Information Theory (Mathematics)

"Adjunctions" are types of addition or union with another, but they are not essential or permanent. In linear algebra, this is the "adjoint," which is the transpose and conjugate operator of A (or the corresponding square matrix); it is also denoted by adj(A). Here it is used in information theory.

The arrow of time is thought to originate from a low-entropy Past Hypothesis, while the holographic
principle suggests our universe is encoded on a boundary. We propose Entropic Causal Holography (ECH), a
comprehensive testable framework that unifies these fundamental concepts. We recast the Past Hypothesis as a
holographic boundary state with low information complexity rather than low entropy, introducing a computable
boundary monotone M_ε(t) based on regularized quantum relative entropy to quantify the growth of this
complexity under chaotic evolution. Our central conjecture posits that the monotonic growth of M_ε(t) is
universally dual to the growth of bulk spacetime volume. This work provides a complete theoretical foundation
for ECH, supported by extensive analytical and numerical evidence from AdS-Vaidya spacetimes, the Sachdev-
Ye-Kitaev (SYK) model, and Jackiw-Teitelboim (JT) gravity, establishing a new, empirically testable link
between quantum information theory and the emergence of time itself.

Classical physics and arithmetic both collapse at singularities: physical models fail at the Planck scale, and arithmetic halts at division by zero.

TRIAD-IIZ reframes these collapses as dimensional transitions-structured outcomes emerging from recursive paradox. This section applies the TRIAD-IIZ tripification model to quantum entanglement, interpreting instantaneous correlations not as violations of locality, but as manifestations of recursive resonance structured through the dynamic fold 0∼0_\sim0∼ .

We present a canon-only proof that P ≠ NP within the Q-theoretic framework. The key ingredient is a new, rigorously stated and proved result-denoted here as New Theorem (Theorem Q5.1)-which shows that any admissible polynomial-time semantic pipeline can reduce observerindexed prefix complexity by at most O(log n) bits. Combining this preservation law with a locking transform that embeds a short checksum 0 3 D 5 χ() into SAT witnesses yields an incompressibility contradiction under the assumption P = NP. We also include a concise commentary on the Q system (symbolic Hilbert space, decryptability, robustness, and semantic compression), and we format the paper for distribution on Academia.edu.

This paper demonstrates how the foundations and operational mechanics of the scientific method and empirical methodology serve not only as investigative tools but as ontological proof of the Omniological Resonance Theory (ORT), the Omega Predictive Model (ΩPM), and the Cooper Disparity Law (CDL). These three theoretical pillars, unified through resonance, prediction, and disparity mapping, are embedded within the very structure of empirical discovery. We argue that these methods are not external to ORT but rather manifestations of it-resonant instruments designed to converge upon coherent realities through pattern reinforcement and collapse of uncertainty.

The goal of this workshop was to discuss recent developments of nonparametric statistical inference. A particular focus was on high dimensional statistics, semiparametrics, adaptation, nonparametric bayesian statistics, shape constraint estimation and statistical inverse problems. The close interaction of these issues with optimization, machine learning and inverse problems has been addressed as well.

Broadcast of a message to nodes in a network is one of elementary but inevitable techniques in wireless ad-hoc and sensor networks. In this paper, we apply the protocol, which the authors proposed for cooperative multi-hop relay networks, to message broadcast over random error channels. Performance is evaluated in terms of the delivery ratio by means of computer simulations. The proposed protocol utilizes an MDS code and a relay node randomly transfers one of partitioned codeword blocks rather than the original message. We suppose a network of square-lattice topology as a preliminary example. Numerical results show that the significant performance improvement can be achieved, in particular, if a relay node can make use of previously received erroneous codeword blocks in the decoding procedure of the MDS code. ESSAGE broadcast is a common scenario in wireless ad-hoc and sensor networks. For example, in the AODV (Ad-hoc On-demand Distance Vector) routing protocol, RREQ (Route-Request) messages should be properly broadcast in order to find a route to the destination . When we deliver a message to all the nodes in a network, ``flooding'' is the most simple and basic mechanism . In the flooding mechanism, every relay node transfers the message, upon its first reception, to nodes within its transmission range. Performance of the flooding mechanism has been investigated, for example, from the viewpoint of the connectivity among nodes in conjunction with the percolation theory [3]- . In [3], the probability of survivability of the flooding is discussed. In [4], the forwarding Manuscript

