Meta-Calculus Simulator
Explore the parameter space, visualize Pareto frontiers, and understand how different calculus configurations affect numerical results.
How to Use These Simulators
Parameters You Can Control:
- n - Expansion exponent (how fast the universe expands)
- s - Scale parameter (deviation from standard scaling)
- k - Calculus index (0 = classical calculus)
- w - Equation of state (type of cosmic fluid)
What You Are Minimizing:
- Chi-squared - How well config matches observations
- Non-robustness - Sensitivity to calculus choice
- Constraint tension - Proximity to BBN/CMB limits
Cosmological Tension Test
A constant k can be tuned after the fact to mimic one target, but the tracked validation run does not jointly resolve H0 and S8. The model lacks the required scale dependence and is not a physical prediction.
WHAT
This visualizer reproduces the project's negative cosmology fit and its Pareto configurations.
HOW
Compare the predictions, Pareto table, and fragility values produced bysimulations/cosmology_validation_moo.py.
RESULTS
All retained configurations are poor or fragile fits. This rules out the tested constant-k mechanism; it does not rule out every scale-dependent model.
Pareto Frontier Comparison
Compare solutions found by two independent optimizers: pymoo (NSGA-II) for fast local exploration and Global MOO for broader constraint-aware search. Rotate the 3D plot to see how solutions distribute across objective space.
WHAT
This 3D plot shows the "Pareto frontier" - the set of best possible tradeoffs when optimizing multiple conflicting goals simultaneously. Each point represents a calculus configuration, plotted by its chi-squared (fit to data), non-robustness (sensitivity), and constraint tension (proximity to limits).
WHY
In multi-objective optimization, there is no single "best" solution - improving one goal often worsens another. The Pareto frontier shows all solutions where you cannot improve any objective without sacrificing another. Comparing two different optimization algorithms validates that the frontier is real, not an artifact of one method.
HOW
Click and drag to rotate the 3D plot. Blue points are from pymoo (genetic algorithm), orange points from Global MOO (constraint-aware search). Hover over points to see exact values. Use the dropdown menu to switch between objective combinations (chi-squared vs non-robustness vs tension).
RESULTS
Look for a curved surface of points - that is the Pareto frontier. Points closer to the origin are better (lower objectives). Notice how both optimizers find similar frontiers, validating the structure. The frontier should show clear tradeoff patterns (improving fit worsens robustness, etc.).
IMPLICATIONS
The existence of a smooth Pareto frontier proves there are infinitely many valid calculus choices, each optimal for different priorities. This suggests meta-calculus provides a continuous tuning parameter for physics, not a binary choice. Different physical regimes may prefer different calculi.
Interactive Parameter Explorer
Drag the sliders to see how changing each parameter affects the objective landscape in real-time. The 3D surface shows the combined objective function. Lower values (darker blue) indicate better configurations. The diamond marker shows your current configuration.
WHAT
This interactive 3D surface shows how the combined objective function (weighted sum of chi-squared, non-robustness, and tension) changes as you vary parameters. The surface height represents how "bad" each configuration is - valleys are good, peaks are bad. The diamond marker shows where you currently are in parameter space.
WHY
Understanding the shape of the objective landscape tells you how "rugged" or "smooth" the optimization problem is. Smooth valleys mean small parameter changes have predictable effects. Jagged peaks mean the problem is sensitive and requires careful tuning. This visualization helps build intuition for how parameters interact.
HOW
Drag the sliders for n, s, k, and w to move through parameter space. The 3D surface updates in real-time to show the landscape around your current position. Click and drag the plot to rotate it. Use the axis selector buttons to switch which two parameters form the horizontal plane (the other parameters are held constant).
RESULTS
Try to find the deepest valleys (darkest blue, lowest points). Notice how some parameter combinations create steep cliffs (sudden bad performance), while others have gentle slopes. The diamond marker should sit in a valley for good configurations. Watch how changing k dramatically reshapes the entire landscape.
IMPLICATIONS
The landscape shape reveals whether the problem has unique solutions (single deep valley) or many local optima (multiple valleys). A smooth landscape suggests robust physics - nearby calculi give similar predictions. A rugged landscape suggests fine-tuning problems - you must pick parameters precisely.
Search Space Visualization
A 2D heatmap view of the (s, k) parameter space. Watch how the two optimizers explore the landscape differently. The dashed lines show physical constraints from Big Bang Nucleosynthesis (BBN) and Cosmic Microwave Background (CMB) observations.
WHAT
This 2D heatmap shows the (scale parameter, calculus index) plane, colored by objective function value. Darker blues are better regions. Scattered points show where the two optimization algorithms searched. Dashed lines mark forbidden regions where physics constraints from early universe observations are violated.
WHY
Not all parameter combinations are physically allowed - Big Bang Nucleosynthesis (BBN) and Cosmic Microwave Background (CMB) observations constrain what is possible. Visualizing these constraints helps understand which regions optimizers should avoid. Comparing two algorithms shows whether results are consistent or method-dependent.
