Four Over Six (4/6): Quantization & Beyond

• Four Over Six (4/6) is a principle defining adaptive quantization ranges in low-precision deep learning, as well as setting sharp thresholds in combinatorial games, QFT renormalization, and elliptic curve theory.
• In deep learning, 4/6 adaptively selects between ±4 and ±6 quantization ranges per block, reducing mean squared error and stabilizing training to achieve loss within 10% of mixed-precision baselines.
• Across combinatorial games, QFT, and arithmetic geometry, 4/6 encapsulates optimal parameters—from winning strategies in pattern avoidance to four-loop corrections in six dimensions and elliptic curves with ℤ/4ℤ torsion and rank 6.

Four Over Six (4/6) is a term with established technical significance in several distinct areas of mathematical and computational research. In modern literature, it most notably refers to an adaptive quantization technique for extremely low-precision arithmetic in deep learning accelerators, as well as to thresholds or parameters in combinatorial game theory, algebraic geometry, and perturbative quantum field theory. This article focuses primarily on the algorithmic and combinatorial origins and implications of "Four Over Six," providing a rigorous analysis across its principal domains of usage.

1. Adaptive Quantization in Low-Precision Deep Learning

In the context of neural network quantization, "Four Over Six" (4/6) denotes an adaptive block-wise quantization strategy designed for the NVFP4 (NVIDIA FP4) numerical format, a block-scaled floating-point representation. The NVFP4 format combines 4-bit FP4 (E2M1) for data storage with 8-bit FP8 (E4M3) and global 32-bit FP32 scales for dynamic range management. This format offers substantial memory and compute efficiency; however, it is prone to large quantization errors for near-maximal values within each quantization block, leading to detrimental accuracy loss and even divergence during LLM training.

The 4/6 modification addresses this deficiency by adaptively choosing, for each block of 16 values, between two quantization dynamic ranges—with endpoints at ±4 or ±6—when casting values to FP4. Concretely, for a block $\mathbf{X}$, the method computes two candidate scales: $$\Delta^{(6)} = \frac{\max|\mathbf{X}|}{6}, \qquad \Delta^{(4)} = \frac{\max|\mathbf{X}|}{4}.$$ Each is quantized and dequantized to reconstruct $\widehat{\mathbf{X}}^{(6)}$ and $\widehat{\mathbf{X}}^{(4)}$. The mean squared error (MSE) is calculated for both variants: $$E^{(6)} = \frac{1}{m}\sum_{i=1}^m\left(\hat X_i^{(6)} - X_i\right)^2, \qquad E^{(4)} = \frac{1}{m}\sum_{i=1}^m\left(\hat X_i^{(4)} - X_i\right)^2.$$ The scale yielding the smaller MSE is selected for quantizing the block. This approach ensures that blocks with most values in moderate ranges use more granular ±4 quantization, while outlier-dominated blocks retain the full ±6 range to avoid saturation.

Efficiently implemented in CUDA kernels on NVIDIA Blackwell GPUs via fused quantization passes and register-resident intermediate storage, 4/6 incurs minimal hardware overhead—less than 2% at inference lengths up to 16,384 tokens, and under 15% during training at lengths up to 131,072 tokens compared to conventional NVFP4 (Cook et al., 1 Dec 2025).

Empirical results indicate that 4/6 prevents divergence during LLM pre-training in fully quantized regimes, reducing final loss to within 10% of the mixed-precision BF16 baseline. For post-training quantization (PTQ), 4/6 improves perplexity and downstream task performance across models (e.g., Llama-3, Qwen 3) and quantization strategies (AWQ, SmoothQuant, RTN, GPTQ).

2. Threshold Phenomena in Combinatorial Games

"Four Over Six" is also established nomenclature for sharp combinatorial thresholds in two adversarial word-building games—the nonrepetitive game and the erase-repetition game—played on finite alphabets. These games feature Ann and Ben alternating appending (possibly erasing) letters to a shared string under avoidance constraints.

• In the nonrepetitive game, Ann wins if she can avoid any square $uu$ with $|u| \geq 2$ indefinitely.
• In the erase-repetition game, occurrence of a square $uu$ triggers automatic erasure of the second $u$.

Rosenfeld proved that the minimal alphabet sizes for Ann to secure a winning strategy are 4 and 6, respectively. Namely,

$$\text{Ann wins the nonrepetitive game for } |A| = 4, \qquad \text{and the erase-repetition game for } |A| = 6$$

(Rosenfeld, 2021). The proof replaces entropy compression and Lovász Local Lemma arguments with a weight-counting technique, assigning rational weights to prefixes of words and establishing exponential lower bounds on the combined weights of playable word sets, using precisely computed constants for $\alpha$ and $\gamma$.