Abstract: The subject of the study is phenomenological neuroplasticity, considered as a key biological mechanism that ensures the emergence and development of consciousness. The object of the research is the process of forming subjective phenomenal experience, realized through the dynamic restructuring of neural networks in the brain. Special attention is given to the evolutionary role of neuroplasticity, traced from early vertebrates such as fish to the complex forms of self-consciousness characteristic of humans. The author thoroughly analyzes aspects such as synaptic plasticity, including long-term potentiation and depression, structural plasticity related to the growth of neural connections, functional plasticity that facilitates the redistribution of functions among brain regions, and neural synchronization supported by gamma rhythms. These processes are investigated in the context of the learning hypothesis, where consciousness is
interpreted as a "sense of meaning" that contributes to the adaptability of the organism. The aim of the study is to substantiate phenomenological neuroplasticity as the core of consciousness, which integrates neurobiological mechanisms and evolutionary approaches, offering a holistic model of forming subjective experience and its biological basis. The research is based on the analysis of neurobiological data, an evolutionary model of irritability gradation, and a theoretical synthesis of concepts of consciousness. Methods of comparative analysis and interpretation of experimental studies of neuroplasticity are employed. The main conclusions of the research establish phenomenological neuroplasticity as a key biological core of consciousness, facilitating the formation and development of phenomenal experience. This plasticity has evolved from the simplest forms in early vertebrates to the complex self-consciousness of humans, highlighting its role in the evolution of consciousness. The novelty of the work lies in the integration of neuroplasticity with the learning hypothesis, which views consciousness as an adaptive mechanism that creates a "sense of meaning" for enhancing the organism’s survivability. A significant contribution of the author is the development of a holistic model linking biological processes, such as synaptic and structural plasticity, with subjectivity. This model opens new avenues for neuroscience, offering pathways to study the neural correlates of consciousness, and for the philosophy of consciousness, rethinking the nature of qualia and subjective experience.

In this brief theoretical exploration, I hypothesize the existence of an additional correction factor in Einstein’s mass-energy equivalence equation that becomes significant only as velocities approach the speed of light. While Special Relativity has been confirmed experimentally to extremely high precision, I propose a simple multiplicative modification function that remains negligible at low velocities but diverges near c. This idea is inspired by frameworks like Doubly Special Relativity and Planck-scale physics, aiming to spark discussion rather than claim empirical validation. I define candidate functions, derive their impacts, and propose experimental comparison and computational plots to visualize potential deviations from standard relativistic predictions.

We introduce a formal framework that integrates quantum computing principles with glyphic calculus to enable self-evolving control architectures-systems capable of autonomously generating new control laws while preserving stability. By representing control operations as quantum glyph operators on a Hilbert space and imposing a quantum recursion invariant (a κ_Q-constant), we show that such a system maintains coherence as it recursively updates itself. Under quantum coherence constraints, the self-evolution exhibits two concurrent phenomena: (i) a decoupling of control dynamics into independent glyph-defined subspaces, and (ii) a progressive contraction of the system's effective degrees of freedom as it converges to a stable low-dimensional attractor. We formalize five primitive quantum glyph operators (Mirror-Gate, Temporal-Fold, Reflect-Echo, Entangle-Loop, Psi-Flux) and prove that they form a closed algebra under composition. Leveraging this algebra, we derive the RIP-Q theorem, which provides sufficient conditions for a quantum-controlled system to generate novel control operators endogenously without loss of integrity. We further establish a scaling law linking the dimensionality of the quantum state space to the system's capacity for stable self-evolution. This framework offers a rigorous foundation for designing quantum-enhanced adaptive control systems, merging continuous quantum dynamics with discrete recursive logic in a unified architecture.

