constraint not present in Theorem 1; whether it closes the gap for thermal states is left open and could be tested by adding the thermal-relaxation constraints to the exact counterexample.
- The symmetry argument suggests that reaching the $d=6$ copositive obstruction would require an asymmetric potential; a natural next search would randomize ring couplings and scan for a matrix outside $K_1$ in the $6\times 6$ circulant family, or drop circulant symmetry entirely.
- If the conjecture holds, the eigenstate bootstrap at fixed $K$ has a feasible set with bounded momentum moments, so the distinction between stationary and eigenstate bootstraps would amount to a genuine compactness property rather than a numerical accident.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This paper studies the gap between two positivity notions in the quantum-mechanical bootstrap: sums of squares, which a finite-level SDP can certify, and pointwise nonnegativity, which every physical state must satisfy. In two or more dimensions these cones differ, and the paper asks whether the difference is realized by a Schrödinger operator. The main result is Theorem 1, which constructs a rational moment vector for a two-dimensional quartic double well that satisfies every level-three stationary constraint of type (S) exactly, together with normalization and energy constraints, has a positive definite moment matrix, but assigns a negative expectation to a rational polynomial p_c that is nonnegative on R^2. Hence this moment vector is feasible for the level-three stationary bootstrap but is the moment sequence of no quantum state. The proof is computational and is certified in exact rational arithmetic by a standalone script. The paper then reports numerical searches for a similar violation in the eigenstate bootstrap, which additionally imposes the constraints (E); no violation is found in five of six settings, and the single apparent exception is attributed to a truncation where essentially no eigenstate constraints survive. The proposed mechanism is that the eigenstate constraints bound high momentum moments, and the paper relates the symmetric obstruction to the copositive cone, locating the first possible symmetric failure at five degrees of freedom.
Significance. If Theorem 1 is correct, it is a significant result for the bootstrap literature: it gives the first exact, low-order realization of the classical nonnegative-versus-SOS gap inside a Schrödinger-operator bootstrap, and it sharply distinguishes the stationary from the eigenstate relaxation. The exact-arithmetic certification is a notable strength: all data are rational, the positive definiteness of the moment matrix is checked by exact LDL^T, and the nonnegativity of the separating polynomial is certified by a rational Gram matrix. The paper also ships a standalone re-verification script, which is the right standard for a computational proof. I did not independently execute that script, so the residual risk is the usual one for machine-checked arguments; nothing in the manuscript suggests an internal inconsistency. The eigenstate side is carefully framed as numerical evidence rather than a theorem, and the copositivity analysis gives a structural explanation for the observed dimension threshold. The paper is likely to be influential in both the bootstrap and the moment/SOS communities.
minor comments (5)
- [II A, Eq. (6)] The derivation of Eq. (6) is compressed: it may not be immediately clear to a reader why the commutator [H, f p_x] produces the cross term ⟨(∂_y f)p_y p_x⟩. A one-line computation of [p_y^2/2, f p_x] and [p_x^2/2, f p_x] would make the relation self-contained and would also clarify the role of taking real parts.
- [IV A, Table I] The d=5, K=2 row is footnoted as having no surviving eigenstate constraints, but the table itself can be misread as an eigenstate-bootstrap violation. The caption or the footnote should state explicitly that this row tests a relaxation with essentially no content beyond ⟨H⟩=E.
- [III A] The rational Gram matrix certificate proving p_c ≥ 0 is only described and referenced to the ancillary files. Printing the certificate, or at least the exact LDL^T pivots and the coefficient-matching projection dimension, would make the main theorem auditable without requiring the reader to run the script.
- [VI] The statement that a scaling theorem of reference [25] implies every unit-diagonal copositive 5×5 matrix lies in K_1 is telegraphic. A precise statement of the theorem and a sentence explaining why dihedral invariance forces the witness matrix to have constant diagonal would improve the readability of the argument.
- [Section headings and references] There are minor formatting artifacts in the extracted text (for example, the Section III heading appears as 'THE ST A TIONAR Y BOOTSTRAP') and some reference entries have irregular formatting. These should be cleaned up in the final version.
