Redundant moments of non-melonic random tensor models: simplifying the tensor bootstrap

Carlos I. Perez-Sanchez

Redundant moments of non-melonic random tensor models: simplifying the tensor bootstrap

T0 review · 0 major / 6 minor · reviewed 2026-07-31 · grok-4.5

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Pith's one-line read Large-N moments of melonic operators depend only on degree, even when the tensor model itself is non-melonic.

desk verdict Solid combinatorial extension: equal-degree melonic moments (and their SDEs) are redundant even when the interactions are non-melonic, which genuinely shrinks the tensor-bootstrap matrices. read the letter →

arxiv 2607.28518 v1 pith:YO2WJ4CH submitted 2026-07-30 math-ph hep-thmath.COmath.MPmath.PR

Carlos I. Perez-Sanchez This is my paper · ORCID

classification math-phhep-thmath.COmath.MPmath.PR MSC 81T1805C1560B2081T27

keywords randomtensorstensorbootstrapmelonicgraphsSchwinger-Dysonequationslarge-Nlimitpositive-semidefinitematricesWickcontractions

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Random tensor models produce huge families of observables, and positivity bootstrap methods need positive-semidefinite matrices whose entries are the large-N moments of those observables. This paper proves that every connected melonic observable is redundant at large N: its moment is fixed solely by how many vertices it has, not by the detailed wiring of its coloured graph. The same redundancy holds term-by-term for the Schwinger–Dyson equations that those moments satisfy. Because the argument never requires the interactions themselves to be melonic, the reduction applies to arbitrary unitary tensor models. The practical payoff is immediate: many entries in the bootstrap matrices can be identified, shrinking the numerical problem before any optimisation is run.

What carries the argument

A canonical vertex labelling of melons, built from dipole-insertion vertex pairs (divps) and an arborescence order on a D-ary tree, that induces a face-preserving bijection between maximal Wick contractions of C and of any other melon of equal degree; the bijection is realised by successive graph swaps that keep the face count additive.

What would settle it

Pick two distinct melonic graphs of equal degree inside a concrete non-melonic model (for example the K3,3 interaction), compute their large-N moments by enumeration of maximal Feynman graphs or by direct Monte-Carlo at large N, and check whether the numerical values agree to leading order.

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Extended reading notes

Core claim

For any connected melonic D-coloured graph C, the large-N moment m(C) exists only as a function of the half-vertex count p(C). Equivalently, if C and C̃ are melonic and have the same number of vertices, their leading-order moments coincide whenever they converge, and the leading-order left-hand sides of their Schwinger–Dyson equations coincide term by term, even when the interaction vertices Bj are non-melonic.

Load-bearing premise

The large-N moments and connected cumulants are assumed to exist and to be dominated by the face-maximising connected Wick graphs once the (often unknown) scaling exponents are chosen correctly.

Editorial extensions

If this is right

All melonic entries of equal degree inside any tensor-bootstrap PSD matrix may be collapsed to a single independent moment.
The left-hand sides of the Schwinger–Dyson equations for melonic observables become identical once degrees match, supplying automatic linear relations among connected and disconnected moments.
Existing melonic and non-melonic PSD constructions can be rescaled so that non-melonic entries remain finite and non-vanishing at large N.
Numerical bootstrap searches for general tensor models need track far fewer independent variables at each truncation order.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same face-counting bijection may extend to other recursively defined families (for example necklaces or melonic-with-handles) once an analogous canonical labelling is found.
If an algorithm for the unknown scalings qB of non-melonic observables is later supplied, the redundancy proved here immediately upgrades those observables into bootstrap-ready data.
The reduction suggests that the leading large-N free energy of a mixed melonic/non-melonic model is insensitive to which particular melons appear, only to their degrees and couplings.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The new content is the extension beyond melonic models. Pérez-Sánchez already had the melonic-only case; here he shows that any two connected melons with the same number of vertices have identical leading large-N moments, and that the melonic pieces of their Schwinger–Dyson equations match term-by-term, even when the bulk interactions Bj are arbitrary. That is Cor. 4.3 and Thm. 5.4, and it is the statement that matters for bootstrap numerics.

What works: the face-counting inequalities for dipole Wick pairings (Lem. 4.1), the canonical arborescence labeling that makes the bijection of maximal Wick sets well-defined (Def. 3.4 + Prop. 4.2), and the swap/union face formula that carries the argument into the SDEs (Lem. 5.1, Prop. 5.3). The non-melonic rescaling of the two PSD matrices in §6 is clean and immediately usable. Tables 1 and 3 make the practical saving obvious. Self-citations are to the author’s own prior combinatorial results and to the bootstrap templates he is improving; nothing circular.

