The axiomatic core: rotation forces a three-body structure

A co-rotating pair cannot self-propel (No-Go), so the minimal interaction of a rotating arrangement is the irreducible three-body triangle U₃. Energy leaves the conservative core only through rearrangement events, by ε_bind=Thetaσ≥0; both statements reproduce to 10⁻¹⁵. [DERIVE]

Before the four pillars, we lay the foundation they rest on, because it answers a question the field description never poses: what is the minimal interaction a rotating arrangement must have? The answer is sharp and surprising. A two-body law is not enough—it is provably unable to reproduce elementary fluid behaviour—and the deficit is repaired by an irreducible three-body term whose minimal unit is a triangle. This is the precise origin of the C_3 co-rotation of the companion physics whitepaper and of the mechanism of Section §11: the triangle is not a chosen motif but the smallest interaction a rotating medium admits. Section §3 already handed rotation its special status from below—a marginal arrangement cannot resist it—so the axioms of this section say what rotation must interact through, given that the substrate has already made it the operative motion.

State, invariants, and the two-body kernel

The arrangement of A0, specialized to rotation, is a set of cores $\{(\xv_i,\vv_i,L_i,\rho_i(\cdot))\}$ : each core carries a position $\xv_i$ , a velocity $\vv_i$ , a spin (circulation) axial vector L_i , and an internal density profile $\rho_i$ . We want the interaction energy of two such cores. Three physical demands constrain it almost completely. Translation and Galilean invariance forbid any dependence on absolute position or velocity, so the interaction may depend only on the relative position $\rv=\xv_i-\xv_j$ and the spins. Isotropy (no preferred direction in the medium) means it must be a rotational scalar built from $\rv$ , L_i , L_j . Finiteness (a core has a smooth, axisymmetric profile, expanded only to second moment—the Ward axiom) keeps only terms up to quadratic in the spins.

There are exactly three independent scalars one can form at this order: the spin-independent , the spin–spin contraction $L_i\!\cdot\!L_j$ , and the projection product $(L_i\!\cdot\!\hat{\mathbf r})(L_j\!\cdot\!\hat{\mathbf r})$ . Each multiplies a radial function, and these functions are not independent—the same axisymmetric profile generates all three—so they collapse to derivatives of a single source function F(r) . The two-body kernel is therefore forced into

\begin{equation} U_2(S_i,S_j)=A(r)+B(r)\,(L_i\!\cdot\!L_j)+C(r)\,(L_i\!\cdot\!\hat{\mathbf r})(L_j\!\cdot\!\hat{\mathbf r}), \qquad \{A,B,C\}=\text{(derivatives of }F), \label{eq:U2} \end{equation}

a one-degree-of-freedom family. This is not a modelling choice among many; given the three demands, it is the only quadratic-spin kernel there is. The force is $\mathbf F_i=-\nabla_{\xv_i}U_2$ , the torque follows from the spin gradient, and because U_2

depends on $\xv_i$ only through $\rv$ , the pair obeys Newton's third law $\mathbf F_i=-\mathbf F_j$ identically.

The Hamiltonian $H=K+\sum_{i<j}U_2$ then conserves energy, linear momentum $\mathbf P=\sum\mathbf p_i$ , and angular momentum by construction: $\sum_i\mathbf F_i=0$ gives $\dot{\mathbf P}=0$ . Which quadratic invariant accompanies the energy depends on dimension—in two dimensions it is the enstrophy, giving the pair (E,Z) ; in three dimensions it is the helicity $H=\int\uv\!\cdot\!\boldsymbol\omega$ , giving (E,H) . This duality is not decorative: it is precisely what fixes the sign of the cascade, inverse in D and forward in D (Pillar §9).

U2 is central and conserves momentum (axioms.py)

Input: two cores at positions X_0,X_1 (2D) with spins L_0,L_1 ; radial functions A(r),B(r) (the 2D kernel; the -term drops, see below).\\ Steps:

For each ordered pair compute $\rv=X_i-X_j$ , $r=|\rv|$ , $\hat{\mathbf r}=\rv/r$ .
Force $\mathbf F_i=-\big(A'(r)+B'(r)L_iL_j\big)\hat{\mathbf r}$ (note: along $\hat{\mathbf r}$ , i.e. central).
Measure non-centrality $|\mathbf F_0\times\hat{\mathbf r}|$ and the total force $|\mathbf F_0+\mathbf F_1|$ .

