Pillar II — Geometric: arrangement fixes length

The rotational core capacity identity N_★ r_eff²=φ s L_core² fixes the effective core length exactly once the arrangement is specified. The published cross-regime audit reproduces to 3×10⁻¹², the identity itself to 10⁻¹⁵. [DERIVE]

Pillar I gives the equations. Pillar II makes the first sharp statement about form: once the arrangement of a rotational core is specified, its effective length is fixed by an accounting identity. We then audit, across four flow regimes, exactly when a single length is an adequate summary of the arrangement and when it is not—turning the question "does this flow have a core radius?" into a measurable, reproducible diagnostic.

The rotational core capacity identity (RCCI)

Let an Okubo–Weiss–type criterion define a core mask of area $A_{\rm core}$ . Let $\{B_k\}_{k=1}^{N_\star}$ be the strong-rotation blobs inside , with areas A_k and equivalent radii $r_k=\sqrt{A_k/\pi}$ . Define the RMS-equivalent radius, the filling fraction, and the shape factor by

$\begin{equation} r_{\rm eff}^2=\frac{1}{N_\star}\frac{1}{\pi}\sum_{k=1}^{N_\star}A_k,\qquad \phi=\frac{\sum_k A_k}{A_{\rm core}},\qquad s=\frac{A_{\rm core}}{\pi L_{\rm core}^2}. \label{eq:rcci_defs} \end{equation}$

RCCI — arrangement fixes the effective length

[LOCK]/[DERIVE]. By construction $\sum_k A_k=\phi A_{\rm core}$ , hence

$\begin{equation} \boxed{\,N_\star\,r_{\rm eff}^2=\phi\,s\,L_{\rm core}^2\,} \label{eq:rcci} \end{equation}$ and, solving for the effective length,

$\begin{equation} r_{\rm pred}=L_{\rm core}\sqrt{\frac{\phi s}{N_\star}}. \label{eq:rpred} \end{equation}$ Equation (eq.) is an exact accounting equality for a given mask and blob partition.

The same identity in any dimension

Written in terms of volumes rather than areas, (eq.) is dimension-free: $N_\star\,v_{\rm eff}=\phi\,V_{\rm core}$ , where $v_{\rm eff}$ is the effective cell volume and $V_{\rm core}$ the core volume in dimensions. This is the form that recurs across the document—it is the capacity identity verified in d=2,3,4,5 in Section §10 (U2), and it is literally the genome-condensate identity $N v_{\rm eff}\simeq\phi V_{\rm core}$ of Section §11 (E1). The geometric pillar is therefore not a two-dimensional accident; it is a single packing law, instantiated here in two dimensions. The fill fraction $\phi$ that enters the identity is, moreover, the same control knob whose marginal value the companion bundle measures for the substrate ( $\phi_{\rm jam}\approx0.633$ – 0.64 at $z\to2d$ ; see E1 of Section §11 and the numeric-adjacency non-claim of Section §14).

Why this is the geometric face of the thesis. The four numbers $(L_{\rm core},\phi,s,N_\star)$ are a complete coarse description of the arrangement of strong rotation inside the core: a length, how full the core is, how non-circular it is, and how many sub-cores pack into it. The identity says that the radius an independent observer would measure is not free: it is fixed by the arrangement. The single number $r_{\rm eff}$ is the arrangement, compressed. When a real measurement $r_{\rm meas}$ departs from $r_{\rm pred}$ , the departure is a quantitative statement that the flow resists compression to one length—i.e. that its arrangement is multiscale.

The cross-regime audit and what it reveals

Embedding (eq.) in four regimes turns the identity into a test of when "one core length" is a faithful summary. The arrangement is supplied by an Okubo–Weiss pipeline (the "A-pipeline"), while $r_{\rm meas}$ is measured by three independent segmenters (swirling strength $\lambda_{ci}$ , a local-circulation $\Gamma_2$ proxy, and a persistence-based method). The normalized residual

$\begin{equation} \mathrm{resid}_i=100\,\frac{|N_\star r_{{\rm meas},i}^2-\phi s L_{\rm core}^2|} {\phi s L_{\rm core}^2} \label{eq:resid} \end{equation}$ measures geometric breakdown sample by sample.

The qualitative picture (respecting the phenomena).

