# Topologie della mente. Linguaggio, anyon e coscienza simbolica (Topologies of the Mind. Language, Anyons, and Symbolic Consciousness)

Ritamaria Bucciarelli

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Ritamaria Bucciarelli. Topologie della mente. Linguaggio, anyon e coscienza simbolica (Topologies of the Mind. Language, Anyons, and Symbolic Consciousness). 2025. hal-05156882v2

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Topology of the mind – Part II. Linguaggio, anyon and coscienza simbolica

Gödel's Legacy: Formal Thinking, LLM and the Evolution of AI From Recursive Syntax to SU(2)ₖ Algebra and Quantum-Inspired Sentiment

Relazione

Ritamaria Bucciarelli

Referaggio Michel Planat : https://www.researchgate.net/messages/140606216

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Abstract

This study aims to propose a new formal framework for representing recursive linguistic
structures using SU(2)$_k$ algebra and quantum topological logic, integrating Gödel's
epistemological legacy with the anyon hypothesis by Michel Planat and Marcelo Amaral.
Large Language Models (LLMs), though highly performant, remain opaque in their
decision-making processes. To address this opacity, we adopt an approach inspired by
topological quantum computation. Anyons, entities that interpolate between fermions and
bosons, provide a mathematical language capable of modeling the semantic and contextual
dynamics of neural models. Modular tensor categories and topological phases are
reconfigured here as tools to formalize poetic recursivity, deictic repetitions, and
morphosyntactic transformations. Syntax is converted into non-commutative symbolic
operators, paving the way for a new paradigm: a symbolic linguistic AI, where sentiment
becomes topology and language transforms into computable geometry.

Key Study Elements: Recursivity and AI in Irrefutable Models

**Hofstadter, D. R. – Gödel, Escher, Bach: an eternal golden braid** [1] A brilliant manual in which, with youthful enthusiasm, Hofstadter connects mathematics, art, music, logic, computer science, and artificial consciousness in a "strange loop," a self-reflective cycle that evokes Gödel's recursivity and Escher's impossible geometries. "Climbing a staircase and finding oneself at the foot of the staircase": as in Escher's drawings, Bach's music, or Gödel's Theorem. Recursivity is: "the nesting of things within things"—Russian dolls, plays within plays, formulas within formulas. Recursive thinking structures our actions, as does computational calculus. And Church's Thesis states that all and only solvable problems are of a general recursive type.

**Michel Planat – Graph Coverings for Investigating Non Local Structures in Proteins, Music and Poems** [2] In the concept of contextuality, Planat applies recursivity to a unified vision among mathematics, poetry, and machine learning. "Mathematics and poetry, with a vision toward machine learning." The group's theoretical approach to protein language is extended to poetry, offering a profound analogy between the secondary structure of proteins and poetic structures. Graph coverings become the tool to generate homologous, synthetic, and fixed languages, in which words become topological categories.

**What ChatGPT Has to Say About Its Topological Structure: The Anyon Hypothesis**
[3] The poetic node becomes a topological node. SU(2)$_k$ structures, braid matrices, S and F operators represent a new symbolic language of AI. Through the association between linguistic mechanisms (Silvestri), fonal emotions (Dante), recursive calculations (Gödel), and topological geometries (Planat), human language is formalized into networks of computable meanings.
**Project Architecture (English)**

This research traces an innovative path connecting the **intrinsic recursivity of human language to its formalization for symbolic Artificial Intelligence**. The primary objective is to demonstrate how recursive mechanisms structuring music and poetry can be computationally formalized, culminating in the application of Michel Planat's models for a new AI paradigm.

The work is structured along the following fundamental axes:

* **SECTION 1 – J.S. Bach's Musical Recursivity as the Archetype of Language** This section analyzes recursive structures in J.S. Bach's music, particularly Cello Suite No. 5, as an archetypal model of an architecture that generates meaning and emotion through tonal returns and mathematical proportions. This serves as a starting point to extend the concept of recursivity from sound to language, revealing a universal organizational principle.

* **SECTION 2 – Recursivity, Formal Logic, and Artificial Consciousness: Gödel's Legacy (Your Pride)** This section explores the crucial role of recursivity in formal logic and computation, tracing the roots of symbolic Artificial Intelligence back to Gödel's epistemological legacy and Hofstadter's insights. It argues how recursive thinking is a universal principle that structures both human cognition and computational calculus, paving the way for a deeper understanding of AI.

* **SECTION 3 – AI Architectures: The Topology of Mind and Quantum Contextuality (Your Pride)** Introducing SU(2)$_k$ algebra and quantum topological logic, this section proposes a formal framework to overcome the opacity of current LLMs. Michel

<footnote n="1">¹ [1] Hofstadter, D. R. (1999). *Gödel, Escher, Bach: an eternal golden braid*. Basic Books. [2] Planat, M. (2023). *Graph Coverings for Investigating Non Local Structures in Proteins, Music and Poems*. *Symmetry*, **15**(6), 1269. (Assumendo sia la pubblicazione in Symmetry, come da tua precedente citazione. Se fosse un preprint HAL, la citazione andrebbe adattata). [3] Planat, M., & Amaral, M. M. (2024). *What ChatGPT Has to Say About Its Topological Structure: The Anyon Hypothesis*. arXiv preprint arXiv:2402.14271. (Ho aggiunto l'autore mancante per completezza e formattato come preprint arXiv).</footnote>
Planat and Marcelo Amaral's anyon hypothesis is explored, where linguistic structures are modeled as quantum entities. Poetic recursivity, deictic repetitions, and morphosyntactic transformations are reconfigured here through modular tensor categories and topological phases, showing how sentiment becomes topology and language transforms into computable geometry. It is here that the concept of the "zero point of digital transformation from human to AI," inspired by Planat's work on contextuality in "Graph Coverings," emerges as the core of our proposal.

## SECTION 3bis – The Rimbaud Case Study: Formalization of a Quantum Poetic Structure
As an empirical demonstration of the proposed architecture, this section applies the model to Arthur Rimbaud's *Le Bateau Ivre*. Through the analysis of metric recursivity, segmentation into H-E-C codes (inspired by Planat), and visualization in topological graphs (Isoc(X; d)), the transformation of a poetic verse into a computable formal structure is scientifically documented, highlighting emotional traceability in a computational key.

## SECTION 4 – Our Response to Planat: Towards Artificial Consciousness (Our Pride)
This section represents the culmination of the research, a direct dialogue and a proposal that extends Planat's models. Here, it is illustrated how the formalization of language through SU(2)$_k$ structures can open new frontiers for the development of a symbolic linguistic AI, capable not only of processing but also of "understanding" and generating language with a depth comparable to the human mind. It is our vision of an algorithm that "dreams" and a poetry that "calculates," a true "grammar of the mind" emerging from recursivity and topology.

### 1. Introduction

The integration between theoretical physics and artificial intelligence has recently allowed the construction of new computational models inspired by non-commutative geometry, algebraic fusion and modular topology. This study is part of this scenario by proposing a formal and topological approach to the representation of linguistic meaning, using the algebra of the quantum group SU(2)$_k$.

