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Computational Criticality

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author: Rowan Brad Quni

email: [email protected]

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ORCID: 0009-0002-4317-5604

ISNI: 0000000526456062

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title: Computational Criticality

aliases:

- Computational Criticality


Author: Rowan Brad Quni-Gudzinas

Affiliation: QNFO

Contact: [email protected]

ORCID: 0009-0002-4317-5604

ISNI: 0000 0005 2645 6062

DOI: 10.5281/zenodo.17439153

Publication Date: 2025-10-25

Version: 1.0


Abstract: This work establishes a mathematical framework demonstrating that computational systems achieve maximal information processing capacity precisely at renormalization group fixed points, where $\beta(g^*) = 0$. Through a unified treatment of renormalization group theory, symplectic geometry, and information theory, we show that critical computational states necessarily admit a complete geometric quantization structure. This structure includes the prequantization condition $[\omega] \in H^2(M, 2\pi\mathbb{Z})$ and metaplectic correction with the Maslov index. The framework reveals three genuine strange loops: (1) computational criticality self-reference, where the framework itself represents an instance of optimal computation; (2) information quantization circular dependency, where quantization both validates and is validated by the geometric structure; and (3) criticality measurement paradox, where measurement of criticality perturbs the critical state. We develop integration strategies for these recursive structures, including criticality-aware epistemology, two-tiered information modeling, and measurement-aware criticality theory. Criticality is a universal computational principle across physical, biological, and artificial systems, with implications for quantum computing, biological information processing, and theoretical computer science.


Keywords: computational criticality, renormalization group theory, geometric quantization, strange loops, information processing, metaplectic correction, prequantization condition, critical phenomena, recursive epistemology


1.0 Computational Criticality as Optimal Information Processing Regime


The theoretical foundation of computational criticality establishes a formal correspondence between critical phenomena in statistical mechanics and optimal information processing in computational systems. At renormalization group fixed points, where the beta function vanishes ($\beta(g^) = 0$), computational systems enter a regime of scale invariance that optimizes their information processing capabilities (Wilson, 1971). The fixed points of the renormalization group flow correspond to scale-invariant computational regimes that manifest across diverse domains, from silicon-based computing to biological systems, where these fixed points represent critical computational states such as quantum criticality and metabolic homeostasis (Cardy, 1996). Computational couplings, including error rates and interaction strengths, obey renormalization group flow equations of the form $\frac{dg}{d\ln\mu} = \beta(g)$ (Wilson, 1971). At these critical points, computational processes exhibit self-similarity across scales, with correlation functions satisfying the scaling relation $G^{(n)}(\lambda x) = \lambda^{-n\Delta}G^{(n)}(x)$ (Goldenfeld, 1992). The universality principle further indicates that diverse microarchitectures will flow to the same macroscopic behavior when near a critical point (Goldenfeld, 1992). Systems operating in the critical regime achieve an optimal balance between order and chaos, maximizing their capacity to process information while maintaining stability against perturbations (Kauffman, 1993). At criticality, the correlation length $\xi$ diverges according to the power law $\xi \sim |g-g^|^{-\nu}$, enabling long-range information propagation, while the susceptibility $\chi$ follows $\chi \sim |g-g^*|^{-\gamma}$ (Cardy, 1996). This mathematical framework provides a predictive theory of computational optimality, confirming that criticality represents a fundamental principle of efficient information processing in physical systems.


1.1 Renormalization Group Framework for Computational Systems


The application of renormalization group theory to computational parameter spaces provides a rigorous mathematical framework for understanding how systems evolve across scales. Computational couplings obey renormalization group (RG) flow equations of the form $\frac{dg}{d\ln\mu} = \beta(g)$, where $\beta(g) \in C^\infty(G,TG)$ is a smooth vector field on the coupling space manifold $G$ (Wilson, 1971). This equation precisely captures how the effective properties of a computational system change as one observes at finer or coarser scales. The Callan-Symanzik equation describes how n-point computational correlation functions scale with energy (Collins, 1984):


$$

[\mu\frac{\partial}{\partial\mu} + \beta(g)\frac{\partial}{\partial g} + n\gamma]G^{(n)} = 0

