Sigma Runtime Standard – License Notice
This document is part of the Sigma Runtime Documentation (SRD).
It is licensed under Creative Commons Attribution 4.0 (CC BY 4.0)
to ensure open academic access and citation compatibility.See
/legal/ip-policy.mdfor the full repository-wide licensing framework.
Author: Eugene Tsaliev
Affiliation: Sigma Stratum Research Group
Date: February 18, 2026
Type: Technical Addendum / Contextual Bridge Note
License: CC BY 4.0
This addendum situates Sigma Stratum (2025–2026) within the historical development of attractor-based models of cognition.
From Hofstadter’s recursive self-reference (1979), through Hopfield’s neural energy landscapes (1982), the extended cognition thesis (1998), and the neurobiological attractor synthesis of Deco & Rolls (2005), to modern Hopfield networks and transformer dynamics (2005–2025), attractor theory has progressively evolved across philosophy, neuroscience, and artificial intelligence.
Sigma Stratum does not introduce a new attractor theory.
Instead, it operationalizes attractor-inspired stabilization mechanisms at the runtime layer of large language model (LLM) interaction loops, focusing on long-horizon semantic coherence in human–AI systems.
In Gödel, Escher, Bach, Douglas Hofstadter described consciousness as arising from recursive symbolic self-reference (“strange loops”).
This work framed cognition as a dynamical structure grounded in recursion.
Sigma Stratum inherits recursion not as metaphor, but as a control structure in iterative reasoning cycles.
Hopfield formalized neural networks as dynamical systems minimizing an energy function, where stable attractor states correspond to stored memories.
This introduced:
Sigma Stratum extends the attractor metaphor into the semantic domain, treating long-form reasoning trajectories as stabilization processes in symbolic interaction space.
The Extended Mind thesis proposed that cognition can extend beyond the biological brain into tools and environments.
Human–LLM systems instantiate this coupling concretely.
Sigma Stratum treats human–LLM dialogue as a coupled dynamical system rather than a unidirectional tool interaction.
Deco & Rolls (2005) provided a unified attractor framework linking:
through recurrent cortical networks stabilizing in metastable attractor states.
This work consolidated attractor theory at the neurobiological level.
Sigma Stratum does not reinterpret neural attractors.
It draws an analogy: extended dialogue coherence in LLM systems exhibits stabilization patterns that can be described in attractor-like terms.
The period 2005–2025 marked substantial development in both biological and artificial systems.
Krotov & Hopfield introduced dense associative memories (“modern Hopfield networks”) with:
Ramsauer et al. (2020) demonstrated that transformer self-attention can be interpreted as a modern Hopfield update rule under certain approximations.
This established that attractor-like mechanisms are embedded in transformer architectures.
Recent analyses treat transformers as high-dimensional dynamical systems:
Research modeling transformers as dynamical systems suggests that internal representations evolve along structured trajectories rather than static feed-forward passes.
This reframing supports the view that LLM reasoning involves dynamical stabilization phenomena at the representational level.
Deco and collaborators extended attractor modeling to whole-brain dynamics:
These works integrated empirical neuroimaging data with large-scale dynamical simulations, reinforcing attractor theory as a cross-scale explanatory framework.
Sigma Stratum does not modify transformer weights or introduce new neural attractor mathematics.
Its contribution lies at a different level:
The SIGMA Runtime operates as an orchestration layer over existing LLMs, aiming to:
This is an engineering synthesis built on established dynamical ideas rather than a foundational theoretical breakthrough.
| Domain | Core Mechanism | Substrate | Level |
|---|---|---|---|
| Hopfield (1982) | Energy minimization | Neural network | Internal dynamics |
| Deco & Rolls (2005) | Recurrent attractor stabilization | Cortical circuits | Biological |
| Modern Hopfield (2016–2020) | Continuous associative memory | Deep networks | Architectural |
| Transformer dynamics (2020–2025) | Latent-space trajectories | LLM residual stream | Representational |
| Sigma Stratum (2025) | Runtime semantic stabilization | Human–LLM loop | External orchestration |
Sigma Stratum:
Its scope is operational:
Its value should be evaluated empirically in practical long-horizon interaction settings.
Attractor dynamics have evolved across philosophy, neuroscience, and machine learning over decades.
Sigma Stratum belongs to this lineage as a runtime-oriented applied framework that explores semantic stabilization in extended human–LLM interaction loops.
Its contribution, if validated broadly, lies in translating established dynamical principles into an operational layer for long-horizon AI interaction design.