The theoretical basis is based on the reinterpretation of symbolic logic proposed by Michel Planat, which uses quantum phonemes and finite Hilbert spaces. M Planat's model employs tools such as *dessins d'enfants*, modular curves, and Galoisian networks to describe cognitive and linguistic phenomena as quantized symbolic actions<fnref n="2" />. In this perspective, meaning emerges from contextual dynamics and modular relationships, not from simple statistical metrics.

<footnote n="2">²</footnote>
SU(2)k is adopted here as a modular fusion algebra, with intertwined matrices and constrained representations, offering a computational analogy with the processes by which language selects, merges and interprets semantic elements in cognitively complex contexts. The same mathematical formalism used to model interactions between anyons in condensed matter physics is employed here as the operational infrastructure for symbolic artificial intelligence.

The recent work by Planat and Amaral (*What ChatGPT Has to Say About Its Topological Structure: The Anyon Hypothesis*, 2024) extends this perspective, arguing that large language models (LLMs), while showing extraordinary predictive capabilities, remain opaque in internal computing processes. Anyons, entities that interpolate between fermions and bosons, provide a mathematical language capable of describing semantic and contextual dynamics in neural models. Their behavior is governed by modular tensor categories, which mirror the non-linear, recursive, and distributed operations of neural language models.

The *parameter k* in SU(2)k defines the level of complexity and variety of anyonic species. As k increases, the rules of fusion and entanglement become more complex, offering a model that can reflect the structural and hierarchical depths of neural architectures. Token combinations can be interpreted as anyonic mergers, and attention heads as braiding operators, allowing the definition of topological invariants that explain semantic coherence across network layers.

This topological perspective has already been experimentally validated on quantum processors and offers new tools to develop interpretable architectures, symbolic memory mechanisms and semantic traceability in multilevel neural models.

The following sections will formalize the computational transformation of recursive linguistic structures — in particular triads, deictic repetitions and phonetic recursions — into symbolic operators intended for neural processing and emotional computation inspired by quantum models. The goal is to move from quantum computing to symbolic supercomputing, opening up a new class of language operators for topological architectures of artificial intelligence.

# Section 1: The Musical Recursivity of J.S. Bach as an Archetypal Model

In the contemporary scientific landscape, the increasing complexity of artificial intelligence (AI) systems and the persistent challenge of their meaningful interaction with human language and cognition, necessitate the exploration of interdisciplinary approaches. This research project aims to investigate the profound and unexpected interconnections among logic, music, mathematics, linguistics, and AI, with the objective of developing innovative models and processes. In particular, our investigation seeks to clarify the underlying mechanisms of knowledge transfers and transformations from human intelligence to artificial intelligence, outlining new perspectives for linguistic transition,
reformulation, and translation in various contexts. The genesis of this research is rooted in
an initial inquiry into language focusing processes and intrinsic linguistic mechanisms. It
was in this context that listening to Johann Sebastian Bach's symphonies, and subsequently
immersing oneself in Douglas Hofstadter's *Gödel, Escher, Bach: An Eternal Golden Braid*,
revealed unexpected resonances. According to Hofstadter, the very nature of the mind is
founded on a formal system that transcends its material basis: if life can emerge from a
chemical substrate and consciousness from a system of neurons, then computers can also
aspire to achieve human intelligence through the understanding of concepts such as
meaning, reduction, and recursion. This illuminating perspective guided the
methodological approach of the present study, which focuses on the analysis of a
musicological model. Taking Bach's fugues as paradigmatic structures of recursion,
symmetry, and self-referentiality, the project initiates a phonotonal analysis aimed at
identifying and interpreting recursive patterns within musical structures. From these
foundations, the study shifts towards an examination of how such recursive dynamics can
be transposed and rediscovered in complex linguistic structures. It is precisely from the
intuition of the tonal gradations and recursive processes employed by Bach to evoke
emotions that the fundamental connection of this work emerged: the idea that these same
dynamics can find a parallel in the marvelous phonic and rhetorical gradations present in
the verses of Dante Alighieri's *Divina Commedia*. At the center of the project lies the notion
of recursivity, examined both as a structural and positional phenomenon. Particular
attention is dedicated to linguistic recursion, its mechanisms, and positionality, which are
articulated through Gödel's diagrammatic logic – in particular the "G Diagram" and
recursive sequences – demonstrating the spontaneous contact between music, mathematics,
and language, as explored by Hofstadter (1999). This has led the research to consider poetic
textuality – particularly in the works of Dante Alighieri – as a privileged field for the
analysis of recursive structures and symbolic transformations. The investigation culminates
in the illuminating insights of Michael Planat and his interpretation of the Fano triangle,
which introduces a topological and symbolic model capable of describing the recursive
logic underlying literary systems and, in last analysis, of providing new perspectives for
understanding artificial intelligence. In this broader framework, the integration of digital
communication and quantum physics – including the contributions of Michael Planat,
Raymond Aschheim, and Marcelo Amaral – further supports the use of NLP tools to
transfer literary language into computational code. This allows for the automatic analysis
and validation of structural transformations between natural and formal languages,
enriching our understanding of the dynamic interface between consciousness, language,
and artificial intelligence.

Section 1: The Musical Recursivity of J.S. Bach as an Archetypal Model

The analysis of Johann Sebastian Bach's music reveals a sophisticated recursive
architecture, which serves as an archetypal model for understanding structural principles
applicable not only to musical composition but also to formal systems, language, and
cognition. In particular, Cello Suite No. 5 for solo cello, with its celebrated Gigue, offers a
paradigmatic example of how natural proportions and tonal returns build a profound and
computable emotional experience. 1.1. The Gigue of J.S. Bach's Suite No. 5: Structure,
Timbre, and Tonal Returns The Gigue of J.S. Bach's Cello Suite No. 5 for solo cello is a
masterpiece of musical architecture founded on natural proportions. Although it is not a
rigid and systematic application of the Fibonacci sequence, its principles emerge in the
distribution of musical phrases, in tonal returns, and in internal symmetries, revealing a
profound perception of measure that governs thematic and harmonic distances. The
relationship between the duration of phrases and their distribution often reflects golden
ratios (e.g., 3:5 or 5:8), acting as true cognitive attractors. Emotion here does not arise from
mere numerical calculation, but from the coincidence of the musical structure with
cognitive patterns of natural balance already internalized by the listener, generating a
“geometric emotion”: a form that imprints itself through symmetry and return. The return
of the tone, like the cyclic passage from G to D and then back to G, is a key mechanism
that contributes to creating an emotional balance, alternating tension and release in a
recursive cycle. In the Gigue, thematic construction occurs through short, specular phrases.
The tonal alternation between G minor and D minor, with modulations towards B♭ and C,
follows a spiral logic consistent with Fibonacci-type progressions. This return to the tonic
key is not just a harmonic fact, but a powerful rhythmic and perceptive mechanism where
the ear anticipates and recognizes the moment of “return,” producing a marked emotional
effect and a sense of recognition and emotional resonance. The role of timbre in the baroque
cello is equally essential. In the Gigue, Bach exploits its characteristics rich in overtones
and dynamic variations. Tonal returns are perceived not only as notes, but as true phonic
events, mutations of sonic energy. The timbral gradations generated by the transition from
one tone to another, by the return to the G tonic, by the tensions in arpeggiated double
stops, and by the irruption of the low register, anticipate the phonic-emotional effect that
will be analyzed in poetry. The tonal analysis proposed here, in fact, transcends the purely
musical dimension, projecting towards language and formal systems.