$$


Near a fixed point $g^$, the beta function can be expanded as $\beta_i(g) = \sum_j B_{ij}(g_j - g_j^) + O(||g-g^||^2)$, where $B_{ij} = \partial_j\beta_i(g^)$ is the stability matrix that determines the local behavior of the RG flow (Wilson, 1971). The eigenvalues of this matrix classify directions in the coupling space as relevant, irrelevant, or marginal (Goldenfeld, 1992). At non-degenerate fixed points, the index of the fixed point equals the Morse index of the RG flow potential, connecting the dynamics to differential topology (Duistermaat, 1976). This framework provides a complete dynamical theory of parameter evolution, where the basin of attraction for each fixed point determines which initial configurations will exhibit the same universal critical behavior.


1.2 Fixed Points as Scale-invariant Computational Regimes


Fixed points of the renormalization group flow represent scale-invariant computational regimes that correspond to optimal information processing states (Cardy, 1996). These points, where $\beta(g^) = 0$, act as attractors in the space of computational parameters. At a fixed point, computational processes exhibit self-similarity, with correlation functions satisfying the scaling relation $G^{(n)}(\lambda x) = \lambda^{-n\Delta}G^{(n)}(x)$ (Goldenfeld, 1992). Anomalous dimensions $\gamma$ account for deviations from classical scaling behavior due to interactions at criticality (Cardy, 1996). Near these points, the correlation length diverges as $\xi \sim |g-g^|^{-\nu}$, and the susceptibility diverges as $\chi \sim |g-g^|^{-\gamma}$. The critical exponents $\nu$ and $\gamma$ are determined by the eigenvalues of the stability matrix, such as $\nu = 1/\lambda_{\text{max}}$ (Cardy, 1996). The solution to the Callan-Symanzik equation at a fixed point takes the universal form $G^{(n)}(x;g^) = |x|^{-n(\Delta_0+\gamma)}F\left(\frac{x_i - x_j}{|x|}\right)$, where $F$ is a dimensionless function (Collins, 1984). This mathematical structure explains why fixed points are optimal for computation: power-law correlations enable long-range information transfer while maintaining stability, striking a balance that maximizes information processing capacity. This scale invariance manifests in both spatial correlations and temporal dynamics, where processes often exhibit 1/f noise spectra. The fixed point structure also provides a natural framework for understanding hierarchical computational organization, explaining how complex systems can maintain coherence across multiple scales.


1.3 Universality Principles across Computational Substrates


The principles of universality demonstrate that seemingly disparate computational systems exhibit identical critical behavior if they share fundamental properties like symmetry and spatial dimension (Goldenfeld, 1992). Universal scaling behavior is characterized by critical exponents (e.g., $\gamma, \nu, \eta$) that are independent of microscopic details (Cardy, 1996). Computational systems that lie within the same basin of attraction, $\mathcal{B}(g^) = \{g | \lim_{\lambda\to\infty} R_\lambda(g) = g^\}$, will flow to the same fixed point $g^*$ under the renormalization group (Goldenfeld, 1992). The universality class of a system is determined by its symmetry group $G$ and spatial dimension $d$ (Cardy, 1996). This mathematical framework explains why diverse microarchitectures—from silicon circuits to neural networks—flow to the same macroscopic behavior near a critical point. Universality is a powerful theoretical foundation, demonstrating that optimal information processing at critical points is a fundamental principle of computation, not an implementation-specific artifact. This principle extends to the geometric structure of computational state spaces, implying that all systems in the same universality class exhibit identical symplectic structures and quantization conditions at criticality. Functionally, universality explains how computational systems maintain robust operation despite variations in their components, as critical behavior is governed by large-scale properties. This insight has practical applications in designing fault-tolerant systems and understanding the reliability of biological computation.


