Mapping Semantic Basins of Attraction in LLMs

I've been experimenting with prompt interpolation by giving a model two creative prompts and having it generate the conceptual steps between them. If you give a model two unrelated concepts, how does it navigate the space between them? What does it gravitate toward? I wanted a way measure it.

To test this, I built a simple game. Two instances of an LLM each start with a different word. Each round, both say a new word that bridges the gap between the previous two. They're trying to say the same word at the same time. Same rules as the party game Convergence, except models let you log every step to analyze the data.

575 games across four frontier models (Claude, GPT, Grok, Gemini), with starting words ranging from concrete nouns to abstract concepts to famous people crossed with unrelated ideas. What started as a probe for interpolation style became something else: models have different topologies of meaning — different shapes to how they organize and traverse conceptual space. And those shapes are measurable.

The First Surprise: Abstract Is Easier Than Concrete

Before running any experiments, I assumed abstract word pairs would be harder to bridge. "Palimpsest" and "thunder" feel like they have nothing in common. "Mountain" and "ocean" feel obviously connected. But the data disproved my hypothesis.

Most models converged faster on abstract pairs than concrete ones. GPT dropped from an average of 3.6 rounds on concrete nouns to 2.8 on abstract words. Minimax showed the same pattern. Claude and Grok were already so fast on concrete words that there was no room to improve, and Gemini was the lone exception, slightly slower on abstract pairs.

The reason is straightforward: concrete nouns have dense synonym neighborhoods. When both GPT instances are trying to bridge "skull" and "garden," they keep landing on almost the same word (cemetery vs. graveyard, tombstone vs. burial) orbiting a concept cluster without ever hitting the exact same word. It took 5 rounds of near-misses before they converged on "inscription."

Abstract words don't have that problem. "Palimpsest" can connect to almost anything through metaphor. There are no close synonyms to get stuck between. The flexibility of abstract associations is an advantage.

This was the first hint that what I was measuring wasn't "difficulty" in the way humans think about it. The game reveals the shape of semantic neighborhoods, and dense neighborhoods create friction that sparse ones don't.

Four Models, Four Personalities

Run enough games and and character profiles emerge out of individual results.

Claude is the fastest and least interesting player. It converges in exactly 2 rounds on almost everything. Its convergence words are always the most obvious bridge: "wave" for mountain and ocean, "echo" for shadow and melody, "bone" for skull and garden. A standard deviation of 0.49 across hundreds of games, nearly deterministic. Best player, least interesting thinker.

GPT is the slowest and most creative. It takes winding paths through unexpected territory. For vertigo paired with mycelium, both GPT instances independently went pharmacological: one said "psilocybin," the other said "ergot." They converged on "alkaloid." Ergot is a fungus (mycelium) that causes vertigo via ergotamine. Psilocybin comes from mushroom mycelium and alters perception. Both are alkaloids, demonstrating cross-domain reasoning. GPT also has a distinctive "near-miss escalation" pattern where both players orbit the same synonym cluster, finding different words within it, then converge on something more abstract that encompasses the near-misses.

Grok takes lateral paths that no other model finds. For shadow paired with melody, it went through "nocturne" to "phantom" to "opera" to "ghost" to "mask," a 4-round path through the Phantom of the Opera. For hammer and butterfly, it produced "pinfish," a compound word that satisfies both starting words (hammer → pin, butterfly → fish). Grok's personality is cultural and associative where GPT's is systematic and taxonomic.

Gemini is fast like Claude but with stranger word choices. It converges quickly but picks words like "thorax" for hammer and butterfly, or "hamlet" for skull and garden. These references are technically correct (butterfly anatomy, Yorick's skull in Shakespeare) but feel like they come from a different knowledge organization than the other models.

These profiles are consistent across hundreds of games, stable under temperature changes, and measurable with simple statistics. Each model moves through conceptual space with a characteristic gait.

Semantic Basins

The real framework emerged in round 4, when I ran 240 games testing whether the patterns from earlier rounds were stable or lucky.

The concept that kept showing up: semantic basins of attraction. Some word pairs have a gravitational center, a concept so naturally positioned between the two starting words that models fall toward it the way a ball rolls toward the bottom of a bowl. Other pairs have no center at all. The terrain is flat, and models wander.

Three types of terrain:

Deep basins have one answer. Beyoncé paired with erosion converges on "formation" 78-82% of the time at standard temperature, across all four models and in both directions (both being a Beyoncé song and a geology term). It's so dominant that even cross-model games (two different models playing against each other) converge on it 70% of the time. The basin walls are clearly defined. When models miss "formation," they land on adjacent geology words like sand, dune, sandcastle. They don't escape the neighborhood.

