The First Loop: The Remnant Fixed Point, the Division Period, and the Self-Tuning Lineage

Abstract

The preceding article of ACF on the first polypeptide established a condensation edge that sequesters the remnant and defined the shared target tgt:cond_rate: what drives the condensation fraction f_c and at what rate constant k? This paper closes that target and derives the full dynamical consequences.

When f_c(W_b) = k · W_b/L_lip, the augmented system has a fixed line F = {(N, W_b) : W_b = β L_lip/k, N > 0}: a family of fixed points parameterised by the steady-state particle count N*, which is free. At any point on F the condensation fraction is exactly β = α/(1 - α) and N is constant.

The fixed line is globally attracting for 0 < k < 2 L_lip/N_0 = 0.1048. The Lyapunov function V = (W_b - W_b^ss)² decreases strictly each cycle in this range. Within the safe-region subset 0 < k < L_lip/N_0 = 0.052 the convergence is monotone; outside it, oscillatory with |λ| < 1. The explicit convergence time to within 10% of W_b^ss is four to five cycles for k near the safe-region lower bound, and one cycle near k = L_lip/N_0.

At any point on F the reduced volume decays by factor Q_α^(-3/2) per cycle while N is constant. The division threshold is v_thr = Q_α^(-3/2). One cycle after each division the vessel reaches v_thr exactly, because v_thr · Q_α^(3/2) = Q_α^(-3/2) · Q_α^(3/2) = 1 for all α ∈ (0, 1). The division period is therefore exactly T = 43,200 s for every α, every N*, and every k in the stability range. This is an algebraic identity from Article 64's definitions, not a numerical coincidence.

Period homeostasis holds unconditionally on F: the period is T regardless of the vessel's particle count, bud fraction, or rate constant. Shape homeostasis (Article 64) holds unconditionally at every division. Size homeostasis does not hold: N* is whatever the vessel had when the loop engaged. The architecture delivers two kinds of homeostasis and withholds a third.

During the transient from post-division to F, the period is two cycles, converging to one as W_b → W_b^ss. Four targets from the two proceeding papers of the journal ACF are simultaneously closed at the fixed line.