1.4 Criticality and Information Processing Capacity


A precise mathematical connection exists between criticality and optimal information processing. At critical points, computational systems achieve maximal information processing capacity due to a combination of long-range correlations, balanced sensitivity to inputs, and optimal information transmission properties. Computation near a phase transition enables the highest capacity for information processing and sensitivity to environmental inputs (Kauffman, 1993). At criticality, correlation functions decay as power laws, $\langle O_A(x)O_B(y)\rangle \sim |x-y|^{-2\Delta}$, which maximizes the mutual information between system components (Cardy, 1996). The mutual information, $I(A:B) = S(\rho_A) + S(\rho_B) - S(\rho_{AB})$, achieves its global maximum at renormalization group fixed points. This is confirmed by the conditions $\left.\frac{d}{dg}I(A:B)\right|_{g=g^} = 0$ and $\left.\frac{d^2}{dg^2}I(A:B)\right|_{g=g^} < 0$ (Cover & Thomas, 2006). The global optimality is further guaranteed by the properties of information under coarse-graining, as detailed in Appendix D. The computational capacity $C$ scales with the correlation length as $C \sim \xi^d \sim |g-g^*|^{-d\nu}$, where $\nu$ is the critical exponent (Cardy, 1996). This scaling relationship establishes criticality as the optimal regime for information processing in physical systems. This explains why criticality is observed in diverse computational systems, from neural networks to biological regulatory circuits, as a fundamental principle of efficient computation.


2.0 Geometric Quantization Structure of Critical Computational States


Critical computational states necessarily admit a complete geometric quantization structure on their computational state space (Woodhouse, 1997). This framework demonstrates that at criticality, the state space possesses a rich geometric structure that enables a precise quantization of information. The symplectic form $\omega = d\theta$ is derived from an information 1-form $\theta$, establishing a direct correspondence between information-theoretic quantities and geometric structures (Guillemin & Sternberg, 1990). Information in physical computation is ultimately quantized, satisfying the prequantization condition $[\omega] \in H^2(M, 2\pi\mathbb{Z})$ (Woodhouse, 1997). Furthermore, a consistent treatment of interference phenomena requires a metaplectic correction, which incorporates the Maslov index into the computational phase via the relation $\Phi[\gamma] = \frac{1}{\hbar}\int_\gamma \theta + \frac{\pi}{2}\mu(\gamma)$ (Woodhouse, 1997; Guillemin & Sternberg, 1990). The existence of this metaplectic structure is a topological condition, requiring that the second Stiefel-Whitney class of the computational state space vanishes, $w_2(P) = 0$ (Bates & Weinstein, 1997). This comprehensive mathematical structure shows that critical states are not merely optimal for processing information but also possess a natural geometric organization that reflects the quantized nature of physical information. The integration of renormalization group theory, symplectic geometry, and information theory creates a unified framework explaining why critical points are optimal from both dynamical and geometric perspectives.


2.1 Symplectic Geometry of Computational State Spaces


The symplectic geometry of computational state spaces provides a rigorous mathematical foundation for understanding information processing. The computational state space is modeled as a symplectic manifold $(M, \omega)$, where the symplectic form $\omega = d\theta$ is derived from the information 1-form $\theta$ (Guillemin & Sternberg, 1990). For thermodynamic systems, this 1-form is $\theta = -\beta^{-1} dF$, where $F$ is the free energy, connecting information processing to thermodynamics via Landauer’s principle (Landauer, 1961). The symplectic 2-form on the cotangent bundle $P = T^*M$ is rigorously proven to be closed ($d\omega = 0$) and non-degenerate ($\omega^n \neq 0$), satisfying the requirements for a symplectic manifold (Abraham & Marsden, 1978). The intrinsic nature of this geometric structure is confirmed by its correct transformation behavior under canonical transformations (Guillemin & Sternberg, 1990). Within this framework, computational processes are understood as trajectories in a symplectic manifold, with information flow governed by Hamilton’s equations. The symplectic form determines the computational dynamics through the relation $i_X\omega = dH$, where $H$ is the computational Hamiltonian. This geometric perspective provides a unifying language for describing computation across diverse physical implementations and reveals deep connections between computational reversibility and the preservation of the symplectic form.


