Volume IV

Fractality

1Fractals are regarded as the geometric wonderworld of mathematics: infinite patterns that repeat at every scale. Yet this description misses what is essential.

2For fractality is not a phenomenon of numbers.

3It is not a feature of forms.

4It is the self-relatedness of distinguishing itself.

5The difference between A and B is not static.

6It is already the beginning of a new distinction:

7How does A differ from B?

8And this question generates the next level – and the one after next.

9These layers do not operate one after another.

10They are simultaneous.

11Fractality is not in the things.

12It is the movement that first lets the things appear as distinguishable.

13For there is only this premise: the difference.

Chapter I

Principia Fractalis

1The Orders of Difference

1.1 Difference of the First Order

2|

3The difference of the first order designates pure distinction without poles.

4It is not a relation between things, but the condition of relation.

5Operational reading:

6An operator without arguments.

1.2 Difference of the Second Order

7A|B

8The difference generates two positions.

9A and B are not entities, but relational poles.

10Fractal reference:

11This is the first explication of the operator |.

1.3 Difference of the Third Order

12The difference distinguishes its own distinctions.

13Several poles arise in a common relational structure.

14Operational reading:

15Fractal reference:

16This is the minimal fractal unfolding:

17Self-related iteration without stabilization.

1.4 Iterative Reading of the Orders

18The orders of difference can be read as iterations:

|              → difference of the 1st order
A|B            → difference of the 2nd order
A|B|C|D        → difference of the 3rd order
A|B|C|D|E|F…   → continued self-distinction

19It is not a matter of an increase of elements,

20but of an increase of relational depth.

1.5 Minimal Transition to Fractal Mathematics

21A fractal structure is present exactly when the following holds:

22The difference distinguishes not only poles,

23but its own order.

24Read formally:

25This is identical with:

26Fractality here is not a property of the set,

27but a property of the order of distinction.

1.6 Memorandum (purely operational)

28The orders of difference are not stages,

29but iterations of the same operator.

Chapter II

Transitions between Orders of Difference (Δ)

2.1 The Transition as a Relation in Its Own Right

1The transition between two orders of difference is not a leap,

2but itself a relational operation.

3It is designated by:

4Δ

5Δ is not a new object, but marks the transformation of one order into the next.

2.2 Minimal Transition

6Example:

|   Δ   A|B

7The difference of the first order unfolds into the second order.

8Δ here designates not the difference itself, but its becoming-effective.

2.3 Transition of Higher Order

9Example:

A|B   Δ   A|B|C|D

10The difference distinguishes anew the poles that have already arisen.

11The transition is self-related here.

12Operational reading:

2.4 Fractal Reading of Δ

13A fractal dynamic is present when transitions themselves are again distinguished:

(|   Δ   A|B)   Δ   (A|B   Δ   A|B|C|D)

14Δ operates here not linearly, but recursively.

15Memorandum:

Fractality arises when transitions do not disappear,

16but become part of the structure.

2.5 Stable and Unstable Transition

17Stable:

A|B   Δ   A|B

18The transition preserves the order.

19Unstable:

A|B   Δ   A|B|C|D

20The transition generates new poles.

21This is the minimal distinction between:

22linear behavior

23nonlinear unfolding

2.6 Preparation of the Fractal Operators

24Δ functions as the base symbol for all later fractal operators:

25recursive difference

26oscillation

27rotation

28coupling

29These operators are specializations of Δ,

30not additional concepts.

2.7 Memorandum

An order is a relation.

31A transition is likewise a relation.

32Fractality begins where relations act upon relations.

Chapter III

Stabilization and Recurrence of Orders of Difference

3.1 Stability as Relational Property

1An order of difference is stable when distinguishing it further

2generates no new order.

3Stability is not rest, but invariant self-relation.

3.2 Minimal Example of Stability

4Example:

A|B|C   Δ   A|B|C

5The order is preserved when it is distinguished further.

6Operational reading:

7The difference operates, but does not change the structure.

3.3 Recurrence (Fixed Point)

8An order that, under Δ, leads back to itself forms a fixed point:

O   Δ   O

9where O designates an order of difference.

10Minimal example:

A|B|C   Δ   A|B|C

11This is the simplest form of fractal self-similarity.

3.4 Unstable Recurrence

12Not every repetition is stable.

13Example:

A|B   Δ   A|B|C|D   Δ   A|B

14The order returns, but not invariantly.

15This is a cyclical self-distinction.

