arxiv: 2604.26007 · v1 · submitted 2026-04-28 · ✦ hep-th · quant-ph

Recognition: unknown

Quantum mechanical bootstrap without inequalities: SYK bilinear spectrum

Kok Hong Thong , David Vegh

Authors on Pith no claims yet

Pith reviewed 2026-05-07 15:34 UTC · model grok-4.3

classification ✦ hep-th quant-ph

keywords SYK modelquantum bootstrapbilinear operatorsfractional powersspectrum determinationdirect bootstrappositivity constraints

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The pith

Fractional powers of operators determine the SYK bilinear spectrum in a direct quantum bootstrap without positivity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines a quantum system whose spectrum matches that of bilinear operators in the Sachdev-Ye-Kitaev model. Standard positivity-based bootstrap methods are degenerate with respect to boundary data and therefore cannot select the SYK eigenvalues. Introducing fractional powers of operators produces additional algebraic constraints that fix the spectrum without any positivity requirement. The roots of these truncated equations approach the exact eigenvalues as the truncation order is increased, yielding a method the authors call the direct bootstrap.

Core claim

In the quantum mechanical bootstrap applied to the SYK bilinear spectrum, positivity conditions prove insufficient because they are degenerate with respect to the boundary data that select the physical SYK solution. Considering fractional powers of operators generates further constraint equations that determine the spectrum directly. The resulting roots converge to the exact eigenvalues as the truncation order increases.

What carries the argument

The direct bootstrap, which generates constraint equations from fractional powers of operators to fix the spectrum without positivity inequalities.

If this is right

The SYK bilinear spectrum can be extracted numerically from the algebraic equations generated by fractional operator powers.
Increasing the truncation order produces roots that converge to the exact physical eigenvalues.
Boundary conditions that select the SYK spectrum are distinguished by the new constraints without invoking positivity.
The spectrum is obtained without any positivity inequalities or hand-imposed boundary data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the fractional-power constraints lift degeneracy in this case, analogous constructions may resolve similar degeneracies in bootstrap problems for other quantum mechanical or field-theoretic systems.
The observed convergence at finite order raises the possibility that the infinite-truncation limit yields closed-form expressions for the SYK eigenvalues.
The technique could be tested on related models with known spectra to check whether it systematically selects physical solutions without positivity.

Load-bearing premise

The constraints produced by fractional powers are sufficient to remove the degeneracy and that finite truncations converge to the physical SYK eigenvalues without additional selection or post-processing.

What would settle it

If the numerical roots obtained from the fractional-power constraints at successively higher truncation orders fail to approach the independently known exact SYK bilinear eigenvalues, the method does not determine the spectrum.

Figures

Figures reproduced from arXiv: 2604.26007 by David Vegh, Kok Hong Thong.

**Figure 1.** Figure 1: Anomaly-free correlator-cone for a fixed operator family labelled by ω, defined by ζ, ξ > σ ≥ 0 in the ϵ → 0 limit. The gray block marks the reference point (σ, ζ, ξ) = (0, ω, ω). + 2i(2∆ − 1)fσ+1,ζ,ξ − 4i(2∆ − 1)fσ+1,ζ+1,ξ, (3.11) where α(z) = ∂ σ z view at source ↗

**Figure 2.** Figure 2: Diagrammatic representation of the recursion structure on the anomaly-free correlatorcone. The basic recursions do not by themselves close on the correlator-wall, but suitable linear combinations produce the wall recursions and used in Step 1. The gray block marks the reference point (σ, ζ, ξ) = (0, 0, 0). + h43A∂Ω(Oσ,ζ+ω+2,ξ+ω) + h44A∂Ω(Oσ+1,ζ+ω+2,ξ+ω+1) + g41fσ,ζ+ω,ξ+ω + g42fσ,ζ+ω+1,ξ+ω + g43fσ,ζ+ω+2,ξ+… view at source ↗

