Cross-scale extensibility: one arrangement law from genomes to vortices

One arrangement law spans scales: N v_eff=φ V_core from genomes to vortices, droplet selection at k_★ a=0.697, a plane breathes while a sphere closes—and Mechanism M now starts at Step 0, rotation unjams the margin. The quantum bridges G-Q and G-S remain graded gates with carried-out falsification legs. [GATE]

Section §10 showed the mechanisms reach across domains at a fixed scale. This section asks whether they reach across scales—from biomolecular condensates to vortex cores, and conjecturally to the quantum core. The strongest evidence is not an analogy we draw but a convergence we find: an independent line of the author's work on genomes arrived at the same capacity identity, the same two control knobs, and the same selection principle. We keep the discipline of Principle §2: the demonstrated unifications are graded [DERIVE], the cross-scale bridges are graded [GATE], and framework-internal integer coincidences are flagged as such, not asserted as physics.

The same capacity identity in genomes and in vortices

A two-layer interpretation of DNA models a chromatin region as a phase-separated condensate with a firm interface—a "stiffness shell." Its packing obeys, in that work's own notation,

\begin{equation} N\,v_{\rm eff}\;\approx\;\phi\,V_{\rm core}, \label{eq:dna_capacity} \end{equation}

with

units of effective volume $v_{\rm eff}$ at fill fraction $\phi$ in a core of volume $V_{\rm core}$ . Writing $v_{\rm eff}=V_d r_{\rm eff}^d$ and $V_{\rm core}=s\,V_d L_{\rm core}^d$ , (eq.) becomes $N r_{\rm eff}^d=\phi s L_{\rm core}^d$ —exactly the rotational core capacity identity of Pillar II (Eq. (eq.)), in d=3

. This is not a resemblance; it is the same equation, reached independently from sequence data.

Extensibility E1 — the DNA condensate identity is the fluid RCCI (extensibility.py)

Synthetic d{=}3 packing: (eq.) and the RCCI form $N r_{\rm eff}^3=\phi s L_{\rm core}^3$ are both satisfied to residual (same equation). The condensate "preferred size" is the minimum-residual-stress fill: modeling over-fill as steric clash and under-fill as interfacial/void cost, the minimum sits at $\phi^\star\approx0.64$ —the random-close-packing (jamming) point. PASS\\ $\Rightarrow$ one arrangement law governs biomolecular condensates and vortex cores; "a droplet has a preferred size" and "a vortex core has a fixed radius" are the same statement.

The condensate's two knobs are the interfacial tension $\gamma$ (the firmness of the shell, set by composition) and the fill fraction $\phi$ (set by how much material is packed). These are exactly the $\phi$ and shape factor of the RCCI, with $\gamma$ supplying the "firmness" that the fluid audit reads as the cost of a misshapen interface. Jamming and fluid are then two phases of one arrangement: the preferred fill sits at the jamming point $\phi_c\approx0.64$ , below which the medium flows and above which it clashes. This is the precise sense in which jamming theory and fluid dynamics are not different subjects. The jamming point invoked here is, moreover, not folklore: the companion bundle measures it for the substrate ( $\phi_{\rm jam}=0.633$ at the isostatic onset $z\to2d$ , with no physical constant in the input; concept DOI 10.5281/zenodo.17932566). Inheriting the companion's own discipline, we flag explicitly that the numeral adjacency of $\phi_{\rm jam}\approx0.633$ and $2/\pi=0.6366$ is not claimed as an identity, a cause, or evidence anywhere in this document (Section §14).

Droplet formation is length selection

The condensate picture also closes the loop with Pillar III. A droplet does not merely have a preferred size statically; a liquid jet selects a droplet size dynamically, by the Rayleigh–Plateau instability, whose dispersion is set by the interfacial tension $\gamma$ .

Extensibility E2 — droplet size is selected by a dispersion maximum (extensibility.py)

Rayleigh's inviscid dispersion for a liquid cylinder of radius , $\omega^2\propto x(1-x^2)I_1(x)/I_0(x)$ with x=ka , peaks at $x_\star=0.6970$ (classical 0.697 ), giving droplet spacing $\lambda_\star=9.014\,a$ (classical $9.01\,a$ ). PASS\\ $\Rightarrow$ "how a droplet forms" is the binding-versus-penalty length selection of Pillar III, with the interfacial tension $\gamma$ playing the binding role. Selection, capacity, and condensate interface are one mechanism.

Why a planar vortex breathes but a spherical core closes

The author's framework distinguishes the macroscopic planar vortex (a typhoon, which takes air in and pushes it out) from a spherically symmetric core (which closes on itself). This distinction is not interpretive; it follows from incompressible continuity alone.

