Our Work

We study null focusing across the amphometric family, a holomorphic interpolation between Euclidean and Lorentzian signature. In the Gaussian-normal class we prove a phase grading of the null screen t…

We study null focusing across the amphometric family, a holomorphic interpolation between Euclidean and Lorentzian signature. In the Gaussian-normal class we prove a phase grading of the null screen tidal operator: an electric part is invariant along the interpolation, a magnetic part carries a half phase, an intrinsic part carries the full phase, and the weights are fixed by curvature block and spatial leg count. The same rule grades the stress tensor, and under the Einstein equations the traces of the two rotating sectors are the constraint densities. Where those sectors vanish the transported focusing data are parameter independent. We prove a finite-interval screen estimate of Myers type, identify its index form with an endpoint-normalised Maslov index, extend it to generators with controlled defect, characterise when the defect can be annihilated by the slicing, and reconstruct the affine normalisation from slicing data. With the standard causal argument these yield a Penrose-type incompleteness theorem whose hypothesis is a sustained energy condition, satisfied by collapsing power-law models and by no expanding one: a collapse statement on future-directed generators. No canonical path-wise integer Maslov index exists on the genuinely complex family; the replacement lives one degree up, as a framed index of two-chains, a boundary winding of the canonical section of a determinant line, equal in the reference framing to the intersection number with the pullback of the complex Maslov cycle under an explicit holomorphic Lagrangian map. An endpoint transfer statement then controls the transport defect by slice sectional curvature alone.

This paper introduces Hidden Phase Mathematics (HPM), a novel theoretical framework designed to model, analyze, and explain abrupt structural changes within mathematical objects. Traditional mathemati…

This paper introduces Hidden Phase Mathematics (HPM), a novel theoretical framework designed to model, analyze, and explain abrupt structural changes within mathematical objects. Traditional mathematical paradigms often assume that the properties of an object are continuously or smoothly governed by its primary formulation. However, many systems exhibit sudden, discontinuous qualitative shifts that resist conventional continuous modeling. HPM posits that any mathematical object does not merely exist in a single static state, but is fundamentally composed of a sequence of latent or hidden phases, denoted as (s_1, s_2, ..., s_n). These hidden phases coexist beneath the surface of the primary manifest presentation and govern internal structural constraints. By defining phase transition operators, multi-phase state spaces, and transition boundaries, HPM provides a rigorous language to decode why and how structural properties transform abruptly. This paper establishes the foundational definitions, algebraic and topological characterizations, dynamic mechanisms of phase shifting, and broad theoretical implications across mathematical domains, omitting experimental validation as the framework is purely foundational and analytical.

Traditional mathematics views proofs as static, immutable artifacts, overlooking the dynamic evolutionary processes by which they are discovered, optimized, and compressed over time. In this paper, we…

Traditional mathematics views proofs as static, immutable artifacts, overlooking the dynamic evolutionary processes by which they are discovered, optimized, and compressed over time. In this paper, we introduce Proof Dynamics Theory (PDT), a novel theoretical framework that models the mathematical proof process as a continuous or discrete dynamic system denoted by P(t). By defining a rigorous state space S for mathematical proofs, evolutionary operators, and proof energy functionals E(P), PDT formalizes how proofs evolve, optimize, and compress. We establish the foundational mathematics of proof trajectories, analyze the stability and convergence of deductive pathways, and propose a metric system for measuring proof entropy and informational density. This framework establishes the foundational groundwork for Proof Science, shifting the paradigm from static verification to dynamic optimization of mathematical knowledge.

This paper introduces the theoretical framework of Mathematical Structural Critical Phenomena (MSCP), a novel paradigm that extends the concepts of phase transitions and critical phenomena from statis…

This paper introduces the theoretical framework of Mathematical Structural Critical Phenomena (MSCP), a novel paradigm that extends the concepts of phase transitions and critical phenomena from statistical physics into the domain of pure abstract mathematics. When a mathematical system, parameterized by a control variable denoted as lambda, approaches a critical threshold denoted as lambda_c, the underlying system undergoes profound structural transformations, giving rise to emergent properties that are absent in non-critical regimes. We formalize the foundational concepts of MSCP, define abstract critical exponents, construct an analogue of the renormalization group for mathematical spaces, and analyze topological transitions near critical boundaries. By establishing a rigorous dictionary between physical criticality and mathematical structural shifts, MSCP offers a unifying perspective on phenomena observed across algebra, topology, graph theory, and number theory.

