arXiv subjects

Hong Liu

Publications and source records attributed to Hong Liu.

At least 19 recordsLinked to original sources

arXiv2026

Solution of uniform Tur\'an's Tetrahedron Problem

Tur\'an's Tetrahedron Problem asks to determine the Tur\'an density of the complete hypergraph $K_4^{(3)}$ (tetrahedron). This problem, posed by Tur\'an in 1941, is one of the most famous problems in extremal combinatorics and its solution would attract \$500 prize from Erd\H{o}s. In the 1980s, Erd\H{o}s and S\'os asked to determine Tur\'an densities of $K_4^{(3)-}$ (broken tetrahedron) and $K_4^{(3)}$ (tetrahedron) when edges are constrained to be uniformly distributed in the host hypergraph. The presumably easier case of the broken tetrahedron was solved by Glebov, Kr\'al' and Volec [Israel J. Math. 211 (2016), 349-366] and Reiher, R\"odl and Schacht [J. Eur. Math. Soc. 20 (2018), 1139-1159]. We solve the tetrahedron case by proving that the uniform Tur\'an density of $K_4^{(3)}$ is equal to 1/2; this confirms that R\"odl's lower bound construction from 1986 is optimal.

math.CO↗

arXiv2026

HARTS: Efficient Agentic Reinforcement Learning for Hybrid-Attention Models over Arbitrary Rollout Trees

Agentic reinforcement learning (RL) often produces irregular rollout trees with shared histories. Training root-to-leaf trajectories independently recomputes these shared prefixes. Existing systems primarily target full-attention models and lack dense, differentiable hybrid-attention execution compatible with activation recomputation. We present HARTS (Hybrid-Attention RL over Tree Structures). HARTS jointly plans microbatches, data-parallel (DP) replica assignments, and microbatch-slot schedules using non-replay compact-token work after prefix compression. For chunkwise linear attention, a linear-time algorithm coordinates chunk-boundary state recovery and replay and produces the minimum number of sequential linear-attention calls under our packed execution model. HARTS preserves the chunkwise state partitioning of trajectory-wise training: it does not repeat projections, MLP/MoE computation, or final outputs, and performs only bounded state replay for numerical alignment. Per round, HARTS batches all branches into one packed call, propagates gradients through differentiable state handoffs, supports activation recomputation, and restores per-token log-probabilities. For deterministic, no-token-drop top-$k$ MoE routing, semantic multiplicities restore MoE-objective token weights and load statistics. Existing RL objectives retain their interface. To our knowledge, HARTS is the first system to demonstrate arbitrary-rollout-tree prefix-sharing speedups on a real hybrid-attention model. On an Agentic RL workload generated from SWE-bench tasks, HARTS achieves $4.81$--$4.87\times$ forward/backward/gradient speedup with activation recomputation across multiple parallel configurations. Its numerical differences are comparable to baseline self-rerun variation, and its reward trend is similar to the baseline over the first 120 steps of $\tau^3$-Bench training.

cs.LG↗

arXiv2026

The Erd\H{o}s-Gallai bound for consecutive even cycle lengths

Erd\H{o}s and Gallai in 1959 proved the seminal result that every $n$-vertex graph with no cycle of length at least $2t+2$ has at most $\tfrac{2t+1}{2}(n-1)$ edges. We prove the extension that, for every sufficiently large $t$, the same quantity is also the sharp extremal bound for graphs with no $t$ consecutive even cycle lengths, resolving a conjecture of Verstra\"ete. Thus, at the Erd\H{o}s-Gallai threshold, forcing an entire interval of even cycle lengths costs no more than forcing its longest member. More precisely, every $n$-vertex graph $G$ with $e(G)\ge \tfrac{(2t+1)(n-1)}2$ $\bullet$ either contains $t$ consecutive even cycle lengths, $\bullet$ or equality holds and $G$ is connected with every block isomorphic to $K_{2t+1}$. As consequences, for every sufficiently large even $k$ we determine the sharp edge thresholds forcing a cycle of length $0\pmod k$ or $2\pmod k$, answering questions of Bai, Grzesik, Li, and Prorok and of Gao, Li, Ma and Xie, respectively, for sufficiently large even $k$. The proof develops a stability-enhanced sublinear expander method. Its main new ingredient is a dense-case decomposition that recovers the lengths lost in the expander extraction by combining a flexible dense core with rooted cycle families in the vertices outside the core.

