CASCADE Local Benchmark
Synthetic toy benchmark: distance to the centre, not a reduction in distance to the singular point for every profile
Reporting a local, non-citable CASCADE benchmark. These are not physics results.
What was actually run
- One 2-D toy problem on [0, 1]^2 per profile. Objective 1 is the chosen profile; objective 2 and the singular set depend on the profile (table below). The reported distance is always the Euclidean distance to the centre (0.5, 0.5), which is the singular point only for some profiles.
- The 21 "simulations" are that toy relabelled with 14 domain names (cosmology, fracture, fusion, ...). No physics model from those domains is evaluated.
- "Improvement" is how much closer the optimizer got to the centre. Getting closer to a blow-up is not a better physical answer, and for the ring, line and exponential profiles the centre is not where the objective blows up.
- Artifact status:
local_benchmark_not_citable, citable: false. Local synthetic benchmark output. Do not cite as independent validation, proof, or experimental evidence.
13/21
cases labelled CASCADE
8
distinct singular profiles
14
domain labels in the file
1
"win" whose distance got larger
Objectives and singular sets by profile
| Profile (objective 1) | Objective 2 | Singular set |
|---|---|---|
| 1/r, 1/r^2, 1/sqrt(r) | r | the centre (0.5, 0.5) |
| 1/r^12 | (r / 0.1)^6 | the centre (0.5, 0.5) |
| ln(r) | abs(r - 0.2) | the centre (0.5, 0.5) |
| 1/(r-r_s) | abs(r - 0.1) | the ring r = 0.1 around the centre |
| 1/tau, tau = abs(0.9 - X[0]) | tau | the line x = 0.9 |
| exp(x): objective 1 is exp(10 x) | abs(X[0] - 0.9) | none: exp(10 x) is finite on [0, 1], so there is no singularity |
Grouped by singular profile
Cases with the same profile and the same k ran the identical problem with the same seed, so their numbers are identical whatever the domain label says.
| Profile | Cases | Distinct outcomes | Domain labels |
|---|---|---|---|
| 1/r | 10 | 2 | qft, quantum_gravity, diffusion, cfd, quantum, quantum, qft, turbulence, cosmology, fusion |
| 1/r^2 | 3 | 2 | cosmology, cosmology, climate |
| 1/(r-r_s) | 2 | 1 | gr, gr |
| ln(r) | 2 | 1 | qft, reservoir |
| 1/sqrt(r) | 1 | 1 | fracture |
| 1/r^12 | 1 | 1 | molecular |
| exp(x) | 1 | 1 | semiconductor |
| 1/tau | 1 | 1 | gr |
Baseline vs CASCADE minimum distance to the centre (0.5, 0.5)
Points below the diagonal: the CASCADE run ended closer to the centre. That is the singular point only for the centred profiles. Overlapping markers are the identical relabelled runs.
All 21 rows from the artifact
| Label | Domain label | Profile | k | Baseline dist | CASCADE dist | Reported | Reported % | Current rule |
|---|---|---|---|---|---|---|---|---|
| numerical relativity | gr | 1/(r-r_s) | -1 | 9.98e-2 | 9.99e-2 | TIE/BASELINE | 0.0 | TIE |
| amplitude invariance | qft | 1/r | -1 | 5.81e-4 | 1.69e-4 | CASCADE | 70.9 | TIE |
| frw scheme robustness | cosmology | 1/r^2 | -1 | 4.20e-4 | 6.88e-5 | CASCADE | 83.6 | TIE |
| factor ordering | quantum_gravity | 1/r | -1 | 5.81e-4 | 1.69e-4 | CASCADE | 70.9 | TIE |
| cosmological tension | cosmology | 1/r^2 | 0.14 | 4.20e-4 | 8.77e-4 | CASCADE | 15.4 | TIE |
| multi geometry diffusion | diffusion | 1/r | 0 | 5.81e-4 | 5.81e-4 | TIE/BASELINE | 0.0 | TIE |
| pde scheme optimizer | cfd | 1/r | -1 | 5.81e-4 | 1.69e-4 | CASCADE | 70.9 | TIE |
| quantum scheme | quantum | 1/r | 0 | 5.81e-4 | 5.81e-4 | TIE/BASELINE | 0.0 | TIE |
| quantum state geometry | quantum | 1/r | 0 | 5.81e-4 | 5.81e-4 | TIE/BASELINE | 0.0 | TIE |
| renormalization invariance | qft | ln(r) | -1 | 2.00e-1 | 2.00e-1 | TIE/BASELINE | 0.0 | TIE |
| anomaly detection | qft | 1/r | -1 | 5.81e-4 | 1.69e-4 | CASCADE | 70.9 | TIE |
| opmflow well placement | reservoir | ln(r) | -1 | 2.00e-1 | 2.00e-1 | TIE/BASELINE | 0.0 | TIE |
| pyfrac crack tip | fracture | 1/sqrt(r) | -0.5 | 2.07e-3 | 1.36e-4 | CASCADE | 93.4 | TIE |
| openmm lennard jones | molecular | 1/r^12 | -1 | 2.33e-3 | 1.81e-3 | CASCADE | 22.3 | TIE |
| climt radiation | climate | 1/r^2 | -1 | 4.20e-4 | 6.88e-5 | CASCADE | 83.6 | TIE |
| devsim avalanche | semiconductor | exp(x) | 1 | 2.35e-2 | 7.07e-1 | TIE/BASELINE | 0.0 | BASELINE |
| spectraldns vortex | turbulence | 1/r | -1 | 5.81e-4 | 1.69e-4 | CASCADE | 70.9 | TIE |
| concept gravity | cosmology | 1/r | -1 | 5.81e-4 | 1.69e-4 | CASCADE | 70.9 | TIE |
| torax toroidal | fusion | 1/r | -1 | 5.81e-4 | 1.69e-4 | CASCADE | 70.9 | TIE |
| power blackhole | gr | 1/(r-r_s) | -1 | 9.98e-2 | 9.99e-2 | TIE/BASELINE | 0.0 | TIE |
| pycbc gw merger | gr | 1/tau | -1 | 1.49e-1 | 4.55e-2 | CASCADE | 69.5 | TIE |
Red distances are larger than the baseline yet the script still labels the row CASCADE (cosmological_tension): the retired rule that labelled this archive also counted danger-zone solution counts and hypervolume, not only distance.
Procedure and winner rules (simulations/run_all_21_singularity.py)
- Build the 2-D toy for the row's profile
- Run baseline pymoo NSGA-II (50 generations x 50 population)
- Run the CASCADE variant at the row's fixed k (same budget)
- Record the minimum distance to the centre (0.5, 0.5), danger-zone counts and hypervolume
Archived labels (retired rule): CASCADE if >10% closer to the centre, OR more solutions inside the danger zone, OR >1% better hypervolume. This rule reproduces 21 of 21 stored labels.
Current producer rule (run_singularity_proof.decide_winner): hypervolume alone, with a 1% tie band. Applied to the stored hypervolumes it labels 0 of 21 rows CASCADE (last table column). The archived outcomes are not outputs of the current rule.