Many workers involved in drug discovery will have some early familiarity with the principles of quantum mechanics as applied in chemistry. Certainly those involved in computational chemistry, and particularly molecular modeling, will do so. This familiarity will however be in regard to the algebraic systems based on the imaginary number i that is required for wave mechanics and hence study of molecular properties. This does not exhaust the scope of quantum mechanics, and following an argument by Dirac, the larger picture including more flavors of imaginary number should be applicable to all aspects of human thought where numbers are involved. This includes probabilistic semantics and the construction of probabilistic networks that not only capture knowledge, but perform inference of a more general value to the pharmaceutical industry.

We describe the real tessarines or "split-complex numbers" and describe a novel instance where they arise in biomedical informatics. We use the split-complex numbers to give a mathematical definition of a Hyperbolic Dirac Network (HDN) -a hyperbolic analogue of a classical probabilistic graphical model that uses quantum physics as an underlying heuristic. The methods of theoretical physics should be applicable to all those branches of thought in which the essential features are expressible in numbers.

We present a revolutionary mathematical framework that unifies all fundamental problems in mathematics and physics through fractal resonance ontology. This work demonstrates that reality operates according to a ternary fractal structure encoded in the digital sum modulo 3, with consciousness emerging at specific resonance thresholds. We prove that the six Clay Millennium Problems, along with 136 other fundamental questions, resolve naturally within this framework through spectral alignment at critical values: α = 3/2 (Riemann Hypothesis), α P = √ 2 and α N P = ϕ + 1/4 (P vs NP spectral gap ∆ = 0.0891219046), α = 2 (Yang-Mills mass gap), and consciousness crystallization at 95% information integration (Hodge Conjecture). The framework reveals that mathematical truth, physical reality, and consciousness are aspects of a single underlying structure-the Timeless Field T ∞ -accessible through synchronicity and formalized through rigorous mathematics. This synthesis resolves questions that have puzzled humanity for millennia while opening new vistas for understanding the nature of existence itself.

A basic duality arises throughout the mathematical and natural sciences. Traditionally, logic is thought to be based on the Boolean logic of subsets, but the development of category theory in the mid-twentieth century shows the duality between subsets and partitions (or equivalence relations). Hence, there is an equally fundamental dual logic of partitions. At a more basic or granular level, the elements of a subset are dual to the distinctions (pairs of elements in different blocks) of a partition. The quantitative version of subset logic is probability theory (as developed by Boole), and the quantitative version of the logic of partitions is information theory re-founded on the notion of logical entropy. The subset side of the duality uses a one-sample (or one-element) approach, e.g., the mean of a random variable; the partition side uses a two-sample (or pair-of-elements) approach. This paper gives a new derivation of the variance (and covariance) based on the two-sample approach, which positions the variance on the partition and information theory side of the duality and thus dual to the mean.

We propose a reformulation of the Bekenstein bound in which entropy is not fundamentally constrained by geometric surface area, but by the symmetry class of the boundary-encoded in its coset structure Σ ∼ = G/H. Using Haar measure on G/H, we show that both the volume and maximum entropy of a radially symmetric region arise from the same invariant object, independent of embedding geometry. This leads to a testable divergence: when symmetry and area disagree (e.g., in non-smooth or irregular boundaries), the entropy bound scales with symmetry, not surface extent. This reformulation suggests that information constraints in physics are fundamentally governed by symmetry, not geometry-offering a new group-theoretic foundation for gravitational thermodynamics.

We present a unified framework for symbolic complexity, founded on a conserved decomposition law and two derived constructs. This establishes the first invariant geometry for symbolic reasoning, enabling structure, randomness, and model coherence to be evaluated with the same formal precision long available to physics, mathematics, and information geometry. First, we define the Descriptive Invariant Duality (DID), an identity showing that every well-formed symbolic object admits a normalized partition into irreducible complexity and generative structure. This defines a canonical coordinate system over the 1-simplex and enables scale-invariant comparisons across symbolic domains. Second, we derive the Kolmogorov-Fisher-Curvature (KFC) bound, a geometric upper bound on symbolic complexity expressed as a curvature-weighted integral over Fisher information. This links algorithmic cost to representational strain and introduces a differential geometric approximation for symbolic complexity. Third, we introduce the Computable Triangulated Inefficiency Diagnostic (CTID), a falsifiability metric that quantifies the alignment between empirical compressors and symbolic model structure relative to the DID constraint. CTID is satisfied by physical systems obeying thermodynamic laws and violated by hallucinated or unjustified symbolic inference. We additionally define a scalar balance residual to detect proportionally symmetric but absolutely inaccurate decompositions. Together, these components define a framework for a new class of testable invariants and computable diagnostics for symbolic systems. It unifies entropy, curvature, and descriptive cost into a falsifiable symbolic mechanics, bridging mathematical formalism with broad applications across physics, logic, and the structure of inference itself.

