Topological Information Residency in Artificial Neural Networks

Original research · Neural networks · 2026

Read the formal paper on Zenodo

Classical connectionism and modern deep learning rest on a shared assumption: the neuron is the primary unit of computation and representation. Edges are treated as passive transmission lines. This framework struggles to explain superposition, where a single neuron participates in representing hundreds of orthogonal concepts.

We invert this posture: nodes are stateless routing buffers, edges are the active computational substrate. Information does not reside in neurons, it resides in the topology of connections.

2. Empirical proof: dispersed vs clustered

On a standard MLP (5→20→20→1, 561 parameters) trained to 100% on sequential XOR, we cut N edges under two protocols. Dispersed: random removal across the graph. Clustered: removal of every edge connected to a specific node.

3. Magnitude pruning: the winning ticket is an edge subgraph

Second proof of information sparsity: we sweep edge removal by magnitude. Removing the weakest edges costs nothing; removing a handful of the strongest destroys computation. The Lottery Ticket Hypothesis' "winning ticket" is not a set of neurons, it is the topological subgraph of high-magnitude edges.

4. The Hexaflake: edge chain with bidirectional capture

First step into edge-centric: replace every scalar weight with a linear chain of 3 capture cells (A, B, C). Forward pass follows State_i = tanh(W_fwd * State_i-1). The middle cell's bidirectional rule Capture_B = tanh(W_cap * [State_A \| State_C]) models local cellular-automaton interactions.

The Hexaflake hits 100% accuracy on benchmarks. Its central cell B aggregates information from the extremities : ablating B must produce a strictly larger drop than ablating A or C (prediction P4).

5. The Dodecaflake: 75% of facets ablatable without loss

Each edge is modelled as a 3D projection space with 12 propagation facets (the twelve pentagons of a regular dodecahedron). A tensor interference mechanism at the centre of the edge combines the signals propagated from both nodes via a gated Hadamard product.

Consequence: 9 of 12 facets can be destroyed and the network still routes signal to its centre with 100% accuracy. Destroying the central interference mechanism, however, kills the model instantly. Computation lives topologically at the centre, peripheral paths act as self-healing waveguides.

The core: tensor interference by phase coherence

The centre is not a learned weight: it is a physical condition. The signals propagated from both nodes are combined by a gated Hadamard product, and the central state is captured only if both wavefronts arrive in phase. Drag the phase difference:

Architectural saturation: on simple boolean tasks (XOR), bidirectional capture yields a marginal -10% contribution. This is not a failure but a signature: forward propagation alone suffices for such linear tasks, and structural interference is only warranted for non-linear, high-dimensional contextual integration, where it replaces learned gating with a non-parametric operation.

Analogy with Majorana fermions (Kitaev 2003): quantum information is protected from local decoherence by being stored in non-local topological relationships between particle pairs.

The slime mould Physarum polycephalum (Nakagaki 2000) solves optimisation and routing problems without any centralised nervous system. Processing emerges from the physical restructuring and rhythmic contraction of a tubule network.

If a Dodecaflake trained on multi-bit parity tasks degrades linearly under symmetric facet pruning, topological protection is falsified.

If any individual node in a fully trained Dodecaflake stably represents a monosemantic feature, the stateless routing hypothesis is falsified.

In a Transformer, removing 50% of low-magnitude weights in W_q/W_k causes 50% degradation.

Ablating the middle cell of a Hexaflake edge must produce a statistically larger performance drop than ablating the entry or exit cells.

If the neuron is not the primary unit, interpretability programs focused on individual neuron activations measure the wrong object. The right object is the high-magnitude edge subgraph. Superposition stops being a paradox: it is what you see when you look at a node (junction) instead of an edge (street).

The multi-facet topology mirrors principles observed in biology (myxomycete tubule self-repair, neuronal redundancy in the adult brain). We obtain a computational model of resilience that is neither ad-hoc regularisation nor an optimisation trick : it is a structural property.

50% of low-magnitude edges can be removed with zero loss = 50% of hardware operations avoidable. Direct target for neuromorphic chips.

9. Perspectives: the edge as a memory substrate

Avenues opened by the result, beyond the experimental scope validated in the paper.

If information resides not in nodes but in edges, then an edge stops being a mere scalar: it becomes a cell able to carry a state vector. This shift suggests using the edge topology itself as addressable storage, rather than externalising embeddings into a vector database.

Today an agent stores its vectors in an external store (for instance Qdrant) and retrieves by cosine similarity. The edge-centric result opens an alternative: retrieval could emerge from a topological resonance within the network itself, the same phase-coherence condition that captures an edge's centre acting as the addressing mechanism. Memory and computation would no longer be separate.

Self-healing storage: the 75% facet redundancy protects content against local loss.

Sparse retrieval: only the strong-edge subgraph participates, the rest is inert.

On-chip memory: a natural substrate for neuromorphic hardware, with no round-trip to a remote store.

Strictly forbidden without an explicit written partnership with Hope 'n Mind SASU.

Download the formal PDF on Zenodo

Limited effect, the network routes around missing edges via neighbouring paths.

Immediate collapse to 51.4% (chance level on XOR). The routing hub is severed, no bypass possible.

Central interference disabled: collapse to 47.1% (chance). This is where computation lives.

All facets cut, centre intact: collapse to 47.1%. The centre alone does not suffice, it needs facets to receive signals.

Up to 9 of 12 facets ablated, the centre still captures the signal at 100%. Absolute topological protection in this regime.

B aggregates memory : removing it cuts bidirectional capture, the network collapses toward chance.

Moderate loss : the end cells contribute to the forward flow but the middle cell absorbs most of the computation.

All paths active : forward flow A->B->C and bidirectional capture B <- {A, C}. The network operates at full capacity.

Up to 50% of connections can be removed with zero loss, provided they are the lowest-magnitude edges. The "winning ticket" is the strong-edge subgraph (an extension of the Lottery Ticket Hypothesis).

Removing just 10% of high-magnitude edges already collapses computation (74.8%), and 20% brings it to chance (48.6%). Computation lives in a minority of strong edges.

Random pruning: degradation is intermediate and grows with the share (98% -> 87% -> 65%), because chance eventually hits strong edges.

Conceptual visualisation of the gated Hadamard product: the central state is captured only if the wavefronts from both nodes arrive in phase. In phase (0 deg) capture is maximal; in opposition (180 deg) interference is destructive and the centre stays silent. It is a strict physical coherence condition.

Exploratory schematic (beyond the paper's experimental scope): if information lives in the edges, each edge becomes a cell able to store a vector. The edge topology could then act as a content-addressable associative memory, retrieval emerging from topological resonance rather than cosine-similarity search in an external vector database (such as Qdrant).