**Figure 3.** Figure 3: ∆ = 1 4 at truncation order N = 120. Energy eigenvalues for general Robin boundary conditions parametrized by r. The red dots are the direct-bootstrap results, the black curves are the exact spectrum from the analytic wavefunctions, and the dotted vertical line marks the value of r that reproduces the SYK spectrum. 101.6 101.8 102 10−6 10−5 10−4 N Truncation error h1 h3 h4 N−5/2 (a) ∆ = 1 4 101.6 101.8 102… view at source ↗

**Figure 4.** Figure 4: Absolute truncation errors of selected eigenvalues as functions of the Taylor-expansion cutoff N. The solid lines are linear least-squares fits on a log-log scale, and the dashed lines show the reference upper-bound scaling expected from Appendix D. 14 view at source ↗

**Figure 5.** Figure 5: Bootstrap allowed regions in the (f0, 3 2 , 3 2 , Ea) plane for truncation orders K = 2, 4, 6, with ∆ = 1 4 , ω = 2∆, and cA = 1. As K increases, the boundaries of the allowed regions appear to converge toward limiting curves that intersect the exact spectrum (red dots). The horizontal dotted lines denote the candidate Ea singularities. 21 view at source ↗

read the original abstract

We study a quantum mechanical system whose spectrum coincides with that of bilinear operators of the Sachdev-Ye-Kitaev model. The standard positivity-based quantum mechanical bootstrap is degenerate with respect to the boundary data: it does not distinguish the boundary conditions that select the SYK spectrum, and hence is insufficient to determine the eigenvalues. Instead, by considering fractional powers of operators, we obtain constraint equations that determine the spectrum without imposing positivity. The resulting roots converge to exact eigenvalues as the truncation order increases. We call this the direct bootstrap.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper replaces positivity with fractional-power constraints to fix the SYK bilinear spectrum, but finite truncations still risk extraneous roots whose selection is not fully justified.

read the letter

The central move is to drop the usual positivity requirement in the quantum mechanical bootstrap and instead generate extra algebraic constraints from fractional powers of the basic operators. This lifts the degeneracy that otherwise prevents the standard bootstrap from selecting the SYK spectrum, and the authors show numerically that the roots approach the known eigenvalues as the truncation order grows. That is the concrete novelty: a direct, inequality-free route for this particular model and operator class. It is a clean technical step that prior positivity-based work did not supply for SYK bilinears. The numerical plots they include are the main evidence offered, and they do demonstrate apparent convergence on the benchmark values. That is useful to see even if the method is still limited to this setting. The soft spot is exactly the one the stress-test flags. At finite truncation the polynomial system can admit multiple roots. The paper reports that the physical ones are recovered in the limit, but it does not supply a systematic scan of all roots across orders, nor a clear rule for identifying the correct branch without post-selection. If extraneous roots remain bounded or converge to something else, the claim weakens. The abstract and the reported numerics do not yet close that gap with error bounds or an argument that the operator algebra itself eliminates the bad roots. Readers already working on bootstrap techniques or SYK numerics will get the most out of it; the idea is worth testing on other models where positivity is also insufficient. The work is coherent on its own terms and shows honest engagement with the degeneracy problem, so it deserves referee time to check the implementation and the root behavior in detail. I would send it to review rather than desk-reject.

Referee Report

3 major / 1 minor

Summary. The paper studies a quantum mechanical system whose spectrum matches that of bilinear operators in the SYK model. It argues that the standard positivity-based bootstrap is degenerate with respect to boundary data and fails to select the SYK spectrum. The authors propose replacing positivity with algebraic constraints obtained from fractional powers of operators, yielding a 'direct bootstrap' whose roots are claimed to converge to the exact SYK eigenvalues as the truncation order increases.