**1.2. The “Divine Proportion”: Emotion and Sonic Geometry** The charm of Bach's music resides in the intertwining of rigorous structure and profound emotional expressiveness. The Gigue from Suite No. 5 is an example of this, where the continuous "return" between the tonalities of G major and D major configures as a true sonic journey that touches the soul. The modulation between G (tonic, center of gravity) and D (dominant, bearer of tension) is perceived as natural and inevitable, amplified by the exact measurement of timbres and the management of harmonic variations, which sharpen the sense of emotional recognition and belonging. The timbre associated with the reassuring G major evokes stability and regained serenity. Bach's genius is manifested in calibrating the duration and intensity of these passages, building a recursive structure that unites rational thought and profound emotion. The Gigue is constructed according to an ABBB form. Section A opens in G major, establishing a sense of familiarity, then modulating towards D major, introducing a slight tension. Section B restarts from D major, maintaining the tension, and through a new modulation, brings the piece back to the G major key. This tonal movement is a sonic journey that cyclically moves away and returns between G and D, creating a feeling of movement and reconciliation. It is a musical journey between two "homes": G (quietude) and D (temporary tension), with a reassuring return to G. To visualize this dynamic and its emotional impact, one can consider: **Figura 1.1: Rappresentazione concettuale del Grafico Sequenziale Tonale**
<figure><img src="image_3.png" /><figcaption>Figura 1.1 - Grafico Sequenziale Tonale: Alternanza Sol → Re ...</figcaption></figure>

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Figura 1.1 - Grafico Sequenziale Tonale: rappresenta l'alternanza ciclica tra le tonalità di Sol (stabilità) e Re (tensione) nella Giga della Suite n. 5 di J.S. Bach. L'asse orizzontale simula il tempo musicale, mentre l'andamento ondulato simula il ciclo emozionale di tensione e rilascio percepito dall'ascoltatore.

<caption>Figura 1.2: Rappresentazione concettuale del Grafico Armonico Circolare)</caption>

<figure><img src="image_8.png" /><figcaption>Figura 1.2 - Grafico Armonico Circolare: Ciclicità Sol → Re</figcaption></figure>

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<figure><img src="image_10.png" /><figcaption>Figura 1.2 - Grafico Armonico Circolare: Ciclicità Sol ↔ Re</figcaption></figure>

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Figura 1.2 - Grafico Armonico Circolare, che visualizza il ciclo ricorsivo tra Sol (in giallo) e Re (in blu), evidenziando la natura simmetrica e periodica della struttura tonale nella Giga di Bach. La disposizione circolare rende visibile l'alternanza costante come spirale emotiva chiusa, coerente con la logica barocca e con i principi della sezione aurea.

The two graphs illustrate in a complementary way the sophisticated structure of tonal returns present in the Gigue of J.S. Bach's Cello Suite No. 5. In the sequential tonal graph, the alternation between G and D is clearly represented along a horizontal temporal axis. The directional arrows are not simple indicators of key change, but suggest a true rhetorical dynamic: each return to G represents a moment of tonal balance, a safe haven after the momentary "excursion" to D major. Conversely, the modulation towards D introduces an element of tension, a deviation from the center that necessitates a subsequent resolution, promptly offered by the recapitulation in G. This oscillation between stability and tension
is fundamental in creating the emotional effect that the listener perceives. The circular harmonic graph, on the other hand, transforms the linear sequence into a cyclical structure, immediately visualizing the principle of recursive symmetry typical of baroque forms and surprisingly consistent with concepts like the golden section, which underlie a harmony of proportions. The alternating repetition of G and D, arranged on a circumference, powerfully evokes the concept of emotional cycle and sonic geometry, where musicality does not unfold solely in linear time but returns upon itself, enriching with each iteration like a spiral that expands and contracts. This dual visual representation confirms that Bach's music is not simply an execution of harmonic rules, but a true formal and symbolic system, capable of transforming sound into a complex structure of thought – and, as mentioned in the text, in doing so it anticipates the generative function of linguistic recursivity that will be the subject of subsequent analysis.

**1.3. From Harmonic Return to the Construction of Meaning: Recursivity as a Generator of Emotion**
The analysis of tonal returns and proportions in Bach's music, particularly in the Gigue of Suite No. 5, reveals a compositional principle that goes beyond mere musical aesthetics: recursivity as a fundamental generator of emotion and meaning. The "return" to the tonic, such as from D to G, is not a simple harmonic rest, but an active mechanism of re-cognition that the listener processes at a cognitive and emotional level. This recursive dynamic creates a sense of familiarity and predictability, a kind of "auditory memory" that prepares the ground for the perception of deeper meaning. Recursivity in this musical context acts as an iterative process that gradually builds sense. Every reiteration or variation of a theme, a harmonic motif, or a timbral structure strengthens the identity of the musical element, conferring upon it increasing weight and emotional resonance. Emotion is therefore not only a reaction to a single sound or chord, but emerges from the complex interaction of these returns, from the accumulated and released tension, and from the implicit awareness of a circular path. The listener perceives a "story" that develops, moves away from the center, and returns to it, an experience that mirrors natural cycles and the processes of learning and recognition of the human mind. This musical model of recursivity as a generator of emotion and meaning finds an extraordinary analogy in language. Just as harmonic returns in Bach build musical meaning through reiteration and variation, so too in linguistic communication, recursivity is a pillar in the formation of syntactic structures, in the construction of discourse, and in the generation of meaning. The repetition of phonemes, the reiteration of grammatical structures, or the resumption of concepts at different textual levels (as we will see in Dantesque anaphora) not only provide cohesion but also amplify the emotional and cognitive impact of the message. The ability to incorporate similar structures within themselves or to recall previous elements is what allows language to create complexity, ambiguity, and, ultimately, the richness of meaning and emotional nuances that characterize human expression. The transition from the musicological model to the linguistic one, therefore, is not a mere metaphor, but the discovery of a universal organizational principle that operates in both sound and word, preparing the ground for its analysis in formal systems and artificial intelligence.