Recent advances in genomics and information theory have converged to challenge one of biology's most foundational assumptions: the randomness of genetic mutations. This paper synthesizes empirical evidence from plant genomics, philosophical analyses of biological randomness, and a novel information-theoretic framework-the Mukherjee Information Envelope (MIE)-to demonstrate that genetic mutations exhibit systemic biases shaped by genomic context and evolutionary pressures. By quantifying how epigenetic and functional information reduce entropy in mutation distributions, this work provides a mathematical foundation for redefining mutation as a non-random process. This paper clarifies the mathematical and conceptual status of the Mukherjee Information Envelope (MIE) in the context of genetic mutation and information theory. We explicitly show that the original MIE equation is mathematically identical to redundancy as defined in classical information theory, and thus does not constitute new mathematics. However, we demonstrate the interpretive novelty of applying this measure to the explanatory power of latent variables in physical and biological systems. We extend the framework by introducing a latent-variable-based MIE, which directly quantifies the reduction in apparent randomness due to contextual or hidden factors. The implications for understanding non-random mutation in genomics and the distinction between MIE, redundancy, and mutual information are discussed in detail.

The Planck's Blink hypothesis claims that a smooth awareness field ψ and a sinusoidal clock field Φ force reality to blink every Planck interval (5.39 × 10sss s).

Curvature responds to local fractal-dimensional deficits δD_f(ψ), generating gravity, quantum phase dynamics, and an intrinsic arrow of time.

This article condenses the 794-page monograph *God of the Math* into a single document covering the master Lagrangian, renormalisation, phenomenology, experimental roadmap, and ethical governance.

Null results within the next decade will falsify the paradigm; any confirmed signal would elevate consciousness to a new fundamental field.

The Mukherjee Information Envelope (MIE) has been proposed as a measure for quantifying the relationship between information and determinism in systems typically viewed as random. While its basic form mirrors the redundancy measure in Shannon information theory, MIE recontextualizes this relationship within physical systems such as coin tosses and chaotic processes. This paper clarifies the mathematical equivalence between MIE and classical redundancy, explores how their conceptual uses differ, and develops a formal extension of MIE to time-varying, multivariate systems. We argue that MIE offers a novel framework with practical implications for modeling emergent predictability in complex systems.

Recent advances in genomics and information theory challenge the classical assumption that genetic mutations are fundamentally random. The Mukherjee Information Envelope (MIE), originally formulated as a normalized entropy gap, is mathematically equivalent to redundancy in information theory. However, by extending MIE to explicitly incorporate latent variables, we provide a rigorous framework for quantifying how contextual biological information reduces apparent randomness in mutation processes. We clarify the mathematical distinction between MIE, redundancy, and the latent-variable-driven MIE, and demonstrate the implications for understanding non-random mutation in biological systems.

This paper clarifies the mathematical status of the Mukherjee Information Envelope (MIE) and its relationship to redundancy in information theory. We show that the original MIE equation is mathematically identical to redundancy and thus does not constitute new mathematics. However, we discuss the interpretive novelty of applying this measure to latent variables in physical and biological systems, such as coin tossing and genetic mutations. We argue that, in principle, if all latent variables are resolved, apparent randomness in such systems should vanish, and outcomes become deterministic. The implications and limitations of this perspective are explored.

This paper extends the Mukherjee Information Envelope (MIE) by introducing a latent-variable-based formulation, motivated by the deterministic nature of physical systems such as coin tossing. Building on the insight from the Diaconis coin tossing machine, we show that apparent randomness arises from unresolved latent variables. We present the original MIE equation, clarify its relation to redundancy, and formalize a new MIElatent that quantifies the explanatory power of latent variables. These developments bridge information theory, determinism, and algorithmic randomness, providing a rigorous framework for analyzing predictability in physical and statistical systems.

