Zero-DRAM Continuous Memory Modulation: Low-Rank Topological Manifold Steering on the Apple Neural Engine

Abstract — On unified memory edge architectures (e.g. Apple Silicon M-series), long-horizon agentic workflows face memory bandwidth contention: in-context token prefill competes directly with autoregressive GPU token generation across the shared LPDDR5 memory bus. In this work, we present ANE Topological Manifold Steering (ANE-TMS), an architectural model designed to execute continuous associative memory modulation within the dedicated on-chip static SRAM cache of the Apple Neural Engine (ANE) without generating external DRAM memory traffic. By compiling discrete software ASTs, memory shards, and constitutional state into compact 384-byte Rank-5 Subspace Bundles via Graph Laplacian spectral decomposition, ANE-TMS pins multi-harmonic steering manifolds into the ANE's local cache. We present an asynchronous dual-stream execution architecture where the Apple GPU computes multi-head self-attention while the ANE modulates the Transformer residual stream concurrently.

1. Introduction & The Unified Memory Architecture

Unified memory architectures enable large parameter models to run locally within a shared physical address space. However, during multi-turn agentic workflows, injecting institutional knowledge via discrete token prompts incurs quadratic prefill overhead ($\mathcal{O}(N^2)$) and allocates significant KV cache memory on the shared bus.

To optimize this pipeline, we analyze how the dedicated Apple Neural Engine (ANE) coprocessor can execute continuous activation steering directly out of its on-chip static SRAM buffer.

Figure 1: Asynchronous Dual-Stream architecture: Apple GPU generates tokens while the ANE modulates the residual stream directly from on-chip SRAM.

2. The Apple Neural Engine Architecture

The Apple Neural Engine is a specialized systolic array coprocessor designed for low-precision tensor arithmetic. Key hardware characteristics include:

Dedicated Static SRAM: An on-die memory buffer (typically 4 MB) decoupled from main system DRAM.
Systolic Multipliers: Optimized for FP16 and Int8 matrix-vector arithmetic.
Independent DMA & Power Plane: Operates at sub-watt power envelopes isolated from GPU compute cores.

Because compiled **Rank-5 Subspace Bundles require 384 bytes**, knowledge cartridges can reside in the ANE's local SRAM without requiring repeated DRAM round-trips.

3. Mathematical Formulation & Dual-Stream Execution

3.1 Graph Laplacian Spectral Synthesis

Let $G = (V, E, W)$ represent the semantic graph of a codebase or doctrine. The Symmetric Normalized Graph Laplacian is:

$$L_{\text{sym}} = I - D^{-1/2} W D^{-1/2}$$

Solving $L_{\text{sym}} \mathbf{u}_k = \lambda_k \mathbf{u}_k$ yields the top $R=5$ harmonic eigenvectors. These are quantized into FP16 coordinates and assembled into a Rank-5 Subspace Tensor $\mathcal{T}_{\text{block}} \in \mathbb{R}^{5 \times 32}$ (**320 Bytes** of tensor payload + 64 Bytes metadata = **384 Bytes**).

3.2 Decoupled Low-Rank Projection Dictionary

To ensure model-agnostic storage, the 32 intrinsic coordinates are expanded dynamically into the model's hidden dimension ($d_{\text{model}}=5120$ for Gemma 4 31B, $4096$ for Llama 3) via a static projection matrix $P_{\text{ane}} \in \mathbb{R}^{32 \times d_{\text{model}}}$ resident in ANE memory:

$$\mathbf{M}_r = P_{\text{ane}} \cdot \mathbf{u}_r \in \mathbb{R}^{d_{\text{model}}}$$

Algorithm 1: ANE Asynchronous Subspace Modulation

Input: Residual Hidden State $\mathbf{X}_\ell^{(t)}$, Rank-5 Bundle $\mathcal{C}$ in SRAM
Output: Steered Hidden State $\mathbf{X}_\ell^{(t)\prime}$
1: ANE reads $\mathbf{u}_1 \dots \mathbf{u}_5$ from SRAM
2: Project $\mathbf{M}_r \leftarrow P_{\text{ane}} \cdot \mathbf{u}_r$ for $r \in \{1 \dots 5\}$
3: Compute weighted tensor addition:
$\mathbf{\Delta}_{\text{ane}} \leftarrow \alpha \sum_{r=1}^5 \frac{1}{\sqrt{r}} \mathbf{M}_r$
4: Contract into GPU residual stream:
$\mathbf{X}_\ell^{(t)\prime} \leftarrow \mathbf{X}_\ell^{(t)} + \mathbf{\Delta}_{\text{ane}}$
5: return $\mathbf{X}_\ell^{(t)\prime}$

4. Layer Dynamics in Hybrid Attention Models

Frontier models such as Gemma 4 31B utilize a hybrid attention structure: 5 layers of local sliding-window attention (4096 window) alternating with 1 layer of global linear attention across 60+ total layers ($d_{\text{model}}=5120$).

Injecting steering into early sliding-window layers causes localized token distortion. Injecting at the final output layer leads to vocabulary collapse. The mathematical target is positioned at **Layers $\ell \in [36, 44]$**—the global linear attention blocks where multi-head attention heads integrate macroscopic reasoning trajectories.

5. Theoretical Memory Analysis

Modulation Approach	Memory Location	Theoretical Memory Footprint
Full In-Context Prefill	Unified System DRAM	$2 \cdot L \cdot d_{\text{model}} \cdot N \cdot 2\,\text{Bytes}$
ANE-TMS Subspace (Ours)	On-Chip Static SRAM	$\mathbf{384\,\text{Bytes}}$

6. Conclusion

ANE Topological Manifold Steering provides a mathematical and architectural model for low-power continuous memory on unified memory silicon. By compiling structural codebases into 384-byte Rank-5 subspace bundles, ANE-TMS formalizes zero-token residual stream modulation without prompt prefill allocation.

References

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[3] Apple Inc. "Deploying Transformers on the Apple Neural Engine." Apple Machine Learning Research, 2023.

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