This closes gaps left by previous results (which gave bounds of 6 and 8, respectively, using entropy compression [Grytczuk–Kozik–Micek, 2013]). The bound 4 for the nonrepetitive game is proven optimal; for the erase-repetition game, potential further improvement to 5 remains open.

3. “Four Over Six” in Higher-Loop Quantum Field Theory

In multi-loop renormalization of field theories, the phrase "four over six" refers to the implementation of four-loop (L=4) calculations in six space-time dimensions ($d=6-2\epsilon$). Specifically, in "Four loop renormalization in six dimensions using Forcer" (Gracey, 1 May 2024), the utility of Tarasov's dimensional recurrence allows the translation of four-dimensional master integrals to six dimensions for automated Feynman diagram analysis in the {\sc Forcer} framework.

Key steps:

1. Tarasov’s Dimensional Recurrence: Four-loop integrals in $d=6-2\epsilon$ are evaluated by relating them to integrals in $d=4-2\epsilon$ using recurrence relations and integration-by-parts reductions.
2. Master Expansions: All three- and four-loop master integrals are evaluated to $\mathcal{O}(\epsilon^{9})$ in $d=6-2\epsilon$, enabling explicit computation of $\phi^3$ and QCD $\beta$-functions.
3. Renormalization Results: Reproduces established MS‾ and MOMt scheme results, including the absence of even zeta values ($\zeta_4$, $\zeta_6$) up to five loops, confirming the so-called "no-even-zeta" property in minimal momentum subtraction.

Thus, “four over six” in this technical context encapsulates the computation of four-loop renormalization quantities in six dimensions.

4. Algebraic Geometry: Elliptic Curves with Torsion $\mathbb{Z}/4\mathbb{Z}$ and Rank 6

In arithmetic geometry, “four over six” has become shorthand for elliptic curves $E$ over $\mathbb{Q}(u)$ with prescribed torsion group $\mathbb{Z}/4\mathbb{Z}$ and maximal possible generic rank (which, as of recent work, is 6) (Dujella et al., 2022). Dujella and Peral constructed explicit families $E_u$ of such curves, using parameterized Weierstrass equations tied to rational points of prescribed order and imposing conditions for independent infinite-order points via quadratic sections.

These constructions refine previous rank-5 records and exploit algebraic identities among torsion structures and point parametrizations. The methods involve:

• Starting from Kubert–Tate normal forms for prescribed torsion.
• Imposing two independent quadratic sections by solving for rational points with specific $x$-coordinates, leading to a genus 0 parametrization.
• Checking the independence and full rank by specialization and 2-descent on the height pairing.

5. Comparative Table of 4/6 Across Domains

Domain	"Four" Refers To	"Six" Refers To
Deep Learning Quantization	FP4 block max ±4	FP4 block max ±6
Combinatorial Game Theory	Alphabet of size 4	Alphabet of size 6
Higher-Loop QFT	Four-loop corrections	Six dimensions in regularization
Elliptic Curve Construction	Torsion order 4 (ℤ/4ℤ)	Generic rank 6 over $\mathbb{Q}(u)$

Each usage encodes a sharp boundary or adaptive choice critical for theoretical guarantees or practical performance.

6. Significance, Impact, and Open Questions

• Quantization: 4/6 achieves state-of-the-art accuracy for aggressive quantized LLMs at block 4-bit precision, is hardware efficient, and is broadly compatible with standard quantization pipelines (Cook et al., 1 Dec 2025).
• Combinatorics: The 4/6 thresholds for Ann's win dramatically refine the known borderlines for pattern-avoidance games, leveraging combinatorial counting in place of probabilistic or entropy-based arguments. The game on 5-letter alphabets remains open.
• Quantum Field Theory: New techniques permit explicit, high-precision four-loop computation in six dimensions, with structural insights on special values (e.g., no-even-zeta) in renormalization group functions (Gracey, 1 May 2024).
• Arithmetic Geometry: Achieving a generic rank 6 elliptic curve with $\mathbb{Z}/4\mathbb{Z}$ torsion over $\mathbb{Q}(u)$ advances the boundary of known constructions and demonstrates the utility of joint quadratic constraints and parametrization techniques (Dujella et al., 2022).

A plausible implication is that further advances in combinatorial game thresholds, quantization efficiency, and high-rank algebraic curve construction will increasingly employ domain-adaptive or locally-optimized strategies reminiscent of the 4/6 principle across mathematics, computer science, and physics.