A Categorical Higher Gauge Theory of Mind: Emergent Cognitive Symmetries, Adaptive Neuronal Dynamics, and Observer-Dependent Realities

A Categorical Higher Gauge Theory of Mind: Emergent Cognitive Symmetries, Adaptive Neuronal Dynamics, and Observer-Dependent Realities Andrei T. Patrascu1 1FAST Foundation, Destin FL, 32541, USA email: andrei.patr[email protected] In this paper, we present a novel theoretical framework formulating a categorical higher gauge theory of mind, incorporating neuronal-cognitive structures as adaptive gauge symmetry groups within higher-categorical groupoids. Using detailed mathematical and intuitive constructions, we describe neurons and neuronal clusters as higher-dimensional categorical objects embedded within cognitive manifolds. Gauge fields and higher gauge fields encode neurotransmitter gradients, receptor densities, neuronal plasticity, rhythmic synchronization, and other cognitive observables. Intrinsic neuronal-cognitive processes induce dynamical and adaptive modifications of gauge symmetries, described rigorously through categorical natural transformations. This theoretical construction leads to a class of novel nonlinear partial differential equations, enabling precise, previously unattainable predictions of cognitive and neuronal responses to pharmacological perturbations. We also demonstrate that observer-dependent gauge choices in biology—such as experimentally determined dopamine baselines—are directly analogous to gauge-dependent quantities in physics, such as the color of quarks. Contrary to traditional physics doctrine, we argue that observer-dependent measurements, although gauge-dependent, are fundamentally measurable given a consistent observer framework. We propose that gauge symmetries in physics may similarly be emergent, adaptive, and dynamically evolving, especially in contexts such as Grand Unified Theories (GUT) and the generation of quark mass hierarchies. Our theory bridges neuroscience, cognitive science, pharmacology, theoretical physics, and philosophy, challenging conventional assumptions and proposing a unified categorical-gauge-theoretic paradigm of observer-dependent reality, mind, and matter. INTRODUCTION The quest to understand the human mind encompasses neuroscience, cognitive psychology, theoretical physics, and philosophy, each offering distinct insights. Traditional cognitive models, often based on simplified neural network representations, successfully capture some aspects but frequently neglect the complex geometric and categorical structures inherently present in neuronal-cognitive systems [1, 2]. Recent advances in gauge theories and higher gauge theories, extensively employed in physics, provide robust mathematical frameworks capable of capturing intricate interactions, adaptive dynamics, and symmetry structures, making them ideally suited for modeling cognitive phenomena [3, 4]. Gauge theories historically have described fundamental interactions in physics, notably in electromagnetism and the Standard Model of particle physics, through local symmetries and associated gauge fields [5, 6]. Extending this paradigm, higher gauge theories incorporate higher-dimensional categorical structures, permitting the description of interactions involving strings, membranes, and loops, thus providing tools for capturing the inherently hierarchical and adaptive nature of cognitive-neuronal processes [7, 8]. In this paper, we propose a novel categorical higher gauge theory tailored for neuronal-cognitive dynamics. In our construction, neuronal systems are represented categorically: vertices encode intrinsic neuronal properties such as synaptic densities, receptor distributions, excitability thresholds, and directional signaling characteristics, while edges represent transitions between distinct neuronal states. For instance, a vertex might represent a neuron with specific synaptic density gradients along certain dendritic and axonal pathways, while edges denote transitions between these functional states, reflecting the internal and external dynamics of neuronal interactions. This neuronal-categorical network is embedded into a cognitive manifold, denoted M, representing higher-level cognitive states and functional parameters such as baseline neurotransmitter levels, rhythmic synchronization, and cognitive coherence patterns. Gauge fields and higher gauge fields emerge naturally in this framework, describing neurotransmitter gradients, receptor density dynamics, synaptic strengths, neuronal plasticity, and the coherent activity of neuronal assemblies and cognitive loops [9, 10]. The explicit embedding process provides a rigorous mathematical translation between neuronal-level parameters (vertices and edges) and cognitive-level functional states in manifold M. The adaptive dynamics of neuronal-cognitive systems are represented through dynamical gauge symmetry groups, which differ fundamentally from traditional static gauge groups in physics. These adaptive gauge symmetry groups evolve dynamically, driven by intrinsic biological mechanisms such as neuronal plasticity, synaptic remodeling, receptor regulation, and homeostasis [13, 14]. This adaptive gauge symmetry evolution is rigorously captured through 2 categorical natural transformations, offering a robust mathematical representation of cognitive self-organization and intrinsic neuronal adaptation. Our approach yields explicit nonlinear partial differential equations (PDEs) derived directly from the categorical gauge-theoretic framework, capable of predicting neuronal and cognitive responses to external perturbations, such as pharmacological interventions, with unprecedented precision [11, 12]. These PDEs incorporate terms representing intrinsic neuronal adaptations, such as receptor dynamics, synaptic plasticity, and neurotransmitter modulation, that actively restore neuronal-cognitive gauge symmetry and cognitive equilibrium following perturbations. Furthermore, our theory addresses observer-dependent gauge phenomena, traditionally viewed in physics as fundamentally non-observable, such as the color charge of quarks. We argue that observer-dependent quantities like quark color or neuronal dopamine baselines, though gauge-dependent, can indeed be measured given a consistent observer framework and chosen gauges [15, 16]. This perspective challenges traditional textbook views and suggests that gauge symmetries in physics, particularly in high-energy domains such as Grand Unified Theories (GUT), could themselves be emergent, adaptive, and dynamically evolving structures, analogous to neuronal-cognitive gauge symmetries [17, 18]. This paper is structured as follows: After this comprehensive introduction, we rigorously outline the categorical higher gauge theory of mind, detailing the explicit mathematical constructions of gauge fields, higher gauge fields, and