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arxiv: 2607.02260 · v1 · pith:WVUIOY6Ynew · submitted 2026-07-02 · 🧮 math.CO

Tensor Amplification and Spectral Transfer for Sidorenko-Type Inequalities

Yuqi Zhao This is my paper

Pith reviewed 2026-07-03 10:32 UTC · model grok-4.3

classification 🧮 math.CO
keywords Sidorenko inequalitygraphontensor amplificationspectral inequalityPerron radiusdoubly nonnegativehomomorphism densityadmissible class
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The pith

For admissible graphon classes, Sidorenko inequalities with v(H) ≤ e(H) are equivalent to the spectral inequality t(H,W) ≥ ρ(W)^{2e(H)−v(H)} p(W)^{v(H)−e(H)} for nonzero W.

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

The paper develops a tensor-amplification framework that applies to any admissible class of graphons, defined as those closed under tensor powers and normalized principal restrictions. Within such classes it proves that the ordinary Sidorenko property is equivalent, as a universal statement, to a spectral lower bound on homomorphism density when the number of vertices does not exceed the number of edges. The equivalence is obtained by combining tensor amplification with a Perron-biased regularization theorem that tracks the spectral radius on an exponential scale. The same machinery shows that equality cases regularize optimally and that relative forcing coincides with regular-forcing for non-matching graphs.

Core claim

For every admissible class C, in the range v(H) ≤ e(H), ordinary C-Sidorenko is equivalent as a universal property over C to the spectral inequality t(H,W) ≥ ρ(W)^{2e(H)−v(H)} p(W)^{v(H)−e(H)} for every non-zero W in C. The spectral transfer follows from a Perron-biased tensor regularization theorem that detects the Perron spectral radius on the exponential scale. Equality cases for non-matching C-Sidorenko graphs regularize optimally, making relative forcing equivalent to relative regular-forcing.

What carries the argument

Tensor-amplification framework relying on closure under tensor powers and normalized principal restrictions, together with the Perron-biased tensor regularization theorem that isolates the spectral radius.

If this is right

  • If H is a non-matching C-Sidorenko graph then every equality case t(H,W)=p(W)^{e(H)} with W in C is regular.
  • Relative forcing is equivalent to relative regular-forcing for every non-matching C-Sidorenko graph.
  • Spectral equivalences hold for Sidorenko-good graphs in the range v(F) ≤ e(F).
  • Sidorenko-good forcing is identified with regular-KNRS forcing for non-matching Sidorenko-good graphs.
  • Quantitative near-equality variants of the inequalities are obtained.

Where Pith is reading between the lines

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

  • The same closure properties may let the method reach other positivity-preserving classes of kernels beyond doubly nonnegative ones.
  • Amplification could convert additional homomorphism-density inequalities into spectral statements.
  • Direct verification of the spectral bound on rank-one or constant graphons would test the claimed equivalence on concrete examples.
  • The isolation of minimal closure axioms suggests the technique can be ported to related extremal problems that rely on positivity.

Load-bearing premise

The graphon class must be closed under tensor powers and normalized principal restrictions.

What would settle it

An admissible class C together with a graph H satisfying v(H) ≤ e(H) and a nonzero W in C such that the Sidorenko bound holds but the spectral bound fails, or the reverse.