Soft spots are ordinary and already flagged by the author. The theorems condition on existence of the large-N limits and on the (still unknown for non-melonic models) scalings qB and sj. There is no algorithm for those scalings, and the D-independence that held in the pure melonic case does not survive. Neither gap breaks the combinatorial claim. Proof-checking risk is the usual one for a long face-counting argument; I did not find an internal contradiction.

This is infrastructure for people already doing tensor bootstrap or colored-graph large-N combinatorics. It will not reorganize a broader field, but it is the right kind of paper: a precise reduction that cuts the size of the positive-semidefinite matrices one actually has to handle. I would send it to referees without hesitation and would cite the redundancy statements when setting up any non-melonic bootstrap.

Referee Report

0 major / 6 minor

Summary. The paper proves that, in arbitrary (not necessarily melonic) unitary-invariant tensor models, the large-N expectation of any connected melonic observable C depends on C only through its degree pp(C). Equivalently, if C and C̃ are connected melons with the same number of vertices, their leading-order moments coincide whenever the moments converge (Cor. 4.3), and the leading-order left-hand sides of the corresponding Schwinger–Dyson equations coincide term by term even when the interaction graphs Bj are non-melonic (Thm. 5.4). The argument is combinatorial: a canonical admissible vertex labelling of melons via arborescences in Λ_D (Def. 3.4, Claim 3.6), face-count inequalities for dipole Wick pairings and multi-propagator attachments (Lem. 4.1), a face-preserving bijection of maximal Wick sets (Prop. 4.2), and an analogous comparison for graph unions appearing in the SDEs (Lem. 5.1, Prop. 5.3). Section 6 adapts existing positive-semidefinite bootstrap matrices to non-melonic observables with correct large-N scalings (Prop. 6.1) and illustrates the resulting reduction in independent melonic entries.

Significance. The result is a clean, load-bearing combinatorial simplification for the emerging tensor bootstrap. It is stronger than a melonic-model statement: the interactions may be non-melonic, yet all connected melonic moments of fixed degree collapse to a single large-N number (and likewise for the melonic parts of the SDEs). Tables 1 and 3 make the numerical payoff concrete. The technical core—canonical labels via lexicographic order on arborescence endpoints, dipole-insertion vertex pairs, and the swap/union face formulae—is carefully set up and is of independent interest in coloured-graph combinatorics. The paper is explicit about its hypotheses (existence of the large-N limit after suitable, often unknown, scalings q_B and s_j) and does not overclaim D-independence outside the melonic case. If the face-counting arguments hold as written, the work immediately reduces the size of the PSD matrices used in tensor bootstrap without changing the physics content of the melonic sector.

minor comments (6)

[§4, Lemma 4.1(a)] Lemma 4.1(a) is only sketched and defers details of dipole maximality under removal/insertion to [Pér26b, Lem. 3.4, Rem. 3.5]. A short self-contained paragraph (or an appendix lemma) reproducing the Δ ≥ D−2 face difference and the inductive step would make the non-melonic extension fully independent of that reference.
[§6.3, Proposition 6.1] Proposition 6.1 is labelled a “sketch of proof.” The face-deficit identity (6.14) and the scaling qp(α∪ν)=q_α+q_ν−1 are central to the claim that the rescaled matrices (6.11) remain finite and non-trivial for non-melonic entries; expanding the argument to a full proof (even if short) would match the standard of §§4–5.
[§4, Corollary 4.3] Corollary 4.3’s passage from cumulants/connected Feynman graphs to normalized moments via Z is brief. A one-sentence reminder that vacuum diagrams factor in both the numerator and the partition function (so the difference E[C̃]−E[C] remains sub-leading) would close the argument more cleanly.
[§3.2, Definition 3.2 / Figure 1] In Definition 3.2 and Figure 1 the admissibility condition is intuitive but dense. A single additional sentence stating the forbidden move (“never re-enter a branch after leaving it”) next to the figure would help readers who skip the arborescence embedding.
[§5, Table 2] Table 2 caption and the surrounding text use both “break-and-weld” and “C∖{x,y}|weld”; consistent notation in the table header would reduce scanning cost when checking the N^{#E(x,y)} powers against (SDE_{C,x}).
[§4 title; §6.3; Introduction] Minor typos and style: “Equiv alent” in the §4 title; “boostrap” in the §6.3 heading; “op. cit.” used for [Pér26b] in the introduction is fine but the forward reference to “Sec. 3, op. cit.” is easy to miss—prefer an explicit citation key.