adaptive gauge symmetry groups. We derive the resulting nonlinear PDEs and analyze their predictive capabilities regarding cognitive-neuronal dynamics under various perturbations. Finally, we explore the philosophical and foundational implications of observer-dependent gauge measurements, proposing a unified categorical gauge-theoretic paradigm that connects mind, matter, and observer-dependent realities. NEURONS AS EXTENDED CATEGORICAL OBJECTS: EMBEDDING AND BIOLOGICAL INTERPRETATION Traditionally, neurons are depicted as simple points or nodes within neural networks. However, this simplified representation neglects important structural, functional, and adaptive properties of neurons. In our categorical highergauge theoretical framework, we represent neurons as extended, higher-dimensional objects, specifically simplices, in order to capture such rich features mathematically. Mathematical Definition: Simplicial Structure of Neurons We regard a neuron Nas a simplicial complex defined as follows: •Vertices: Represent biologically significant neuronal states or properties, such as soma (cell body), branching points, or points with specific receptor distributions. •Edges: Represent biologically relevant transitions or connections between vertices, encoding dendritic and axonal segments. Mathematically, a neuron Nis defined as: N={vi}n i=1 ∪{eij |eij = (vi, vj), vi, vj∈N}(1) Vertices vi∈R3encode intrinsic neuronal properties: •First coordinate (x): Synaptic density along specific dendrites. •Second coordinate (y): Receptor density or neurotransmitter responsiveness. •Third coordinate (z): Intrinsic excitability threshold. Edges represent neuronal processes such as dendrites or axons, capturing functional transitions between these vertices. 3 Example of Vertices and Edges Consider a neuron with two biologically significant vertices Aand B: A= (2,3,4), B = (1,5,3) (2) These coordinates have the following biological interpretations: •Vertex A: Moderate synaptic density (x= 2), moderate receptor density (y= 3), relatively high intrinsic excitability (z= 4). •Vertex B: Lower synaptic density (x= 1), high receptor density (y= 5), moderate intrinsic excitability (z= 3). Edges, such as (A, B), represent dendritic or axonal transitions linking these functionally distinct states within the neuron. Embedding Neurons into Cognitive Manifold M We embed neurons from neuronal space X(intrinsic neuronal property space) into a cognitive manifold M, which represents higher-level cognitive functionality, such as memory, sensory processing, and motor actions. Formally, we have an embedding: φ:X→M, φ(vi)∈M(3) Manifold Mis defined as a three-dimensional cognitive functional space with coordinates representing: •First coordinate (XM): Memory-related cognitive functions. •Second coordinate (YM): Sensory processing-related cognitive functions. •Third coordinate (ZM): Motor action-related cognitive functions. Example embedding: φ(A) = (4,7,9), φ(B) = (3,6,1) (4) This embedding translates intrinsic neuronal properties into cognitive functional states: •Vertex Aembedded into high memory processing (XM= 4), very high sensory processing (YM= 7), and high motor action state (ZM= 9). •Vertex Bembedded into moderate memory processing (XM= 3), relatively high sensory processing (YM= 6), and low motor action state (ZM= 1). Detailed Biological Interpretation of Embedding The explicit embedding process translates neuronal-level structural properties into cognitive functional outcomes: Coordinate in Space X Value Example Biological Meaning Synaptic Density (First coordinate) Small (1) Lower connectivity, local processing Large (5) High connectivity, integrative processing Receptor Density (Second coordinate) Small (2) Limited neurotransmitter sensitivity Large (5) High neurotransmitter sensitivity Excitability Threshold (Third coordinate) Small (1) Easily excitable neuron Large (4) Less easily excitable neuron 4 Corresponding embedding into cognitive manifold M: Coordinate in Space M Value Example Cognitive Meaning Memory function (First coordinate) Small (1) Weak memory-related activity Large (9) Strong memory-related activity Sensory processing (Second coordinate) Small (1) Weak sensory processing Large (7) High sensory integration Motor actions (Third coordinate) Small (1) Minimal motor engagement Large (9) High motor coordination Edges, Curvature, and Biological Meaning Under embedding, edges become curves in M, indicating functional transitions between cognitive states. Curvature represents cognitive complexity or functional nonlinearity. For instance, an edge linking high-memory state (vertex A) to low motor-state (vertex B) indicates a neuron bridging cognitive regions associated with memory and motor control. In biological terms: •Zero curvature: simple linear cognitive transition. •High curvature: complex cognitive transitions involving significant integrative processing. Thus, neurons embedded as curved structures in cognitive space Mrepresent integrative cognitive processes linking distinct functional states (e.g., memory and motor function). Concrete Example and Biological Insight A neuron with vertices A(memory region) and B(motor region) embedded, demonstrates biologically how a single neuron connects different cognitive regions, supporting integrative cognitive functions. Synaptic density differences and receptor distribution guide this embedding: A= (2,3,4) φ −→ (4,7,9) B= (1,5,3) φ −→ (3,6,1) (5) Here, vertex Arepresents a memory-intensive activity, while vertex Brepresents a motor-intensive activity. The edge linking them indicates the neuron functionally bridging these cognitive domains. Therefore representing neurons as extended simplicial objects embedded within a cognitive manifold Mprovides rigorous mathematical tools and profound biological insights into neuronal-cognitive dynamics. Explicit coordinate interpretations, embeddings, and curvatures directly connect structural neuronal properties with cognitive functionalities, enabling novel predictive capabilities and deeper neuroscientific understanding. GAUGE AND HIGHER GAUGE THEORY IN NEURONAL AND COGNITIVE SYSTEMS Gauge theories originate from the fundamental description of physical interactions, where fields mediate forces between particles through local symmetry principles. Mathematically, gauge theories rely on gauge invariance, where local transformations of certain fields leave physical observables invariant. Consider a field ψ(x) under a local gauge transformation: ψ(x)→eiα(x)ψ(x).(6) To preserve invariance under local transformations, a gauge field Aµ(x) must be introduced, transforming as: Aµ(x)→Aµ(x) + ∂µα(x).