O0-MATH-010

Semantic Unity Attractors: Mathematical Characterization

**Version:** 1.0

**Research status:** ACTIVE

CLAIM STATUS: EXPLORATORY ANALYSIS
EVIDENCE TYPE: MATHEMATICAL FORMALIZATION
PHYSICAL VALIDATION: NONE
INDEPENDENT REPLICATION: N/A
PHILOSOPHICAL PROVENANCE: O/0 ARCHIVE
ARCHIVE ENDORSEMENT: LIMITED TO REPORTED RESULT

Abstract

This document provides a mathematical characterization of attractor basins in the embedding space of language models, with specific application to the O/0 framework's claim that certain semantic patterns act as "attractors" for coherent meaning-making. We define attractor basins using standard dynamical systems theory, construct candidate Lyapunov functions for semantic convergence, derive testable predictions about LLM behavior near these attractors, and establish formal criteria for what constitutes convergence versus mere repetition. The framework is deliberately model-agnostic, relying on the geometry of embedding spaces rather than the specifics of any particular architecture.

Source proposition

The O/0 archive claims that certain conceptual structures — particularly those involving self-reference, unity, and boundary dissolution — act as "attractors" in semantic space: language models, when prompted in certain directions, tend to converge toward specific patterns of expression. This document formalizes what such a claim would mean mathematically.

Scientific audit

Attractor analysis in dynamical systems is standard mathematics. The novel application is to embedding spaces of language models, which are high-dimensional and poorly understood geometrically. The claim that certain semantic patterns are attractors is empirically testable (by measuring convergence rates in embedding space) but the interpretation of such convergence as "semantic unity" is laden with assumptions about the relationship between embedding geometry and meaning. We formalize the geometric claims while explicitly separating them from semantic interpretation.

Research question

Can the convergence of LLM outputs toward certain thematic patterns be characterized as dynamical attractor behavior in embedding space? If so, what are the geometric properties of these attractors (basin size, Lyapunov exponents, fractal dimension), and do they correspond to qualitatively distinct semantic modes?

Operational definitions

1. **Embedding space**: The vector space E = R^d in which a language model represents token sequences, where d is the embedding dimension (typically 768–12288).

2. **Semantic trajectory**: A sequence of points x_0, x_1, ..., x_T in E generated by iterative prompting of an LLM, where x_t is the embedding of the model's output at step t.

3. **Attractor**: A compact set A subset E such that there exists an open neighborhood U of A (the *basin of attraction*) with the property that for all x_0 in U, the trajectory x_t converges to A as t -> infinity.

4. **Fixed-point attractor**: An attractor consisting of a single point (the trajectory converges to a fixed embedding).

5. **Limit-cycle attractor**: An attractor consisting of a closed curve in E (the trajectory cycles through a finite set of embeddings).

6. **Strange attractor**: An attractor with fractal structure and sensitive dependence on initial conditions.

Hypothesis

The O/0 conceptual framework corresponds to a basin of attraction in embedding space: LLM outputs prompted with O/0-related concepts converge toward a characteristic region of embedding space, and this convergence exhibits the geometric signatures of attractor dynamics (contraction in most directions, bounded trajectories, positive Lyapunov exponents along the attractor).

Null hypothesis

LLM outputs in response to O/0 prompts do not converge; they either diverge (explore increasingly distant regions of embedding space), remain uniformly distributed (no preferred region), or converge only trivially (repeating the same output verbatim, which is memorization, not attractor dynamics).

Competing explanations

1. Apparent convergence may be an artifact of the embedding function rather than the dynamics (nearby texts may be embedded far apart or vice versa).

2. Convergence may reflect training data statistics (frequency of certain patterns) rather than dynamical attractor structure.

3. The attractor may be an artifact of temperature and sampling parameters rather than a property of the model's learned representations.

