Open research questions in Advanced Mathematical Theories and Applications

What the literature leaves open

• Future research can apply the coupled Leonardo sequence to real-world problems. The sequence can be used to model complex phenomena. Further analysis of the sequence's properties is needed.

  Some properties of the coupled Leonardo sequence · 2026 · DOI

• There is a need for new generalizations of the Leonardo sequence. The coupled Leonardo sequence fills this gap. The paper provides a new tool for mathematical analysis.

  Some properties of the coupled Leonardo sequence · 2026 · DOI

• Current systems fail to strictly distinguish the Universal Cosmos from the 4-Dimensional Universe. Current systems rely on external parameters and artificial assumptions.

  General Outline of PFUS Primitive Field Unified Steady‑State System: Dynamic Evolution, Structural Fusion & Expansion, Information Coding and Origin of Time · 2026 · DOI

• The mathematical friction encountered when attempting to unify the prevailing models of cosmology and theoretical physics. The need for a new framework that abandons problematic assumptions. The challenge of providing a strictly mechanical, deterministic universe governed by spatial translation and structural density.

  The Mechanics of Simultaneity: Gravity as Spatial Inflow and Time as Localized Kinematic Friction within the Infinite Fractal Descent Framework · 2026 · DOI

• The paper suggests further study of the IFD framework and its implications for cosmology and theoretical physics. The framework provides a new perspective on the mechanics of spatial inflow and topological drag.

  The Mechanics of Simultaneity: Gravity as Spatial Inflow and Time as Localized Kinematic Friction within the Infinite Fractal Descent Framework · 2026 · DOI

• The model's behavior is complex and nonlinear. The model has multiple parameters that need to be estimated. The model's behavior can exhibit bistability and periodic orbits.

  The SIS model in the chemostat, with general increasing growth functions · 2026 · DOI

• The theory is based on a specific lattice structure, which may not be applicable to all physical systems. The theory is still in the early stages of development and requires further testing and validation.

  Discrete Geometric Physics. Tetralogy: Ontology of the Lattice, Vacuum, Fields and Cosmology, Algebraic Structure and Topological Dynamics of the Vacuum, Cosmological Consequences of D-Graphs · 2026 · DOI

• Further development and testing of the theory of Discrete Geometric Physics. Application of the theory to complex problems in physics and engineering. Exploration of the potential implications of the theory for our understanding of the universe.

  Discrete Geometric Physics. Tetralogy: Ontology of the Lattice, Vacuum, Fields and Cosmology, Algebraic Structure and Topological Dynamics of the Vacuum, Cosmological Consequences of D-Graphs · 2026 · DOI

• Application of the UHS framework to various fields, including physics and chemistry. Further development of the UHS framework to include additional features and capabilities.

  Unified Harmonic Shells (UHS) · 2026 · DOI

• The paper identifies a gap in the current understanding of shells and their behavior. The UHS framework aims to address this gap by providing a scale-invariant radial quantization law.

  Unified Harmonic Shells (UHS) · 2026 · DOI

• The paper suggests further research on the remaining open items, including one observational measurement for the absolute DGUT calibration. The identified geometric calculations for the slip-time evaluation, the electroweak VEV, the torsion-interval sequence, and the nuclear intervals are suggested as future research directions.

  The Cohesion UFT Research Calculator, Version 6 · 2026 · DOI

• The previous version of the calculator lacked some central results. The new version addresses this gap by incorporating new results, including the exact R(Dst) solution and the forward fine-structure chain.

  The Cohesion UFT Research Calculator, Version 6 · 2026 · DOI

• Future research may focus on applying the formula to other contexts. Future research may investigate the implications of the formula for the study of matter, energy, and local conscious order.

  The Federico Maya Eternal Information Formula v.4 A Mathematical Consequence of the Dual-Lock Architecture (with a protected technological outlook) · 2026 · DOI

• The paper identifies a gap in the understanding of the Riemann Hypothesis and the Dual-Lock Architecture. The paper aims to fill this gap by introducing the Federico Maya Eternal Information Formula.

