The Nine Axes
Each axis is independent. Each takes a finite set of values. A running system’s coordinate is the tuple of its values across all nine — read off its structure, not its name.
c = ⟨ F, K, C, I, R, A, T, V, τ ⟩
The cell coordinate · nine axes, one tuple
01 · THE KIND OF MACHINE
model of state
The formal model the state is held in. Each value contracts a body of theory and a vocabulary of operations. The choice of formalism decides what is expressible, what is checkable, and what is awkward — every other axis lives on top of it.
02 · HOW STATES COMBINE
compositional structure
How simple states assemble into larger ones. Atomic states stand alone. Sequential composes them in succession. AND-parallel runs orthogonal regions concurrently. OR-hierarchy nests sub-states inside super-states — the structure the cascade was a name for. Dynamic-structure rebuilds the composition while running.
03 · HOW TRANSITIONS RUN
execution model
How transitions advance in time. Sequential takes one step at a time. Cooperative yields between steps. Preemptive can be interrupted at safe points. Parallel runs many at once on shared resources. Distributed runs across independent hosts with no shared clock.
04 · WHERE THE BOUNDARY IS
state visibility
The boundary around state. Shared memory exposes it to anyone in the address space. Encapsulation hides it behind an interface. Message-passing requires explicit transfer across a channel. Process and machine isolation enforce the boundary with hardware. The boundary is the architecture.
05 · WHAT DRIVES THE STEP
transition trigger
What causes the system to advance. A closed loop steps on its own clock. Event-driven advances on an external signal. Reactive runs continuously over a stream. Autonomic chooses its own next move from a model of itself and its environment.
06 · WHAT THE MACHINE KNOWS OF ITSELF
reflection
Whether the running system can see and change its own structure. Opaque states are not inspectable from inside. Introspectable states expose their structure as data. Runtime-modifiable states can replace components without stopping. Reflective states model themselves — the structure is itself a value the machine can operate on.
07 · LANDAUER FLOOR
transition cost
The cost floor of a transition, inherited from the Periodic Stack of Computation. L₀ is free — a deterministic move into a cached cell. L₁ is sustained — paying to hold the state stable. L₂ is the Landauer floor — every resolved difference pays. L₂max is the wide-mismatch regime where a transition asks the substrate to do work it was not built for.
08 · WHAT CAN BE CHECKED
correctness surface
The kind of guarantee a transition carries. Model-checked transitions are exhaustively verified against a specification. Typed transitions are checked at compile time. Invariant-tested transitions are sampled at runtime. Contract-enforced transitions hold under stated pre- and post-conditions. Unverified transitions carry no formal guarantee.
09 · HOW LONG IT HOLDS
state persistence
How long the state lives, and what it survives. Ephemeral lasts a frame. Checkpointed can be resumed after a crash. Snapshot is a captured point-in-time copy. Versioned carries a full history. Durable persists across machine boundaries — the workflow continues even if the host does not.
The Cell
The product of the cardinalities is the discrete subspace — the addressable map of state-space.
10 · 5 · 5 · 5 · 4 · 4 · 4 · 5 · 5 = 2,000,000
Discrete cell count · the addressable map of state-space
discrete cells
orthogonal axes
compute primitives below
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