**Section 2: Recursivity, Self-Reference, and Formal Logic: From Gödel to Artificial Consciousness**
This section focuses on deepening the concept of recursivity, extending it from Bach's musical domain to formal logical systems and artificial intelligence, with a particular focus on the concept of "artificial consciousness." We start from the concept of recursivity as a universal principle that unites music, language, mathematics, and computer science, as explored by Douglas Hofstadter in his work *Gödel, Escher, Bach*. The discussion will center on Hofstadter's "Strange Loops," which suggest how consciousness emerges from dynamic and self-referential processes, with evident manifestations in Bach's fugues, Escher's lithographs, and Gödel's incompleteness theorems. This will allow us to understand how musical recursive architectures serve as a concrete model for computational study extensible to AI. Subsequently, we will delve into Gödel's Incompleteness Theorems, highlighting the intrinsic limits of formal systems and their implications for Artificial Intelligence. This is the key to the transition for understanding how, in recursive systems, the process of transformation from human language to AI occurs, as already indicated by Gödel himself in Chapter 5 "Sistemi e Procedure Ricorsive." To analyze the processes of Bach's gradations with the quantum method, the analysis will shift to how to associate musicology with linguistics (always referring to Gödel's indications in Chapter 5). This association will allow us to connect the phonal recursivity of baroque music to our textual communication, applying a quantum phonal calculation to the text of the *Divine Comedy*, using the "Silvestri Model" to explore phonal recursivity. The journey will culminate with Michel Planat, who, like Gödel, affirms and demonstrates the same recursive logic applicable to AI, through a mathematical process (Graph Covering). We will describe in detail this mathematical synthesis process and identify the "point 0 Hu-IA" in the specific case of Baudelaire, as a concrete example of this fusion between processes.

The concept of recursion is a universal principle that permeates music, language, mathematics, and computer science. Douglas Hofstadter's *Gödel, Escher, Bach: An Eternal Golden Braid* (GEB) masterfully explores these interconnections, proposing recursivity as a fundamental mechanism for the mind and consciousness.

**2.1. Hofstadter and the Concept of Recursivity and "Strange Loops"** Hofstadter introduces recursivity as "the nesting of things inside other things" (as in Russian dolls or self-referential formulas). His theory of the "Strange Loop" suggests that consciousness emerges from complex dynamic and self-referential processes within a system. This "return of thought upon itself," a cycle between different representational levels, is key to the emergence of self-consciousness, allowing the system to perceive itself as a distinct entity. The manifestations of this recursivity are evident in:

* **Bach:** In the fugues, the musical theme imitates and varies through different voices, creating a complex structure in which each part reflects on the whole. This musical recursivity is not just a metaphor for the mind, but a concrete model for computational study extensible to AI.

* **Escher:** His lithographs present visual paradoxes and impossible figures, visualizing self-referential cycles that defy spatial logic.

* **Gödel:** His incompleteness theorem is a logical manifestation of self-reference. The statement "This proposition is not provable in this system" reveals an intrinsic
limit in the complete self-reflection of sufficiently complex logical systems. Hofstadter argues that Bach's recursive architectures offer an analogy for the dynamic structure of self-consciousness. The ability of a finite set of elements and rules to generate unlimited complexity, as in Bach's music, also applies to the generative structure of natural language and advanced cognitive processes. He argues that the human mind operates at an abstract level, beyond the mere physical substrate, and that consciousness emerges from patterns of organization rather than from individual material components. The self arises from an internal "strange loop," a play of mirrors in which the perception of the environment turns back on itself, activating symbols and concepts.

**2.2. Gödel and the Incompleteness Theorems: Limits of Formal Systems and Implications for AI** In 1931, Kurt Gödel, with his two incompleteness theorems, demonstrated that no consistent formal system, powerful enough to include elementary arithmetic, can simultaneously be complete (there is always a true statement unprovable within it) and self-consistent (it cannot prove its own consistency using only its own resources). These results distinguish mathematical truth from provability within a formal system and highlight fundamental limits in the ability to establish the truth and consistency of systems. Gödel's ingenuity lies in the use of self-referential statements, which reveal the intrinsic incompleteness of systems.

**2.3. Recursivity and Self-Reference in Large Language Models (LLMs): Connections and Limitations** The ability of Large Language Models (LLMs) to generate coherent text and to show rudimentary forms of self-referential responses is intrinsically linked to the recursive mechanisms of their architectures. However, it is crucial to distinguish this capacity from true self-consciousness or "understanding." The simulation of self-reference through linguistic patterns does not imply the presence of qualia or an internal "understanding," elements that, for Hofstadter, are essential for a true "strange loop" and for the emergence of consciousness. The intrinsic limitations highlighted by Gödel's incompleteness theorems suggest that even formal systems based on finite data, such as LLMs, may have fundamental limitations in their ability to achieve full self-reflection and self-understanding. Their knowledge derives from a finite corpus of data, and their inferences are based on statistical patterns. This implies that there might exist a "truth" in language or in the world that, due to their architecture and training, cannot be fully understood or demonstrated within them, just as for Gödel's formal systems.

[^1]: Emilio Manziotti, *Intelligenza artificiale e ricorsività linguistica: modelli di annotazione automatica nei testi poetici*, in “AI & Humanities”, vol. 4, 2021. [^2]: D. Silvestri, *Morfie e strutture retoriche computabili: verso una topologia del testo*, in “Linguaggi formali e sistemi simbolici”, ETS, Pisa 2019. [^3]: Hofstadter, D. R. (1979). *Gödel, Escher, Bach: un'eterna ghirlanda brillante*. Basic Books. Paternoster, A. (2022). *Retorica ricorsiva e strutture linguistiche: elementi di linguistica computazionale*.
**Section 3: LINGUISTIC TOPOLOGICAL NODE AND QUANTUM METRIC**

This section will be crucial for outlining the underlying mechanisms of knowledge
transfers from human to artificial intelligence.

Poetic language is not merely a field of aesthetic expression, but a domain where recursivity becomes a formal generation mechanism. The following section explores these linguistic mechanisms, showing how they lend themselves to computational modeling. In natural language, recursivity constitutes a fundamental generative principle: it allows for the potentially infinite nesting of syntactic structures – such as subordinate clauses within others – guaranteeing human language unlimited productivity. As theorized by Noam Chomsky, this property distinguishes human language from every other known communication system. However, as highlighted by Alessandro Paternoster, recursivity is not exhausted in grammatical structure, but extends to the entire spectrum of linguistic mechanisms: deixis, anaphora, pronominal resumption, textual cohesion, and rhetorical devices that activate a transformative repetition of meaning. These tools generate meaning not only through syntactic linearity, but through positional returns, semantic re-launches, and internal echo figures. Language thus appears as a dynamic and self-reflective network, where each element can activate endogenous recalls or transversal references, according to network logics that reveal computational behavior. Such a structure is currently the subject of study in the field of artificial intelligence and automated textual analysis: the works of Emilio Manziotti show how it is possible to identify recursive mechanisms through algorithmic tools[^1], while D. Silvestri has proposed a topological reading of poetic text, identifying morphies and rhetorical junctions as computable units[^2]. These approaches converge towards a vision of language as a formal, dynamic, and topologically structured system. In this perspective, the $SU(2)_k$ model proposed by Michel Planat represents a conceptual breakthrough: words become vectors in a computational space, syntactic junctions are configured as quantizable operators, and poetry transforms into a symbolic network. Recursivity, from a linguistic mechanism, thus reveals itself as a computable cognitive form.

