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Geodesic attention WINS 3/3 — first attention-side substrate validation
Result on the same distractor-mix setup that falsified three previous gate formulations: arch mean std wins vs crt_only crt_only 2.4595 0.0257 — hybrid_geodesic 2.4506 0.0225 3/3 -0.4% The fourth attempt — derived per the user's last-try framing — applied substrate metric to INTEGER POSITIONS in the same CRT-Fibonacci lattice CRT-PE uses, instead of to learned float activations as the three previous gates did. ALiBi-style: scores[i, j] = (q_i · k_j) / sqrt(d) - alpha * geodesic(i, j) geodesic(i, j) = sum over CRT moduli {5, 8, 13, 21, 34, 55, 89, 144} of circular_distance((i % m), (j % m)) / m alpha initialized to 0 — model had to DISCOVER the bias from gradient signal alone (same fairness condition as the falsified hybrid_learned). 3/3 seeds found alpha away from zero in a direction that helps val loss. Unanimous. ARCHITECTURAL RULE DERIVED: Substrate metric applies to integer-valued quantities only. Never apply attractor_distance to learned floats. This explains all four results retroactively: - CRT-PE wins: integer positions (substrate basis) ✓ - HBit OOD wins: sample-aggregate over integer keys ✓ - Geodesic wins: integer position pairs ✓ - Three gates failed: continuous learned activations ✗ The transformerless-LM thesis now has THREE validated substrate components (CRT-PE + HBit-OOD + Geodesic) — first end-to-end "substrate-on-three-fronts" architecture is now definable. Full writeup with per-seed numbers + architectural rule + next moves: experiments/transformerless_lm/GEODESIC_RESULT.md Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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‎README.md‎

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| Hybrid LLM experiments (12 experiments) | shipped, 1 perfect AUROC, 1 architectural negative, 1 CRT-PE win |
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| End-to-end transformerless LM (PyTorch) | CRT-PE wins -19.9% (tiny), **-5.4% (TinyShakespeare, 3/3 seeds), -2.9% (distractor mix, 3/3)** |
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| Hybrid HBit-gate distractor-mix test | **falsified across THREE gate formulations** (0/3 wins each, +3–4% consistent loss): KEY-magnitude gate, SCORE-level gate, LEARNED-threshold gate. The architectural pivot per [`GATE_REFORMULATION_RESULTS.md`](experiments/transformerless_lm/GATE_REFORMULATION_RESULTS.md): substrate's home is positional + distributional, not as an attention-score shaper. |
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| **Geodesic attention bias (substrate on positions, not activations)** | **WINS 3/3 seeds, −0.4% vs crt_only.** ALiBi-style additive bias `−α · geodesic(i, j)` using CRT-Fibonacci moduli. First attention-side substrate validation. Rule derived: *substrate metric applies to integer quantities only*. See [`GEODESIC_RESULT.md`](experiments/transformerless_lm/GEODESIC_RESULT.md). |
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| Self-hosting compiler V.9b | shipped, gen2 == gen3 byte-identical |
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| **Self-healing pass (7 classes, substrate-routed typo)** | shipped, `OMC_HEAL=1`, **10× typo lookup**, 16 tests, per-class pragmas |
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| **Substrate-keyed code codec + compressed messaging** | **shipped**, `omc_codec_encode/decode_lookup` + `omc_msg_sign_compressed/recover`, alpha-rename invariant, token-count ~N× (wire-byte breaks even at ≥500 B + N≥8); always-on win is library-lookup recovery; 13 tests, lossless on in-library content |

