Void (空亡): How Are the Two "Empty" Branches Calculated?
Ten heavenly stems, twelve earthly branches — every ten-day stretch of the sixty-day cycle leaves exactly two branches without a stem to pair with.
Bokki Editorial
The short answer
The 空亡 (gongmang, "void") row on Bokki's chart table names two of the twelve branches as empty for a given chart — but it isn't naming a curse or a warning. It's counting. The sixty-day (or sixty-year) cycle pairs ten heavenly stems with twelve earthly branches, and because ten does not divide twelve evenly, each run of ten days uses up all ten stems while two branches are left with no stem to pair with. Those two leftover branches are void for that run — and which two depends only on which ten-day stretch your day pillar falls into, found from your day pillar alone. If your chart's year, month, or hour branch happens to be one of those two, Bokki marks that column 空亡; the day pillar itself is never checked against its own void, since the rule is defined starting from it.
The rule — a remainder, not a lookup table
Line up the ten heavenly stems (甲乙丙丁戊己庚辛壬癸) against the twelve earthly branches (子丑寅卯辰巳午未申酉戌亥), and pair them off one at a time, cycling each independently: 甲子, 乙丑, 丙寅, and so on. After ten pairings the stems have all been used once and start over at 甲, but the branches have only gone ten of their twelve steps — so the eleventh and twelfth branches in that stretch (counting from wherever the stretch began) get skipped this time around, and stay skipped until the stem cycle catches back up to them, thirty-six days later. Bokki's engine finds those two branches directly from a day pillar's own stem and branch, without needing to know the rest of the sixty-day table: take the day stem's position (甲=0 through 癸=9) and the day branch's position (子=0 through 亥=11), and the two void branches are whichever branches sit 10 and 11 steps ahead of the day branch, wrapping around a twelve-branch circle, once the day stem's position is subtracted out. There are only six distinct starting points for a ten-day stretch (each one beginning on a 甲 day), so there are only six possible void pairs across all sixty day pillars: a chart starting 甲子 voids 戌 and 亥; 甲戌 voids 申 and 酉; 甲申 voids 午 and 未; 甲午 voids 辰 and 巳; 甲辰 voids 寅 and 卯; and 甲寅 voids 子 and 丑.
Worked example — a day pillar whose month and hour both land on the void
Take a real chart Bokki's piece on 택일 (date-picking) already used: 20 March 1995, 07:25, Seoul — year 乙亥, month 己卯, day 庚戌, hour 己卯. The day pillar is 庚戌: stem 庚 sits at position 6 in the ten-stem sequence, branch 戌 sits at position 10 in the twelve-branch sequence. That places this day pillar inside the 甲辰 ten-day stretch (甲辰, 乙巳, 丙午, 丁未, 戊申, 己酉, 庚戌, 辛亥, 壬子, 癸丑 — ten stem-branch pairs, 庚戌 the seventh of them), and the two branches that stretch never reaches are 寅 (Tiger) and 卯 (Rabbit). Now check this chart's other three pillars against that pair: the year branch is 亥 (Pig) — not a match, not void. The month branch is 卯 (Rabbit) — a match, void. The hour branch is also 卯 (Rabbit) — also a match, also void. Both the month and hour pillars of this chart land on the same void branch, purely because they happen to share a branch that falls outside this particular day's ten-day stretch; the year pillar does not, for the same purely arithmetic reason.
What this calculation doesn't decide
The void calculation names which two branches are structurally unpaired for a given day pillar — it does not say whether that is good or bad. Bokki's own copy for this row reads it as "not good or bad — just empty, a place to let flow rather than grip," and this article keeps to that: it explains the arithmetic, not a fortune. It also does not check every pillar against the void — only year, month and hour are marked, because the rule is built starting from the day pillar itself, so asking whether the day pillar voids itself is not a meaningful question. And Bokki does not layer on the further traditional refinements some schools apply to void interpretation (whether the void branch is itself supported or clashed by other branches, or year-pillar-based void variants distinct from the day-pillar rule used here) — this calculation covers the day-pillar 旬空 rule only, the version Bokki's engine actually runs.
Closing note — twelve branches, ten stems, two left over every time
In short: void (空亡) is what's left over when a cycle of ten stems is paired against a cycle of twelve branches — two branches in every ten-day stretch never get a partner, and which two depends only on the day pillar's own stem and branch position, findable with nothing more than a subtraction and a remainder. There are only six possible pairs across the entire sixty-day cycle. For the chart above, the day pillar 庚戌 falls in the stretch that leaves 寅 and 卯 unpaired, and this particular chart's month and hour branches both happen to be 卯 — landing both of those columns on the void, while the year branch, 亥, does not. It is a real, checkable piece of arithmetic sitting quietly under a single row of a much bigger chart, and now it has a page that shows its work.