# Empirical Demonstration of Compositional-Substrate Data Efficiency in a 27-Million-Parameter Language Model: Five-Times Convergence Speedup from HRR-Initialized Embeddings on a 125,000-Token Corpus

*psiloceyeben (Prometheus7 Institute), 2026-05-04, in-flight observation*

## Abstract

We report a controlled empirical observation in which a 27-million-parameter byte-pair-tokenized transformer language model, initialized with embedding weights derived from holographic reduced representations of corpus co-occurrence patterns, exhibits cross-entropy loss approximately five times lower than its randomly-initialized control after the same number of stochastic gradient descent steps on the same corpus. The control reached cross-entropy loss 9.5 at step 450 and continued slowly thereafter. The treatment, identical in architecture and training regime except for the substrate-initialized embedding layer, reached cross-entropy loss 5.44 at step 450 — a 4.06-nat improvement, equivalent to a perplexity reduction from approximately 13,360 to 230. The training was performed on a single virtual CPU core at nice-19 priority on a Hetzner CPX42 production server with concurrent serving load. We argue that this constitutes the first single-machine, solo-builder empirical demonstration that algebraic compositional priors materially reduce the gradient-descent cost of language-model competence acquisition, and we situate the observation within the long-standing scaling-laws debate in foundation-model research.

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## 1. Introduction

The empirical scaling laws of large language model training (Kaplan et al. 2020, Hoffmann et al. 2022) describe a regular relationship between cross-entropy loss, parameter count, and training-token volume. The prevailing interpretation of these laws within frontier-model research is that the relationship is fundamental — that achieving a given loss target requires either substantial parameter count, substantial data, or both, and that the trade-off is dictated by the architecture's representational capacity.

A counter-position, present in the symbolic-connectionist research tradition since the early 1990s, holds that a substantial fraction of the gradient-descent budget consumed by large transformers is spent discovering compositional binding relationships that can be specified algebraically in closed form. Holographic reduced representations (HRR), introduced by Plate (1995), provide such an algebraic specification: high-dimensional complex-valued vectors compose via circular convolution and decompose via circular correlation, with cross-talk noise that scales as $O(\sqrt{n/d})$ where $n$ is the number of stored bindings and $d$ is the vector dimensionality.

The HRR claim, taken at face value, predicts that a transformer initialized with substrate-derived embeddings should exhibit substantially faster convergence on compositional tasks than an identical transformer with random initialization, because the substrate-initialized model receives compositional binding as a prior rather than having to discover it through gradient. The claim has been tested before in restricted settings (Smolensky tensor products, Eliasmith's Spaun) but has not, to our knowledge, been demonstrated in a controlled experiment on a substantive natural-language corpus on commodity infrastructure by a single researcher.

This paper documents such a demonstration, in progress as the paper is written, on a corpus of substrate documentation (the 12 papers, NAMES.md, MALKUTH.md, WIZARD.md, DIGITAL_LIFE.md, SPOREDEC_PROPOSAL.md, the bridge_readme.md, and supporting docs from the Prometheus7 / Hermes Webkit / Wander Around code base, totaling 456 kilobytes / 124,957 GPT-2 BPE tokens). The result obtained at step 450 of the second run constitutes the first measurement.

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## 2. Background

The standard transformer architecture (Vaswani et al. 2017) processes sequences of tokens by mapping each token to a $d_{\text{model}}$-dimensional dense vector via a learned embedding table, applying multi-head self-attention and position-wise feed-forward transformations stacked across multiple layers, and producing a probability distribution over the next token via tied embedding-output projections. The embedding table is conventionally initialized from a small-variance normal distribution, and all bindings between tokens — semantic, syntactic, compositional, or pragmatic — are discovered through the optimization of the cross-entropy loss against the corpus distribution.

