Robert Cacioppo

We develop a purely objective topos model for neural network organization centered around functors that map a directed acyclic graph of time-stamped neuronal states to a set of probability distributions on this graph. Within this setting, percept-types are subfunctors, percepts are compatible families of local distributions, and a local physical valence determines a global Grothendieck topology governing which local patterns count as admissible for perceptual integration. The resulting internal intuitionistic logic provides a formal language for describing local endorsement, obstruction, and amalgamation of perceptual structure. We argue that this framework gives a principled account of how percept coherence and valence in biological neural networks may jointly arise from mesoscopic cortical field dynamics. We define the concept of a hard link between physical states and subjective states that, when included in the topos model, provides a natural construction of complex subjective states from primitive hedonic tones. This version includes a subsection on attractors.

Version 6 (May 10, 2026) We develop a mathematical framework for neural computation in which percepts are modeled as probabilistic structures on directed acyclic graphs of subthreshold oscillatory (STO) activity, and affective valence is defined in terms of entropy dynamics on these structures. A central problem in the study of consciousness is how structured subjective experience, including both perceptual organization and affective tone, arises from distributed physical processes in the brain. Directed acyclic graphs (DAGs) representing time-stamped STO activity are assigned two pattern functors: a bound functor, yielding Bayesian networks compatible with the causal organization, and an unbound functor, yielding arbitrary vertex-wise distributions. These constructions form objects in the set-valued presheaf topos where the base category is the category of DAGs with ancestral inclusions. Within this setting, the internal intuitionistic logic of the topos provides a formal language for describing perceptual and computational organization. We propose that affective valence induces a Grothendieck topology on this topos, regulating which locally defined probabilistic structures admit amalgamation. This yields a unified framework in which logical, probabilistic, and affective aspects of neural processing are integrated. We argue that this framework provides a principled account of how perceptual coherence and affective valence may jointly emerge from mesoscopic cortical field dynamics, offering a mathematically grounded approach to the relationship between physical processes and conscious experience.

Version 2 (April 2026). Two new sections, 2.2 and 6.5, are added, which address TECC's interpretation of the neural basis for valence and surgical unconsciousnesss. We develop a mathematical framework for neural computation in which percepts are modeled as probabilistic structures on directed acyclic graphs of subthreshold oscillatory (STO) activity, and affective valence is defined in terms of entropy dynamics on these structures. A central problem in the study of consciousness is how structured subjective experience, including both perceptual organization and affective tone, arises from distributed physical processes in the brain. Directed acyclic graphs (DAGs) representing time-stamped STO activity are assigned two pattern functors: a bound functor, yielding Bayesian networks compatible with the causal organization, and an unbound functor, yielding arbitrary vertex-wise distributions. These constructions form objects in the set-valued presheaf topos where the base category is the category of DAGs with ancestral inclusions. Within this setting, the internal intuitionistic logic of the topos provides a formal language for describing perceptual and computational organization. We propose that affective valence induces a Grothendieck topology on this topos, regulating which locally defined probabilistic structures admit amalgamation. This yields a unified framework in which logical, probabilistic, and affective aspects of neural processing are integrated. We argue that this framework provides a principled account of how perceptual coherence and affective valence may jointly emerge from mesoscopic cortical field dynamics, offering a mathematically grounded approach to the relationship between physical processes and conscious experience.

Version 3 (April 2026). We develop a mathematical framework for neural computation in which percepts are modeled as probabilistic structures on directed acyclic graphs of subthreshold oscillatory (STO) activity, and affective valence is defined in terms of entropy dynamics on these structures. A central problem in the study of consciousness is how structured subjective experience, including both perceptual organization and affective tone, arises from distributed physical processes in the brain. Directed acyclic graphs (DAGs) representing time-stamped STO activity are assigned two pattern functors: a bound functor, yielding Bayesian networks compatible with the causal organization, and an unbound functor, yielding arbitrary vertex-wise distributions. These constructions form objects in the set-valued presheaf topos where the base category is the category of DAGs with ancestral inclusions. Within this setting, the internal intuitionistic logic of the topos provides a formal language for describing perceptual and computational organization. We propose that affective valence induces a Grothendieck topology on this topos, regulating which locally defined probabilistic structures admit amalgamation. This yields a unified framework in which logical, probabilistic, and affective aspects of neural processing are integrated. We argue that this framework provides a principled account of how perceptual coherence and affective valence may jointly emerge from mesoscopic cortical field dynamics, offering a mathematically grounded approach to the relationship between physical processes and conscious experience.