Tertius Interveniens: The Attribution of Change and the Preservation of Relation in Saju Charts — A Functorial Formalization Modeled on the Sexagenary Cycle (六十甲子) and Daeun (大運)

The sentence "Change is attributed to one side, and the relation is preserved across those changes" is intuitively clear, yet what it asserts is not, for it is not even settled whether attribution and preservation state the same fact twice or are two distinct judgments. Taking the Daeun (大運, ten-year luck periods) of two charts as its input structure, this paper rewrites that sentence in the language of category theory. The common refinement obtained by merging the two persons' Daeun boundaries is taken as a totally ordered category T, each person's Daeun flow as a functor from T to a state category S, and the relation between the two persons as a candidate natural transformation between the two functors. The attribution of a step is then defined as the set of persons whose collapse functor does not send that step to an identity, and the preservation of the relation is defined as the commuting of the naturality squares. There are six main results. First, if all the local squares commute, global preservation follows, and if global preservation does not hold, there is at least one step at which a discontinuity occurs, the earliest of which is uniquely determined. Second, at a step attributed to one side only, the preservation condition reduces to a form in which the relation absorbs that person's transition. Third, if the state category is thin, preservation becomes vacuously true, so for preservation to have content there must be distinct morphisms between two states. Fourth, if the state category is not thin, attribution and preservation are logically independent, and neither can be derived from the other. Fifth, the preservation judgment does not depend on the choice of refinement. Sixth, in a Gunghap (宮合, compatibility of two charts) that reads the partner's Daeun relative to the day stem of one chart, the attribution of the transition and the attribution of the reading frame come apart, and the preservation judgment at the level of readings reflects the judgment at the level of states only when the reading functor is faithful. What a reading loses is not the number of states but the way the states are connected, which is why a Gunghap that reads only names, such as the Ten Gods (十神) of the heavenly stems or the nayin (納音), cannot in principle detect the failure of naturality at the antipode. As a model, the sexagenary cycle (六十甲子) of 60 stem-branches is constructed as a groupoid of displacements, and it is shown that the shortest-displacement reading loses naturality only at steps that pass through or jump over the antipode, that is, the position at which the heavenly stems are the same and the earthly branches clash; the result is then applied to the six pairs formed by the four charts of a fictitious family, selected from randomly generated charts so as to contain the cases to be described (Appendix C). The three pairs among the three persons whose Daeun progress forward are preserved over the whole interval, while in two of the three pairs involving the one person whose Daeun progresses backward a local discontinuity occurs, depending on the initial displacement, either at the first boundary or at a boundary several decades later. The step at which the discontinuity is placed divides, according to the direction of reading, into the entry and the exit of the antipodal cell, and what is invariant under both directions is not the step at which the discontinuity occurs but the antipodal cell and the attribution of its two ends. When two persons turning in the same direction meet boundaries alternately beside the antipode, a reading in one direction may be preserved at no step at all; yet as long as the reading is determined anew for each cell, there is no such thing in this formalism as the relation coming to a halt. On an interval with fewer than sixty boundaries, the number of discontinuities in a reading in one direction is at most one for a pair turning in opposite directions, whereas for a pair turning in the same direction it falls, while the two persons' boundaries alternate, into only two cases: none at all, or one at every step. Relaxations of the discontinuity condition are also examined. Because the defect arises only in units of one full turn of the cycle, a relaxation that allows a tolerance either changes nothing or renders the judgment vacuous, and among the relaxations examined in this paper the one that does not become vacuous is topological preservation, which asks whether the winding number is 0. Topological preservation is determined solely by the first reading and the sum of the differences between the two persons' transitions, so it depends neither on the order and attribution of the steps nor, provided neither end cell is antipodal, on the reading direction. Even if the reading direction and the antipode convention are fixed by a judgment of Myeongni (命理) computed from the natal chart (原局) and the luck, such as the success or failure (成敗) of the structure pattern (格局), the judgment of topological preservation does not change on an interval neither of whose end cells is antipodal, and for a pair that crosses an antipodal cell once and otherwise never enters one, all that such a judgment can move is the place where the local discontinuity falls. In this way, preservation at the individual layer, at which a structure is formed or broken in the luck, and preservation at the relational layer, which is attributed to no one, are separated within a single formalism. The starting-age reckoning and the forward or backward progression of Daeun follow the rules of Yuanhai Ziping (淵海子平) and Sanming Tonghui (三命通會), and the shape of a reading that sets the luck against the natal chart is taken from the treatment of luck progression (行運) in Ziping Zhenquan (子平眞詮). The treatment of annual luck in Ditian Sui (滴天髓), the doctrine of climatic adjustment in Qiongtong Baojian (窮通寶鑑) and the judgment of favorability in Ziping Zhenquan are each assigned their place, together with the reasons for not admitting them into the formalism. The position this paper takes is the one Kepler took toward astrology in Tertius Interveniens (1610). Intervening between those who believed astrology wholesale and those who threw it out wholesale, Kepler sought to strip away the interpretation of symbols while retaining the geometrical harmony that resides in the angles formed by the planets. This paper, likewise, neither verifies nor denies Myeongni. Having stripped away the layer of interpretation, it formalizes only the structure that remains, namely the cycle of the 60 stem-branches and the forward and backward progression of Daeun, and the result that the antipode coincides with the position at which the stems are the same and the branches clash is arithmetic that comes to light only after that stripping away and that was already contained in the tradition. That the two stem-branches of an antipode share even their nayin phase was already recorded by Shen Kuo (沈括) in Mengxi Bitan (夢溪筆談); what this paper adds is the role that this arithmetic plays within the formalism. This paper does not address the empirical truth or predictive power of propositions of Myeongni. A chart is not a ground of proof but one model of the formal system, and through that model this paper formally tests the distinction between attribution and preservation on the input structure that Myeongni provides in Daeun. This record contains both the English and the Korean edition.