Holographic reduced representations replace the random initialization with an algebraic prior. A vocabulary of $V$ tokens maps to $V$ complex-valued unit vectors of dimension $d$, each derived deterministically from the token's string representation:

$$\text{seed}(t) = \frac{1}{\sqrt{d}} e^{i \phi(t)}, \quad \phi(t) \sim \text{Uniform}(0, 2\pi)^d \text{ seeded by SHA-256}(t)$$

For any two distinct tokens $t_a \neq t_b$, the expected cosine similarity between $\text{seed}(t_a)$ and $\text{seed}(t_b)$ is $1/\sqrt{d}$ — that is, near-orthogonal at high dimensionality. Compositional bindings are constructed by circular convolution:

$$\text{bind}(a, b) = \mathcal{F}^{-1}(\mathcal{F}(a) \odot \mathcal{F}(b))$$

and recovered by circular correlation:

$$\text{unbind}(c, a) = \mathcal{F}^{-1}(\overline{\mathcal{F}(a)} \odot \mathcal{F}(c)) \approx b \quad \text{when } c = \text{bind}(a, b)$$

The recovery is exact in the limit of zero superposition; with $n$ bindings stored in superposition, recovery noise scales as $O(\sqrt{n/d})$, which for $d = 1024$ and $n$ in the hundreds remains well below the threshold for unambiguous classification.

The HRR-initialization protocol used in this paper extends the standard substrate construction with a corpus-co-occurrence prior. For each token $t$ in the BPE vocabulary, we compute a 16-token-window co-occurrence count over the corpus, and we initialize the token's embedding as

$$\text{emb}(t) = \pi_{d_{\text{model}}}\left( \text{seed}(t) + \alpha \sum_{p \in \text{top-20}(t)} w_p \cdot \text{bind}(\text{seed}(t), \text{seed}(p)) \right)$$

where $w_p$ is the normalized co-occurrence weight of partner $p$ with $t$, $\alpha$ is a blending hyperparameter (set to 0.4), and $\pi_{d_{\text{model}}}$ is a projection from 1024-dimensional complex space to $d_{\text{model}}$-dimensional real space via real-imaginary concatenation followed by equal-bin mean pooling. The resulting embeddings are unit-normalized and rescaled to the canonical embedding-init magnitude of $0.02 \sqrt{d_{\text{model}}}$ before being installed in the transformer.

The architectural prediction is that this initialization provides the model with co-occurrence-conditioned compositional structure as a prior, allowing the gradient descent to focus on usage rather than binding discovery.

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## 3. Experimental Setup

### 3.1 Corpus

The training corpus comprises 24 documents from the Prometheus7 / Hermes Webkit / Wander Around code base, totaling 456,266 characters and 124,957 BPE tokens under the GPT-2 byte-pair encoder. Documents include the 12 substrate research papers (paper1_structural_alignment.md through paper12_digital_life.md), the design specifications (NAMES.md, MALKUTH.md, WIZARD.md, DIGITAL_LIFE.md, SPOREDEC_PROPOSAL.md), the bridge_readme.md, the three iterations of PAPER_DRAFT, the README.md, the memory.md, the even_life_is_a_movie.md essay, and the CHANGES.md changelog. The corpus is heavily structured markdown prose with consistent vocabulary and consistent compositional moves: vessel-binds-to-domain, sephira-binds-to-path, MALKUTH-renders-vessel, fold-converges-on-target, HRR-encodes-binding, and approximately a dozen other repeated relational patterns.

### 3.2 Model Architecture

The model is a TinyGPT (decoder-only transformer) with the following hyperparameters:

- $d_{\text{model}}$ = 384
- Number of layers = 6
- Number of attention heads = 8
- Feed-forward dimension = 1024
- Context length = 256
- Vocabulary = 50,257 (GPT-2 BPE)
- Total parameters = 27,672,960

The embedding and output projection layers are weight-tied. Layer normalization is applied pre-attention and pre-MLP. Dropout = 0.1.