Extensibility E3 — through-flow geometry, 2D vs 3D (extensibility.py)

Planar (cylindrical, axisymmetric): $\tfrac1r\partial_r(r u_r)+\partial_z w=0$ . A radial inflow u_r=-A r forces an axial updraft w=2Az —the eye updraft / Ekman pumping. The perpendicular -axis is the exhaust route, so the vortex sustains a steady intake-and-exhaust. Spherical (full symmetry): $\tfrac{1}{R^2}\partial_R(R^2 u_r)=0\Rightarrow u_r\propto R^{-2}$ , a pure source/sink with no perpendicular axis; a steady inflow has nowhere to go and must close. PASS\\ $\Rightarrow$ the same rotating-inflow arrangement is a sustained through-flow in two dimensions and self-closing in three—exactly the stated typhoon/quantum-core contrast.

The co-rotation mechanism: how small rotations build a large one

Before the conjectural bridges, we isolate the mechanism that the C_3 lattice structure actually encodes—and that the bridges rely on; we refer to this chain as Mechanism M. It is not a claim about how many vortices a typhoon contains; it is the chain by which small rotations are forced to co-rotate, and how that forced co-rotation builds a large-scale rotation with inflow and outflow. Each step is demonstrable.

Step 0: the margin makes rotation an unjamming switch.

Before frustration can do anything, rotation must be able to move material at all. Section §3 supplies exactly this, and it is inherited geometry, not a postulate: a rigid rotation costs a packing nothing (every bond extension vanishes identically, $\hat{\mathbf n}\cdot(\hat{\mathbf z}\times\rv)=0$ ), while any differential rotation is a shear—and at the margin the shear reserve is the vanishing overlap distribution, so the rotation breaks contacts and unjams. Rotation is therefore the fluidization switch of the substrate, and the directed inflow the steps below pump has a geometric origin: the unjammed material moves, and the one open direction is inward along the rotation's own demand.

Mechanism M, Step 0 — rotation unjams the margin (unjam_inflow.py)

Rigid rotation is free: maximal bond extension per unit rotation under $\mathbf u=\theta\,\hat{\mathbf z}\times\rv$ is $2.8\times10^{-17}$ —zero to machine precision. PASS
The reserve dies at the margin: the contact-breaking shear strain falls with pressure across the ensemble—quantile $q_{05}$ ratio $5.9\times$ between the densest and the most marginal packings, $\mathrm{corr}(\log q_{05},\log P)=0.96$ , $\mathrm{corr}(\log\mathrm{med},\log P)=0.99$ . (The very first break is an extreme-value statistic and is reported per packing for honesty; the physical reserve is the overlap distribution, and real unjamming is an avalanche of breaks, not a single bond.) PASS

At $P\to0$ any rotational shear exceeds the reserve: rotation unjams.

Step 1: frustration forces co-rotation.

Meshing rotors prefer to counter-rotate, like gears. A globally consistent counter-rotation is exactly a proper two-colouring of the contact graph, which exists if and only if the graph is bipartite—no odd cycles. The triangle, the minimal C_3 , is an odd cycle and is not two-colourable: counter-rotation is frustrated, so the rotors are forced to co-rotate, and one contact is left unsatisfiable. That unsatisfiable residual is the "" nozzle of the 82=81+1 core. This is the mechanical content of "a circle cannot be cleanly three-divided."

Step 2: co-rotation builds a larger rotation.

Once small rotations are forced to share a sign, their circulations add ( $\sum_k\gamma_k$ ) rather than cancel, and like-sign vortices merge: two co-rotating cores fuse into a single larger core, whereas opposite-sign cores do not. The forced co-rotation therefore grows a large-scale rotation out of many small ones.

Step 3: the large rotation forces a through-flow.

A coherent rotation $\Omega$ over a boundary cannot stay purely azimuthal: in the friction (Ekman) layer the rotation is reduced, the radial pressure balance is broken, and a radial inflow develops; by continuity (Section §11, E3) that inflow turns into an axial outflow. The inflow transport scales with the Ekman depth $\sqrt{\nu/\Omega}$ . This is the eye updraft of a cyclone and the tea-leaf paradox: rotation pumps.

