CT-Optimal Hardware
ASIC Design for Hodge-Decomposition Neural Networks
Level 3 (includes competing scaffolds). This seed analyzes the mismatch between C-Former's mathematical structure and current GPU architecture. The binder is: the computational scaffold should match the mathematical scaffold (Element I). Competing patterns: GPU transformers, Cerebras wafer-scale, Graphcore IPU mesh, Google TPU systolic arrays.
Irreducible leakage (A9): this analysis derives what the hardware SHOULD be. Whether fabrication cost, yield, and ecosystem support make it viable is a separate question requiring empirical input from semiconductor engineering.
The TD6 tile works and beats standard transformers. But GPUs are not the ideal scaffold for this architecture — C-Former training is 14x slower despite having 40% fewer parameters. The reason: GPUs are optimized for dense matrix multiply (standard transformers). C-Former's operations are sparse, structured, and graph-local. The mismatch IS the 14x slowdown.
The Hodge decomposition is fixed linear algebra on a 13-node, 24-edge graph. The three projectors (gradient, cycle, boundary) are constant matrices derived from the tile topology. They never change during training. Multi-tile chains have regular, predictable communication patterns: each tile only talks to its neighbors. This is the opposite of what GPUs are designed for.
From Element I (scaffold stability): the computational scaffold should match the mathematical scaffold. When there is a mismatch, you pay B_th for the translation overhead at every operation. The 14x slowdown is the cost of running graph-local sparse operations on hardware designed for dense global operations.
GPU vs CT-Optimal: The Scaffold Mismatch
| DIMENSION | GPU (CURRENT) | CT-OPTIMAL ASIC | CT SOURCE |
|---|---|---|---|
| Compute unit | CUDA core: scalar multiply-accumulate | TD6 tile unit: 13 nodes, 24 edges, 3 hardwired projectors in registers | Element I, Hodge |
| Memory hierarchy | Global VRAM + shared memory + registers (designed for large tensors) | Tile-local state in registers + boundary exchange buffer with neighbors | A4, B4 |
| Communication | All-to-all via NVLink/PCIe (token-parallel) | Mesh topology: each tile talks only to left/right neighbors (tile-parallel) | A4, Polycrystalline |
| Data flow | Token-parallel: same operation on all tokens simultaneously | Tile-parallel: each tile processes its own Hodge decomposition independently | B4, Element I |
| Projectors | Computed dynamically via attention weights | Hardwired constant matrices derived from tile topology (never change) | Hodge, TD6 |
| Sparsity | Dense matrix multiply (wasted FLOPs on zero entries) | Structured sparsity: only 24 edges in the graph are non-zero | A4, B_th |
| Batch strategy | Large batches to amortize kernel launch overhead | Small batches sufficient: no kernel overhead when operations are hardwired | B_th, B_cx |
| Power profile | High power (300-700W) for massive parallel FP ops | Low power: sparse structured ops, most silicon idle at any moment | A7, B_th |
The 14x training slowdown despite 40% fewer parameters is the direct consequence of this scaffold mismatch. Every C-Former operation pays a B_th tax for translating between graph-local sparse structure and dense GPU tensor cores.
Research Questions
Q1: What is the ideal compute unit for a TD6 tile?
13 nodes, 24 edges, 3 fixed projectors. The entire tile state fits in registers. The Hodge projectors are constant 24x24 matrices. A single tile's forward pass is: (1) project edge flow onto three subspaces, (2) apply learned mixing weights, (3) reconstruct. This is three matrix-vector multiplies with constant matrices plus a learned linear combination.
Q2: What memory hierarchy matches multi-tile chains?
Multi-tile chains have a specific communication pattern: each tile maintains local state and exchanges boundary information with its immediate neighbors. This is a 1D mesh topology — the exact opposite of all-to-all attention. The memory hierarchy should be: tile-local registers (fast, small) + neighbor exchange buffer (medium) + chain-level aggregation (slow, rare).
Q3: What does the data flow look like?
GPU transformers are token-parallel: the same attention operation runs on all tokens simultaneously. C-Former should be tile-parallel: each tile in the chain runs its own Hodge decomposition independently, then exchanges boundary information. The parallelism axis is tiles, not tokens.
Q4: Can the Hodge projectors be hardwired?
Yes. The three projectors are derived from the tile topology and never change. They are constant matrices: , , . On a GPU, these are stored in memory and loaded every forward pass. On custom silicon, they can be baked into the circuit — zero memory access, zero latency. This alone eliminates the dominant B_th cost of the current implementation.
Q5: What throughput improvement over GPU is theoretically achievable?
The 14x training slowdown on GPU suggests at least 14x headroom from scaffold alignment alone. But the improvement could be much larger: hardwired projectors eliminate memory bandwidth bottleneck, tile-local computation eliminates communication overhead, structured sparsity eliminates wasted FLOPs on zero entries. Conservative estimate: 14–50x. Optimistic: 100x+.
Q6: What existing ASIC architectures are closest?
Three candidates share structural features with CT-optimal hardware:
Q7: Is there an FPGA prototype path?
A single TD6 tile (13 nodes, 24 edges, three 24x24 constant projectors) fits trivially on any modern FPGA. A multi-tile chain of 8–16 tiles fits on a mid-range FPGA board ($500–$2000). This would provide the first empirical measurement of CT-optimal hardware performance without fabrication cost.
Q8: Implications for edge/mobile deployment
C-Former already achieves competitive results with 40% fewer parameters (2.2M vs 3.8M). On CT-optimal silicon, the combination of fewer parameters + hardwired projectors + tile-local computation could make C-Former viable on edge devices where standard transformers cannot run. The structured sparsity means power consumption scales with actual computation, not silicon area.