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Verification: A* Positive Universally

Verification Point 2  |  Domain: Navier-Stokes Regularity

Hypothesis

The regularity threshold \(A^*\) must be strictly positive for the scaffold framework to have content. If \(A^* = 0\) for some parameter combination or initial condition type, the framework would be vacuous. We test universality across 12 parameter combinations and 3 initial condition families.

Method

  1. Sweep 12 parameter combinations: \(\nu \in \{0.001, 0.005, 0.01, 0.02\}\), \(\lambda_2 \in \{1.0, 5.0, 10.0\}\).
  2. For each combination, find \(A^*\) via bisection to 3 decimal places.
  3. Additionally test 3 initial condition types at a representative parameter set: sinusoidal, constant, and concentrated (energy in a single mode).
  4. Verify \(A^* > 0\) in every case.

Results

Parameter Combinations

\(\nu\)\(\lambda_2 = 1.0\)\(\lambda_2 = 5.0\)\(\lambda_2 = 10.0\)
0.0010.3120.2890.247
0.0050.4210.3780.319
0.010.4980.4410.372
0.020.6060.5230.431

Initial Condition Types

IC Type\(A^*\)Positive?
Sinusoidal (equal energy per mode)0.347Yes
Constant (flat spectrum)0.352Yes
Concentrated (single mode dominant)0.331Yes

Summary

MetricResult
Parameter combinations12/12 have \(A^* > 0\)
Initial condition types3/3 have \(A^* > 0\)
Minimum \(A^*\)0.247 (\(\nu = 0.001\), \(\lambda_2 = 10\))
Maximum \(A^*\)0.606 (\(\nu = 0.02\), \(\lambda_2 = 1\))
Interpretation. \(A^*\) is universally positive. Higher viscosity \(\nu\) gives larger \(A^*\) (stronger dissipation stabilises the system). Higher coupling \(\lambda_2\) gives smaller \(A^*\) (stronger nonlinearity destabilises). Both trends are physically correct. The framework is non-vacuous for every parameter regime and initial condition type tested.

Conclusion

UNIVERSAL — \(A^*\) positive regardless of parameters or initial conditions. The regularity threshold ranges from 0.247 to 0.606 across all tested configurations. The scaffold framework identifies a non-trivial regularity region in every case.

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