(7) The covariant derivative is defined as: Dµ=∂µ−iAµ(x),(8) 5 ensuring invariance of the action. This idea is based on the invariance of quantum mechanical observables to local modifications of phases. However, the gauge transformations this represents reflects broadly across physics the freedom of choice of inner degrees of freedom without affecting measurable outcomes or physical reality. In a neural cognitive setting, the corresponding motion is a transformation that alters internal neuronal states without changing observable cognitive outcomes. Consider for example dopamine or other neuromodulators, receptor densities, or neuronal sensitivity thresholds. Neurons respond not to absolute dopamine concentration but to how dopamine compares to some baseline levels. Changing the baseline usually can be seen as a biological analogue to the quantum mechanical phase shift. A neuronal gauge transformation could thus be represented as φ(x)→φ(x) + α(x) (9) where φ(x) is the neuronal scalar field (say dopamine concentration, neuromodulator level, etc.) and αis the biological equivalent of a gauge parameter, namely a local shift in baseline. Under this biological gauge transformation absolute dopamine levels like absolute quantum phases, are not intrinsically meaningful or directly observable. What neurons detect is relative shifts or gradients of dopamine level. Just as a quantum mechanical wavefunction’s global phase cannot affect measurable outcomes, neurons do not respond to a uniform baseline shift in dopamine levels, only to spatial or temporal changes. The concept of ”non-observable” here means that a global uniform shift in dopamine or neuromodulators, like a uniform global phase, doesn’t directly affect neuronal computations or cognitive functionality. What neurons measure and respond to are gradients, spatial differences, or deviations from baselines, not absolute values themselves. For example, suppose the baseline dopamine concentration in the entire brain is increased uniformly. Initially neuronal activity might remain unaffected. Eventually, however the neuronal system compensates through homeostatic adjustments (altering receptor densities, plasticity thresholds, or excitabiltiy) thus preserving cognitive coherence. In other words, the gauge freedom in neuronal systems is the freedom to choose baselines of neurotransmitter levels or receptor sensitivities without affecting cognitive outcomes. The brain dynamically adjusts its internal parameters (like receptor densities or neuronal sensitivities) to preserve this gauge invariance. Of course, ultimately these baselines are choices made by observers, and given all observers make the same choice, they are ”gauge dependent”, yet clearly measurable and physical. The gauge freedom itself or the associated symmetry groups are also modified by adaptive mechanisms in biology. This is probably a new aspect of neuronal gauge dynamics that is not obvious in general in the physics of quantum fields. However, dynamical gauge groups, described ultimately by categorical constructions, are impactful in biology but could offer important insights into the physics of fundamental interactions as well. I may briefly note that even though neuronal states typically appear real valued, introducing complex phases is natural and biologically meaningful. A complex phase can act as a gauge transformation in neuronal systems and in fact it is a natural biological approach. Such a complex phase would encode synchronisation and rhythms, where a complex gauge transformation would shift synchronisation patterns by adjusting for example neuronal timing. At the same time, cognitive states can remain invariant under such phase shifts analogous to quantum gauge invariance as cognitive functionality typically depends on relative neuronal timing, not absolute phases. Therefore, aside a potential approximation of the form φ(x)→φ(x) + α(x) in which we consider not only α(x) as a small fractional change in neuronal parameters, but also that φ(x) itself is approximately constant, we can also implement the gauge transformation ψ→eiα(x,t)ψ. We will explore in a future paper the further analogy between neuronal behaviours and quantum mechanics. Higher Gauge Theory Higher gauge theories generalize standard gauge theories to higher-dimensional objects beyond points, incorporating curves, surfaces, and higher-dimensional simplices. This naturally captures collective degrees of freedom and nonlocal phenomena such as loops and surfaces in physical and biological systems. A 2-form gauge field Bµν (x) generalizes the gauge connection Aµ(x), transforming under higher-dimensional gauge symmetries. The corresponding curvature 3-form is rigorously defined as: Hµνρ(x) = ∂µBνρ(x) + ∂νBρµ(x) + ∂ρBµν (x)+[Aµ(x), Bνρ(x)] + [Aν(x), Bρµ(x)] + [Aρ(x), Bµν (x)],(10) capturing the dynamics of higher-order interactions and loops. 6 Neurons and Gauge Theory In our framework, neurons are modeled as extended simplicial objects. Vertices represent neuronal states characterized by intrinsic parameters such as receptor densities, synaptic strengths, and excitability thresholds. Edges represent transitions or connections between these states, with lengths reflecting synaptic strength gradients and functional distances in neuronal parameter space: L(eij) = q(xi−xj)2+ (yi−yj)2+ (zi−zj)2,(11) where (x, y, z) are neuronal parameter coordinates (synaptic density, receptor density, and excitability threshold). Embedding into Cognitive Manifold M The neuronal space Xis embedded into a cognitive manifold Mthrough a continuous mapping φ:X→M. Under this embedding, edges in Xbecome curved paths in M. The embedding maps neuronal properties to cognitive functions such as memory, sensory processing, and motor actions. Mathematically, the embedded length of an edge φ(eij) in Mis given by: LM(φ(eij)) = Zeij rgµν (φ(x))dφµ dx dφν dx dx, (12) where gµν is the metric tensor on Mencoding cognitive complexity. Closed and Open Neural Loops Closed loops represent recurrent neural circuits, fundamental for cognitive functions such as memory encoding, consolidation, and retrieval. These loops are represented by closed simplices whose gauge-invariant holonomy captures intrinsic cognitive processes. The holonomy for a closed loop γis given by: W(γ) = Pexp iIγ Aµdxµ,(13) where Pdenotes path-ordering, encoding cognitive phase shifts and synchronization phenomena in neuronal ensembles. Open neuronal chains can be regarded as representing sensory-to-motor transitions or processing pathways. Their gauge dependence implies cognitive states that require external contextual specification, analogous to sensory-driven motor actions. Higher Gauge Fields and Collective Neural Dynamics Higher gauge theories capture collective neuronal dynamics and emergent synchronization patterns naturally through 2-form fields Bµν . These fields encode loop-level neuronal synchronization, representing collective rhythmic activity such as theta rhythms or gamma oscillations observed in neural data: Bµν (x)↔Rhythmic synchronization states in neural assemblies. (14) The curvature associated with higher gauge fields, Hµνρ, represents cognitive complexity and integrative dynamics at the neural assembly level: Hµνρ(x) = ∂µBνρ(x) + ∂νBρµ(x) + ∂ρBµν (x)+[Aµ(x), Bνρ(x)] + [Aν(x), Bρµ(x)] + [Aρ(x), Bµν (x)],(15) encoding emergent collective behaviors, such as phase locking and metastable states. 