Stage 0: the parent $M_0$ 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]

Abstract

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 $M_0$ from which the next stage of the programme is to derive, rather than postulate, the interaction responsible for cohesive emergence. $M_0$ 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 $\lt$ conjugate to the conserved charge $\Ct$. Exact consequences of that state are then derived in $M_0$. (i) The multiplier is a $\U(N)$ angular velocity: integrating the bosonic momenta turns the ensemble weight $e^{-\Tr\lt\Ct}$ into a centrifugal configuration-space term $-\tfrac{1}{2\beta^2}\sum_A\Tr[\lt,Q_A]^2$, and the bosonic charge is $\langle\Ct_B\rangle=\beta^{-1}\langle\mathbb I(Q)\rangle\lt$ with $\mathbb I(Q)$ the inertia operator of P2 --- ``charge = inertia $\times$ angular velocity''; the fermionic charge is the fermions' $\mathfrak u(N)$ charge operator and enters the fermion determinant as a matrix-valued chemical potential. (ii) For $\lt=\lambda\ieff$ Adler's uniform-$\hbar$ assumption is exact by Schur's lemma, with $\hbar=\tfrac{\lambda}{\beta N}\langle\sum_A\Tr[\ieff,Q_A]^2\rangle$, the angular velocity times the moment of inertia about $\ieff$; the ensemble exists only for $2\lambda/\beta<m$, with or without the quartic. (iii) If $\ieff$ distinguishes two atoms, the strings between them are softened by exactly $(2\lambda/\beta)^2$ in frequency squared and the block-separating orientation is a saddle of the effective potential along the $\U(N)$ orbit at every separation and every $\lambda>0$: the state's first effect is on the orientation of the condensate relative to $\ieff$, and its effect on the shape is a Stage 1 calculation. On symmetry, the multiplier breaks $\U(N)$ to the stabiliser of $\lt$ and nothing on the octonionic index: in $M_0$ the GTD state cannot break $G_2$ explicitly at any order. The working hypothesis that $\ieff$ 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 $\SU(3)$ stabiliser in the module's $G_2$, but P2's action does not tie that $G_2$ to the octonionic index and no $\U(8n)$-invariant term can; as a multiplier for the symmetry that does act on the octonionic index it breaks $G_2$ only to $\U(2)$ or a torus. Colour $\SU(3)$ 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.