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Level 9: Physics Coupling as Dual Number
Experiment: exp_ns_level9.sx | Domain: Navier-Stokes Regularity | Hierarchy Level: 9
Overview
Level 8 made simulation parameters (\(A\), \(\nu\), \(\text{dt}\)) into dual numbers. Level 9 goes deeper: the physics coupling constants \(\lambda\) (linear stretching), \(\lambda_2\) (quadratic self-amplification), and \(\sigma\) (velocity-vorticity feedback) become dual numbers. This yields the full gradient \(\nabla T = (\partial T / \partial A, \partial T / \partial \nu, \partial T / \partial \lambda, \partial T / \partial \lambda_2, \partial T / \partial \sigma)\) and the Hessian diagonal \(\partial^2 T / \partial x_i^2\) for each parameter.
Key Results
Full Gradient (Sensitivity Ranking)
| Parameter | \(\partial T / \partial x\) | Sign | Interpretation |
|---|
| \(A\) (amplitude) | Largest magnitude | Negative | More amplitude → earlier blow-up |
| \(\nu\) (viscosity) | Second largest | Positive | More viscosity → later blow-up |
| \(\lambda\) (stretching) | Third | Negative | More stretching → earlier blow-up |
| \(\lambda_2\) (quadratic) | Fourth | Negative | More self-amplification → earlier blow-up |
| \(\sigma\) (feedback) | Smallest | Weakly negative | Feedback effect is secondary |
Hessian Diagonal (Curvature)
| Parameter | \(\partial^2 T / \partial x^2\) | Curvature | Meaning |
|---|
| \(\lambda\) | Positive | Convex | Self-amplifying: more stretching accelerates its own effect |
| \(\lambda_2\) | Positive | Convex | Quadratic term compounds super-linearly |
| \(\sigma\) | Negative | Concave | Self-limiting: feedback saturates at high coupling |
| \(\nu\) | Negative | Concave | Diminishing returns from added viscosity |
| \(A\) | Positive | Convex | Large amplitudes are disproportionately destabilising |
Cross-Derivatives
| Cross-Derivative | Bounded? | Significance |
|---|
| \(\partial^2 T / \partial \lambda \, \partial A\) | Yes | Regularity is robust to stretching geometry |
| \(\partial^2 T / \partial \lambda_2 \, \partial A\) | Yes | Quadratic stretching does not break amplitude regularity |
Interpretation. The Hessian reveals a fundamental asymmetry: destabilising parameters (\(\lambda\), \(\lambda_2\), \(A\)) are convex (self-amplifying), while stabilising parameters (\(\nu\), \(\sigma\)) are concave (self-limiting). This means the system has a built-in bias toward instability — but the bounded cross-derivatives show that the regularity signature is robust even when physics couplings are varied. The sensitivity ranking confirms amplitude is the dominant control parameter.
Reproducibility
../simplex/build/sxc exp_ns_level9.sx -o build/exp_ns_level9.ll
OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_ns_level9.ll \
../simplex/runtime/standalone_runtime.c \
-o build/exp_ns_level9 \
-lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib
./build/exp_ns_level9
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