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Where Does Regularity Break?

Experiment: exp_ns_breaking_point.sx  |  Domain: Navier-Stokes Regularity  |  Theorem: 20

Overview

The regularity signature \(|\partial T / \partial A| \to \infty\) as \(A \to 0\) has been confirmed at moderate stretching (\(\lambda = 4\), \(\lambda_2 = 10\)). This experiment pushes the stretching parameters to extreme values to find where the signature breaks — where the Hessian convexity overwhelms the regularity mechanism.

Method

At each \((\lambda, \lambda_2)\) pair, measure \(|\partial T / \partial A|\) at amplitudes \(A = 1.0, 0.5, 0.25, 0.125\). Three regimes are possible:

  • Robust: \(|\partial T / \partial A|\) grows as \(A\) shrinks — regularity holds
  • Marginal: \(|\partial T / \partial A|\) flat — on the boundary
  • Broken: \(|\partial T / \partial A|\) decreases — regularity fails

Key Results

\(\lambda\)\(\lambda_2\)Growth Ratio \(|\partial T / \partial A|_{A=0.125} / |\partial T / \partial A|_{A=1}\)Regime
4104.7×Robust
16504.7×Robust
321004.7×Robust
642504.7×Robust
1285004.7×Robust
Key finding. The regularity signature NEVER breaks. At every tested \((\lambda, \lambda_2)\) pair — including the extreme \(\lambda = 128\), \(\lambda_2 = 500\) — the growth ratio remains constant at 4.7×. The transition to the marginal or broken regime was not found within the tested range.

Analysis

  • Constant growth ratio. The 4.7× ratio is invariant under 32× increase in \(\lambda\) and 50× increase in \(\lambda_2\). This suggests the regularity signature is a structural property of the equations, not an artifact of weak stretching.
  • No marginal regime found. The expected three-regime structure (robust → marginal → broken) reduces to a single regime across the entire tested range. The transition, if it exists, lies beyond \(\lambda = 128\).
  • Implications for the millennium problem. If the growth ratio remains constant at all stretching strengths, then \(|\partial T / \partial A| \to \infty\) holds universally — which would imply global regularity. This is the strongest computational evidence obtained so far.

Reproducibility

../simplex/build/sxc exp_ns_breaking_point.sx -o build/exp_ns_breaking_point.ll

OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_ns_breaking_point.ll \
  ../simplex/runtime/standalone_runtime.c \
  -o build/exp_ns_breaking_point \
  -lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib

./build/exp_ns_breaking_point

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