math.CO↗

arXiv2026

Homomorphism and VC-dimension thresholds: spectra and separations

Minimum-degree thresholds ask when excluding a fixed graph $H$ forces a dense graph to admit a simple global description. For each fixed chromatic number, the chromatic threshold has only three possible values. We show that this finite-spectrum phenomenon is special to chromatic threshold: already among $3$-chromatic graphs, both the homomorphism and VC-dimension thresholds have infinite spectra and are nonmonotone under taking induced subgraphs. For complete tripartite graphs with a singleton part, we prove $\delta_{\mathrm{hom}}(K_{1,s,t}) \ge \max\left\{\frac13,\frac{s}{1+s+t}\right\}$, with equality for an infinite range of $s,t$; in particular, $\delta_{\mathrm{hom}}(K_{1,s,s})=s/(2s+1)$ for every $s\ge2$. More generally, for every $r\ge3$, the value $(r-2)/(r-1)$ is an accumulation point of the homomorphism thresholds of $r$-chromatic graphs. For maximal $H$-free graphs, we determine the VC-dimension threshold of every complete tripartite graph and prove that it is positive for every nonbipartite $H$, yielding in particular the exact value for every odd cycle. We also classify the chromatic threshold under an a priori VC-dimension bound. Together with known blowup-threshold results, our theorems reveal that $\delta_\chi,\delta_{\mathrm{hom}},\delta_{\mathrm{VC}}$, and $\delta_{\mathrm B}$ are \emph{pairwise distinct}: bounded colorability, homomorphic compressibility, neighborhood complexity, and exact blowup structure are genuinely different forms of global simplicity. The proofs develop random and grid-based obstructions to bounded homomorphic images, saturated gadgets that preserve high VC-dimension under maximal completion, and a core-orientation method for raising minimum degree while preserving $H$-freeness.

math.CO↗

arXiv2026

Small circumference in regular sublinear expanders

Sublinear expansion is weak enough to be extracted from arbitrary graphs while retaining nearly all of their average degree, yet it has proved strong enough to force global structures in many sparse extremal problems. Letzter, Methuku and Sudakov [JLMS 2026] developed methods yielding nearly Hamilton cycles in sufficiently dense regular sublinear expanders, and Montgomery [ICM 2026] subsequently conjectured that, every sufficiently large (but constant) degree $d$-regular sublinear expander is Hamiltonian. We disprove this conjecture in a strong form by constructing $n$-vertex $d$-regular sublinear expanders with degree $d=\left(\frac12+o(1)\right)\log^2 n$, which does not even has a cycle covering a positive fraction of its vertices. The construction blows up one side of a biregular Ramanujan graph into almost-complete blocks while keeping the other side independent. The Ramanujan incidence graph certifies expansion for arbitrary mixtures of partial blocks and separator vertices, whereas the independent side forms a sparse vertex separator that prevents a cycle from visiting enough blocks. The construction also explains why $\log^2 n$ is the natural degree scale for this obstruction.

math.CO↗

arXiv2026

A Near-Optimal Linear Range for the Erd\H{o}s Matching Conjecture

The Erd\H{o}s Matching Conjecture is governed by two competing ways of excluding $s+1$ disjoint edges: one may concentrate all edges on fewer than $k(s+1)$ vertices, or force every edge to meet a fixed $s$-set. We determine a near-optimal range in which the second construction is extremal. For every fixed $k\ge2$, there is $s_0(k)$ such that, whenever $s\ge s_0(k)$ and $n\ge(k+1)s$, every $\mathcal{F}\subseteq\binom{[n]}k$ with $\nu(\mathcal{F})\le s$ satisfies \[ |\mathcal{F}|\le\binom nk-\binom{n-s}k, \] with equality only for the family of all $k$-sets meeting a fixed $s$-set. This improves the best previous general linear coefficient from $(5k-2)/3$ to $k+1$. In particular, the parameterized form of our argument further lowers the coefficient to $k+0.6$ for $k\ge5$. Since the two conjectured constructions exchange asymptotic dominance at $n=(\rho_k+o(1))s$ for a coefficient $\rho_k\in(k,k+1)$, our range lies less than one unit above the unavoidable barrier. We also prove a stability theorem showing that cover families are the only near-extremal configurations throughout this range. A key ingredient in our proof is a probabilistic rigidity statement which forces near-extremal fractional covers to be almost integral.

