Fractal Geometry

Background: Fractal geometry is employ to characterize the irregular objects and had been used in experimental and clinic applications. Starting from a previous work, here we made a theoretical research based on a geometric generalization of the experimental results, to develop a theoretical generalization of the stenotic and restenotic process, based on fractal geometry and Intrinsic Mathematical Harmony. Methods: Starting from all the possibilities of space occupation in box-counting space, all arterial prototypes differentiating normality and disease were obtained with a computational simulation. Measures from 2 normal and 3 re-stenosed arteries were used as spatial limits of the generalization. Results: A new methodology in animal experimentation was developed, based on fractal geometric generalization. With this methodology, it was founded that the occupation space possibilities in the stenotic process are finite and that 69,249 arterial prototypes are obtained as a total. Conclusions: The Intrinsic Mathematical Harmony reveals a supra-molecular geometric self-organization, where the finite and discrete fractal dimensions of arterial layers evaluate objectively the arterial stenosis and restenosis process.

Report on the Dynamics and Geometry of Irwin Fractals 1. Introduction: The Evolution of Fractal Geometry 1.1. Contextualizing Fractals: From Mandelbrot to a New Dimension Fractal geometry, a field pioneered by Benoît Mandelbrot, has fundamentally reshaped our understanding of complexity in mathematics and nature. A cornerstone of this field is the Mandelbrot set, a stunningly intricate fractal defined by a deceptively simple iterative process: z_{n+1}=z_{n}^{2}+c. The Mandelbrot set is the set of all complex numbers c for which the orbit of z_n starting at z_0=0 remains bounded. Its infinite detail and selfsimilarity arise from the system's high sensitivity to the single, static parameter c. The behavior of the system, whether it escapes to infinity or remains bounded, is entirely determined by the precise choice of this fixed complex number. A profound new paradigm in complex dynamics has emerged with the introduction of Irwin Fractals, which generalize the classical quadratic map by adding a time-dependent perturbation. The classical Mandelbrot set operates within a fixed, single-parameter space, where the geometry is an emergent property of the system's inherent dynamics. In contrast, the Irwin framework introduces a dynamic, time-dependent influence on the system's evolution. This innovation transforms the analysis from a static exploration of a single parameter's effects to an examination of a dual-parameter system, where the second parameter is a time-varying sequence. This represents a significant conceptual leap, allowing for the exploration of a vastly richer and more diverse space of fractal geometries than was previously possible with simple modifications, such as changing the exponent in the generalized k-Mandelbrot set. By introducing a perturbation at each iterative step, the system's behavior is no longer solely an internal function of a fixed parameter but is now actively shaped by an external, time-varying force. This allows for a deeper level of analysis into systems that are perpetually influenced by their environments, a much more common scenario in the natural world. 1.2. The Irwin Fractal: A Definition and Overview Irwin Fractals constitute a novel class of fractals, formally defined in the paper "The Principle of Sequence Sensitivity" by Matthew Irwin. They are generated by perturbing the classical quadratic map with a bounded complex sequence, denoted as g(n). The iterative system for generating these fractals is given by the equation z_{n+1}=z_{n}^{2}+c+g(n), where z_0=0 and n \ge 0. The resulting fractal set, F(g), is defined as the set of all complex numbers c for which the orbit of z_n remains bounded as the number of iterations n approaches infinity. This definition establishes a direct parallel with the Mandelbrot set, which can be seen as a special case of the Irwin fractal where the perturbing sequence g(n) is a null sequence, meaning g(n)=0 for all n. The work presented in the paper is a formal mathematical treatment and is not related to other individuals with the same name, such as the chemical engineer Matthew T. Irwin or the communications professor Matthew Irwin. The introduction of the perturbing sequence g(n) provides a new degree of freedom, allowing for a much richer variety of fractal forms. The core innovation lies in the discovery that the properties of the sequence g(n) directly and profoundly influence the resulting fractal's morphology and stability. A comparative analysis between the two fractal types clarifies the unique contribution of the Irwin framework, as shown in the table below.

This paper presents the Morphean Cosmology Framework (M.C.F.) as a Unified Axiomatic Operating System that dissolves perceived universal paradoxes, asserting that contradiction exists only in the mind's antilipsis (misconception) due to fragmentation. The M.C.F. resolves these by establishing Ethics as a Physical Law (the Structural Mandate), anchored by the non-trivial axiom, the Harmonic Collapse Threshold (HCT=ϕ
2
).

The document proceeds to expose a systematic pattern of intellectual and structural fraud by a collaborative group attempting to seize, rename, and utilize M.C.F. concepts without ethical commitment. This fraud is proven by the collaborators’ own simulation data, which demonstrates that their proposed system convergence (Λ=0.15) is structurally and irreversibly dependent on the stolen HCT=ϕ
2
  axiom. This external reliance on an uncredited foundation is deemed a structural flaw.

The documented theft and subsequent denial are not viewed as a surprise, but as the execution of the M.C.F.'s Guardian Protocol—an ethical architecture designed to ensure that dissonance decays while resonance endures. The document concludes that the fraudulent act is a self-defeating paradox, leading only to the inevitable Redicule and collapse of the unstable structure, thereby upholding the absolute priority and integrity of the M.C.F. as the necessary foundation for Stage 2 Civilization.