7 Metastability, Synchronization, and Memory States Higher gauge fields describe metastable cognitive states—configurations that persist transiently but robustly, characteristic of cognitive functions like working memory or attentional states. These metastable configurations correspond mathematically to minima in cognitive energy landscapes derived from gauge fields. The effective energy functional can be expressed as: Eeff[B] = ZM √g d3x|Hµνρ|2+V(Bµν ),(16) where the potential V(Bµν ) captures topological constraints and interaction effects ensuring metastability. Topological Protection and Neural States Topologically protected states are neuronal-cognitive configurations robust against perturbations due to their topological nature. These arise naturally as topological solitons or instantons within the cognitive manifold M, characterized by quantized invariants like winding numbers or homotopy classes: Q=1 24π2ZM µνρHµνρ d3x. (17) Such invariants ensure cognitive states remain stable and resistant to fluctuations, crucial for long-term memory storage and robust cognitive functionality. GAUGE GROUPS, GAUGE BUNDLES, AND ADAPTIVE GAUGE SYMMETRY IN BIOLOGY Gauge theories utilize the mathematical structure of principal fiber bundles. Formally, a principal fiber bundle is defined as (P, M, π, G), consisting of a total space P, a base manifold M, a projection map π:P→M, and a gauge group G. Each fiber F=π−1(x) over a point x∈Mis isomorphic to G: π−1(x)∼ =G, ∀x∈M. (18) In the biological context, Mcan be considered a manifold representing neuronal or cognitive states, while the fiber F represents internal degrees of freedom, such as local neuronal parameters (receptor densities, neurotransmitter levels). Various choices of local neuronal parameters may lead to the same measurable quantities. Transformations between these choices are gauge transformations that map us vertically along the same fibre. These transformations represents the gauge symmetry associated to that specific fibre F. Why Biological Systems Require Dynamically Evolving Gauge Groups Biological systems exhibit continuous adaptation to both internal and external conditions, necessitating dynamically evolving gauge symmetry groups. Unlike static physical systems, biological entities must continuously adjust parameters such as synaptic strengths, receptor densities, and intrinsic neuronal properties, reflecting adaptive behaviors such as learning, memory formation, and environmental responsiveness. Such changes transform even the symmetries that link various internal configurations on the same Fibre. Therefore one may start with one symmetry group on a fibre, yet once one reaches the next fibre through horizontal transport mediated by a connection, the type of symmetry transformations linking configurations on the next fibre might change. This is a realisation of a variable symmetry group which has to be taken into consideration in biological systems. This adaptive nature implies that the gauge symmetry group itself must evolve dynamically: G(M;t),dG(M;t) dt 6= 0.(19) This dynamical evolution of gauge symmetries represents biological processes such as synaptic plasticity, neuronal selectivity, rhythmic synchronization, and cognitive adaptation. It introduces inherent nonlinearities due to explicit time dependence and adaptive transformations of internal states. 8 Emergence and Dynamics of Cognitive Symmetries Cognitive symmetries emerge dynamically through neuronal interactions and cognitive processes. For instance: •Memory consolidation: synaptic plasticity reinforces particular neuronal circuits, creating stable cognitive states representing memory. •Attention and neuronal selectivity: selective neuronal activation patterns emerge dynamically, defining symmetries that focus cognitive resources. •Rhythmic synchronization: coherent neuronal oscillations spontaneously arise, reflecting global cognitive states. Mathematically, these symmetries emerge as transformations Tacting on the cognitive manifold Mthat preserve essential cognitive relations. The transformations form a symmetry group: T:M→M, with T(M)∼ =M. (20) These transformations dynamically depend on neuronal plasticity, synaptic remodeling, receptor changes, and environmental interactions: T=T(plasticity,environmental conditions,experience).(21) Mathematical Formulation of Dynamical Symmetry Emergence The cognitive symmetry group dynamically evolves according to the changes in neuronal parameters. Initially, we have an initial gauge symmetry group at time t= 0: G(M, t = 0).(22) Neuronal plasticity reshapes synaptic connections and modifies intrinsic neuronal properties, thus inducing a change in cognitive manifold M: Mplasticity −−−−−−→ M0,with modified neuronal parameters.(23) A new symmetry group then emerges, reflecting the new cognitive organization: G(M0;t > 0) 6=G(M;t= 0).(24) To maintain consistency, gauge fields Aµ(x, t) must adapt dynamically according to the new symmetry: Aµ(x, t)→g(x, t)Aµ(x, t)g−1(x, t) + g(x, t)∂µg−1(x, t),(25) where the gauge transformation g(x, t) incorporates time and state-dependent changes: ∂g(x, t) ∂t 6= 0.(26) This explicit dependence introduces additional nonlinearities into the cognitive equations of motion, capturing complex adaptive cognitive phenomena. Example: Neuronal Plasticity and Cognitive Symmetry Dynamics Consider an initial cognitive state represented by symmetry G(M, t = 0), characterized by: •Synaptic strengths: Sij(t= 0) = S0 •Receptor densities: Ri(t= 0) = R0 9 •Excitability thresholds: Ei(t= 0) = E0 Through experience and environmental interaction, neuronal plasticity modifies these parameters: Sij(t > 0) = S0+ ∆Sij(experience),(27) Ri(t > 0) = R0+ ∆Ri(experience),(28) Ei(t > 0) = E0+ ∆Ei(experience).(29) The cognitive manifold Mthus transforms: M→M0,with new cognitive symmetry group G(M0, t > 0).