Circularity Check

1 steps flagged · score 1.0 of 10

Self-contained combinatorial face-counting proof; only minor non-load-bearing self-citation for a sketched melonic inequality

self citation load bearing [Lemma 4.1(a), proof paragraph]
"This one has been proven in [Pér26b], but we sketch the proof. ... (details are in [Pér26b, Lem. 3.4, Rem. 3.5])."

The pure-melon face-maximality inequality that seeds the later swap arguments is justified in part by a same-author citation. The paper does sketch the argument (Δ ≥ D−2 at dipoles, preserved under dipole removal), so the step is not load-bearing for the non-melonic extension, but it is the only place where a central combinatorial ingredient is not fully self-contained.

full rationale

The central claims (Cor. 4.3, Thm. 5.4) are derived from Wick’s theorem and face-counting inequalities for connected melonic graphs. Prop. 4.2 builds an explicit face-preserving bijection ϕ between maximal Wick pairings of two melons of equal degree via the canonical arborescence label of Def. 3.4 and the swap formula (4.1); Cor. 4.3 and Thm. 5.4 follow directly. No parameter is fitted and then re-presented as a prediction; no uniqueness theorem is imported to force the result; the bootstrap matrices in §6 are applications, not inputs to the proof. The sole self-citation of substance is Lem. 4.1(a), whose pure-melon face inequality is sketched in-place and only defers details to [Pér26b, Lem. 3.4]; that prior result is a parameter-free combinatorial lemma by the same author and is not the novel non-melonic content. Circularity burden is negligible.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

Pure combinatorial/probabilistic derivation. No fitted parameters. Background is standard Wick calculus and colored-graph large-N asymptotics; the paper’s own definitional machinery (canonical melon labels, divps) is bookkeeping, not new physics. The main external vulnerability is the existence of the large-N limit with suitable scalings for non-melonic interactions.

assumptions (5)

standard math Wick’s theorem evaluates Gaussian tensor integrals as sums over pairings of black/white vertices, with amplitude N^{f−(D−1)p}.
Used throughout §2.3 and all face-counting arguments.
domain assumption Leading large-N order of a moment is controlled by face-maximizing connected Feynman graphs once observables and interactions are scaled by suitable powers of N.
Stated in §2.1; for non-melonic models the optimal scalings sj, qB are not known algorithmically.
domain assumption Unitary U(N)^D invariance forces observables to be D-colored bipartite graphs.
Setup of §2; standard in the Gurău–Ryan colored-tensor framework.
domain assumption A connected melon admits a complete list of dipole-insertion vertex pairs obtained by iterative dipole removal down to the 2-vertex graph.
Definition and construction in §3.1; used to build canonical labels and the Wick bijection.
standard math The swap operation on (D+1)-colored graphs satisfies f(G 7_v^w H)=f(G)+f(H)−D.
Quoted from prior work and used as Eq. (4.1) in Lem. 4.1 and later proofs.

invented entities (2)

Canonical admissible vertex labeling of a melon via lexicographic order on arborescence endpoints in Λ_D
purpose: Gives a unique way to identify dipole-insertion pairs so one melon can replace another of equal degree inside a maximal Wick graph.
Defined in Def. 3.2–3.4 and Claim 3.6; pure combinatorial bookkeeping with no external empirical handle.
Dipole insertion vertex pair (divp)
purpose: Marks the vertex pairs that must be Wick-contracted to maximize faces in a melon.
§3.1; standard melonic lore refined for the replacement map.

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Pith. "Pith review of Redundant moments of non-melonic random tensor models: simplifying the tensor bootstrap." pith.science (2026). https://pith.science/paper/YO2WJ4CH

@misc{pith2026260728518,
  author       = {Pith},
  title        = {Pith review of: Redundant moments of non-melonic random tensor models: simplifying the tensor bootstrap},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YO2WJ4CH}},
  note         = {Machine review of arXiv:2607.28518}
}

abstract

We prove the redundancy of a family (the so-called melonic family) of large-$N$ moments in arbitrary tensor models. This reduces the number of independent entries in the positive semidefinite matrices that build the core of the tensor bootstrap (for tensor models there are already several generalizations of the positive semidefinite Toeplitz and Hankel matrices of lattice gauge theory and random matrix bootstrap, respectively). More concretely, we prove that any melonic operator $C$ at large-$N$ has an expectation value that depends on $C$ exclusively through its degree. A similar statement is shown here to hold for the Schwinger-Dyson equations of melonic moments. The strength of our result is also its scope, which is not limited to melonic tensor models.

