cyber hierarchy

This proposal studies how the cybergraph could scale toward Avogadro numbers — 10^23 particles, 10^15 neurons — by deriving graph partitions from the tri-kernel's output. Counts, hop costs, proof costs and timescales below are design estimates whose assumptions must be tested.

shards hold regions of graph state. neurons remain protocol subjects, and progs retain their work under those subjects. Shard routing IDs, network destinations, token books and neuron IDs have separate meanings under the domain model. Partitioning state preserves its validation, availability, spending and finality obligations.

the insight

the tri-kernel that computes focus also reveals the natural hierarchy. all three operators contribute:

springs provides the eigenvectors that define fold lines. heat controls the resolution — which level of the hierarchy you read. diffusion reveals the flow patterns that validate the folds. the three together give robust community detection that no single operator provides alone

no administrator assigns structure. the tri-kernel computes it as a side effect of computing focus. the same operators that rank particles also partition the graph for scaling

four dimensions

the cybergraph has four dimensions — the four primitives themselves. particles that are close in any dimension should share a shard

particles — semantic

particles with high mutual focus flow — many cyberlinks between them, strong axon weights — form semantic clusters. the tri-kernel reveals these through spectral decomposition (springs) and multi-scale smoothing (heat)

neurons — social

neurons that frequently interact can supply social-locality signals. Public links and explicitly disclosed coordination may support placement decisions. Private UTXO ownership and spending relationships retain the selected protocol's privacy policy. A neuron may act across several shards; co-location is a routing optimization subject to disclosure and capacity constraints.

tokens — economic

each token supplies an economic grouping. particles priced in $CYB can cluster in $CYB regions; trading $CYB for $H crosses books in the token dimension. A new token creates a book under oikos. The assignment of one or more books to physical shards is a separate partitioning decision, preserving each book's complete obligations and issuer rules.

The economic dimension supplies explicit labels alongside computed clustering. A signal names its target network under the selected protocol profile. That destination does not determine a shard, token issuer or signing subject. An application may display networks as cards and may offer a private network convention such as $H(\texttt{"network:"} \| \nu)$; using that convention requires an explicit network profile. A neuron can address several compatible networks, and a network can contain many neurons and shards. Joining or creating a network requires its own admission and initialization rules. See network.

locations — geographic

latency matters for interactive use. neurons in the same physical region want low-latency access to their neighborhood. location proof provides this dimension. validators in a region preferentially serve that region's shards

the 4×4 matrix

each dimension has four scales. a particle has a coordinate in each dimension at each scale

shards are the base operational level in this proposal — they hold state, process transactions and run the tri-kernel. Zones, domains and the root aggregate shard topology at different heat kernel temperatures. The selected consensus profile must define the stake, validator assignment, authenticated summaries and validation duties at every level. Graph clustering alone grants no authority to finalize another region's state.

a particle's shard = the intersection of its coordinates across all four dimensions. two particles sharing more coordinates → cheaper to move tokens between them. sharing all four → same shard, zero cross-shard cost

shard(particle) = (semantic_shard, social_shard, token_shard, geo_shard)

the root shard

the root shard is where all four dimensions meet at their global level — the origin (0,0,0,0)

it holds two things:

1. the crystal — the 5,040 particle seed that defines the foundational ontology. these particles are maximally general, referenced by everything, naturally highest focus

2. the routing table — maps particle hash → domain. not shard-level routing — that is each domain's job

root    → knows domains
domain  → knows zones
zone    → knows shards
shard   → knows particles

four hops to find any particle among 10^23. the root shard is the first hop

Before the graph has enough structure to fold, this model places its state in one root region. An unpartitioned bootloader such as bostrom is the reference case. The proposed phase transition threshold $|P^*| \sim \rho^2$ is a candidate signal for considering division; the protocol must also authorize and verify the split. The root continues to coordinate domain routing.