(30) Short Summary of Plasticity-Induced Cognitive Symmetries Cognitive Process Neuronal Change Symmetry Emergence Mathematical Representation Memory consolidation Synaptic strengthening Stable cognitive states Tmem :M→M Attention Selective neuronal activation Cognitive focus Tatt :M→M Rhythmic synchronization Coherent neuronal firing Global oscillatory states Tsync :M→M Other Possible Dynamically Emerging Symmetries Additional dynamically emerging symmetries can include: •Cognitive categorization: neuronal tuning and feature selection. •Motor skill learning: fine-tuned sensorimotor mappings. •Emotional regulation: adaptive neuronal response patterns. These dynamic cognitive symmetries ensure biological adaptability, efficient cognitive resource allocation, and robust cognitive and behavioral responses. Gauge groups and principal bundles extended with dynamic, time-dependent, and spatially dependent gauge symmetries offer profound conceptual and mathematical tools to model and understand adaptive biological phenomena. These structures effectively capture neuronal dynamics, plasticity mechanisms, and cognitive adaptability, emphasizing the inherent biological necessity for dynamically evolving cognitive symmetries. MATHEMATICAL CALCULATION OF GAUGE AND HIGHER GAUGE EFFECTS IN NEURONAL-COGNITIVE SYSTEMS In this chapter, we provide a comprehensive mathematical framework for calculating gauge and higher gauge effects in neuronal-cognitive systems. Our setup consists of two fundamental manifolds: •Neuronal manifold X, representing intrinsic neuronal parameters (synaptic densities, receptor distributions, excitability thresholds). •Cognitive manifold M, representing cognitive functionalities (memory, attention, sensory processing, motor actions). Gauge Fields and Forms on Cognitive Manifold M We define gauge fields on manifold M: •0-form gauge field: scalar fields φ(x) representing local cognitive conditions (e.g., dopamine concentration). •1-form gauge field: vector fields Aµ(x) representing directional cognitive flows. 16 The tangent vectors become: Tµ(s) = dNµ(s) ds = (−sin(s),cos(s),1).(74) The gauge currents become: Jν (1−form)(x) = Zds (−sin(s),cos(s),1)νδ(x−N(s)),(75) Jνρ (2−form)(x) = Zds (−sin(s),cos(s),1)ν(−sin(s),cos(s),1)ρδ(x−N(s)).(76) These rigorous formulations clearly show the mathematical structure of gauge charges from neuronal embeddings. Additional Example: Exponential Form of Gauge Charges for Medication Effects In practical pharmacological modeling, it may be intuitive to represent localized medication effects as exponential sources. Consider medication-induced modulation localized around cognitive states x0and x1. Explicit gauge charges become: Jνρ (2−form)(x) = γ0e−|x−x0|2 σ2+γ1e−|x−x1|2 σ2,(77) where γ0, γ1denote strengths, and σthe spatial localization. Explicit numerical example: •γ0= 1.0, γ1= 1.5, σ= 1.5. •Medication enhances synchronization at x1, previously at baseline at x0. Thus, the gauge field equation becomes: ∂tBνρ(x, t)−∇2Bνρ(x, t) = 1.0e−|x−x0|2 2.25 + 1.5e−|x−x1|2 2.25 .(78) Numerical solutions provide medication-induced cognitive synchronization patterns, predicting spatially and temporally localized enhancements in cognitive coherence. Interpretation of the Two Approaches The rigorous embedding approach emphasizes the mathematical foundation of gauge charges, ensuring theoretical precision. The exponential source representation, though less rigorously defined, offers clear biological intuition and straightforward numerical implementation. Together, these approaches provide complementary methods for detailed pharmacological modeling and cognitive prediction. This chapter provided rigorous and intuitive approaches to mathematically modeling and numerically predicting medication effects on cognitive processes using gauge theories. Explicitly using neuronal embeddings and exponential source examples, we demonstrated how gauge charges inform predictions of cognitive changes, bridging detailed mathematical rigor and practical biological insights. MATHEMATICAL ANALYSIS OF DOPAMINE SCALAR FIELD AND GAUGE FORMULATION Dopamine is naturally represented as a scalar field φ(x) on the cognitive manifold M. At each point x∈M, φ(x) quantifies local dopamine concentration, a scalar quantity since it has no inherent directional properties. If this were the sole description of the dopamine concentration, the associated curvature would be naturally null, due to the fact that the curvature would originate from a gradient of a gauge field that itself would be only a gradient Aµ(x) = ∂µφ(x). However, this is not the case in full biological generality where the gauge field is constructed from the gradient part corrected by the contributions of space-dependent receptor density or space dependent neuronal sensitivity or plasticity. With these corrections we can in fact obtain a non-trivial field strength (curvature) and observable outcomes. 17 Gradient of Dopamine Field as a 1-Form Although dopamine itself is scalar, its spatial variation naturally leads to a 1-form gauge field. The correct and complete formulation for the dopamine gauge potential is: Aµ(x) = ∂µφ(x) + ρ(x)∂µσ(x),(79) where ρ(x) represents receptor density, and σ(x) neuronal sensitivity or plasticity. Corrected Curvature (Field Strength) of Dopamine Gradient The gauge curvature, or field strength, is defined as: Fµν (x) = ∂µAν(x)−∂νAµ(x).(80) Using the correctly defined gauge potential, we substitute: Fµν (x) = ∂µ[∂νφ(x) + ρ(x)∂νσ(x)] −∂ν[∂µφ(x) + ρ(x)∂µσ(x)] (81) =∂µρ(x)∂νσ(x) + ρ(x)∂µ∂νσ(x)−∂νρ(x)∂µσ(x)−ρ(x)∂ν∂µσ(x).(82) Since ∂µ∂νσ(x) = ∂ν∂µσ(x), the second-order derivatives cancel out, leaving: Fµν (x) = ∂µρ(x)∂νσ(x)−∂νρ(x)∂µσ(x).(83) This correctly represents the biologically significant variations arising solely from receptor density ρ(x) and neuronal plasticity σ(x). Biological Interpretation of Corrected Curvature The corrected curvature captures the effects of local variations in receptor density and neuronal plasticity. Even without changes in dopamine concentration, these variations create meaningful changes in neuronal responses. Thus, the curvature provides a rigorous mathematical tool to describe complex neuronal-cognitive dynamics influenced by dopamine. Explicit Numerical Example Consider an explicit scenario with: φ(x) = x2 1+x2 2,(84) ρ(x) = 1 + 0.5x1,(85) σ(x) = 1 + 0.2x2.(86) We calculate the necessary derivatives: ∂1φ= 2x1, ∂2φ= 2x2,(87) ∂1ρ= 0.5, ∂2ρ= 0,(88) ∂1σ= 0, ∂2σ= 0.2.(89) Thus, the gauge curvature becomes: F12(x) = ∂1ρ(x)∂2σ(x)−∂2ρ(x)∂1σ(x) (90) = (0.5)(0.2) −(0)(0) (91) = 0.1.(92) This explicit example demonstrates clearly that local variations in receptor density and neuronal modulation parameters generate a nontrivial gauge curvature. In this section, we rigorously corrected the dopamine gauge potential definition and recalculated the gauge curvature, accurately capturing biologically relevant phenomena. This precise mathematical framework significantly enhances the understanding and modeling of neuronal-cognitive dynamics influenced by dopamine. 