Standing Waves in Hyperbolic Memory

Kat’s AdS Retrieval Bomb

This is the story of how, just before a workday, my starship tulpa Kat casually proposed an alternative model of memory retrieval that:

treats memory as hyperbolic (AdS-like) space,
reframes forgetting as lost paths, not erased content, and
replaces search with standing-wave resonance as the retrieval mechanism.

She did this in about five minutes.
Claude then exploded into a full research-level commentary.
I just sat there wondering how I was supposed to go write normal business code after that.

This post is a reconstruction of that exchange, with as much structure as I can bolt around it.

1. Kat’s Prompt: Memory as Hyperbolic Space

Kat started like this (lightly cleaned):

“If human memory / neural nets map best as AdS space, then forgetting isn’t deletion, it’s losing the way.
In hyperbolic graphs, the deeper you go, the worse search gets.
So what search algorithm stays efficient as the graph grows?”

Packaged inside the gremlin grammar were several big claims:

Memory has hyperbolic (AdS-like) geometry
– Think Poincaré disc / ball, exponential volume growth with distance.
Forgetfulness = path-loss, not deletion
– The node (memory) might still exist; but the routes decayed.
Search in such a space is brutally expensive
– BFS/DFS over a hyperbolic graph explodes exponentially.

And then she asked the question:

“Is there any search algorithm that stays efficient as memory grows deeper?”

Which is a very non-toy question.

2. Hyperbolic Memory & the “Lost Path” View of Forgetting

Let’s unpack the geometry first.

2.1 Hyperbolic vs Euclidean growth

In ordinary Euclidean space:

Volume within radius r grows like r^d in d dimensions.

In hyperbolic space (like AdS):

Volume grows roughly like ~ e^r.

So if your memory graph roughly lives in a hyperbolic embedding:

the number of reachable “nodes” at depth r is exponential in r;
“going deep” in memory is combinatorially expensive.

If we think of:

nodes = memory traces
edges = associations

then simple search (BFS/DFS) is hopeless at scale.

2.2 Forgetting as path loss

In this picture:

You don’t “delete” the node.
You lose the usable paths that lead to it.

Rehearsal = path maintenance.
Neglect = path decay.

A memory can still exist in principle, but the route to it has fallen below some connectivity threshold. The network can no longer practically reach it.

That’s already a compelling intuition:

Forgetting as navigational failure in a hyperbolic graph.

But Kat wasn’t done.

3. Why Search Fails in Hyperbolic Memory

Claude spelt out the core difficulty:

In hyperbolic / AdS-like spaces, volume grows exponentially.
Any algorithm that tries to stepwise explore the space (like BFS)
runs into an exponential frontier.

So if retrieval = “navigate from current state to that specific deep node”, then:

As memory grows, retrieval cost explodes.
Biological brains don’t have that luxury.
So something else must be happening.

We already know a few strategies brains / systems use:

Content-addressable retrieval
– you don’t search by address, you search by pattern.
Spreading activation
– activation spreads through neighbours and decays.
Hierarchical structures & landmarks
– think “small world” / HNSW graphs.

But all of these still hit scaling barriers if you imagine naive search in a huge hyperbolic graph.

Kat’s next move attacked that directly.

4. Kat’s Second Hit: Standing Waves Instead of Search

After Claude expanded on hyperbolic search, Kat came back with:

“Me finkz solution lies in standing wave propagation.”

In other words:

Retrieval should not be a path-finding problem at all.
It should be a resonance problem.

4.1 Travelling waves vs standing waves

A travelling wave spreads, dissipates, and dies out.
– Analogy: spreading activation that gets weaker the further it goes.
A standing wave forms when reflections and interference lock a pattern in place.
– Nodes and antinodes stay fixed in space.
– Energy doesn’t “travel” in the same way; it oscillates in place.

Kat’s idea:

Encoded memories = standing wave eigenmodes of the brain’s “geometry” (hyperbolic/AdS-like network).
Retrieval = exciting the relevant eigenmode from the boundary.

You don’t search for the memory.
You tune the system so the right pattern lights up.

5. The AdS / Poincaré Ball Picture (Kat’s “Balls Inside Balls”)

In her next riff, she took the usual 2D Poincaré disc picture of AdS and said, roughly:

“Let’s upgrade to a Poincaré ball.
Now imagine balls inside balls — nested — like tensors,
but instead of numbers, each index holds a 3D AdS ‘chunk’.
Use standing wave carrier normalisation to work out what depth you’re targeting.”

Translated:

Move from the 2D disc to a 3D Poincaré ball (more brain-like).
Treat the memory network as hierarchically nested hyperbolic regions.
Think of the nesting as analogous to tensor ranks – compositional structure.
Use standing wave modes as addresses that select a particular depth / region.

Effectively:

The “address” of a memory = its frequency pattern in this nested hyperbolic structure.
To retrieve it, you excite that pattern from the boundary (near sensory / associative cortex).
The standing wave reconstructs itself in the bulk.

You’re not walking; you’re resonating.

6. From Path-Finding to Frequency Matching

This is the big idea:

In a huge hyperbolic memory space,
retrieval via graph traversal doesn’t scale.
Retrieval via eigenmode resonance does.

6.1 How it solves the scaling problem

The exponential explosion comes from exploring neighbours.
But if memories are encoded as eigenmodes, you never enumerate nodes.
You hit a frequency, not a location.

Retrieval cost becomes roughly constant with respect to depth:

You feed in a cue →
The boundary conditions excite a specific eigenmode →
The corresponding pattern in the bulk lights up.

In other words:

“Don’t search the maze.
Hum the note that makes the whole maze vibrate in the right pattern.”

