Dimensionless diagnosis: the $Pi$-invariants and the event RG

A selected arrangement is diagnosed by three invariants (Π_T,Π_L,Π_ST) and explained by an event RG whose fixed point, (1+α_★)=ψ+χ, is where Π_L stops flowing. The one-half law of Pillar III is this plateau read in real space.

A selected arrangement is diagnosed by three invariants (Π_T,Π_L,Π_ST) and explained by an event RG whose fixed point, (1+α_★)=ψ+χ, is where Π_L stops flowing. The one-half law of Pillar III is this plateau read in real space.

The axiomatic core says what the interaction is; the present section says how to recognize a selected arrangement and how it flows under coarse-graining. Both are dimensionless statements, and both make the thesis operational: the form a flow settles into is the fixed point of an event-driven renormalization, and one can read off from data whether a given structure sits at that fixed point.

The \Pi-invariants

A quasi-stationary arrangement at "optimal contact"—the configuration the medium relaxes into and holds—is diagnosed by three dimensionless invariants: a temporal \Pi_T, a spatial \Pi_L, and a spatio-temporal coupling \Pi_{ST}. At an optimal arrangement the first two are pinned, \Pi_T\approx1 and \Pi_L\approx1, while the coupling invariant carries the criterion. The operational test (ST1') is a line: in the log plane the coupling \Pi_{ST} holds a straight line of slope -a/b along an optimal trajectory (transitions between arrangements included), and the orthogonal departure from that line is a single defect, the departure of \Pi_{ST} from that line:

\begin{equation} \delta_{ST}=\big(\text{departure of }\Pi_{ST}\text{ from the line of slope }-a/b\big), \label{eq:pi_line} \end{equation}
and the arrangement is judged optimal when \Pi_T\approx1, \Pi_L\approx1, and \delta_{ST}\le0.05. The defect is the trigger of the document's reproducibility discipline (a fail-fast rule): a structure with \delta_{ST} above tolerance is flagged for splitting, deferral, or exclusion rather than reported as a clean selection. This is what makes "form follows arrangement" falsifiable in practice—an arrangement either holds the line or it does not.

The event renormalization group

Under coarse-graining by a factor \lambda, the event-dissipation law (eq.) flows together with the selected length: the binding release weakens as \eps_{\rm bind}\mapsto\lambda^{-\chi}\eps_{\rm bind}, the effective stiffness as \sigma_{\rm eff}\mapsto\lambda^{\psi}\sigma_{\rm eff}, the length as L\mapsto\lambda L, and the cascade exponent \alpha may run. The spatial invariant

\begin{equation} \Pi_L=\frac{L^{\,1+\alpha}\,\eps_{\rm bind}}{\sigma_{\rm eff}} \qquad\Longrightarrow\qquad \frac{d\log\Pi_L}{d\log\lambda}=(1+\alpha)-\psi-\chi \label{eq:event_rg} \end{equation}
therefore stops flowing exactly at the fixed point. The selected arrangement is that fixed point:
\begin{equation} \boxed{\;(1+\alpha_\star)-\psi-\chi=0\;\Longleftrightarrow\;\Pi_L\ \text{plateau}\;} \label{eq:fixed_point} \end{equation}
and the sign of (1+\alpha)-\psi-\chi gauges whether a structure is strengthening (>0) or weakening (<0) as it is coarse-grained—the same monotonicity that the special-edition typhoon protocol uses to call intensification versus dissipation.

Event-RG fixed point =\Pi_L plateau (event_rg.py)

Iterating the recursion L\mapsto\lambda L, \eps_{\rm bind}\mapsto\lambda^{-\chi}\eps_{\rm bind}, \sigma_{\rm eff}\mapsto\lambda^{\psi}\sigma_{\rm eff} and tracking \Pi_L:

The 1/2 length law of Pillar §8 is this plateau read off in real space; the JFM exponent is the fixed-point exponent, and \delta_{ST} is the measured departure from it.

So the layers close on each other: the substrate (\S§3) makes the medium a fluid and hands rotation the unjamming switch, the axioms (\S§4) fix the interaction, the \Pi-invariants diagnose when an arrangement is selected, and the event RG explains why that arrangement is the one selected—it is the only scale-free configuration of the binding-release and stiffness flow. (Section §3 names the physical candidate for how a driven medium reaches and holds that fixed point: self-organized marginality, gate G-SOC.) The four pillars now follow as the in-domain verification of this structure.