Part II: Operator Families¶
What does the transforming
The Principle¶
Operator radicals encode what happens to substrates. The same operator applied to different substrates produces related but distinct transformations—this consistency proves the algebra is real.
刂/刀 (dāo) — The Blade Operator¶
Domain: Cutting, arc-sweeping, boundary-making, distinction-creating
Geometric Principle: The π-operation—the curved sweep that traverses what straight lines cannot.
Critical Distinction: This is the SCYTHE blade, not the knife blade. - Knife = linear push (A→B), one thing at a time - Scythe = arc sweep (O→G→π→P), swath per stroke
Transformations by Substrate¶
| Character | Pinyin | Substrate | Transformation | DDJ Chapters | Guodian? |
|---|---|---|---|---|---|
| 利 | lì | 禾 (grain) | Scythe arcs through field → harvest/path-cutting | 8, 11, 22, 56, 81 | ✓ A01 |
| 則 | zé | 貝 (value) | Blade cuts through value → consequent pattern | 3, 22, 64 | ✓ |
| 剛 | gāng | 岡 (ridge) | Blade meets ridge → rigid/inflexible (failed arc) | 76, 78 | ✓ |
| 別 | bié | 另 (other) | Blade separates → distinguish/separate | — | — |
| 判 | pàn | 半 (half) | Blade halves → divide/judge | — | — |
| 制 | zhì | 未 (not-yet) | Blade on potential → control/restrain | — | — |
| 割 | gē | 害 (harm) | Blade harms → sever/cut off | — | — |
| 解 | jiě | 角+牛 (horn+ox) | Blade unhorns ox → untangle/dissolve | 56 | ✓ A28 |
| 列 | liè | 歹 (death) | Blade death → arranged/split | 39 (as 裂) | — |
Pattern Analysis¶
When 刂 meets organic substrate (禾, 角): Result = productive cutting, harvest, untangling - 利: Grain falls in swaths - 解: Complexity untangles
When 刂 meets value substrate (貝): Result = consequent pattern, rule established - 則: Exchange cut into predictable form
When 刂 meets rigid substrate (岡): Result = hardening, rigidity (failed transformation) - 剛: Ridge + blade = inflexible (what resists the arc)
The algebra reveals: The blade operator either completes the π-operation (利, 解) or fails against rigidity (剛). The text's preference for 柔 over 剛 is geometric, not moral—the flexible blade arcs; the rigid blade breaks.
口 (kǒu) — The Opening Operator¶
Domain: Distribution, naming, articulation, passage-creation
Geometric Principle: Creates channels for flow, articulates distinctions, enables passage.
Transformations by Substrate¶
| Character | Pinyin | Substrate | Transformation | DDJ Chapters | Guodian? |
|---|---|---|---|---|---|
| 和 | hé | 禾 (grain) | Grain through mouths → harmonious distribution | 2, 4, 42, 55, 56 | ✓ A15, A28 |
| 名 | míng | 夕 (evening) | Evening/darkness given mouth → naming | 1, 32, 37 | ✓ |
| 可 | kě | 丁 (nail) | Fixed given mouth → expressible/sayable | 1, 32 | ✓ |
| 知 | zhī | 矢 (arrow) | Arrow + mouth → knowing (targeted articulation) | 1, 16, 33, 56 | ✓ throughout |
| 同 | tóng | 冂 (enclosure) | Enclosed opening → unified/merged | 4, 56 | ✓ A28 |
| 吾 | wú | 五 (five) | Five + mouth → I/self (articulated position) | throughout | ✓ |
| 言 | yán | — | Mouth extended → speech/words | throughout | ✓ |
| 告 | gào | 牛 (ox) | Ox + mouth → announce/report | — | — |
| 問 | wèn | 門 (gate) | Gate + mouth → ask/inquire | — | — |
Pattern Analysis¶
When 口 meets discrete substrate (禾, 矢): Result = distribution, targeted communication - 和: Grain shared among mouths → harmony - 知: Arrow + mouth → precise knowing
When 口 meets darkness/undefined (夕): Result = naming, articulation of the obscure - 名: Evening given mouth → name emerges
When 口 meets fixedness (丁): Result = expressibility, rotation into sayable form - 可: Fixed nail given mouth → can be articulated
When 口 meets enclosure (冂): Result = unification, merging - 同: Enclosed opening → sameness/unified
The algebra reveals: The 口 operator consistently creates passage/distribution. Whether grain through mouths (和) or darkness through articulation (名), the operator enables flow where there was separation.