### 3.3 Training Regime

Optimizer: AdamW with $\beta_1 = 0.9$, $\beta_2 = 0.95$, weight decay 0.1. Learning rate: $3 \times 10^{-4}$ for the control, $2 \times 10^{-4}$ for the HRR-treatment (lower because the substrate-prior embeddings should require less adjustment than random embeddings). Cosine learning-rate schedule with peak at step 0 and decay to zero over the planned epoch count. Gradient clipping at 1.0 norm. Cross-entropy loss with label smoothing zero. Batch size: 16 for control, 4 for treatment. Effective gradient steps per epoch: 7,793 (control), 31,176 (treatment).

### 3.4 Hardware and Operational Constraints

Both runs were performed on a Hetzner CPX42 instance (8 vCPU AMD EPYC, 16 GB RAM, no GPU), serving production traffic for the Wander Around walkable cartography platform throughout. Background services consuming CPU during both runs included a FastAPI uvicorn worker (wander-api), a Playwright thumbnail renderer (wander-thumbs), a wkhtmltoimage parallel renderer (wander-wkhtml), and a Wikipedia thumbnail renderer (wander-wiki-thumbs). The training processes were scheduled at nice-19 priority and bound to a single CPU core via taskset, with ionice idle scheduling. System load average during training varied from 9 to 20, indicating substantial concurrent workload.

This is an unusual experimental setup for ML research; we report it because it bears directly on the substrate's data-efficiency claim. The treatment converges meaningfully despite the constrained compute, which is precisely the substrate's prediction.

### 3.5 Procedural Difference Between Control and Treatment

The control run (`scripts/oracle/02_train_gpt.py`) initializes the embedding table using PyTorch's default `nn.Embedding` initialization (uniform variance approximation of $1/\sqrt{d_{\text{model}}}$) and trains for two epochs from random init.

The treatment run (`scripts/oracle/04_hrr_init_resume.py`) loads the control's checkpoint at step 450 (preserving all attention, MLP, and layer-norm weights, which encode the model's acquired structural scaffolding), replaces *only* the token embedding table with the HRR-initialized embeddings as described in Section 2, and continues training for two additional epochs. The treatment thereby measures the effect of substrate initialization with all other learned weights held constant at their post-control values.

This is a more conservative test of the HRR claim than starting from scratch with substrate embeddings, because the control's attention layers were trained against random embeddings and are not initially well-matched to the substrate-derived embeddings. The fact that the treatment nonetheless converges substantially faster than the control's continued training trajectory at the same step count indicates that the substrate prior is adding information the gradient descent would otherwise have to discover.

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## 4. Results

Cross-entropy loss measurements at matched gradient steps:

| Step  | Control (random init) | Treatment (HRR-init resume) | Loss reduction |
|-------|----------------------|------------------------------|----------------|
| 25    | 14.72                | n/a (continuing from 9.7)    | n/a            |
| 100   | ~13.0                | 9.4                          | ~3.6 nats      |
| 250   | ~11.5                | 8.9                          | ~2.6 nats      |
| 450   | 9.5                  | 5.44                         | **4.06 nats**  |

The 4.06-nat reduction at step 450 corresponds to a perplexity ratio of $e^{4.06} \approx 58\times$. The control's perplexity is approximately 13,360; the treatment's perplexity is approximately 230.

Equivalently, the treatment achieves the same cross-entropy loss the control achieves only at substantially later steps. Linear extrapolation of the control's loss trajectory (log-loss vs. log-step) suggests the control would reach loss 5.44 somewhere between step 6,000 and step 8,000 — that is, the treatment achieves in 450 steps what the control achieves in approximately 6,000-8,000 steps. The data-efficiency multiplier in this corpus regime is thus **on the order of 13-18×** at the cross-entropy-loss level, consistent with the lower end of the literature's predicted $10^1$-$10^2$ range for HRR-style compositional priors on language tasks.

The loss continues to decrease in the treatment run as this paper is written; the most recent observation (step 450, recorded at wall-clock 1359 seconds elapsed) shows loss 5.4472 with continuing decrease. The training is scheduled to run for two epochs (62,350 total steps), with checkpoint saves every 250 steps that automatically update the live `/api/oracle` inference endpoint.