Figures

Figures reproduced from arXiv: 2607.28518 by the authors.

**Figure 1.** We show a 3-ary rooted tree, a branch (0,2,6) and a non-admissible numeration. ‘Last-parents’ are 4, 6, 14, 22 and 24. All labels up to (incl.) 20 are placed legally, as it is possible to leave a branch, as in the transition 18 Ñ 20, but never allowed to come back (forbidden is in red/dashed). Consider the set ΛD “ tpl, αq : l “ 0, 1, 2 . . . and α “ 0, 1, . . . , Dp´1u. Since ΛD will index all possible D-ary trees,… view at source ↗

**Figure 2.** A melon C with vertex label λ (a) can be constructed from a tree (b), from which also an arborescence is constructed and depicted as embedded in Λ3 in (c). When (b) is turned upside down and the leaves of the tree pruned, we obtain (c). The first coordinate of pl, αq P Λ3 are denoted by black numbers l “ 0, 1, 2, 3. The second coordinate α is the dot’s number right to left at depth l. The label (a) of the melon yiel… view at source ↗

**Figure 3.** The swap for H and G pD ` 1q-coloured graphs (in gray, we represent propagators or 0 colour). theoretical counterpart2 of a connected sum of topological spaces, but we shall extensively exploit only that fpG 7v wHq “ fpGq ` fpHq ´ D. (4.1) This formula can be proven by noticing that the D faces that contained the propagator at v and those D faces that contained the propagator w merge after applying the swap. Conside… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: In gray, the map ϕpπq that yields a new Wick contraction defined by permuting V pC˜q by σ and pairing in such a way, that the diagram commutes (i.e. τ˜ ˝ σ “ τ ). The map that consists in the replacement of C by C˜ is possible thanks to Definition 3.4 and Claim 3.6. ‚ …

**Figure 5.** Figure 5: The union of graphs exemplified for D “ 3 (above) and D “ 6 (below). The vertex parity is not depicted, as the bipartiteness of the graphs is obvious. Lemma 5.1. For any connected D-coloured graphs B, C and B1 , the following holds: (a) For Wick contractions π of B and…

**Figure 6.** Figure 6: (a) Two graphs B below and B1 above, and their vertices x and w, (b) shows B Yx,w B1 . In dashed gray, we depict a selection of edges from a Wick contraction that plays a role in (c) and (d) (in solid gray). In particular, (c) depicts the colour-c faces [in (a) and (b)…

**Figure 7.** Figure 7: The two first graphs show the constraints imposed by the determinant’s positivity below it with green (red is where the determinant is negative). The last graph is the superposition of both constraints. The moments m2p of the quartic ensemble are expressed as function …

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 8 linked inside Pith

[2]

[BL17] Valentin Bonzom and Luca Lionni

arXiv:1201.0176. [BL17] Valentin Bonzom and Luca Lionni. Counting gluings of octahedra. Electron. J. Comb., P 3–36,

arXiv
[3]

Some structural and enumerative aspects of discrete surfaces and PL-manifolds

[Bon22] Valentin Bonzom. Some structural and enumerative aspects of discrete surfaces and PL-manifolds. Habilitation thesis (HDR), arXiv:2212.12200 [math.CO] (2022),

arXiv 2022
[9]

Pérez-Sánchez

[Pér24a] Carlos I. Pérez-Sánchez. Bratteli networks and the Spectral Action on quivers. arXiv:2401.03705 [math.RT] (2024) [To appear inAdv. Theor. Math. Phys. ],

arXiv 2024
[12]

Pérez-Sánchez

[Pér26b] Carlos I. Pérez-Sánchez. Twofold universality of large- N melonic random tensors. arXiv:2607.08677 [math.CO] (2026),

arXiv 2026
[13]

[PPS26] Nathan Pagliaroli, Carlos I

(Software written in SageMath.)10.5281/zenodo.19661197. [PPS26] Nathan Pagliaroli, Carlos I. Pérez-Sánchez, and Brayden Smith. Bootstrapping Tensor Integrals. arXiv:2604.19714 [hep-th],

arXiv
[1999]