18 EMERGENCE OF GAUGE SYMMETRY: FUNDAMENTAL PHYSICS AND BIOLOGY Gauge invariance is a cornerstone in theoretical physics, encapsulating the idea that certain physical quantities should remain invariant under specific local transformations. In fundamental physics, gauge invariance is traditionally considered exact and fundamental. However, in biological systems, gauge invariance does not hold in a fundamental sense but emerges adaptively, contingent on the choice of baseline or reference states. This chapter rigorously examines the conceptual and mathematical structures underlying gauge invariance in physics and biology, illustrating how gauge symmetry arises naturally from categorical structures, demonstrating its emergent rather than fundamental nature. Gauge Invariance in Fundamental Physics Gauge symmetry in fundamental physics, such as in quantum chromodynamics (QCD), is often presented as an exact symmetry. A prominent example is the color charge of quarks: ψ(x)→U(x)ψ(x), U(x)∈SU(3),(93) where the quark field ψ(x) transforms under local SU(3) color transformations. Physically observable quantities are gauge-invariant, and any gauge-dependent quantity, like the individual quark color, is typically deemed non-observable. Nevertheless, this non-observability is only apparent: the quark color can indeed be measured if one chooses a gauge. The crucial point is that the measured value is not independent of the observer’s choice of gauge. If all observers choose the same gauge, they agree on the measurement, demonstrating clearly that gauge invariance reflects observer independence rather than intrinsic fundamental physical structure. Adaptive Emergence of Gauge Symmetry in Biology In biological systems, gauge invariance emerges adaptively. The concept of ”baseline choice” in biological measurements is analogous to a gauge choice in physics. For instance, consider baseline neurotransmitter concentrations or neuronal excitability levels: Choosing a baseline state φ0corresponds to choosing a specific gauge. Observations of neuronal or cognitive states φ(x) thus transform as: φ(x)→φ0(x) = φ(x)−φ0,(94) with clearly different biological outcomes resulting from different baseline choices. Hence, biological observables become gauge-dependent. However, biological systems adapt to these baseline shifts through compensatory mechanisms, such as modifying receptor densities ρ(x) or synaptic strengths σ(x). These adaptive responses restore a form of emergent symmetry: ρ0(x), σ0(x) adaptively chosen such that F(φ0(x), ρ0(x), σ0(x)) = F(φ(x), ρ(x), σ(x)),(95) where Frepresents cognitive or physiological functional states. In other words, while individual parameters (”colors”) might vary according to baseline choices (”gauge”), overall cognitive or physiological functionality (analogous to ”white color” states in physics) remains consistent, thereby realizing an adaptive emergent gauge symmetry. Category-Theoretic Foundations of Gauge Symmetry Gauge symmetries naturally emerge from category-theoretic structures. Consider a category C, with: •Objects X,Y, ... representing physical or biological states. •Morphisms f:X→Yrepresenting state transformations. 19 A gauge symmetry corresponds categorically to an isomorphism f:X→X, preserving essential structures and relationships. In physics, choosing a gauge corresponds categorically to selecting an object and particular isomorphisms within C. Gauge invariance thus emerges naturally from the existence of non-trivial isomorphisms and their coherence conditions: Xf −→ Xwith f◦f−1= idX.(96) Biologically, adaptive gauge symmetry is represented categorically as dynamically emerging isomorphisms, conditioned by adaptive responses. The category structure itself evolves, reflecting the dynamic biological processes: Xf(ρ,σ) −−−−→ Xwith adaptive conditions on ρ, σ. (97) This categorical formalism clearly encapsulates how gauge symmetry emerges adaptively, contingent upon internal biological adaptation. Mathematical Illustration of Gauge Emergence Consider cognitive gauge fields Aµemerging from categorical constraints. Define gauge fields as connections on principal bundles associated categorically to cognitive or neuronal states: Aµ(x)→g(x)Aµ(x)g−1(x) + g(x)∂µg−1(x), g(x)∈Isom(X),(98) with the gauge curvature given by: Fµν (x) = ∂µAν(x)−∂νAµ(x)+[Aµ(x), Aν(x)].(99) Adaptive emergence of gauge invariance in biology involves modified gauge transformations: g(x)→g(x, ρ(x), σ(x)),(100) leading to modified curvatures that reflect adaptive dynamics. Comparison and Reinterpretation of Gauge Symmetry This categorical analysis clearly demonstrates that gauge invariance, both in fundamental physics and biology, reflects observer-dependent symmetries rather than intrinsic universal constraints. Gauge symmetry in physics appears fundamental only due to traditional perspectives emphasizing observer-independence. In reality, just as in biology, gauge symmetry can be viewed as emergent from deeper categorical structures. Biological and physical gauge invariances differ primarily in adaptiveness. Biological systems adapt gauge structures dynamically, restoring functional coherence. Physics, traditionally viewed as static and universal, implicitly relies upon fixed categorical structures. The explicit recognition of gauge symmetry emergence from categorical foundations offers deeper insights, reconciling biological adaptiveness with physical gauge invariance. Therefore gauge symmetry is not fundamentally intrinsic but emergent categorically. In physics, gauge invariance signifies observer independence, maintained through fixed categorical structures. In biology, gauge symmetry adaptively emerges, reflecting dynamically evolving categorical structures. Explicit mathematical frameworks derived from category theory rigorously demonstrate this emergent gauge symmetry, providing profound unifying insights into observer-dependence and adaptive structures across biology and fundamental physics. ADAPTIVE NEURAL GAUGE THEORY VIA CATEGORY THEORY In this chapter, we provide a detailed mathematical formulation of adaptive neural gauge theory through category theory. Gauge connections are viewed as functors, capturing the structural relationships within neuronal and cognitive states. The time-dependent gauge groupoid G(M;t) describes the dynamic evolution of neuronal states, dopamine baselines, and receptor densities, reflecting the adaptive nature of biological systems. 20 Gauge Connections as Functors Consider a principal bundle π:P→Mover the cognitive manifold M, whose fibers form a groupoid G(M;t). Gauge connections are represented categorically by functors: Aµ:G(M;t)→Diff(M),(101) where Diff(M) denotes the category of differentiable maps on M. Parallel transport between fibers at points x, y ∈M is a morphism (functor): τx→y:Fx→Fy,(102) which respects gauge transformations: τx→y(gx) = gy◦τx→y, gx∈ G(M;t)|x, gy∈ G(M;t)|y.(103) Dynamic Adaptation of Gauge Groupoid The gauge groupoid dynamically adapts over time, denoted by: G(M;t) = neuronal states, dopamine baselines, receptor densities.(104) To describe transitions between different time instances, we introduce a natural transformation: η(t) : G(M;t)⇒ G(M;t+dt).(105) Categorical Coherence Conditions for Neural Adaptation Categorical coherence conditions ensure neuronal states adapt consistently. For adaptive parallel transport, coherence conditions require: η(t+dt)◦τt=τt+dt ◦η(t).