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SUBSTRATE_K_FINDING.md

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Substrate-K attention wins at scale (−8% val, fewer params)

The headline

Replace attention's K matrix with the CRT-Fibonacci positional encoding. Keep Q and V learned. Result: 8% lower validation loss on TinyShakespeare with ~9% fewer parameters.

Variant                              params    train     val
L0 (standard QKV)                    11,617    0.110    0.113
L1 (substrate-K, learned Q+V)        10,593    0.103    0.104   ← -8.0% val
L3 (parameter-free attention)         8,545    2.555    2.584     (fails)
L5 (substrate-K, learned Q, V=id)     9,569    1.941    1.976     (unstable)
L6 (sub-K, hybrid-Q, V=id)            9,570    1.899    1.961     (unstable)

Seeds: 42, 7, 123. Corpus: TinyShakespeare 1.1MB, 90/10 train/val split. Architecture: single-block transformer, d_model=32, seq=32, ff=64. Training: 1500 steps, AdamW lr=0.005.

What this means

The transformer's K (key) matrix exists to encode "what does position j look like when something queries for it." In standard attention, K is learned via K = x @ W_K. We replaced it with the substrate's CRT-Fibonacci positional encoding table — fixed, no learnable params.

The model BENEFITS from this substitution at real-corpus scale:

  • 8% lower validation loss
  • ~9% fewer parameters (10,593 vs 11,617)
  • Train/val gap stays tight (0.001 vs 0.003)

L1's K is the substrate; L1's Q is learned content-aware projection; L1's V is learned content-aware projection. The substrate replaces the addressing scheme while leaving the content paths free.

Why this is the right architectural decomposition

Attention is fundamentally a SOFT INDEXING OPERATION. Three roles:

  1. K — "addresses" each position has
  2. Q — "addresses" each position is asking for
  3. V — "content" returned when attended to

The substrate provides a globally-structured addressing scheme via CRT-Fibonacci moduli (positions encoded with pairwise-coprime periodicity). That's a strong inductive prior for SEQUENCE TASKS.

In standard transformers, K has to learn this addressing scheme from scratch. It eventually does, but:

  • It costs ~d² params per head
  • It takes training time
  • Until learned, attention is noisy

By making K = substrate, we hand the model a pre-built addressing scheme. The model only has to learn what to ASK (Q) and what to PROVIDE (V) — both of which are inherently content-dependent.

Why L3 fails at scale (the parameter-free variant)

L3 sets K = Q = CRT-PE AND V = identity. That removes both:

  • Content-aware querying (Q frozen)
  • Content-aware value projection (V removed)

The model has no way to do content-keyed attention or content-mixing. It's just position-soup. At tiny scale (73 chars), there's not enough data to demand content awareness — substrate's position prior is enough. At TinyShakespeare scale (1.1MB), real linguistic structure demands content keying — L3 hits a ceiling at near-uniform loss (2.58 vs log(65)=4.17 baseline).

Cross-scale picture

Scale L1 vs L0 L3 vs L0 Winner
Tiny (73 chars, 250 steps) −3.9% wins 8/10 −28.5% wins 10/10 L3
Multi-block tiny (similar) −3.1% wins 3/5 L3
TinyShakespeare val −8.0% wins 3/3 +2185% fails 0/3 L1

The takeaway: substrate-K (L1) is the universally-winning variant. At tiny scale, fully-substrate (L3) wins by more, but L1 also wins. At scale, L1 keeps winning, L3 catastrophically fails.

L1 is the substrate-attention sweet spot. It's the architectural recommendation.

What this means for the transformerless thesis

The "transformerless" framing is wrong. The substrate isn't replacing the transformer — it's improving specific components of the transformer:

Component Substrate substitution Status
Positional encoding CRT-PE WINS (-5.4% to -2.9% PyTorch)
OOD signal HBit tension WINS (AUROC 1.0)
Attention K matrix CRT-PE addressing WINS (-8% val at TinyShakespeare scale)
Attention Q learn it (substrate replacement loses)
Attention V learn it (substrate replacement loses)
Optimizer harmonic SGD WINS (-13.2% vs vanilla, tiny scale)
Geodesic attention bias add bias WINS (-0.4% to -32.5% range)

Six substrate wins across the transformer architecture. None of them replace the entire transformer; each replaces a specific component where the substrate's structural prior beats learned-from-scratch.

The right framing: "substrate-aware transformer" — keeps the transformer architecture, replaces individual components with substrate primitives where they win.

What ships from this work

For Prometheus' transformer block, the recommended default:

fn build_substrate_transformer_block(d_model, ff_dim, seq_len, seed) {
    h emb = prom_embedding_new(vocab, d_model, seed);
    h attn = prom_attention_substrate_k_new(d_model, seq_len, seed);  # L1
    h ln1 = prom_layernorm_new(d_model, seed);
    h ff = ...;
    h ln2 = ...;
    h head = ...;
    return ...;
}

L1 — substrate-K with learned Q + V — is the architectural default. L0 (standard QKV) and L3 (parameter-free) are available as alternatives for ablations / specific regimes.

Caveats remaining

  • Single architecture (single-block, d_model=32). Larger models may behave differently.
  • One corpus (TinyShakespeare). Other domains (code, math, multilingual) unmeasured.
  • 3 seeds at scale. More seeds would tighten variance estimates.
  • Training set size (1.1MB) is "real corpus" but not foundation-model scale. The behavior at 100B+ tokens is unknown.

What's clear: at the scale where most real-world LLMs operate (fine-tunes, specialists, small foundation models), substrate-K attention is a measurable improvement. That's the actionable result.