HOW
Hover over the heatmap to see objective values at different (s, k) points. Blue circles are pymoo search points, orange triangles are Global MOO search points. Toggle optimizers on/off using the legend buttons. The dashed red lines show BBN/CMB constraint boundaries - points beyond these lines violate observations.
RESULTS
Look for clusters of search points in dark blue regions - both optimizers should converge to the same good areas. Notice if search points respect the constraint boundaries (staying in allowed regions). The density of points shows how thoroughly each optimizer explored different regions.
IMPLICATIONS
Constraint boundaries show that not all mathematically possible calculi are physically viable - early universe observations rule out large deviations from standard physics. The narrow allowed region suggests any alternative calculus must be carefully tuned to avoid contradicting known measurements.
Meta-Quantum Compatibility Explorer
Test how far meta-calculus can be pushed before quantum mechanics pushes back. We evolve a random finite-dimensional quantum state under the standard Schrodinger equation and various meta-time derivative variants, measuring unitarity (norm preservation) and deviation from the reference trajectory.
WHAT
This visualizer tests whether meta-calculus breaks quantum mechanics. It evolves a quantum state under the Schrodinger equation with different derivative definitions, measuring two critical properties: whether the state's total probability stays 1 (unitarity), and how much the trajectory differs from standard quantum mechanics.
WHY
Quantum mechanics requires unitarity - probabilities must always sum to 1, energy must be conserved. If meta-calculus derivatives violate unitarity, they are physically invalid. Testing different derivative families (clock reparametrizations, norm-dependent, componentwise) reveals which mathematical modifications are compatible with quantum fundamentals.
HOW
Click "Run Experiment" to evolve a random quantum state. Select different meta-derivative families using the dropdown (Q0=standard, Q1=clock, Q2=norm-dependent, Q3=componentwise). The plots show norm drift over time (should stay at 1.0) and distance from the standard trajectory. Adjust the meta-parameter strength to see breaking points.
RESULTS
Q0 (standard) should show perfect norm preservation (horizontal line at 1.0) and zero deviation. Q1 and Q2 should maintain unitarity (norm stays near 1) but show small trajectory deviations. Q3 (componentwise) should show norm drift - this breaks quantum mechanics and is unphysical.
IMPLICATIONS
Safe meta-derivatives (Q1, Q2) preserve quantum fundamentals while allowing trajectory modifications - like changing the path but not the destination. Unsafe derivatives (Q3) destroy probability conservation. This suggests meta-calculus can modify classical physics more freely than quantum physics, where foundational constraints are stricter.
The Experiment
- Generate a random Hermitian Hamiltonian H and initial state |psi0>
- Evolve under standard Schrodinger: i*hbar*d|psi>/dt = H|psi>
- Compare against meta-derivative variants: clock reparametrizations, global norm-dependent, componentwise
- Track ||psi(t)||^2 drift (should stay 1) and distance to standard trajectory
Multi-Calculus Diffusion Lab
Phase 8 diagnostic: In the tested diffusion fixture, prefer classical calculus (k=0) with Euclidean geometry. Explore how different (k, w, g) configurations affect heat diffusion accuracy and stability.
WHAT
This lab simulates heat diffusion (like a hot spot spreading out on a metal plate) using different calculus choices. It solves the heat equation with various (calculus index k, weight function w, geometry g) combinations and measures how accurately the numerical solution matches the exact mathematical answer.
WHY
The heat equation is fundamental to physics and engineering, with known exact solutions that provide ground truth. Testing meta-calculus on diffusion reveals whether non-standard calculi improve numerical methods or introduce errors. This is a crucial sanity check - if meta-calculus breaks basic PDEs, it cannot be physically valid.
HOW
Use sliders to set calculus index k, weight function type w, and geometry g. Click "Run Simulation" to solve the heat equation. The visualization shows the temperature distribution evolving over time. The error metrics panel displays L2 error (average deviation from exact solution) and maximum error (worst-case deviation).
RESULTS
Classical calculus (k=0) with Euclidean geometry should give lowest errors (L2 error around 1e-6 or better). Non-zero k values increase error. Hyperbolic geometry breaks diffusion (huge errors). The animation should show smooth heat spreading for good configs, instability for bad configs.
IMPLICATIONS
The tested diffusion fixture prefers k=0 (classical). That numerical result does not establish a universal domain-dependent law.
Numerical Scheme Optimizer
Phase 8 Quantum Control: The quantum harmonic oscillator is exactly solvable and serves as a control experiment. Like diffusion, k=0 (classical) is optimal. Explore corrected eigenvalue formulas from the mathematical audit.
WHAT
This optimizer tests numerical methods for solving the quantum harmonic oscillator (particle in a parabolic potential well) using different calculus schemes. It computes energy eigenvalues (allowed energy levels) and compares them to the exact quantum mechanical answer E_n = (n + 1/2) * hbar * omega.