反 (fǎn) — The Reversal Operator¶
Domain: Return, oscillation, cycle completion, 2π closure
Geometric Principle: The arc that closes the circle. If 曲 (curving) is the π-operation, 反 is what happens when the arc comes back.
Core Character¶
| Character | Pinyin | Components | Meaning | DDJ Chapters | Guodian? |
|---|---|---|---|---|---|
| 反 | fǎn | 厂 + 又 | Cliff + hand reaching back → reversal | 25, 40, 65, 78 | ✓ A22, A37 |
Related Transformations¶
| Character | Pinyin | Relationship | Meaning | DDJ Chapters | Guodian? |
|---|---|---|---|---|---|
| 返 | fǎn | 反 + 辶 (movement) | Reversal in motion → return journey | — | — |
| 復 | fù | 彳 + 复 | Step + return → restore/return to root | 16, 28, 52, 64 | ✓ A24, B06 |
| 歸 | guī | 帚 + 止 | Broom + stop → return home | 16, 34, 52 | ✓ A24 |
The 反者道之動 Formula¶
Chapter 40: 反者道之動，弱者道之用
Structural translation: - 反者道之動 = "Reversal IS pattern's movement" - 弱者道之用 = "Yielding IS pattern's function"
The 反 operator doesn't just describe return—it IS the engine. The pattern moves BY reversing. This is the 2π completion: every arc implies its return.
Guodian validation: The formula appears in slip A37, confirming this is ~300 BCE core doctrine, not later systematization.
Pattern Analysis¶
反 as cycle completion:
Outward arc (π) → 反 → Return arc (π) → Full cycle (2π)
反 in the recursion cycle (Chapter 25):
大 (great) → 逝/潰 (overflows) → 遠 (extends far) → 反 (returns)
↓ ↓
O₁ ───────────── expansion ─────────────────────→ O₂
The algebra reveals: 反 isn't "reversal" as going backward. It's the completion of the circle. The scythe swings out (π) and returns (π) = full cycle (2π). Without 反, there's no completion. The "return" is not failure or retreat—it's the arc closing on itself.
氵 (shuǐ) — The Water Operator¶
Domain: Flow, descent, finding path, continuous transformation
Geometric Principle: Does the π-operation naturally. Water arcs around obstacles without forcing. Water IS 無為 in physical form.
When Water Operates on Substrates¶
| Character | Pinyin | Substrate | Transformation | DDJ Chapters | Guodian? |
|---|---|---|---|---|---|
| 治 | zhì | 台 (platform) | Water channeled → governance | 3, 17, 60, 64 | ✓ |
| 淵 | yuān | — (deep) | Water extended deep → profound source | 4, 8 | ✓ A23 |
| 沖 | chōng | 中 (center) | Water at center → hollow/concentrated flow | 4, 42 | ✓ A23 |
| 清 | qīng | 青 (clear/green) | Water clarified → transparency | 15, 45 | ✓ A09 |
| 深 | shēn | — (probe) | Water probed → depth | 4, 15 | — |
| 渾 | hún | 昆 (mingled) | Water mingled → murky/primordial | 15, 25 | ✓ A08 |
| 漸 | jiàn | 斬 (cut) | Water cutting → gradual | — | — |
| 泮 | pàn | 半 (half) | Water halving → dissolve/melt | 64 | ✓ A25 |
| 法 | fǎ | 去 (remove) | Water removes → law/method | 25 | ✓ A23 |
Pattern Analysis¶
When 氵 meets structure (台, 中): Result = channeled flow, organized but not forced - 治: Water channeled by platforms → governance (self-organizing) - 沖: Water at center → hollow core, concentrated flow
When 氵 meets depth indicators: Result = profound, reaching source - 淵: Deep water → the generative abyss (Chapter 4) - 深: Probed water → depth itself
When 氵 meets clarity/obscurity: Result = transparency or primordiality - 清: Clarified water → clear/transparent - 渾: Mingled water → murky/undifferentiated
The algebra reveals: Water doesn't force transformation—it FINDS transformation. The 氵 radical shows substrates being acted upon by flow-finding, not by cutting or forcing. This is why Chapter 8 uses water as the supreme demonstration: water does 利 (cuts paths) without 為 (forcing).
宀 (mián) — The Roof/Enclosure Operator¶
Domain: Covering, housing, containing, creating interior space
Geometric Principle: Creates the void-space (無) where function (用) can operate. The roof makes the room.
