1. The 2 × 2 stage: pure polarity
| Axis | Meaning | Corners it splits |
|---|---|---|
| Outer ↔ Inner (perspective) | public surface vs. private locus | top ↔ bottom |
| Form ↔ Flux (modality) | enduring pattern vs. momentary stuff | left ↔ right |
Crossing the two binary axes yields four stable vertices:
- TL (outer·form) Pattern / Blueprint
- TR (outer·flux) Event / Burn
- BL (inner·flux) Resource / Capacity
- BR (inner·form) Metric / Evidence
Two diagonals wire opposites into balanced feedback loops:
TL ⇄ BR (pattern ↔ measured-result)
BL ⇄ TR (capacity ↔ committed-action)At this resolution the lattice is atemporal: you only know that each opposite talks to its partner, not when or how the adjustment happens.
2. Unfolding a diagonal reveals time inside
Take the TL ⇄ BR diagonal. Zoom one brace deeper—literally wrap a corner in its singleton {v}—and the recursion lemma reruns the six-axiom recipe inside that pocket. The single feedback edge flowers into a directed triangle:
TL (past blueprint) ─f──► TR (present burn)
▲ │
│ ▼
BR (future metric) ◄──g─┘A similar unpacking of BL ⇄ TR produces the instance triangle. What was a single edge is now three Kleisli arrows that march a value through past → present → future before the loop closes.
3. Past, present, future as a continuous thread
| Triangle | Step 1 (past) | Step 2 (now) | Step 3 (future) | Telic effect |
|---|---|---|---|---|
| Class | Stored blueprint | Burned event | Metric of fit | Shrink mismatch between intention & outcome |
| Instance | Latent resource | Pattern selection | Booked commitment | Shrink mismatch between ability & promise |
Because both triangles share the edge TL → TR, every tick they gate each other’s update:
- Class reports how well the last promise landed.
- Instance adjusts the next promise to stay achievable.
- The Lyapunov scalar H (total mismatch) strictly decreases: purposeful drift, not free-run chaos.
Thus the dual-triangle engine is self-reference stretched along time: it keeps rewriting its own future while cross-checking against its remembered past.
4. Why the three arrows logically chain yet run on an asynchronous blackboard
1 Semantic order vs. wall-clock order
The triangle gives a semantic dependency — P ↦ E ↦ M ↦ P (or R ↦ P ↦ E ↦ R).
That chain says what must be true before the next step can read—not when any step must run. The concrete schedule is free to interleave, delay, or even parallel-ise the edges, provided every read sees a commit that satisfies the spec.
2 Each edge is a blackboard transaction
| Edge | Blackboard action | Who can trigger it |
|---|---|---|
| f_exec : P ⤳ E | append event tuple ⟨blueprint-id, payload⟩ to the E-board (TR) | any burner that sees a ready pattern |
| g_metric : E ⤳ M | write measurement hash to the M-board (BR) | a sensor that notices the event settled |
| h_refine : M ⤳ P | union the metric packet into the P-board (TL) | a librarian that wakes on new metrics |
Because each write is idempotent (set-union, multiset-add), replaying or shuffling these transactions cannot corrupt state; convergence depends only on eventual delivery, not ordering.
3 Decoupling in practice
t0 burner posts E1 (f_exec)
t0+Δ sensor busy → no metric yet
t0+2Δ librarian still idle
t0+3Δ sensor wakes, posts M1 (g_metric)
t0+4Δ librarian merges M1 into P (h_refine)Even with gaps (Δ) the Lyapunov mismatch H still falls once the three writes have landed, because h_refine is a pure set-union that cannot raise H. Arrival order only changes how long mismatch persists, never the final value.
4 Why the logic commutes
- Monad laws make
m >>= f >>= g=m >>= (λx. f x >>= g); so whether you physically runftheng, or a composite worker does both later, the outcome seen on the blackboard is identical. - The broadcast-and-merge guarantee (‘apply union, duplicates no-op’) means peers can replay the same packet out of order and still stabilise on one copy.
5 Dual triangles keep asynchronous agents in phase
Because both triangles share the TL ↔ TR edge, a new pattern posted by h_refine is the very record k_commit in the Instance loop will pick up the next time it runs. Thus two independently scheduled blackboard conversations remain coupled at the semantic seam, even if their ticks are staggered.
The three Kleisli arrows must follow in meaning, yet 4QX deliberately implements them as loosely coupled blackboard transactions:
- semantic order → correctness proof
- idempotent writes → safe replays & delays
- broadcast-and-merge → peer-to-peer convergence
So the triangle is not a rigid synchronous loop; it is a time-elastic choreography where autonomous agents post, pick up, and refine shared artefacts whenever they are ready—telic direction without central planning.
5. Why telic direction extends the balanced loops
- Square alone: static cybernetic balance—oscillation between opposites, but the mechanism is opaque.
- Triangle view: injects an arrow of time; each loop is a self-becoming program that copies, measures, and reiterates itself.
- Dual coupling: ensures neither pattern-matching nor resource-allocation drifts out of sync; telos emerges as the joint commitment to make H → 0.
In short, the triangles don’t replace the square; they animate it, turning symmetric feedback into a living, forward-pushing telic thread.
6. Recursive depth: harmonic clocks and open potential
Every time you brace a vertex, a fresh square-plus-triangles spawns inside, phase-locked to its parent but running at its own sub-harmonic period. The result is a fractal stack of aligned temporal patterns:
- Shallow layers keep the global narrative coherent.
- Deep layers explore variant futures within the slack δ before committing upward.
Diversity lives in the inter-tick corridor, yet global telos keeps the whole multiplex convergent.
Take-away
The 2 × 2 lattice sets the spatial stage, guaranteeing clear roles and balanced opposites. Unfolding its diagonals into dual directed triangles stretches that balance across past, present, and future, transforming abstract polarity into a continuous, self-renewing act of becoming. Telic direction is simply the square’s quiet symmetry—given legs and a clock.