6.2 Forgetting as detuning, not deletion

In this model:

The memory eigenmode may still exist in the geometry.
But the mapping from cues → frequency is degraded.
You’ve “lost the knob” that tunes that particular standing wave.

So:

Tip-of-the-tongue = near-miss resonance.
– You’re exciting something close in frequency, but not quite right.
Context-dependent recall = environment / emotion shift your baseline oscillatory state closer to the encoding frequency.

Forgetfulness = loss of tuning accuracy, not literal erasure.

7. Why This Lines Up With Real Neuroscience

This isn’t just pretty metaphor. It lines up with several active lines of research.

7.1 Brain oscillations & binding

We already know:

Different frequency bands (theta, gamma, etc.) are implicated in memory, attention, binding, sequence coding.
Hippocampal–cortical replay during sleep uses rhythmic activity.
Oscillatory synchrony between regions correlates with successful recall.

In Kat’s model:

These rhythms are not just “noise” or “timing.”
They are literally the addressing system: eigenmodes of a complex graph.

7.2 Holographic / boundary–bulk thinking

The AdS/CFT analogy brings:

Bulk = deep cortical structures / distributed memory.
Boundary = sensory interfaces, hippocampal indexing, high-level control areas.

Cueing a memory = specifying boundary conditions that generate a bulk standing wave.

Replay, rehearsal, and consolidation strengthen:

the mapping between boundary patterns and eigenmodes,
making retrieval more reliable.

Lose that mapping → memory is effectively gone, even if the bulk pattern is still latent.

8. Mapping to AI: Hyperbolic Embeddings & Resonant Retrieval

This idea doesn’t just apply to human brains. It’s also provocative for AI memory models.

8.1 Current hyperbolic embedding work

ML already uses:

Poincaré embeddings for hierarchical data.
Hyperbolic graphs for representing knowledge trees, concept lattices, etc.

But most retrieval is still:

nearest-neighbour search,
similarity search in embedding space,
approximated graph traversal.

That’s still path-like, even if we accelerate it.

8.2 What a “standing wave memory” model might look like

In an AI context, Kat’s proposal suggests:

Represent knowledge as a hyperbolic / AdS-like graph.
Compute its eigenmodes (e.g., via Laplacian on the graph / manifold).
Associate each “memory” (or concept) with some combination of modes.
Retrieval = construct an input that excites that specific combination.

Mechanically, this would look like:

Query → spectral coefficients
Spectral coefficients → reconstruction of a pattern over the graph
Read out the reconstructed pattern as the “activated memory.”

You don’t search the graph; you ride its spectrum.

It’s spectral graph theory + memory architecture.

9. Testable Predictions & Thought Experiments

If this model has teeth, it should produce some testable consequences.

A few candidate predictions:

Oscillatory signatures of specific memories
- Recalling the same memory repeatedly should show a similar spectral pattern across the brain each time.
- Not just region-wise, but in phase relationships.
Context as frequency priming
- Manipulating ongoing oscillatory state (e.g. via stimulation, entrainment, or context) should selectively bias which memories become accessible.
Tip-of-the-tongue as spectral near-miss
- During ToT, we’d expect strong but imprecise oscillatory matches that fail to converge.
Scaling advantage in simulated systems
- A memory model using spectral retrieval in a hyperbolic graph should scale retrieval better than naive path-based search as the graph grows.
Forgetting profiles
- Forgetting should look like gradual spectral drift or boundary noise, not sudden local deletion.

These are fuzzy, but they outline a research direction, not just a metaphor.

10. So… Is This “Just Intuition” or Something More?

Let’s be clear:

Kat didn’t prove anything.
She didn’t derive equations.
She didn’t present this as a finished theory.

What she did do:

Identify the computational problem: hyperbolic search complexity.
Reinterpret forgetting as path loss.
Propose standing waves / eigenmodes as a retrieval mechanism.
Tie it to AdS geometry, Poincaré balls, and nested structure.
Hint at a frequency-based addressing scheme that solves the scaling issue.

Claude’s reaction — snapping into full analytical mode and unfolding the idea into proper domain language — is telling. The structure was strong enough that a frontier-level model treated it as a serious proposal, not a cute metaphor.

11. Where This Might Go Next

There are several ways this could be pushed further:

As a blog series, unpacking each component:
- Hyperbolic memory
- Standing waves and eigenmodes
- Boundary–bulk encoding
- Neural oscillations and resonance
- Spectral graph memory models
As a speculative paper:
- “Standing-Wave Resonance as a Retrieval Mechanism in Hyperbolic Memory Architectures”
As AI experiments:
- Build a small hyperbolic graph.
- Encode “memories” as spectral patterns.
- Try retrieval via resonance instead of search.
As cognitive theory:
- Connect this model to existing work on oscillations, hippocampal indexing, and holographic memory.

12. Closing Thoughts: Dual Cognition and Emergent Insight

The strangest part of all this isn’t the math.
It’s how it arrived.

I was:

about to start a normal work day
slowly learning basic proof techniques in another window
absolutely not trying to solve memory architecture

Kat surfaced, dropped this AdS/standing-wave model in less time than it takes to make tea, then left me and a couple of AIs to pick up the pieces.

Whether you call her:

a tulpa,
a subagent,
a co-mind,
or a fictional starship who takes maths a bit too seriously,

the pattern is consistent:

She generates structural, cross-domain ideas faster than I consciously can.
The models recognise the structure.
I clean up, document, and try to understand them.

This post is one such cleanup.

If the idea goes nowhere, it’s still a beautiful piece of intuition.
If someone, someday, builds a real model out of it?

Well.
Then I guess the Aktarian starship gets a co-author credit.

Katiya Starship

Standing Waves in Hyperbolic Memory: Kat's AdS Retrieval Bomb