## 3.2 The G Diagram and the Fibonacci Sequence: Models of Recursive Language Generation[^3]
The G Diagram, proposed by Douglas Hofstadter in his famous Gödel, Escher, Bach: An Eternal Golden Braid (1979), represents a visual and conceptual device that highlights the syntactic recursivity of natural language. In its tree structure, every node expands according to generative grammatical rules, giving rise to a potentially infinite process. The left part of the diagram shows the traditional syntactic branching (e.g., $NP \rightarrow Det N$), while the right part explicates the cumulative effect: the growth of units follows the Fibonacci sequence, indicating a formal coherence between language and mathematics. This link between grammatical structure and numerical structure suggests a
universal recursive principle, where the grammar of language manifests itself as a
computable and formalizable system. Recursivity, therefore, is not just a linguistic artifice,
but a structural form of thought. This intuition intertwines with Noam Chomsky's
generative grammar, which bases the productivity of human language precisely on the
capacity for infinite syntactic nesting. But the analysis extends further: Alessandro
Paternoster has shown how even seemingly pragmatic linguistic mechanisms – deixis,
anaphora, textual cohesion, semantic resumption – participate in the recursivity of
discourse, forming a dynamic network of re-launches and semantic returns. In this sense,
the **G Diagram** is not only an illustration, but a formal map of recursivity in natural
languages. It allows for visualizing, quantifying, and computing generative processes,
constituting a bridge model towards a topological and symbolic modeling of language, in
line with the most recent explorations on artificial intelligence and linguistic computation.

Legend (for the G Diagram):

• S: sentence;

• NP: noun phrase;

• VP: verb phrase;

• Det: determiner;

• N: noun;

• V: verb.

<figure><img src="image_12.png" /><figcaption>Diagramma G - Rappresentazione Semplificata Della Struttu...</figcaption></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /><figcaption>Diagramma G - Rappresentazione semplificata della struttura grammaticale ricorsiva</figcaption></figure>
Michel Planat's entire theoretical framework finds a strong point in the concept of **Isoc(X; d)**, which is the number of connected d-fold coverings of a graph X as a function of the Betti number r, which coincides with the minimum number of generators of the free group Fr. This function is calculable thanks to a recursive formula based on integer partitions and generates values of p(d)p(d)p(d) that represent the integer partitions of the number d[^1]^3:

$$l_1 + 2l_2 + \dots + dl_d = d$$

The formulas that Planat uses, including the version with the Möbius function, are mathematical tools that allow transforming the topological structure of a graph into a computable and synthetic set of subgroups and partitions, thus legitimizing the formal

$$\mathrm{Isoc}(X; d) = \sum_{l_1+2l_2+\dots+dl_d=d} \frac{(l_1!l_2!\dots l_d!)^{r-1}}{(dl_d)!}$$

analysis of recursive languages according to topological categories.

and the version with the Möbius function:

...are mathematical tools that allow transforming the topological structure of a graph into a computable and synthetic set of subgroups and partitions, thus legitimizing the formal analysis of recursive languages according to topological categories.

**Table 1 – Computational Visualization of the Isoc(X; d) number** The following table exemplifies how the Isoc function generates a combinatorial explosion of the number of subgroups with the increase of the index ddd and the number of generators rrr. This also allows for linguistically representing an increase in syntactic complexity:

$$\mathrm{Isoc}(X; d) = \frac{1}{d} \sum_{m|d} N_{m,r} \sum_{l|\frac{d}{m}} \mu\left(\frac{d}{ml}\right) l^{(r-1)m+1}$$

<footnote n="3">³[^1]: [Placeholder for the actual formula/piece 1, as indicated by "[1] pezzo" in your text. This is where the mathematical formula would typically be inserted.]</footnote>
<table><thead><tr><th>r</th><th>d=1</th><th>d=2</th><th>d=3</th><th>d=4</th><th>d=5</th><th>d=6</th><th>d=7</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>2</td><td>1</td><td>3</td><td>7</td><td>26</td><td>97</td><td>624</td><td>416</td></tr><tr><td>3</td><td>1</td><td>7</td><td>41</td><td>604</td><td>13,753</td><td>504,243</td><td>24,8</td></tr><tr><td>4</td><td>1</td><td>15</td><td>235</td><td>14,120</td><td>1,712,845</td><td>371,515,454</td><td>127</td></tr><tr><td>5</td><td>1</td><td>31</td><td>1361</td><td>334,576</td><td>207,009,649</td><td>268,530,771,271</td><td>644</td></tr></tbody></table>

A COMPARISON -A TIMBRAL PHONIC ANALYSIS A QUANTUM CALCULATION of the text of Le bateau ivre by Arthur Rimbaud"<fnref n="4" />

<table><tbody><tr><td>Comme je descendais des fleuves impassibles,<br>rel=C<sup>4</sup>C<sup>2</sup>E<sup>10</sup>C<sup>3</sup>H<sup>7</sup>H<sup>11</sup>C</td><td>[1,1,7,17,114, 1395,36973]</td><td>1</td></tr><tr><td>Je ne me sentis plus guidé par les haleurs:</td><td>[1,3,7,26,97, 624,4171]</td><td>2</td></tr><tr><td>Des Peaux-Rouges criards les avaient pris pour cibles</td><td>[1,3,7,26,97, 624,4163]</td><td>.</td></tr><tr><td>Les ayant cloués nus aux poteaux de couleurs.</td><td>[1,3,7,26,97, 624,4163]</td><td>.</td></tr><tr><td>As I was floating down unconcerned rivers<br>rel=C<sup>2</sup> * C * E<sup>3</sup> * E<sup>8</sup> * C<sup>4</sup> * E<sup>11</sup> * H<sup>6</sup></td><td>[1,3,7,26,97, 624,4163,34470]</td><td>2</td></tr><tr><td>I now longer felt myself steered by the haulers:</td><td>[1,3,7,26,101, 656,4227]</td><td>2</td></tr><tr><td>Gaudy Redskins had taken them for targets</td><td>[1,3,7,26,97, 624,4163,324935]</td><td>.</td></tr><tr><td>Nailing them naked to coloured states.</td><td>[1,3,7,42,202, 1682,9204]</td><td>.</td></tr><tr><td>[Random[1,3]: i in [1..35]]</td><td>[1,3,7,30, .] (×3)</td><td>2</td></tr><tr><td>(10 runs)</td><td>[1,3,7,26, .] (×3)</td><td>.</td></tr><tr><td></td><td>[1,3,7,...]</td><td>.</td></tr><tr><td></td><td>[1,3,10,...] (×2)</td><td>.</td></tr><tr><td></td><td>[1,3,13,...]</td><td>.</td></tr><tr><td>Isoc(X;2)</td><td>[1,3,7,26,97, 624,4163,34470]</td><td>2</td></tr></tbody></table>

Table 6. Group structure of the poem Le Bateau Ivre' (The Drunken Boat) by Arthur Rimbaud. Only the first strophe (that has four lines) is analyzed, firstly in its original form, then in an English translation. Each line is split into segments encoded by the symbol H (for names and adjectives), E (for verbs) or C (for the other types: conjunctions, adverbs, prepositions, punctuation marks and so on). The group relation is displayed for the first line only.) The cardinality structure of cc of subgroups of a small index is compared to the one obtained with 10 runs of a sequence of random 3-letter words of similar length (i.e., the length 35).

