Xia Gua / Ti Gua 下卦／替卦 Glossary

The two chart types of Xuan Kong Flying Star: a reading within ±4.5° of a mountain's centre yields a normal (lower-trigram) chart; the outer 3° on each side of the mountain triggers a replacement-star (ti gua) chart.

Xiàguà / tìguà · 下卦／替卦 (simplified) · 下卦／替卦 (traditional)

Source credibility (S/A/B/C/X): A B    Rule computability (A/B/C): A thresholds and charts; the replacement-table variant choice is disclosed, not resolved

What it is

Each of the 24 Mountains spans 15°, subdivided into five fenjin of 3°. When the compass reading sits near the mountain's centre, the chart is built from the mountain's own palace star — this is the lower-trigram chart （下卦盤）. When the reading drifts into the outer strips — a "leaning" orientation （兼向） — the chart must substitute a replacement star （替星） for the palace star, producing a different flying-star chart entirely: the ti gua chart （替卦盤）.

The thresholds are precisely definable, stated from the mountain's centre line c: the lower-trigram zone is |θ − c| ≤ 4.5° (9° wide, the middle three fenjin), and the ti-gua zone is 4.5° < |θ − c| ≤ 7.5° (3° on each side, the outer two fenjin). Beyond 7.5° the reading belongs to the neighbouring mountain. A checkable property: every boundary line's degree value ends in 2.5 or 7.5. Two further thresholds must not be confused with these: the ±3° "leave-trigram" lines （出卦界） sit at the midpoints of mountain boundaries, not mountain centres — the 16 inauspicious lines of the "Eight Killings Needle Degree" rhyme are exactly those eight midpoints ±3°.

School differences

Two separate disagreements live here. First, where the boundary sits: sources differ on how far from the mountain centre the off-grid threshold sits, but the unified reading — lower-trigram ±4.5°, ti-gua 3° per side, off-grid beyond 7.5° — is the one consistent with the five-fenjin structure of each mountain, and the project standardises on it (dispute register entry D-06). Second, how the replacement is computed: the Kun-Ren-Yi rhyme （坤壬乙訣） gives replacement stars for 12 of the 24 mountains, and for the remaining 12 two readings produce different charts — take the mountain's own trigram number, or take the star reaching that palace in the period chart (entry D-03, both shipped). Relatedly, the Shen family lists replacement stars for all 24 mountains while the Zhongzhou line holds that "only thirteen stars replace" — a direct conflict about which mountains replace at all. These are collated in the School Dispute Register, not voted on.

How this engine computes it

• The two thresholds are two independent named constants in the kernel — the leave-trigram boundary (3.0°, based at mountain-boundary midpoints) and the leaning-orientation boundary (4.5°, based at the mountain centre) — deliberately never merged.
• Trigger condition for ti gua: 4.5° < |θ − c| ≤ 7.5°. Replacement charts follow the Shen-family table by default, with the Zhongzhou table available as a switch; where the selected reading matters, the choice is recorded in ruleset_id and both variants are shown for the disputed 12 mountains.
• Because a phone magnetometer's error (±3°–5°) is the same order as the whole 9° lower-trigram zone, ordinary compass input never silently decides between the two chart types — boundary readings are flagged and higher-precision measurement is recommended.
• All 5,184 lower-trigram and ti-gua charts are covered by the golden-set replay tests.

Related entries

Xuan Kong Flying Star · 24 Mountains · Luopan · Magnetic Declination

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