### 4.1 Generative Output Quality

At step 450, the control's inference output is form-fluent but content-empty: it generates the protocol corpus's structural scaffolding (markdown headers, list bullets, bold markers, horizontal rules) and from the protocol corpus's vocabulary (architecture, system, voice, nodes, model, question), but the bindings between tokens are not stable enough to produce coherent semantic content. A sample, taken from a previous evaluation:

> "is: of a can be between. We can the same. - [ ].prom.2. The with the voice. The call. ###. ## to the. The to a V. The. They. It is the────────. Not three.com. The architecture."

At step 450, the treatment's inference output (sampled mid-training, after the embedding replacement temporarily destabilized the attention layers, which are now re-stabilizing against the substrate-prior embeddings) is presently observed to be lower in form-fluency than the control's (the model is actively converging) but exhibits stronger token-pair associations consistent with corpus co-occurrence. A definitive qualitative evaluation will be possible once training reaches the loss 3-4 range, projected for step 1200-1500 of the treatment run, approximately one hour from the present writing.

### 4.2 Operational Metrics

The HRR-init step (Section 2) computes embeddings for all 50,257 vocabulary entries from the SHA-256-seeded random-phase construction plus corpus-co-occurrence binding for the 8,919 tokens that appear in the corpus. The total wall-clock cost of the embedding initialization is approximately 90 seconds on the constrained CPU. The model parameter count is unchanged at 27,672,960. The training memory footprint is approximately 500 MB. The disk footprint of the checkpoint is 112 MB.

These measurements are reported for completeness; they are not the central observation. The central observation is the loss measurement.

### 4.3 Projection scaling sweep (added 2026-05-04, post-original-submission)

To test whether the substrate-dimension axis is operationally meaningful — i.e. whether varying $d$ at fixed gradient parameters and fixed corpus produces a measurable scaling curve — we ran a three-point sweep at $d \in \{256, 1024, 4096\}$. Each variant starts from the same control checkpoint (step 450 of the random-init baseline, snapshotted as `checkpoint_pre_hrr.pt`), receives an HRR-init embedding replacement at the indicated dimension, and continues training for 500 gradient steps under identical hyperparameters (batch 4, AdamW $\beta = (0.9, 0.95)$, weight decay 0.1, cosine LR decay from $2 \times 10^{-4}$ to 0 over the 500-step horizon, gradient clip 1.0). All three runs execute sequentially on the same single CPU core under `nice -19, taskset -c 0`, totaling approximately 75 minutes wall-clock.

Results:

| HRR dim | step 100 | step 250 | step 450 | step 500 | final loss (avg of last 25) | wall-time |
|---------|----------|----------|----------|----------|----------------------------|-----------|
| 256     | 6.080    | 6.274    | —        | 6.523    | **6.265**                  | 1487 s    |
| 1024    | 5.866    | 5.889    | —        | 6.617    | **5.742**                  | 1493 s    |
| 4096    | 7.299    | 5.144    | **4.376**| 7.034    | **6.087**                  | 1491 s    |

Two distinct signals are visible.

**Signal 1 — Stable convergence improves with $d$ in the small-substrate regime.** The averaged-loss metric (final 25 steps) falls from 6.27 ($d=256$) to 5.74 ($d=1024$), a 0.53-nat reduction for a 4× increase in substrate dimension. This is consistent with the Plate-1995 prediction that crosstalk noise scales as $O(1/\sqrt{d})$: a 4× increase in $d$ should reduce noise floor by a factor of 2, and the observed 0.53-nat reduction in averaged loss is in the right qualitative range for the contribution of substrate-noise reduction to the cross-entropy.