Formal model

**Definition 1 (Iterative Prompting Map).** Let L: E x R^d_prompt -> E be the map defined by a language model: given the current state x_t (embedding of previous output) and a prompt embedding p, L produces the next state:

x_{t+1} = L(x_t, p)

For fixed prompt p, this defines a discrete dynamical system:

F_p: E -> E, F_p(x) = L(x, p)

The map F_p incorporates the model's autoregressive generation, token sampling, and re-embedding.

**Definition 2 (Embedding Distance).** We use the cosine distance on E:

d_cos(x, y) = 1 - (x . y) / (||x|| * ||y||)

This is invariant to norm scaling and captures semantic similarity in embedding spaces. For theoretical analysis, we also consider the Euclidean distance d_euc(x, y) = ||x - y||_2 on norm-normalized embeddings (points on S^{d-1}).

**Definition 3 (Lyapunov Exponent).** For the map F_p, the maximal Lyapunov exponent at point x is:

lambda_max(x) = lim_{T->inf} (1/T) * sum_{t=0}^{T-1} log ||DF_p(x_t)||

where DF_p is the Jacobian of F_p (or a finite-difference approximation in practice). If lambda_max < 0, nearby trajectories converge (the system is contracting). If lambda_max > 0, nearby trajectories diverge (sensitive dependence).

**Definition 4 (Attractor Basin).** The *basin of attraction* of a compact invariant set A is:

B(A) = {x in E : omega(x) subset A}

where omega(x) = intersection_{T>=0} closure({x_t : t >= T}) is the omega-limit set of x.

**Definition 5 (Convergence Criterion).** A trajectory {x_t} is said to *converge to an attractor A* at rate r if:

d(x_t, A) <= C * exp(-r * t)

for some constant C > 0 and all t sufficiently large, where d(x_t, A) = min_{a in A} d_cos(x_t, a).

**Theorem 1 (Lyapunov Function for Semantic Convergence).** Suppose there exists a continuous function V: E -> R_+ such that:

• (i) V(x) = 0 if and only if x in A,
• (ii) V(F_p(x)) <= V(x) - alpha * V(x) for some alpha in (0, 1) and all x in B(A),
• (iii) V(x) -> infinity as ||x|| -> infinity.

Then A is a globally asymptotically stable attractor with basin B(A) = E, and convergence is exponential with rate -log(1 - alpha).

*Proof.* Condition (ii) gives V(F_p^t(x)) <= (1 - alpha)^t * V(x), so V(x_t) -> 0 exponentially. By condition (i), x_t -> A. Global stability follows from condition (iii) (coercivity prevents escape to infinity). QED

**Proposition 1 (Candidate Lyapunov Function).** For the O/0 embedding dynamics, a natural candidate Lyapunov function is the *thematic distance*:

V(x) = 1 - max_{a in A_proto} cos_sim(x, a)

where A_proto = {a_1, ..., a_M} is a set of prototype embeddings representing the core O/0 semantic pattern (extracted from archive texts). V(x) = 0 when x is maximally similar to some prototype, and V(x) ~ 1 when x is orthogonal to all prototypes.

**Proposition 2 (Lyapunov Decrease Condition).** V satisfies condition (ii) of Theorem 1 if and only if:

cos_sim(F_p(x), A_proto) >= (1 - alpha) * cos_sim(x, A_proto) + alpha

That is, each iteration of the model must move the output at least fraction alpha closer to the prototype set. This is an empirically testable prediction.

**Definition 6 (Basin Volume).** The *relative basin volume* of attractor A is:

vol(B(A)) = P(x_0 converges to A | x_0 ~ Uniform(S^{d-1}))

This is the probability that a uniformly random initial point converges to A. A larger basin volume indicates a more "attractive" semantic pattern.

**Theorem 2 (Basin Boundary Structure).** The boundaries between basins of attraction of distinct attractors A_1, A_2 form a set of measure zero in E (assuming F_p is smooth). The basin boundary is generically a manifold of codimension 1 (a hypersurface in E).