Dataset and relevant documents for "Investigating a Question-Driven Design Process in Designing Explanations for AI-Supported Video-Based Learning" journal paper

Enrichment Factor data, and processed Harmonized target log1pEF data of CHAIR, an antigen-resolved antibody-reactome resource for healthy adult immune aging  

Modern mathematics has traditionally been characterized by structural specialization, where disparate branches such as algebra, geometry, topology, and number theory develop independent languages and …

Modern mathematics has traditionally been characterized by structural specialization, where disparate branches such as algebra, geometry, topology, and number theory develop independent languages and methodologies. Although categorical frameworks have successfully bridged various subfields, a comprehensive foundational paradigm that systematically extracts the deepest common origins of seemingly unrelated mathematical structures remains incomplete. This paper introduces the Deep Homology of Mathematical Objects (DHMO) theory. The core thesis of DHMO is that complex, specialized mathematical objects descend from a shared, abstract universal precursor space, denoted as the Z-space. By formalizing the relational dynamic through the universal mapping schema A <- Z -> B, DHMO provides a rigorous methodology for tracing distinct mathematical entities back to a common ancestral source. This framework unifies diverse mathematical domains, reveals hidden structural symmetries, and establishes a new epistemological foundation for the unity of mathematics without relying on empirical validation.

Traditional mathematics relies fundamentally on discrete categorization, classifying entities into rigid taxonomic boxes such as groups, rings, fields, and topological spaces. This paper introduces th…

Traditional mathematics relies fundamentally on discrete categorization, classifying entities into rigid taxonomic boxes such as groups, rings, fields, and topological spaces. This paper introduces the Mathematical Structure Continuum Spectrum Theory (MSCS), which posits that mathematical structures are not isolated, discrete entities, but rather points, trajectories, and regions within a continuous structural spectrum. By introducing a continuous spectrum parameter lambda, denoted as S(lambda), we formalize the notion that mathematical structures can undergo continuous deformations, revealing hidden bridges between seemingly disparate fields of mathematics. This theory challenges classical discrete taxonomy, offering a unified topological and algebraic framework for understanding structural evolution, degeneration, and phase transitions in abstract mathematical space.

This paper establishes a rigorous mathematical synthesis between two foundational results in nonlinear dynamics: the Center Manifold Theorem (CMT) and the Trojan Universality Class (TUC). The Center M…

This paper establishes a rigorous mathematical synthesis between two foundational results in nonlinear dynamics: the Center Manifold Theorem (CMT) and the Trojan Universality Class (TUC). The Center Manifold Theorem provides dimensional reduction, asserting that near a nonhyperbolic equilibrium, all long-term dynamics are captured by a low-dimensional, locally invariant center manifold to which nearby trajectories are exponentially attracted. However, the CMT is structurally blind: it provides no information about the qualitative nature of the reduced flow. The Trojan Universality Class fills this gap by providing a structural diagnostic for the dynamics on the center manifold when the reduced system is Hamiltonian with two dominant oscillatory modes separated by a persistent spectral gap. Under these conditions, the system admits a near-integrable Birkhoff normal form with an exponentially small remainder, yielding Nekhoroshev-type exponential confinement of action variables. The central result is the Interaction Theorem, which establishes a quantitative relationship between the contraction rate of the center manifold and the effective perturbation strength of the Trojan normal form. Faster contraction of the stable modes reduces the residual coupling from the stable directions, which in turn amplifies the protective effect of the spectral gap. The stability time then depends multiplicatively on both quantities: larger gaps and faster contractions produce exponentially longer confinement. The theorem also yields a hierarchy of structural failure modes—separatrix splitting, resonance overlap, spectral-gap collapse, and mode activation—each of which provides observable spectral diagnostics for progressive loss of Trojan protection. The framework is local, requires Hamiltonian reduced dynamics, and applies only when a further reduction to two dominant modes is valid. Its limitations are explicit: strong dissipation, three or more comparable modes, and large-amplitude global dynamics lie outside its scope. The unified CMT-TUC pipeline transforms stability analysis from a case-by-case endeavor into a transferable science: any two systems sharing the same spectral gap, effective perturbation strength, and nondegeneracy conditions inherit identical qualitative bounds on action drift and the same hierarchy of breakdown, regardless of physical substrate.