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Tree suspensions turn one single-forbidden hypergraph density into another with an explicit formula, forcing infinitely many accumulation points and algebraic numbers of unbounded degree.

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2026-07-11 00:11 UTC pith:WGHWNZVI

load-bearing objection Clean transfer mechanism that multiplies known seeds into infinite accumulation points and unbounded algebraic degree for single-forbidden degree spectra.

arxiv 2607.06518 v1 pith:WGHWNZVI submitted 2026-07-07 math.CO

Tree suspensions and transfer functions for single degree Tur\'an spectra

Jiangdong Ai , Laihao Ding , Hong Liu , Haotian Yang This is my paper
classification math.CO MSC 05C6505C35
keywords Turán spectrumdegree Turán densitytransfer functiontree suspensionaccumulation pointsalgebraic degreehypergraph Lagrangiansingle-forbidden densities
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper builds a systematic way to manufacture new single-forbidden degree Turán densities from old ones. For any forbidden k-uniform hypergraph F, a tree-suspension operation produces a new single forbidden hypergraph F* whose ℓ-degree Turán density is given by an explicit continuous function of the density of F. Three such transfer maps are obtained: a universal map that works for every degree parameter ℓ, a second map that works when ℓ is at least half the uniformity, and a third map for ordinary Turán density. Because the universal map is strictly increasing and continuous, every accumulation point is sent to a new accumulation point, so the spectrum has infinitely many accumulation points once a single seed is known. Combining two independent maps produces nested radical extensions whose algebraic degrees grow without bound, so single-forbidden densities already realize the same arithmetic complexity that was previously known only for finite forbidden families.

Core claim

There exist explicit continuous transfer functions of the single-forbidden ℓ-degree Turán spectrum: for every F one can construct another single k-graph F* whose density is a prescribed algebraic function of the density of F. The maps are realized by tree suspensions and give matching lower and upper bounds in the regimes needed for the applications.

What carries the argument

Tree suspensions: recursive constructions that attach choice-tree patterns of copies of F to root copies of F and finish with connecting edges. Their robustness property yields a two-part lower-bound construction; recursive embedding lemmas (or a Lagrangian-plus-Sidorenko argument) supply the matching upper bounds that turn the operations into transfer functions.

Load-bearing premise

The upper bounds rest on the claim that sufficiently dense links always contain the required choice trees or that the Lagrangian contribution through a fixed vertex is maximized exactly at the predicted point; if either embedding or the Lagrangian calculation fails, the density equalities collapse.

What would settle it

Exhibit a single k-graph F for which the ℓ-degree density of its (ℓ,0)-tree suspension is strictly larger than Φ_{k,ℓ}(π_ℓ(F)), or for which the corresponding (ℓ,s-1) suspension exceeds Ψ or Υ_k; any such example would break the claimed transfer equalities.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The paper introduces transfer functions on the single-forbidden ℓ-degree Turán spectra Π^k_ℓ: continuous maps f such that for every k-graph F there exists a single k-graph F* with π_ℓ(F*)=f(π_ℓ(F)). These maps are realized by a new family of (ℓ,t)-tree suspensions T^{(k)}_{ℓ,t}(F). Matching lower and upper bounds are proved for three regimes, yielding the universal transfer Φ_{k,ℓ} (all 1≤ℓ<k), the map Ψ when ℓ≥ k/2, and the ordinary-Turán map Υ_k when ℓ=1 (Theorem 1.1). As applications, Φ propagates accumulation points, so Acc(Π^k_ℓ) is infinite for every k≥3 and 1≤ℓ<k (Theorem 1.2), and the composition of two independent transfers produces algebraic numbers of arbitrarily large degree over ℚ for ℓ∈{1,⌈k/2⌉,…,k-2} (Theorem 1.3).

Significance. The work supplies a systematic, constructive mechanism for generating new single-forbidden densities with exact control of the value, something previously available only for finite or infinite families. The infinitude of accumulation points recovers and unifies recent results of Conlon–Schülke and of Li–Liu–Schülke–Sun, while the algebraic-degree theorem shows that the arithmetic complexity known for Π^k_{fin} already appears inside the single spectrum for ordinary Turán density and a broad range of degree densities. The tree-suspension construction is explicit, the lower bounds rest on a clean robustness lemma, and the upper bounds use only standard embedding and Lagrangian tools; the arguments are fully written out and appear self-contained.

minor comments (5)
  1. In the definition of γ (just before Lemma 2.5) the three cases are written with slightly different algebraic forms; a short remark that they all solve the same equation Q_{s,t+1}(x)=Q_{s,t+1-ℓ}(x)-(1-β)x^s would make the subsequent density formulae easier to verify at a glance.
  2. Figure 1 is helpful but the caption is long; a one-sentence summary of what is being illustrated (the connecting edges for a single root vertex) would improve readability.
  3. In the proof of Theorem 2.10 the family M is defined via the existence of certain homomorphisms; a brief parenthetical example (already present later) placed at the definition would clarify that M is non-empty.
  4. The algebraic lemmas (3.1–3.2) are standard but the citation to Capelli’s criterion could be expanded by a one-line statement of the precise form used, for readers less familiar with binomial irreducibility.
  5. A few minor typographical inconsistencies appear (e.g., spacing around Π^k_ℓ versus Π^k_{ℓ,fin}); a light copy-edit pass would remove them.