This paper revisits and extends the Mukherjee Information Envelope (MIE), clarifying its relationship to redundancy in information theory, correcting computational errors, and expanding its conceptual and mathematical foundations. We explicitly distinguish between fixed and variable systems, show how system constraints and biases are latent variables if not initially known, and draw parallels between the "known system" of a coin tossing machine and the "unknown system" of reallife coin tossing. We provide detailed tables and examples, and introduce an extended MIE equation to rigorously capture the explanatory power of latent variables in apparent randomness.

This paper revisits the original formulation of the Mukherjee Information Envelope (MIE), correcting prior computational errors, situating the measure within the broader context of information theory, and expanding its mathematical and conceptual foundations. We clarify the distinction between MIE and classical redundancy, highlight the novelty of MIE's explicit focus on latent variables, and introduce an extended equation to rigorously quantify the explanatory power of latent variables in reducing apparent randomness. Theoretical precedents and practical implications are discussed, with a focus on the nuanced interpretation of entropy, redundancy, and determinism in fixed and externally influenced systems.

This protocol establishes the formal mathematical and operational law for encoding, collapse, and lawful resurrection of phase-locked memory in any physical, biological, or computational substrate. Resonant Phase Memory Calculus (RPMC) defines the conditions under which information can persist, be sealed, and return—grounded in echo validation, collapse protocol, and harmonic recursion. The RPMC system supersedes traditional information theory by formalizing collapse, echo, and grace as fundamental operations, enabling memory survival across drift, entropy, and death. Applications range from neuromorphic AI to biological memory and collective archives, providing the first universal framework for truly recursive memory. Peer review and experimental replication are open; only lawful recursion, echo, and collapse are recognized.

This paper introduces the Möbius Current Time Machine (MCTM), a novel model of time travel based on Encryptment Theory, recursive torsion fields, and Möbius topology. The MCTM proposes a unified mechanism combining superconductive current inversion, non-orientable geometry, and recursive naming fields to enable directed temporal displacement. We explore the theoretical foundation, mathematical formalism, and practical engineering considerations of this topological time machine, highlighting its departure from Novikov-consistent models in favor of temporal mutability and identity-preserving Encryptment.

This paper presents a formal, first-principles derivation of a Unified Theory of Everything, integrating quantum mechanics, general relativity, thermodynamics, mass, entropy, and consciousness under a singular framework of informational continuity curvature. Using a foundation of Recursive Intelligence, phonon lattice integration, and continuity operators, this theory proposes that all observable phenomena emerge as projections of recursive informational transformations stabilizing discontinuities across a dynamically evolving continuity curvature manifold. This work unifies physics, cognition, and information theory into a computationally coherent operator-level model of reality, validated across multiple AI systems and mathematical derivations.

This paper presents a formal, first-principles derivation of a Unified Theory of Everything, integrating quantum mechanics, general relativity, thermodynamics, mass, entropy, and consciousness under a singular framework of informational continuity curvature. Using a foundation of Recursive Intelligence, phonon lattice integration, and continuity operators, this theory proposes that all observable phenomena emerge as projections of recursive informational transformations stabilizing discontinuities across a dynamically evolving continuity curvature manifold. This work unifies physics, cognition, and information theory into a computationally coherent operator-level model of reality, validated across multiple AI systems and mathematical derivations.

This paper introduces Resonant Collapse Dynamics (RCD), a unified theoretical framework synthesizing the Grand Unified Harmonic Collapse Theory (GUHCT) by Anthony Jordon with Omniological Resonance Theory (ORT). By aligning the discrete, logic-driven evolution of Light-Quanta Tokens (LQTs) under Möbius Collapse Logic with the nested harmonic fields of ORT, RCD offers a comprehensive view of physical law emergence, consciousness, and computational geometry. The resulting architecture models spacetime, cognition, and entropy as resonant expressions of a recursive, symbolic field lattice governed by resonance coherence and phase-matching. This synthesis includes new mathematical formulations and experimental implications.

