=== Paper 1 Controllability v3: Numerical Verification ===

System: n=3, n^2=9, T=2pi

Eigenvalues of M:
  Columns 1 through 7

   0.7877 + 0.6160i   0.7877 - 0.6160i  -1.5883 + 0.0000i  -0.6296 + 0.0000i   0.9597 + 0.2812i   0.9597 - 0.2812i   1.0000 + 0.0000i

  Columns 8 through 9

   1.0000 + 0.0000i   1.0000 + 0.0000i

||M - Phi_B*xPhi_A||_F = 4.6869e+02  (MATLAB-verified)

|eig(M)| =     1.0000    1.0000    1.5883    0.6296    1.0000    1.0000    1.0000    1.0000    1.0000

Three distinct moduli -> k = 3, horizon = 18.8496

--- Step-size sensitivity ---
  1e-3 vs 1e-4: ||DeltaM||_F = 9.6591e-02
  1e-4 vs 1e-5: ||DeltaM||_F = 8.3716e-03

min sigma_min(K(t)) = 0.2254  (MATLAB-verified: 0.2254)

--- Computing Wt_1 by quadrature (N=1000) ---

Table 1 — MATLAB-verified Gramian Statistics:
i       lambda_min      lambda_max              kappa                 PD
------------------------------------------------------------------------
1       0.5211          4.0840e+05              7.8374e+05            true
2       0.5681          8.7585e+06              1.5416e+07            true
3       0.5705          1.5160e+08              2.6573e+08            true

Recursion residual = 0.0000e+00  (MATLAB-verified: 0.0000e+00)

--- Gramian scaling study: n=2, 3 ---
(n=4 needs >5GB; excluded from MATLAB Online run)

n=2  n^2= 4  kappa = 2.1280e+10 +/- 6.6477e+10
n=3  n^2= 9  kappa = 4.9528e+11 +/- 1.3567e+12

--- Minimum-energy control ---
J* = x0^T * Wt_3^{-1} * x0 = 0.292468
lambda_min(Wt_3) = 0.5705  -> Wt_3 invertible, control exists

--- H-controllability ---
min_j integral = 1.8270  (>0 -> H-controllability confirmed)

=== Paper 1 v3 complete. All MATLAB-verified. ===
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Fig1_Ctrl_EigModuli.pdf saved.
Fig2_Ctrl_GramianSpectrum.pdf saved.
Fig3_Ctrl_ScalingStudy.pdf saved.
Fig4_Ctrl_StepSizeSensitivity.pdf saved.
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