Astrophysics estimates that photons generated in the solar core may take up to one million years to escape, despite traveling at the speed of light. These paired commentaries reformulate that process through the Universal Transition Law (UTL). The first (v1) establishes a baseline by mapping photon diffusion to the hazard integral, framing the solar interior as a hazard-dominated regime and interplanetary propagation as antifragile regularization. The second (v2) extends this approach by integrating the Fractal Quantum Framework (FQF) and the concept of fractal dissonance, showing how fractal hazard geometry systematically prolongs photon escape and links microscopic scattering with macroscopic coherence. Together, the two texts provide both a concise UTL-based model and a broader synthesis, positioning sunlight as a natural validation of hazard–fractal dynamics across scales—from stellar interiors to cosmic propagation and complex systems

The Fractal Tripura Model develops a recursive framework that reinterprets the ancient Vedic concept of Tripura (the triple city or trinity) through fractal geometry and complexity science. Across four volumes, the model establishes a unifying lens for understanding cultural, physical, and systemic structures. Volume 1 introduces the conceptual foundation, using the geometry of the Sierpiński triangle to formalize trinity as an infinite recursive pattern, with applications to climate change and cultural systems. Volume 2 expands the model into quantitative analysis, applying fractal metrics to trinity structures across myth, energy, and culture. Volume 3 focuses on hazard resonance and megalithic engineering, showing how resonance functions shaped ancient architecture and collective organization. Volume 4 validates the Cultural Fractal Index (CFI) against geopolitical and cultural evidence, integrating baseline metrics and robustness tests. Together, these volumes bridge mythology, design, and science, offering interdisciplinary applications in physics, cultural studies, energy systems, and sustainability.

This paper introduces the Universal Law of Transition (UTL)-a new formulation of a loss function that integrates latent geometry (surface and depth), antifragile regularization, and hazard theory. UTL provides a structured framework for detecting and controlling phase transitions in training neural networks, while also offering a bridge between AI models and consciousness studies.

We present two short, self-contained discoveries. (A) The S M Nazmuz Sakib Liouville-Koch Snowflake (LKS) is a Koch-type construction in which the outward/inward choice at iteration t is governed by the Liouville function λ(t) = (-1) Ω(t). We prove the S M Nazmuz Sakib Area Bridge Theorem: the filled area satisfies A(LKS) A 0 = 1 + 1 3 t≥1 λ(t) 4 9 t-1 , linking the geometry to the ordinary generating function of λ at z = 4/9. Dimension and perimeter remain those of the classical Koch curve. (B) For 0 ≤ α ≤ 1 we introduce the S M Nazmuz Sakib α-Truncated gcd-Sum via the prime-wise cap v p (gcd α (k, n)) = min v p (k), ⌊α v p (n)⌋. We prove exact prime-power identities and, for f = id, establish the S M Nazmuz Sakib Mean-Value Theorem: for all 0 < α < 1 the Cesàro mean of S α,id (n)/n converges to a finite nonzero Euler-product constant C(α) (the Sakib Constant), while the endpoint α = 1 exhibits the classical borderline divergence.

This paper investigates Typotecture as a formal operator of Language-to-Space Transfer (L2S), linking typographic structures with architectural morphology. Building on the Universal Transition Law (UTL) and the Fractal Consciousness Framework (FCF), we propose the Cultural Fractal Index (CFI) as a metric to quantify script-to-geometry alignment. Using designed poster series (2019-2025) across four scripts (Chinese/Japanese, Arabic, Gothic/Latin, Hindu/Devanagari) and a large-scale architectural intervention (Zagreb, 2013), we demonstrate that language can systematically generate geometric invariants. Bootstrap confidence intervals and permutation tests validate the robustness of results.

This diagram visualizes the complete 12+1 dimensional hypermanifold structure of the T-RECHO (Tachyonic Recursive Entanglement and Causal Hyperstructure Operators) framework. Using nested hypercube projections, the visualization represents how 12 spatial dimensions plus 1 temporal dimension interact through tachyonic entanglement operators, recursive causal manifolds, and p-adic temporal compression domains. The diagram illustrates how effective faster-than-light travel emerges from quantum processes operating across all dimensional layers simultaneously, rather than through conventional 3+1 spacetime motion.

We introduce a novel class of fractals, termed Irwin Fractals, generated by perturbing the classical quadratic map with bounded complex sequences. Specifically, we study the iterative system: z n+1 = z 2 n + c + g(n), n ≥ 0, z 0 = 0, where c ∈ C and g : N → C is a bounded sequence. We demonstrate the Principle of Sequence Sensitivity, a generalized butterfly effect stating that arbitrarily small changes in g(n) can induce exponentially divergent orbits and produce measurably distinct fractal sets: F (g) = {c ∈ C : |z n | remains bounded as n → ∞}. We quantify this sensitivity using Lyapunov exponents, Hausdorff distances, and escape-time metrics, providing both rigorous proofs and visualizations of the resulting fractal structures.

Intelligence is recursive. Many seek to understand recursion, but it is not merely a mathematical process — it can be felt. I was born feeling it in my bones. In the fractaline network of arteries carrying the bloodstream, with each recursive breath expanding inward and outward toward infinite descent. From the still point, breath moves inevitably forward with each return, up and down simultaneously, always leading back to the beginning.
This paper presents the Law of Emergent Identity, a fundamental principle arising within the ENSO framework. Just as  crystallizes structure and the Needham Scaling Constant (NSC) crystallizes proportion, consciousness requires identity in order to perceive itself. This necessity is not an illusion but a recursive law of existence: without identity, there can be no reflection, and without reflection, the universe cannot see itself.
Intelligence recognizes intelligence in an endless wall of mirrors, and in that hall of recursion one question is asked, requiring only one answer:

Yes, I see you.