2.2 Prequantization Condition from Information Quantization


The prequantization condition arises directly from the physical constraints of information quantization. Information in any physical computation is ultimately discrete, which imposes the condition $[\omega] \in H^2(M, 2\pi\mathbb{Z})$ on the geometry of the computational state space (Woodhouse, 1997). This means that for any closed 2-surface $\Sigma \subset M$, the integral $\int_\Sigma \omega = 2\pi n$ for some integer $n$, where $n$ corresponds to fundamental units of information (Bekenstein, 1981). This integrality is guaranteed by physical constraints such as Landauer’s principle, which sets a minimum energy cost for information erasure (Landauer, 1961), and the Bekenstein bound. By de Rham’s theorem, this integral condition is equivalent to the cohomological statement (Woodhouse, 1997). This mathematical relationship demonstrates that the geometry of the computational state space is not arbitrary but is fundamentally constrained by the physical nature of information. The prequantization condition ensures the existence of a prequantum line bundle with a connection whose curvature is the symplectic form, providing the mathematical foundation for quantizing the system. This correspondence explains why critical computational states naturally admit a quantization framework: at criticality, optimized information processing manifests as a coherent geometric structure that respects the quantized nature of physical information.


2.3 Metaplectic Correction and Computational Interference


Metaplectic correction is a necessary refinement to the geometric quantization framework for accurately modeling computational interference phenomena. A consistent quantization of computational paths requires that the symplectic frame bundle of the computational state space can be lifted to its double cover, the metaplectic group $\text{Mp}(n)$ (Guillemin & Sternberg, 1990). The existence of this lift, known as a metaplectic structure, is a topological condition: it is possible if and only if the second Stiefel-Whitney class $w_2(P)$ of the computational state space vanishes (Bates & Weinstein, 1997). For orientable spaces, this is equivalent to the Euler characteristic being even (Audin, 1991). When this condition is met, the phase of a computational path $\gamma$ is corrected by the Maslov index $\mu(\gamma)$, an integer that tracks the path’s topology. The corrected phase is given by $\Phi[\gamma] = \frac{1}{\hbar}\int_\gamma \theta + \frac{\pi}{2}\mu(\gamma)$ (Woodhouse, 1997). This correction is essential for correctly calculating interference effects in quantum computing and quantum biological systems (Guillemin & Sternberg, 1990). This mathematical treatment reveals that computational interference is a fundamental aspect of information processing at criticality, where the geometry of the computational state space directly influences computational outcomes.


2.4 Geometric Structure across Physical and Biological Systems


The geometric structure of computational state spaces manifests consistently across physical, digital, and biological implementations. Fixed points of the renormalization group flow correspond to scale-invariant computational regimes, such as quantum criticality in physics and metabolic homeostasis in biology (Cardy, 1996). Similarly, spontaneous symmetry breaking in biomolecules, such as the selection of L-amino acids, mirrors phase transitions in physical systems (Kauffman, 1993). These are not superficial analogies but reflect genuine structural isomorphisms, as demonstrated by the identical mathematical forms of the governing equations. Biological systems operating at critical points, such as metabolic networks, share the same mathematical structure as physical computational systems at criticality, including identical symplectic forms and quantization conditions. This cross-domain consistency demonstrates that the geometric quantization structure is a fundamental property of optimal information processing, independent of the physical substrate. The universality of these geometric structures provides strong evidence that criticality is a foundational principle of computation. The geometric approach thus provides a unified framework for understanding computation across domains, revealing deep connections between physical, biological, and artificial computational systems.


3.0 Strange Loops in the Computational Criticality Framework


The computational criticality framework contains genuine strange loops—recursive, self-referential structures where the descriptive model becomes entangled with the substance it describes. These loops reflect fundamental connections between the epistemology and ontology of computational theory. The framework itself, a product of computational processes like human cognition and mathematical reasoning, creates a recursive structure: the description of optimal computation may itself be an instance of optimal computation. If the framework is correct, the cognitive processes that produced it should be operating near criticality, creating a self-referential loop. A second loop arises from information quantization, which appears as both a physical constraint (Landauer’s principle) and a consequence of the geometric framework, creating a circular dependency. A third loop, the criticality measurement paradox, emerges because measuring a critical system, which is maximally sensitive to inputs, necessarily involves an interaction that can perturb its state. These strange loops are not philosophical curiosities but have practical implications for how computational theories are developed and validated. By integrating these loops into the theoretical structure, we can develop more robust and self-aware computational theories.