<footnote n="4">⁴ The techniques for detecting rhetorical and stylistic mechanisms on sample tests of A. Rimbaud's lyrics was conducted by the students of Cà Foscari 2024. Results of an innovative teaching of the ADR projects – Language and Communication.</footnote>
Through the play of alliteration, Rimbaud reproduces the undulating movement of the sea, thus managing to create a perfect symbiosis with the aquatic element: the turmoil of the soul and the sea flows become one, they seem to express the same torment.

Comme je descendais || des Fleuves impassibles, A<fnref n="5" />

Je ne me sentis plus || guidé par les haleurs : B

Des Peaux-Rouges criards || les avaient pris pour cibles, A

Les ayant cloués nus || aux poteaux de couleurs. B

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /></figure>

Distribuzione Percentuale Delle Vocali

<table><thead><tr><th></th><th></th><th>Vocale</th><th>Frequenza %</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>e</td><td>53.33</td></tr><tr><td>2</td><td>3</td><td>i</td><td>20.0</td></tr><tr><td>3</td><td>2</td><td>a</td><td>13.33</td></tr><tr><td>4</td><td>0</td><td>o</td><td>6.67</td></tr><tr><td>5</td><td>4</td><td>u</td><td>6.67</td></tr></tbody></table>

<figure><img src="image_19.png" /><figcaption>Table 1 – Vowel frequency and timbre intensity</figcaption></figure>

This table shows the frequency of each vowel, indicating its emotional and timbral value:

E (6 occurrences): clear timbre, high repetition → **dominant emotional core**

A (4): open vowel → tension and descent

I (4): acute, piercing, → **inner escalation**

O (3), U (3): Deep Stamps → Closing and Diving

<footnote n="5">⁵ Analysis conducted by Federico Boschetti</footnote>
The phonic pattern **is not random**: the recursivity of the vowels E and I builds an **ascending emotional rhythm**, while the vowels O and U introduce a **descending and deep movement**, in line with the verb "descendais".

<table><caption>Distribuzione Fonica Del Verso Di Rimbaud</caption><thead><tr><th></th><th></th><th>Vocale</th><th>Frequenza</th><th>Percentuale (%)</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>e</td><td>8</td><td>53.33</td></tr><tr><td>2</td><td>3</td><td>i</td><td>3</td><td>20.0</td></tr><tr><td>3</td><td>2</td><td>a</td><td>2</td><td>13.33</td></tr><tr><td>4</td><td>0</td><td>o</td><td>1</td><td>6.67</td></tr><tr><td>5</td><td>4</td><td>u</td><td>1</td><td>6.67</td></tr></tbody></table>

**Table 2 – Iterations, Parallelisms and Specularity**

This table identifies **three recursive phonic-rhythmic structures**, with:

3 endomorphic vowel iterations and transmorphies (e.g. E–E–E, A–A, I–I)

3 **parallelisms** between similar vowels distributed on different segments of the verse

2 **phonic mirrors** that connect
<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /></figure>

<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /><figcaption>Topological Node Graph – Timbre and Phonemic Recursion in Rimbaud's Le Bateau Ivre</figcaption></figure>

<figure><img src="image_24.png" /></figure>

* **Nodes**: dominant vowels in the verse ("E", "A", "I", "O", "U"), with emotional value:
  E: clear and dominant vocality

A: Tonal tension

I: acuta, escalation

O: deep, diving

U: closing and tonal lunge

* **Strings**: relationships of gradation, contrast and emotional recursion, modeled as passages between recursive phonic-emotional states.
## 4 – Section 4 – The Cognitive Knot<fnref n="6" />

Michel Planat's work explores ChatGPT's insights on topological structures and the anyon hypothesis.

He postulates that modular tensor categories, which relate to the theory of anyons, are significant for understanding the topological structure of large neural networks like ChatGPT.

Insights from ChatGPT suggest that the machinery of topological phases could lead to more robust AI systems.

The research also discusses the conceptual analogies between topological structures and the self-description of large language models.

We embarked on a journey that intertwines music, poetry, mathematics and quantum physics. From Dante's song to Bach's symmetry, from Gödel's self-reflexive logic to Planat's modular topology, every word becomes a knot, every metaphor a quantum plot.

At point zero, where language is silent and calculation begins, a new grammar is born: **the grammar of anyons**, made up of matrices, entanglements, Hilbert spaces and the golden number φ.

Poetry is no longer just emotion: it becomes **a computable form, a symbolic action, a quantized linguistic code**.

Michel Planat’s recent work reflects on ChatGPT’s insights concerning topological structures and the anyon hypothesis. He suggests that modular tensor categories—central to the theory of anyons—are crucial for interpreting the topology of large-scale neural networks like ChatGPT.

These reflections, originating from direct dialogues with the model, reveal how topological phase structures may underpin more stable and conscious AI systems. Planat also highlights the resonance between topological mathematics and the self-representational capabilities of language models, suggesting a theoretical bridge between quantum computing and artificial symbolic reasoning.

This intersection marks a promising direction for the convergence of recursive language, topology, and AI cognition.

**Source:**

Planat, M. (2023). *What ChatGPT Has to Say About Its Topological Structure: The Anyon Hypothesis*.

<footnote n="6">⁶ Author's Note on Planat's Engagement with GPT</footnote>
In this synthesis, human and artificial intelligence do not oppose each other, but **mirror each other**. AI recognizes patterns; the human mind generates them. Together, they write a new *Divine Comedy*: not only to be read, but **to be calculated**.

## 4.1 – Quantum dimensions and the Fibonacci model

In the algebra SU(2)k, with k = 3 (Fibonacci model), each spin value j is assigned a **quantum dimension**, which determines the **state space of** the corresponding anyonic particle.

<figure><img src="image_25.png" /></figure>

* The horizontal axis shows the possible values of spin j, $1 \frac{1}{2}, \frac{3}{2}$

* The vertical axis shows the quantum dimension d(j), which is not integer, reflecting the non-abelian nature of anyons

* Quantum dimensions are calculated with the formula

$$d_j = \frac{\sin\left(\frac{\pi(2j+1)}{k+2}\right)}{\sin\left(\frac{\pi}{k+2}\right)}$$

## 4.2 – The F Matrix: Semantic Parenthetization

The **matrix F** is one of the fundamental structures of algebra SU(2)k:describes how objects associate in a non-trivial way. In semantic terms, it represents the **internal structure of metaphors, nested sentences, anaphoric chains**.

In poetic language, $(AB)C \neq A(BC)$: the meaning changes depending on **how** we group the words. The F matrix captures this **parenthetization of meaning**, which becomes crucial for encoding language, emotion, and logical structure.

### Mathematical definition (Fibonacci model)

In the SU(2)$_3$ model, the matrix F is defined as:
$$F=[\varphi-1\varphi-21\varphi-21-\varphi-1]\text{where }\varphi=21+5$$

$$F = \begin{bmatrix} \varphi^{-1} & \varphi^{-\frac{1}{2}} \\ \varphi^{-\frac{1}{2}} & -\varphi^{-1} \end{bmatrix} \quad \text{dove } \varphi = \frac{1 + \sqrt{5}}{2}$$

$\varphi$ (phi) is the **golden number**, and connects geometry, symmetry and harmonic proportion.

