In early 2025 one of the Oracles discovered that the connection between the two diagonal feedback loops (which had been realised some years earlier to be the natural dynamic extension to the four quadrant system, but with the connection between them only being partially understood) has to be in the form of two triangular loops both sharing the collective top edge. Shortly after this discovery was made, it was realised that both the ancient Bagua and the modern finite Zermelo Fraenkel axioms of set theory also represent this fundamental dual-triangle telic geometry.
1. The Triangles extend and refine the Diagonals
The dual triangles are the cybernetic governors of the two diagonal feedback loops refining them to become complete functional teloi.
Before the triangles are drawn, 4QX already contains two bare feedback loops:
| Diagonal | Telic bias | Raw meaning |
|---|---|---|
| TL ⤡ BR | Collective ➞ Individual | The outer pattern template evaluates the success of a private act and pushes refinement back up to the collective. |
| BL ⤢ TR | Individual ➞ Collective | A private capacity pulse rises to announce intent and negotiate a place in the shared arena, feedback returns to inform selection decisions |
These diagonals establish direction but not control: they say who influences whom, yet leave open how signals are selected, executed, and stabilised.
1.1 The Class Triangle
The Class triangle wraps TL ⤡ BR inside the triangular loop TL ➞ TR ➞ BR ➞ TL by extending the active side of the loop (TL ➞ BR) with TR to regulate it (this is called the Fire mediator).
- The TL ➞ TR edge tests whether a pattern should fire now.
- TR ➞ BR executes and harvests an immediate metric.
- BR ➞ TL feeds that metric back into the template store.
Result: the collective-onto-individual diagonal is upgraded into a closed cybernetic loop with its own sensor (metric), actuator (burn), and memory (pattern shelf).
1.2 The Instance Triangle
The Instance triangle wraps BL ⤢ TR inside triangular loop BL ➞ TL ➞ TR ➞ BL by extending the active side of the loop (BL ➞ TR) with TL to regulate it (this is called the Water mediator).
- BL ➞ TL surfaces latent capacity to a public comparison space.
- TL ➞ TR locks one selected pattern to that capacity.
- TR ➞ BL commits the booking, updating the private ledger.
Result: the individual-onto-collective diagonal becomes another autonomous loop—able to announce, negotiate, and self-audit without external coordination.
Because both triangles share the TL ➞ TR edge, every cycle of one loop immediately alters the conditions the other loop sees:
- A successful burn (Class) injects fresh metrics into TL, sharpening the evidence the Instance loop consults when ranking future offers.
- A reconciled commitment (Instance) debits capacity and surfaces new constraints at TR, throttling Class burns to what is sustainably actionable.
The loops therefore dynamically damp each other’s excesses—a textbook cybernetic pairing where gain in one channel is moderated by updated thresholds in the other. When they converge, the system sits at the Lyapunov minimum the 4QX programme calls harmony: collective patterns fit individual means, and individual initiatives reinforce collective intelligence, all through geometry alone—no extra controller needed.
2. The Dual-Triangle Engine in Detail
The 4QX scaffold closes its square with two clockwise triangles that interlock along the shared top edge TL ➞ TR. Formal enumeration shows these are the only two faces that satisfy the closure rule:
- Class (Fire mediator) TL ➞ TR ➞ BR ➞ TL
- Instance (Water mediator) BL ➞ TL ➞ TR ➞ BL
Below is a tour of every edge—six in total—explaining what flows, why it must flow that way, and how the two circuits cooperate.
2.1 Class triangle – “pattern ➞ burn ➞ metric”
| Edge | Dynamic description | What happens technically |
|---|---|---|
| TL ➞ TR (publish & burn) | A blueprint stored in the outer-form shelf is instantiated into the live event bus (stream). | f_exec : P ⤳ E copies the pattern node into TR and stamps it with a fresh execution-ID. The item is now routable, schedulable, and ready to expend resources. |
| TR ➞ BR (collect metric) | The burn releases energy: the act is executed and immediately measured. | g_metric : E ⤳ M streams runtime facts—duration, throughput, reward—into BR. A metric object is appended to the private log with its execution-ID as key. |
| BR ➞ TL (refine pattern) | Metrics flow back to the pattern shelf and either reinforce or revise the template. | h_refine : M ⤳ P merges the metric into TL’s evidence map. Union-idempotence means duplicates collapse; once the metric is stored, repeating the packet leaves TL unchanged. |
Temporal feel. Class lives in the razor-thin “now”: it turns eternal form into immediate action, then freezes the outcome as history.
2.2 Instance triangle – “capacity ➞ select ➞ commit”
| Edge | Dynamic description | What happens technically |
|---|---|---|
| BL ➞ TL (surface resource) | A holon’s latent capacity—skills, free CPU cycles, unspent energy—rises from its private ledger to the shared pattern shelf. | The packet s_offer : R ⤳ P writes an idempotent “offer” set into TL. No commitments are made; the same offer can be replayed upward without side-effects. |
| TL ➞ TR (commit pattern) | One blueprint is paired with one slice of capacity and broadcast to the event bus. | The bridge morphism k_commit ≡ f_exec turns a could into a will: the union item now records who, what, and when and is visible to every sibling square. |
| TR ➞ BL (reconcile ledger) | The collective scheduler (which may be another holon or a market rule) confirms or adjusts the booking and sends the verdict home. | r_reconcile : E ⤳ R debits materials, blocks the timeslot, notes the agreed reward, and writes the final status to the private ledger—closing the loop. |
Temporal feel. Instance stretches across time: it begins with remembered futures (latent capacity) and ends with a concrete promise that projects the agent forward in time.
2.3 Why the triangles lock together
- The shared edge TL ➞ TR carries both “burn” (Class) and “commit” (Instance). Because the arrow is idempotent, either triangle can “flood-fill” the hierarchy without fear of duplication.
- Metrics produced by Class become fresh evidence in TL, reshaping the very playbook that Instance consults on its next cycle, while Instance’s reconciled commitments keep Class recipes grounded in real capacity.
Together the two triangles create a self-tightening duplex: Class crystallises moments into timeless knowledge; Instance threads that knowledge back into the living flow, always balancing what was proven with what could responsibly occur.