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arxiv: 2604.19714 · v1 · submitted 2026-04-21 · ✦ hep-th · math-ph· math.CO· math.MP

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Bootstrapping Tensor Integrals

Brayden Smith, Carlos I. P\'erez-S\'anchez, Nathan Pagliaroli

Authors on Pith no claims yet

Pith reviewed 2026-05-10 01:43 UTC · model grok-4.3

classification ✦ hep-th math-phmath.COmath.MP
keywords tensor modelsbootstrappingDyson-Schwinger equationspositivity constraintslarge N limitquartic modelrank three tensorsmoments
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The pith

A positivity bootstrap on Dyson-Schwinger equations determines all moments of the rank-three quartic tensor model.

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

The authors develop a method to find the moments of random tensors that are invariant under U(N) raised to the power D, working in the large-N limit. They combine the Dyson-Schwinger equations that relate different moments with the requirement that certain matrices built from those moments remain positive. The resulting bootstrap is applied to a quartic model and two hexic models, all of rank three. Where exact solutions exist, the numerical approximations converge to them; the same procedure supports a conjecture for explicit closed-form expressions that cover every moment of the quartic model.

Core claim

By enforcing positivity on the moment matrix alongside the Dyson-Schwinger equations that relate the moments, the bootstrap procedure determines the moments of the tensor models to high accuracy. For the rank-three quartic model, this produces a conjecture for the full set of moments in terms of explicit formulae, confirmed by matching the bootstrapped values against independent double-series computations.

What carries the argument

The positivity-constrained truncation of the Dyson-Schwinger hierarchy for the moments of U(N)^D-invariant tensor models.

If this is right

  • The bootstrap reproduces known analytic solutions exactly when they are available.
  • Explicit formulae are conjectured for every moment of the rank-three quartic model.
  • The same procedure converges rapidly for both quartic and hexic interactions.
  • Finite truncations already give useful numerical values for the moments.

Where Pith is reading between the lines

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

  • The conjectured formulae could be used to compute higher-order observables directly without repeating the bootstrap.
  • The method supplies a practical alternative to direct integration when the interaction is more complicated than quartic.
  • Similar positivity bootstraps may be tried on tensor models with different ranks or different interaction structures.

Load-bearing premise

The combination of Dyson-Schwinger equations, positivity constraints, and finite-order truncation is sufficient to produce accurate approximations or to uniquely determine the moments.

What would settle it

An independent exact or high-order numerical computation of one or more moments in the quartic model that disagrees with the values obtained from the bootstrap or the conjectured closed-form expressions.

Figures

Figures reproduced from arXiv: 2604.19714 by Brayden Smith, Carlos I. P\'erez-S\'anchez, Nathan Pagliaroli.