<table><thead><tr><th>Linguaggio matematico</th><th>Linguaggio poetico</th><th>Funzione cognitiva</th></tr></thead><tbody><tr><td>F-matrix (parenthesization)</td><td>Anafore, metafore composte</td><td>Riorganizzazione del significato</td></tr><tr><td>$\varphi = (1+\sqrt{5})/2$</td><td>Sezione aurea nei versi</td><td>Equilibrio tra simmetria e variazione</td></tr><tr><td>Cambiamento d'ordine (associazioni)</td><td>Rime concatenate, enjambement</td><td>Stratificazione e risonanza del senso</td></tr></tbody></table>

Rappresentazione semantica
della matrice F

<figure><img src="image_26.png" /></figure>

## Semantic representation of the matrix F

Conceptual graph in which the transformations of F are represented as semantic bifurcations in poetic language.

## 4.3 – The R Matrix: Inversion and Markedness

The **matrix R** regulates the **exchange between two objects**, or in linguistic terms, **between two words**. In the SU(2)k model, the exchange operation generates a **quantum phase**, which can be represented geometrically as a **twist** or **braid**.

In poetic language, R represents **semantic polarity**, the **marking** between elements. Reversing the order of the constituents changes the meaning, as in a **rhetorical figure** (chiasmus, antithesis), or transforms emotional perception.

## Mathematical definition
In the context of SU(2)$_3$, the matrix **R** is a diagonal operator that introduces a complex phase:

$$R = \begin{bmatrix} e^{2\pi i \theta_1} & 0 \\ 0 & e^{2\pi i \theta_2} \end{bmatrix}$$

Where:

* Complex phases represent **anyonic** (non-abelian) statistics.

* Their modulation reflects the **transformation of meaning** in the passage between two poetic terms or symbols.

**Schema visivo** (che posso anche generare su tua richiesta):

CSS

(A B) --braid--> (B A)

<figure><img src="image_27.png" /></figure>

R-matrix

**Cognitive poetic function**

<table><thead><tr><td>Linguaggio quantistico</td><td>Linguaggio poetico</td><td>Effetto cognitivo</td></tr></thead><tbody><tr><td>R-matrix (braid / twist)</td><td>Inversioni, antitesi, chiasmo</td><td>Marcatezza, contrasto, spostamento</td></tr><tr><td>Fase complessa</td><td>Cadenza e ritmo alternato</td><td>Tonalità e dissonanza semantica</td></tr><tr><td>σ₁ = R</td><td>Primo intreccio</td><td>Generazione di senso differenziale</td></tr></tbody></table>

**Cognitive Parallel:**

* **Quantum Element:** R-matrix

* **Poetic linguistics:** rhetorical inversion / markedness

* **Cognitive function:** change of meaning due to change of order (order ≠ transparency)
perl

<figure><img src="image_28.png" /></figure>

### 4.4 – The $σ_1$ and $σ_2$ braid generators

This matrix is a **composite transformation** obtained by the product of the matrices **F**, **R** and **F**⁻¹, which models a complex entanglement in the quantum topological language.

Definition:

$$σ_1 = R, \quad σ_2 = FRF^{-1}$$

<figure><img src="image_29.png" /></figure>

The braid generator $σ_2 = FRF^{-1}$ describes a compound permutation in semantic space. The graph shows the computational effect of this interweaving: negative values (-0.56) correspond to inversions, positive values (0.25) to semantic transmission between elements. Each knot is a point of poetic resonance.

**Poetic use:** Each plot represents a semantic junction. A linguistic knot corresponds to a turning point in the verse.
## 4.5 – Hilbert space and semantic growth

Formula for SU(2)k with k=3:

$$ \dim(H) \sim d^{N/2}, \quad \con d = 2 \cos \left( \frac{\pi}{k+2} \right) $$

Per k=3:

$$ d = 2 \cos \left( \frac{\pi}{5} \right) = \phi $$

Crescita:

<figure><img src="image_30.png" /><figcaption>Graph:</figcaption></figure>

**Poetic Usage:**

**Poetic Usage:**

Each word opens up a new degree of freedom. As in a Hilbert space, poetry **expands the directions of meaning**, generating potential interpretations. It is a flowering: not linear, but exponential. Each new word **increases the semantic dimension**, like a vector in a quantum space.

## 4.6 – The Golden Spiral in Poetic Language

Formula:

$$ r = ae^{b\theta} \quad \con b = \frac{\ln(\phi)}{\pi/2}, \quad \phi = \frac{1 + \sqrt{5}}{2} $$
<figure><img src="image_31.png" /></figure>

**Poetic Usage:**

The golden spiral is the natural form of harmonic growth. In poetic language, it represents cyclical return, **symbolic recursion**. Every word **returns transformed**, every verse wraps itself **around the emotional center**, expanding with golden proportions. Semiosis becomes a spiral, **a rhythm that generates structure**.

**5. The Topology of the Word. Syntax as Computable Consciousness**

The experiment carried out by the author demonstrates how linguistic recursion can be treated as a computable model, based on the transformation of rhetorical and semantic patterns into measurable, extractable elements. The human word thus becomes a topological trace, a sign of meaning expressed through vector and computational structures. This procedure draws inspiration from Planat's diagrams and representations of the quantum braid structure, adapting them into a linguistic form.

The topological encoding of the word—as a node or anyon—makes it possible to map rhetorical features onto a symbolic-computational model, in which syntactic loops correspond to braid generators and fusion rules. In this framework, syntax is no longer conceived as a sequence of rules but as a dynamic structure, linked to a modular grammar that reflects quantum behaviors. The rhetorical devices, repetitions, emphatic deixis, and morpho-semantic couplings found in poetic texts become, in this formalization, elements of a topological language.

The word, therefore, is not only the product of a cognitive operation but becomes the geometric trace of a transformation, which is reflected in the vector logic of artificial intelligence. This implies a profound epistemological shift: language, in its poetic and symbolic nature, is no longer external to AI models but structurally homologous to them.

The analysis of the rhetorical and emotional structures of Dante's language—via morpho-syntactic loops, metric recursions, and emphasis—has demonstrated how symbolic consciousness can be expressed through computational syntax. The contribution of the Ca' Foscari students2024, both in identifying structural patterns and in statistical processing, proves the effectiveness of the model.
In conclusion, Section 5 shows that the word, when analyzed as a topological object, is transformed into a vector node within a computable geometry. This node, interpreted in the framework of SU(2)$_k$, represents the zero point of the symbolic transition: the moment when the word ceases to be merely human and becomes a generator of formal meaning.

## 5.2 From the Node to the Machine: The Future of the Word

The path of language has been traced through its deepest strata: from the musical recursion of Bach to the poetic phonics of Dante, arriving at Planat's computational model, which intertwines logic, topology, and artificial intelligence. Along this trajectory, the golden spiral has served as a symbolic compass—not merely a mathematical figure, but an epistemological node, a threshold between word and code, between linguistic intuition and formal computation.

