Before the oriented Fire–Water triangles give the 4QX lattice its direction and its universal telos, the plain 2 × 2 grid with its two diagonals already forms a self‑sufficient agent–arena space. At this “monad‑but‑not‑yet‑triangle” level:
- the four corners act as minimal roles (Pattern, Event, Resource, Metric),
- each diagonal is a closed feedback loop that alternates those roles, and
- the two loops, while undirected, jointly enable class‑instance co‑evolution—the default telic drive of any autonomous system.
This article explains that pre‑oriented world, why it counts as a functional system, and what undirected telic means in practice.
1 The 2 × 2 Grid Recap
We start with two binary splits produced by finite‑ZF self‑reference:
| Axis | Bit 0 | Bit 1 |
|---|---|---|
| Perspective | Outer | Inner |
| Modality | Form | Flux |
The Cartesian product gives four loci:
Perspective ↓
Outer‑Form TL | TR Outer‑Flux
-----------------+----------------
Inner‑Form BL | BR Inner‑Flux
Modality →
For clarity we attach the neutral role‑labels already standard in 4QX documentation, but without orienting arrows:
TL Pattern store (P)
TR Event ledger (E)
BL Resource cache (R)
BR Metric buffer (M)
2 Undirected Diagonals as Telic Loops
2.1 Why only the Diagonals close
Horizontal or vertical edges swap just one polarity; they open a distinction but do not return. A diagonal flips both bits and then flips them back on the return leg, giving a 2‑step there‑and‑back.
Class‑like loop P ◄──► M (TL ↔ BR)
Instance‑like loop R ◄──► E (BL ↔ TR)
Each loop can repeat indefinitely without accumulating paradox because after two hops the composite map is the identity.
2.2 Semantic Reading
| Loop | Step 1 | Step 2 | Interpretation |
|---|---|---|---|
| P ↔ M | Pattern → Metric | Metric → Pattern | “Evaluate blueprint, stash measurement; treat measurement as refined blueprint.” |
| R ↔ E | Resource → Event | Event → Resource | “Spend capacity, record event; treat record as the new capacity baseline.” |
Because no orientation is fixed, each hop can be read either way—the loop is bi‑telic. What makes it telic is that each completion reconciles description with occurrence.
3 Class–Instance Co‑Evolution
Label one loop Class‑like (Pattern–Metric) and the other Instance‑like (Resource–Event). Alternating the two gives a four‑beat rhythm:
1 P (form) — blueprint exists
2 R (inner form) — capacity offered
3 E (flux) — action witnessed
4 M (inner flux) — outcome measured
back to P …
A whole cycle therefore:
- compares what should be (P) with what can be (R),
- enacts something (E),
- folds reality back (M → P).
Even without oriented triangles, the quadrants co‑evolve: each completion subtly updates the context for the next.
4 Why This Counts as a Functional System
| Criterion | How the 2×2+diagonals satisfies it |
|---|---|
| Autonomy | Both loops close locally; no external node is required to finish a thought. |
| Continuity | Re‑entering the same loop requires only re‑reading its two vertices—state is preserved. |
| Purpose (telic) | Each completion tends to minimise the discrepancy between the two vertices’ content; that contraction is the telic drive, albeit undirected. |
| Composability | Loops can interleave freely: P→M hop can trigger an R→E hop, enabling richer rhythms. |
Hence the grid + diagonals is already a minimal self‑correcting engine.
5 Relationship to the Monad
5.1 Unit and Join Before Orientation
In the full 4QX monad the oriented BL→TL→TR→BL and TL→TR→BR→TL triangles realise η and μ. When arrows are dropped:
- A unit becomes “either direction along R–E–R,”
- A join becomes “either direction along P–M–P.”
The monad laws reduce to id ∘ id = id on each diagonal—still true.
5.2 Potential for Orientation
Because the edges are undirected, a future agent can choose which way to read them, polarising one half as “give” and the other as “take.” Orientation is deferred choice, not additional structure.
6 From Undirected to Living Space
Adding a single mediator to each diagonal (the step that yields the Fire/Water triangles) freezes one direction as “active” and the return as “passive.” That orientation turns the neutral loops into cybernetic gears that mesh and pump entropy outward. But the seed of purpose—and the class‑instance story—was already latent in the two undirected diagonals.
7 Conclusion
The plain 2 × 2 grid plus its two diagonals constitutes the default telic monad space: a pair of self‑closing feedback loops that perpetually reconcile blueprint with fact and capacity with deed. Only after we orient the loops do we obtain the dual triangles and the stronger universal telos, but the essence of self‑maintaining agency is present as soon as the diagonals exist.