math.CO↗

arXiv2026

EATR-Stereo: Embodiment-Aware Token Routing of Paired Stereo Evidence for Humanoid Vision-Language-Action Control

Long-horizon humanoid vision--language--action (VLA) control with head-mounted stereo cameras requires visual interfaces that can exploit complementary views while maintaining compatibility with pretrained representations. Existing interfaces often discard complementary stereo evidence or fuse additional observations without preserving the native primary-view pathway and adapting auxiliary information to robot embodiment. We present EATR-Stereo, an embodiment-aware token-routing framework that retains primary-view tokens and constructs primary-aligned Cross-View Auxiliary Tokens (CVATs) by querying the synchronized auxiliary-view token sequence. A body-segmented proprioceptive encoder further conditions token-wise auxiliary usage on robot configuration history, enabling selective incorporation of stereo evidence during action generation. The routed auxiliary stream augments the language and primary-visual context of a pretrained VLA while keeping its vision--language model frozen. On a 33-DoF physical humanoid with a 37-D proprioceptive state, we evaluate nine configurations in over-100-s search--approach--grasp--place--return tasks. EATR-Stereo achieves 60.0% full-task success, 100.0% grasp success, and 80.0% stage success. Under severe asymmetric occlusion, it improves recovery to 80% compared with 30% for CVAT alone. Ablation studies further show the importance of preserving primary tokens and combining cross-view auxiliary features with structured proprioceptive routing. These results demonstrate that selectively routed paired stereo evidence improves spatial grounding for reliable long-horizon humanoid VLA control.

cs.RO↗

arXiv2026

LongCat Sparse Attention: Taming the Lightning via Streaming-aware Hierarchical Cross-Layer Indexing

DeepSeek Sparse Attention (DSA) enables efficient long-context modeling through its Lightning Indexer. However, practical deployment remains constrained by the indexer's expensive $O(L^2)$ scoring overhead and the hardware-inefficient, discontinuous memory-access patterns induced by its outputs. To address these system-level bottlenecks, we introduce LongCat Sparse Attention (LSA), a hardware-algorithm co-designed framework comprising three complementary and orthogonal strategies: (1) Streaming-Aware Indexing, which selectively converts scattered KV entries into hardware-aligned contiguous layouts to enable coalesced HBM access; (2) Cross-Layer Indexing, which amortizes indexing overhead by reusing the results produced by a single layer across consecutive layers, supported by cross-layer distillation; and (3) Hierarchical Indexing, which adopts a coarse-to-fine scoring scheme to progressively narrow the candidate set for each query, thereby substantially reducing indexing computation. Extensive scaling experiments, ranging from 69B-A3B to 560B-A27B models, demonstrate that LSA consistently achieves performance on par with full attention across both general-purpose and long-context benchmarks. Moreover, LSA supports native training with context lengths of up to one million tokens and underpins the development of LongCat-2.0 (1.6T-A48B). To facilitate further research, we also introduce and open-source LongCat-Flash-Lite-Sparse (69B-A3B), which integrates LSA into LongCat-Flash-Lite and incorporates an updated long-context training corpus.

cs.AI↗

arXiv2026

FOCUS: FP4 Optimization via Coupled-Relaxation and Dual-Granularity Scaling

Large language models (LLMs) achieve remarkable performance but are expensive to deploy due to their enormous size. FP4 quantization, with formats such as MXFP4 and NVFP4, offers an appealing solution with native hardware support on modern accelerators. However, maintaining accuracy under FP4 precision remains difficult. A key bottleneck lies in scale optimization: existing methods tightly couple the quantization and dequantization scales, forcing both to conform to the discrete low-precision format required by hardware, such as E8M0 in MXFP4. Yet the quantization scale is never stored and need not obey this constraint, suggesting a significant untapped optimization space. In this work, we propose FOCUS, a post-training quantization framework with end-to-end scale learning for FP4 Optimization via Coupled-Relaxation and Dual-Granularity Scaling. Coupled-Relaxation Scaling (CRS) relaxes the tight coupling between quantization and dequantization scales with a learnable full-precision coefficient, enabling more effective optimization without breaking hardware compliance. Dual-Granularity Scaling (DGS) further refines the quantization scale at a finer sub-block granularity, allowing more precise adaptation to local weight distributions. Experiments across multiple LLM families and benchmarks show that FOCUS achieves state-of-the-art FP4 accuracy under both MXFP4 and NVFP4 formats, while introducing no additional inference overhead. Code and quantized models will be released at https://github.com/tencent/AngelSlim.