Within one hierarchy, new shards derive from an authenticated split or merge of existing coverage. The biological metaphor is growth from a single fertilized cell; its protocol counterpart requires explicit state handoff. See spectral cell division for the proposed split/merge mechanism and its open questions. Registration of a foreign domain establishes a separate route and verification profile rather than inventing ancestry in this tree.

two information flows

subjective (neuron-driven)

tokens, cyberlinks, attention allocations. neurons choose where to move these. a neuron decides to send $CYB from shard A to shard B — that is a subjective decision, costs a proof relay

direction: horizontal and downward. neurons push information into shards

objective (shard-computed)

focus aggregations, rank summaries, community structure, routing updates. no neuron moves these — each shard computes them deterministically from its local state and propagates upward

direction: upward only. shards push truth to zones, zones to domains, domains to root

root     ← receives domain summaries (objective)
domain   ← receives zone summaries (objective)
zone     ← receives shard summaries (objective)
shard    ← receives cyberlinks, tokens (subjective from neurons)
         → computes local focus, propagates upward (objective)

a neuron cannot push a fake rank summary upward — the shard computes it deterministically from the tri-kernel and proves it via zheng. the proof propagates with the summary. each level verifies the level below

the subjective layer (what neurons want) and the objective layer (what the graph computes) flow in different directions through the same structure. tokens flow wherever neurons send them. truth flows wherever the math says it goes

hop cost

moving tokens between shards costs hops. the cost depends on how many dimensions differ and at what level:

small world models motivate an average path length of order O(log N) under their topology assumptions. Correlation between dimensions may shorten routes, but graph diameter and authenticated routing cost require separate measurements. The proposed six-to-seven-hop target at Avogadro scale remains a hypothesis. Each cross-shard hop must carry the zheng evidence required by its profile.

UTXOs and spending-state profiles

The private-UTXO profile represents notes by commitments and requires a ZK proof for spending. Its disclosure contract specifies which public transition metadata remains visible. Public balance projections and explicit public-credit profiles retain their separate semantics.

The mutator-set research variant assigns each shard an AOCL for creation and SWBF for spending. That variant proposes bloom-filter positions as the spending representation and MMR compaction for retained witnesses. Its privacy, witness-update and storage bounds need their own proof and measurements. Profiles using a nullifier set retain their nullifier uniqueness and finality rules. A split or merge must preserve spending uniqueness across old and new boundaries for either representation; renaming the partition changes no ledger encoding or monetary rule.

within-shard transfers are cheap — local state update, no cross-shard coordination. cross-shard transfers require zheng proof relay. the social dimension co-locates frequent transactors in the same shard

see cyber/state for transfer mechanics. see AOCL and SWBF for the mutator set

folding the tri-kernel

the tri-kernel has a locality radius: h = O(log(1/ε)) hops. each particle's focus depends only on its h-hop neighborhood

within a shard: the tri-kernel runs at full resolution. every cyberlink, every axon weight, every market price is visible

within a zone: shards communicate aggregated focus vectors. each shard exports its boundary particles' focus values to neighboring shards

across zones: zones exchange coarse-grained focus summaries. the error is bounded:

$$\|\phi^*_{\text{folded}} - \phi^*_{\text{global}}\| \leq C \cdot e^{-\alpha h}$$

more communication → smaller error → closer to global focus

timescales

The fast timescale assumes fixed shard boundaries. The slow timescale adjusts them using accumulated statistics. Stability analysis must bound convergence relative to boundary changes, delayed messages and adversarial load.

rebalancing

when a shard grows too large: split it along the Laplacian eigenvector boundary (spectral bisection via springs)

when two shards have become tightly coupled (high cross-shard focus flow): merge them

when a zone's internal connectivity drops below threshold (springs eigengap shows it is really two zones): split the zone

State migration (particles and UTXOs change serving partitions) is amortized over the slow timescale. Token obligations remain in their home books. A handoff must authenticate old/new coverage and checkpoints, preserve available history and duplicate-spend protection, and fence writers before new partitions admit effects. Open cross-shard operations keep their original identities, conditions and unknown outcomes through migration.

shard count

at Avogadro scale — estimated count at each level per dimension:

most of the 4D space is empty — dimensions correlate. shards exist only where particles actually cluster

comparison

address space:

the key difference: every other system designs the hierarchy. cyber computes it. the tri-kernel is simultaneously the probabilistic engine, the folding oracle, and the routing advisor. one computation serves all three purposes

open questions

shard boundary latency: how many blocks of cross-shard latency is acceptable before UX degrades? this determines the minimum shard size

privacy and routing: publishing which shards a neuron uses can disclose interaction patterns. Which placement and subscription records can remain private while coverage and routing remain verifiable?

incentive alignment: validators specialize in shards. what prevents a validator from refusing to serve a low-value shard?

cold-to-hot reactivation: when an archived particle gets new cyberlinks, it must rejoin a shard. which shard? the semantic dimension may have shifted since it was archived

see cyber/architecture for the five-primitive resource model. see tri-kernel architecture for the locality filter. see cyber/state for the bbg world state. see cyber/network for the narrowcast relay protocol. see forgetting for the hot/cold tier separation

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