December 20247 Experiments Completed

MOO-First Experiments

Systematic experiments using Multi-Objective Optimization to validate, refine, or bound framework claims. Each experiment produces honest results regardless of outcome.

CASCADE VALIDATED

Singularity-Skirting MOO

61.9%

Win Rate

CASCADE enables access to "danger zones" near physics singularities where standard MOO fails. The bigeometric derivative D*[1/r] = -1 (constant) allows evaluation where classical calculus diverges.

13/21

Wins

93.4%

Best Gain

Domains

k=-1

Optimal

View full CASCADE singularity proof →

Executive Summary

EXP-1: Shock

VERIFIED

Best L2

EXP-2: Conv

VERIFIED

2nd order

EXP-3: k(L)

CORRECTED

R2=0.82

EXP-4: Chaos

NEUTRAL

Matches RK4

EXP-5: Gap

REFINED

BC-dep

EXP-6: Quantum

INVALID

0% unit

EXP-7: Cosmo

CANNOT

H0: 0/9

Methodology: Build on existing infrastructure (37 result files, 8 MOO scripts). Use pymoo NSGA-II for optimization. Document honest outcomes for each experiment.

EXP-1: Shock Tube CFD

VERIFIED

Visual engineering demonstration on the Sod shock tube. Tests meta-calculus flux limiting against classical limiters (minmod, superbee, Van Leer, MC).

CORRECTED: What Really Happened

+ Sod: 10/30 solutions strictly dominate Van Leer
+ Shu-Osher: 5.4% L2 improvement over minmod
+ Statistical: 12.1% +/- 3.1% across 5 seeds
+ Adaptive k switching works automatically

Critical Failure

- Parameters are problem-specific (cross-validation)
- Sod params on Shu-Osher: -3.9% WORSE
- Only 1D benchmarks tested
- Real CFD much more complex

Data: results/shock_tube_dual_moo.json, results/shu_osher_moo.json | Script: simulations/shock_tube_moo.py, simulations/shu_osher_moo.pyFull validated results →

EXP-2: Convergence Order Verification

VERIFIED

Lax equivalence test on 1D linear advection. Measures convergence order via log-log error analysis.

CORRECTED: What Really Happened

+ Bigeometric achieves 2nd order (order = 1.99)
+ Matches Lax-Wendroff reference exactly
+ Meta schemes interpolate via k parameter
+ All fits have R^2 > 0.99

Critical Failure

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

Data: results/convergence_order_verification.json | Script: simulations/convergence_order_verification.py

EXP-5: Spectral Gap Boundary

REFINED

Extended from 9 tests to 200+ configurations. Discovered boundary conditions matter critically.

What Still Holds

+ Dirichlet BCs: 100% pass rate
+ Periodic BCs: ~95% pass rate
+ Symmetric operators: reliable
+ FRW cosmology cases: safe

Counterexamples Found

- Neumann BCs: ~80% pass (20% fail)
- Mixed operators + Neumann: gap_ratio = 0.69
- First derivatives + Neumann: gap ~ 0
- High condition numbers correlate with failures

Data: results/spectral_gap_boundary_moo.json | Script: simulations/spectral_gap_boundary_moo.py

EXP-3: k(L) Mechanism Discovery

BUG FIXED

Tested 14 derived features (8 affine + 6 non-affine). Found intercept bug that caused spurious R^2 differences. All affine features are mathematically equivalent.

Overfitting Analysis

Cross-validation reveals that complex features (sigmoid) achieve high R^2 on interpolated data but catastrophically fail on the 8 true baseline points. Simple log-linear is robust.

Corrected Results

+ Log-linear: k(L) = -0.015 * log10(L) + 0.19
+ All affine features have R^2 = 0.8206 (identical)
+ "Energy density beats log_L" was BUG ARTIFACT
+ Physical meaning: gradual quantum-classical crossover

Non-Affine Analysis

- Sigmoid R^2=0.94... but OVERFITS (8M% LOO error)
- Complex features fit interpolation, not truth
- Only 8 true baseline points
- 18% unexplained variance is genuine noise

Data: results/k_mechanism_discovery.json | Script: simulations/k_mechanism_discovery.py

EXP-4: Lorenz Chaos Preservation

NEUTRAL

Tests whether meta-calculus can preserve Lorenz attractor geometry longer than classical RK4. Measures divergence time, energy drift, and Lyapunov exponent estimation.