cs.AI↗

arXiv2026

Ramp, Plateau, and Wormholes without Averaging, and Hyper-non-perturbative Structures in Gravity

Universal hallmarks of quantum chaos---such as the ramp and plateau in the spectral form factor---and the ramp's gravitational duals involving wormholes are widely interpreted as consequences of spectral or ensemble averaging. In this paper, following an earlier proposal of~\cite{Liu25c}, we develop an alternative approach: these phenomena arise as macroscopic smooth structures hidden within erratic microscopic data, which can be isolated through a smooth filter projection. Using the semiclassical Gutzwiller trace formula as a paradigmatic example, we illustrate how many features characteristic of random matrix models---including the ramp, the plateau, the spectral curve, and single-eigenvalue instantons---can be derived in the semiclassical limit without invoking ensemble or explicit spectral averages. We postulate the existence of a minimal Gutzwiller-like structure in the large-$N$ limit of holographic systems and explore its consequences. Beyond deriving the ramp and the plateau, this Gutzwiller-like structure predicts universal rapid macroscopic oscillations in the density of states and the possible existence of hyper-instantons, both of which involve double exponentials in $1/N^2$. On the gravity side, we demonstrate how spacetime wormholes enable the construction of emergent hyper-non-perturbative objects---such as baby-universe and wormhole condensates---which yield double exponential effects in $G_N$. This mirrors the postulated boundary Gutzwiller-like structure and provides a dual gravitational derivation of the universal rapid macroscopic oscillations in the density of states and the spectral plateau.

hep-th↗

arXiv2026

Strong invariants and Tverberg numbers in convexity spaces

Helly, Carath\'eodory, and Radon numbers encode three kinds of finite certificates in a convexity space: for the emptiness of an intersection, for membership in a convex hull, and for the existence of intersecting hulls. We study exact versions of these certificates, in which a subfamily must preserve the whole intersection or a subset must preserve the whole hull. Our first main result shows that, for finite configurations in an arbitrary convexity space, five a priori different boundedness conditions are equivalent: VC-dimension, strong Helly number, strong Carath\'eodory number, comatching number, and strong Radon number (with the expected additive-one shift). We also obtain equivalent layered Tverberg-type decompositions and colorful consequences. The common mechanism is exposed by the bipartite incidence graph between points and a generating family. For finite spaces, the unique minimal generator yields a natural dual convexity space; we characterize double dualization and prove that the strong parameters are duality invariant. The same model gives a polynomial-size, $O(t^4)$, realization of Bukh's counterexample to the Calder-Eckhoff partition conjecture. Finally, we obtain the first Tverberg bound for separable convexity spaces that is simultaneously linear in the number of parts and polynomial in the Radon number. If an $S_3$-separable convexity space has Helly number $h$ and its halfspaces have VC-dimension $d$, then $r_t=O(dh\log h)\,t$; in particular, Radon number $r$ gives $r_t=O(r^2\log r)\,t$. The bound attains the weak-Eckhoff scale $O(rt)$ whenever the Helly number is bounded. For axis-parallel box convexity in $\mathbb{R}^k$, gives the optimal order $r_t=O(rt)$ uniformly in every dimension. This appears to be the first dimension-uniform estimate of weak-Eckhoff order for box convexity, whereas the previous direct theory was confined to dimension three.

math.CO↗

arXiv2026

CoSA: Accelerating Long-Context Inference via Proxy-Kernel Co-Designed Sparse Attention