  The Federico Maya Eternal Information Formula v.4 A Mathematical Consequence of the Dual-Lock Architecture (with a protected technological outlook) · 2026 · DOI

• The lack of a high-precision numerical coefficient, - The need for a fully analytic Hadamard expansion of the current two-point function under modular flow

  Condition NL for Finite Balls: A Spectral Reduction, the Unconditional Metric Sector, and a Universal Kinematic Residue · 2026 · DOI

• The computational results are consistent across every λ and L tested, but methodological choices have not been varied. The paper notes that several aspects deserve further investigation.

  Curvature Caustics and Exact Vacuum Rigidity in a Discrete Space-Time Network · 2026 · DOI

• Further investigation of the paper's methodological choices is needed. The paper's findings may be applied to other networks with Chebyshev connectivity. The development of new models of discrete space-time may be informed by the paper's results.

  Curvature Caustics and Exact Vacuum Rigidity in a Discrete Space-Time Network · 2026 · DOI

• The present version documents the best controlled numericalevidence (order-unity relative residue) and states with precision what remains to be done fora high-precision extraction, so that the public record is complete and honest

  Condition NL for Finite Balls: A Spectral Reduction, the Unconditional Metric Sector, and a Universal Kinematic Residue · 2026 · DOI

• Table 1: The iterative spatial compression of a collapsing solar mass core utilizing the fundamental IFD scaling constant of approach to zero without ever achieving total singularity..4. The table demonstrates the asymptotic This mathematical progression provides a second-order insight into the preservation of physical complexity. In the IFD cosmology, because the geometry of space itself shrinks proportionally with the matter, the relative distances between internal particles scale down at the exact same exponential rate as the particles themselves. To an internal observer undergoing the descent (an observer located at the compression is felt or perceived. Their local measuring sticks, their atomic structures, and their biological mechanisms shrink concurrently with their surrounding environment. Therefore, an entire macroscopic universe can be folded into a microscopic region without crushing its internal informational integrity. state), absolutely no spatial Furthermore, as the spatial footprint continuously compresses inward, gravitational time dilation reaches extreme, asymptotic values relative to the outside universe. What an outside macroscopic observer (the observer) perceives as a frozen, static star hovering indefinitely at the event horizon is, internally, a hyper-accelerated, highly dynamic environment. Billions of years of external, macroscopic time may translate to mere fractions of a second in the initial stages of the internal descent. Conversely, internally, the micro-universe experiences its own vast, seemingly endless cosmological epochs.

  The Infinite Fractal Descent Cosmology: An Exhaustive Analysis of Micro-Macro Equivalence and the Grand Universe Hierarchy · 2026 · DOI

• The paper does not claim a dynamical model. The geometric origin of the integer 31 remains under investigation. The paper does not provide a physical interpretation of the geometric representation.

  Platonic G: Gravitation from Geometry — A Four‑Part Empirical Observation · 2026 · DOI

• To investigate the geometric origin of the integer 31. To explore the physical significance of the geometric representation of the gravitational constant. To apply the high-precision numerical identity in numerical calculations.

  Platonic G: Gravitation from Geometry — A Four‑Part Empirical Observation · 2026 · DOI

• Further development of PVM for astrophysical applications. Investigation of the implications of the IFD framework for our understanding of the universe.

  The Prime Core Black Hole and the Epoch of Sub-Quantum Rupture: Topological Stability within the Infinite Fractal Descent Framework · 2026 · DOI

• The classical paradox of orbital decay is unresolved in standard General Relativity. The IFD framework addresses this gap by introducing a prime core black hole as a topological anchor.

  The Prime Core Black Hole and the Epoch of Sub-Quantum Rupture: Topological Stability within the Infinite Fractal Descent Framework · 2026 · DOI

• The paper identifies the challenge of developing a novel mathematical framework to understand the Yang-Mills mass gap and cosmological balance. The research notes the difficulty of concentrating Frobenius signatures on a specific alphabet of integers. The paper highlights the need to restrict the sample to the bitoroidal subset of 1272 inner-torus states obtained by quotient by SU (3).

  Gravitational Balance from Galois Negation: A Structural Theorem on Multiplicative Manifolds with Cross-Domain Verification · 2026 · DOI

• The paper suggests that future research should focus on developing new mathematical frameworks for understanding complex systems. The research recommends exploring the applications of the framework in fields such as physics and engineering. The paper proposes investigating the concentration of Frobenius signatures on a specific alphabet of integers in other contexts.

