theorem
proved
eight_tick_forces_D3
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The 8-tick cycle forces space dimension D = 3.
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IndisputableMonolith.Foundation.DimensionForcing on GitHub at line 155.
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Derivations using this theorem
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EightTickFromDimension -
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formal source
152 norm_num at h16
153
154/-- The eight-tick cycle forces D = 3. -/
155theorem eight_tick_forces_D3 (D : Dimension) :
156 EightTickFromDimension D = eight_tick → D = 3 := by
157 intro h
158 unfold EightTickFromDimension eight_tick at h
159 exact power_of_2_forces_D3 D h
160
161/-! ## The Clifford Algebra / Spinor Argument
162
163The spinor argument for D=3 is grounded in Clifford algebra theory:
164
1651. **Clifford algebras Cl_D**: The algebra generated by {e₁, ..., e_D} with
166 eᵢ² = -1 and eᵢeⱼ = -eⱼeᵢ for i ≠ j.
167
1682. **Dimension dependence**:
169 - Cl₁ ≅ ℂ (complex numbers)
170 - Cl₂ ≅ ℍ (quaternions)
171 - Cl₃ ≅ M₂(ℂ) (2×2 complex matrices) ← UNIQUE: gives 2-component spinors
172 - Cl₄ ≅ M₂(ℍ) (2×2 quaternionic matrices)
173
1743. **Spin groups**: Spin(D) ⊂ Cl_D is the universal double cover of SO(D).
175 - Spin(1) ≅ ℤ/2ℤ (discrete)
176 - Spin(2) ≅ U(1) (abelian)
177 - Spin(3) ≅ SU(2) ← UNIQUE: simplest non-abelian compact Lie group
178 - Spin(4) ≅ SU(2) × SU(2) (product structure)
179
1804. **Bott periodicity**: Cl_{D+8} ≅ Cl_D ⊗ Cl_8, so the period is 8 = 2³ = 2^D.
181
182D = 3 is special because it's the unique dimension where:
183- Spinors are 2-component complex vectors
184- Spin(D) is SU(2) (non-abelian but simple)
185- The Bott period 8 equals 2^D
papers checked against this theorem (showing 11 of 11)
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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"
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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"
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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."
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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."
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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"
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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."
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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"
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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."
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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)"
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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"
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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"