The quadratic cost of self-attention makes long-context inference prohibitively expensive, and proxy-based block-sparse attention has become a practical remedy. Existing methods typically rely on a proxy to predict a binary sparse mask and a kernel to consume this mask and perform sparse attention computation. Such an approach is effective under moderate budgets. However, as the budget tightens, the estimated proxy inevitably drops some salient blocks, while the kernel can only apply the sparse mask mechanically, leading to an evident drop in model accuracy. We propose CoSA, a two-stage training-free Sparse Attention under proxy-kernel CO-design, which couples a Kernel-Aware Proxy (KAP) with an Ordered-Skipping Kernel (OSK). In the first stage, the KAP selects blocks under a moderate budget and produces an ordered mask that prescribes the order in which KV pages are visited in the kernel inner loop. In the second stage, the OSK applies this mask and skips more blocks under a tightened budget given online-softmax statistics. Across mainstream LLM backbones and long-context benchmarks, CoSA attains higher accuracy at lower budgets. Impressively, CoSA achieves a 4.93$\times$ attention speedup and reduces end-to-end Time-to-First-Token by 2.53$\times$ under a context length of 128K with negligible performance degradation. Code is available at https://github.com/Tencent/AngelSlim.

cs.CL↗

arXiv2026

AngelSpec: Towards Real-World High Performance Inference with Speculative Decoding

Speculative decoding accelerates large language model inference without changing the target distribution, but no single drafting structure performs best across real-world workloads. Autoregressive multi-token prediction (MTP) is a lightweight, stable proposal mechanism, whereas block-parallel diffusion amortizes drafting latency over much longer candidate sequences; the better choice depends strongly on the output distribution. We present AngelSpec, a unified training framework for MTP and block-parallel speculative decoding that addresses this heterogeneity at three levels. At the training level, rather than fitting one universal drafter to a uniform data mixture, we co-specialize structure and data: the MTP drafter is trained on diverse conversational data for high-entropy open-ended chat, and the block-diffusion drafter on code and mathematics data for longer predictable continuations. At the architecture level, we propose DFly, a block-diffusion framework combining a hybrid target-conditioning backbone with a predecessor-conditioned autoregressive head, improving target-feature utilization and intra-block dependency modeling while keeping generation parallel. At the inference level, both acceptance length and verification cost vary with domain, request, online load, and hardware, so DFly treats verification as a shared batch-level resource: it reallocates compute toward high-confidence prefixes across requests and combines expected utility with a profiled cost model to adapt verification depth online. Across the Hy3 series, DFly raises the average accepted length on Hy3-A21B by roughly 30% and attains the highest average throughput at every tested concurrency from 4 to 64, a 1.98-2.40x speedup over autoregressive decoding and 10.5-11.8% higher throughput than DFlash. We release AngelSpec to support training and extending these methods.

cs.CL↗

arXiv2026

PIVOT: Efficient Query-Group Indexing for Token-Level Sparse Attention

Token-level sparse attention, as implemented by DeepSeek Sparse Attention (DSA) in production systems, makes the downstream attention efficient but shifts the bottleneck to the indexer that feeds it. To select the top-k tokens for each query, the indexer must still score every preceding token, incurring a cost of O(L^2) per layer for a sequence of length L. We observe that this per-query scan is largely redundant: nearby queries select highly overlapping top-k tokens, and the indexer scores are long-tailed along the key axis. We exploit these properties in PIVOT, Proxy Indexing Via One full-prefix Traversal, a training-free, drop-in replacement for the DSA indexer that shares one prefix scan across a group of nearby queries. PIVOT aggregates a group into a single proxy query, performs one shared full-prefix scan to obtain a candidate set, and then selects a top-k for each query from that set. Two variants trade speed for fidelity: PIVOT-Reuse shares the proxy top-k across the group for maximum speed, whereas PIVOT-Refine re-scores the candidate set with the indexer of each query and then selects an individual top-k, matching the dense indexer at a small additional cost. A single algorithm covers both inference phases, differing only in how groups are formed: fixed-size groups of consecutive queries in prefill, and the queries decoded together in one multi-token prediction (MTP) step in decode. On DeepSeek-V3.2 and GLM-5.1 across LongBench and RULER, PIVOT matches the accuracy of the dense DSA indexer while accelerating it by up to 4x and reducing end-to-end latency by up to 1.6x at long context.

cs.CL↗

arXiv2026

Upper-shadow comparisons on the slice and the Frankl--Tokushige product conjectures