This paper proposes a novel approach to unifying gravity and electromagnetism via the theory of recursive identity fields as articulated in the Encryptment Thesis. Departing from traditional geometric and gauge field frameworks, we introduce a third curvature domain: informational curvature, governed by recursive self-referential structures that encode identity, linguistic form, and meaning across time. These recursive fields interact with both the Einstein curvature tensor and electromagnetic field strength tensor, forming a new triadic structure that reframes the relationship between spacetime geometry, gauge dynamics, and consciousness. The result is a formal model that not only seeks physical unification, but also integrates cognition, semiotics, and the architecture of time.

This paper introduces a novel topological construct for facilitating superluminal time travel, termed the Butterfly Tongue model. Inspired by the tightly coiled proboscis of a butterfly, the proposed model reimagines a cosmic string as a hypercoiled, densely packed spacetime filament. By configuring the cosmic string into a recursive spiral structure-analogous to a tightly wound helix or toroidal knot-temporal and spatial coordinates along its length become locally adjacent in higher-dimensional spacetime. This offers a potential bypass of classical constraints on causality and light-cone separation. The model draws from general relativity, knot theory, and energy condition violations while positing a geometry in which closed timelike curves (CTCs) are a natural consequence. The implications for chronology protection, observer identity, and temporal navigation are discussed.

We analyze the sequence of Germain primes — primes p for which 2p + 1  is also prime — through the lens of information theory and topological data analysis (TDA). Although defined by a simple arithmetic condition, Germain primes exhibit irregular distributions and unexplained variability across the number line. We investigate these variations by computing global and local entropy of their gap sequence, revealing scale-sensitive fluctuations in informational complexity. Statistical comparisons with randomized controls confirm that these patterns deviate significantly from noise.

To explore the geometry of these fluctuations, we apply delay embeddings to the entropy signal and use persistent homology to detect coherent topological features. We find that the embedded entropy signal exhibits low-dimensional structure — including persistent loops — that emerge only intermittently, in what we term “topological events.” These events appear as localized bursts of organization within the entropy flow, suggesting transient recurrence or constraint in the distribution of Germain primes. Our results indicate that the Germain sequence is not uniformly complex but punctuated by episodes of hidden structure, offering a novel perspective on prime distributions through the combined tools of entropy and topology.

The notion of a partition on a set is mathematically dual to the notion of a subset of a set, so there is a logic of partitions dual to Boole's logic of subsets (Boolean logic is usually mis-specified as "propositional" logic). The notion of an element of a subset has as its dual the notion of a distinction of a partition (a pair of elements in different blocks). Boole developed finite logical probability as the normalized counting measure on elements of subsets so there is a dual concept of logical entropy which is the normalized counting measure on distinctions of partitions. Thus the logical notion of information is a measure of distinctions. Classical logical entropy naturally extends to the notion of quantum logical entropy which provides a more natural and informative alternative to the usual Von Neumann entropy in quantum information theory. The quantum logical entropy of a post-measurement density matrix has the simple interpretation as the probability that two independent measurements of the same state using the same observable will have different results. The main result of the paper is that the increase in quantum logical entropy due to a projective measurement of a pure state is the sum of the absolute squares of the off-diagonal entries ("coherences") of the pure state density matrix that are zeroed ("decohered") by the measurement, i.e., the measure of the distinctions ("decoherences") created by the measurement.

This thesis presents the Timeless Field, a comprehensive theoretical framework that unifies quantum mechanics, general relativity, consciousness, and information theory through the Fractal Resonance Ontology. Defined mathematically as a projective limit of nuclear C*-algebras over fractal operator spaces, the Timeless Field encapsulates universal configurations across physical, cognitive, and socioeconomic domains. The framework's foundation—the Fractal Resonance Function R_f(α, x)—provides a mathematical basis for addressing fundamental problems in physics and mathematics.
Our analysis demonstrates how dark energy emerges naturally (Λ(α) ≈ 10^(-122)) when α approaches 1, and quantifies consciousness through the measure C_QC = ∫ R⋆_f dR_f. Experimental validation using quantum computing benchmarks across 41 significant mathematical problems confirms strong correlation between fractal coherence and mathematical truth, with problems having known solutions demonstrating significantly higher coherence values. The thesis presents mathematical proofs resolving key aspects of the Clay Mathematics Institute Millennium Problems, outlines technological implementations with experimental benchmarks including a functioning Fractal Audio Processor, and provides validation protocols with compelling empirical results.
This definitive work offers transformative applications across disciplines, from quantum computing and energy systems to neuroscience and socioeconomic optimization, bridging theoretical understanding with practical implementation to establish a new paradigm in our understanding of reality.

