Lemma Note: \kappa_u As The Unique Exchange-Odd Cross-Coupling Slot

This note isolates and formalizes the symmetry-descent statement used in the two-branch transport dynamics: once the selected octonionic direction u is fixed (pointwise in u), the reduced dynamics admit a unique exchange-odd quadratic cross coupling between the two branches. The coefficient of that coupling is the scalar \kappa_u (which is separately tracked as K-invariant data descending from the parent reduction).

This is a pointwise-in-u statement: it does not decide whether u is kinematic or dynamical, and it does not fix the orientation/sign choice \kappa_u > 0.

Setup (Branch Space And The Exchange Involution)

Fix a unit imaginary u with u^2=-1 and write the selected complex line as \mathbf C_u = \mathrm{span}_{\mathbf R}\{1,u\}.

Treat branch amplitudes as a pair \((A,B)\in \mathbf C_u^2.\)

The reduced transport kernel uses an exchange-with-conjugation involution \(\mathcal C(A,B) := (\bar B, \bar A).\) In real coordinates, \(A=a_1+ua_2,\qquad B=b_1+ub_2,\) this acts as \((a_1,a_2,b_1,b_2)\mapsto (b_1,-b_2,a_1,-a_2).\)

Lemma (Uniqueness Of The Exchange-Odd Quadratic Invariant)

Consider real-valued quadratic functionals on \mathbf C_u^2. The exchange-odd condition is the constraint \(Q(\mathcal C(A,B))=-Q(A,B).\)

The transport coupling term is a cross quadratic (bilinear in the A and B coordinates). Write the most general real cross quadratic as \(Q_{\times}(A,B)=c_1\,a_1b_1+c_2\,a_1b_2+c_3\,a_2b_1+c_4\,a_2b_2.\)

Imposing exchange-oddness under (a_1,a_2,b_1,b_2)\mapsto (b_1,-b_2,a_1,-a_2) forces \(c_1=c_4=0,\qquad c_2=c_3,\) so the exchange-odd cross sector is one-dimensional: \(Q_{\times}(A,B)\propto a_1b_2+a_2b_1=\mathrm{Im}_u(AB).\)

Equivalently: up to an overall scalar, the only exchange-odd cross quadratic is \mathrm{Im}_u(AB).

Corollary (The Coupling Slot And Moment-Map Interpretation)

In the reduced Hamiltonian description, an exchange-odd coupling must therefore enter (at quadratic order) as \(H_{\mathrm{cpl}}(A,B) = -\kappa_u\,\mathrm{Im}_u(AB),\) for a scalar coefficient \kappa_u.

This identifies \kappa_u as the coefficient of the unique exchange-odd quadratic coupling slot on branch space. In symplectic terms: the quadratic \mathrm{Im}_u(AB) is the moment map for the corresponding exchange-mixing generator on the reduced branch space (up to normalization), and \kappa_u is its moment-map coordinate.

What This Does Not Do

This lemma does not derive:

  • why the preferred direction u is selected (kinematic vs dynamical bracket remains open);
  • why the physical readout orientation should pick the constructive class \kappa_u > 0 (this is the orientation axiom / selector gate);
  • any quantitative matching to Efimov/Faddeev kernels (that remains a separate proof obligation).

It is also worth recording a simple degeneracy: at the level of the quadratic Hamiltonian-Rayleigh generator, changing \kappa_u \mapsto -\kappa_u is equivalent to flipping the branch label B\mapsto -B, so purely linear stability data cannot fix the sign.