Interactive Visualization
Scheme Invariance Visualizer
Applying tensor invariant visualization techniques to meta-calculus scheme invariance
What Is Scheme Invariance?
WHAT:
Testing which physics predictions stay the same when we change calculus frameworks. We run the same calculations in classical, geometric, and bigeometric calculus to see what changes and what doesn't.
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.
RESULTS:
Some quantities are scheme-invariant (eigenvalues, probabilities, cross-sections) - they stay the same. Others are scheme-dependent (wave function phase, intermediate steps) - they change with our calculus choice.
IMPLICATIONS:
We can separate physical reality from mathematical scaffolding. What remains invariant is "real physics." What varies is either mathematical artifact - or potentially new physics at extreme boundaries!
Inspired by: "The First Ever Visualization of Tensor Invariants" by Visual Tensor. The key insight: invariants have circular/ellipsoidal contour lines because their values don't change under rotation in component space.
What You're Looking At: Finding Physics That Doesn't Depend on Your Choice of Math
What is Scheme Invariance?
In meta-calculus, there are infinitely many calculi you can use (controlled by parameters alpha, beta, k). Some observables (like the Hubble parameter today) give the same answer regardless of which calculus you use. These are "scheme-invariant" and represent real physics. Others (like the Hubble parameter near the Big Bang) change depending on your calculus choice - these are mathematical scaffolding, not physical reality.
The Tensor Analogy
This is like tensor invariants in physics: the magnitude of a vector is the same in any coordinate system (rotation invariant), but individual components change. Here, "rotation" means changing calculus parameters (alpha, beta). Circular contours = invariant observables. Non-circular contours = scheme-dependent scaffolding.
Tensor Invariant-Inspired Visualization
This visualization applies concepts from tensor invariant theory to scheme invariance in meta-calculus.
Tensor Invariants
- Rotation in physical space
- Circular contours = invariant
- Eigenvalues = independent invariants
- Characteristic polynomial basis
Scheme Invariants
- Rotation in generator space
- Circular contours = physical observable
- Scheme-invariant quantities = basis
- Physical predictions independent of calculus
Contour Interpretation
Circular contours = invariant (like tensor magnitude). Non-circular = scheme-dependent.
Rotation Path
Flat line = value unchanged during rotation = invariant. Varying = scheme-dependent.
Breaking Regions
High variance (red) = schemes disagree. This is where new physics might hide!
Observable Categories:
Key Insight (from Tensor Invariants)
Just as tensor eigenvalues and trace are independent invariants that form a basis for all tensor invariants, our scheme-invariant observables(H_0, T_CMB, cross-sections) form a basis for physical predictions. Quantities that vary with scheme choice (like H(t) near Big Bang) are mathematical scaffolding, not physical reality - or they signal new physics at the boundary.
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.Circular contours: For vectors, invariants have circular contour lines. For 2nd-order tensors, they have ellipsoidal contour surfaces.
- 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 generator space: When you change the calculus system (alpha, beta, k parameters), different quantities behave differently. Some remain constant, others don't.
- 2.Circular contours = invariant: Scheme-invariant observables (H0, temperature, cross-sections) have approximately circular contours in generator space.
- 3.Independent invariants: Physical observables (measured quantities) form a basis for predictions. Scheme-dependent quantities are scaffolding.
- 4.Physical meaning: What remains invariant under scheme change is "real physics." What varies might be mathematical artifact - or new physics at the boundary!
Scheme-Breaking Regions
The most exciting application of this visualization: finding where schemes disagree.
Low Variance (Green)
All schemes agree. This is "normal physics" where our calculus choice doesn't matter. Standard GR, QM, and QFT predictions live here.
Medium Variance (Yellow)
Schemes start to disagree. Typically numerical artifacts or boundary effects. Worth investigating but usually explainable.
High Variance (Red)
Strong disagreement! Either: (1) singularity/coordinate artifact, or (2) new physics hiding here. The Planck scale, black hole interiors, and Big Bang are red zones.
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
Just as all tensor invariants can be expressed as functions of the eigenvalues (or equivalently, the trace and determinant), all physical predictions should be expressible as functions of scheme-invariant observables. This is the meta-calculus analog of representation independence in physics.