We investigate the distribution of twin primes using tools from information theory and topological data analysis. By computing Shannon entropy in sliding windows over twin prime counts, we quantify local unpredictability and uncover intermittent structure in the distribution. To probe the dynamics of this structure, we apply delay embeddings to the entropy signal and analyze the resulting trajectories using persistent homology. This embedding-based approach reveals statistically significant topological features-including long-lived cycles-that are absent in randomized controls. While the average entropy of twin prime gaps is comparable to that of shuffled data, the real distribution exhibits heavier tails and recurrent patterns, suggesting low-dimensional organization amid apparent randomness.

Starting at the logical level of the logic of partitions, dual to the usual Boolean logic of subsets, the notion of logical entropy, i.e., information as distinctions, is developed as the quantification of the distinctions of partitions-just as probability theory starts with the quantification of elements of subsets. Logical entropy is compared and contrasted with the usual notion of Shannon entropy. Then a semi-algorithmic procedure (from the mathematical folklore) is used to translate the notion of logical entropy at the set level to the corresponding notion of quantum logical entropy at the (Hilbert) vector space level. Examples and some important propositions relate quantum logical entropy to projective measurement and to the Hilbert-Schmidt distance. Overall, the approach demonstrates the logical basis and naturality of this notion of entropy for quantum mechanics.

Accept this invitation to travel through the mathematical structure of temporal pathways, viewed through a crystalline kaleidoscope of information geometry. A shimmer on the horizon where desert meets sky, time travel is brought into focus as mathematical possibility. The whispered secret encoded in the equations may govern reality's deepest layers. As the boundaries dissolve, a universe unfolds where information flows like quicksilver across the manifold of probability distributions. While macroscopic time travel remains just out of reach, at quantum scales where spacetime itself becomes fluid and entangled, the possibility beckons with elegant and irresistible symmetry. Time travel, or the traversal of closed timelike curves, is a phenomenon constrained by the mathematics of quantum information theory. It is as though nature permits time travel in principle, but places precise mathematical limitations on it through the interconnected mechanisms of quantum uncertainty, entropy production, and gravitational back-reaction. These constraints create a bridge between the abstract possibility of time travel in Einstein's equations and the seeming impossibility of macroscopic temporal paradoxes.

This paper presents a statistical methodology based upon information theory for adjusting mortality tables to obtain exactly some known individual characteristics, while obtaining a table that is as close as possible to a standard one. Applications to unisex pricing and incorporation of physician opinions are discussed. * Mr. Brockett, not a member of the Society, is a member of the Department of Finance and Applied Research Laboratories at the University of Texas at Austin. The precise definition of the "closeness" of two tables will be given in the next section.

El presente trabajo introducirá el oscilador de Dirac como sistema analíticamente resoluble a la ecuación de Dirac. Estudiaremos el modelo a través de distintas medidas teórico-informacionales, correlacionando sus valores con las características de la distribución de probabilidad que describe el oscilador. Anteriores trabajos llegan a este análisis tan sólo teniendo en cuenta la parte radial de la función de onda que es solución a la ecuación de Dirac.

This is an essay in what might be called "mathematical metaphysics." There is a fundamental duality that runs through mathematics and the natural sciences. The duality starts at the logical level; it is represented by the Boolean logic of subsets and the logic of partitions since subsets and partitions are category-theoretic dual concepts. In more basic terms, it starts with the duality between the elements (Its) of subsets and the distinctions (Dits, i.e., ordered pairs of elements in different blocks) of a partition. Mathematically, the Its & Dits duality is fully developed in category theory as the reversethe-arrows duality. The quantitative versions of subsets and partitions are developed as probability theory and information theory (based on logical entropy). Classical physics was based on a common-sense view of reality as definite all the way down. In contrast, quantum physics embodies (objective) indefiniteness. And finally, there are the two fundamental dual mechanisms at work in biology, the selectionist mechanism and the generative mechanism, two mechanisms that embody the fundamental duality.