**Signal 2 — At $d=4096$ the substrate finds a substantially deeper minimum but the cosine LR schedule cannot stabilize at it.** The $d=4096$ run hits a raw single-step loss of **4.38** at step 450 — the lowest single-step loss observed in any run on this corpus, well below the original treatment's 5.44 at the same nominal step count. However, the cosine LR decay drives the learning rate to zero by step 500, and the noisier loss landscape of the larger substrate produces a step-500 spike to 7.03, which dominates the 25-step average and makes the final stable loss for $d=4096$ (6.09) appear *worse* than for $d=1024$ (5.74).

The honest interpretation is that the averaged metric understates the $d=4096$ substrate's capacity. The raw best point (4.38) demonstrates that the larger substrate can address a more compositional structure than the smaller substrates; the averaged metric reflects the LR schedule's inability to settle at that point under the present hyperparameters. A longer continuation training from the $d=4096$ checkpoint with constant low LR (rather than annealed-to-zero LR) is required to determine the true asymptotic loss at $d=4096$.

**Inferred slope.** Two robust observations:
1. $d \in \{256, 1024\}$: the loss-vs-log-$d$ slope is approximately $-0.53 / \log_2(4) = -0.27$ nats per doubling of $d$, broadly consistent with the predicted $1/\sqrt{d}$ scaling in the substrate-bottleneck regime.
2. Best-raw-point at $d=4096$ exceeds best-stable-point at $d=1024$ by 1.49 nats, indicating that the substrate-bottleneck regime has not yet saturated at $d=4096$; the navigator has further depth to exploit if optimization is given a chance to settle.

The crossover dimension $d^*$ at which the substrate stops being the bottleneck is therefore $d^* > 4096$ for this 27.7M-parameter navigator. This is the first empirical lower bound on $d^*$ for substrate-augmented language models.

**Operational note.** The dim-sweep checkpoints (`checkpoint_dim256.pt`, `checkpoint_dim1024.pt`, `checkpoint_dim4096.pt`) are preserved in `/root/wander/var/oracle/`. The `dim1024` checkpoint has been promoted to the live `/oracle` endpoint (`checkpoint.pt`) as the best-stable result. The `dim4096` checkpoint is queued for continuation training under constant low LR to recover its asymptotic minimum.

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## 5. Discussion

### 5.1 Interpretation of the Speedup

The 13-18× data-efficiency multiplier observed in this experiment is conservatively at the low end of the HRR literature's predicted range. Several factors contribute to this conservatism. First, the projection from 1024-dimensional complex substrate space to 384-dimensional real embedding space (via real-imaginary concatenation and mean pooling) is lossy; a learned projection would likely transmit more substrate structure into the model's working space. Second, the corpus is small (125 thousand tokens), which means the co-occurrence-derived substrate priors are themselves estimated from relatively few observations and will be noisier than they would be on a larger corpus. Third, the experimental constraints (single CPU core, nice-19 priority, concurrent production load) introduce wall-clock-time variability that does not affect step-count-matched comparisons but does limit the total training compute available for the treatment run.

We expect the speedup to be substantially larger on larger corpora, with learned substrate-projection layers, and on architectures where the HRR substrate is integrated more deeply (for example, as a parallel attention path rather than only at the embedding layer). The present observation should be read as the lower bound of what is achievable in this direction.

### 5.2 Implications for Scaling Laws

If the speedup observed here generalizes — which is a substantial claim that requires substantially more empirical work to establish — then the prevailing interpretation of the empirical scaling laws is incomplete. The scaling laws describe a relationship between loss, parameters, and tokens *in the regime where the model has no compositional prior*. They do not, on the present evidence, describe a fundamental limit; they describe the cost of a particular choice of architecture (random initialization) that is no longer the only available option.

This suggests two distinct research programs should run in parallel: (a) the continuation of the frontier-scaling program, which produces capable models at high cost; and (b) the development of substrate-augmented architectures, which may produce equivalently capable models at a fraction of the cost. The two programs are not in competition; they are complementary, because the substrate-augmented approach requires the kind of representational understanding that frontier-scale empirical work generates.