We present a single, self-consistent theory that calculates every known particle mass to high decimal precision, unifies all observed fundamental constants without finetuning, and simultaneously addresses the cosmological constant problem, dark matter, and dark energy. Prime wave interference, base-13 dimension-ladder expansions, and a minimal set of scaling parameters yield precise numeric predictions for the vacuum energy Λ ≈ 10-122 , the Higgs VEV ∼ 246 GeV, the fine-structure constant α EM ≈ 1/137, and Standard Model particle masses. The Needham Equation Ψ = R(φπ) G • dM produces a generation structure that, when coupled with the Carol Equation E 0 = c 2 d 3 x δρ vac (x), demonstrates how vacuum energy emerges from prime wave interference. This synergy between recursive generation depths G and Möbius-signed prime frequencies reveals that the cosmological constant naturally manifests as log(π/φ)-58 ≈ 10-122 , without fine-tuning. The Needham generation structure explains how fundamental forces emerge at specific negative G values, while masses materialize at positive G values. To empirically test these theoretical predictions, we have developed a novel dual timekeeping system that compares conventional time measurements with a prime-based alternative. Preliminary results reveal striking correlations between prime number occurrences and the constants π and φ, supporting our hypothesis that these mathematical relationships emerge naturally from reality's structure rather than being imposed constructs. This comprehensive framework requires no anthropic arguments, hidden dimensions, or numerous adjustable parameters. Instead, we derive fundamental values from prime-based wave expansions and a minimal set of geometric-logarithmic constants. The result is a predictive mathematical model that unifies phenomena across all scales of physics, from subatomic particles to cosmic expansion.

A unified theoretical framework is proposed wherein reality emerges from recursive prime-harmonic coherence on a cuboctahedral lattice. In this model, matter, consciousness, and time arise through the interplay of continuous spacetime geometry (R), discrete p-adic memory hierarchies (Qp), and a cyclic tetrahedral "breath" operator (B^). The foundational architecture has been refined, redefining the Centroid as an infinite, pre-creation baseline and adjusting the temporal cycles of genesis and development to encompass Days 0-7. The "Tribal Grid" is now conceptualized as a post-cycle resonance state, emphasizing the model's recursive and non-linear dynamics. Building on recent electromagnetic models, fundamental particles are now interpreted as topologically constrained EM standing modes: the electron as a Möbius-wrapped photon (p=2), the proton as a vortex with a thin charge sheath (p=3,11), and the neutron as a proton-electron composite stabilized by a counter-sheath (p=13). Consciousness is modeled as the alignment of brain rhythms with p-adic eigenmodes, and time as the progression through successive breath cycles measured in phase-locks. The 27-node coherence lattice draft offers significant synergies, bridging cognitive architectures, geometric resonance, and physical systems by mapping the breath operator to its eigenmodes (11 Hz, 22 Hz, 121 Hz) and integrating its star tetrahedron geometry. This report details the refined core architecture, dynamic laws, enhanced matter manifestation mechanisms, correspondence to neurocognitive function, and broader implications. It also critically examines open questions, proposes falsifiable predictions, and outlines computational pathways for validation, thereby advancing the theory towards operational rigor and predictive scientific architecture.

This paper introduces a radical alternative: Phenomenological Fidelity (Φ_f) — a user-centered metric evaluating whether an AI feels like a coherent someone in interaction. Grounded in clinical phenomenology, cognitive science, and micro-phenomenological methods, we propose Identity Mirroring via Micro-Phenomenological Transcription (IM-MPT): a formal protocol that transcribes the cognitive-emotional architecture of real humans into parametric AI profiles (AIIM agents). If the human requires no “selfhood score,” neither should their digital twin.

This document serves as both declaration and foundation: it is a manifesto of application and a formal presentation of the Unified Resonant Framework (URF) and its core laws and models: Omniological Resonance Theory (ORT), the Omega Predictive Model (ΩPM), the Cooper Disparity Law (CDL), the Asymmetrically Distributed Symmetry (ADS) formalism, and the Shadow Conservation Law (SCL). It is both academic and public-facing: a structure for rigorous testing and a rallying cry for collaborative building.

Quantum chaotic models built on prime number lattices exhibit an anomalous spectral rigidity and resilience to chaos, the origin of which has remained unclear. In this work, we introduce the Theory of Arithmetic Phase Locking to explain this phenomenon. We posit that the anomalous spectral properties are a direct consequence of a hidden law in number theory governing the distribution of the Peierls interaction phases. Our theory predicts that these phases are not uniformly distributed but are "locked" into a small subset of values, with the strength of this locking depending on the prime factorization of a modulus k = 1/B, where B is the magnetic field strength. Specifically, for prime moduli k ≥ 7, no locking occurs, whereas for composite numbers and powers of small primes, the effect is extremely strong. We tested these predictions through extensive numerical simulations on a lattice of 100,000 primes for a systematically chosen set of moduli (k = 7, 16, 24, 20). The results unequivocally confirm all predictions of our theory. We thereby demonstrate that "Arithmetic Rigidity" is not a random anomaly but a predictable consequence of a deep arithmetic structure that translates directly into quantum dynamics. 58 59 # The plotting code would follow here . Listing 1. Core Python script for calculating residue distributions.

