eight_tick_forces_D3
Why this theorem is linked from GWTC-4.0: Tests of General Relativity. I. Overview and General Tests unclear
Pith linked this Lean declaration because the review connected a specific passage in the paper to this theorem. The relation tag says how strong that connection is; it is not a generic placeholder.
The final mass and final spin as inferred from the low- and high-frequency parts of the waveform are consistent with each other.
Relation between the paper passage and the cited Recognition theorem.
The 8-tick cycle forces space dimension D = 3.
plain-language theorem explainer
If the eight-tick period computed from spatial dimension D equals the fixed eight_tick, then D equals 3. Researchers closing the T7-T8 step in the Recognition Science forcing chain cite this to derive three spatial dimensions from the self-similar octave. The proof is a one-line wrapper that unfolds the power-of-two definition and applies the power_of_2_forces_D3 lemma.
Claim. If $2^D = 8$, then $D = 3$, where $D$ is a natural number denoting spatial dimension and 8 is the fixed eight-tick period.
background
The DimensionForcing module proves that spatial dimension D equals 3 is forced by the RS framework. Dimension is the abbrev for natural numbers. EightTickFromDimension D is defined as 2^D, the period required for ledger coverage in D dimensions. The module_doc states that the 8-tick cycle synchronizes with the 45-tick cumulative phase via lcm(8,45)=360, which uniquely identifies D=3 through the relation 2^D=8.
proof idea
This is a one-line wrapper. It introduces the hypothesis, unfolds EightTickFromDimension and eight_tick at the hypothesis, then applies the sibling lemma power_of_2_forces_D3.
why it matters
The theorem supplies the eight-tick forcing step used by spinor_eight_tick_forces_D3 and why_D_equals_3. It fills the T7 eight-tick octave (period 2^3) to T8 D=3 step in the unified forcing chain. The result shows that the self-similar fixed point and eight-tick period select three spatial dimensions, consistent with Bott periodicity and the unique spinor structure at D=3.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.
papers checked against this theorem (showing 24 of 24)
-
Tweezer array traps 6100 atoms with 12.6s coherence
"we experimentally realize an array of optical tweezers trapping over 6,100 neutral atoms in around 12,000 sites, simultaneously surpassing state-of-the-art performance for several metrics that underpin the success of the platform. Specifically, while scaling to such a large number of atoms, we demonstrate a coherence time of 12.6(1) seconds, a record for hyperfine qubits in an optical tweezer array."
-
Compressive memory lets transformers reach SOTA on long-range language tasks
"The Compressive Transformer uses the same attention mechanism over its set of memories and compressed memories, learning to query both its short-term granular memory and longer-term coarse memory."
-
XLNet beats BERT on 20 tasks via permutation pretraining
"XLNet integrates ideas from Transformer-XL... into pretraining"
-
Temporal Graph Networks beat prior models on evolving graphs with lower cost
"a generic, efficient framework for deep learning on dynamic graphs represented as sequences of timed events"
-
7B model matches GPT-4V on vision tasks with long context
"Trained with 24K interleaved image-text contexts, it can seamlessly extend to 96K long contexts via RoPE extrapolation."
-
TimesNet reshapes 1D series into 2D tensors for time modeling
"Based on the observation of multi-periodicity in time series, we ravel out the complex temporal variations into the multiple intraperiod- and interperiod-variations. To tackle the limitations of 1D time series in representation capability, we extend the analysis of temporal variations into the 2D space by transforming the 1D time series into a set of 2D tensors based on multiple periods."
-
VMamba scans images along four routes for linear vision scaling
"By traversing along four scanning routes, SS2D bridges the gap between the ordered nature of 1D selective scan and the non-sequential structure of 2D vision data, which facilitates the collection of contextual information from various sources and perspectives."
-
Hybrid CNN-state space model tops CNNs and Transformers on medical segmentation
"Inspired by the State Space Sequence Models (SSMs), a new family of deep sequence models known for their strong capability in handling long sequences, we design a hybrid CNN-SSM block that integrates the local feature extraction power of convolutional layers with the abilities of SSMs for capturing the long-range dependency."
