Six-Line Hexagram Matrix: Deterministic Liu Yao Engine

How Deterministic Six-Line Hexagram Matrix Works After the Cast

Most digital I Ching tools stop once six lines appear. In traditional six-line hexagram matrix modeling (Liu Yao or Najia), casting is only the input; the compiler combines the 6-bit figure with the exact timestamp to derive a deterministic chart.

1. The Cast Is Only the Input: Three Entry Paths

Every cast reduces to six values ordered bottom-up (Line 1 to Line 6, order=bottom-up):

• Four-State Line Values (6, 7, 8, 9): ?hex=789876 encodes Old Yin (6, moving), Young Yang (7, static), Young Yin (8, static), and Old Yang (9, moving), fixing both primary and resulting figures.
• Two-State Binary Input (0, 1): ?hex=101010 maps 1 → 7 and 0 → 8 for a static chart with zero moving lines.
• Two-Number Modulo Cast (num1, num2) & Seeded Yarrow (seed): Matching HexSage's numerical consultation, ?num1=168&num2=42 computes upper trigram (num1+num2) mod 8, lower |num1-num2| mod 8, and moving line (num1+num2) mod 6 + 1. Passing ?method=yarrow18&seed=hn_audit_01 derives 18 yarrow splits via SHA-256. When hex and seed coexist, hex wins and logs seed: unused (hex provided).

2. The Moment Matters: Complete Casting Time and True Solar Time

In the six-line hexagram matrix, t records the second Line 6 settles. Two identical figures cast on different days keep the same internal stem-branch coordinates and Self/Other pointers, but their Six Relatives receive different prosperity scores and Day Void states. Any t missing an explicit UTC offset returns HTTP 400. Toggle day_boundary between 23:00 (Zi hour) and midnight, or pass lon for True Solar Time meridian offset (4 min/deg, excluding Equation of Time).

3. Four Pillars Computed from a Timestamp (Solar-Term Anchored)

Classical six-line hexagram matrix analysis anchors its Month Pillar to astronomical solar terms (Jie Qi, every 30° of solar longitude from Li Chun at 315°). Verify ten reference timestamps below:

Locating Self, Other, and Five-Element Scores in Liu Yao Najia

Han scholar Jing Fang (77–37 BC) organized the 64 figures into Eight Palaces (Ba Gong), equipping each hexagram with mechanical pointers and Five-Element coordinates.

4. Self (Shi) and Other (Ying): Locating the Ruling Line Mechanically

In the six-line hexagram matrix, Self and Other are exact line indices (self_line and other_line, separated by 3 steps) determined by the bitwise XOR difference delta = lower_trigram XOR upper_trigram in GF(2)^3:

5. Five Elements and Six Relatives: Quantitative Scoring

Comparing each line's Najia branch element against the Palace Element yields the Six Relatives: Same = Siblings; produces Palace = Parents; produced by Palace = Offspring; controlled by Palace = Wealth; controls Palace = Officer. In ?hex=101010&t=2026-10-28T14:08:00%2B08:00 (#63 Ji Ji, Kan Water Palace), Line 4 carries Wu-Shen Metal → Metal produces Water = Parents → Day Yi-Hai places Shen, You in Day Void → Month Xu supports Shen (+2), scoring +2 (Prosperous, Void).

▶ Watch 2K Lecture: Jing Fang 8-Palace XOR Engine (2:47)

EP.02 · Najia & Shi-Ying Coordinates YouTube ↗

What the Hexagram Says (and Why "Six Lines" Isn't Human Design)

English readers exploring a six-line hexagram matrix (Liu Yao) often encounter "Human Design"—a modern personality typology that borrows hexagram numbers from birth time alone without casting.

6. Classical Hexagram Text vs. Modern Bodygraph Systems

Classical six-line hexagram matrix analysis requires both a freshly cast hexagram and the completion timestamp, pairing Jing Fang's stem-branch mechanics with the Zhouyi judgment text. Every chart above links directly to our How to Read Changing Lines guide, I Ching Yes or No matrix, and I Ching Career Guidance.

7. Worked Example, End to End: Moving Lines in ?hex=789876

Open /six-lines-najia?hex=789876&t=2026-10-28T14:08:00%2B08:00 (aca0ede1): bottom-up [7,8,9,8,7,6] yields Primary #63 Ji Ji with moving lines at Line 3 (Old Yang 9, Self) and Line 6 (Old Yin 6, Other), transforming into #42 Yi (Line 3 Ji-Hai Siblings → Yi-Mao Offspring; Line 6 Wu-Zi Siblings → Wu-Xu Officer).

Prove It Yourself: Modulo-4 Invariance and the 36:11 Boundary Bound

In debates over digital yarrow stalks vs. three-coin generators, everything in yarrow stalk casting reduces to modulo-4 arithmetic.

8. Support Invariance {6, 7, 8, 9} and the 36:11 Boundary Proof

In Change 1 (49 stalks, 1 set aside, 48 in play), splitting into L and 48 - L and counting by fours (where divisible by 4 yields remainder 4) removes r(L) + r(48 - L) + 1. Since 48 ≡ 0 (mod 4), only residue pair (0,0) removes 9 while (1,3), (3,1), (2,2) remove 5. Changes 2 and 3 operate on 44 or 40 stalks and remove 4 or 8, locking the three-change support strictly to {6, 7, 8, 9} under any split distribution.

Why does uniform splitting over L ∈ [1, 47] give 36:11 (≈ 3.27:1) instead of 3:1? Both excluded boundaries (L=0, 48) lie in residue class 0 (mod 4), leaving 11 points vs. 12 each in 1, 2, 3. Smooth hand splits restore 3:1. Audit this in the six-line hexagram matrix:

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