16Operational reading:

17The structure oscillates between orders.

3.5 Cycle Formation

18A cycle is present when several orders of difference

19are connected with one another by Δ:

O₁   Δ   O₂   Δ   O₁

20Example:

A|B   Δ   A|B|C|D   Δ   A|B

21Cycles are not errors,

22but regular forms of nonlinear dynamics.

3.6 Preparation of Oscillation

23The repeated, unstable recurrence of an order

24forms the basis for fractal oscillation.

25Oscillation here is:

26not periodic movement in space

27but a periodic change of the order of difference

3.7 Memorandum

28Stability does not mean standstill.

29It means that the difference distinguishes itself,

30without changing its order.

Chapter IV

Oscillation as Alternation of the Order of Difference

4.1 Concept of Oscillation (operational)

1Oscillation designates no temporal course,

2but a regular alternation between orders of difference.

3It is present when no order is permanently stabilized,

4but the relation between orders swings back and forth.

4.2 Minimal Oscillation Schema

O₁   Δ   O₂   Δ   O₁   Δ   O₂   …

5where O₁ and O₂ are different orders of difference.

6Example:

A|B   Δ   A|B|C|D   Δ   A|B   Δ   …

7The structure returns,

8but not to the same relational depth.

4.3 Oscillation without External Parameters

9Oscillation requires:

10no time parameter

11no frequency

12no movement in space

13It arises exclusively from the self-distinction of the transitions.

14Operational reading:

15The transition Δ itself becomes unstable.

4.4 Order-Preserving vs. Order-Changing Oscillation

O   Δ   O   Δ   O

16This is not oscillation, but stability.

O₁   Δ   O₂   Δ   O₁

17This is the minimal fractal oscillation.

4.5 Nested Oscillation

18Oscillation can itself be distinguished:

(O₁   Δ   O₂)   Δ   (O₃   Δ   O₄)

19Here not only orders alternate,

20but patterns of alternation.

21This is the entry into higher fractal dynamics.

4.6 Preparation of the Fractal Operators

22Oscillation forms the basis for:

23fractal difference

24rotation

25multidimensional coupling

26All these operators describe structured oscillations of orders of difference.

4.7 Key Sentence

27Oscillation is not a course in time,

28but a periodic change of the relational order.

Chapter V

Coupling of Oscillations – Relational Networks

5.1 Coupling as Simultaneous Relation

1A coupling is present when several oscillations are not independent,

2but act on one another.

3It is not a connection of objects,

4but a relation of transitions.

5.2 Minimal Coupling of Two Oscillations

5Schema:

(O₁   Δ   O₂)   ‖   (P₁   Δ   P₂)

6The two alternations do not take place one after the other,

7but relationally at the same time.

8Example:

(A|B   Δ   A|B|C|D)   ‖   (E|F   Δ   E|F|G|H)

9The orders influence one another,

10without merging.

5.3 Cross-Coupling

11In a cross-coupling, the transition of one oscillation acts back

12on the order of the other:

(O₁   Δ   O₂)   Δ   (P₁   Δ   P₂)

13Operational reading:

14A transition is itself distinguished by another transition.

5.4 Network Formation

15A relational network arises when several coupled oscillations

16can no longer be unfolded linearly:

17The structure is no longer readable as a chain,

18but only as simultaneous dynamics of difference.

5.5 Order in the Network

19A network has no central order.

20Stability arises locally through compatible transitions.

21Minimal example:

22O₁ Δ O₂

│     │
Δ     Δ
│     │

23P₁ Δ P₂

24The orders stabilize themselves relationally,

25not hierarchically.

5.6 Preparation of Nonlinear Mathematics

26With network formation, the transition is completed from:

27linear unfolding

28cyclical dynamics

29to:

30nonlinear, fractal structure

31From here on, no global order can still be presupposed.

32Mathematics becomes self-referentially distributed.

5.7 Key Sentence

33A network is not a mesh of things,

34but a coupling of transitions.

Chapter VI

Measure, Depth, and Fractal Dimension of Orders of Difference

6.1 Measure as Relational Property

1A measure describes no magnitude,

2but the depth of the self-distinction of an order.

3Measure is therefore not an external scale,

4but arises in the difference itself.

6.2 Depth of the Order of Difference

5The depth of an order corresponds to the number of effective self-relations.

6Examples:

|                 → depth 0
A|B               → depth 1
A|B|C|D           → depth 2
A|B|C|D|E|F|G|H   → depth 3

7The depth does not grow through new elements,

8but through recursive ordering of the order.

