The cubic against the mirror: does the parity-conjugate pair of the Adler--Millard ensemble survive the cubic term of the octonionic matrix model? :Paper P6 of a ten-part series P1-P10

Paper P6 of a ten part series P1-P10 on Interaction, Cohesive Emergence, and Fundamental Constants in Generalised Trace Dynamics P6 — Mirror configurations and parity tests The Stage~1 paper of this series [1] found that, in the trap-plus-sea scheme of the constants paper [2], the state of trace dynamics selects a parity-conjugate pair --- two spin-$\tfrac12$ fuzzy spheres that are mirror images of each other, $Q^{(2)}=-Q^{(1)}$ --- as the ground state of the seven-matrix model for $\omega/m\gtrsim0.1$--$0.2$, and left as the first open question whether that pair survives the cubic term $(\ii\mu/3)\varphi_{ABC}\Tr Q_AQ_BQ_C$ of the companion paper \CP, which is odd under $Q\to-Q$ and which the companion paper's own geometric phase needs. This note answers it, from both ends. In the companion paper's model, where the spheres are cubic-driven, \tagP\ a sphere of the opposite handedness has no stationary radius at any angular velocity inside the bound, any $f=1$ two-sphere configuration must consist of blocks of opposite handedness and so forgoes the cubic energy of a same-handed pair, and at the benchmark that cost ($19.9$) exceeds the state's largest possible gain ($1.5$) by an order of magnitude; \tagD\ the mirror-pair start relaxes to the configuration the Stage~1 paper found for two atoms --- two vacuum-handed spheres with their frames as anti-parallel as handedness allows (overlap $-0.51$, planes $59^\circ$ apart, against the handedness-limited optimum $-\tfrac12$, $60^\circ$; on the spheres the anticommuting fraction is $(1-\text{overlap})/2=0.76$, the missing quarter being what handedness costs). In the trap-plus-sea scheme, where the spheres are sea-driven, \tagD\ the mirror pair is the ground state only at $\mu=0$, where it is an $\SO(7)$-degenerate orbit (a rigid mirror pair is cubic-blind on every plane); the cubic lifts the degeneracy at second order and selects, as $\mu\to0^+$, the pair on a plane on which the cubic vanishes, from which the two sphere planes tilt in opposite senses toward the direction of maximal associativity gain. The ground state is this explicit one-parameter family: both spheres keep equal radii, opposite centres and near-anticommutation with $\ieff$, and both acquire the vacuum handedness, $\varphi_b=\sin(3\,\text{tilt}/2)$, with the tilt linear in $\mu$ for $\mu/(gm)\lesssim0.6$ ($\varphi_b=0.23$, $0.43$, $0.58$ at tilts $9^\circ$, $17^\circ$, $24^\circ$ for $\tilde\mu/m_0=0.01$, $0.02$, $0.03$) and the energy gain exactly half the cubic energy, as linear response requires; the far end of the family, tilt $60^\circ$, $\varphi_b=1$, overlap $-\tfrac12$, is the two-atom configuration of the companion paper's model. The exact two-block form is lost at $\mu/(gm)\approx0.6$--$1$, the mirror placement of the centres at $1$--$2$, and the companion paper's cubic-driven sphere, in the state's orientation ($f=0.80$), is the ground state at the first scan point above its bosonic existence threshold $\mu/(gm)=\sqrt{8(1-4\hat\omega^2f)}$, far below the benchmark $\mu/(gm)=4$. The pair is therefore the $\mu\to0$ end of a family whose members at any $\mu>0$ have a common handedness, and the regime in which it is the ground state does not overlap the regime in which the companion paper's geometry exists. The $\mathbb Z_2$ the state selects is broken by any cubic, perturbatively by a small one; what survives of the state's preference, with the cubic, is the family itself --- two same-handed atoms at opposite centres, with frames tilted away from anti-parallel by as much as the cubic can pay for --- and the enlarged merger window of the Stage~1 paper.