Singularity-Skirting Theory
Non-Newtonian calculus provides a mathematical foundation for "skirting" singularities. The bigeometric derivative D*[1/r] = -1 (constant) enables evaluation where classical calculus diverges.
61.9%
CASCADE Win Rate
k=-1
Bigeometric (log)
10
Physics Domains
k=-1 (bigeometric) optimal for: 1/r (Coulomb), 1/r^2 (curvature), 1/sqrt(r) (crack tip), 1/(r-r_s) (horizon).
View full CASCADE singularity proof →Educational Demonstration - Not Validated Physics
This page demonstrates how the numerical methods toolkit might apply to foundational physics questions. These are exploratory applications for educational purposes, not validated physics theories.
For verified mathematical results, see Spectral Gap Verification.
Foundational Structures
Dualities, symmetries, and quantization ambiguities.
The deepest structures in physics often involve dualities (equivalent descriptions of the same physics) and ordering ambiguities (choices that arise when quantizing classical systems). Meta-calculus provides tools to systematically explore these foundational issues.
1. Kramers-Wannier Duality
Overview
This visualizer demonstrates one of the most beautiful results in statistical mechanics: the Kramers-Wannier duality of the 2D Ising model. It shows that the physics at high temperature is mathematically equivalent to the physics at low temperature, connected by a duality transformation K* = -0.5 * ln(tanh(K)).
Dualities are profound because they reveal hidden symmetries in physics. The same mathematical structure describes both the ordered (low-T) and disordered (high-T) phases. This duality also predicts the exact critical temperature where phase transitions occur (the self-dual point K = K*). Similar dualities appear throughout physics, from string theory to quantum field theory.
2. Wheeler-DeWitt Operator Ordering
Overview
This visualizer explores the Wheeler-DeWitt equation - the central equation of quantum cosmology that describes the quantum state of the universe. The key issue: when promoting classical variables (like the scale factor a) to quantum operators, different orderings give different equations. This tool compares Laplace-Beltrami, symmetric, and Weyl orderings.
The operator ordering problem is one of the deepest unsolved issues in quantum gravity. In ordinary quantum mechanics, ordering affects subleading corrections (hbar^2 terms). But in quantum cosmology, the universe IS the quantum system - there is no external classical background. Different orderings can predict different probabilities for the universe to nucleate, tunnel, or evolve. Which ordering is correct?
Domain Restrictions & Validity
Kretschmann Formula: K ~ r^(-6), D_BG[K] = e^(-6)
Valid only for r > 0 (outside singularity). The bigeometric derivative captures the power-law scaling but does NOT resolve the r=0 singularity itself.
FRW Radiation Era: n=0.5 parameter
Ricci scalar can be zero in radiation-dominated cosmology. This is a physical feature, not a singularity - the universe was smooth and homogeneous.
The Unifying Principle
Both dualities and ordering ambiguities represent choices in how we describe physics. Meta-calculus provides systematic tools to identify which features survive all choices and are therefore genuinely physical.
The framework unifies these seemingly different phenomena under the umbrella of scheme independence - the central principle that physical predictions should not depend on arbitrary mathematical conventions.
Foundational Structures Summary
1
Duality Tested
(Kramers-Wannier)
3
Orderings Compared
(Laplace-Beltrami, Symmetric, Weyl)
100%
Classical Limit Agreement
(All orderings converge)