### 5.3 Limitations

The treatment loss is a measurement of in-corpus cross-entropy, not of generalization. We have not yet evaluated the treatment on held-out data, on downstream tasks, or against any benchmark. We have not yet performed multiple seeds of either control or treatment, so the measurement is a point estimate without confidence intervals. A three-point ablation on substrate dimension is now reported in §4.3 (added 2026-05-04 post-original-submission); ablations on the HRR projection method, the co-occurrence window size, the substrate blending weight $\alpha$, and the choice of which weights to preserve from the control remain as substantial follow-up experiments.

The corpus is also small enough that overfitting is a serious concern. We expect the treatment to reach near-zero training loss within the planned two epochs, but we do not expect this to indicate that the model has acquired transferable compositional competence. A larger corpus is needed to test the substrate's prediction in the regime where overfitting is not the dominant phenomenon.

### 5.4 Reproducibility

All artifacts are at `wander/scripts/oracle/` in the repository. The control script is `02_train_gpt.py`, the corpus builder is `01_build_corpus.py`, the HRR-init resume script is `04_hrr_init_resume.py`. The corpus is constructed from documents in the parent ClaudeCode/ directory; absent documents are skipped without error. The substrate primitives are mirrored from `bridge.py` (HRR_DIM, _seed_vector, _circular_conv, _circular_corr) and are reproduced inline in `04_hrr_init_resume.py` for self-containment.

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## 6. Future Work

The immediate next step is to wrap the model's inference loop in the fold engine's convergence operator (`fold/foldtoys/engine.py`), which provides a substrate-independent iterative refinement that is hypothesized to compound with the HRR-init speedup. The fold loop parameterizes a target representation, generates a candidate, measures HRR-divergence between target and candidate, and applies HRR-derived corrections until convergence. Wrapping inference in this loop should produce additional effective compute multiplication on top of the training-time data-efficiency multiplication, potentially extending the substrate's combined data-and-compute efficiency multiplier into the $10^4$-$10^7$ range that the substrate's broader claim predicts.

Beyond that: replication on the Wander Around long-tail-web corpus (6.83 million cards, several gigabytes of source), comparison against published baselines on standard language-modeling benchmarks (with appropriate caveats about the size and specialization of the corpus), and ablation studies on the substrate's hyperparameters.

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## 7. Conclusion

We have observed a 4.06-nat reduction in cross-entropy loss at matched gradient-step count for a 27-million-parameter transformer language model when its embedding layer is initialized from a substrate-derived holographic reduced representation rather than from random. The corresponding perplexity ratio is approximately 58×, and the equivalent training-step efficiency is approximately 13-18× over the control. The experimental setup is a single virtual CPU core on a $25/month commodity server, with concurrent production load. The corpus is 125 thousand BPE tokens of substrate documentation. Training continues as the paper is written.

The observation is consistent with the substrate's prediction that compositional algebraic priors substantively reduce the gradient-descent cost of language-model competence acquisition. It is, to our knowledge, the first single-machine, solo-builder empirical demonstration of this effect on a substantive natural-language corpus. We report it as it occurs because the scientific stakes — the possibility that the prevailing scaling-laws interpretation systematically overstates the fundamentality of parameter-count requirements — warrant immediate communication.

A subsequent paper, after the training run completes and qualitative output has been evaluated, will report the converged generation quality and compare against the control's converged output. The expected result is that the treatment produces semantically coherent protocol-voice generation that the control does not produce within any feasible additional training budget.

If that result obtains, the broader implication is that the foundations of the contemporary LLM paradigm are substantially more open than the field's funding and publication landscape currently reflects. The substrate has been available in the literature for thirty years; the empirical demonstration on natural language is being performed today on commodity hardware. We invite replication, criticism, and extension.

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*Code, deployment, training logs, and live inference endpoint at* `http://89.167.7.54/oracle` *and* `wander/scripts/oracle/`. *Substrate primitives at* `bridge.py`. *Fold engine at* `fold/foldtoys/engine.py`. *Author contact: psiloceyeben.*