In coherent single-core regimes (tropical cyclone, ocean eddy) the prediction tracks the measurement to a few percent and the residuals stay small: a one-length description is adequate, so the arrangement is well summarized by a single scale. In dense two-dimensional turbulence the errors become order unity, the rank correlation between $r_{\rm pred}$ and $r_{\rm meas}$ turns negative, and the residuals develop heavy tails: no single core length is adequate, because the arrangement is genuinely multiscale (filaments and merging events). The multi-vortex surrogate sits in between, with a $\sqrt{N_\star}$ penalty that is visible and method-dependent. This ordering—coherent $\to$ multi-core $\to$ filamented—is exactly the ordering a configurational view predicts: the more the arrangement refuses to collapse to one cell, the more the one-length summary fails.

The cross-regime audit, swirling-strength ( $\lambda_{ci}$ ) segmenter (the strongest of the three; the shear-sensitive $\Gamma_2$ proxy is uniformly worse and is retained as a stress test). Median relative error of $r_{\rm pred}$ versus the measured radius, coverage within $\pm10\%/\pm20\%$ , and the $r_{\rm pred}$ – $r_{\rm meas}$ rank correlation. The ordering coherent (– $7\%$ , $\rho>0$ ) $\to$ multi-core ( $18\%$ ) $\to$ filamented ( $55\%$ , $\rho<0$ ) is the audit's robust conclusion. All reproduced by `verify_rotcore.py`.
Regime	rel. med.	$C_{10\%}/C_{20\%}$
Tropical cyclone	$4.1\%$	$93\%/100\%$
Ocean eddy	$6.9\%$	$71\%/98\%$
TC multi-vortex	$18\%$	$29\%/56\%$
2D turbulence	$55\%$	$10\%/19\%$

The single most telling entry is the negative rank correlation in two-dimensional turbulence. It is not noise: it concentrates in frames with elongated, strain-dominated filaments and vortex mergers, where the Okubo–Weiss mask fuses several interacting vortices. In other words, the geometric breakdown of the one-radius description is the spatial signature of the enstrophy cascade—the same fragmentation of coherent cores into filaments that, in Pillar §9, carries the forward enstrophy flux onto thin vorticity-gradient events. The geometric pillar and the dissipative pillar are thus reading one physical process from two angles: when the arrangement stops being a single cell (Pillar II's residual blows up), it is precisely because it is cascading enstrophy to small scales through filamentary events (Pillar §9's flux concentration). That two independent diagnostics locate the same transition is a coherence check on "form follows arrangement."

Pillar II — the geometric identity and its audit (verify_rotcore.py)

Recomputation from the sample-level table (4800 samples; 12 case $\times$ method groups of 400) against the published audit tables.

Identity check. $r_{\rm pred}=L_{\rm core}\sqrt{\phi s/N_\star}$ holds to a maximum relative error of $9.9\times10^{-16}$ across all 4800 samples. PASS (Equation (eq.) is satisfied to floating-point.)
Robust metrics. Median relative error, MAPE, MAE, NRMSE and Spearman $\rho$ for every group reproduce the published table to a worst-case relative mismatch of $2.8\times10^{-12}$ . PASS
Identity residuals. Median/mean of (eq.) per group reproduce the published residual table (e.g. 2D turbulence: median $\approx 85$ – $91\%$ , mean $\approx 127$ – $141\%$ ; coherent cores: single to low tens of percent). PASS

Run python verify_rotcore.py on the frozen sample table.

Reading the residuals as a configuration diagnostic.

The gap between the median and the mean residual is a measure of how heavy the tail is—i.e. how often the arrangement produces a frame in which a single length is badly wrong. Small gap (coherent cores) means the arrangement is reliably single-cell; large gap (2D turbulence under the shear-sensitive $\Gamma_2$ proxy) means rare but extreme multiscale events. This is the sense in which the residual is not a "model error" so much as a quantitative complexity parameter of the arrangement.

What Pillar II does not claim

The RCCI is a kinematic sum rule, not a dynamical law: it does not predict how a core evolves, and it introduces no constitutive relation. Its content is exactly that the effective length is fixed by the arrangement, and that departures are measurable. The audit's conclusions are stated as a regime/method ordering, which we expect to be robust to reasonable changes of error metric, rather than as absolute percentages.

Pillar II, in one line. Once the rotational arrangement $(L_{\rm core},\phi,s,N_\star)$ is fixed, the effective core length is fixed by an exact identity ([DERIVE]), reproduced to $10^{-15}$ ; the cross-regime audit (reproduced to $10^{-12}$ ) measures exactly when one length is enough and when the arrangement is irreducibly multiscale.