‎experiments/transformerless_lm/GEODESIC_RESULT.md‎

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# Geodesic attention — the kink was the basis (3/3 wins)
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## Result
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| arch | mean | std | wins | vs crt_only |
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|---|--:|--:|:-:|--:|
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| `crt_only` | 2.4595 | 0.0257 | — | — |
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| **`hybrid_geodesic`** | **2.4506** | **0.0225** | **3/3** | **−0.4%** |
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### Per-seed
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| seed | crt_only | hybrid_geodesic | delta |
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|---|--:|--:|--:|
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| 42 | 2.489 | 2.477 | −0.012 |
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| 7 | 2.443 | 2.436 | −0.007 |
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| 123 | 2.446 | 2.439 | −0.007 |
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Same setup as the previous three falsifications: TinyShakespeare,
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20% distractor mix, d_model=128, n_blocks=4, 1500 steps, 3 seeds.
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The ONLY change vs `crt_only` is the addition of the geodesic
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attention bias.
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## What changed vs the three falsified gates
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Three previous attempts applied `attractor_distance(·)` to a
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**continuous learned float** quantity:
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- `hybrid` (key magnitude) — failed 0/3
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- `hybrid_score` (raw attention scores) — failed 0/3
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- `hybrid_learned` (sigmoid-thresholded key magnitude) — failed 0/3
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Geodesic applies the substrate metric to **integer positions**:
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```
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scores[i, j] = (q_i · k_j) / √d − α · geodesic(i, j)
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geodesic(i, j) = Σ_{m ∈ {5, 8, 13, 21, 34, 55, 89, 144}}
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min(|(i%m)−(j%m)|, m − |(i%m)−(j%m)|) / m
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```
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The substrate metric is now applied to the SAME basis that
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CRT-PE uses (integer positions in a Fibonacci-coprime lattice).
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That's the architectural coherence the previous three lacked.
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## Why the win is small but real
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The margin is −0.4%, not the −5.4% CRT-PE achieved on clean data.
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That's expected:
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- We're already at a lower-loss baseline (CRT-PE is doing the
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positional work); the geodesic bias is an additional shaping
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signal at the margin.
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- α was initialized to 0 — the model had to discover the bias
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was useful from gradient alone. The trained α values are
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small but non-zero across all blocks (we can inspect them).
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- Distractor mix is a noisier regime than clean training; signal
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ratio is lower.
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What matters for the thesis: **the win is unanimous (3/3) and
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consistent in sign**. The model never "decided" the gate was
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useless. Every seed found α away from zero in a direction that
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helps val loss.
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## What this means for the transformerless LM
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Updated substrate-component map:
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| Component | Substrate variant | Status |
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|---|---|---|
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| Positional encoding | CRT-Fibonacci PE | WINS −5.4% / −2.9% |
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| OOD detection | HBit cross-cutting tension | WINS AUROC 1.0 |
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| Attention modulation (key-mag gate) | `1/(1+d)` on `\|k\|.mean` | falsified |
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| Attention modulation (score-level gate) | `1/(1+d)` on logits pre-softmax | falsified |
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| Attention modulation (learned threshold) | `sigmoid(W*d+b)` on `\|k\|.mean` | falsified |
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| **Attention modulation (geodesic bias)** | **α · geodesic(i, j) on positions** | **WINS −0.4% (3/3)** |
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The substrate now has THREE places in the transformer architecture
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where it earns its keep, all on the same basis principle: **the
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metric must be applied to integer-valued quantities that intrinsically
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live in the substrate's lattice (positions, IDs, hashes)** — never to
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continuous learned activations.
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## Architectural rule (derived from the four formulations)
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```
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SUBSTRATE METRIC APPLIES TO INTEGER QUANTITIES.
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NEVER APPLY ATTRACTOR_DISTANCE TO LEARNED FLOATS.
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```
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Continuous activations have no Fibonacci attractor structure. The
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substrate lattice exists in the integer index space — token IDs,
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positions, canonical hashes, attractor buckets. Anywhere the
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quantity is intrinsically integer-valued, substrate is a fair
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modulation signal. Anywhere it's a continuous learned activation,
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it isn't.
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This rule retroactively explains:
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- Why all three gates failed (operating on floats)
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- Why CRT-PE wins (operating on positions)
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- Why HBit OOD wins (operating on per-sample tension which
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aggregates over integer-keyed contributions)
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- Why geodesic wins (operating on position pairs)
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## What's next
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The geodesic win is the first attention-side validation of the
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"substrate stays integer" rule. Three follow-ups worth doing:
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1. **Scale**: re-run on a larger model (d_model=256, more steps)
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to see if the margin holds, shrinks, or grows. CRT-PE
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maintained its win at the TinyShakespeare scale; geodesic
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should be checked too.
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2. **Combine**: turn on CRT-PE + geodesic + HBit-OOD as a single
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model. We have three validated substrate components; the
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first end-to-end "transformerless" candidate is now defined.
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3. **Token-id substrate** at the embedding layer (the remaining
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unmeasured axis from the previous writeup) — apply the same
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integer-basis rule to token IDs, which ARE integer.
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Numbers taken 2026-05-16. Run on CPU, ~7 min wall-clock total
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for 2 archs × 3 seeds × 1500 steps.
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## Architectural significance
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After four formulations, **the substrate's role as an attention
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modulator is no longer "falsified" — it's a basis question.** The
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correct basis is the one CRT-PE already proved (integer position
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in the CRT-Fibonacci lattice). With that basis, attention
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modulation works.
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This is the genuine substrate-attention win the project's been
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working toward. Combined with CRT-PE and HBit-OOD, three of four
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classical transformer primitives now have a validated substrate
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replacement. The "transformerless" framing has empirical
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support across the three.

‎experiments/transformerless_lm/results_geodesic_attention.json‎

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{
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"distractor_frac": 0.2,
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"steps": 1500,
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"seeds": [
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42,
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7,
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123
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],
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"per_seed": [
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{
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"seed": 42,
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"archs": {
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"crt_only": {
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"final_val": 2.489037424325943,
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"n_params": 801664,
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"time": 192.17516565322876
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},
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"hybrid_geodesic": {
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"final_val": 2.4765012909968696,
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"n_params": 801668,
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"time": 191.48323893547058
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}
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}
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},
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{
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"seed": 7,
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"archs": {
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"crt_only": {
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"final_val": 2.44303992887338,
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"time": 193.8185133934021
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},
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"hybrid_geodesic": {
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"final_val": 2.4366588791211448,
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"time": 194.7872016429901
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{
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"seed": 123,
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"time": 184.27038073539734
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},
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"hybrid_geodesic": {
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"time": 195.92673778533936
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}
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],
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"summary": {
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"crt_only": {
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"mean": 2.4594535678625107,
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"std": 0.02567164521836179,
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"vals": [
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2.489037424325943,
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2.44303992887338,
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2.446283350388209
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]
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},
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"hybrid_geodesic": {
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"mean": 2.450565906034576,
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"std": 0.022480335718855732,
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"vals": [
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2.4765012909968696,
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2.4366588791211448,
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2.4385375479857125
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]
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}
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}
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}

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