6.3 Measure through Transitions

9The measure of an order shows itself in the effect of Δ:

10slight change → slight depth

11strong change of order → greater depth

12Minimal example:

A|B   Δ   A|B        → slight depth
A|B   Δ   A|B|C|D    → greater depth

6.4 Fractal Dimension (operational)

13The fractal dimension of a structure results from the ratio:

14order of distinction

15to

16order of transition

17Formally noted (purely relational):

18D := ratio(order, Δ)

19This is not a numerical definition,

20but a comparative relation between depths.

6.5 Minimal Comparison

21Example:

22O₁ = A|B

23O₂ = A|B|C|D

24If:

O₁   Δ   O₂

25then O₂ possesses a higher fractal dimension than O₁,

26since the transition itself has a structure-forming effect.

6.6 Dimension without Number

27The fractal dimension here is:

28no real value

29no measured quantity

30no constant

31It is an order-relation between structures.

32Key sentence:

33Dimension is the answer to how deeply the difference distinguishes itself.

6.7 Conclusion of the Linear Part

34With measure, depth, and dimension, the linear introduction is concluded.

35From here on:

36transitions act on transitions

37oscillations couple themselves in a network-like way

38measures are no longer globally comparable

39Mathematics becomes nonlinear.

6.8 Transitional Sentence

40What follows is not a continuation,

41but a self-unfolding of the system.

Chapter VII

Fractal Difference (\(\circleddash\))

7.1 Positing

1Fractal difference designates a transition

2in which not only the order of difference,

3but the transition itself is effectively distinguished.

4In Principio notation:

5Fractal difference is therefore a recursive transition.

7.2 Minimal Example

A|B   ⊝   A|B|C|D

6Reading:

7The order changes through a change in the transition structure.

8Comparison:

A|B   Δ   A|B|C|D    → transition
A|B   ⊝   A|B|C|D    → transition of the transition

7.3 Iteration of Fractal Difference

9Fractal difference is iterable:

10With growing iteration, the depth of self-relatedness increases.

7.4 Key Sentence

11Fractal difference describes not how much changes,

12but how transitions change their own structure.

Chapter VIII

Fractal Linkage (\(\circledast\))

8.1 Positing

1Fractal linkage designates the simultaneous effectiveness of several transitions.

2In Principio notation:

3\(\circledast\) is neither addition nor connection,

4but a relational coupling.

8.2 Minimal Example

(A|B   Δ   A|B|C|D)   ⊛   (E|F   Δ   E|F|G|H)

5The orders of difference do not unfold independently,

6but act on one another as they unfold.

8.3 Nested Linkage

⊛¹₁  → simple coupling
⊛²₁  → nested coupling
⊛ⁿₘ  → fractal network

7The indices designate depth and breadth of the coupling,

8not dimensions or coordinates.

8.4 Key Sentence

9Linkage connects nothing.

10It causes transitions to structure one another.

Chapter IX

Fractal Oscillation (\(\widetilde{\circleddash}\))

9.1 Positing

1Fractal oscillation is present

2when a transition generates no stable order,

3but leads into an unstable return.

4In Principio notation:

9.2 Minimal Example

A|B   ⊝̃   A|B|C|D   ⊝̃   A|B   …

5The order returns,

6but not invariantly.

9.3 Oscillation of Higher Order

7Here oscillations themselves oscillate.

8This is the entry into multilayered fractal dynamics.

9.4 Key Sentence

9Oscillation is the form

10in which the difference cannot be stabilized.

Chapter X

Fractal Rotation (\(\mathcal{R}\))

10.1 Positing

1Fractal rotation describes a directed transition.

2It is not the order that changes, but its orientation.

3In Principio notation:

4\(\mathcal{R}\) := directed Δ

10.2 Minimal Example

A|B   ℛ   B|A

5The poles are preserved,

6their relational reading direction changes.

10.3 Iteration

7\(\mathcal{R}\)² → return of orientation

8\(\mathcal{R}\)³ → new cyclical order

9Rotation is therefore a cyclical transition structure,

10not movement in space.

10.4 Key Sentence

11Rotation changes not the order,

12but the direction of the distinction.

Chapter XI

Summary of the Fractal Operators

1All introduced operators are specializations of the difference:

2Operator

3Relational Meaning

4\(\circleddash\)

5transition of the transition

6\(\circledast\)

7coupling of transitions

8\(\widetilde{\circleddash}\)

9unstable return

10\(\mathcal{R}\)

11directed transition

12Network

13nested Δ-structure

14Closing Key Sentence

15The fractal operators do not extend the system.

