The Chronos Stability Coordinate x: Historical Provenance, Ratio-Share Reconciliation, and a Minimal Primitive Shell Closure

This working preprint consolidates the development of the Chronos stability quantity historically denoted by x or chi and clarifies how its mathematical meaning changed across earlier stages of the research program. Earlier Chronos work used the same notation for more than one type of quantity, including a direct active-to-restoring ratio and a bounded stability selector. This paper separates those roles explicitly. It reserves rho for the positive direct response ratio and x for the bounded normalized stability coordinate derived from that ratio. This resolves an important notation collision and shows why several earlier numerical values near 0.55128 should not be treated as identical versions of the same coordinate. The paper then develops a minimal four-phase shell construction. Four equally spaced phase channels are decomposed into one uniform mode and a three-dimensional nonuniform active sector. A canonical transition operator derived from the four-cycle supplies both the unit active-channel strength and the normalized active rank used in the reduced model. The four-phase symmetry also determines the equal phase allocation used in the overlap term. Combining this phase structure with the first nonzero spherical response produces a primitive connected stability response through the minimal quartic shell-action class. A self-consistency condition then selects a unique attracting fixed point for the bounded stability coordinate, giving a current value near 0.5512827409. The paper proves uniqueness and rapid convergence of this fixed point within the stated reduced selector model. The work also establishes a clear model boundary. Repeated return operations do not generate new reduced return classes, and the primitive selector closes through quartic order within the stated minimal dimension-four shell-action construction. However, more general higher-dimensional effective operators could introduce additional primitive terms, so the result is not presented as a universal theorem for every possible microscopic theory. A further section translates the shell result into the spectral and Wheeler-DeWitt language used elsewhere in the Chronos program. The algebraic translation is exact once the shell and spectral ratios are identified, but the physical identification between those two constructions remains a model bridge rather than an empirical result. The paper therefore serves both as a provenance record and as a mathematical consolidation. It preserves the earlier history of the Chronos stability quantity while establishing a stricter modern convention: rho denotes the direct response ratio, x denotes its bounded representation, and the present value of x is the selected root of the explicit minimal shell closure rather than a decimal retroactively imposed on earlier formulas.