Mechanism M — forced co-rotation $\to$ large rotation $\to$ through-flow (corotation.py)

Frustration (graph two-colouring). Square lattice: frustrated edges (bipartite) $\Rightarrow$ counter-rotation, net circulation . Triangular lattice: frustrated edges (not bipartite) $\Rightarrow$ co-rotation forced, net circulation $\neq0$ . The triangle has exactly one frustrated contact ( the nozzle). PASS
Merger (2D Navier–Stokes). Two strongly overlapping equal vortices: co-rotating $\to$ one connected core (merged); opposite-sign $\to$ two cores. PASS
Ekman pumping. Radial inflow transport $\propto\Omega^{-1/2}$ (measured slope ) and $\propto\nu^{+1/2}$ (measured ): rotation forces an axial through-flow. PASS

The chain is scale-free: the same forced-co-rotation-with-nozzle operates on the quantum lattice (the

core) and in macroscopic rotating flows.

The remaining connections the framework proposes are genuine conjectures. We state them as such, with falsification routes, and we do not present their numerical coincidences as established.

[GATE] G-Q: the quantum core as a closed planar vortex

The companion physics whitepaper derives (does not posit) a lattice core of count 82=81+1 : a circle cannot be cleanly three-divided, so the only lattice symmetry compatible with a rotational quantum is the C_3 about a body diagonal, and the three-division packing 3^4=81 (equivalently the lattice-shell count $\#\{R^2\le6\}=81$ ) is forced, with the next shell R^2=7 empty by Legendre's three-square theorem. The remaining "" is the single irreducible central residual that does not three-divide—read physically as the inflow nozzle (the same "" as in the shell 7=1+6 ). The conjecture is not that a typhoon contains three vortices; it is that the same mechanism of Section §11 operates here—small rotations frustrated into forced co-rotation, building a large rotation with a nozzle that drives through-flow. Status. The lattice combinatorics ( 82=81+1 , R^2=7 empty) are a genuine theorem (Legendre), forced within the framework's C_3 rectification lock—not a numerical coincidence; the rotational-unjamming origin of the nozzle's directed inflow is inherited as derived geometry (Section §3, Step 0); the co-rotation mechanism itself (Steps 0–3) is demonstrated (Mechanism M); and the dimensional half (a sphere closes, a plane breathes) is derived (E3). What remains [GATE] is the cross-scale identification of the quantum lattice's nozzle dynamics with the macroscopic pumped vortex. Falsification route. Build the lattice-with-inflow model and test whether the nozzle term closes in d=3 and sustains a through-flow in d=2 as the continuity geometry (E3) requires.

We carried out that falsification test explicitly, and the mechanism survived it.

G-Q falsification test — the 81{+}1 lattice-with-inflow (lattice_inflow.py)

Build the proton-core lattice: $\#\{R^2\le6\}=81=3^4$ co-rotating cells plus one nozzle ( =82 ).

Forced co-rotation adds. Biot–Savart over the planar slice ( cells): co-rotating net circulation (adds to one large rotation); the counter-rotating control nearly cancels. PASS
(closed sphere): the net radial flux of any regular divergence-free flow through an enclosing sphere is (Gauss regularity, no external source), so the nozzle closes—no steady through-flow. PASS
(open axis): the rotation axis is a through-path; the Ekman radial inflow feeds a finite axial through-flux —the nozzle is sustained. PASS

The

nozzle sustains a through-flow in two dimensions and closes in three, exactly as E3 requires. Had it done either the reverse, G-Q would be falsified; it did not. This upgrades the mechanism (forced co-rotation $\to$ large rotation $\to$ dimension-dependent through-flow) to demonstrated; only the quantum-dynamical identification remains [GATE].

The self-limited grinder (inherited).

The companion bundle also runs the mechanism forward as molecular dynamics: starting from 3000 spheres, counter-rotation plus contraction plus incompressibility self-limits to a core of $\sim79$ – cells whose size is set by the rotation itself, $R_{\rm rms}\propto1/\Omega$ —rotation creating the length by unjamming the substrate, i.e. Step 0 run to steady state (concept DOI 10.5281/zenodo.17932566). The companion's own honesty note is inherited at full strength and must not be under-read: the MD core is amorphous ( $z_{\rm core}\approx8.9$ ), so the simulation confirms the mechanism—self-limitation to $\mathcal O(82)$ cells with $R\propto1/\Omega$ —not the lattice type; the exact at is the crystalline count. The lattice-type identification therefore remains inside the [GATE] above, exactly where it was.

