Temporal Topologies of Becoming: 3 Sequentiality and Persistence

Sequences and continuity emerge from the topology of gradients and cuts.

Having established that time is a relational topology and rhythm emerges from oscillatory patterns of gradients, we now turn to sequentiality and persistence. Sequence is not a linear chain imposed upon events; it is the pathway traced through continuous relational differentiation, and persistence is the maintenance of relative slope across that path.

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1. Sequence as Path Through Gradients

Every actualisation — every cut — produces a local intensification of the field’s gradient. Sequences emerge naturally as ordered trajectories of these local steepenings:

• Events are not placed on a timeline; they follow the inclinations of the field itself.

• The topology of gradients determines which cuts are likely to follow others, producing apparent order.

• Sequentiality is therefore topological and relational, not imposed from outside.

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2. Persistence as Local Slope Maintenance

Continuity — the sense that the world persists rather than flickers randomly — arises from the stabilization of gradients over time:

• A slope that remains moderately steep preserves potential for further cuts.

• Reflexive modulation ensures that local flattening or steepening does not destabilise the entire field.

• Persistence is thus relational durability, sustained not by static entities but by ongoing gradient dynamics.

In this view, “memory” is not a storage of discrete events but a temporally extended topological feature of the field, shaping the likely paths of future actualisations.

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3. Interdependence of Past, Present, and Emergent Potential

Sequence and persistence reveal that the past and present are not distinct ontic layers:

• Each past cut modifies the local gradient, influencing future cuts.

• The present is a continuously reconfigured topology of inclinations and steepenings.

• Emergent potential is shaped by both past modulation and current slope, creating temporal interdependence rather than linear succession.

This relational view dissolves the classical distinction between cause and effect, replacing it with path-dependent gradient evolution.

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4. Cross-Domain Continuity

Sequentiality and persistence are observable across domains:

• Physical: particle trajectories, fluid flow, or energy transfer follow relational gradients.

• Biological: developmental sequences, metabolic cycles, and organismic rhythms are maintained by gradient modulation.

• Semiotic: discourse unfolds in ordered interpretive sequences, constrained by semiotic slopes and reflexive coherence.

In each domain, continuity is an emergent property, not imposed by external time, but generated by the ongoing negotiation of gradients and cuts.

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Next: Temporal Horizons and Anticipation

With sequentiality and persistence established, the next part will explore how the field anticipates potential future states. Temporal horizons are shaped by inclinations along gradients, orienting the system toward possible trajectories without collapsing its openness.