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 on the evaluated part of the grid are poor or fragile fits. The run is inconclusive about the tested mechanism: its H0 pass criterion (within 2 sigma of both SH0ES and Planck) cannot be met by any single H0, and the model already varies with redshift.
Pareto Frontier Comparison
The stored pymoo NSGA-II front frompublic/moo_results.json, plus a historical comparison of pymoo NSGA-II and GlobalMOO on the same problem (equal total evaluation budgets are not verified: GlobalMOO's initial cases are outside its iteration budget and the file records no totals).
WHAT
The 26 solutions written by one pymoo NSGA-II run of meta_calculus/moo_integration.py, plotted by their stored objectives (chi2_total, 1 - spectral_gap_mixed, 1 - min_individual_gap) or by their parameters (s, k, w).
WHY
A multi-objective run returns a set of trade-offs rather than one answer. Showing the stored front lets you see what the optimizer actually returned instead of a synthetic picture of it.
HOW
Drag to rotate, hover for each solution's parameters, and switch between objective space and parameter space. The table below the plot gives the historical optimizer comparison (one seed; equal total budgets not verified).
RESULTS
All 26 stored points are mutually non-dominated. Most have k within 1e-3 of zero because the chi2 objective is a function of k alone. In the historical comparison GlobalMOO reached a lower best chi2 than pymoo; the invariance and spectral-gap objectives tied. Its total evaluation counts were not recorded, so it does not establish an equal-budget win. It is history, not a GlobalMOO advantage: a preregistered rerun with equal measured budgets, on the June problem with its optimum moved off the box centre, read inconclusive, and its descriptive direction favours pymoo (the fair rerun).
IMPLICATIONS
This is one unseeded front and one seeded baseline on a single toy objective set. It does not show that many calculus choices are physically valid, and the optimizer comparison is not a general ranking.
Interactive Parameter Explorer
Drag the sliders to see how each parameter changes an illustrative toy objective, computed live in your browser. The diamond marker shows your current configuration.
WHAT
A 3D surface over (s, k) of a toy objective: the fit of n to the meta-Friedmann prediction n_act(s, w), plus a constructed penalty on k and s. No observational data enters it.
WHY
It builds intuition for how the meta-Friedmann relation responds to s and w. It is not the objective used by the stored optimizer runs.
HOW
Drag the sliders for n, s, k and w, or pick a preset. The surface is recomputed for the current n and w. Click and drag the plot to rotate it.
RESULTS
With the matter-era default (n = 2/3, w = 0, s = 0) the fit term is zero. k does not enter the fit term; it only moves the surface through the constructed penalty, which is lowest at k = 0 by design.
IMPLICATIONS
Because the k dependence is constructed, this panel cannot say anything about which calculus nature prefers.
Search Space Visualization
A 2D heatmap of an illustrative toy objective over (s, k), with the stored NSGA-II solutions overlaid and the optimizer's search-box bounds drawn as a dashed rectangle.
WHAT
A heatmap of the selected toy objective over (s, k), the 26 stored pymoo NSGA-II solutions as cyan points, the search-box bounds from moo_integration.py, and a star at the grid minimum.
WHY
It shows where the stored solutions sit relative to the search box. No optimizer trajectories were recorded, so none are drawn.
HOW
Pick an objective, move n and w, and toggle contours or the stored points. Hover for values.
RESULTS
Every stored solution lies inside the search box, as it must: the optimizer samples only inside those bounds. The star marks the minimum on the displayed grid of the toy objective; the continuous minimum can fall between grid points.
IMPLICATIONS
The box bounds are labelled BBN and CMB in the code, but here they act as fixed search limits, not as a fit to those observations.
Meta-Quantum Compatibility Explorer
A synthetic illustration of the meta-time derivative families in meta_calculus/quantum/. The browser panel does not evolve a state; its bars come from a seeded formula. The Python module is what runs the real RK4 evolution.
WHAT
Bars for four derivative families (Q0 standard, Q1 clock, Q2 norm-dependent, Q3 componentwise), labelled synthetic: they are generated by a seeded formula, not by evolving a quantum state.