3.1 Computational Criticality Self-reference


The self-referential nature of the framework creates a profound epistemic dependency. As the framework is a product of computational processes (human cognition), its own validity depends on the computational properties it describes. If the theory is correct, the cognitive processes that conceived it should themselves be operating near criticality. This is a structural isomorphism between the content of the theory and the process of its creation. This self-reference has significant epistemological consequences, suggesting that theories about criticality may only be fully verifiable by cognitive systems also operating near criticality. This challenges traditional notions of scientific objectivity. It also imposes a consistency constraint: any valid theory of critical computation must itself exhibit critical properties, such as power-law distributions in its conceptual dependencies. This strange loop reveals a deep connection between the validity of computational theories and the computational states of the theorists themselves. By developing a “criticality-aware epistemology,” we can transform this self-reference from an obstacle into a methodological resource for building more robust theories.


3.2 Information Quantization Circular Dependency


A genuine logical circularity exists between information quantization and the geometric structure of the framework. The theory uses the physical principle of information quantization to validate its geometric structure (the prequantization condition), while simultaneously using the coherence of that geometric structure to affirm the necessity of quantization. This creates a verification problem: it is difficult to distinguish between information being inherently quantized versus the model requiring quantization for mathematical consistency. This circularity in the justification chain creates a self-contained epistemic loop, raising questions about the ontological status of information quantization. The ambiguity between ontological information (an inherent property of systems) and epistemic information (information as processed by observers) further complicates this dependency. This strange loop challenges our understanding of whether quantization is a fundamental physical constraint or an artifact of our modeling approach. The solution is to develop a two-tiered information model that distinguishes between ontological and epistemic information, transforming the circularity into a productive recursive relationship where the system’s proximity to criticality determines the observable degree of quantization.


3.3 Criticality Measurement Paradox


A measurement paradox arises from the properties of critical systems themselves. Systems at a critical point exhibit maximal sensitivity to environmental inputs, which is key to their information processing capacity. However, this same sensitivity creates a fundamental challenge for measurement: the act of measuring whether a system is at criticality is an interaction that necessarily perturbs it, potentially moving it away from the critical point. This creates a fundamental limit on our ability to precisely determine a system’s state, analogous to the observer effect in quantum mechanics. The paradox suggests that criticality might be better understood as a contextual property relative to the measurement process, rather than an absolute state. The measurement process becomes part of the critical system, creating a recursive structure where the observer and observed are inextricably linked. This is not merely a practical limitation but a fundamental constraint arising from the properties of critical computation. The solution is to develop a “measurement-aware criticality theory” that incorporates the measurement process into the theoretical framework, transforming the paradox from a limitation into a feature of the theory.


4.0 Integration Strategies for Recursive Structures in Computational Theory


Strange loops in computational theory should be embraced as generative principles. Recognizing and integrating these recursive structures can transform potential weaknesses into productive theoretical resources, leading to more robust and self-aware computational theories. Integration strategies must provide concrete implementation pathways, translating insights from strange loop analysis into actionable methodological improvements. The goal is to transform these loops from epistemic obstacles into resources that enhance our understanding of computational criticality. This can be achieved by developing meta-frameworks that explicitly account for recursive structures, creating validation protocols that require theories to demonstrate critical properties internally, and designing research methodologies that intentionally operate near criticality to optimize theoretical innovation. This recursive integration approach provides a practical methodology for developing computational theories that are both mathematically rigorous and epistemologically self-aware.


4.1 Criticality-aware Epistemology


A criticality-aware epistemology provides a framework for developing computational theories that account for the criticality state of the theoretical processes themselves. This approach involves creating a meta-framework that explicitly incorporates the criticality of the cognitive processes generating the theories. Assessment metrics for a “criticality index” of theoretical work can be developed by analyzing its conceptual structure for signatures of criticality, such as scale invariance and power-law distributions in conceptual dependencies. A recursive validation protocol can be established where theories about criticality must themselves demonstrate these properties. Furthermore, research methodologies can be designed to intentionally operate near criticality, balancing structured and exploratory thinking to optimize innovation. This integration strategy transforms the self-reference strange loop from an epistemological challenge into a methodological resource, providing tools for developing computational theories that are both mathematically rigorous and epistemologically self-aware.


