One-shot ZF axioms → the 4QX names you use every day
| Single-use axiom (Finite-ZF) | Geometry move it performs | 4QX place / action it brands | Everyday wording |
|---|---|---|---|
| Empty ∅ exists | creates the very first vertex TL | Pattern bucket | “blank blueprint” |
| Pair {x,y} | flips inside / outside once, giving TR | Event stream | “first thing that actually happens” |
| Replacement f(x)= {x} | nests TR one brace deeper → BL | Resource ledger | “capacity copied inside for later use” |
| Union ⋃ | flattens a one-item family {{∅}} → BR | Metric evidence | “ground truth rolled up” |
| Power-set 𝒫 | BL → TL edge δ | Offer step (surface slack to planning) | “everything I could spare” |
| Separation {s⊆x | φ} | TL → TR edge ε | Commit step (filter pattern into an executable slice) |
The first four axioms carve the four quadrants once; the last two axioms wire the two feedback diagonals exactly once.
Appendix B proves each mapping is unique: every multiplex edge corresponds to one—and only one—constructor, and vice-versa .
How the mapping hangs together
- Empty ⇒ Pattern.
Starting from nothing, Empty selects ∅; that set is the TL vertex, so the very notion of “a blueprint that can later change” is born here. - Pair ⇒ Event.
Pairing the void with itself wraps ∅ once, giving {∅}. That new locus lives on the “outer + flux” corner (TR), the place where the organisation surfaces its live acts. - Replacement ⇒ Resource.
Applying f(x)= {x} to TR pushes its contents one brace inward, producing {{∅}} (BL). Whatever you still hold inside but haven’t spent yet is exactly what we call Resource. - Union ⇒ Metric.
Folding the singleton family {{∅}} leaves the brace count odd, anchoring the result in BR (inner + flux). Union is literally “roll-up many small facts into one digest” → Metric. - Power-set ⇒ Offer.
𝒫(∅) = {∅}. Surfacing that subset is the BL → TL hop where hidden slack becomes a candidate pattern. In ops terms: “here’s every team-hour we could allocate.” - Separation ⇒ Commit.
Filtering the TL bucket with a true predicate keeps only the slice you’re ready to execute; the arrow lands back in TR as the concrete work order. That’s the commit step every sprint depends on.
Big picture
- Four nouns (Pattern, Event, Resource, Metric) = the four objects created by the first four axioms.
- Two verbs (Offer, Commit) = the two process edges installed by the last two axioms.
Run once, these six moves finish the square plus its two diagonal feedback paths; nothing else is needed to ignite the telic engine .