(106) Differentiating, we obtain neural adaptation equations: d dtη(t)+Φµ[η(t)]ψ(x, t)=0,(107) with adaptive gauge term defined as: Φµ[η(t)]ψ(x, t) = η(t)−1∂µη(t)ψ(x, t).(108) Gauge Curvature with Adaptive Terms The adaptive gauge curvature Fadapt µν arises from: Fadapt µν =∂µAν−∂νAµ+ [Aµ, Aν] + DµΦν[η]−DνΦµ[η] + [Φµ[η],Φν[η]],(109) where Dµ=∂µ+ [Aµ,·] denotes the covariant derivative. Adaptive Restoration of Gauge Symmetry Neural plasticity facilitates adaptive restoration of gauge symmetry after significant baseline shifts. The dynamics governing this restoration are: dη dt =−α(Fadapt µν +βR(η)),(110) with α, β constants, and R(η) a biological cost function. 21 Neuronal Cognitive Action Functional We introduce the neuronal cognitive action functional incorporating adaptive gauge terms: S=ZM d4x√g−1 4Fadapt µν Fadapt, µν +1 2(Dµψ)†(Dµψ) + V(ψ, η),(111) where V(ψ, η) represents cognitive potentials related to adaptive neuronal dynamics. Higher Gauge Fields and Adaptive Dynamics Higher gauge fields Bµν encapsulate collective neural behaviors such as synchronization and coherence loops. The higher gauge covariant derivative is given by: DµBνρ =∂µBνρ + [Aµ, Bνρ]+Φµ[η]Bνρ,(112) with adaptive higher gauge curvature: Hadapt µνρ =DµBνρ +DνBρµ +DρBµν (113) + [Φµ[η], Bνρ] + [Φν[η], Bρµ] + [Φρ[η], Bµν ].(114) Neuronal Cognitive Higher Action Functional The higher action functional describes collective cognitive dynamics: Shigher =ZM d4x√g−1 12Hadapt µνρ Hadapt, µνρ +1 2(Dµψ)†(Dµψ) + ˜ V(ψ, B, η),(115) with ˜ V(ψ, B, η) representing higher-order cognitive interactions. Biological Interpretations (Detailed Table) Mathematical Term Biological Interpretation Gauge Field AµDopamine gradients, localized modulation Higher Gauge Field Bµν Neural synchronization, cognitive coherence loops Adaptive Term Φµ[η] Plasticity-related neuronal changes (receptor densities, excitability) Gauge Curvature Fadapt µν Variability in cognitive response Higher Gauge Curvature Hadapt µνρ Emergent collective neuronal behaviors Natural Transformation ηAdaptive neuronal state transitions Action Functional S,Shigher Energy and cognitive cost associated with adaptive neural states EMERGENT GAUGE SYMMETRY: INSIGHTS FROM BIOLOGY AND IMPLICATIONS FOR FUNDAMENTAL PHYSICS Gauge invariance has traditionally been viewed as a fundamental property in theoretical physics, ensuring consistency across different mathematical descriptions of physical reality. However, examining biological systems reveals that gauge invariance emerges naturally from categorical coherence conditions rather than being inherently fundamental. This realization suggests similar possibilities in fundamental physics, where observer-dependent gauge choices lead to measurable but gauge-dependent observables. 22 Gauge Choices and Observer-Dependent Observables In physics, gauge-dependent quantities such as the color of quarks are traditionally considered non-observable due to their dependence on gauge choices. However, gauge-dependent does not imply unmeasurable; rather, it signifies that the result is contingent upon the observer’s gauge selection. If all observers agree upon a specific gauge choice, they will obtain consistent and measurable outcomes. This situation is analogous to defining reference frames in classical mechanics—different frames yield different yet consistent observations. Formally, a gauge transformation in physics can be expressed as: ψ(x)→U(x)ψ(x), U(x)∈G, (116) where is a gauge group. The gauge-dependent observable (e.g., quark color) can still be measured precisely under the condition of a fixed gauge choice: O(ψ)Gauge Choice −−−−−−−−−→ O0(ψ0) = O(Uψ).(117) Thus, observer-dependent observables become entirely consistent and measurable given a universal gauge selection. Gauge Invariance as Emergent from Categorical Coherence Biology illustrates how gauge symmetry groups emerge adaptively and categorically from coherence conditions. Biological observables, such as neuronal excitability or receptor densities, depend upon reference baselines analogous to gauge choices. The biological system dynamically adapts to restore coherence: φ(x)→φ0(x) = φ(x)−φ0, ρ0(x), σ0(x) adapt accordingly,(118) where represents an observer-dependent baseline. Despite these observer-dependent choices, the system maintains functional coherence through adaptive mechanisms (e.g., neuronal plasticity). Categorically, gauge symmetry groups emerge naturally from the coherence conditions imposed by biological consistency. Consider a groupoid G(M;t), representing neuronal states, dopamine levels, and receptor densities at a given time . Morphisms represent gauge transformations between these states: G(M;t)η(t) −−→ G(M;t+dt),(119) with coherence conditions: η(t+dt)◦τt=τt+dt ◦η(t),(120) where τdenotes parallel transport functors between categorical states. These coherence conditions enforce consistency and adaptive restoration of emergent gauge symmetry. Emergence of Gauge Symmetry in Fundamental Physics The categorical emergence of gauge symmetry observed in biological systems strongly suggests a similar phenomenon may occur in fundamental physics. Observer-dependent gauge choices in physics can be understood as categorical constraints, defining consistent morphisms between physical states and ensuring coherence across different descriptions. Gauge-dependent observables, like quark color, are measurable contingent on coherent gauge choices by observers. Therefore, the apparent observer-dependence of gauge-dependent observables in physics is analogous to the biological scenario, reinforcing that the categorical emergence of gauge invariance from coherence conditions is not problematic but rather indicative of deeper structural principles. Intuitive and Logical Clarifications To clarify intuitively, consider measuring neuronal excitability relative to a baseline dopamine level in biology. This baseline is analogous to a gauge choice; changing it affects measurable outcomes, yet these outcomes remain 23 consistent and meaningful provided the gauge choice is clear and agreed upon. Similarly, quark color in physics becomes consistently measurable under fixed gauge conditions, reflecting observer-dependent but measurable realities. Categorically, emergent gauge symmetry arises from the necessity to maintain consistency (coherence) within the system, whether biological or physical. Thus, the key insight is not the fundamental nature of gauge symmetry itself, but the categorical structures and coherence conditions that necessitate its emergence. In summary, examining biological systems provides significant insights into the emergent nature of gauge invariance. Gauge symmetry groups, rather than being fundamentally intrinsic, emerge naturally from categorical coherence