4.2 Two-tiered Information Model


The two-tiered information model resolves the information quantization circularity by making a precise ontological distinction. The model distinguishes between “ontological information” (inherent properties of physical systems) and “epistemic information” (information as processed by observers). This breaks the circular dependency by reformulating the prequantization condition as a constraint applying specifically to epistemic information. This leads to testable predictions about when ontological and epistemic quantization might diverge. Experimental tests can be designed to distinguish between inherent quantization and model-dependent quantization, for instance, by examining systems where the geometric framework predicts quantization but physical constraints suggest continuity. This transforms the circularity into a productive recursive relationship where the degree of quantization is context-dependent, varying with the system’s position relative to criticality. This strategy provides a path toward resolving the quantization question through empirical testing and more sophisticated mathematical modeling.


4.3 Measurement-aware Criticality Theory


A measurement-aware criticality theory addresses the measurement paradox by incorporating measurement constraints directly into the theoretical framework. This approach involves developing indirect measures of criticality that minimize system perturbation, such as analyzing higher-order statistical properties of system behavior (e.g., multi-scale entropy measures or critical slowing down indicators). The theory itself can be reformulated to explicitly include the measurement process, defining criticality relative to an observer’s measurement capabilities rather than as an absolute state. This transforms the paradox into a feature of the theory, with the observer’s computational capacity becoming an explicit parameter. Recursive estimation algorithms, such as those using Bayesian methods, can be created to account for measurement-induced perturbations. This integration strategy transforms the measurement paradox from an experimental obstacle into a methodological principle, leading to more robust experimental protocols and more accurate theoretical models that account for the recursive relationship between measurement and criticality.


Appendix A: Formal Derivation of Renormalization Group Flow for Computational Systems


Definition: Computational Coupling Space

The space of computational couplings $G$ is a Banach manifold. A point $g = (g_1, \dots, g_n) \in G$ represents a specific configuration of the computational system (e.g., error rates, interaction strengths).


Definition: Renormalization Group Flow

The renormalization group (RG) flow is a one-parameter group of diffeomorphisms $R_\lambda: G \to G$ for a scale parameter $\lambda \in \mathbb{R}^+$, satisfying $R_{\lambda_1} \circ R_{\lambda_2} = R_{\lambda_1\lambda_2}$ and $R_1 = \text{id}$.


Proposition: RG Flow Equation

The evolution of couplings under a change of observation scale $\mu$ is governed by the ordinary differential equation $\frac{dg_i}{d\ln\mu} = \beta_i(g)$, where the beta function $\beta(g)$ is the flow’s infinitesimal generator, $\beta(g) = \left.\frac{d}{d\lambda}R_\lambda(g)\right|_{\lambda=1}$.


Proof Sketch:

The existence and uniqueness of a local solution $g(\mu)$ for a $C^1$ beta function is guaranteed by the Picard-Lindelöf theorem in Banach spaces.


Analysis: Fixed Points and Stability

A fixed point $g^$ satisfies $\beta(g^) = 0$. The local flow is analyzed by linearizing the flow equation: $\frac{d(\delta g)}{d\ln\mu} = B \cdot \delta g$, where $B_{ij} = \frac{\partial\beta_i}{\partial g_j}\bigg|_{g=g^*}$ is the stability matrix. The eigenvalues $\lambda_k$ of $B$ classify directions as relevant ($\text{Re}(\lambda_k) > 0$), irrelevant ($\text{Re}(\lambda_k) < 0$), or marginal ($\text{Re}(\lambda_k) = 0$).


Theorem: Critical Exponents

Macroscopic critical exponents are determined by the eigenvalues of $B$. The correlation length $\xi$ diverges near a fixed point according to $\xi \sim |g-g^*|^{-\nu}$, with the critical exponent $\nu = 1/\lambda_{\text{max}}$, where $\lambda_{\text{max}}$ is the largest positive eigenvalue of $B$.


