How bottom-up aggregation automatically yields Gödel-style codes, hash-table semantics and associative referencing—without any extra namespace layer
1 From Dual-Triangle Flow to “Addresses”
- Top-down (TD) drive —Pattern → Event → Metric → Pattern executes forecasts, measures δ, and refines the plan.
- Bottom-up (BU) return —Resource → Pattern → Event → Resource aggregates concrete outcomes back into the shared store.
Because every BU hop must re-enter the same four-corner lattice, the path a packet travels is fully described by the binary choices it makes at each recursion step:
| Choice | 0-bit | 1-bit |
|---|---|---|
| Perspective (outer / inner) | outer | inner |
| Modality (state / flux) | state | flux |
Concatenate those two bits each time a vertex spawns a child and you obtain a prefix-free binary string—the unique construction trace of that vertex.
The integer whose binary expansion equals that string is the vertex’s identity.
No separate “encoder oracle” is required; the multiplex geometry is the encoder.
2 Gödel-style Codes Emerge for Free
- Encode\textbf{Encode} Take the 2 × depth bits emitted while descending the lattice; read that as a base-2 numeral kk.
- Decode\textbf{Decode} Given kk, peel two bits at a time to walk back up the lattice, reconstructing each ancestor set.
This pair of functions γ\gamma / γ−1\gamma^{-1} is total, computable, and mutually inverse—identical to classic Gödel numbering, yet produced by nothing more than routine vortex recursion.
3 Why the Codes Behave Like Hash Keys
Bucket selection A holon hosting RR buckets chooses one by k mod ∣R∣k \bmod |R|.
Collisions only occur if two different paths happen to share that remainder; load metrics recorded in M track such collisions, triggering automatic re-hash via the Class loop whenever δ grows.
Because:
- paths are prefix-free (no key is a prefix of another),
- every key is a plain integer,
lookup cost is O(1)O(1) on average after each refine step. The directory you obtain is a self-healing hash-table whose maintenance is handled entirely by ordinary 4QX feedback—no external GC, no DNS-style registry.
4 Associative Semantics via BU Aggregation
Key–value pairs enter the system as Events ⟨op, (k,v)⟩
The BU return stores them in Resource as Kuratowski pairs {{k}, {k,v}}.
Pattern then holds directory schemas; Metric counts hits, misses, and load.
Because every holon carries its own filtered cache of neighbours’ buckets, the fabric forms an eventually-consistent, peer-to-peer associative array:
- recent keys replicate outward (high hit counts → low δ everywhere);
- stale keys age out naturally (zero hits → eviction in refine);
- missing keys reroute to neighbours, raising δ just long enough for copies to spread.
5 “Names” as Living Objects, Not Labels
A so-called name (e.g. “Explorer-Probe-7”) is itself a small Pattern holon that models another holon’s key. When shared, it arrives as an HF set whose path-code becomes its own key. Thus:
- reference is performed by structural inclusion of one holon inside another;
- no external symbol table can drift—the Gödel key is the object;
- identity collisions are impossible (injectivity of the encoding).
Names sit in an intermediate stratum: more concrete than the timeless Platonic lattice, more abstract than moment-to-moment sensor flux. They function as dynamic, first-class citizens that travel, cache and age exactly like any other holon.
6 Garbage Collection & Fallback—Side-effects, Not Sub-systems
- BU hit counters decay; refine evicts buckets whose δ-contribution outweighs their utility.
- When hardware drops a bucket, δ spikes locally, prompting peers to copy or recreate the lost content—graceful fallback with no global coordinator.
- The same mechanism balances storage cost against access latency, automatically tuning cache size to resource constraints.
7 Implications for Architects & Developers
| Benefit | What You Don’t Need |
|---|---|
| Canonical serialization → integer | Custom marshalling formats |
| Global uniqueness by construction | UUID services or naming authorities |
| Self-optimising directory | External KV stores and eviction daemons |
| Proof-friendly arithmetic keys | Out-of-band audits of data provenance |
Implementers merely embed the two-bit encoder/decoder (a few lines of code) and expose the standard 4QX select / commit / settle and execute / assess / refine hooks. The hash-table, naming, garbage collection, and replication appear automatically.
8 Take-away
When the bottom-up aggregation path returns through the same four-corner vortex, the binary choices it must record become an intrinsic Gödel address. Those addresses act as perfect hash keys, enabling a self-healing associative array to blossom at the very first layer of manifestation—long before any human-style labels arise.
In a 4QX world, therefore, naming isn’t bolted on; it is the inevitable by-product of multiplex geometry itself.