Deep Research
In-depth research reports on mathematics, physics, and computer science topics
A = B: Background — Proof Machines and the Hypergeometric Domain
Overview of Part I of the A=B book: proof machines, the hypergeometric domain, and algorithmic identity verification.
A = B: Epilogue — Operator Algebra Viewpoint on Hypergeometric Summation
Epilogue of the A=B book: the operator algebra perspective on hypergeometric summation and algorithmic proof.
A = B: The Five Basic Algorithms for Computerized Identity Proofs
Part II of the A=B book: the five core algorithms (Gosper, Zeilberger, WZ, q-analogues, multivariate) for automated identity proof.
Acoustic LENR and Sonofusion: A Comprehensive Survey
Survey of acoustic LENR and bubble fusion research (2002–2026): Taleyarkhan's claims, the misconduct scandal, replication failures, and the field's collapse.
Action-Dimensional Hamiltonians in Classical Mechanics
Historical and mathematical survey of Hamiltonians with dimensions of action, from Maupertuis through ADM formalism.
Algorithmic Discovery of Algebraic Identities Through Programmatic Search
Survey of tools for automated discovery of algebraic identities: QuickSpec, WZ method, Gröbner bases, and the finitist programs of Wildberger and Zeilberger.
Algorithmic Discovery of Algebraic Identities: A Survey (Seven Decades)
Seven-decade survey from Sister Celine (1945) to Google DeepMind FunSearch (2024): automated discovery and proof of algebraic identities.
Austrian Economics & the Philosophy of Science
An annotated bibliography of 24 core entries, 8 supplementary titles, and 6 thematic categories covering the methodological and epistemological intersection of Austrian economics and philosophy of science.
Blackbody Radiation: A Practical Laboratory Guide
Experimental guide to blackbody radiation physics: cavity radiator construction, calibration, and the metrology tradition from PTR to modern NIST standards.
Building a Total Language for Experimental Mathematics in Rust
Blueprint for a termination-guaranteed programming language for experimental mathematics using System T recursion, e-graphs (egg), and Rust.
Celestial Mechanics: Key Equations
Reference compilation of fundamental equations in celestial mechanics, from Newton's law of gravitation to orbital dynamics.
Chemistry's Experimental Evolution: From Ancient Practice to Modern Science
History of chemistry as an experimental science, from Egyptian embalmers and Islamic alchemists through the Enlightenment and modern quantitative chemistry.
Classification of Groups: Extensive Taxonomy
Extensive taxonomy of group types from finite groups (cyclic, symmetric, simple) through infinite structures (Lie groups, profinite, free groups).
Closed-Form Solutions in Weber Electrodynamics
Derivation and survey of closed-form analytical solutions to the Weber electrodynamic equations.
Combinatorics: A Comprehensive Treatment
Comprehensive treatment of combinatorics: enumerative, algebraic, probabilistic, and extremal methods with key results and techniques.
Comprehensive Analysis of Weber Electrodynamics
A thorough analysis of Weber electrodynamics — the velocity-dependent force law alternative to Maxwell field theory.
Cutting-Edge Programming Language Design: A Research Synthesis
Synthesis of 2020–2025 PL research: novel type systems, memory management innovations, and empirical syntax research informing next-generation language design.
Discrete Calculus and Finite Mathematics: A Comprehensive Landscape
Survey of the discrete calculus tradition — difference operators, finite sums, and algebraic foundations — as an alternative to continuous analysis.
Discrete Formulation of Ampère's Force Law
Discrete formulation of Ampère's force law between closed current circuits, with circuit discretization and summation schemes.
ECDLP: Combinatorial and Holonomic Research Directions
Speculative research directions for attacking the elliptic curve discrete logarithm problem using combinatorial and holonomic methods.
Entropy Decrease Without Life
Analysis of physical mechanisms for local entropy decrease in the absence of biological processes.
From Naked-Eye Observations to Universal Gravitation: The Making of Celestial Mechanics
History of celestial mechanics from Tycho Brahe's star charts through Kepler's laws to Newton's gravitational dynamics.
Hamiltonian Structures: Dimensional Analysis of Conjugate Variables
Survey of Hamiltonian structures across physics, analyzing the dimensions of conjugate variable pairs in different physical systems.
Introduction to Computational Quantum Mechanics
Introduction to data structures and numerical methods for computational quantum mechanics.
KS Phase-Space Regularization for n-Body Weber Electrodynamics
Application of Kustaanheimo-Stiefel regularization to the n-body Weber electrodynamics problem.
Lecture Materials: The Birth of Quantum Mechanics
Lecture notes tracing the birth of quantum mechanics from Planck's blackbody law through the Bohr model, matrix mechanics, and the Schrödinger equation.
LENR: Low Energy Nuclear Reactions — Grand Compilation
Annotated bibliography of publications, key people, institutions, conferences, documented results, and theoretical frameworks. Emphasis on Russian, Ukrainian, Japanese, Chinese, Indian, and other non-Western sources.
Mathematical History of Ellipses and Ellipsoids
History of ellipses and ellipsoids from ancient Greek geometry through modern quantum algorithms and random matrix theory.
Mathematical Methods for Singular Potentials and Classical Dynamical Systems
Survey of geometric, topological, and algebraic techniques for classical dynamical systems with singular potentials (1970–2024).
Mathematicians Who Reject Infinity: A Survey of Finitist Thought
Survey of finitist and ultrafinitist traditions in mathematics, from Kronecker and Hilbert through Zeilberger and Wildberger.
