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.

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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.

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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.

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:

@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