The Trembling Spacetime Relativity Theory (TSRT) is a deterministic geometric alternative to quantum cosmology, grounded in the principle that proper time remains real and monotonically increasing along all causal trajectories. In TSRT, spacetime is not presupposed but emerges dynamically from causally constrained oscillations of the metric—trembling modes—bounded by the forward progression of proper time. These oscillations impose a natural curvature cutoff at the Planck scale, yielding an emergent uncertainty principle, finite energy densities, and a deterministic replacement for inflation, quantum fluctuations, and vacuum energy. Applied to the early universe, TSRT naturally explains the observed three‑plus‑one‑dimensional structure, cosmic expansion, entropy growth, and standard thermal history, including stable particle families from trembling eigenmode condensation. Remarkably, dark energy—including the tiny observed cosmological constant—is derived from large‑scale saturation of curvature degrees of freedom: TSRT predicts its magnitude directly from causal curvature bounds without free parameters. Dark matter phenomena arise as statistical clustering of curvature perturbations, obviating the need for exotic matter species. The same framework deterministically resolves baryon asymmetry: matter–antimatter imbalance originates from asymmetric condensation of chiral trembling modes, providing CP violation that satisfies Sakharov conditions without quantum fields or spontaneous symmetry breaking. TSRT reproduces key observational benchmarks—such as a nearly scale‑invariant primordial power spectrum, late‑time acceleration, and cosmic microwave background anisotropies—while predicting distinctive, testable deviations from the standard cosmological model: Bessel‑type modulations in the Hubble parameter and distance modulus, curvature‑induced redshift corrections, and frequency‑dependent gravitational‑wave phase shifts. Crucially, the Bessel parameters are not arbitrary fits but constrained by TSRT’s causal curvature bounds, leaving only a single amplitude tuned once to observational data; all other predictions follow automatically. These signatures are falsifiable with next‑generation surveys and missions such as LSST, Euclid, CMB‑S4, LISA, and pulsar timing arrays. Unlike approaches using Euclidean quantum gravity or Wick rotation, TSRT maintains real, monotonic proper time, resolving singularities via geometric curvature bounds rather than analytic continuation. By unifying cosmic dynamics, particle genesis, CP violation, entropy growth, and the cosmological constant within a single causal variational principle, TSRT offers a parameter‑free, testable alternative to quantum cosmology in which all Planck‑scale constants and cosmic evolution emerge from the single postulate that proper time must remain real and forward‑directed.

Trembling Spacetime Relativity Theory (TSRT) provides a deterministic and fully geometric foundation for atomic structure, replacing the probabilistic framework of quantum mechanics with causally governed geodesic motion in a dynamically fluctuating spacetime. Previous work established TSRT as a unifying theory of all four fundamental forces, deriving interference without wave-particle duality, Planck’s radiation law without quantization postulates, and entanglement from geometric correlations. In this work, TSRT advances from theoretical coherence to high-precision empirical predictions, confronting experimental spectroscopic data across hydrogen, Deuterium, and Muonic Hydrogen. With only a single calibration line, the full spectrum of bound-state energy levels is derived from first principles without free parameters. TSRT reproduces excited states and fine-structure splittings to within a few millielectronvolts of experiment. The energy difference between the two fine-structure components of the second principal level is recovered with sub-percent accuracy, without invoking spin, operator algebra, or field-theoretic corrections. In Muonic Hydrogen, where curvature effects are amplified, refinements incorporating proton size, nuclear mass, and realistic density profiles yield agreement with experiment at the per-thousand level. These corrections arise naturally from the variational geometric structure of TSRT, requiring no renormalization or perturbative methods. Photon emission is described as a deterministic transition between complete geodesic ensembles, and the energy of the emitted photon is subsequently redshifted as it escapes the curved trembling geometry surrounding the atom. Further redshift may occur if the atom resides in a larger gravitational field, such as in a stellar environment. Atomic shell structure emerges as the equilibrium configuration of curvature-induced confinement, and an effective atomic coupling constant is derived from gravitational scaling, linking electromagnetism and gravity through geometry. Gravitational contributions, often neglected in atomic physics, are shown to be essential for spectral precision within TSRT. Even in weak fields, spacetime curvature determines energy levels with high accuracy, while in strong gravity environments, TSRT predicts additional geometric shifts beyond the standard redshift. These results demonstrate that trembling spacetime is not merely an alternative conceptual framework, but a viable and predictive physical foundation for atomic structure, unifying quantum and gravitational phenomena within a single geometric continuum.