16They unfold what is already laid out in the difference.

Chapter XII

Indeterminacy \(δ\) as Instability of the Order of Transition

12.1 Positing

1Indeterminacy designates no inaccuracy of description,

2but a non-fixability of transition.

3δ is not a disturbance factor,

4but a relational quality of its own.

12.2 Operative Definition

5In Principio notation:

6δ := Δ without stabilization

7That means:

8A transition acts,

9without committing itself to an unambiguous order of difference.

12.3 Minimal Example

A|B   Δ   ?

10The transition generates no fixed order,

11but a range of possible orders.

12Written operatively:

A|B   Δδ   A|B|C|D …

12.4 Indeterminacy of Higher Order

13Indeterminacy can itself be distinguished:

(A|B   Δδ   A|B|C|D)   Δδ   …

14Here not only the order is indeterminate,

15but the structure of the transition itself.

16This is fractal indeterminacy.

12.5 Relation of δ to the Fractal Operators

17\(\circleddash\) without δ → structured recursion

18\(\circleddash\) with δ → unstable recursion

19\(\widetilde{\circleddash}\) with δ → chaotic oscillation

20Indeterminacy is therefore no exception,

21but a mode of unfolding.

12.6 Memorandum

22Indeterminacy does not mean absent order,

23but absent fixation of the order.

Chapter XIII

Observer as Relational Position

13.1 Positing

1An observer is not an external subject,

2but a position within the order of difference.

3It is noted as a relation of its own.

13.2 Observer Notation

4\(\circledcirc\)

5\(\circledcirc\) designates a marked position of distinction.

13.3 Observer Difference

6The difference between observer and order is written as:

7Reading:

8The order A|B is distinguished from the standpoint of a particular relational position.

13.4 Effect of the Observer

9The observer generates no new order,

10but fixes transitions.

11Example:

A|B   Δδ   A|B|C|D
⊚ | (A|B   Δ   A|B|C|D)

12What previously was indeterminate

13is determined by the observer relation.

13.5 Several Observers

14Several observers generate no objectivity,

15but coupled perspectives:

16Objectivity here is no agreement,

17but compatibility of the relations.

13.6 Memorandum

18An observer is not a standpoint outside the structure,

19but a fixation within the relation.

Chapter XIV

Conclusion: Transition to the Nonlinear System

1With the operators \(\circleddash\), \(\circledast\), \(\widetilde{\circleddash}\), \(\mathcal{R}\) and the introduction of δ and \(\circledcirc\)

2the system is fully operable.

3From here on:

4Transitions act on transitions

5Indeterminacy is structural

6Observation is relational

7Networks cannot be unfolded linearly

Chapter XV

Overview of the Principio Fractal Operators

1(with minimal examples of application)

1. Basic Notation of the Orders of Difference

|                Difference of 1st order
A|B              Difference of 2nd order
A|B|C|D          Difference of 3rd order
A|B|C|D|…        higher self-distinction

2Application (minimal):

3Structuring

4Classification

5Basic relation without sets

2. Transition (Δ)

6Meaning:

7Transformation of one order of difference into another.

O₁   Δ   O₂

8Application:

9Modeling of change

10Structural development

11Transition between states

12Example:

A|B   Δ   A|B|C|D

3. Fractal Difference (\(\circleddash\))

13Meaning:

14Recursive transformation of the transition itself.

15Application:

16multi-stage development

17nonlinear change

18structural growth

19Example:

A|B   ⊝   A|B|C|D

20(the transition generates new transition structure)

4. Fractal Linkage (\(\circledast\))

21Meaning:

22Simultaneous coupling of several transitions.

23Application:

24Networks

25Parallel processes

26coupled dynamics

27Example:

5. Fractal Oscillation (\(\widetilde{\circleddash}\))

28Meaning:

29Unstable recurrence between orders of difference.

30Application:

31cyclical processes

32alternating states

33resonances

34Example:

6. Fractal Rotation (\(\mathcal{R}\))

35Meaning:

36Directed transition with change of orientation.

37\(\mathcal{R}\) := directed Δ

38Application:

39change of perspective

40phase shift

41inversion of relations

42Example:

A|B   ℛ   B|A

7. Indeterminacy \(δ\)

43Meaning:

44Non-fixability of an order of transition.