Circularity Check

0 steps flagged

No significant circularity: transfer equalities are proved by explicit two-part constructions plus independent embedding/Lagrangian upper bounds; seeds are external or trivial.

full rationale

The derivation chain for Theorems 1.1–1.3 is self-contained. Tree suspensions T^{(k)}_{ℓ,t}(F) are defined recursively (Defs. 2.1, 2.3); robustness (Lemmas 2.2, 2.4) is proved by induction on the tree depth and does not presuppose any density value. The lower bound π_ℓ(T) ≥ Q_{s,t+1}(γ) follows from an elementary two-part host (Lemma 2.5) that uses only the definition of π_ℓ(F)=β. Matching upper bounds are supplied case-by-case: recursive choice-tree embedding (Lemmas 2.7–2.8) for Φ, support-set embedding for Ψ when ℓ≥k/2, and a finite-family Sidorenko-plus-Lagrangian argument (Prop. 2.11 + Claim 2.12) for Υ_k; each step works with explicit density margins and standard external tools. Algebraic-degree growth (Thm. 1.3) starts from the trivial single-edge seed π_ℓ=0 and iterates the two independent transfers via field-theoretic lemmas (3.1–3.2) proved in the paper. Accumulation (Thm. 1.2) merely applies the continuous map Φ to known external seeds (0 for ℓ≥2, Conlon–Schülke for ℓ=1). No equation reduces to its own input by definition, no parameter is fitted and re-predicted, and no uniqueness theorem is imported from the authors’ prior work to force the construction. The single self-citation [1] supplies only a seed accumulation point that is then mapped forward; the transfer mechanism itself is independent.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 1 invented entities

The paper is pure mathematics. It relies on standard existence of Turán densities, known seed accumulation results, Capelli’s criterion and Chebotarev’s density theorem for the algebraic-degree argument, and Sidorenko’s extension theorem for the ordinary upper bound. No free parameters are fitted; the only invented objects are the tree-suspension hypergraphs themselves, which are explicitly constructed.

axioms (5)
  • domain assumption Existence of the ℓ-degree Turán density π_ℓ(F) for every finite F (Lo–Markström).
    Used throughout to define the spectra Π^k_ℓ.
  • domain assumption 0 is an accumulation point of Π^k_ℓ for every ℓ≥2 (Ai–Ding–Liu–Yang).
    Seed for the accumulation-point transfer when ℓ≥2.
  • domain assumption Acc(Π^k)∩[0,1) is non-empty for k≥3 (Conlon–Schülke).
    Seed for the ordinary case ℓ=1.
  • domain assumption Finite-family form of Sidorenko’s extension theorem: π(Ext(F))=π_λ(F).
    Invoked in the proof of the ordinary transfer function Υ_k (Proposition 2.11).
  • standard math Capelli’s criterion for irreducibility of binomials X^s-a over a number field, and Chebotarev’s density theorem.
    Used in Lemmas 3.1–3.2 to force full algebraic degree of successive radical extensions.
invented entities (1)
  • (ℓ,t)-tree suspension T^{(k)}_{ℓ,t}(F) independent evidence
    purpose: Explicit single forbidden hypergraph whose degree Turán density equals a prescribed algebraic function of π_ℓ(F).
    Defined recursively via choice-tree patterns; the robustness property (Lemma 2.4) is proved by induction and supplies the lower bound.

pith-pipeline@v1.1.0-grok45 · 23969 in / 2745 out tokens · 31689 ms · 2026-07-11T00:11:46.407244+00:00 · methodology

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read the original abstract

For integers $1\le \ell<k$, let $\Pi^k_\ell$ denote the single-forbidden $\ell$-degree Tur\'an spectrum of $k$-uniform hypergraphs. We introduce transfer functions for this spectrum: explicit functions $f$ such that, for every $F$, there is another single $k$-graph $F^*$ with $\pi_\ell(F^*)=f(\pi_\ell(F))$. This gives a mechanism for producing new single-forbidden densities while retaining full control of the resulting value. Our transfer functions are realized by a new family of suspension-type operations, called tree suspensions. From these operations we obtain three explicit maps: one acting on $\Pi^k_\ell$ for every $1\le\ell<k$, a second acting when $\ell\ge k/2$, and a third acting in the ordinary Tur\'an case $\ell=1$. The common feature is a robust tree structure which gives the lower bound by a two-part construction and, in the regimes above, admits a matching embedding or Lagrangian upper bound. As a first application, the universal transfer function propagates accumulation points. Using the recent zero-accumulation results for $\ell\ge2$ together with the ordinary Tur\'an accumulation result of Conlon and Sch\"ulke, we prove that $\Pi^k_\ell$ has infinitely many accumulation points for every $k\ge3$ and every $1\le\ell<k$. This recovers, in particular, the known infinitude of accumulation points in the ordinary and codegree spectra. As a second application, combining two independent transfer functions forces algebraic degrees to grow. For every $k\ge3$ and every $\ell\in\{1,\lceil k/2\rceil,\ldots,k-2\}$, the spectrum $\Pi^k_\ell$ contains algebraic numbers of arbitrarily large degree over $\mathbb Q$. Thus the arithmetic complexity previously known for finite forbidden families already occurs in the single-forbidden spectrum, both for ordinary Tur\'an density and for a broad range of degree Tur\'an densities.

Figures

Figures reproduced from arXiv: 2607.06518 by Haotian Yang, Hong Liu, Jiangdong Ai, Laihao Ding.

Figure 1
Figure 1. Figure 1: The (1, 0)-tree suspension T (3) 1,0 (F) for F a single edge. As ℓ = 1 there is a single root copy X1 = {x 1 1 , x1 2 , x1 3 } (top), so the construction is a rooted tree. For each a ∈ [3] an attached choice-tree pattern T a 2 (F) hangs below; it consists of a root copy of F (middle, with vertices written y1, . . . , yf ) each of whose vertices carries a leaf copy T1(F) (bottom, vertices z1, . . . , zf ). … view at source ↗

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