Circularity Check
No significant circularity: Theorem 1 is verified by an independent SOS certificate, and the paper's explanatory claims are supported by external theorems and clearly labeled numerical evidence.
full rationale
Theorem 1 does not reduce to its own inputs. The moment vector m is produced as an exact rational point in the affine solution space of the level-three stationary constraints, and the separating polynomial p_c is then optimized against m; but the negative expectation is not an artifact of that construction because p_c ≥ 0 on R^2 is certified independently by the exact rational Gram certificate showing (x^2 + y^2 + w^2)P_c is a sum of squares. The paper explicitly notes that p_c is not itself a sum of squares, so M3 ⪰ 0 cannot force L_m(p_c) ≥ 0; the nonnegativity proof stands apart from the moment-matrix feasibility test. The group-averaging argument justifying restriction to invariant witnesses is a valid lemma, not a restatement of the desired conclusion. The eigenstate-bootstrap searches are presented as numerical evidence with a stated limitation, and the copositivity explanation invokes external results (Diananda's theorem, Parrilo's hierarchy, Dickinson et al.'s scaling theorem) rather than a self-citation chain. The only self-citation is Ref. [22], cited as a strategy analogy and not load-bearing for the main theorem. The exact-arithmetic verification pipeline is a correctness risk if not independently reproduced, but that is a verification concern, not circularity. No fitted parameter is renamed as a prediction, and Conjecture 1 is explicitly labeled a conjecture.
Assumptions & free parameters
free parameters (3)
- Potential coefficient 20 (well depth) =
20
- Cross-coupling coefficient -1 =
-1
- Energy E =
12
assumptions (6)
- standard math Hilbert's theorem: in two or more variables, not every nonnegative polynomial is a sum of squares.
- standard math Reznick's theorem: if f is positive definite, then (sum x_j^2)^N f is a sum of squares for some N.
- standard math Diananda's theorem: the copositive cone equals P+N for d <= 4 and differs at d = 5.
- domain assumption The Weyl algebra satisfies [q_j, p_j] = i and has no finite-dimensional or bounded representation.
- domain assumption Every physical state gives nonnegative expectation to every pointwise nonnegative polynomial in the position operators.
- domain assumption The level-K bootstrap constraint set is exactly the degree-truncated family (5) of (S) and (E) relations.
Cite this review
Pith. "Pith review of Stationarity is not enough: tightness of the quantum mechanical bootstrap and the copositive cone." pith.science (2026). https://pith.science/paper/FTS44DRN
@misc{pith2026260807047,
author = {Pith},
title = {Pith review of: Stationarity is not enough: tightness of the quantum mechanical bootstrap and the copositive cone},
year = {2026},
howpublished = {\url{https://pith.science/paper/FTS44DRN}},
note = {Machine review of arXiv:2608.07047}
}
read the original abstract
The numerical bootstrap for quantum mechanics tests a candidate state's positivity only against sums of squares, whereas every physical state assigns nonnegative expectation to every pointwise nonnegative polynomial. In one dimension the two coincide; in two or more they do not. Whether this gap is realized depends sharply on which constraint set is imposed. For the stationary bootstrap, which imposes <[H,O]> = 0 and is the relaxation appropriate to thermal and mixed states, we exhibit a two-dimensional quartic double well and a moment vector that satisfies every level-three stationary constraint exactly, has a positive definite moment matrix, and yet assigns a negative expectation to a polynomial nonnegative on R^2; it is therefore the moment sequence of no state. All data are rational, and every step is verified in exact arithmetic. For the eigenstate bootstrap, which additionally imposes <OH> = E<O>, the same search finds no violation in any of five settings spanning two, three and five degrees of freedom and truncation levels three and four, tested against complete families of separating polynomials. The single exception occurs at a truncation so low that only one eigenstate constraint survives, and the theory presented here accounts for it. We identify the mechanism: the eigenstate constraints bound the high momentum moments, otherwise unbounded on the feasible set, and it is those unbounded directions that reach the region between the two cones. Finally, under the reflection symmetries of a typical potential the relevant obstruction is copositivity rather than nonnegativity, which, for the quartic witnesses available at the lowest truncation, places the first possible failure at five degrees of freedom. We conjecture that the eigenstate constraints imply an Archimedean-type bound on the momentum moments, and formulate the corresponding tightness statement.
Reference graph
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every level-three constraint of type(S)for(7)holds exactly, together with⟨1⟩= 1and⟨H⟩=E
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< 0, wherep c is a polynomial with rational coefficients satisfyingp c ≥0on all ofR 2
all four blocks ofM 3 are positive definite; 3.L m(pc) =− 10649385831371 122880000000000 =−0.0866649238. . . < 0, wherep c is a polynomial with rational coefficients satisfyingp c ≥0on all ofR 2. Consequentlymis not the moment sequence of any posi- tive measure, hence of no quantum state, pure or mixed, yet it is feasible for the level-three stationary bo...
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