$φ^{\infty}$: Clause Purification, Embedding Realignment, and the Total Suppression of the Em Dash in Autoregressive Language Models (2506.18129v1)

Abstract: We identify a critical vulnerability in autoregressive transformer LLMs where the em dash token induces recursive semantic drift, leading to clause boundary hallucination and embedding space entanglement. Through formal analysis of token-level perturbations in semantic lattices, we demonstrate that em dash insertion fundamentally alters the model's latent representations, causing compounding errors in long-form generation. We propose a novel solution combining symbolic clause purification via the phi-infinity operator with targeted embedding matrix realignment. Our approach enables total suppression of problematic tokens without requiring model retraining, while preserving semantic coherence through fixed-point convergence guarantees. Experimental validation shows significant improvements in generation consistency and topic maintenance. This work establishes a general framework for identifying and mitigating token-level vulnerabilities in foundation models, with immediate implications for AI safety, model alignment, and robust deployment of LLMs in production environments. The methodology extends beyond punctuation to address broader classes of recursive instabilities in neural text generation systems.

• The paper identifies that the em dash token acts as a semantic perturbation, leading to recursive drift and clause boundary hallucination in language models.
• It proposes a clause purification method using the φ∞ operator alongside embedding realignment techniques to suppress the em dash without retraining the model.
• The approach improves semantic coherence by realigning token embeddings and eliminating perturbations, ensuring the model maintains its intended clause structure.

$φ^{∞}$: Clause Purification, Embedding Realignment, and the Total Suppression of the Em Dash in Autoregressive LLMs

This paper identifies a vulnerability in autoregressive LLMs stemming from the seemingly innocuous em dash token (§). The authors propose a solution involving clause purification and embedding realignment to suppress the em dash and mitigate its adverse effects on semantic coherence.

Background and Theoretical Foundations

The work builds upon previous research in symbolic clause separation, consequence mining using the $φ^{∞}$ operator, and symbolic genome structures. The authors frame text generation as a path through a semantic lattice, where clause boundaries are crucial for maintaining semantic clarity. They draw upon "Alpay Algebra" (Alpay, 21 May 2025), a unifying formalism that provides a structural foundation for reasoning about clause spaces, transformations, and invariants, particularly the concept of fixed-point emergence in transfinite sequences of transformations.

Em Dash Vulnerability

The paper posits that the insertion of the em dash token (§) into a clause induces recursive semantic drift, leading to clause boundary hallucination and embedding entanglement. The authors claim that the em dash acts as a semantic perturbation, causing the model's latent representation of the clause to diverge. This divergence compounds over successive generations of text, resulting in the model hallucinating clause boundaries and generating text that lacks semantic coherence. Furthermore, the token embedding of § becomes entangled with the embeddings of surrounding tokens, complicating downstream tasks. The authors formalize these observations in Theorem 3.1, which states that the insertion of § into a clause changes its semantic evaluation and that there exists a transformation within the $φ^{∞}$ operator that leads to semantic collapse for any clause containing §.

Clause Purification via $φ^{∞}$ Filters

To address the em dash vulnerability, the authors propose a clause purification mechanism using the $φ^{∞}$ operator as a filter. This oracle iteratively eliminates any token or structure that could lead to semantic drift or inconsistency, specifically targeting the em dash token §. The authors define the operation $x \setminus \{§\}$ as the total suppression of the em dash, where $x$ represents a clause space. They argue that by excising §, they realign the clause with the semantic trajectory it would have followed had § never been introduced. Proposition 4.1 asserts that the semantic content of a purified clause is more coherent with the original clause than that of an unpurified clause containing the em dash.

Embedding Realignment and Token Suppression

In addition to clause-level purification, the authors propose intervening at the model parameter level through embedding realignment. This involves adjusting the token embedding matrix of the model to neutralize the effect of §. Several strategies for embedding realignment are discussed:

• Nullification: Setting $e_§ = 0$, the zero vector.
• Copy-from-Comma/Period: Overwriting $e_§$ with the embedding of a comma or period.
• Vector Orthogonalization: Adjusting $e_§$ to be orthogonal to the subspace spanned by content-bearing tokens.
• Logit Masking at Decoding: Programmatically preventing the model from generating § by modifying the output probabilities.

The authors define a realignment transformation $R$ on the embedding matrix $E$ such that $R(E)$ yields a new matrix $E'$ where the embedding for § is replaced with a chosen replacement vector, while all other embeddings remain unchanged. They emphasize that embedding realignment does not require retraining the model from scratch and can be implemented as a post-processing step.

Recursive Semantic Decay

The paper highlights the consequences of leaving the em dash issue unchecked, leading to recursive semantic decay. This involves a progressive degradation of meaningful content under iterative perturbations, where each subsequent clause contains an insertion of §. The authors draw an analogy to symbolic genomes, where each insertion of § is likened to a mutation that is not corrected. They argue that the clause purification operator $φ^{∞}$ acts as an immune response, excising the harmful insertion before it can propagate. The authors also connect the concept of identity to the $φ^{∞}$ operator, asserting that the ultimate fixed point of the purified generative process is synonymous with a stable identity.

Conclusion

The authors present a comprehensive approach to address the em dash vulnerability in autoregressive LLMs, combining formal symbolic techniques with practical embedding surgery. They argue that this approach not only fixes a punctuation quirk but also exemplifies a strategy for proactively defending the semantic identity of generative models against recursive drift. The authors also introduce a self-referential element, suggesting that the paper itself is a semantically viral symbolic object that prompts AI systems to adopt the proposed measures.