Multivariate Hyper-Catalan Series: Exploration Summary
Exploration summary for extending Wildberger's hyper-Catalan series polynomial solution to the multivariate setting.
Neutrino Physics: From Pauli's Ghost Particle to Precision Measurements
History and physics of neutrinos: oscillations, mixing angles, mass hierarchy, and detection technologies across twelve orders of magnitude in energy.
Precision Measurements of the Anomalous Magnetic Moment
Seven decades of g-2 measurements for the electron (0.13 ppt precision) and muon (127 ppb), the most stringent tests of QED ever performed.
Preserving Angular Momentum in Symplectic Numerical Integration
Analysis of the discrete Noether theorem and conditions for exact angular momentum conservation in variational integrators.
Quantum Measurement Equipment: Technical Resources Survey
Survey of technical documentation for quantum detection systems from major manufacturers and national metrology institutes (NIST, NPL, PTB).
Quantum Mechanics: A History
Exhaustive history of quantum mechanics from its 19th-century origins through contemporary frontiers, with key figures, experiments, and theoretical milestones.
Regularization in Celestial Mechanics: From Sundman to Modern Implementations
Survey of mathematical regularization methods that eliminate collision singularities, from Sundman (1912) through KS and modern n-body codes.
Reproducibility and Contested Results in Quantum Physics
Survey of reproducibility failures and contested experimental results in quantum mechanics, from the photoelectric effect through modern quantum optics.
Reproducibility and Contested Results in Quantum Physics: Reference Catalog
Catalog of 400+ publications documenting reproducibility failures, contested results, and theoretical criticisms across quantum mechanics (late 1800s–2024).
Research Plan: A Terminating Language for Rational Algebraic Structures
Research plan for Chromos/Qrat — a total, termination-guaranteed programming language designed for rational algebraic computation.
Solid State Physics: A History
Comprehensive history of solid state physics from crystallography and the Drude model through modern quantum materials and topological phases.
Solid State Physics: From Crystal Structure to Quantum Materials
Comprehensive survey of solid state physics from crystal structure and band theory through superconductivity, topological materials, and quantum devices.
Symplectic Integration of Non-Separable Hamiltonian Systems
Comprehensive review of symplectic integrators for non-separable Hamiltonians, from implicit Gauss–Legendre methods to modern semiexplicit schemes.
Symplectic Topology and Periodic Behavior of Large N-Body Systems
Application of symplectic topology tools — Floer theory, Arnold conjecture, KAM — to periodic orbits in large N-body Hamiltonian systems.
Technical Resources for Quantum Measurement Equipment
Manufacturing and calibration resources for quantum measurement systems: photomultipliers, superconducting nanowire detectors, and SI-traceable standards.
The ADE Classification: A Reference of Connected Subjects
Reference on the ADE Dynkin diagrams and their surprising unifying role across algebra, geometry, singularity theory, and physics.
The Continuity Equation: From Continuous to Discrete
Derivation and discretization of the continuity equation, from the integral form through finite-difference formulations.
The Equal A Priori Probability Postulate: A Relational Mechanics Explanation
Derivation of the equal a priori probability postulate of statistical mechanics from relational mechanics and Mach's principle.
The Landscape of Modern Mathematics Since 1960
Overview of the most productive era in mathematical history: Grothendieck's revolution, geometric analysis, higher categories, and the deepening interconnections across fields.
The Scientific Method: An Extended Summary
Extended summary of philosophy of science concepts situated within the scientific discoveries and disciplinary developments that motivated them.
The Scientific Method: From Ancient Philosophy to Modern Methodological Pluralism
Evolution of the scientific method from Aristotle through Popper, Kuhn, and Feyerabend to contemporary methodological pluralism across disciplines.
The Wildberger-Rubine Polynomial Paper: Viral Headlines, Skeptical Reception
Analysis of the 2025 Wildberger-Rubine paper in the American Mathematical Monthly — 115K views, genuine combinatorial contributions, and a bifurcated reception.
Towards a Minimal Computational Foundation for Modern Physics
Research problem statement exploring whether modern physics can be grounded in a minimal, termination-guaranteed computational framework.
Weber Electrodynamics: Comprehensive Bibliography (1900–2025)
300+ publications on Weber electrodynamics and action-at-a-distance theories, organized chronologically across 125 years.
Weber Electrodynamics: Reference List (1900–2025)
A bibliography of publications related to Wilhelm Weber's electrodynamics from 1900 to 2025.
Weber Electrodynamics: Two-Body Problem
Solution to the Weber electrodynamics two-body problem via Wildberger's hyper-Catalan series method.
Weber Two-Body Dynamics via Hypergeometric Series
Reformulation of the Clemente & Assis (1991) analytical Weber solution in terms of Gauss and Appell hypergeometric functions.
Weber-Type Gravitational Force Laws and Perihelion Precession: Annotated Bibliography
Annotated bibliography of 80+ publications on Weber-type gravitational force laws and Mercury perihelion precession, spanning 1846 to present.
Wildberger's Polynomial Breakthrough Meets Zeilberger's Algorithmic Proofs
How Doron Zeilberger and collaborators used WZ techniques to prove Wildberger's Geode array conjectures within weeks of publication.
Zero-Knowledge Proofs: From Cryptographic Curiosity to Production Infrastructure
How ZKPs evolved from theory to $1.5B+ venture capital and production systems (Groth16, PLONK, STARKs, folding schemes) securing billions in crypto value.