45Δδ

46Application:

47open systems

48unstable transitions

49ambiguity without chance

50Example:

A|B   Δδ   A|B|C|D …

8. Observer (\(\circledcirc\))

51Meaning:

52Fixed relational position within the structure.

53\(\circledcirc\) | O

54Application:

55perspectives

56measurement

57stabilization of transitions

58Example:

59\(\circledcirc\) | (A|B Δ A|B|C|D)

9. Networks

60Meaning:

61Nested transition structures that cannot be unfolded linearly.

62Application:

63complex systems

64distributed dynamics

65self-referential structures

66Example:

10. Summary Memoranda

67Differences generate orders.

68Transitions generate dynamics.

69Fractal operators describe self-relatedness.

70Nothing is measured.

71Everything is put into relation.

11. Minimal Path of Application

72Linear:

73Nonlinear:

74Fractal:

75Example Process 1: Stabilization from a Simple Transition

76Goal: show how a fixed point arises from Δ.

77Start:

78A|B

79Step sequence:

A|B   Δ   A|B|C|D
A|B|C|D   Δ   A|B|C|D

80Reading:

81First expansion of the order, then stabilization (fixed point).

82Result form:

O   Δ   O

83Example Process 2: Oscillation between Two Orders

84Goal: show how \(\widetilde{\circleddash}\) functions as an alternation of orders.

85Start:

86A|B

87Step sequence:

A|B   ⊝̃   A|B|C|D   ⊝̃   A|B   ⊝̃   A|B|C|D   ⊝̃   …

88Reading:

89No order stabilizes itself; the structure oscillates.

90Result form:

O₁   Δ   O₂   Δ   O₁   Δ   O₂   …

91Example Process 3: Fractal Difference (\(\circleddash\)) as “Transition of the Transition”

92Goal: show that \(\circleddash\) is not “more Δ,” but Δ on Δ.

93Start:

94A|B

95Step sequence (recursive):

A|B   Δ   A|B|C|D
(A|B   Δ   A|B|C|D)   Δ   (A|B|C|D   Δ   A|B|C|D|E|F|G|H)

96Short notation:

A|B   ⊝   A|B|C|D|E|F|G|H

Chapter XVI

1Reading:

2Not only does the order grow, but the structure of growth is distinguished anew.

3Example Process 4: Network Formation through Coupling (\(\circledast\))

4Goal: show how two processes structure each other reciprocally.

5Start (two subprocesses):

6P: A|B

7Q: E|F

8Individual steps:

A|B   Δ   A|B|C|D
E|F   Δ   E|F|G|H

9Coupling:

(A|B Δ A|B|C|D)   ⊛   (E|F Δ E|F|G|H)

10Continuation as network:

((A|B Δ A|B|C|D) ⊛ (E|F Δ E|F|G|H))   Δ   R|S|T|U|…

11Reading:

12The coupling generates a new order that is not reducible to either of the two strands.

13Example Process 5: Indeterminacy \(δ\) and Observer (\(\circledcirc\)) as Fixation

14Goal: show that δ is structural and \(\circledcirc\) fixes transitions.

15Start (indeterminate transition):

A|B   Δδ   A|B|C|D …

16The observer establishes a fixation:

⊚ | (A|B   Δ   A|B|C|D)

17Consequence:

A|B   Δδ   …
⊚ | (…)  ⇒  A|B   Δ   A|B|C|D

18Reading:

19δ means: transition not fixed.

20\(\circledcirc\) means: transition is drawn as a determinate relation.

21Cookbook of Principio Fractal Mathematics

22(Operative Patterns)

23Recipe A – Stabilizing

24Goal:

25Fix an order of difference.

26Ingredients:

27Order O

28Transition Δ

29Procedure:

O   Δ   O

30Example:

A|B|C   Δ   A|B|C

31Result form:

32Fixed point, stable structure

33When to use:

34Classification

35Invariance

36Conclusion of a development

37Recipe B – Oscillating

38Goal:

39Force alternation between two orders.

40Ingredients:

41Two orders O₁, O₂

42Oscillation \(\widetilde{\circleddash}\)

43Procedure:

O₁   ⊝̃   O₂   ⊝̃   O₁   ⊝̃   …

44Example:

A|B   ⊝̃   A|B|C|D   ⊝̃   A|B   …

45Result form:

46Periodic alternation of order

47When to use:

48Resonances

49boundary states

50rhythmic dynamics

51Recipe C – Growing (fractal difference)

52Goal:

53Not only expand structure, but deepen transitions.