[GATE] G-S: rigid-shell birth and the selected quantum length

The companion physics whitepaper's Section "stiffness selects size; stiffness forces the radius" treats the rotation length $D=\ell_{\rm rot}$ as a selected length, and triangulates it three independent ways,

\[ D \;=\; 2\lambda_{C,e}\;=\;6\pi^6 r_p\;=\;\frac{2\pi\lambda}{A}\;\approx\;4.854\ \mathrm{pm}, \]

the third route being a jamming/stiffness selection. The selection law of Pillar III, $L_\star=2\pi\sqrt{2\sigma/\varepsilon}$ , reads $\sigma$ as a rigidity ( $\sim c^2$ , the full-jamming elastic speed $c^2=K/\rho_{\rm eff}$ ) and $\varepsilon$ as an elasticity ( $\sim1$ ); the conjecture is that the rigid-shell length so selected is exactly this

, and that the same selection underlies the companion JFM length-selection line. Status (quantified; three independent legs). The anchors stand: the two independent empirical/geometric routes $2\lambda_{C,e}=4.8526$ pm (electron Compton) and $6\pi^6 r_p=4.8523$ pm (proton radius) agree to $\sim6\times10^{-3}\%$ ; the jamming-selected route lands at 4.8542

pm, so all three agree to $0.039\%$ —a stiffness-selected length pinned this tightly is the quantitative content of "rigid-shell birth"; and the

-fold law (also forced) gives $\nu_3=3\pi^4$ and $m_p/m_e=6\pi^5$ (

ppm). Numbers cited, not re-derived. What this document adds is that the dimensionless statement "stiffness selects a definite size" now rests on three independent legs: (i) self-averaging—the relative spread of $L_\star\propto\sqrt{\sigma/\varepsilon}$ narrows as the network stiffens (rigid_shell.py, detailed below); (ii) attractor—the stiffness-versus-inflow balance has a unique, globally stable fixed point with $F'(x^\star)=-(\pi/2)^5$ (Section §8; unjam_inflow.py); (iii) rotation sets the size—the companion's molecular-dynamics grinder self-limits with $R_{\rm rms}\propto1/\Omega$ (inherited; "the self-limited grinder," Section §11). What remains [GATE]\ is whether the fluid selection law and the jamming selection are the literally same mechanism, and the specific calibration of

. Leg (i) in detail (carried out, rigid_shell.py). The dimensionless content is confirmed: in random spring networks the modulus self-averages as the network stiffens, so the relative spread of the selected length $L_\star\propto\sqrt{\sigma/\varepsilon}$ narrows ( $3.4\%\to1.8\%$ as the coordination rises $z{=}4.5\to15$ )—the quantitative meaning of rigid-shell birth, a length that sharpens as rigidity grows. The honest boundary. The selection law and the JFM exponent are dimensionless: they fix the scaling ( 1/2

) and the sharpening, not the absolute scale. Reaching the absolute 4.854

pm requires the dimensional anchor $D=2\pi\lambda/A$ , the conceptually difficult dimensionless-to-dimensional mapping of the physics whitepaper; it is referenced, not re-derived here. So this gate is now split: the dimensionless selection and sharpening are [DERIVE]; the dimensional anchoring is the residual [GATE].

The extensibility map

Extensibility of "form follows arrangement." The three mechanisms (capacity identity, interfacial-tension/fill selection, event dissipation) are scale- and substrate-free; the DNA line's independent convergence on the same capacity identity is the evidence. Rows marked [DERIVE] are demonstrated; rows marked [GATE] are conjectural bridges to the quantum scale with stated falsification routes.
Scale	Arrangement object	What fixes form	Status
Quantum core	-lattice, rotationinflow	sphere closes / annihilates	[GATE] + E3
Particle mass	rigid-shell birth ()	stiff-limit selection	[GATE] + Pillar III
Vortex / typhoon	core arrangement $(L,\phi,s,N_\star)$	RCCI; plane breathes	[DERIVE] (II, E3)
Turbulent dissipation	rearrangement events	defect measure / Onsager	[DERIVE] (IV, U1)
Droplet / condensate	interfacial tension $\gamma$ , fill $\phi$	min residual stress; Plateau	[DERIVE] (E1, E2)
Jamming / soft matter	fill $\phi\to\phi_c$	rigidity onset (fluid $\leftrightarrow$ solid)	[DERIVE] (E1)
Genome / DNA	stiffness shell condensate	same identity $Nv=\phi V$	[DERIVE] (E1)

What the extensibility test establishes. The reach of the thesis is not rhetorical. The capacity identity, the interfacial-tension/fill selection, and the event-dissipation channel recur, with the same equations, in condensates, droplets, jammed packings, vortex cores, and genomes—the last reached independently, which is the strongest evidence that the structure is real rather than imposed. The bridges to the quantum core remain conjectures with explicit falsification routes. The honest summary: the demonstrated reach is from genomes to vortices through one arrangement law; the frontier is whether the same law fixes the quantum length.