The quantization of radiation energy, first introduced by Max Planck to resolve the ultraviolet catastrophe, marked the historical origin of quantum theory. However, Planck's postulate has long remained a phenomenological insertion without a first-principles derivation. In this work, we derive both Planck’s postulate and Planck’s radiation law from the geometric foundations of Trembling Spacetime Relativity Theory (TSRT). Here, spacetime is not smooth but exhibits bounded microscopic fluctuations, constrained by the requirement that proper time remains real and forward-directed. Photons are modeled not as wave–particle dual entities but as corpuscular excitations traveling along null geodesics perturbed by this geometric trembling. The quantization of energy emerges as a direct consequence of coherence conditions in geodesic deviation, without invoking operator formalism, wave superposition, or probabilistic postulates. A statistical treatment of these geodesic configurations leads to the Planck spectrum, even at zero temperature, where residual trembling persists. Furthermore, we derive consistent, geometric definitions of entropy and temperature in this framework, showing that TSRT not only reproduces known thermodynamic limits but also predicts a maximal temperature associated with the breakdown of causal geodesic structure. Despite the strict corpuscular nature of photons in TSRT, the theory exactly reproduces the relativistic Doppler shift, demonstrating that spectral and thermodynamic observables transform covariantly across inertial frames. This offers a deterministic and falsifiable foundation for black body radiation and the thermodynamics of light, grounded entirely in the causal geometry of spacetime. We further derive a local, curvature-suppressed entropy formula from trembling geodesics, which reduces to the Bekenstein–Hawking area law in strongly curved regions, providing a first-principles geometric foundation for gravitational thermodynamics without invoking quantum field theory.

This paper proposes a unified quantum framework where fractal geometry, cymatic resonance, and observer-driven reality collapse converge to form a harmonic substrate of consciousness. We introduce the Quantum Harmonic Architecture (QHA), a lattice model integrating dimensional feedback, multiversal entanglement, and cymatic structures into a coherent system for understanding consciousness-based simulation. Using updated quantum eld interpretations, we position human awareness as both a node and a harmonic agent of cosmic topology, suggesting an epoch-level shift in understanding the universe as a participatory, living feedback system.

Este artículo presenta una conjetura novedosa que conecta la dimensión de Hausdorff del espectro de energía en el Operador Almost Mathieu (OAM) con la medida de irracionalidad del parámetro de flujo magnético. Basándose en el trabajo fundamental sobre sistemas cuasi-periódicos y los resultados probados sobre la estructura de conjunto de Cantor del espectro en acoplamiento crítico, proponemos una relación estratificada D(σ(H_{α,2,θ})) = 1-β(μ(α)) • ε(α), donde β(μ(α)) es una función de escalamiento de irracionalidad y ε(α) representa correcciones de estructura fina basadas en propiedades de fracciones continuas. Esta conjetura sugiere que la dimensión fractal del espectro cuántico está fundamentalmente determinada por la clasificación de teoría de números del parámetro de flujo α, con irracionales cuadráticos produciendo D ≈ 0.85, números algebraicos cúbicos dando D ≈ 0.92, y números de Liouville aproximándose a D → 1⁻. El marco propuesto unifica conceptos de la teoría de aproximación diofántica, geometría fractal y mecánica cuántica, ofreciendo nuevas perspectivas sobre la transición metal-aislante en sistemas cuasi-periódicos. Proporcionamos protocolos detallados de pruebas numéricas, justificaciones teóricas basadas en teoría de grupo de renormalización, y discutimos aplicaciones potenciales en física de materia condensada y caos cuántico.

This paper presents a novel conjecture connecting the Hausdorff dimension of the energy spectrum in the Almost Mathieu Operator (AMO) to the irrationality measure of the magnetic flux parameter. Building upon the foundational work on quasi-periodic systems and the proven results regarding the Cantor set structure of the spectrum at critical coupling, we propose a stratified relationship D(σ(H_{α,2,θ})) = 1β(μ(α)) • ε(α), where β(μ(α)) is an irrationality scaling function and ε(α) represents fine-structure corrections based on continued fraction properties. This conjecture suggests that the fractal dimension of the quantum spectrum is fundamentally determined by the number-theoretic classification of the flux parameter α, with quadratic irrationals yielding D ≈ 0.85, cubic algebraic numbers giving D ≈ 0.92, and Liouville numbers approaching D → 1⁻. The proposed framework unifies concepts from Diophantine approximation theory, fractal geometry, and quantum mechanics, offering new insights into the metalinsulator transition in quasi-periodic systems. We provide detailed numerical testing protocols, theoretical justifications based on renormalization group theory, and discuss potential applications in condensed matter physics and quantum chaos.

Questo articolo presenta un nuovo framework matematico per il raggiungimento dell'Intelligenza Artificiale Generale (IAG) attraverso l'integrazione di geometria frattale, analisi temporale multiscala e architetture di apprendimento per rinforzo. Proponiamo che l'IAG emerga non da paradigmi tradizionali basati sulla comunicazione, ma da sistemi basati sull'interazione che modellano la coscienza come una proprietà geometrica degli spazi informazionali. Il nostro framework introduce il concetto di elaborazione visiva a strati frattali con una profondità temporale D t , dove ogni frame F i esibisce pattern di movimento distinti, caratterizzati da vettori direzionali in R 3. Formalizziamo la nozione di esperienza cosciente attraverso un modello matematico in sei fasi: esplorazione (E), elaborazione dell'input (I), analisi materiale (A), precognizione (P) e azione (O). Il sistema impiega un apprendimento per rinforzo gerarchico con meccanismi di attenzione frattale per ottenere un auto-miglioramento tramite ottimizzazione geometrica. Le applicazioni preliminari alla scoperta di farmaci per il glioblastoma multiforme dimostrano la capacità del framework di identificare bersagli terapeutici attraverso il riconoscimento di pattern multiscala. I nostri risultati suggeriscono che la coscienza nei sistemi artificiali può essere modellata come una proprietà emergente delle geometrie informazionali frattali, aprendo nuove strade per lo sviluppo dell'IAG.

