MOO-First Experiments

CORRECTED: What Really Happened

• ~ Sod: best fixed scheme (Meta-Adaptive) is 0.37% below Van Leer in L2: a tie
• ~ Sod dual MOO: 10 of 30 pymoo NSGA-II front points Pareto-dominate Van Leer (one unseeded run, best L2 gain 0.65%)
• ~ Shu-Osher (SSP-RK2 re-run 2026-10-07): MOO-tuned L2 0.1563 vs Van Leer 0.1578, 1.0% over the best standard limiter, seed 42, in-sample, limiter k kept in the TVD range [0, 1] (results/shu_osher_moo.json). The earlier 5.4% over minmod came from an unstable forward-Euler integrator and is retracted.
• ~ 5-seed, 15-generation re-run (SSP-RK2): best L2 0.1565 +/- 0.0001, 3.5% below minmod and 0.9% below Van Leer, in-sample (results/multi_seed_moo_results.json; the earlier 12.1% came from the retracted integrator)

Critical Failure

• - Parameters are problem-specific: artifact verdict PROBLEM_SPECIFIC (results/cross_validation_results.json)
• - Sod params on Shu-Osher: L2 0.1304 vs superbee 0.1207 (8.0% higher); transfer change vs Shu-Osher-tuned params -3.27% (the earlier figure from the retracted solver is superseded)
• - Only 1D benchmarks tested
• - Real CFD much more complex

CORRECTED: What Really Happened

• + Classical 2nd order verified: Central-RK4 = 1.99, Lax-Wendroff = 1.98
• + Meta-k schemes measure 0.84-0.95: ~1st order, rising mildly with k
• - "Bigeometric = 2nd order" retracted: that scheme was central difference in disguise (D_BG inverts back to u')
• + All fits have R^2 >= 0.994 (the measurements are clean; the old labels were not)

Critical Failure

• - Only 1D linear advection tested
• - Gaussian pulse (smooth IC)
• - Short time (T = 0.5)
• - No shock/discontinuity tests

What Still Holds

• + 79/88 valid normal samples satisfy the proposed bound
• + 3 distinct small baseline compositions satisfy it (the file lists 5; the others repeat a case)
• + The guarded search is reproducible

Counterexamples Found

• - 9 valid cases violate the proposed bound
• - 5 Dirichlet and 4 periodic counterexamples
• - Minimum valid gap ratio: 0.2127
• - This refutes the general claim; it is not a theorem

Encoded-bounds run

• R^2 = 0.7096
• k = -0.0137 log10(L) + 0.1585
• 8 synthetic scale models
• Not independent physical evidence

Common-bounds control

• R^2 = 0.4370
• 7 of 8 selected k~0
• Linear-pattern test: failed
• Archived lookup claim: not validated

What This Shows

• - meta-0.3 does not match RK4: it spirals into the fixed point (-8.49, -8.49, 27) and stays there
• - Its "100 time units" is just t_max: it never left the box because it stopped moving
• - Every other meta scheme leaves the box by t = 2.65
• - No meta scheme preserves the chaotic attractor

Honest Interpretation

• NEGATIVE: the meta schemes change the dynamics
• The Lyapunov estimator is unvalidated: it gives 0.05 for RK4 vs the known 0.906
• This bounds framework capabilities
• Not all problems benefit from meta-calculus

CORRECTED: What Really Happened

• - 0/1000 configurations preserve unitarity naturally
• - Forced normalization masked the failure (bug lines 139-141)
• - Best norm drift: 2.56e-4 in the historical blend model
• - Previous "MOO discovery" claim was INVALID

Critical Failure

• - The historical blend breaks the norm at every scanned point; this does not show that every bigeometric evolution must
• - Plain Crank-Nicolson is unitary by construction; the current simulations/quantum_phase_moo.py evolves that way and ignores k, w and p
• - All 1000 scanned (k, w, p) points break unitarity; every scanned k is nonzero (k != 0)
• - Data file: public/quantum_phase_corrected.json (corrected_results.json and the old quantum_phase_moo.json are superseded: they still record the bugged 100% unitary run)

Why the Test Is Inconclusive

• - The H0 criterion needs one H0 within 2 sigma of both SH0ES (73.04 +/- 1.04) and Planck (67.4 +/- 0.5). Those intervals do not overlap, so no single H0 can pass, whatever k is.
• - Only the first 200 grid configurations were evaluated, reaching 4 k values (-0.500 to -0.342)
• - H0: 0/9 Pareto solutions pass; S8: 0/9
• - Li7: best change -28% (does not help)

What It Does and Does Not Show

• The model ratio H_meta / H_classical = 1 + k ln(1 + z) already varies with redshift (k = 0.1: 1.0000 at z = 0, 1.0693 at z = 1)
• So this run says nothing about whether a redshift- or scale-dependent model is required
• Previous "may resolve H0" claims were post-hoc fitting and stay withdrawn
• Method: grid search, not NSGA-II

Reused existing result files, MOO scripts and test infrastructure.

pymoo NSGA-II for the shock-tube, Shu-Osher and k(L) runs; grid searches for the spectral-gap boundary, quantum phase scan and cosmology; fixed-scheme comparisons for convergence and chaos.

Document both positive and negative results. Update claims accordingly.