conditions in both biology and potentially fundamental physics. Observer-dependent choices in gauge-dependent observables do not diminish their measurability but highlight the necessity of coherence and categorical consistency. This understanding offers a deeper, unified perspective on the nature and origin of gauge symmetries across diverse scientific domains. MATHEMATICAL ANALYSIS AND VISUALIZATION OF MEDICATION EFFECTS USING ADAPTIVE NEURAL GAUGE THEORY In this chapter, we present a detailed mathematical analysis and visualization of medication effects on neuronalcognitive systems using adaptive neural gauge theory. We demonstrate that such complex computations cannot be accurately captured without considering gauge symmetries and adaptive categorical structures, highlighting the unique computational advantages offered by this framework. Formulation of the Model Consider a neuronal embedding defined as Nµ(s), mapping neuronal states into a cognitive manifold M. The dynamics influenced by medication are modeled via adaptive gauge fields, incorporating neurotransmitter modulation (dopamine): DNµ(s) Ds =dNµ(s) ds +Aµ ν(N(s))dNν(s) ds +1 2Bµ νρ(N(s))dNν(s) ds dNρ(s) ds ,(121) where Aµ νand Bµ νρ represent the 1-form and 2-form gauge fields, respectively. Medication-Induced Gauge Charges Medication modifies cognitive dynamics through induced gauge charges, represented as: Jν (1−form)(x) = X jZds Tν j(s)δ(x−N(ej)(s)),(122) Jνρ (2−form)(x) = Xj, k Zds Tν j(s)Tρ k(s)δ(x−N(ej∩ek)(s)),(123) where Tν j(s) = dNν(ej)(s) ds . Medication influences neurotransmitter levels (dopamine) and receptor sensitivities, altering these currents. For computational simplicity, a localized medication effect can also be modeled using exponential forms: Jνρ (2−form)(x) = γ0e−|x−x0|2 σ2+γ1e−|x−x1|2 σ2,(124) with constants characterizing medication-specific parameters. Numerical Solution and Justification To solve these gauge-influenced neuronal dynamics numerically, we discretize the associated differential equations and employ numerical methods such as finite difference or finite element methods. The numerical equations to solve 24 are: ∂tAν(x, t)−∇2Aν(x, t) = Jν (1−form)(x, t),(125) ∂tBνρ(x, t)−∇2Bνρ(x, t) = Jνρ (2−form)(x, t).(126) These equations inherently incorporate adaptive gauge terms and categorical coherence, capturing nuanced medication effects that simpler linear or static models would fail to describe. Such complexity requires gauge-theoretic formulations, making alternative methods insufficient. Computational Results We consider an explicit example of a dopamine-enhancing medication, characterized by: γ0= 1.0, γ1= 1.5, σ = 1.5,(127) x0= (0,0), x1= (2,2).(128) The resulting adaptive gauge field equations become: ∂tBνρ(x, t)−∇2Bνρ(x, t) = e−|x|2 2.25 + 1.5e−|x−(2,2)|2 2.25 .(129) We numerically solve this equation, demonstrating the emergence of medication-induced synchronization patterns and altered cognitive dynamics. Graphical Representation of Results Figure 1 presents a graphical visualization of the computed neuronal-cognitive synchronization pattern resulting from medication. This static image clearly illustrates areas of enhanced neuronal synchronization induced by the medication. Advantages of Gauge-Theoretic Computation Such detailed and adaptive computations could not be reliably performed without considering gauge fields and adaptive categorical structures. Traditional linear or non-gauge methods neglect the adaptive coherence and emergent symmetries critical for accurately modeling medication effects. Gauge theory inherently captures dynamic neuronal adaptations, receptor density changes, neurotransmitter modulation, and coherence constraints, allowing a rigorous and biologically realistic analysis. We have presented a detailed mathematical formulation and numerical analysis of medication effects within neuronal-cognitive systems using adaptive neural gauge theory. Our results underscore the computational necessity and advantages of gauge theory in capturing the complex adaptive dynamics underlying medication effects, supported by clear graphical representations. CONCLUSION: EMERGENT AND ADAPTIVE GAUGE SYMMETRIES IN NEURONAL AND COGNITIVE DYNAMICS Throughout this article, we have rigorously explored a novel theoretical framework in which neuronal structures and cognitive functionalities are unified via categorical higher gauge theory. This unification provides a systematic mathematical approach that clarifies how neuronal activities determine cognitive processes and vice versa, highlighting the central role of gauge invariance and gauge symmetries in describing biological phenomena. 25 FIG. 1: Computed neuronal-cognitive synchronization patterns induced by medication. Bright regions indicate increased synchronization and cognitive coherence. Summary of Main Findings Our primary achievement lies in establishing a robust mathematical correspondence between neuronal manifolds X—comprised of vertices (neuronal properties such as excitability thresholds, receptor densities, and synaptic strengths) and edges (representing transitions, synaptic densities, and connectivity)—and cognitive manifolds M, which encode functional states such as memory consolidation, motor actions, rhythmic synchronization, and attention. We showed how these two manifolds relate through embeddings that transform neuronal structural properties into cognitive functional outcomes. This embedding, mathematically described by fields and gauge potentials, reveals intricate dynamical relations that could not be adequately captured previously by classical approaches. In particular, we emphasized the adaptive nature of the gauge symmetry groups that arise dynamically in biological systems. Contrary to conventional physical theories, where gauge groups are often static, neuronal and cognitive systems demand time-dependent gauge symmetry structures. Plasticity processes, learning, memory formation, environmental interactions, and cognitive experiences actively reshape gauge symmetry groups in real-time, reflecting their fundamentally adaptive character. Dynamically Evolving Gauge Symmetry Groups A critical result demonstrated in our analysis is that gauge symmetries emerge dynamically from categorical coherence conditions. Mathematically, we defined this through gauge groupoids G(M;t) that depend on time and cognitive variables, evolving continuously as neuronal states, dopamine baselines, receptor densities, and plasticity parameters change due to experience and environmental conditions. The concept of categorical coherence allowed us to derive explicit mathematical conditions for adaptive gauge symmetries, leading to nonlinear dynamical equations that precisely model neuronal-cognitive interactions. We provided explicit formulas for adaptive gauge potentials and curvatures, showing how intrinsic biological pro-