Stage 0: the parent M₀ as a dynamical system: choices and their sources, exact consequences of the Adler--Millard ensemble, and the Stage 1 calculations with their kill criteria [Paper 5 of a ten-part series P1-P10]

Paper P5 of a ten part series P1-P10 on Interaction, Cohesive Emergence, and Fundamental Constants in Generalised Trace Dynamics P5 — Statistical state, conserved charge and inter-atom exchange This document writes down, choice by choice and with the source of each choice, the parent dynamical system M₀ from which the next stage of the programme is to derive, rather than postulate, the interaction responsible for cohesive emergence. M₀ is the seven-matrix model of the companion paper (P2) placed in the setting the programme's own papers prescribe: atoms as diagonal blocks of one master matrix with the off-diagonal blocks as the only coupling (the master extension of GTD, which the finite-measure test shows is an extension and not a rewriting), the harmonic trap identified --- on P1's chosen adjoint branch --- with the covariant-velocity term of the gauge audit, and the state taken to be the Adler--Millard canonical ensemble with its matrix multiplier < conjugate to the conserved charge . Exact consequences of that state are then derived in M₀. (i) The multiplier is a (N) angular velocity: integrating the bosonic momenta turns the ensemble weight e-< into a centrifugal configuration-space term -1/2β²∑A[<,QA]², and the bosonic charge is B=β⁻¹ I(Q)< with I(Q) the inertia operator of P2 --- ``charge = inertia × angular velocity''; the fermionic charge is the fermions' u(N) charge operator and enters the fermion determinant as a matrix-valued chemical potential. (ii) For <=λ Adler's uniform- assumption is exact by Schur's lemma, with =λ/β N∑A[,QA]², the angular velocity times the moment of inertia about ; the ensemble exists only for 2λ/β 0: the state's first effect is on the orientation of the condensate relative to , and its effect on the shape is a Stage 1 calculation. On symmetry, the multiplier breaks (N) to the stabiliser of < and nothing on the octonionic index: in M₀ the GTD state cannot break G₂ explicitly at any order. The working hypothesis that is an imaginary octonionic unit is examined under both of its readings: as the Adler--Millard multiplier on an octonionic module ($N=8n$) it is admissible and has (3) stabiliser in the module's G₂, but P2's action does not tie that G₂ to the octonionic index and no (8n)-invariant term can; as a multiplier for the symmetry that does act on the octonionic index it breaks G₂ only to (2) or a torus. Colour (3) is also the stabiliser of a vector-type condensate, the collinear commuting configuration, and the centrifugal term prefers centrally symmetric condensates over the regular simplex that the fermion sea of P3 selects, so the colour-versus-geometry competition in the state is two-sided. Stage 1 is four calculations on existing machinery, each with a kill criterion. Five choices remain the author's; defaults are stated.