What This Shows

- Meta-0.3 matches RK4 exactly (100.0 time units)
- No improvement over classical method
- Higher k values cause early divergence
- Chaotic systems resist scheme modifications

Honest Interpretation

NEUTRAL: Neither helps nor hurts
Exponential error growth dominates
This bounds framework capabilities
Not all problems benefit from meta-calculus

Data: results/chaos_preservation_moo.json | Script: simulations/chaos_preservation_moo.py

CORRECTED December 2025

Fatal Bug: Forced Normalization Invalidated Results

The original P05 simulation had a critical bug: forced normalization after each timestep (lines 139-141). This artificially made all configurations appear unitary-preserving when they were not.

Honest result after bug fix: Bigeometric quantum evolution does NOT naturally preserve unitarity. 0/1000 configurations preserve norm without explicit correction. Best norm drift: 0.000256 (still requires normalization). Previous claim that "MOO discovers unitarity-preserving parameters" was INVALID.

EXP-6: Quantum Phase Diagram

CORRECTED - BUG FOUND

CORRECTED December 2025: Original P05 simulation had forced normalization bug. After bug fix, bigeometric quantum evolution does NOT preserve unitarity. 0/1000 configurations preserve norm naturally. MOO discovery claim was INVALID.

CORRECTED: What Really Happened

- 0/1000 configurations preserve unitarity naturally
- Forced normalization masked the failure (bug lines 139-141)
- Best norm drift: 0.000256 (still diverges without correction)
- Previous "MOO discovery" claim was INVALID

Critical Failure

- Bigeometric evolution requires explicit normalization to stay unitary
- Classical Crank-Nicolson (k=0, w=0, p=0) preserves unitarity
- ALL meta-calculus parameters (k u003E 0) break unitarity
- Data file: public/corrected_results.json

Data: results/quantum_phase_moo.json | Script: simulations/quantum_phase_moo.py

EXP-7: Cosmology vs Real Data

CANNOT RESOLVE

Tests meta-calculus predictions against SH0ES, DESI, KiDS, and Planck data. Critical validation: can the framework resolve cosmological tensions?

Why It Cannot Resolve Tensions

- H0: 0/9 Pareto solutions (k is uniform)
- S8: 0/9 Pareto solutions (same reason)
- Li7: -28% improvement (does not help)
- Tensions need SCALE-DEPENDENT physics

Honest Assessment

k produces UNIFORM corrections
Previous claims were post-hoc fitting
Framework is numerical tool, not physics
This confirms HONEST-STATUS.md

Data: results/cosmology_validation_moo.json | Script: simulations/cosmology_validation_moo.py

Methodology

Step 1

Build on Existing

Reused 37 result files, 8 MOO scripts, existing test infrastructure.

Step 2

MOO Optimization

pymoo NSGA-II with 30-50 population, 50-100 generations per experiment.

Step 3

Honest Documentation

Document both positive and negative results. Update claims accordingly.

What Changed

Claim	Before	After
Shock Tube CFD	"Not implemented"	"5.4% improvement (validated 12.1% +/- 3.1%)"
Spectral Gap Preservation	"Always holds (9/9)"	"Holds for Dirichlet/Periodic BCs"
k(L) Pattern	"Energy density better (R^2=0.73)"	"BUG FIXED: All affine features R^2=0.82"
Convergence Order	"Unknown"	"Bigeometric = 2nd order (verified)"
Chaos Preservation	"Not tested"	"NEUTRAL: Matches RK4 (0% improvement)"
Quantum Unitarity	"Not tested (forced normalization bug)"	"CORRECTED: 0% unitary (bug fixed Dec 2025)"
Cosmology Tensions	"May resolve H0"	"CANNOT resolve (uniform k)"
HONEST-STATUS.md	Version 3.0	Version 3.2 (updated)

All 7 Experiments Complete

DONE

MOO-First Validation Complete

All 7 experiments implemented with honest outcomes documented. 2 VERIFIED, 1 CORRECTED, 1 REFINED, 1 NEUTRAL, 1 INVALID (bug), 1 CANNOT RESOLVE. 3 VERIFIED, 1 PARTIAL, 1 REFINED, 1 NEUTRAL, 1 CANNOT RESOLVE.

Dec 2024

Key Insight

Meta-calculus is a numerical tool, not a physics theory. It helps with regularization, scheme consistency, and parameter optimization - but cannot resolve physics problems that require scale-dependent modifications.