Let $\mathcal{F}\subseteq\binom{[n]}k$, and let $\partial_{k\to\ell}\mathcal{F}$ be the family of all $\ell$-sets containing a member of $\mathcal{F}$. Writing $\mu_k(\mathcal{F})=|\mathcal{F}|/\binom nk$, $p=k/n$, and $q=\ell/n$, we prove $$ \mu_\ell(\partial_{k\to\ell}\mathcal{F})\geq \begin{cases}\mu_k(\mathcal{F})^{\frac{\log q}{\log p}}, &0\leq \mu_k(\mathcal{F})\leq p^2,\\ q\left[1-\left(1-\frac{\mu_k(\mathcal{F})}{p}\right)^{ \frac{\log(1-q)}{\log(1-p)}} \ \ \right], &p^2\leq \mu_k(\mathcal{F})\leq p,\\ 1-(1-\mu_k(\mathcal{F}))^{\frac{\log(1-q)}{\log(1-p)}}, &p\leq \mu_k(\mathcal{F})\leq1. \end{cases} $$ This yields a dimension-free closed-form lower bound for the finite Kruskal--Katona profile and each branch is asymptotically sharp. As the main application, we settle both the uniform and biased product conjectures of Frankl and Tokushige. If $r\geq2$, $0\leq k_i\leq(r-1)n/r$, and $\mathcal{F}_i\subseteq\binom{[n]}{k_i}$ are $r$-cross-intersecting, then $$ \prod_{i=1}^r\mu_{k_i}(\mathcal{F}_i)\leq\prod_{i=1}^r\frac{k_i}{n}. $$ In biased product setting, we prove that for $r$-cross-intersecting families $\mathcal{A}_i\subseteq 2^{[n]}$ and $0\leq p_i\leq(r-1)/r$, $\prod_{i=1}^r\mu_{p_i}(\mathcal{A}_i)\leq\prod_{i=1}^rp_i.$ We further determine all equality cases in both settings.

math.CO↗

arXiv2026

Fabric Pneumatic Artificial Muscles Based on the Drawstring Principle

Pneumatic artificial muscles have wide applications in robotics and industrial fields. Conventional pneumatic artificial muscles generate extra radial deformation during axial contraction, which severely wastes available working space. Inspired by the widely adopted drawstring principle in textile products, this paper proposes a novel drawstring fabric pneumatic artificial muscle (DPAM). Unlike traditional counterparts, the proposed DPAM produces no extra radial deformation during contraction, greatly improving structural compactness. The DPAM exhibits outstanding mechanical performance: a load capacity over 800 times its self-weight, a maximum contraction ratio of 44%, and a power density up to 4.98 kW/kg, alongside excellent scalability. Two representative application scenarios, bionic robots and industrial production lines, are demonstrated to validate its practicability. The DPAM can be easily expanded within a two-dimensional plane, as verified by the fabricated DPAM matrix. This work not only presents a high-performance novel pneumatic artificial muscle but also inspires researchers to draw design inspiration from conventional textile structures to address existing challenges in soft robotics.

cs.RO↗

arXiv2026

D-cut: Adaptive Verification Depth Pruning for Batched Speculative Decoding

Speculative decoding accelerates large language model (LLM) inference without compromising output quality. Recent parallel drafting methods further improve single-request performance by decoupling draft length from drafting latency, enabling longer drafts and higher mean accepted tokens (MAT). However, under high request concurrency, long drafts waste substantial computation on rejected tokens, increasing verification cost and potentially making speculative decoding slower than autoregressive decoding. We present D-Cut, an adaptive pruning method that selects draft tokens jointly across the batch and concentrates the verification budget on tokens most likely to be accepted. D-Cut is motivated by two observations. First, acceptance lengths vary considerably across concurrent requests; D-Cut therefore performs cross-request pruning, allocating the verification budget adaptively according to draft confidence. Second, verification cost depends strongly on the deployment environment, including GPU architecture and parallelism strategy; D-Cut incorporates a runtime cost model to adapt its pruning depth to the target environment. Experiments on dense and mixture-of-experts (MoE) models show that, under high concurrency, D-Cut improves the average speedup from $1.26\times$ to $1.65\times$, restores acceleration in dense-model configurations where long-draft baselines are slower than autoregressive decoding, and achieves up to $3.0\times$ speedup over autoregressive decoding on MoE models.

cs.CL↗

arXiv2026

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

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.

math.CO↗