  Gravitational Balance from Galois Negation: A Structural Theorem on Multiplicative Manifolds with Cross-Domain Verification · 2026 · DOI

Most-cited papers in Advanced Mathematical Theories and Applications

• ΛCDM and MOND: A debate about models or theory? · Studies in History and Philosophy of Science Part A · 2021 · 8 citations
• Reference dependence, expectations and anchoring in the Becker-DeGroot-Marschak mechanism · Theory and Decision · 2024 · 6 citations
• Low Seesaw Scale in the Grimus–Neufeld Model · Acta Physica Polonica B Proceedings Supplement · 2022 · 3 citations
• Advances in Fractal Cosmology and Singularity Resolution: An Update on the Infinite Descent and Subatomic Self-Similarity · Zenodo (CERN European Organization for Nuclear Research) · 2026 · 0 citations
• The Planck-electron hierarchy derived as a discrete topological distance: $ln(m_P/m_e) = d - 2\sqrt{5}$ · Zenodo (CERN European Organization for Nuclear Research) · 2026 · 0 citations
• The Infinite Fractal Descent (IFD) Framework: A Geometric Unification of Quantum Mechanics and Cosmology · Zenodo (CERN European Organization for Nuclear Research) · 2026 · 0 citations
• Empirical Verification of Infinite Fractal Descent: Transitioning from Theoretical Modeling to Observable Phenomena · Zenodo (CERN European Organization for Nuclear Research) · 2026 · 0 citations
• The Infinite Fractal Descent Cosmology: An Exhaustive Analysis of Micro-Macro Equivalence and the Grand Universe Hierarchy · Zenodo (CERN European Organization for Nuclear Research) · 2026 · 0 citations
• Complete Mass-Energy Hierarchy from E8 Geometric Framework: From Leptons to Planck Scale · Zenodo (CERN European Organization for Nuclear Research) · 2026 · 0 citations
• How Pattern Algebra Derives the Planck Mass M_P Without Free Parameters · Zenodo (CERN European Organization for Nuclear Research) · 2026 · 0 citations

Most recent work

• Advances in Fractal Cosmology and Singularity Resolution: An Update on the Infinite Descent and Subatomic Self-Similarity · Zenodo (CERN European Organization for Nuclear Research) · 2026
• The Planck-electron hierarchy derived as a discrete topological distance: $ln(m_P/m_e) = d - 2\sqrt{5}$ · Zenodo (CERN European Organization for Nuclear Research) · 2026
• The Infinite Fractal Descent (IFD) Framework: A Geometric Unification of Quantum Mechanics and Cosmology · Zenodo (CERN European Organization for Nuclear Research) · 2026
• Empirical Verification of Infinite Fractal Descent: Transitioning from Theoretical Modeling to Observable Phenomena · Zenodo (CERN European Organization for Nuclear Research) · 2026
• The Infinite Fractal Descent Cosmology: An Exhaustive Analysis of Micro-Macro Equivalence and the Grand Universe Hierarchy · Zenodo (CERN European Organization for Nuclear Research) · 2026
• Complete Mass-Energy Hierarchy from E8 Geometric Framework: From Leptons to Planck Scale · Zenodo (CERN European Organization for Nuclear Research) · 2026
• How Pattern Algebra Derives the Planck Mass M_P Without Free Parameters · Zenodo (CERN European Organization for Nuclear Research) · 2026
• Scale Resonance: From Graphene Lattice to Saturns Hexagon: An Indicator for the Universal Geometric Order of the π-Revolution · Zenodo (CERN European Organization for Nuclear Research) · 2026
• NOMOS: A Categorical Framework for High-Energy Physics and Information Compression (v1-v409) · Zenodo (CERN European Organization for Nuclear Research) · 2026
• The Universal Mass Stratification Gradient: Reevaluating Electromagnetic Charge and the Structural Role of the Neutron in the Infinite Fractal Descent Cosmology · Zenodo (CERN European Organization for Nuclear Research) · 2026

Related topics in Physics and Astronomy