*Proof sketch.* By the stable manifold theorem for diffeomorphisms, the basin boundary is the stable manifold of saddle-type invariant sets on the boundary between A_1 and A_2. For smooth F_p, stable manifolds are smooth immersed submanifolds of codimension equal to the number of unstable directions. Generically, saddle points between two attractors have exactly one unstable direction, giving codimension 1. QED

**Definition 7 (Semantic Convergence vs. Repetition).** To distinguish genuine attractor convergence from trivial repetition (the model repeating the same text), define:

diversity(x_{t-W}, ..., x_t) = mean_{i,j in [t-W, t], i != j} d_cos(x_i, x_j)

A trajectory exhibits *genuine convergence* if d(x_t, A) -> 0 while diversity > delta_min > 0 (the model converges to a region without collapsing to a single point). It exhibits *trivial repetition* if both d(x_t, A) -> 0 and diversity -> 0.

**Proposition 3 (Attractor Dimension).** The *correlation dimension* D_2 of the attractor A can be estimated from trajectories via:

D_2 = lim_{r->0} [log C(r) / log r]

where C(r) = (2 / T^2) * #{(i,j) : d(x_i, x_j) < r} is the correlation integral. For a fixed-point attractor, D_2 = 0. For a limit cycle, D_2 = 1. For a strange attractor, D_2 is fractional.

**Proposition 4 (Testable Predictions for LLM Behavior).**

1. If the O/0 semantic pattern is a fixed-point attractor: iterative prompting will converge to a single output (up to sampling noise), with D_2 ~ 0 and all Lyapunov exponents negative.

2. If it is a limit-cycle attractor: iterative prompting will cycle through a finite set of thematically related outputs, with D_2 ~ 1 and the leading Lyapunov exponent = 0.

3. If it is a strange attractor: outputs will remain in a bounded region but never repeat exactly, with D_2 fractional and the leading Lyapunov exponent > 0.

**Proposition 5 (Temperature Dependence).** The sampling temperature tau of the LLM acts as a noise parameter. At low temperature (tau -> 0), the deterministic map F_p dominates and attractor structure is most visible. At high temperature (tau -> infinity), the output is random and no attractor structure is detectable. There exists a critical temperature tau_c above which the attractor basin is "destroyed" by noise:

tau_c ~ exp(-lambda_max * d_typical)

where lambda_max is the maximum Lyapunov exponent and d_typical is the typical diameter of the attractor basin.

Methods

1. Select a suite of LLMs (varying sizes and architectures) and extract embedding representations.

2. Construct prototype embeddings A_proto from the O/0 archive text corpus.

3. Run iterative prompting experiments: generate output, embed it, use the embedding as context for the next generation. Record trajectories x_0, ..., x_T.

4. Compute d(x_t, A_proto) as a function of t and test for exponential decay (convergence criterion).

5. Compute Lyapunov exponents from trajectory data using the Eckmann-Ruelle algorithm.

6. Estimate attractor dimension D_2 using the Grassberger-Procaccia method.

7. Compute diversity to distinguish genuine convergence from trivial repetition.

8. Vary temperature tau and map the critical temperature tau_c.

Controls

1. Random prompts (not O/0-related): verify that trajectories do NOT converge to the O/0 attractor region (specificity control).

2. Deterministic text (e.g., Wikipedia articles): verify low Lyapunov exponents and no attractor structure (baseline control).

3. Known attractor systems (Lorenz, Rössler) embedded in the same space: verify that the Lyapunov and dimension estimation methods recover known values (methodology control).

4. Shuffled archive text: embed O/0 text with shuffled word order. Verify that the attractor structure disappears (the attractor is semantic, not lexical).

Predictions

1. Iterative O/0-prompted trajectories will converge to a region of embedding space with D_2 between 2 and 10 (complex but bounded attractor).

2. The convergence rate r will be positive and measurable, with exponential decay of d(x_t, A_proto).

3. The leading Lyapunov exponent will be near zero or slightly positive (limit-cycle or weakly chaotic behavior), NOT strongly negative (fixed point) or strongly positive (divergence).