Bootstraps to strings: solving random matrix models with positivity

[Lin20] Henry W Lin. Bootstraps to strings: solving random matrix models with positivity. JHEP, 2020(6):1–28,

2020
[2013]

[BGS13] Joseph Ben Geloun and Dine Ousmane Samary

arXiv:1111.4997. [BGS13] Joseph Ben Geloun and Dine Ousmane Samary. 3D Tensor Field Theory: Renormalization and One-loop β-functions. Ann. Henri Poincaré , 14:1599–1642,

arXiv
[2014]

Melons are branched polymers

[GR14] Răzvan Gurău and James P Ryan. Melons are branched polymers. Ann. Henri Poincaré , volume 15, 2085–2131. Springer,

Show all 13 references

[2019]

Additional constraints for the tensor bootstrap

[LT26a] Samuel Laliberté and Reiko Toriumi. Additional constraints for the tensor bootstrap. arXiv:2606.24859 [hep-th],

arXiv
[2022]

Pérez-Sánchez

[Pér22b] Carlos I. Pérez-Sánchez. On multimatrix models motivated by random noncommutative geometry. II: A Yang- Mills-Higgs matrix model.Ann. Henri Poincaré , 23(6):1979–2023,

1979
[2024]

Random Tensors 2024

[Pér24b] Carlos I. Pérez-Sánchez. Open problem formulation: Positivity constraints for tensor integrals . IHP, Workshop “Random Tensors 2024”, Paris, October 202410.5281/zenodo.20742506,

2024
[2025]

Pérez-Sánchez

[Pér26a] Carlos I. Pérez-Sánchez. Critical curve of two-matrix models ABBA, AtB, AuB and ABAB, Part I: Monte Carlo. arXiv:2603.25715 [math-ph] (2026),

2026
[2026]

Finite- N bootstrap constraints in matrix and tensor models

[LT26b] Samuel Laliberté and Reiko Toriumi. Finite- N bootstrap constraints in matrix and tensor models. arXiv:2603.17364 [hep-th],

arXiv

Pith tools

Reviewed July 31, 2026 · model on record in the stance chip above.

[1] [2]

[BL17] Valentin Bonzom and Luca Lionni

arXiv:1201.0176. [BL17] Valentin Bonzom and Luca Lionni. Counting gluings of octahedra. Electron. J. Comb., P 3–36,

arXiv

[2] [3]

Some structural and enumerative aspects of discrete surfaces and PL-manifolds

[Bon22] Valentin Bonzom. Some structural and enumerative aspects of discrete surfaces and PL-manifolds. Habilitation thesis (HDR), arXiv:2212.12200 [math.CO] (2022),

arXiv 2022

[3] [9]

Pérez-Sánchez

[Pér24a] Carlos I. Pérez-Sánchez. Bratteli networks and the Spectral Action on quivers. arXiv:2401.03705 [math.RT] (2024) [To appear inAdv. Theor. Math. Phys. ],

arXiv 2024

[4] [12]

Pérez-Sánchez

[Pér26b] Carlos I. Pérez-Sánchez. Twofold universality of large- N melonic random tensors. arXiv:2607.08677 [math.CO] (2026),

arXiv 2026

[5] [13]

[PPS26] Nathan Pagliaroli, Carlos I

(Software written in SageMath.)10.5281/zenodo.19661197. [PPS26] Nathan Pagliaroli, Carlos I. Pérez-Sánchez, and Brayden Smith. Bootstrapping Tensor Integrals. arXiv:2604.19714 [hep-th],

arXiv

[6] [1999]

Bootstraps to strings: solving random matrix models with positivity

[Lin20] Henry W Lin. Bootstraps to strings: solving random matrix models with positivity. JHEP, 2020(6):1–28,

2020

[7] [2013]

[BGS13] Joseph Ben Geloun and Dine Ousmane Samary

arXiv:1111.4997. [BGS13] Joseph Ben Geloun and Dine Ousmane Samary. 3D Tensor Field Theory: Renormalization and One-loop β-functions. Ann. Henri Poincaré , 14:1599–1642,

arXiv

[8] [2014]

Melons are branched polymers

[GR14] Răzvan Gurău and James P Ryan. Melons are branched polymers. Ann. Henri Poincaré , volume 15, 2085–2131. Springer,