###################################################################### # EMPIRICAL VALIDATION SUITE # The Davis Duality of Approximation and Obstruction # Testing: do ML errors concentrate where the TARGET is locally complex? ###################################################################### --- TEST 1: Errors track function Hessian norm --- ====================================================================== TEST 1: Errors track function Hessian norm ✓ PASS ρ(||Hess f||, |error|) = 0.4140, p = 1.1850e-83 Q5/Q1 error ratio: 5.09x Quintile errors: ['0.0204', '0.0313', '0.0395', '0.0455', '0.1040'] ====================================================================== --- TEST 2: Smooth vs oscillatory regions --- ====================================================================== TEST 2: Oscillatory regions have higher error ✓ PASS Mean error in smooth region: 0.0301 Mean error in oscillatory region: 0.1885 Ratio (oscillatory/smooth): 6.27x ρ(K_analytic, |error|) = 0.6466, p = 3.5194e-237 ====================================================================== --- TEST 3: Frequency gradient (continuous curvature variation) --- ====================================================================== TEST 3: Errors increase with local frequency (curvature) ✓ PASS ρ(ω², |error|) = 0.4321, p = 9.1645e-92 Q5/Q1 error ratio: 5.06x Quintile errors: ['0.0481', '0.0962', '0.1774', '0.2561', '0.2435'] ====================================================================== --- TEST 4: Architecture independence (intrinsic curvature) --- ====================================================================== TEST 4: Architecture independence (intrinsic curvature) ✓ PASS Arch ρ(ω²,|error|) p-value GBT 0.4172 4.74e-85 *** RF 0.4068 1.45e-80 *** KNN 0.4790 2.96e-115 *** MLP 0.5054 3.99e-130 *** ====================================================================== --- TEST 5: Error decreases with capacity, modulated by curvature --- ====================================================================== TEST 5: High-curvature regions consistently worse across capacities ✓ PASS Capacity err_lowK err_highK ratio 50 0.2536 0.3318 1.31x 100 0.2101 0.3095 1.47x 200 0.1465 0.2848 1.94x 400 0.1149 0.2557 2.23x 800 0.1037 0.2462 2.37x Average high-K/low-K error ratio: 1.86x ====================================================================== --- TEST 6: Wrong-target discrimination (intrinsic) --- ====================================================================== TEST 6: Curvature discriminates true vs wrong targets ✓ PASS |r| with true target: 0.9600 |r| with best wrong target: 0.0375 Discrimination ratio: 25.63x ====================================================================== --- TEST 7: Neural network errors track function curvature --- ====================================================================== TEST 7: Neural network errors track function curvature ✓ PASS ρ(ω², |error|) = 0.6275, p = 2.1054e-219 Q5/Q1 error ratio: 6.67x Quintile errors: ['0.0438', '0.0388', '0.1027', '0.2124', '0.2919'] ====================================================================== --- TEST 8: Regime transition at curvature threshold --- ====================================================================== TEST 8: Performance degrades with increasing frequency ✓ PASS ρ(frequency, R²) = -0.9762, p = 3.3144e-05 ω κ~ω² R² 0.5 0.2 0.9957 1.0 1.0 0.9778 2.0 4.0 0.7619 3.0 9.0 0.2614 5.0 25.0 -0.1063 8.0 64.0 -0.2521 12.0 144.0 -0.2279 16.0 256.0 -0.3079 ====================================================================== --- TEST 9: Error is coordinate-invariant --- ====================================================================== TEST 9: Rotation invariance (isometry preserves error) ✗ FAIL Error in (x,y) coordinates: 0.2952 Error in rotated coordinates: 0.1425 Error in polar coordinates: 0.1782 Ratio rotated/original: 0.483 Ratio polar/original: 0.604 ====================================================================== --- TEST 10: Error scales with local spectral complexity --- ====================================================================== TEST 10: Error scales with local spectral complexity ✓ PASS ρ(spectral complexity, |error|) = 0.5416, p = 8.0042e-153 Q5/Q1 error ratio: 8.20x Quintile errors: ['0.0276', '0.0676', '0.1293', '0.1491', '0.2264'] ====================================================================== --- TEST 11: High-K regions need more parameters to reach same error --- ====================================================================== TEST 11: High-K regions consistently need more capacity ✓ PASS N_est err_lowK err_highK ratio 30 0.0535 0.1968 3.68x 60 0.0462 0.1943 4.21x 120 0.0414 0.1924 4.65x 240 0.0387 0.1959 5.06x 480 0.0374 0.1974 5.28x Average ratio: 4.57x ====================================================================== ###################################################################### # FINAL SCORECARD ###################################################################### PASSED: 10/11 FAILED: 1/11 PASS RATE: 91% Test Result Name ---------------------------------------------------------------------- 1 ✓ Errors track function Hessian norm 2 ✓ Oscillatory regions have higher error 3 ✓ Errors increase with local frequency (curvature) 4 ✓ Architecture independence (intrinsic curvature) 5 ✓ High-curvature regions consistently worse across capacities 6 ✓ Curvature discriminates true vs wrong targets 7 ✓ Neural network errors track function curvature 8 ✓ Performance degrades with increasing frequency 9 ✗ Rotation invariance (isometry preserves error) 10 ✓ Error scales with local spectral complexity 11 ✓ High-K regions consistently need more capacity ====================================================================== NON-CIRCULARITY GUARANTEES: Curvature = function Hessian norm (INTRINSIC), not surface shape ML sees flat coordinates only — curvature is HIDDEN in the function Local frequency (omega) serves as analytically tractable curvature proxy Tests include actual neural networks (MLP), not just tree models Coordinate invariance tested explicitly ====================================================================== Davis Geometric — C = tau/K The Davis Duality of Approximation and Obstruction ======================================================================