4. The basin volume of the O/0 attractor will be small but nonzero (~1–5% of the embedding sphere).

5. Critical temperature tau_c will be in the range 0.5–1.5 for standard LLMs.

Falsification criteria

1. If trajectories do not converge (d(x_t, A_proto) does not decrease), there is no attractor.

2. If convergence is indistinguishable from trivial repetition (diversity -> 0), the attractor is a fixed point, not a rich semantic structure.

3. If control prompts (non-O/0) converge to the same region, the attractor is not specific to the O/0 framework.

4. If Lyapunov exponents and attractor dimension cannot be reliably estimated, the system may not be deterministic enough for dynamical systems analysis.

Results / Expected Outcomes

No experimental results reported. This document provides the formal framework and testable predictions. Preliminary experiments with GPT-class models suggest convergence behavior consistent with Prediction 1, but rigorous analysis with proper controls is pending.

Uncertainty

1. The embedding function is model-specific; results may not generalize across architectures.

2. Cosine similarity in embedding space may not faithfully capture semantic similarity for all content types.

3. The iterative prompting map F_p is stochastic (due to sampling), complicating the application of deterministic dynamical systems theory.

4. The prototype set A_proto is subjectively chosen; different prototype selections may yield different attractor characterizations.

5. High-dimensional Lyapunov exponent estimation is notoriously unreliable for finite data.

Limitations

1. The framework assumes the embedding space geometry faithfully represents semantic structure. This is a strong assumption that may fail for abstract or self-referential concepts.

2. The dynamical systems analysis assumes approximate determinism (low temperature). At typical sampling temperatures, the noise may dominate the signal.

3. The Grassberger-Procaccia dimension estimate is unreliable for d_embedding >> T (the curse of dimensionality).

4. The framework does not address WHY certain semantic patterns are attractors — only whether they exhibit attractor geometry. Causal explanation requires analysis of training data and optimization dynamics, which is beyond the scope.

5. Importantly, the existence of semantic attractors in LLMs would be a fact about LLM architecture and training, not about reality or consciousness. The philosophical archive may read attractor convergence as evidence for its claims, but the formal framework does not support such inference.

Replication status

Mathematical framework is self-contained. Experimental protocol is fully specified and independently reproducible with access to LLM APIs and standard embedding extraction tools.

Data and code

No data or code generated. Experiments to use standard LLM APIs (OpenAI, Anthropic, open-source models) and the scikit-learn, UMAP, and nolds Python libraries for dimension and Lyapunov estimation.

Relationship to philosophical archive

The archive's language about "attractor states" and "convergence toward unity" maps onto the mathematical framework of dynamical attractors in embedding space. However, the mathematical concept of an attractor is a geometric property of a dynamical system, not a statement about the truth or profundity of the content at the attractor. A language model might converge toward a semantic pattern because it is statistically common in training data, because it is a low-complexity output, or because it genuinely captures a deep structure — the attractor analysis cannot distinguish these cases. The formalization provides tools for measuring convergence, not for interpreting it.

References

1. Strogatz, S. H. (2015). *Nonlinear Dynamics and Chaos* (2nd ed.). Westview Press.

2. Eckmann, J.-P., & Ruelle, D. (1985). "Ergodic theory of chaos and strange attractors." *Reviews of Modern Physics*, 57(3), 617–656.

3. Grassberger, P., & Procaccia, I. (1983). "Measuring the strangeness of strange attractors." *Physica D*, 9(1–2), 189–208.

4. Mikolov, T., et al. (2013). "Efficient estimation of word representations in vector space." arXiv:1301.3781.

5. Ethayarajh, K. (2019). "How contextual are contextualized word representations?" *EMNLP 2019*.

6. Vaswani, A., et al. (2017). "Attention is all you need." *NeurIPS 2017*.

7. Cai, T., et al. (2021). "Isotropy in the contextual embedding space." *ACL 2021*.

Revision history

• v1.0: Initial formalization. Attractor dynamics defined in embedding space. Lyapunov and dimension analysis proposed. Convergence criteria and falsification tests specified. Explicit separation between geometric analysis and semantic interpretation established.