The derivation unfolds in a forced sequence. Each step produces something whose own nature forces the next step. This document records the sequence and, at each step, the canonical name for what is produced.
Given: the impossibility of nothing
Nothing requires thing, equivalence, and negation to be nothing. These are already something. Something exists. What exists has dynamics that must be derived.
Produced: thing, negation, equivalence (not derived — they are what nothing needed to fail)
1. Existential coherence
What must be the case for something to exist?
- Including — the act: something includes itself in what it is
- Excluding — the act: something excludes what it is not
- Inclusion — the condition: what something is
- Exclusion — the condition: what something is not
- Coherence — the structure: the minimal unit stabilizing the interplay of including and excluding
Undetermined: how coherence is sustained.
2. Relational coherence
What sustains coherence?
- Relating — the act that sustains the interplay of including and excluding
- Relation — the condition: inclusion and exclusion coherently maintained
- Relational form — the structure: the minimal configuration formalizing the interplay of relating and relation
Properties introduced:
- Reflexive sequence — relating can relate to itself, producing indefinitely deep structures
- Positional invariance — inclusion and exclusion stay distinct and ordered through deepening
- Composition — relational forms can be placed side by side into ordered wholes
Undetermined: what holds relational form together.
3. Self-sustaining closure
What holds form together?
- Sustaining — the act that mediates between relating and relation, ensuring their joint persistence within form
- Self-coherence — the condition: relating and relation stabilized as a unified dynamic
- Closure — the structure: the self-maintenance of relation through relating
Produced: a self-sustaining relational unit.
Undetermined: the unit has no boundary. It hasn’t distinguished itself from its outside.
4. Boundary
What distinguishes the unit from its outside?
- Bounding — the act of excluding what lies beyond the unit
- Distinction — the condition maintaining the division between relation and non-relation
- Boundary — the structure stabilizing inside vs. outside
Produced: a bounded relational unit.
Undetermined: the boundary is inert — the unit hasn’t engaged it.
5. Reflexion
What happens when the unit engages its own boundary?
- Folding — the act of turning relation upon the boundary itself
- Self-relation — the condition: the boundary is now part of the system
- Reflexive form — the structure: boundary folded into the system, stabilized
Produced: a reflexive relational unit — self-sustaining, bounded, self-engaging.
Undetermined: in distinguishing itself from what it’s not, the unit has implied there may be others.
6. Multiplicity
What about the other side of the boundary?
- Differentiating — distinguishing from other units
- Co-presence — multiple units in differentiated yet shared existence
- Tension — the structure stabilizing distinct units in interplay
Structures that arise from tension:
- Chain — a linear sequence of units at shared depth
- Network — non-linear configurations of chains
- Node / Edge — units as positional elements, connections as relational bindings
Undetermined: local and global coherence of the multirelational unit.
7. Field coherence
How do many units cohere, both locally and globally?
Twin dynamics (local and global are co-determined):
- Inter-unit relating + Field-integrating — twin acts
- Mutual relation + Field coherence — twin conditions
- Mutual form + Field form — twin structures
Properties:
- Depth alignment — units align reflexive sequences for coherent engagement
- Closure criteria — the field is closed when all units are engaged and no further differentiation is induced
- Reflexive equilibrium — further deepening doesn’t change the form
Produced: an integrated relational field.
Undetermined: the field has no meta-boundary.
8–9. Meta-boundary and meta-reflexion
The field recapitulates the unit’s journey: it acquires a boundary (8), then folds it in (9). This recurrence produces:
- Recursive domain unfolding — each closure exposes the next domain
- Predictive determination — each closure forecasts its successor’s shape
10. Terms
The relational apparatus hardens into manipulable expressions.
- Term — a position that refers to something within a context
- Variable — a named position
- Function — a body that returns to its context
- Application — function meeting argument
- Fixed point — self-application finding stable configuration
- Reduction — simplification by applying functions to arguments
- Value — a term that can’t be reduced further
11. Observation and judgement
The system witnesses what it has produced.
- Observation — what can be seen from a term within a context
- Judgement — a term, in a context, observed to have a property (the triadic assertion)
12. Order and algebra
Judgements stand in relations to each other. Those relations force:
- Order — some judgements subsume others
- Meet — greatest common refinement of two judgements
- Join — least common coarsening of two judgements
- Implication — if this judgement, then that one
- Negation — implying the bottom (the trivially empty judgement)
Produced: a Heyting algebra — constructive logic. Not classical logic: proof of existence requires construction.
Properties forced by the algebra on the syntax:
- Soundness — well-typed terms don’t go wrong
- Confluence — different reduction paths reach the same result
- Normalization — every term reaches a value
13. Stability
Before dynamics, the system must freeze what it has.
- Stability — fixing traces, terms, judgements, observations in place so they can be revisited without drift
14. Flow and nucleus
Two dimensions open.
- Flow — how things move through contexts over time
- Nucleus — how things settle under closure (the smallest closed structure containing something)
15. Geometry
Flow and nucleus commute.
- Geometry — the relational space where dynamics (flow) and closure (nucleus) cohere
- Residuation — the law governing the interplay of flow and nucleus
16. Disciplines, filters, profiles
Geometry carves out internal universes.
- Discipline — a structural pattern that respects both flow and nucleus
- Regime — the set of structures a discipline stabilizes
- Filter — a discipline that commutes with all nuclei and all flows
- Profile — what a filter carves out: a complete relational universe inside the larger one, reconstructing the entire derivation internally
Profiles nest: filters within profiles yield sub-profiles, each containing the full tower.
17. Physics
Geometry acts on the contents of profiles.
- Observable — witnessing applied to a term within a profile
- State — recognitions fixed under a profile’s flow
- Evolution — flow applied to a state
- Measurement — nucleus applied to a state
Evolution and measurement interact through residuation.
18. Grand closure
The structure, applied to itself, yields itself. Profiles reconstruct the full tower. The full tower produces profiles. Fixed point.