WHY
Unitarity is the property to check: for Hermitian H, real scalar weights (Q0-Q2) preserve the norm of the continuous equation, while componentwise weights (Q3) generally do not.
HOW
Set the evolution time and seed, and switch between the norm-drift and distance-to-standard views. There is no run button; the bars update immediately.
RESULTS
Read the bars as an illustration of the expected pattern, not as a measurement. Measured numbers come from run_meta_quantum_experiment in Python.
IMPLICATIONS
These are project-defined modifications of the Schrodinger equation, not established quantum dynamics.
The Python experiment (meta_calculus/quantum/meta_quantum_compatibility.py)
- 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 the Euclidean norm drift abs(||psi(t)|| - ||psi(0)||) (should stay 0) and distance to standard trajectory
Multi-Calculus Diffusion Lab
A browser heat-diffusion panel. See the panel itself for what is computed live and how the error is measured.
WHAT
A heat-diffusion panel (a hot spot spreading on a plate) with controls for the calculus configuration.
WHY
The heat equation has exact solutions, so any numerical scheme can be checked against ground truth.
HOW
Use the controls inside the panel. The panel states which quantities are computed live and which are reference values.
RESULTS
Compare what the panel reports with the exact heat kernel. Treat any value the panel does not mark as computed as illustrative.
IMPLICATIONS
A single diffusion fixture does not establish a domain-dependent law for k.
Numerical Scheme Optimizer
The quantum harmonic oscillator is exactly solvable, E_n = (n + 1/2) hbar omega, so it serves as a control case for numerical schemes.
WHAT
A panel about numerical schemes for the quantum harmonic oscillator, whose exact energy levels are E_n = (n + 1/2) hbar omega.
WHY
A problem with a known answer is the right place to test whether a scheme adds error.
HOW
Use the controls inside the panel; it states which values are computed and which are reference values.
RESULTS
Compare any computed level against the exact E_n. A standard scheme should converge to it as the grid is refined.
IMPLICATIONS
One control problem says nothing general about which calculus physics prefers.
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 compare Gaussian random-walk spectral gaps under several coordinate embeddings of synthetic sample states.
WHAT
Synthetic sample states on the 2-simplex of diagonal qutrit states, and the spectral gap 1 - lambda_2 of a Gaussian random-walk operator built in each coordinate embedding.
WHY
Different coordinates put different distances between the same states, so the random walk mixes at different rates.
HOW
Set the number of states, kernel width and seed, and switch between the simplex view and the spectral-gap bars. Everything is computed live.
RESULTS
Each bar is one embedding's gap on these synthetic states. The gap of a composed multi-calculus operator is not computed here.
IMPLICATIONS
This is a local toy comparison. The broader audit contains counterexamples, so no amplification result is claimed.
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)) (coordinatewise chart, not the Bures metric)
Key Findings
Stored NSGA-II Front
Most stored solutions have |k| below 1e-3. The chi2 objective is a function of k alone, so this follows from how the objective is built; it is not an independent discovery.
No Joint H0/S8 Resolution
A constant k can be tuned after the fact to one target, but the tracked validation run does not resolve H0 and S8 together (see the cosmology section above).
Quantum Extensions
Norm-Preserving Families
For Hermitian H, real scalar weights (Q0-Q2) preserve the norm of the continuous equation; componentwise weights (Q3) generally do not. The browser panel only illustrates this.
Local Gap Comparison
This simplified qutrit visualization compares local embeddings; it is not evidence for a general spectral-gap bound.
Technical Note
Cosmology (01-03): Panel 01 plots stored artifacts (public/moo_results.json,public/globalmoo_vs_fallback_baseline.json). Panels 02 and 03 are illustrative toy objectives computed in the browser; only the overlaid cyan points in 03 are stored data.
Quantum (04-07): The meta-quantum compatibility explorer is a synthetic illustration; the numerical experiments live in the Python module meta_calculus/quantum/. The diffusion visualizer is a local toy comparison. The broader guarded audit contains counterexamples, so no amplification theorem is claimed.