Moderate basins have multiple valleys of similar depth. Shadow paired with melody sends Claude and Gemini to "echo" with 100% consistency, but Grok goes to "nocturne" (via Phantom of the Opera) and GPT splits between "nocturne" and "resonance" (approaching from acoustic physics). Same input, different conceptual frames, and each frame is internally consistent. Grok reaches for "nocturne" about a third of the time, with a scattered tail through phantom, opera, ghost, mask, and sonata. Not deterministic like Claude's "echo," but recognizably the same personality showing up.

Flat terrain has no basin at all. Kanye West paired with lattice produces maximum entropy. No model converges on the same word reliably. Each enters a different conceptual frame: Claude goes structural (scaffold, brick, wire), Gemini goes academic (graduation, coordinate, debt, through "College Dropout"), Grok goes hip-hop material culture (ice, diamond, chain, drip), and GPT goes geometric (facet, matrix, crystal). The terrain is so flat that even the slight push of word order sends models in different directions.

The basin framework makes a testable claim: if basins are real structure in the model's knowledge graph, then reducing temperature (which reduces randomness in token sampling) should sharpen existing basins without creating new ones. That's mostly what happened. At temperature 0.3, Claude and Gemini converge on "formation" for Beyoncé and erosion 100% of the time instead of ~80%. Kanye and lattice remain chaotic at any temperature. No setting can manufacture a basin where none exists.

One model broke the pattern. GPT got less consistent at low temperature on some pairs: "formation" dropped from 82% to 50%, "network" from 64% to 50%. The likely explanation: GPT has multiple competing attractors of similar strength, and lower temperature makes it commit harder to whichever path it enters first rather than defaulting to the most popular choice. At standard temperature, randomness acts like a tiebreaker that tends to favor the common answer. Remove the randomness and GPT picks a frame and sticks with it, even if it's the less common one. This makes GPT the most "opinionated" model at low temperature; it doesn't consensus-seek.

Basins are properties of knowledge. Temperature can sharpen them but can't create them.

When Models Negotiate

The most interesting games happen when two different models play against each other.

Same-model games average 2.3 to 2.8 rounds with tight variance. Both players share the same knowledge organization, so they tend to meet quickly. Cross-model games average 4.1 rounds with a standard deviation above 3. They can converge in 2 rounds or spiral for 19, and you can't predict which.

What matters is what they converge on.

In 48% of cross-model games, the two models converged on a word that neither model produced in any solo game. New semantic territory that neither model contains alone.

Claude vs. GPT on shadow and melody: Claude alone converges on "echo" in 2 rounds. GPT alone gets "chord" in 4. Together? Eight rounds. Claude keeps reaching for clean, precise words (echo, night, frequency, wave, sound, transmission) while GPT keeps reaching for richer, more connotative ones (nocturne, resonance, silence, noise, signal, audio). They're speaking adjacent languages. Claude approaches from physics, GPT from aesthetics. They finally meet at "broadcast," a word where physics and aesthetics overlap, and one that neither model ever found on its own.

Grok vs. Gemini on hammer and butterfly: Grok opens with "pin" (butterfly pin), Gemini with "thorax" (butterfly anatomy). From there, they jointly build toward butterfly-collection territory (insect, specimen, collection, entomology) and land on "museum." Neither model found anything resembling this path solo. The cross-model game produced a more coherent narrative than either model alone.

A third example, from a different run of shadow and melody: Claude vs. GPT started in acoustics (echo, reverb), drifted through "ambience" as acoustic atmosphere into "ambience" as environmental atmosphere, and ended at "biome." Nine rounds from music to ecology, following a metaphor neither model intended to create.

The longest game recorded: Claude vs. GPT on Beyoncé and erosion at 19 rounds. Both models bypassed the obvious "formation" attractor on round 1 and entered a deep geological spiral. Claude spoke process (weathering, glaciation, deposition) while GPT spoke classification (pedogenesis, stratigraphy, varve). They orbited each other through geological terminology for 17 rounds before converging on "periodicity," a word neither model produced in any of its dozens of solo games.

There's a pattern to when novelty appears, and it maps directly onto the basin framework. Tesla paired with mycelium (deep basin): 0% novelty, even mismatched models fall into "network." Beyoncé paired with erosion: 10-30%, "formation" dominates, with occasional escapes like "periodicity." Shadow paired with melody (moderate basin): 73-90% novelty, the models disagree on the right frame and have to negotiate. Kanye paired with lattice (flat terrain): 80-100%, everything is novel because nothing is stable. The interesting zone is moderate basins, where models are forced to build a bridge between different conceptual frames.