read the original abstract

We develop a tensor-amplification framework for Sidorenko-type inequalities in graphon classes. The framework applies to any admissible class, meaning a class closed under tensor powers and normalized principal restrictions. These two closure properties isolate the structural input needed for the amplification arguments, while preserving natural positivity constraints such as the doubly nonnegative constraint. For every admissible class $\mathcal{C}$, we prove two transfer principles. First, equality cases regularize optimally: if a non-matching graph $H$ is $\mathcal{C}$-Sidorenko, then every equality case $t(H,W)=p(W)^{e(H)}$ with $W\in\mathcal{C}$ is regular. Consequently, relative forcing is equivalent to relative regular-forcing for every non-matching $\mathcal{C}$-Sidorenko graph. Second, in the range $v(H)\le e(H)$, ordinary $\mathcal{C}$-Sidorenko is equivalent, as a universal property over $\mathcal{C}$, to the spectral inequality $t(H,W)\ge \rho(W)^{2e(H)-v(H)}p(W)^{v(H)-e(H)}$ for every non-zero $W\in\mathcal{C}$. The spectral transfer is obtained from a Perron-biased tensor regularization theorem detecting the Perron spectral radius on the exponential scale. We also prove quantitative near-equality variants and apply the framework to doubly nonnegative graphons and bounded doubly nonnegative kernels. This yields spectral equivalences for Sidorenko-good graphs in the range $v(F)\le e(F)$, and identifies Sidorenko-good forcing with regular-KNRS forcing for non-matching Sidorenko-good graphs.

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

0 major / 3 minor

Summary. The manuscript develops a tensor-amplification framework for Sidorenko-type inequalities that applies to any admissible class C of graphons (closed under tensor powers and normalized principal restrictions). It proves two transfer principles: (i) for non-matching H that is C-Sidorenko, equality cases t(H,W)=p(W)^{e(H)} with W in C are regular, so relative forcing is equivalent to relative regular-forcing; (ii) when v(H)≤e(H), the C-Sidorenko property is equivalent as a universal statement over C to the spectral inequality t(H,W)≥ρ(W)^{2e(H)−v(H)}p(W)^{v(H)−e(H)} for every nonzero W in C, obtained via a Perron-biased tensor regularization theorem. The framework is applied to doubly nonnegative graphons and bounded doubly nonnegative kernels, producing spectral equivalences for Sidorenko-good graphs and identifying Sidorenko-good forcing with regular-KNRS forcing for non-matching cases.

Significance. If the derivations hold, the work isolates the two closure properties as the precise structural hypotheses needed to carry out amplification while preserving positivity constraints such as double nonnegativity. This yields clean equivalences between combinatorial Sidorenko statements and spectral inequalities in the stated range, together with regularization results for equality cases. The explicit identification of minimal closure assumptions is a strength. The potential circularity concern (reliance on external closures rather than fitted parameters) does not land, because admissibility is defined directly by those closures and the equivalences are proved inside that class. No machine-checked proofs or code are present, but the results are parameter-free once admissibility is fixed.

minor comments (3)
  1. [Abstract] Abstract: the phrase 'quantitative near-equality variants' is mentioned without any indication of the form of the error term or the dependence on the distance to equality; a single sentence describing the quantitative statement would improve readability.
  2. [Abstract] The symbols t(H,W), p(W), and ρ(W) appear in the abstract before any definition; a brief parenthetical reminder of their standard meanings (homomorphism density, edge density, spectral radius) would help readers outside the immediate subfield.
  3. [Introduction] The manuscript should state explicitly, perhaps in the introduction, whether the two closure properties are also necessary for the equivalences or only sufficient.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the clear summary of the tensor-amplification framework and its applications, and the recommendation for minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines admissible classes C via two explicit closure properties (tensor powers and normalized principal restrictions) treated as external structural assumptions on C. It then derives the two transfer principles and the equivalence between C-Sidorenko and the spectral inequality for v(H) ≤ e(H) as theorems that hold precisely when those closures are present. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-definition, or load-bearing self-citation; the central claims are proved from the stated closures while preserving positivity constraints. The derivation is therefore self-contained against the given inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework depends on the definition of admissible classes and the existence of the Perron-biased regularization theorem; no free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption A class C is admissible if it is closed under tensor powers and normalized principal restrictions.
    Explicitly stated as the structural input needed for the amplification arguments.

pith-pipeline@v0.9.1-grok · 5815 in / 1220 out tokens · 38798 ms · 2026-07-03T10:32:08.563324+00:00 · methodology

discussion (0)

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Reference graph

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