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 = E, 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.
The eigenstate quantum-mechanical bootstrap can converge to the physical moment set while remaining nonexact at every finite order. We prove this for a one-dimensional quasi-exactly solvable sextic oscillator at explicitly known eigenvalues, including excited levels. At every finite order, the lower and upper bounds on
u2 remain strictly below and above its physical value, even after adjoining arbitrary finite polynomial Gram tests and all eigenstate-annihilator relations visible in the retained moment space. Hence neither side of the centered observable admits a finite sum-of-squares certificate modulo the eigenstate annihilator. The gap persists under any finite set of additional localizing constraints whose weights are strictly positive on polynomial vectors. We then prove convergence for polynomial Schrödinger operators satisfying explicit sum-of-squares confinement and derivative bounds. The eigenstate relations give moment estimates uniform in the truncation order; these yield a regular Schrödinger representation supported on the prescribed eigenspace. Finally, a four-mode extension combines this fixed-observable obstruction with feasible position moments having no positive representing measure at every finite order. At level three, this construction yields nonphysical feasible points with strictly positive definite moment matrices.
The many-body bootstrap certifies ground-state properties by minimizing energy over correlators constrained only by positivity, without any variational wavefunction. Its cost, however, grows exponentially with the size of the operators involved, placing the long-range string correlators that diagnose topological phases out of reach. We overcome this by treating string operators as primary objects: bare and endpoint-dressed strings satisfy a closed operator algebra with one another and with local words, yielding a semidefinite program whose cost grows only polynomially with string length. A single computation then yields estimates of string correlators of all lengths. Rigorous two-sided bounds are obtained for each target string once the ground-state energy is pinned within a window. When benchmarked against density-matrix renormalization group computations on the cluster Ising and spin
−1 Heisenberg chains, the method certifies the nonlocal string order of the cluster and Haldane phases directly from the Hamiltonian. Our rigorous certification of string order parameters reveal that the sharpness of the bounds is set by which operators enter the calculation, not by their length alone. By exploiting algebraic closure of strings and local words, our framework lifts spatially extended observables into polynomially tractable bootstrap variables, opening up a route to wavefunction-free certification of nonlocal order in quantum many-body systems.
We give exact rational certificates for two results in the bipartite Bell scenario with three two-outcome measurements per party. First, the maximum of the
I3322 (Froissart) functional over the almost-quantum set -- the
1+AB level of the Navascues-Pironio-Acin (NPA) hierarchy -- is certified to lie between
0.2514708838 and
0.2514709016 (width
<2×10−8), in the normalization with classical bound
0 and quantum maximum
≈0.2508754; to our knowledge no value for this landmark had been published. Second, we exhibit a single exact rational behavior
T∗ carrying certificates in both directions: a rational moment matrix proving
T∗ almost-quantum, and a rational dual object proving that every NPA level-2 completion of
T∗ requires slack at least
q=4.5934×10−4, so that
T∗ is not quantum. The strict inclusion
Q⊊Q~ in the minimal three-setting scenario is thereby established end-to-end with no reliance on floating-point computation. Anatomically, the witness is locally quantum: all nine of its
2×2 correlator subtables pass the Tsirelson-Landau-Masanes criterion, so its non-quantumness is irreducibly a three-setting phenomenon, concentrated in ordering-dependent pseudo-moments coupling each party's third setting to the other's first two; the level-2 certificate is equivalently an exact sum-of-squares identity that any quantum state values nonnegatively and the witness is forced to value at
−q. The certificates follow the rational rounding-projection-lifting programme recently developed for noncommutative optimization; for NPA moment matrices in canonical form the required projection reduces to a per-moment uniform shift, making certification essentially free at these sizes, and we work the programme in both the feasibility and infeasibility directions on one witness. Machine-checkable certificates and verifiers requiring only the Python standard library accompany the paper. Files in this record: the paper (PDF); exact rational certificates for both theorems (i3322_1AB_certificate.json, i3322_level2_infeasibility_certificate.json); standalone verifiers requiring only the Python 3 standard library (verify_i3322_certificate.py, verify_level2_certificate.py); and the generating scripts (certify_landmark.py, certify_level2.py). See README_ancillary.txt for which artifact proves which theorem and how to run the verifiers. v2: corrects the mechanism attribution in Sec. 5 (anatomy) and adds the archive reference to Data Availability; theorems, certificates, and verifiers unchanged.
The spectral weight with which a many-body eigenstate contributes to a given channel is not fixed by the smooth one-point ETH envelope, and its higher-order correlations---the connected cumulants of three and more channel-resolved spectral functions---have lacked a systematic organization. We construct one. In random free fermions, where every channel overlap is a squared Slater minor of a Haar-orthogonal one-body eigenvector matrix, the
r-resolvent moment admits an exact replica-sector decomposition: a sum over the double-coset sectors of the orthogonal Weingarten algebra, in which the permutation sectors are the replica contraction classes and each sector value is determined by the overlap pattern of the channels and of the eigenstates. The sector coefficients follow from an exact occupancy-matrix assembly---a signed convolution over the slot permutations, a decomposition into connected blocks, and a finite-state chain transfer matrix whose local transition rules are independent of
r and
k---so the sector theorem holds at arbitrary finite multiplicity, realized exactly at
r=3 (
k=2,3,4),
r=4 and
r=5 (
k=2). The structure carries the physical hierarchy: the fully coincident sector is the projection that yields the Selberg moment ratios
qr, the levels are nested by exact marginalization, and the sectors are the multi-energy spectral cumulants behind the resolvent cumulants, with the irreducible self-energy vertices of the Feshbach ladder as the connected kernel. The multi-resolvent correlations of random free fermions are therefore organized, level by level, by one replica-sector algebra---an exact microscopic benchmark for the higher-order fluctuation sector of eigenstate thermalization, with its microscopic sector geometry made explicit.