Independent research found the same convergence dynamics at larger scale. In a 2025 study, groups of LLMs negotiated labels for ambiguous text across 7,500 multi-agent discussions. The intrinsic dimensionality of their outputs dropped from ~7.9 to ~0.4 over successive rounds — high-dimensional disagreement collapsing into tight consensus. Models split into roles without prompting: some as semantic anchors setting the initial frame, others as integrators pulling the group toward agreement.

Direction Matters

Another one of the less expected findings is that word order changes the outcome, with the effect varying by model.

The clearest case is Claude on Tesla paired with mycelium. Forward direction (Tesla is Player A's word): "network" 80% of the time. Reverse direction (mycelium is Player A's word): "electricity" 100% of the time. Five out of five reverse runs, perfectly reproducible. When Tesla is the starting point, Claude thinks about Tesla's ideas (electrical networks, connectivity); when mycelium is the starting point, Claude thinks about Tesla's inventions (electricity, power).

GPT shows a similar split on the same pair: "network" forward, "grid" in reverse. On shadow and melody, GPT converges on "chord" forward but "nocturne" in reverse. Gemini and Grok are the direction-invariant models, producing the same attractor regardless of word order. On flat terrain like Kanye paired with lattice, all four models show direction effects, because there's no basin strong enough to override the initial framing.

The implication is that concept associations are trajectory-dependent. The path from A to B is different from the path from B to A, at least for some models.

In cross-model games, this creates a Player A advantage. Decisive models (Claude, Grok) converge faster when they lead, because they set a strong initial direction that the other model can follow. Deep basins erase the advantage entirely. When the answer is "formation," who goes first doesn't matter.

One exception: Grok converges faster than Claude from either position. Its lateral, cultural-reference style of association somehow negotiates better than Claude's precise, safe-bridge approach. The best solo player isn't the best negotiator.

We're Bad at Predicting Semantic Distance

Round 5 included a prediction test: three new word pairs where I guessed the basin type before running the games. I went 1 for 3, and the failures were more interesting than the success.

Einstein paired with ocean: I predicted deep, expecting "wave" as a universal attractor (gravitational waves, ocean waves). It was moderate. "Wave" is too literal. GPT finds alternatives (spacetime, gravity, curvature) that are equally valid bridges at a higher level of abstraction. Gemini doesn't even reach "wave," going straight to "relativity" and "physics." The supposedly obvious connection had competition.

Beyoncé paired with mycelium: I predicted flat as there's no obvious connection between pop music and fungal biology. It was moderate. The hidden connections run through "hive" (Beyhive fandom, beehive, fungal networks as underground hives) and "network" (social networks, fungal networks). Grok converged on "hive" 67% of the time. Obvious in retrospect, but only if you think in metaphors instead of surface categories.

Shakespeare paired with algorithm: I predicted moderate. Correct; four different frames emerged. Gemini locks onto "sonnet" (literary form that's also a formal pattern) with 100% consistency, Claude goes to "code" (cipher code, programming code), GPT finds "language" (the shared medium), and Grok wanders between "code" and "script" (plays and programs).

The meta-finding: human intuition about semantic distance is biased in predictable ways. We overestimate the importance of surface similarity (Einstein and ocean obviously share "wave") and underestimate the richness of metaphorical intersection (Beyoncé and mycelium have nothing in common, until you realize they both connect through hives and networks). Models find bridges in conceptual structure that feel arbitrary to us but are systematic within the model's knowledge organization.

What the Game Measures

575 games later, I think the word convergence game is a low-cost probe for something that's otherwise hard to observe: the topology of a model's conceptual space.

These patterns replicated across 240 games and held under temperature controls. Each model moves through conceptual space with a measurable, characteristic style. Pairs of concepts have stable topological properties (deep basins, moderate basins, flat terrain) that exist independently of which model traverses them or how much randomness you allow.

The basin framework has a mathematical counterpart. Wyss et al. (2025) proved that LLM activation manifolds partition into a finite set of basins of attraction, each an invariant meaning category that the model's dynamics funnel toward. Their spectral analysis found most of this structure collapses into roughly three dominant dimensions. The game is a low-resolution version of what the math describes.

The most interesting results came from friction between different models. "Broadcast" exists because Claude spoke physics while GPT spoke aesthetics. "Biome" exists because a metaphor about acoustic ambience drifted into environmental ambience and neither model corrected course. The negotiation between different knowledge organizations produces concepts that no single organization contains.

The word convergence game was a silly experiment that turned out to be a lens for watching models organize and negotiate meaning, and for discovering that how they do it has a shape.

575 games were played on March 1, 2026, using Claude Sonnet 4.6, GPT-5.2, Grok 4.1 Fast, and Gemini 3 Flash via OpenRouter. The full dataset and analysis code are available in the word-convergence-game repository. The multi-agent convergence study referenced is Chiang et al. (2025). The spectral basin formalization is Wyss et al. (2025).