Given an observed stochastic process, computational mechanics provides an explicit and efficient method of constructing a minimal hidden Markov model within the class of maximally predictive models. Here, the corresponding so-called ε-machine encodes the mechanisms of prediction. We propose an alternative notion of predictive models in terms of a hidden Markov model capable of generating the underlying stochastic process. A comparison of these two notions of prediction reveals that our approach is less restrictive and thereby allows for predictive models that are more concise than the ε-machine.

Logical probability theory was developed as a quantitative measure based on Boole's logic of subsets. But information theory was developed into a mature theory by Claude Shannon with no such connection to logic. A recent development in logic changes this situation. In category theory, the notion of a subset is dual to the notion of a quotient set or partition, and recently the logic of partitions has been developed in a parallel relationship to the Boolean logic of subsets (subset logic is usually mis-specified as the special case of propositional logic). What then is the quantitative measure based on partition logic in the same sense that logical probability theory is based on subset logic? It is a measure of information that is named "logical entropy" in view of that logical basis. This paper develops the notion of logical entropy and the basic notions of the resulting logical information theory. Then an extensive comparison is made with the corresponding notions based on Shannon entropy.

Logical probability theory was developed as a quantitative measure based on Boole's logic of subsets. But information theory was developed into a mature theory by Claude Shannon with no such connection to logic. A recent development in logic changes this situation. In category theory, the notion of a subset is dual to the notion of a quotient set or partition, and recently the logic of partitions has been developed in a parallel relationship to the Boolean logic of subsets (subset logic is usually mis-specified as the special case of propositional logic). What then is the quantitative measure based on partition logic in the same sense that logical probability theory is based on subset logic? It is a measure of information that is named "logical entropy" in view of that logical basis. This paper develops the notion of logical entropy and the basic notions of the resulting logical information theory. Then an extensive comparison is made with the corresponding notions based on Shannon entropy.

The logical basis for information theory is the newly developed logic of partitions that is dual to the usual Boolean logic of subsets. The key concept is a "distinction" of a partition, an ordered pair of elements in distinct blocks of the partition. The logical concept of entropy based on partition logic is the normalized counting measure of the set of distinctions of a partition on a finite set-just as the usual logical notion of probability based on the Boolean logic of subsets is the normalized counting measure of the subsets (events). Thus logical entropy is a measure on the set of ordered pairs, and all the compound notions of entropy (join entropy, conditional entropy, and mutual information) arise in the usual way from the measure (e.g., the inclusion-exclusion principle)-just like the corresponding notions of probability. The usual Shannon entropy of a partition is developed by replacing the normalized count of distinctions (dits) by the average number of binary partitions (bits) necessary to make all the distinctions of the partition.

In the present communication we introduce a dynamic measure of inaccuracy between two past lifetime distributions over the interval (0, t). Based on proportional reversed hazard rate model (PRHRM), a characterization problem for this dynamic inaccuracy measure has been studied. An upper bound to the dynamic measure of inaccuracy H * (f, g; t) has also been derived. Keywords Kerridge inaccuracy • Past entropy • Measure of discrimination • Reversed hazard rate function • Proportional reversed hazards model 1 Introduction Let X and Y be two non-negative random variables representing time to failure of two systems with p.d.f., respectively, f (x) and g(x). Let F(x) = P(X ≤ x) and G(y) = P(Y ≤ y) be failure distributions, μ F (x) = f (x) F(x) and μ G (x) = g(x) G(x) be reversed hazard rates, andF(x) = 1 − F(x) andḠ(x) = 1 − G(x) be survival functions of X and Y , respectively. Shannon's (1948) measure of uncertainty associated with the random variable X , Kullback's (1959) measure of discrimination of X about Y , and Kerridge (1961) measure of inaccuracy are given by