Conventions and principal distinctions

The unconditioned ground has no mathematical symbol in this account. $\unsplit$ denotes the unsplit prior to the operational source/readout distinction, rather than the ground itself or the quality attributed to it. The different typeface distinguishes this ontological notation from locally defined evolution operators such as $U_t$ in the mathematics. $S$ is the working source or structure-side cut; $T$ is the readout or record-side cut. The schematic map is $p:S\to T$. The descent theorems use $X$ for that readout codomain; this is a local notation convention, not another ontological level. Symbols such as $T$ for a time horizon or $T_u$ for a transition kernel retain their expressly defined mathematical meanings.

The older ontological notation $\source$ is interpreted here through $\unsplit$, not through a source state in $S$. Neither symbol names the closure operator $\mathcal O$ in $L=\mathcal O\Delta\partial[L]$. No theorem in this book treats $\unsplit$ as a set, a Hilbert space, a field, or an argument of a computable transition law.

For probability laws $P,Q$ on the same measurable space,

For density operators $\rho,\sigma$, $D(\rho,\sigma)=\tfrac12\norm{\rho-\sigma}_1$. Complete-path distance, endpoint distance, quantum-state distance, and faithfulness to a past event are different claims. A bound in one space is transferred to another only through a specified common channel or map.

An exact theorem is a mathematical implication on a declared domain. A physical model supplies laws of motion, preparation, and interaction. A psychophysical constitutive law assigns experiential significance to a physical structure. A phenomenological report describes what a subject reports as experienced; its metaphysical interpretation is a separate claim. In particular, a mathematically complete assignment need not be an empirically correct assignment.

The phrase candidate internal resolution has one controlled meaning in this monograph: all maps and laws required for the nominated finite constitutive model are specified, and its conclusions follow under their hypotheses. It does not mean that the actual-vessel realization problem, empirical adequacy, universal source selection, or external verification has already been settled.

Terminology