HELIX-Ω

Dense neural projections store and apply a full matrix. HELIX-Ω asks whether some of those matrices can be replaced by compositions of much smaller transforms along tensor axes, reducing physical parameters, estimated arithmetic, and memory on edge-class hardware while retaining competitive validation quality.

The program treats that as a hypothesis, not a promised breakthrough. Physical parameters, active parameters, dense operator dimensionality, estimated FLOPs, wall time, throughput, peak memory, validation loss, and perplexity are measured separately so an inexpensive operator is never mistaken for an equally capable model.

The first primitive reshapes each token vector into a small tensor and applies learned matrices one axis at a time: Y = X ×₁ A₁ ×₂ A₂ … ×ₘ Aₘ. It never constructs the implied giant dense matrix. Attention stays dense; only the three gated-MLP projections change, isolating the structured operator from unrelated architectural changes.

The first rescue added an identity-initialized learned diagonal after each structured projection. That adds only O(d) parameters and starts functionally identical to the pure TensorAxis arm, making it a narrow test of whether coordinate conditioning repairs the quality gap.

The dense control and structured arms shared width, depth, attention, tokenizer, token stream, optimizer policy, precision, sequence length, validation corpus, and hardware. The first real-data pilot used three order-swapped seed pairs. The rescue used two order-reversed dense / pure / diagonal triads with hash-pinned shared initialization: all non-MLP tensors matched exactly, pure and diagonal factors matched exactly, and their initial logits were byte-identical.

Every optimizer step printed timestamps, loss, train PPL, validation freshness, learning rate, gradient norm, elapsed time, ETA, tokens per second, memory, checkpoint state, and structural accounting to both the terminal and durable logs. Final checkpoints were reloaded and evaluated over the entire 217,645-target validation stream.

The pure TensorAxis MLP saved 22.37% of physical parameters, 20.68% of estimated FLOPs per token, and 32.19% of mean peak MLX memory in the three-seed pilot. But mean validation loss was 5.72141 versus 5.56445 for dense, and the structured arm lost the validation-loss-versus-wall-clock comparison in every seed.

The matched diagonal rescue preserved similar resource savings but recovered only 0.025% of the pure-to-dense loss gap on average. Its compute throughput was 10.40% below pure TensorAxis. The registered stop rule therefore fired: this one-stage branch will not be scaled or optimized with custom kernels.

The experiment separated an operator that is cheaper from an operator that is better. The structured math demonstrably reduced storage, arithmetic estimates, and memory, but the saved resources did not compensate for the quality penalty. Matching initialization also showed the original loss gap was not merely an unlucky random draw.

HELIX-Ω remains a research platform, but the tested single-stage pure and diagonal TensorAxis variants are closed. Any future direction must begin with a materially different, preregistered hypothesis rather than silently adding complexity until a favorable result appears.