jsut 200 more words

EEMLA.tex

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 A critical aspect of the methodology is the mechanism responsible for determining which rule is applicable at each position within the input matrix $\mathbf{X}$. Rather than relying on stochastic selection, this approach implements a deterministic, location-based rule selection strategy that leverages the contextual information encoded within the model's representations. A major advantage of this fixed-index approach is the minimization of dynamic surfaces in the model's cost function, thereby reducing the amount of noise in output, as well as reducing the required amount of training as over fitting is a non-issue without randomness.
 
+{\raggedright \normalsize \textit{Fixed-Index Architecture}}
+
+The implementation architecture utilizes predetermined index relationships within the tensor space. For example, indices $(0,1)$ and $(1,2)$ might have a fixed relationship where their values are added together and output at index $(0,4)$. This fixed-index approach creates explicit computational pathways within the neural network architecture, allowing for deterministic mathematical operations without disturbing the stochastic nature of the remaining network. The fundamental advantage of this approach lies in its compatibility with modern hardware acceleration techniques, particularly Single Instruction Multiple Data (SIMD) operations that commonly take place on GPUs. These GPUs are already leveraged for matrix multiplication in the vast majority of existing AI/ML runtimes. 
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+Each mathematical operation type is assigned specific input and output indices within the tensor, creating a predictable computational graph that can be optimized during compilation using the CUDA compiler, \mintinline{c}|gcc|, and manual assembler optimization like with DeepSeekV3 \parencite[16]{deepseekai2025deepseekv3technicalreport}. Addition operations, for instance, use indices $(i,j)$ and $(i+1,j)$ as inputs, with results stored at $(i,j+d/2)$, effectively partitioning the embedding space into operand and result regions. Multiplication operations utilize indices $(i,j)$ and $(i,j+1)$ as inputs, with results projected to $(i+1,j+d/2)$, maintaining a consistent pattern of spatial relationships within the tensor. More complex operations like matrix determinant calculations employ a $3\times3$ submatrix starting at index $(i,j)$ with results consolidated at $(i+3,j)$. This systematic approach to index mapping enables highly efficient computation on GPU architectures, as the fixed patterns allow for optimized memory access patterns due to hard-coded indexing at compile time, and reduced cache thrashing during tensor operations. Modern GPUs excel at these fixed-pattern operations, particularly when they can be expressed as fused operations within CUDA kernels or optimized through tensor cores designed specifically for matrix multiplication\parencite{cuda_programming_guide_2025}.
+
+The architecture maintains parallel processing paths that preserve the dual nature of the system's capabilities. The standard language processing path continues to leverage the probabilistic, statistical nature of the transformer architecture, preserving the original LLM capabilities that have proven effective for natural language understanding and generation. Simultaneously, the mathematical computation path applies fixed-index transformations for specific operations, creating a deterministic subsystem within the larger stochastically variant network. These parallel streams capitalize on the inherent parallelism of GPU architectures, allowing different CUDA cores and cache regions to process distinct streams simultaneously. The fixed-index nature of the mathematical operations enables compiler optimizations that can allocate dedicated tensor cores for these operations, maximizing throughput and minimizing latency. Existing models, as shown in Figure \ref{tb:model-sizes} tend to use far more VRAM than cores, leading to an allocation inefficient in terms of performance per millisecond of inference. The paths are later merged through concatenation and a projection layer, a process that similarly benefits from the warp-level primitives available in modern GPU architectures for efficient tensor manipulation.
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+The attention mechanism serves as a noise filter and integration component, allowing the model to selectively focus on either standard language representations or mathematically transformed representations based on input context. This selective focusing behavior effectively routes information through the appropriate pathway based on the input's semantic requirements. From a hardware acceleration perspective, this mechanism benefits from the recent advancements in GPU architecture specifically designed for transformer models. The attention operations leverage dedicated tensor cores in NVIDIA's Ampere and Hopper architectures, which provide specialized hardware acceleration for matrix multiplication and accumulation operations at various precisions. The fixed-index nature of the approach enables further optimization of these operations through persistent CUDA kernels that maintain tensor data in high-bandwidth on-chip memory (L3-L4 cache), reducing expensive global memory access operations during the attention computation phase.  
 
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