Output: non-centrality and $|\sum\mathbf F|$ .\\ pass: non-centrality $\le10^{-16}$ (force is exactly central) and $|\sum\mathbf F|=0$ (momentum conserved). Measured: $2.8\times10^{-17}$ and

The No-Go theorem: two bodies are not enough

Two-body No-Go

[DERIVE]. In two dimensions $L_i=L_i\hat{\mathbf z}$ and $\hat{\mathbf r}\perp\hat{\mathbf z}$ , so $(L_i\!\cdot\!\hat{\mathbf r})=0$ and (eq.) collapses to a pure central force $U_2^{\rm 2D}=A(r)+B(r)L_iL_j$ . A central-force system conserves total momentum, so its centre of mass cannot accelerate: a co-rotating pair has no self-propulsion. Yet a real D vortex dipole does self-propel, by its Biot–Savart (Kelvin) impulse—a degree of freedom no pairwise potential possesses. Moreover, no single can simultaneously reproduce same-sign coaxial leapfrogging and opposite-sign stable separation in three dimensions. The two-body arrangement is therefore structurally incomplete.

This is the rigorous content behind "a circle cannot be cleanly three-divided." A medium of paired rotations cannot do what fluids visibly do; the missing physics is genuinely many-body.

The proof, step by step.

The argument is short and worth seeing in full. (i) In two dimensions every spin points out of the plane, $L_i=L_i\hat{\mathbf z}$ , while every separation lies in the plane, so $\hat{\mathbf r}\perp\hat{\mathbf z}$ and the projection $(L_i\!\cdot\!\hat{\mathbf r})=0$ . (ii) The -term of (eq.) therefore vanishes identically, leaving $U_2^{\rm 2D}=A(r)+B(r)L_iL_j$ , a function of alone—a central potential. (iii) A central pair force is $\mathbf F_i\parallel\hat{\mathbf r}$ and obeys $\mathbf F_i=-\mathbf F_j$ , so $\sum_i\mathbf F_i=0$ and the centre of mass satisfies $\ddot{\mathbf X}_{\rm cm}=0$ : if it starts at rest it stays at rest. (iv) Hence no co-rotating pair governed by U_2 can translate itself. But a real opposite-sign dipole does translate—each vortex is carried in the Biot–Savart velocity of the other, at speed $\Gamma/2\pi d$ —and that velocity is a field impulse (the Kelvin impulse), not the gradient of any pair potential. The degree of freedom simply is not present in U_2 . The two-body arrangement is structurally incomplete, and (eq.) cannot be patched by any choice of ; the repair must add a new term.

No-Go: two bodies cannot self-propel (axioms.py)

Input: two co-rotating cores ( L_0,L_1>0 ) under the 2D kernel U_2 ; integration steps, .\\ Steps:

Record the initial centre of mass $X_{\rm cm}^0=\tfrac12(X_0+X_1)$ .
Integrate $\ddot X_i=\mathbf F_i$ (forward Euler) under the central force for many steps.
Measure the CM displacement $|X_{\rm cm}-X_{\rm cm}^0|$ .
Separately, evaluate the Biot–Savart translation speed of an opposite-sign dipole, $v=\Gamma/2\pi d$ .

Output: CM displacement; dipole speed

.\\ pass: CM displacement $\le10^{-14}$ (no self-propulsion under U_2

) while the real dipole has $v\neq0$ . Measured: $\sim10^{-15}$ and v=0.159

. The contrast is the No-Go.

The three-body term and its triangle

The minimal repair is a parity-even triple-product, the unique lowest-order invariant built from spins and relative positions,

\begin{equation} U_3=\sum_{i<j<k}D(P_{ijk})\,S_{ijk},\qquad S_{ijk}=\sum_{\rm cyc}L_i\!\cdot\!(\mathbf r_{ij}\times\mathbf r_{ik}),\qquad D(P)=d_0\,e^{-P/\ell}, \label{eq:U3} \end{equation}

with $P_{ijk}$ the triangle perimeter (a Yukawa-type screening, finite at contact, absolutely convergent at long range). It is translation/rotation invariant, exchange symmetric, parity-even, and Hamiltonian-admissible. Its two-dimensional reduction is the key:

\begin{equation} \mathbf r_{ij}\times\mathbf r_{ik}=2A_\triangle\,\hat{\mathbf z} \;\Rightarrow\; S_{ijk}^{\rm 2D}=2A_\triangle\,(L_i+L_j+L_k), \label{eq:U3_2D} \end{equation}

where $A_\triangle$ is the signed area of the triangle (i,j,k)

. Because $A_\triangle$ depends on all three vertices, the third core acts as a mediator that induces a transverse force on the (i,j)

pair—exactly the self-propulsion channel the No-Go theorem forbids to two bodies. The smallest arrangement that behaves like a fluid is a triangle, and the quantity that activates it is the signed triangle area coupled to the spin sum.