-
One multi-output state space model matches S4 on long sequences
"We build on the design of the S4 layer and introduce a new state space layer, the S5 layer. Whereas an S4 layer uses many independent single-input, single-output SSMs, the S5 layer uses one multi-input, multi-output SSM."
-
7B models process million-token videos and texts
"we leverage recent advancements in scaling context window size, particularly Blockwise RingAttention (Liu et al., 2024; Liu and Abbeel, 2023), a technique that scales context size without approximations or overheads, enabling efficient training on long sequences... progressively increase the effective context length of the model across 5 stages: 32K, 128K, 256K, 512K, and 1M"
-
Chunked prefills let decodes piggyback for up to 10x faster LLM decode
"Chunked-prefills allows constructing multiple decode-maximal batches from a single prefill request, maximizing coverage of decodes that can piggyback."
-
91 GW events pass GR consistency tests
"The final mass and final spin as inferred from the low- and high-frequency parts of the waveform are consistent with each other."
-
Robot hand solves Rubik's cube after simulation-only training
"ADR automatically generates a distribution over randomized environments of ever-increasing difficulty. Control policies and vision state estimators trained with ADR exhibit vastly improved sim2real transfer."
-
Gated DeltaNet beats Mamba2 on retrieval and long sequences
"We introduce the gated delta rule and develop a parallel training algorithm optimized for modern hardware... preserves the benefits of chunkwise parallelism"
-
Equal-area pixel grid on the sphere speeds up large astronomical map processing
"HEALPix – the Hierarchical Equal Area iso-Latitude Pixelization – is a versatile data structure... Originally developed to address the data processing and analysis needs... In this paper we consider the requirements and constraints to be met in order to implement a sufficient framework for the efficient discretization and fast analysis/synthesis of functions defined on the sphere"
-
DeepSpeed-Ulysses trains 4x longer sequences 2.5x faster
"DeepSpeed-Ulysses at its core partitions input data along the sequence dimension and employs an efficient all-to-all collective communication for attention computation. Theoretical communication analysis shows that whereas other methods incur communication overhead as sequence length increases, DeepSpeed-Ulysses maintains constant communication volume when sequence length and compute devices are increased proportionally."
-
Linear bias lets models train short and test long
"We show that this method trains a 1.3 billion parameter model on input sequences of length 1024 that extrapolates to input sequences of length 2048, achieving the same perplexity as a sinusoidal position embedding model trained on inputs of length 2048 but training 11% faster and using 11% less memory."
-
Automatic sharding scales MoE models past 600 billion parameters
"We demonstrate that such a giant model can efficiently be trained on 2048 TPU v3 accelerators in 4 days"
-
Sparse attention scales transformers to 100,000-step sequences
"We introduce sparse factorizations of the attention matrix which reduce this to O(n√n)... We call networks with these changes Sparse Transformers, and show they can model sequences tens of thousands of timesteps long using hundreds of layers... setting a new state of the art for density modeling of Enwik8, CIFAR-10, and ImageNet-64."
-
Rotary embeddings add relative position signals to transformer attention
"the proposed RoPE encodes the absolute position with a rotation matrix and meanwhile incorporates the explicit relative position dependency in self-attention formulation"
-
One model, two minds: Qwen3 fuses fast answers and slow reasoning
"The Qwen3 MoE models have 128 total experts with 8 activated experts per token."
-
Five-color map of three quarters of the sky goes public
"3π Steradian Survey covers the sky north of Dec=−30 in five filters (grizy_P1)"
-
Gravity as a broken gauge symmetry, with no Big Bang singularity
"this solution can be thought of as a pre-geometric de Sitter spacetime, which also provides a novel solution for the problem of the Big Bang singularity"
-
A √N detuning rule shields collective quantum batteries from decay
"g_eff = g√N drives system into ultra-strong coupling; provides ceiling N_max on Tavis-Cummings validity"