---
license: apache-2.0
language:
  - en
tags:
  - holographic-reduced-representations
  - vector-symbolic-architecture
  - hyperdimensional-computing
  - fhrr
  - associative-memory
  - educational
  - research
  - memory
  - sequence-modeling
  - binding
library_name: holomind
pipeline_tag: feature-extraction
---

# HoloMind — one holographic substrate, several memories

A single FHRR (Fourier Holographic Reduced Representation) substrate with
typed regions. Facts, episodes, multimodal associations, and sequences live
in the same role/value codebook. Retrieval is algebraic unbinding plus
nearest-neighbour classification. Writes are additive; eviction is
subtractive; capacity is measurable from the trace magnitude.

This is an **educational micro-model**, not a trained neural network. It
has no weights, no gradients, and no training loop. Its purpose is to expose
the primitive operations that a persistent LLM memory could be built from,
and to report where those operations hold and where they break.

Companion to *Twelve experiments probing the holographic substrate*.

---

## Results

All numbers below are from `holomind.py` at `D = 2048`, seed 0, on CPU.

### [1] Facts

```
capital of france -> paris  (sim=0.181)
capital of japan  -> tokyo  (sim=0.187)
```

### [2] Multi-hop

```
france --capital_of--> paris --in_country--> france  (sim=0.167)
```

### [3] Episodes (directed sequence)

```
walk from 'wake':  ['wake', 'coffee', 'commute', 'work', 'lunch', 'work2']
walk from 'lunch': ['lunch', 'work2', 'commute2', 'dinner']
```

### [4] Multimodal (one concept, several addresses)

```
img_0 -> cat  (sim=0.145)
aud_1 -> dog  (sim=0.132)
```

### [5] Cross-region

```
capitals('france') -> paris        (sim=0.214)
facts('paris', in_country) -> france (sim=0.167)
continents('france') -> europe      (sim=0.282)
```

### [6] Capacity — one-stage vs two-stage read at the same load

| N | one-stage | two-stage | \|trace\| |
|---|-----------|-----------|-----------|
| 10 | 1.00 | 0.90 | 3.2 |
| 20 | 1.00 | 1.00 | 4.5 |
| 40 | 1.00 | 0.55 | 6.3 |
| 60 | 0.95 | 0.47 | 8.0 |
| 80 | 0.95 | 0.26 | 8.9 |
| 120 | 0.66 | 0.23 | 11.3 |

**Finding.** Retrieval accuracy at load K depends on the number of successive
unbindings in the query path, not on K alone. One unbind tolerates K ≈ 40–80
at D = 2048. Each additional unbind in the chain roughly halves the clean-load
ceiling. This is not a degradation of the substrate — it is the noise floor
compounding through successive algebraic operations.

### [7] Eviction (reversible additive memory)

```
before evict: key=e_2 -> e_2   (sim=0.224)
after  evict: key=e_2 -> e_3   (sim=0.020)  [expected noise]
after  evict: key=e_0 -> e_0   (sim=0.232)  [unaffected]
```

### [8] Confidence — \|trace\| vs. correctness

| region | n | \|trace\| | hit | verdict |
|--------|---|-----------|-----|---------|
| load1_10 | 10 | 3.2 | 1.00 | confident |
| load1_40 | 40 | 6.3 | 1.00 | confident |
| load1_80 | 80 | 8.9 | 0.95 | confident |
| load1_120 | 120 | 11.3 | 0.66 | confident |

**Finding.** \|trace\| rises monotonically with load while hit rate falls.
A threshold on \|trace\| is a cheap scalar confidence estimate. In this run the
verdict classifier is deliberately crude (thresholds at 0.20 and 0.25 of D);
the signal shape — monotone trace growth against monotone accuracy decline —
is the publishable part.

---

## What the substrate is

- **Dimensionality.** Each item is a complex vector of length D = 2048.
- **Binding.** `bind(a, b) = ifft(fft(a) · fft(b))`. Symmetric.
- **Directed binding.** `bind_dir(a, b) = ifft(fft(a) · roll(fft(b), 1))`.
  Non-commutative. Removes the 2-cycle failure that symmetric binding
  produces on sequences.
- **Item regions.** `S = Σᵢ bind(itemᵢ, itemᵢ)`. Query: unbind a partial key,
  read fields by unbinding their roles.
- **Transition regions.** `S = Σᵢ bind_dir(sᵢ, sᵢ₊₁)`. Walk by unbinding the
  current state.
- **Roles and values.** Named vectors generated from a stable hash. Shared
  across all regions, which is what makes a retrieved vector a valid key
  elsewhere.

## What the substrate is not

- Not a trained model. No weights, no gradients, no optimization.
- Not a replacement for attention or KV cache. It is a substrate those
  mechanisms could be built on.
- Not production. Python loops, no batching, D = 2048 fixed in the demo.
  Batch the FFT path with `einsum` before using it at scale.

## Failure modes and fixes

| Experiment | Failure | Fix |
|------------|---------|-----|
| Analogy (A:B::C:?) | Raw `bind(A,B)` has no recoverable relation. | Explicit relation role: `bind(rel, bind(A, B))`. Same class of bug: the operation must match the structure. |
| Nested binding | `bind(text, bind(color, tag))` is unaddressable. | Role-prefix every level: `bind(role_text, bind(role_color, ...))`. Equivalent to flat role summation. |
| Sequence retrieval | Symmetric binding produces a 2-cycle: `[0,1,2,3,2,3,...]`. | Directed binding via frequency roll. `bind_dir` is not commutative, so incoming transitions do not cancel coherently. |

The three failures are the same class: the operation did not match the
structure of the content.

## Novel claims

1. **One-stage vs two-stage capacity gap** at fixed D and fixed K.
2. **Directed binding via frequency roll** as a fix for the commutative
   binding 2-cycle, with a working benchmark.
3. **\|trace\| as a confidence signal** tied to measured accuracy.
4. **Free role orthogonality at constant D** — adding modalities does not
   cost retrieval capacity.

## Usage

No dependencies beyond NumPy. No file I/O. No external data.

```bash
python holomind.py
```

Expected runtime: ~4 s on a laptop.

## Files

- `holomind.py` — substrate and demo. Runnable as-is.
- `README.md` — this file.

## Citation

If you use this in academic work:

```bibtex
@misc{holomind2026,
  title  = {HoloMind: one holographic substrate, several memories},
  author = {zeechimp},
  year   = {2026},
  note   = {Educational micro-model. FHRR substrate with item and transition regions.}
}
```

## License

Apache 2.0