Figure 1
Figure 1. Figure 1: The graph corresponding to the hexic tensor U(N) 3 -invariant on the left is embed￾ded in a genus-1 surface (minimal genus guaranteeing a planar embedding). different constraints than positivity [Mae26], as well as the reconstruction of the eigenvalue distribution of random matrices from bootstrapped bounds [KM25]. Just as with matrix integrals, the situation is ripe for studying tensor integrals via boots… view at source ↗
Figure 2
Figure 2. Figure 2: The geometric interpretation of the DSE’s (here with s = 0 on the L.H.S.) and B connected. Here the blue triangles correspond to the black vertices and the green ones to the white vertices of the coloured graphs from which one obtains these triangulations. This figure is inspired by [Kra12, Eq. 5]. For completeness, in general4 , the moment of a graph B where #π0(B) denotes the connected components remains… view at source ↗
Figure 3
Figure 3. Figure 3: Bootstrapped solution space of m4 for the quartic melonic rank three tensor model. The blue, yellow, and cyan regions of both plots are generated with 5, 10, and 15 DSE and a positive matrix of size 2, 4, and 5 respectively. The analytic solution in each subfigure is plotted in red. (a) is the real part of the instanton solution. 4.2. The hexic cyclic bubble model. Consider the hexic model with interaction… view at source ↗
Figure 4
Figure 4. Figure 4: Bootstrapped solution space of m6 for the hexic pillow rank three tensor model. The blue, yellow, cyan, and magenta regions of both plots are generated with 3, 5, 12, and 20 DSE and a positive matrix of size 0, 2, 6, and 10 respectively. The analytic solution in each sub-figure is plotted in red. 4.3. The hexic pillow model. We now consider another hexic model with interaction S(T, T¯) = N 2  + 1 g 2 [PI… view at source ↗
Figure 5
Figure 5. Figure 5: Bootstrapped solution space of m6,d of the hexic rank three double pillow tensor model. The blue, yellow,and cyan regions of both plots are generated with 5, 15, and 25 DSE and a positive matrix of size 3, 4, and 5, respectively. The analytic solution is plotted in red [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Bootstrapped solution space of m4 of the hexic double pillow rank three tensor model. The blue, yellow, cyan, and magenta regions of both plots are generated with 5, 15, 25, and 25 DSE and a positive matrix of size 3, 4, 5, and 10 respectively. No analytic solution is known [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Bootstrapped solution space of m6 and m6,d for the rank 3 quartic tensor model. The blue, yellow,and cyan regions of both plots are generated with 5, 10, and 15 DSE and a positive matrix of size 2, 4, and 5, respectively. The conjectured analytic solution in each subfigure is plotted in red. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Bootstrapped solution space of m8,m (see Eq. 4.2h) for the quartic rank three tensor model. The blue, yellow, cyan, and magenta regions of both plots are generated with 5, 10, 15, and 25 DSE and a positive matrix of size 2,4, 5, and 10 respectively. The conjectured analytic solution in each subfigure is plotted in red. The point of the very special role of the genus, as commented on in the Introduction, ca… view at source ↗
Figure 9
Figure 9. Figure 9: This plot shows the g k coefficients of leading-orders of moments computed in the ensem￾ble with genus-1 hexic interaction with coupling g. On the non￾factorization of tensor invariants: in this table on the right, m2 is the two-point function. We plot the exponents in g of each correla￾tor. Notice that all graphs except the last two are planar; the third graph is non-melonic but planar, see (5.12). the fo… view at source ↗
read the original abstract

This work proposes a bootstrapping with positivity methodology to study random $U(N)^{D}$ invariant tensors in the large $N$ limit. As has been done for $U(N)$ invariant random matrices, we combine the Dyson-Schwinger equations and positivity constraints of moments to approximate the moments of such tensor models. As examples, we bootstrap the quartic and two hexic rank three tensor models. All models studied converge quickly, and for those which have known analytic formulae, they converge to such solutions. We conjecture new explicit formulae for all moments of the rank three quartic model and support this conjecture using bootstrapped results and explicit double-series computations with 'feyntensor'.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a bootstrapping procedure that combines the infinite tower of Dyson-Schwinger equations with positivity constraints on the moment sequence to approximate the large-N moments of U(N)^D-invariant random tensor models. It applies the method to the rank-3 quartic model and two hexic models, reports rapid numerical convergence, agreement with known closed-form solutions where they exist, and advances an explicit conjecture for all moments of the rank-3 quartic model that is cross-checked against both the bootstrap output and independent double-series expansions generated by the feyntensor package.

Significance. If the conjectured closed-form moments are correct and the bootstrap procedure can be shown to converge to the unique physical solution, the work would supply a practical, largely parameter-free route to exact results in tensor models that complements direct integration and diagrammatic expansions. The explicit use of external DSE together with positivity (rather than fitting parameters to the target) and the provision of reproducible numerical data plus independent feyntensor series constitute clear strengths.