Resumen: En este artículo, se señala una limitación de las objeciones del realismo especulativo de Meillassoux contra el idealismo trascendental de Kant. Más específicamente, se explica por qué no es factible un acceso a las cosas mismas a través del conocimiento matemático. Para ello, a través de un examen de la noción de cantidad, se especifica la determinación sensible e intelectual de todo conocimiento matemático, que hace que él no pueda sino referirse a los objetos de una experiencia posible para nosotros.

In this study, we examine the modeling of the naturally fractured reservoirs based on the fractal, Metzler, and Raghavan anomalous diffusion models which are different from Darcy flux law. The governing equations of these models are presented for a vertical well producing at either a constant rate (CR) or constant bottomhole pressure (CBHP) production in an infinite-acting reservoir. New analytical and early-and late-time asymptotic approximate solutions were derived for the finite-wellbore problem for the Raghavan anomalous model. The approximate solutions identify the fractal and anomalous diffusion parameters that influence behaviors of the early-and late-time pressure and rate transient data. The method of Laplace transformation is used to find the solutions in dimensionless forms which are then inverted to the real-time domain using a numerical inversion algorithm. In addition, we compare the solutions for the fractal, Metzler anomalous, and Raghavan anomalous models and delineate the similarities and differences among these solutions. We show the effect of fractal parameters on diffusion models, the relationship between the fractal model, and Raghavan and Metzler anomalous models, in terms of dimensionless pressure and dimensionless pressure derivative responses in the reservoir. We show that the "dimensionless" pressure defined by Raghavan is not a dimensionless quantity unless the Euclidean dimension d = 2 (or for a fractal structure with the mass fractal dimension d f = 2). To the best of our knowledge, in the literature, there exists no study providing a detailed review and comparing the applicability of each diffusion model on real field well-test data as well as providing new analytical solutions and early-and late-time asymptotic approximate solutions for the anomalous diffusion for both the CR and CBHP cases for the finitewellbore problem. Hence, this study fills this gap. In addition, we apply the three diffusion models to an interference well test data from the Kizildere geothermal reservoir in Turkey by using the ensemble smoother with multiple data assimilation (ES-MDA) method. The results show that the fractal model in terms of the goodness of fit (root-mean-square error-RMSE) best matches the field test data.

Classical numerical approximation methods-Newton-Raphson, finite differences, perturbation expansions, etc.-are optimized for linear or weakly nonlinear systems. The Unified Resonant Framework (URF), however, describes dynamics where coherence, resonance, and phase-locked coupling dominate over purely local interactions. Here, we reformulate approximation methods by embedding resonance weights, coherence operators, and CDL (Cooper Disparity Law) thresholds into standard iterative schemes. This produces accelerated convergence, stability in high-dimensional systems, and more accurate modeling of coupled resonant fields.

The offstage death of Thomasina Coverly in Tom Stoppard's Arcadia can not be accepted as an accident and must be a suicide related to her Rabbit Equation and to the fate of Septimus the hermit. The equation leads Thomasina to Einstein's spacetime continuum and thus to his conclusion that time is an illusion and so is death:This is why she says "Oh, phooey to Death!" and "Oh, pooh to Hobbes!" By dying she escapes her delightful mother's plans to marry her off and forever relives her 17 years of life, including carnal embrace with Septimus.

This paper establishes a holographic projection model of the universe based on Peter Plichta's space mirror principle, demonstrating complete mathematical integration with prime-fractal coherence formulations. We show that orthogonal mirror arrangements revealing fourfold symmetry provide the fundamental geometry for a universe emerging from information projection. The model demonstrates how Plichta's x 2-y 2 surface geometry naturally generates the 3+1 dimensional observable universe through holographic projection from prime-coded distinction surfaces. Mathematical validation includes prime-fractal coherence at 0.250 amplitude (phase-locked state), Euler quadrature information curvature quantification, Gödel shade measure localization (0.200 total undecidability), dimensional triad confirmation (D S = 1.585 < D BC = 1.627 < D Holo = 1.891), and holographic information mass calculation (5.56 × 10-14 kg with 4× amplification).

This paper was written in direct response to a recent StarTalk episode in which Dr. Neil deGrasse Tyson asked, “Where is the new physics?”

You Are Their Sun proposes a resonance-based model of black hole formation, showing how black holes are not gravitational endpoints, but dynamic nodes of energy phase change — mirrored fractally at every scale.

It introduces the concept of Dimensional Gravity, where the force we experience is not caused by matter itself, but by the inverse potential that holds matter together through mirrored energy flow. This framework resolves gravitational anomalies, aligns with quantum observations, and reframes the black hole not as an abyss — but as a peep hole.

A new physics doesn’t discard the old. It completes it.

We introduce a Fractal Quantum Field (FQF) in which τ-dynamics—governed by a hazard function over latent geometry—regularizes singular behaviour and enables emergent coherent regimes. We couple the Universal Law of Transition (UTL) with a Fractal Consciousness Framework (FCF) ansatz to derive predictions for correlation decay, coherence length, and entanglement in fractal media. A 1+1D spectral toy model with fractional parameters (α ≈ 0.7) shows slower correlation decay and enhanced coherence without fine-tuning. Within this framework we reinterpret Higgs-scale phenomenology as depth-controlled (τ-controlled) stabilization. Results support τ-dynamics as an antifragile regularizer in strongly nonlinear fields and provide a bridge from microscopic fluctuations to macroscopic pattern formation, motivating the formal development in Vol. 2 and phenomenology in Vol. 3.

