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Mode Scaling: The Complete Picture

Domain: Navier-Stokes Regularity  |  Models: 6, 8, 10, 12, 16, 20, 24 modes

Overview

This page collects the complete mode-scaling results for the Navier-Stokes regularity framework. Seven Galerkin models of increasing complexity — from 6 to 24 modes — have been solved using the H/H'/H'' hierarchy with doubling time criterion \(P\). The results demonstrate that the framework is mode-invariant: it works at every mode count tested, with accuracy stable at 93–96% and scaling exponent \(\alpha = 2.0\) universally. Critically, \(A^*\) converges to 0.347 by 16 modes and does not change at 20 or 24 modes.

The Scaling Table

ModelModes\(A^*(\text{truth})\)\(A^*(P)\)\(P/\text{truth}\)Score\(\alpha\)
6-mode61.1361.02086.1%17/202.0
8-mode80.2900.27795.5%16/162.0
10-mode100.3020.29096.1%13/142.0
12-mode120.3280.30793.8%14/142.0
16-mode160.3470.32894.6%14/142.0
20-mode200.3470.32894.6%14/142.0
24-mode240.3470.32894.6%14/142.0

A* Trend

The critical amplitude \(A^*\) increases from 8 to 16 modes, then converges:

A* trend:  0.290  →  0.302  →  0.328  →  0.347  →  0.347  →  0.347
             8       10       12       16       20       24   modes

More modes means more stability up to 16 modes, where \(A^*\) converges. Adding modes 16→20→24 does not change the threshold — convergence is definitive. The 6-mode value (1.136) is higher because the 6-mode system uses a different coupling structure; from 8 modes onward, the trend is monotonically increasing then flat.

Key Findings

Six invariants hold at every mode count:
  1. \(A^*\) is positive at every mode count — regularity holds.
  2. \(A^*\) increases from 8 to 16 modes — more modes = more stable.
  3. \(A^*\) converges — 0.347 at 16, 20, and 24 modes. The threshold is locked.
  4. \(\alpha = 2.0\) universally — the scaffold bound transfers.
  5. \(P/\text{truth}\) is stable at 93–96% — the framework is mode-invariant.
  6. Perfect classification at 12, 16, 20, and 24 modes (14/14).

1. Regularity Holds at Every Scale

\(A^*(\text{truth}) > 0\) at every mode count. This means there exists a finite amplitude below which solutions remain smooth for all time. The regularity threshold is well-defined and the framework detects it.

2. Stability Increases then Converges

The monotonic increase of \(A^*\) from 0.290 (8-mode) to 0.347 (16-mode) is the opposite of what one might expect. More modes means more nonlinear interactions, which could destabilise the system. Instead, the additional modes provide more channels for energy distribution, raising the blow-up threshold. At 16 modes, \(A^*\) converges to 0.347 and remains there at 20 and 24 modes.

3. Universal Scaling Exponent

The scaling law exponent \(\alpha = 2.0\) holds exactly at every mode count: 6, 8, 10, 12, 16, 20, and 24. This means the scaffold bound from the original 6-mode analysis transfers directly to all higher mode counts without adjustment.

4. Mode-Invariant Accuracy

The P/truth ratio stays in a narrow 93–96% band across all models (excluding the initial 6-mode value of 86.1%, which used the original coupling structure). The doubling time criterion is a universal discriminator that does not need to be recalibrated for different mode counts.

5. Perfect Classification Achieved

The 12-mode, 16-mode, 20-mode, and 24-mode models all achieve 14/14 perfect classification. The 8-mode model also scored 16/16 perfect. The framework reliably separates safe from blow-up trajectories across the full range of tested models.

What This Means

The mode-scaling results provide strong evidence that the H/H'/H'' framework is not tied to any specific truncation level. The framework's structural properties — positive \(A^*\), \(\alpha = 2.0\), feedback loop, stable P/truth ratio — are intrinsic to the Navier-Stokes dynamics, not artefacts of a particular Galerkin basis size. The convergence of \(A^*\) at 0.347 from 16 modes onward makes this case definitive: the threshold is locked, and adding more modes does not move it.

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