Octonionic matrix dynamics and conditional cohesive emergence: exact vacua, competing geometries and finite-time records [Paper P2 of a ten-part series P1-P10]

Paper P2 of a ten part series P1-P10 on Interaction, Cohesive Emergence, and Fundamental Constants in Generalised Trace Dynamics P2 — Collective matrix dynamics and geometric ordering This paper works out, as completely as we can, one specific matrix model in which geometry, matter excitations and records of the geometry all come from the same action. The model is a G₂-invariant matrix mechanics for seven Hermitian N× N matrices with an octonionic cubic term, a commutator quartic and a positive mass term, together with fermions coupled through octonionic Clifford multiplication. We present it as a benchmark for the spacetime--matter programme, not as a derivation of that programme from first principles, and we are explicit about which of its ingredients are assumptions. Within the model we prove, using an octonionic completion of the square, that the configuration $Q=0$ is the unique classical minimum for t=μ²/(g²m²)<9, and that at $t=9$ nonzero global minima, among them a fuzzy sphere, become degenerate with it. We derive the complete bosonic and fermionic fluctuation spectra of the sphere, and we analyse a known seven-direction competitor configuration, including an exact certificate for its full Hessian. When the second quartic invariant allowed by G₂ is added, a second completion of the square gives a global lower bound on the energy; sphere blocks and seven-direction blocks saturate it on different coefficient surfaces, and an explicit energy gap shows that in an open region of coupling space no configuration built from spheres is a global minimum. The sphere's own stable modes provide a finite-time information protocol for two prescribed values of the radius, with an exact overlap formula, an explicit commuting readout, several independent spectral record fragments and a coloured force kernel; a transverse-mode version of the protocol runs on a sphere that is a proven global minimum of the same positive-second-quartic action. We estimate the preparation cost and the back-reaction, and show how the result depends on the ultraviolet tower of modes. A susceptibility bound valid for every N≥3 shows that, at one loop, the fermion sea alone cannot drive a continuous instability of the light bidoublet modes once the radius is allowed to respond; the bosonic self-energy of those modes, computed numerically on the same normal slice, makes the complete one-loop coefficient positive for N≤8 (the tadpole terms alone are marginally negative for N≥6 at the example point), while the bosonic model alone would have the opposite sign. Harmonic comparisons at the coexistence point show how quantum fluctuations can lift a classical degeneracy between geometries; there the potential is the square of a gradient, the bosonic Hessian of the sphere is the square of a first-order operator built from the model's own Dirac operator, and the lifting is a property of the chosen fermion content, which the supersymmetric completion of the same bosonic action would not share. The model is the enlarged master-matrix theory of the programme's relational proposal: we show that the charge-induced interaction of that proposal is contained in this model, exactly as its single-matrix sector and in a constrained form on its commuting configurations; that the strings between atoms carry and exchange Adler--Millard charge with the atoms; and that the gapped record bath fails the zero-frequency test of the programme's own classicality audit. None of this derives the effective quantum ensemble, autonomous record formation, physical spacetime or the Standard-Model constants; an analysis of the free parameters and of the Albert-algebra spectrum says exactly what a microscopic calculation would have to supply. The exposition is aimed at a graduate student, with the calculations carried out step by step, proofs given, and reproducibility programs supplied.