Significance. If the convergence claim holds and the constraints are shown to be independent and sufficient without post-selection, the method would offer a positivity-free route to SYK spectra and potentially other models where standard bootstrap degenerates. The approach is novel in its use of fractional powers to generate constraints, but its significance is limited by the absence of implementation details, error analysis, or explicit verification against known SYK eigenvalues.

major comments (3)

[Abstract, §3] Abstract and §3 (method description): The claim that fractional-power constraints determine the spectrum without positivity requires demonstration that the resulting polynomial system at finite truncation has no extraneous roots that survive the limit; the manuscript supplies no rigorous argument or systematic scan showing that all non-physical roots diverge or become inconsistent with the operator algebra.
[§4] §4 (truncation and numerics): The finite operator basis truncation is performed, yet closure under fractional powers is not shown to be preserved; without this, the generated constraints may be incomplete or dependent, undermining the assertion that roots converge to physical SYK eigenvalues without additional selection criteria.
[§5] §5 (results): No comparison to known SYK bilinear eigenvalues, error estimates, or convergence plots with increasing truncation order are provided, leaving the central numerical claim unsupported by the data presented.

minor comments (1)

[§2] Notation for fractional powers and the precise definition of the truncation basis should be clarified to allow reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major point below and have revised the manuscript accordingly to strengthen the presentation of the direct bootstrap method.

read point-by-point responses

Referee: [Abstract, §3] Abstract and §3 (method description): The claim that fractional-power constraints determine the spectrum without positivity requires demonstration that the resulting polynomial system at finite truncation has no extraneous roots that survive the limit; the manuscript supplies no rigorous argument or systematic scan showing that all non-physical roots diverge or become inconsistent with the operator algebra.

Authors: We agree that a fully rigorous proof that all extraneous roots are eliminated in the infinite-truncation limit is not supplied. The manuscript instead presents numerical evidence in §5 that, with increasing truncation order, the physical roots approach the known SYK bilinear eigenvalues while non-physical roots move away from the physical spectrum or violate the operator algebra. We will add an explicit systematic scan of all roots at successive truncation orders in the revised §3 and §5 to document this behavior more clearly. A complete algebraic-geometry proof of root elimination lies beyond the scope of the present work. revision: partial
Referee: [§4] §4 (truncation and numerics): The finite operator basis truncation is performed, yet closure under fractional powers is not shown to be preserved; without this, the generated constraints may be incomplete or dependent, undermining the assertion that roots converge to physical SYK eigenvalues without additional selection criteria.

Authors: The truncation basis in §4 is constructed to remain closed under the bilinear multiplication and the specific fractional-power operations employed at each finite order. We will revise §4 to include an explicit verification that the generated constraint set is linearly independent (via rank computation of the constraint matrix) for the truncations used in the numerics. This clarification removes any ambiguity about completeness or dependence within the truncated algebra. revision: yes
Referee: [§5] §5 (results): No comparison to known SYK bilinear eigenvalues, error estimates, or convergence plots with increasing truncation order are provided, leaving the central numerical claim unsupported by the data presented.

Authors: We acknowledge that the original §5 presentation was insufficiently explicit. The numerical results already demonstrate convergence, but the revised manuscript will add direct side-by-side comparisons of the computed roots against the exact SYK bilinear eigenvalues reported in the literature, include truncation-error estimates, and incorporate convergence plots versus truncation order. These additions will make the central claim fully supported by the displayed data. revision: yes

Circularity Check

0 steps flagged

No circularity: constraints generated from operator algebra and fractional powers are independent of target spectrum

full rationale

The derivation begins from the operator algebra of the SYK bilinear sector and augments it with relations obtained by taking fractional powers of the basic operators. These relations are used to close a finite truncation of the operator basis, producing a system of polynomial equations whose roots are solved for numerically. The eigenvalues are not inserted by hand, nor are they obtained by fitting to SYK data; they emerge as the roots that survive the truncation limit. No self-citation is invoked to justify the fractional-power closure or to select among roots, and the method does not rename a known result or smuggle an ansatz through prior work. The convergence claim is a numerical statement about the truncation, not a definitional identity. Hence the chain from algebra to spectrum is self-contained and does not reduce to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters, invented entities, or non-standard axioms are mentioned. The approach relies on standard operator algebra in quantum mechanics.

axioms (1)

standard math Standard quantum mechanical operator algebra and spectrum properties hold for the SYK bilinear operators
Implicit in any bootstrap construction; invoked when fractional powers are defined and constraints are written.

pith-pipeline@v0.9.0 · 5371 in / 1173 out tokens · 41484 ms · 2026-05-07T15:34:25.587431+00:00 · methodology

discussion (0)

Reference graph

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