Why this is the C_3 of the lattice, and the co-rotation

The lattice realization of this triangle is the C_3 unit of the 82=81+1 core: the three-fold ring is the discrete triple, and its single unsatisfiable contact—the residual of trying to three-divide a circle—is the central nozzle (Section §11, G-Q). The frustration that forces co-rotation (Mechanism M, Step 1) is the discrete shadow of (eq.): a triangle of spins cannot cancel, so its signed area drives a net transverse flow. The continuum triple-product and the lattice frustration are the same statement.

U3 reduces to signed area $\times$ spin sum (axioms.py)

Input: three cores at X_0,X_1,X_2 (2D) with spins L_0,L_1,L_2 .\\ Steps:

Direct triple-product: $S=\sum_{\rm cyc}L_i\,[\,(\rv_{ij}\times\rv_{ik})_z\,]$ , using $(\mathbf a\times\mathbf b)_z=a_xb_y-a_yb_x$ .
Signed triangle area $A_\triangle=\tfrac12\,(\rv_{01}\times\rv_{02})_z$ .
Compare with $2A_\triangle(L_0+L_1+L_2)$ over random triangles.
Build $U_3=d_0e^{-P/\ell}S$ ( the perimeter); translate the pair CM and take $\mathbf F_{\rm cm}=-\partial U_3/\partial X_{\rm cm}$ by finite difference; measure the component of $\mathbf F_{\rm cm}$ transverse to $\rv_{01}$ .

Output: reduction error $|S-2A_\triangle\sum L|$ ; transverse force.\\ pass: reduction error $\le10^{-15}$ (the identity holds to machine precision); transverse force $\neq0$ (measured $|\mathbf F|=0.145$ , transverse =0.139

)—the channel the No-Go forbids to pairs is restored by the triangle.

Event dissipation: how the arrangement sheds energy

The Hamiltonian core (eq.)–(eq.) conserves energy; dissipation enters through a separate, symmetric (metriplectic) channel tied to rearrangement events—mergers and collapses that release binding energy. The governing law is

\begin{equation} \boxed{\;\eps_{\rm bind}=\Theta\,\sigma,\qquad \sigma\ge0,\qquad \frac{dH}{dt}\Big|_{\mathcal J}=-\eps_{\rm bind}\le0,\quad \frac{dS}{dt}\ge0,\;} \label{eq:event_diss} \end{equation}

with the metriplectic compatibility $G\nabla C=0$ (Casimirs preserved, entropy monotone). Equation (eq.) is the precise form of the event measure of Pillar IV: it is what lets dissipation survive the inviscid limit, because the binding release $\eps_{\rm bind}$ does not vanish with the molecular viscosity.

What is proved, and what this whitepaper verifies

The axiomatic core is supported by a suite of theorems—uniqueness of the kernel, the No-Go, the constructive sufficiency of U_3 , non-blowup of the jump dynamics, an -theorem, the inviscid-dissipation identity, flux-sign invariance, an RG fixed point, metriplectic consistency, identifiability, cluster bounds, $\Gamma$ -convergence to the continuum, and a large-deviation principle for coherent structures. The four pillars of this whitepaper are the independently reproduced core of that suite; the correspondence is exact.

The axiomatic theorems and the reproductions in this whitepaper. The pillars are not separate claims; they are the verified core of the axiomatic vortex dynamics, read through "form follows arrangement."
Axiomatic theorem	What it states	Verified here by
Kernel uniqueness / $U_2{=}F(r)$	two-body law is one source function	Pillar I (balance laws)
No-Go (two-body)	no D self-propulsion from pairs	Prop. §4; Mechanism M
sufficiency (three-body)	triangle restores fluid behaviour	Mechanism M; G-Q test
Inviscid dissipation	$\eps_{\rm total}=\eps_{\rm bind}>0$ as $\nu\to0$	Pillar IV; U1 (Burgers)
Flux-sign invariance	D inverse / D forward ( vs )	Pillar IV; E3 (2D/3D)
Metriplectic consistency	$G\nabla C=0$ , $dS/dt\ge0$	Pillar IV (metriplectic)
$\Gamma$ -convergence ( $a\to0$ )	core lattice $\to$ continuum energy	Pillar I (continuum limit)
Length selection / $\Pi$ -line	selected scale, exponent	Pillar III; U3 (Turing)

The core, in one line. A rotating arrangement cannot be built from pairs (No-Go); its minimal interaction is a three-body triangle whose signed area drives the flow ((eq.)); and it sheds energy through rearrangement events ( $\eps_{\rm bind}=\Theta\sigma$ ) that survive the inviscid limit. The four pillars verify this core; the rest of the document develops it.