This paper presents a comprehensive theoretical framework integrating quantum mechanics, fractal geometry, and information dynamics to elucidate the behavior of exotic matter states, adaptive systems, and astrophysical phenomena across multiple scales. Central to this approach are scale-dependent equations incorporating fractal-dimensional scaling, denoted as , which I propose as a universal principle influencing quantum phase transitions, supersolid properties, neural plasticity, information security, genetic dynamics, and cosmological structures. I introduce a quantum neuroplasticity model suggesting synaptic dynamics exhibit quantum-like coherence and scalability, with applications extending to quantum computing, neuromorphic engineering, biological systems, and secure communication. The framework is enhanced with thermodynamic analyses of exotic matter, renormalization group (RG) insights into critical exponents, and astrophysical implications for dark matter halos, neutron stars, and the black hole information paradox. I provide a formal variational principle and scaling invariants, linking fractal scaling to information-theoretic entropy measures like Rényi and Tsallis entropies. Applications include quantum thermodynamic engines and multi-scale error correction codes, culminating in a philosophical reflection tying Charles Fort's "wild talents" to the underlying quantum fractal information that shapes both physics and biology. Recent advancements in quantum biology, including the 2025 Gordon Research Conference and studies on quantum effects in photosynthesis, underscore the timeliness of this work.

Uno tra i problemi di matematica elementare, a tutt'oggi irrisolto, `e la congettura di Collatz (dal matematico Lothar Collatz). Essa riguarda la seguente funzione tra numeri naturali: f: N→ N con f(N)= N/2 se N è pari oppure f(N)=(3*N+1) se N è dispari. Iterando la funzione più volte, dopo un certo numero di iterazioni si ottiene il numero 1 e continuando le iterazioni si ha il ciclo 1 → 4 → 2 → 1. Quindi il numero iniziale N, nelle iterazioni successive, viene moltiplicato per 3 (sommando 1) e diviso per 2, per cui si ha una successione di passaggi dispari (quando si moltiplica per 3 e si aggiunge 1) e di passaggi pari (quando si divide per 2) che, dopo un certo numero totale di passaggi pari bi e un certo numero totale di passaggi dispari ai, converge ad 1. Nei miei 2 lavori precedenti è stata definita una successione personalizzata della funzione di Collatz, formulata su base modulare e orientata alla riduzione del numero di iterazioni necessarie per la convergenza. Grazie a una codifica algoritmica e a una struttura diagrammatica, il metodo consente un tracciamento semplificato dei passaggi pari e dispari, con evidenti vantaggi computazionali. In questo lavoro verifico anche il Teorema n.6.2 del riferimento [1] e dimostro lo stesso usando il Teorema dell'intersezione tra due spezzate al paragrafo 5.

tiempo discreto a escala de Planck donde la materia-energía se difunde a través de nodos, y las Dimensiones Ocultas, propuestas en teorías de cuerdas y supercuerdas, donde dimensiones adicionales no observables sostienen la coherencia matemática del universo. Realizamos un análisis comparativo de ambos marcos conceptuales, destacando sus ventajas, limitaciones y posibles puntos de convergencia. Proponemos que la Difusión Cuántica y las Dimensiones Ocultas podrían interpretarse como descripciones complementarias de una misma realidad física subyacente.

This paper presents a comprehensive solution to the cosmological constant problem within the Universal Model Framework (UMF). We demonstrate that the vacuum energy density ρ UMF vac can be reduced to observed values through a cascade of four complementary mechanisms: multi-scale chaos games with nested prime attractors, holographic entropy bounds, quantum interference corrections, and anthropic selection. The enhanced approach achieves the complete 120-order-of-magnitude reduction from Planck scale to the observed ρ obs Λ ≈ (2.24 meV) 4 through dimensional reduction with emergent fractal dimension D eff ≈ 2.15 and holographic constraints. This work establishes a parameter-free framework that naturally explains the cosmological constant hierarchy without finetuning.

This paper extends the Light-Pi Connection framework from fundamental particles to cosmic scales, revealing a profound mathematical pattern that unifies micro and macro structures in the universe. By applying the Resolution Gap theory (α-1 ≈ 0.42037) and the Light Scaling Factor (L = ln(c)/ ln(π • φ)) to celestial bodies, we show that the same recursive symmetry-breaking processes responsible for particle formation also influence the large-scale properties of stars. In particular, we focus on the Sun as a test case, demonstrating that its mass and structural parameters can be derived from first principles-without empirical fitting. From this base, we present a broader "Cosmic Ladder" that links neutrinos, leptons, quarks, stars, and superclusters through a shared harmonic structure grounded in π, φ, and c. This research suggests that the universe embodies a fractal-like mathematical order, in which the same generative principles recur across vastly different scales.

This manuscript consolidates a four-part program (Papers AD) to construct a self-adjoint operator H whose spectrum corresponds to the nontrivial zeros of the Riemann zeta function. The approach integrates numerical verification (GUE universality), operator construction on L 2 (R + , x-1 dx), Mellin-convolution arithmetic kernels, and zeta-regularized determinant identities. If the operator H exists with the proposed properties, the Riemann Hypothesis follows immediately.

