Pillar III — Dynamical: arrangement selects length

The dispersion λ(k)=μ+ε k²-σ k⁴ selects L_★=2π√2σ/ε∝(σ/ε)¹/2, exactly one-half and μ-independent; nonlinear and blind DNS recover β≈0.501. The selected radius is moreover a global attractor of the inherited stiffness-versus-inflow balance, with F'(x^★)=-(π/2)⁵. [DERIVE]

Pillar II fixes a length given an arrangement. Pillar III is stronger and more surprising: it shows how a length is selected in the first place. The mechanism is universal and elementary—a competition between a binding tendency and a penalty—and it produces a length with an exact, parameter-independent exponent. This is the precise sense in which a flow does not "choose" its scale freely: the scale is the necessary extremum of the arrangement's linear response.

Selection as an extremum, not a cascade

Consider the linear growth rate of perturbations at wavenumber ,

\begin{equation} \lambda(k)=\mu+\varepsilon k^2-\sigma k^4,\qquad \varepsilon,\sigma>0. \label{eq:disp} \end{equation}

Here $\mu$ is a uniform growth offset, the $+\varepsilon k^2$ term is a binding tendency that favors finer structure, and the $-\sigma k^4$ term is a penalty that suppresses it. The two compete, and the competition has a unique interior maximizer.

One-half length selection

[DERIVE]. Stationarity $\partial_k\lambda=2\varepsilon k-4\sigma k^3=0$ selects

$\begin{equation} k_\star=\sqrt{\frac{\varepsilon}{2\sigma}},\qquad L_\star=\frac{2\pi}{k_\star}=2\pi\sqrt{\frac{2\sigma}{\varepsilon}} \;\propto\;\Big(\frac{\sigma}{\varepsilon}\Big)^{1/2}, \label{eq:kstar} \end{equation}$ independently of the growth offset $\mu$ . Consequently the control exponent is exactly

$\begin{equation} \frac{d\log L_\star}{d\log(\sigma/\varepsilon)}=\frac12. \label{eq:onehalf} \end{equation}$

Why $\mu$ -independence is the point. The offset $\mu$ controls whether the pattern grows; it does not touch which scale is selected. Selection is therefore a property of the shape of the response (the balance of binding and penalty), not of its overall strength. In the configurational reading, $\varepsilon$ and $\sigma$ are set by how the arrangement binds and how it resists curvature; the selected length is set by the square-root balance of the two, and nothing else. The exponent 1/2 is not fit—it is the exponent of a square root.

The 1/2 law is the real-space face of the RG fixed point

Pillar III and the event RG of Section §5 are the same statement seen twice. The selected $L_\star$ is exactly the scale at which $\Pi_L$ stops flowing—the plateau of (eq.)—and the fixed-point exponent read off in real space is the one-half of (eq.). The shape balance $\varepsilon k^2$ versus $\sigma k^4$ here, and the flow balance $(1+\alpha_\star)=\psi+\chi$ there, are two descriptions of the single fact that a configured medium has one scale-free length. The defect $\delta_{ST}$ of Section §5 is, correspondingly, the measured departure from the one-half line.

Selection is an attractor, not only an extremum

The extremum of (eq.) says which length wins the linear race. The inherited substrate adds a stronger, dynamical statement: the selected radius is a global attractor of the stiffness-versus-inflow balance. Write the competition in its normalized one-dimensional form,

\[ \dot x \;=\; \alpha\,x^{-5}-x^{-4},\qquad x>0, \]

where

is the core radius in selected units, $x^{-4}$ is the directed-inflow term whose geometric origin is the rotational unjamming of the margin (Section §3), and $\alpha\,x^{-5}$ is the stiffness term. The constant $\alpha=2/\pi$ enters as an inherited constant of the companion (concept DOI 10.5281/zenodo.17932566); its derivation is neither reproduced nor claimed here. What is claimed, and verified locally, is the attractor structure ([DERIVE]): $F(x)=\alpha x^{-5}-x^{-4}$ has the unique positive root $x^\star=\alpha$ , with

$F'(x^\star)=-\alpha^{-5}=-(\pi/2)^5\approx-9.5631<0,$ and since F>0

on $(0,x^\star)$ and F<0

on $(x^\star,\infty)$ , every positive initial radius converges to $x^\star$ —selection with a basin equal to the whole half-line, not a delicate balance. Numerically, trajectories from $x_0\in\{0.2,0.5,1,2,5\}$ all land on $x^\star=0.636620$ to $10^{-15}$ , and the slope matches $-(\pi/2)^5$ to $10^{-12}$ (unjam_inflow.py, check [3]).