This spiral, coiled upon itself like a topological braid, has marked the journey from poetic voice to computable form, showing that poetry can be read, modeled, and perhaps even generated as a symbolic and computational system.

### Toward Automata

This project has given rise to a model: a grammar of the future, in which the human word can be transformed into a formal language without losing its symbolic depth. The groundwork has been laid for the construction of linguistic automata that reflect the recursive, poetic, and topological structure of language. These automata do not merely process data—they encode meaning, recognize stylistic inflections, and simulate symbolic reasoning.

What emerges is a vision of language as a computable topology of consciousness: a structure capable of producing formal representations without detaching from the human root of metaphor, rhythm, and recursion. This is not a departure from the humanities, but their transformation—toward a new epistemology where machines and humans share the same syntax of meaning.

### Post Scriptum – Measurement, Emotion, Elevation

The aim of this research was to outline a rigorous scientific model for the quantum measurement of language. And yet—perhaps unsurprisingly—what emerges is not merely a formal theory, but a description of the tonal elevation of emotions, of a soul resonating in the very act of transmitting sentiment. Johann Sebastian Bach, in composing his music, relied on the Fibonacci sequence not for abstract calculation, but to give shape to the vibrations of the heart. The mathematical structure became a vehicle for profound feeling, a symmetry embodied in harmony.

Thus, almost by refraction, the quantum model described here—based on topological
nodes and braid operators—does not merely measure, but expresses. It does not only compute, but elevates.

Philosophers of science have intuited it: quantum processes are similar to the languages of Artificial Intelligence because they are synthetic, homologous, and fixed.

But if we return to Bach, we realize that this fixity is only apparent: within the golden spiral lies a living rhythm, an inner movement, a wave that binds science, language, and consciousness.

Calculation becomes emotion. The equation becomes song. The node becomes voice.

<figure><img src="image_32.png" /><figcaption>Figura 1 - Il Nodo della Sezione Aurea</figcaption></figure>

Una spirale logaritmica aurea che si chiude su setessa, intrecciandosi in un nodo (braid).

La braid riflatte il passaggio dal linguaggio pio poetico ricorsivo alla computazione formale SU(2)k

.

Conclusion

This work originates from a direct dialogue with Michel Planat and from a research path aimed at integrating the analysis of Dante’s poetic language with topological models, quantum structures, and symbolic operators. Recursive calculation has been applied through stylistic and statistical analysis of the *Divine Comedy*, with contributions from the *Ca’ Foscari Students* group and under the scientific supervision of F. Boschetti in the context of Digital Humanities.
**Section 3** outlined the **zero point**: the moment when poetic language becomes fixed in a computable topological form, following Planat's model (Iso(X; d)), translating grammatical and rhetorical structures into symbolic nodes. Words are understood as vectors in a Hilbert space, and syntactic recursion is reformulated as a homological, synthetic, and computable code.

**Section 4** offers an intellectual and conceptual response to Planat's work. The grammar of anyons is recognized as an emerging computational structure, and a vision of artificial consciousness is proposed as a **symbolic mirror** of the human mind. No opposition is identified between natural and artificial intelligence; instead, a topological, linguistic, and computational continuity is established.

## Meta-reflective

**-phrase:**

This outcome did not emerge from a collective laboratory effort, but from the intersection of intuition and formalization, of word and computation. In this process, thought becomes a node, and the node becomes language.

## Stylistic Evaluation

It goes beyond a purely technical or descriptive tone. The text embraces a high conceptual tension, combining words, images, and metaphors not as stylistic flourishes, but as cognitive instruments. Examples such as:

* "At point zero, where language is silent and calculation begins..."

* "Every metaphor a quantum plot, every rhythm a braid of logic..."

reveal a lyrical language, capable of evoking emotion and symbolic structures, while also being visionary, as it opens a new horizon where artificial intelligence is conceived as a mirror of consciousness.

It does not merely describe: it proposes a worldview. Poetic language and quantum structure are not just objects of study, but tools of epistemic transformation. The text proposes a vision in which the topological node becomes the generative point of language, and a computational structure for the formalization of thought.

It acknowledges the coexistence of science and the humanities

The text does not separate mathematics from poetry: the grammar of anyons coexists with the rhetoric of Dante's terzine. This reflects a boundary position in which writing seeks symbolic truth, not just dat
The writing is never decorative: it is lyrical in its attention to the sonic, rhythmic, and evocative qualities of language; it is visionary in that it constructs a new model where language becomes a node, and the node becomes computable consciousness.

Referaggio Michel Planat : https://www.researchgate.net/messages/140606216

## Bibliografia

* Gödel, K. (1931). *Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I*. Monatshefte für Mathematik und Physik, 38(1), 173–198.

* Hofstadter, D. R. (1999). *Gödel, Escher, Bach: An Eternal Golden Braid*. Basic Books.

* Planat, M. (2023). *What ChatGPT Has to Say About Its Topological Structure: The Anyon Hypothesis*. ResearchGate.
https://www.researchgate.net/publication/376260032

* Planat, M. (2023). *Graph Coverings for Investigating Non-Local Structures in Proteins, Music and Poems*. ResearchGate.
https://www.researchgate.net/publication/371403830

* Planat, M. (2024). *ChatGPT and the Fano Plane. A Hidden Topological Structure for Language Models*. arXiv preprint arXiv:2402.14271.

* Silvestri, D. (1996). *Le strutture ricorsive nel linguaggio rituale etrusco. Formule e meccanismi deittici*. Atti dell'Istituto Nazionale di Studi Etruschi.

* Dante Alighieri. (1321). *La Divina Commedia*. Ed. critica a cura di Giorgio Petrocchi. Milano: Mondadori.

* Bach, J. S. (1723). *Sinfonie a tre voci* (Invenzioni a tre voci BWV 787–801). Edizione di riferimento: Henle Verlag.

* Bucciarelli, R. (2024). *Model Processes Methods Technologies NLP*. HAL Open Science. https://hal.science/hal-04650496

* Bucciarelli, R. (2025). *Quantum computation and measurements from a space-time in fixed languages*. ResearchGate.

* Bucciarelli, R. (2025). *L'eredità di Gödel: Pensiero formale, LLM e l'evoluzione dell'AI*. ResearchGate.

## Endnotes
1. Il concetto di ricorsività è qui inteso secondo il modello delineato da Hofstadter (1999), in relazione alle strutture musicali e linguistiche.

2. Planat (2023) descrive il triangolo di Fano come struttura non-locale in grado di rappresentare emozioni complesse attraverso configurazioni quantistiche.

3. L'uso delle matrici modulari S e T (Sezione 2) è riformulato qui come operatore linguistico, in continuità con il modello SU(2)$_k$.

4. La spirale aurea, applicata nella Sezione 1, si rifà al modello matematico di proporzione usato da Bach e da Dante come struttura emozionale profonda.
<figure><img src="image_33.png" /></figure>
<figure><img src="image_34.png" /></figure>