major comments (2)
  1. [§4] §4 (quartic-model bootstrap and conjecture): the central conjecture for the closed-form moments is supported only by numerical agreement up to a finite truncation order. The manuscript provides no proof or additional numerical test that the truncated DSE-plus-positivity system uniquely determines the infinite sequence, nor does it supply a truncation-error bound or convergence-rate analysis beyond the statement of “rapid convergence.” This is load-bearing because, as the skeptic notes, other sequences could satisfy the same finite-order conditions yet deviate at higher orders.
  2. [§3.2] §3.2 (positivity implementation): positivity is enforced only on moments up to a finite order. The text does not demonstrate that the resulting Hankel matrices remain positive at all higher orders or that the finite truncation suffices to guarantee the existence of a positive measure whose moments solve the full DSE tower.
minor comments (2)
  1. [Notation] The notation for the tensor invariants and the precise definition of the moment-generating function could be accompanied by an explicit low-order example to improve readability.
  2. [Results] A short table comparing bootstrap values, conjectured closed forms, and feyntensor series at several orders would make the numerical support more transparent.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive major comments. We address each point below, indicating the revisions we will make to strengthen the presentation of the numerical evidence and the scope of the method.

read point-by-point responses

  1. Referee: [§4] §4 (quartic-model bootstrap and conjecture): the central conjecture for the closed-form moments is supported only by numerical agreement up to a finite truncation order. The manuscript provides no proof or additional numerical test that the truncated DSE-plus-positivity system uniquely determines the infinite sequence, nor does it supply a truncation-error bound or convergence-rate analysis beyond the statement of “rapid convergence.” This is load-bearing because, as the skeptic notes, other sequences could satisfy the same finite-order conditions yet deviate at higher orders.

    Authors: We agree that the conjecture for the closed-form moments of the rank-3 quartic model rests on numerical agreement between the bootstrap output and the independent feyntensor double-series expansions up to the orders we have computed. The manuscript does not claim or provide a rigorous proof that the finite-order DSE-plus-positivity truncation uniquely fixes the entire infinite sequence, nor does it include a formal truncation-error bound or convergence-rate analysis. We will revise §4 to include additional tables or figures that display the stability of the extracted coefficients as the truncation order is increased, and we will explicitly discuss the possibility that other sequences could agree at low orders but differ later. This additional numerical test addresses part of the concern while acknowledging that a full uniqueness proof lies outside the scope of the present work. revision: partial

  2. Referee: [§3.2] §3.2 (positivity implementation): positivity is enforced only on moments up to a finite order. The text does not demonstrate that the resulting Hankel matrices remain positive at all higher orders or that the finite truncation suffices to guarantee the existence of a positive measure whose moments solve the full DSE tower.

    Authors: The positivity constraints are applied to the Hankel matrices constructed from moments up to the truncation order of the bootstrap system, after which the DSE relations determine the higher moments. In the numerical results presented, the solved moment sequences produce positive-semidefinite Hankel matrices at all orders we check. We do not demonstrate that positivity persists for the infinite sequence or that the truncation guarantees the existence of a representing positive measure for the complete DSE tower; such a guarantee would require additional analytic work. We will revise the text in §3.2 and the discussion of the method to state clearly that the procedure is a truncated approximation whose practical success is evidenced by rapid convergence and agreement with known exact solutions where they exist. revision: yes

Circularity Check

0 steps flagged

No circularity: moments obtained from independent DSE + positivity; conjecture is post-hoc pattern guess verified by separate series computation.

full rationale

The derivation begins from the model definition to write exact Dyson-Schwinger equations for the moments, then imposes positivity of the moment matrix (a property of any actual measure) and truncates at finite order to obtain a numerical system solved for approximate moments. For cases with known closed forms the output matches those forms exactly in the limit. The new conjecture for the rank-3 quartic moments is an extrapolation from the numerical pattern and is cross-checked by an independent double-series expansion performed with the external tool feyntensor. No equation equates a derived quantity to a fitted parameter or to a self-cited uniqueness result; the DSE tower is model-derived rather than assumed, and the conjecture is not fed back into the bootstrap. The method is therefore self-contained against external benchmarks and exhibits no reduction of outputs to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard Dyson-Schwinger equations for tensor models and the domain assumption that moments satisfy positivity constraints; no free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption Dyson-Schwinger equations hold for the U(N)^D invariant tensor models
    Invoked as the starting point for the bootstrap in the abstract.
  • domain assumption Certain combinations of moments are non-negative
    Positivity constraints are combined with the equations to close the system.

pith-pipeline@v0.9.0 · 5416 in / 1289 out tokens · 44157 ms · 2026-05-10T01:43:01.353161+00:00 · methodology

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