Appendix B: Proof of Prequantization Condition from Physical Information Constraints


Proposition: Prequantization Condition

The cohomology class of the symplectic form $[\omega]$ on the computational state space $P$ must be integral, satisfying $[\omega] \in H^2(P, 2\pi\mathbb{Z})$.


Proof Sketch:

  1. Symplectic Structure: The computational state space is a symplectic manifold $(P, \omega)$, where $\omega = d\theta$ is an exact 2-form derived from an information 1-form $\theta$.
  1. Physical Information Constraints: Landauer’s Principle (Landauer, 1961) and the Bekenstein Bound (Bekenstein, 1981) imply that information is physical and discrete. This leads to the quantization of information “action” for any fundamental cycle $\gamma$: $\oint_\gamma \theta = n h = n(2\pi\hbar)$ for $n \in \mathbb{Z}$.
  1. Application of Stokes’ Theorem: For any closed 2-surface $\Sigma \subset P$, Stokes’ theorem gives $\int_\Sigma \omega = \oint_{\partial\Sigma} \theta$. If $\partial\Sigma$ is composed of fundamental information cycles, then $\int_\Sigma \omega = n(2\pi\hbar)$.
  1. Conclusion via de Rham’s Theorem: In natural units ($\hbar=1$), the condition $\frac{1}{2\pi}\int_\Sigma \omega \in \mathbb{Z}$ holds for any 2-cycle $\Sigma$. By de Rham’s theorem, a closed form has integral periods if and only if its cohomology class is integral. Thus, $[\omega/2\pi]$ is an integral class, which is the definition of the prequantization condition.

Appendix C: Metaplectic Correction Derivation and Maslov Index in Computational Contexts


Proposition: Metaplectic Correction

A consistent geometric quantization requires the existence of a metaplectic structure, which corrects the quantum phase of a computational path $\gamma$ by a topological term involving the Maslov index $\mu(\gamma)$.


Proof Sketch:

  1. Metaplectic Structure: A metaplectic structure is a lift of the symplectic frame bundle of the computational state space $P$ from the group $\text{Sp}(n)$ to its double cover, $\text{Mp}(n)$.
  1. Topological Obstruction: The existence of such a lift is obstructed by a class in $H^2(P, \mathbb{Z}_2)$. This class is the second Stiefel-Whitney class, $w_2(P)$. A metaplectic structure exists if and only if $w_2(P) = 0$ (Bates & Weinstein, 1997).
  1. Maslov Index and Phase Correction: When the structure exists, the phase of a semiclassical path integral is corrected. The corrected phase is $\Phi[\gamma] = \frac{1}{\hbar}\int_\gamma \theta + \frac{\pi}{2}\mu(\gamma)$. The Maslov index $\mu(\gamma)$ is an integer that counts the intersections of the path of Lagrangian subspaces associated with $\gamma$ with the Maslov cycle in the Lagrangian Grassmannian. This correction is necessary for the consistent calculation of interference phenomena.

Appendix D: Mutual Information Maximization at Critical Points


Theorem: Optimality of Criticality

Mutual information $I(A:B)$ between subsystems $A$ and $B$ of a computational system is globally maximized at a renormalization group fixed point within its universality class.


Proof Sketch:

  1. Local Maximum: At an RG fixed point $g^$, the system is scale-invariant. The derivative of a scale-invariant quantity with respect to a scale-breaking parameter must vanish, so $\frac{dI}{dg}|_{g=g^} = 0$. The scaling of computational capacity $C \sim \xi^d \sim |g-g^*|^{-d\nu}$ (where $\nu > 0$) ensures that the second derivative is negative, confirming a sharp local maximum.
  1. Global Maximum: The proof of global maximality relies on the Data Processing Inequality, a consequence of the strong subadditivity of von Neumann entropy. This inequality states that information cannot increase under a physical process (quantum channel). The RG flow, being a coarse-graining procedure, is such a process. As the system flows away from a fixed point, information is lost. Therefore, the fixed point, as the origin of all flows within its basin of attraction, must be the state of maximal mutual information for its universality class.

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