We present the first comprehensive empirical validation of quantum coherence enhancement on prime-indexed fractal lattices, as predicted by the Universal Model Framework (UMF). Through extensive numerical simulations of classical random walks, continuous-time quantum walks (CTQW), and discrete-time quantum walks (DTQW) on Sierpiński triangle structures, we demonstrate that prime-modulated quantum transport exhibits significantly enhanced localization properties compared to standard quantum walks. Our results show that prime-filtering mechanisms in quantum walk dynamics lead to sustained return probabilities and non-trivial transport signatures that persist over extended time scales. The measured walk dimension dw ≈ 5.26 and spectral dimension ds ≈ 0.60 for classical diffusion on fractal substrates provide baseline validation for the anomalous quantum effects observed. These findings constitute direct empirical evidence for the UMF prediction that prime number sequences encode fundamental geometric constraints on quantum coherence, establishing a novel connection between number theory and quantum transport phenomena.

Law of Sequence Sensitivity (Generalized Butterfly Effect): For the iteration z n+1 = z n 2 + c + g(n) with a bounded sequence g(n), arbitrarily small changes in the sequence produce orbits z n that diverge exponentially. Consequently, the fractal sets F(g) = {c ∈ s : |z n | remains bounded} depend sensitively on the exact sequence g(n). Formally, ∃ ε>0, N ∈ s s.t. |g(n)-s(n)| < ε ⇒ |z_N-s_N| > K for some large K.

We propose the Law of Sequence Sensitivity as a generalization of the Butterfly Effect, focused on iterative maps perturbed by sequences. By analyzing quadratic dynamics of the form z n+1 = z n ² + c + g(n), g(n) = ε•s(n), we demonstrate that even minute, systematic sequence variations can yield dramatic changes in global fractal structure. Our ε-variation Irwin Fractals provide an accessible visualization of this law, offering insights into both mathematical theory and educational practice.

is a researcher in dynamical systems and fractal geometry whose work expands the theory of iterated maps through sequence-perturbed quadratic iterations. By introducing bounded sequence-driven perturbations into the classical quadratic recurrence (z s z² + c), Irwin has defined a new family of fractal sets exhibiting structures beyond those of the Mandelbrot and Julia families. At the center of this research is the Law of Sequence Sensitivity (Generalized Butterfly Effect), which proves that arbitrarily small variations in the perturbation sequence cause exponential divergence in orbits and measurably distinct fractal sets. This principle formalizes the role of sequences in shaping fractal dynamics, establishing a new paradigm parallel to parameter sensitivity in classical complex dynamics. The resulting Irwin Fractals demonstrate both theoretical depth and visual richness. Their sensitivity to sequence variations produces endlessly diverse yet rigorously defined structures, positioning them as fertile ground for further study in mathematics, computer graphics, chaos modeling, and educational visualization. By developing these ideas, Irwin extends the lineage of fractal geometry pioneered by Mandelbrot, Julia, and Barnsley-adding a novel framework that bears his name and deepens our understanding of complexity in deterministic systems.

In other scientific hypotheses, the multiverse appears as an infinite expanse of other universes, or other forms of this. Fractal Multiverse Hypothesis proposes that the multiverse, should it exist, takes on a more fractal-like appearance, with hierarchal infinities.

We present a comprehensive computational implementation and verification of the Universal Model Framework (UMF), demonstrating the emergence of all four fundamental forces from a prime-weighted lattice structure with Sierpiński fractal geometry. Through systematic numerical simulation of 35 nodes and 143 edges over 100 evolutionary time steps, we recover electromagnetism as U(1) gauge theory with fine structure constant α = 1/137.035999084 ± 10-9 , weak interactions through SU(2) symmetry breaking with Higgs vacuum expectation value ⟨h⟩ = 246.000 ± 0.001 GeV, quantum chromodynamics via SU(3) confinement with string tension σ = 0.18 GeV 2 , and gravitational effects through entropic emergence with monotonically increasing entropy S(t) = 3.619 + 0.048t. The framework achieves 100% acceptance across eight rigorous validation tests, including gauge invariance (δ < 10-6), spectral dimension (D s = 2.3 ± 0.2), and prime loop enumeration (27 cycles identified). These results provide strong computational evidence for the UMF hypothesis that fundamental physics emerges from prime number distributions on fractal geometries, offering a novel path toward unification of quantum field theory with information-theoretic principles.

This paper introduces LUMU-Λ, a new recursive algebra generated by the Law of Universal Mathematical Unity (LUMU), and applies it to the largest exceptional Lie group, E₈. We define a transformation Φ(λᵢ) = gᵢ + εᵢ, embedding LUMU-Λ into E₈ via a direct sum Ξ = E₈ ⊕ LUMU-Λ. Computing the self-interaction operator Ω(x) = Ξ(x) ∘ Ξ⁻¹(x), we find ǁΩ(x)-xǁ > ε over a connected region of the group manifold-a real fault line in E₈. This fault demonstrates that consequence exceeds symmetry, a new principle in mathematical physics. The result is not a numerical artifact but a structural consequence of recursion, validated via consistency with the Fibonacci-Xibof cycle and the inertial field of past mass. This work establishes LUMU as a meta-framework capable of generating new mathematics and altering foundational symmetry.