54Ingredients:

55Initial order O

56Fractal difference \(\circleddash\)

57Procedure:

O   ⊝   O'

58Example:

A|B   ⊝   A|B|C|D|E|F|G|H

59Result form:

60Self-similar expansion

61When to use:

62Development

63Scaling

64nonlinear growth

65Recipe D – Networking (coupling)

66Goal:

67Make several processes effective simultaneously.

68Ingredients:

69Two transitions

70Linking \(\circledast\)

71Procedure:

(O₁ Δ O₂)   ⊛   (P₁ Δ P₂)

72Example:

73Result form:

74Nonlinear network

75When to use:

76Parallel processes

77complex systems

78coupled dynamics

79Recipe E – Allowing Indeterminacy

80Goal:

81Keep transition open.

82Ingredients:

83Order O

84Transition with indeterminacy Δδ

85Procedure:

O   Δδ   O' …

86Example:

A|B   Δδ   A|B|C|D …

87Result form:

88Range of possible orders

89When to use:

90open systems

91exploration

92unstable processes

93Recipe F – Fixing (inserting observer)

94Goal:

95Determine an indeterminate transition.

96Ingredients:

97Observer \(\circledcirc\)

98Transition

99Procedure:

100Example:

101\(\circledcirc\) | (A|B Δ A|B|C|D)

102Result form:

103Determinate transition

104When to use:

105Measurement

106decision

107perspective fixation

108Recipe G – Orienting (rotation)

109Goal:

110Reverse or reread relation.

111Ingredients:

112Order O

113Rotation \(\mathcal{R}\)

114Procedure:

O   ℛ   O⁻¹

115Example:

A|B   ℛ   B|A

116Result form:

117Oriented relation

118When to use:

119change of perspective

120inversion

121phase shift

122Short Rule (for Practice)

123Δ changes orders.

124\(\circleddash\) changes transitions.

125\(\circledast\) couples dynamics.

126\(\widetilde{\circleddash}\) keeps systems in motion.

127δ opens.

128\(\circledcirc\) fixes.

129Example Run 1: Stabilizing, Growing, Oscillating (with Numbers)

130Start: number 2 as order

131Step A: Transition (Δ) – “one order further”

0|1|2   Δ   0|1|2|3   = O(3)

132Step B: Stabilization (fixed point)

133We assume that it remains stable under further Δ (fixed-point pattern):

0|1|2|3   Δ   0|1|2|3

134Step C: Fractal Growing (\(\circleddash\)) – “transition of the transition”

135Instead of merely increasing the order by +1, we make a recursive leap:

0|1|2   ⊝   0|1|2|3|4   = O(4)

136(Interpretation: not “+2,” but Δ is structurally deepened; the result appears as a greater unfolding.)

137Step D: Oscillation (\(\widetilde{\circleddash}\)) between 3 and 4

138O(3) \(\widetilde{\circleddash}\) O(4) \(\widetilde{\circleddash}\) O(3) \(\widetilde{\circleddash}\) O(4) …

139Example Run 2: Coupling (\(\circledast\)) of Two Number Processes

140We let two “number orders” run simultaneously:

141Both make a transition:

P: 0|1|2|3   Δ   0|1|2|3|4
Q: 0|1|2     Δ   0|1|2|3

142Now we couple the transitions:

(0|1|2|3 Δ 0|1|2|3|4)  ⊛  (0|1|2 Δ 0|1|2|3)

143Concrete effect (minimally visible):

144The new order takes up both lines of development as simultaneously effective, e.g. as a nested order:

0|1|2|3|4  ⊛  0|1|2|3   ⇒   0|1|2|3|(4|3)

145(“4|3” stands here as a coupling node: two ends that do not merge into one final value, but persist as relational tension.)

146Example Run 3: Indeterminacy \(δ\) and Observer (\(\circledcirc\)) — “the number is not fixed”

147Indeterminate transition from O(3)

0|1|2|3   Δδ   0|1|2|3|4  …

148Here δ means: The transition is not fixed (e.g. it could also unfold otherwise).

149We show this as two possible subsequent branches:

0|1|2|3   Δδ   { 0|1|2|3|4   or   0|1|2|3|5 }

150Observer fixes (\(\circledcirc\))

151Thus the indeterminacy becomes a determinate path:

0|1|2|3  →  0|1|2|3|4