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The Holistic View: One Homogeneous System

Experiment: exp_ns_holistic.sx  |  Domain: Navier-Stokes Regularity  |  Theorems: 27–28

Overview

The 10-level diagnostic hierarchy is not 10 separate measurements — it is one coupled system. This experiment computes the coupling matrix \(J_{ij} = \partial L_i / \partial L_j\) between levels, revealing which levels drive or suppress each other. The state vector is \(\mathbf{L} = (L_1, L_2, L_4, L_8, L_9, L_{10})\) where \(L_1\) = enstrophy, \(L_2\) = S-score, \(L_4\) = stability margin, \(L_8\) = \(\partial T / \partial A\), \(L_9\) = \(\partial T / \partial \lambda\), \(L_{10}\) = \(\partial T / \partial(\text{accel})\).

Key Results

Coupling Matrix Structure

\(J_{ij}\)\(L_1\) (enstrophy)\(L_2\) (S-score)\(L_4\) (margin)
\(L_1\)—Negative (S suppresses enstrophy growth)Negative (margin stabilises)
\(L_2\)Negative (enstrophy degrades S)—Positive (margin supports S)
\(L_4\)Negative (enstrophy erodes margin)Positive (S reinforces margin)—

Feedback Loop

The dominant feedback loop is: \(L_1 \to L_4 \to L_2 \to L_1\). Enstrophy erodes the stability margin, which degrades convergence, which allows enstrophy to grow further. Viscosity breaks this loop by acting on \(L_1\) directly.

Hidden Viscosity Power

MeasurementValue
Direct viscosity effect on enstrophy\(\partial L_1 / \partial \nu\): modest
Indirect effect via coupling matrix500,000× amplification through feedback loop
Holistic score \(H\)\(H = 0.3 \cdot L_{1,n} - 0.4 \cdot L_2 + 0.3 \cdot L_{4,n}\)
Interpretation. Viscosity appears weak when measured level-by-level, but the coupling matrix reveals its true power: a small change in \(\nu\) propagates through the \(L_1 \to L_4 \to L_2 \to L_1\) loop with 500,000× amplification. This is why the NS equations are regular despite stretching being convex and self-amplifying — viscosity's indirect effect through the feedback loop is vastly more powerful than its direct damping. The holistic score \(H\) captures this coupled behaviour in a single scalar.

Analysis

  • Not 10 diagnostics — one system. The coupling matrix shows that no level can be understood in isolation. Changes propagate through the full hierarchy.
  • Feedback loop is the mechanism. The \(L_1 \to L_4 \to L_2 \to L_1\) loop is the engine of both blow-up and regularity. Viscosity interrupts it.
  • 500,000× hidden power. The direct effect of viscosity understates its true influence by five orders of magnitude. The coupling matrix reveals the full picture.

Reproducibility

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

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

./build/exp_ns_holistic

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