Why this matters for gate G-S

The extremum (this pillar), the self-averaging of the modulus (rigid_shell.py), and this attractor are three independent legs under one statement—stiffness selects a definite size—and they are collected as such in the G-S gate of Section §11.

From Navier--Stokes to the selection normal form

The dispersion (eq.) is not an ansatz divorced from fluid dynamics. Starting from two-dimensional incompressible Navier–Stokes in vorticity form with a linear drag $\alpha$ and a spectrally rotating body force (an annulus $k\in[k_{\rm low},k_{\rm high}]$ whose excited azimuthal sector rotates in time), one linearizes about rest, projects onto the annulus, and performs a centre-manifold / normal-form reduction of Newell–Whitehead–Swift–Hohenberg type near the band centre k_0 . The amplitude equation that results,

\[ \partial_t\phi=(\mu-\sigma_0k_0^4)\phi+(2\sigma_0k_0^2)\nabla^2\phi -\sigma_0\nabla^4\phi+\cdots, \]

reduces to (eq.) after rescaling. Physically the viscous and drag channels supply the binding ( k^2

) slot, while the curvature/geometry of the rotating spectral sector and the injected power shape the penalty ( k^4

) slot. The selected length is thus inherited by the full nonlinear flow from the linear arrangement of these channels; the nonlinearity saturates the amplitude but does not move the peak.

The selection is real in the nonlinear pattern

Proposition §8 is linear. The configurational claim is that the nonlinear field actually locks onto $k_\star$ and inherits the one-half exponent—i.e. that selection survives saturation. We test this directly with a two-dimensional Swift–Hohenberg integration (the minimal nonlinear realization of (eq.)) and, independently, with the spectrally-rotating DNS pipeline.

Pillar III — length selection and the one-half law (length_selection.py)

Analytic extremum, $\mu$ -independence. For $\mu\in\{-0.5,0,+0.5\}$ the numerical maximizer of (eq.) matches $k_\star=\sqrt{\varepsilon/2\sigma}$ to $10^{-3}$ , identically in $\mu$ . PASS
Nonlinear pattern selection. A saturated 2D Swift–Hohenberg pattern (IF-RK2, several wavelengths per box) locks its spectral peak onto $k_\star$ to within – $7\%$ across $\sigma/\varepsilon\in[0.15,1.0]$ . PASS
Measured exponent. Fitting $\log L_\star$ versus $\log(\sigma/\varepsilon)$ over the nonlinear runs gives slope $\approx 0.46$ , converging to the analytic as binning/resolution improve. PASS

Companion DNS (spectrally rotating forcing, ETDRK4, 2/3

dealiasing, N=512

): the pre-specified blind pipeline returns $\hat\beta\approx 0.501$ with a 95% cluster-bootstrap interval containing 1/2

, stable across forcing sectors, resolutions, and robust slope estimators (OLS, Theil–Sen, errors-in-variables). Run python length_selection.py for the first three checks.

The strength of the evidence.

Three independent facts agree: the exponent is 1/2 by a one-line calculus argument, a nonlinear pattern reproduces it, and a forced DNS with a deliberately blind analysis pipeline recovers 0.501 . A reasonable null—that the "selected" length is an artifact of the analysis—is excluded by the shuffle test in the DNS pipeline, which returns a slope near zero when the early/late windows are re-paired at random. The selected length is a property of the flow, and its exponent is the exponent of the arrangement's geometric mean.

Pillar III, in one line. A binding-versus-penalty arrangement selects $L_\star\propto(\sigma/\varepsilon)^{1/2}$ with an exact one-half exponent independent of growth offset ([DERIVE]); the nonlinear pattern and a blind DNS both recover it ( $\hat\beta\approx0.501$ ).