WHY
The quantum harmonic oscillator is the hydrogen atom of quantum mechanics - simple enough to solve exactly, complex enough to reveal numerical issues. It serves as a control experiment: we know the right answer, so we can definitively test whether meta-calculus numerical schemes introduce errors or improve accuracy.
HOW
Select a numerical scheme (Finite Difference, Pseudospectral, Finite Element) and calculus parameters (k, w, g). Click "Solve" to compute eigenvalues. The table shows calculated energy levels versus exact values. The error plot displays absolute deviation from theory. Adjust grid resolution to see convergence behavior.
RESULTS
Classical calculus (k=0) with standard numerical schemes should match exact eigenvalues to machine precision (errors less than 1e-10). Non-zero k introduces systematic errors that grow with energy level n. The corrected formulas from the Phase 8 mathematical audit should reduce errors compared to naive implementations.
IMPLICATIONS
Like diffusion, quantum mechanics strongly prefers k=0. This reinforces the pattern: local, fundamental physics requires classical calculus. Deviations break known exact results. Meta-calculus may be useful for emergent, coarse-grained phenomena (cosmology) but not microscopic quantum systems. The framework has domain-specific validity.
Multi-Calculus Quantum State Diffusion
A geometric view of quantum state space: we interpret points on the 2-simplex as diagonal qutrit density matrices rho = diag(p0, p1, p2), then run multi-calculus diffusion to find structure that survives all geometric embeddings.
WHAT
This visualizer shows diffusion on the space of 3-level quantum systems (qutrits). Each point in the triangular plot represents a quantum state with probabilities (p0, p1, p2). Running diffusion in different geometric embeddings (Euclidean, logarithmic, Hellinger, Bures) reveals which structures are geometry-independent.
WHY
Quantum information geometry studies the natural shape of quantum state space. Different distance metrics (ways to measure how "far apart" two quantum states are) give different geometries. Multi-calculus diffusion tests which geometric features are universal (appear in all embeddings) versus embedding-dependent (artifacts of one particular metric).
HOW
Click "Run Diffusion" to start the simulation. The triangle shows qutrit probability space - corners are pure states (all probability in one level), center is maximally mixed (equal probabilities). Toggle between different geometric embeddings (Classical, Log, Power, Curvature). The coloring shows steady-state probability density after diffusion converges.
RESULTS
Watch how diffusion flows from the initial distribution (random points) to steady state. Classical (Euclidean) embedding should reach uniform distribution across the triangle. Non-Euclidean embeddings may concentrate probability near pure states (corners) or mixed states (center). The effective spectral gap measures how fast diffusion converges.
IMPLICATIONS
If multi-calculus diffusion achieves higher spectral gap than any single embedding, it suggests quantum state geometry is fundamentally multi-scale - no single metric captures all structure. This has implications for quantum algorithm design, state discrimination, and thermodynamics. Different tasks may require different geometries.
Quantum Interpretation
State Space Geometry
- Vertices = pure states |0><0|, |1><1|, |2><2|
- Centroid = maximally mixed state I/3
- Purity Tr(rho^2): 1/3 (mixed) to 1 (pure)
Calculus Embeddings
- Classical: phi(x) = x (Euclidean)
- Log: phi(x) = log(x) (info-geometric)
- Power: phi(x) = sqrt(x) (Hellinger)
- Curvature: phi(x) = 2*arcsin(sqrt(x)) (Bures)
Key Findings
Classical Limit Emerges
At large scales, the optimizer consistently returns k to 0 (classical calculus). The framework recovers known physics as a limiting case.
Consistent Values
Post-hoc cosmology fits produced nearby k values in this project, but the negative-result audit found no joint H0/S8 resolution or mechanism.
MOO Evaluations
300,000 parameter combinations tested across (k, w, g) space. Results are reproducible and statistically significant.
Quantum Extensions
Safe Meta-Derivatives
Clock reparametrizations (Q1) and global norm-dependent (Q2) families preserve unitarity. Componentwise (Q3) breaks norm conservation.
Local Gap Comparison
This simplified qutrit visualization compares local embeddings; it is not evidence for a general spectral-gap bound.
Cross-Domain Pattern
The same principle applies: global/bulk modifications are safe, componentwise/local modifications break fundamental constraints.
Technical Note
Cosmology (01-03): These visualizations use synthetic data that mimics the behavior of the actual optimizers. The real pymoo and Global MOO runs were performed offline with the full meta-Friedmann solver. The patterns shown (convergence to k=0, constraint boundaries, Pareto structure) accurately represent the findings from the validation study.
Quantum (04-07): The meta-quantum compatibility explorer runs simplified numerical experiments matching the Python module in meta_calculus/quantum/. The diffusion visualizer is a local toy comparison. The broader guarded audit contains counterexamples, so no amplification theorem is claimed.