Interactive Visualization
Scheme Invariance Visualizer
Tensor-invariant visualization applied to illustrative toy functions
What Is Scheme Invariance?
WHAT:
Scheme invariance asks which predictions stay the same when the calculus or numerical scheme changes. This page does not run physics: it illustrates the idea with toy functions on a two-parameter plane and measures which ones are unchanged by rotating that plane.
WHY:
"Real" physics should not depend on arbitrary mathematical choices. If a prediction is truly physical, it should be the same no matter which calculus we use to compute it.
WHAT THE TOY SHOWS:
Only functions that ignore direction in the plane are constant on circles. Every other toy varies, and the page reports the measured spread instead of a label chosen in advance.
LIMITS:
Invariance under a chosen transformation is a useful filter for artifacts, but it is a heuristic: a quantity that varies across schemes usually signals a scheme-dependent definition, a numerical artifact, or a singular chart, and the toy cannot tell these apart.
Inspired by: "The First Ever Visualization of Tensor Invariants" by Visual Tensor. For a vector, a rotation-invariant function has circular contour lines because its value does not change under rotation in component space.
What You're Looking At: An Illustrative Toy
The toy functions
Each curve is a hand-written function of two plane parameters (alpha, beta) and an index k. Earlier versions named them after physical observables (H0, T_CMB, |M|^2); they were never computed from cosmology or field theory, so they are now labelled Toy A to Toy F, plus one control.
The tensor analogy
A function of a vector that is unchanged by rotation has circular contours. Here we rotate the (alpha, beta) plane and measure each toy's spread along circles. Only toys that ignore direction (the squared-radius control, and Toy B which ignores alpha and beta entirely) pass.
Contour Interpretation
Contours that coincide with the dotted circles mean the toy is rotation-invariant. Other contour shapes mean it depends on direction in the plane.
Rotation Path
Each toy is plotted as relative deviation from its own circle mean, so a flat line is a real zero spread, not a wide y-axis.
Variation Map
Bright regions are where the toy functions change most around small circles, mostly the branch cut of Toy F on the negative alpha axis. It says nothing about physics.
Toy functions (status computed at the current k):
The Tensor Analogy
Tensor Invariants (from video)
- 1.Rotation in physical space: When you rotate a tensor, its components (u11, u22, u12) change, but certain functions (eigenvalues, trace, determinant) stay constant.
- 2.Contour shapes: For vectors, the length invariant has circular contour lines. For 2nd-order tensors the contour geometry depends on the chosen invariant: in (u11, u22, u12), constant trace is the plane u11 + u22 = c, while constant determinant u11 u22 - u12^2 = c is an indefinite quadric (a hyperboloid or a cone).
- 3.Independent basis: The eigenvalues form a basis from which all other invariants can be derived. Trace and determinant are combinations of eigenvalues.
- 4.Physical meaning: Kinetic energy (function of velocity magnitude) and strain energy (function of deformation invariants) are invariant because they represent real physical quantities.
Scheme Invariants (our framework)
- 1.Rotation in a parameter plane: the toy rotates (alpha, beta) about the origin. This is an analogy; a change of calculus is not a rotation of this plane.
- 2.Circular contours = rotation-invariant: true for the toy functions that ignore direction. The toys formerly named H0, temperature and cross-section are not rotation-invariant, which is why they are no longer labelled that way.
- 3.Measured, not assumed: the spread of each toy along circles is computed and shown; an invariance label is never set by hand.
- 4.Physical meaning: none for the toys themselves. In real models, invariance under a scheme change is a heuristic filter, not a criterion that decides what is physical.
Reading the Variation Map
The variation map shows where the toy functions change most around small circles. High values come from the toys' own formulas (for example the atan2 branch cut of Toy F); they do not locate any physical region. In real calculations, disagreement between schemes near a singularity usually reflects a singular chart or a numerical artifact and has to be explained case by case.
Mathematical Foundation
Characteristic polynomial analogy:
Tensor: det(U - lambda*I) = 0 → eigenvalues lambda_1, lambda_2
Scheme: G_scheme groupoid → scheme-invariant observables O_1, O_2, ...
Key: I_1 = tr(U) = sum(lambda_i) ↔ Physical observable = scheme-independent combination
Tensor invariants can be expressed through the eigenvalues (or trace and determinant). The project's working hypothesis is that physical predictions should likewise be expressible through scheme-invariant quantities. That is a modelling heuristic, not a result demonstrated on this page.