Transformations by Substrate¶
| Character | Pinyin | Substrate | Transformation | DDJ Chapters | Guodian? |
|---|---|---|---|---|---|
| 室 | shì | 至 (arrive) | Roof + arrival → room (space entered) | 11 | ✓ A07 |
| 家 | jiā | 豕 (pig) | Roof + pig → household | throughout | ✓ |
| 安 | ān | 女 (woman) | Roof + woman → peace/settled | 64, 80 | ✓ A25 |
| 宗 | zōng | 示 (altar) | Roof + altar → ancestral/source | 4 | ✓ A23 |
| 害 | hài | 口 + 丯 | Roof + obstructed → harm | 81 | ✓ |
| 定 | dìng | 正 (correct) | Roof + correct → settled/fixed | 37, 57 | ✓ A13, A30 |
Pattern Analysis¶
When 宀 meets arrival/presence (至): Result = functional interior space - 室: The room you can enter—void under roof enables dwelling
When 宀 meets source indicators (示): Result = ancestral, origin-connected - 宗: Altar under roof → ancestral source
When 宀 meets stability (正, 女): Result = settledness, peace - 定: Correctness under roof → settled - 安: Woman under roof → peace/security
When 宀 meets obstruction: Result = harm (blocked void) - 害: Obstructed under roof → harm (the void not functioning)
The algebra reveals: The 宀 operator creates functional void-space. But it can also create obstruction (害). The difference is whether the interior remains 虛 (empty/functional) or gets blocked. Chapter 11's teaching: the room works BECAUSE of its emptiness. 宀 + proper void = function. 宀 + obstruction = harm.
彳 (chì) — The Step/Path Operator¶
Domain: Movement, progression, way-following, behavioral pattern
Geometric Principle: Continuous motion along path. The step that maintains direction.
Transformations by Substrate¶
| Character | Pinyin | Substrate | Transformation | DDJ Chapters | Guodian? |
|---|---|---|---|---|---|
| 德 | dé | 直+心 (straight+heart) | Stepping with straight heart → alignment | 38, 51, etc. | ✓ throughout |
| 復 | fù | 复 (return) | Stepping back → restore/return | 16, 28, 52, 64 | ✓ A24, B06 |
| 徼 | jiào | 敫 (boundary) | Stepping to boundary → edge/perimeter | 1 | ✓ |
| 得 | dé | 寸+貝 (measure+value) | Stepping to measured value → obtain | throughout | ✓ |
| 行 | xíng | — | Double step → walk/conduct | 27, 41, 64 | ✓ |
| 道 | dào | 首 (head) | Stepping with head → way/pattern | throughout | ✓ throughout |
Pattern Analysis¶
When 彳 meets interior state (心): Result = alignment, integrated movement - 德: Heart-stepping straight → virtue/alignment with pattern
When 彳 meets return (复): Result = restoration, coming back - 復: Step-return → restore to root
When 彳 meets boundary (敫): Result = edge-orientation - 徼: Step-boundary → perceiving from form-edge
When 彳 meets direction (首): Result = way, path, pattern - 道: Step with head leading → the way itself
The algebra reveals: The 彳 operator is about continuous aligned motion. Not single action but sustained walking. This is why 德 (alignment) contains it—德 isn't a one-time choice but continuous stepping in pattern-alignment. And 道 itself contains 彳—the pattern IS the walking, the continuous movement.
Summary: The Operator Algebra¶
| Operator | Domain | Action Type | Key Transformation |
|---|---|---|---|
| 刂 (blade) | Cutting/arcing | π-operation | Substrate → harvested/patterned |
| 口 (opening) | Distribution/naming | Passage-creation | Substrate → articulated/flowed |
| 反 (reversal) | Return/completion | 2π closure | Arc → full cycle |
| 氵 (water) | Flow/finding path | Natural 無為 | Substrate → channeled/clarified |
| 宀 (roof) | Covering/containing | Void-creation | Space → functional interior |
| 彳 (step) | Walking/patterning | Continuous motion | Position → aligned movement |
The Consistency Test¶
If this algebra is real, then: 1. Same operator on similar substrates → similar transformations 2. Different operators on same substrate → different transformations 3. Predictions should match text usage
Test case: What does 刂 do? - 刂 + organic (禾) → harvest (利) - 刂 + value (貝) → consequent (則) - 刂 + rigid (岡) → failure (剛)
Prediction confirmed. The blade arcs through organic material, cuts value into pattern, but fails against rigidity.
Test case: What does 口 do? - 口 + discrete (禾) → distributed (和) - 口 + darkness (夕) → articulated (名) - 口 + fixed (丁) → expressible (可)
Prediction confirmed. The opening enables passage/distribution across all substrates.
The algebra is real. The radicals are operands. The characters are equations.
This is Part II of the Structural Lexicon. Part I (Substrate Families) documents what receives operations. Part III (Structural Positions) documents the coordinate system.