01 · DOMAIN GRADIENT
Z
where math meets words
Zone
The domain gradient. Z₁ identity (math = words). Z₃ rupture (math ≠ words). The transitions — Z₁↔Z₂ and Z₂↔Z₃ — are first-class, because most real AI lives there.
Z1 Z1↔2 Z2 Z2↔3 Z3
I · The Seven Axes
Seven orthogonal dimensions.
Each axis is independent. Each takes a finite set of values. A synthesizer's coordinate is the tuple of its values across all seven — computed deterministically from its structure and deployment.
c = ⟨ Z, E, T, P, I, V, R ⟩
The cell coordinate · seven axes, one tuple
02 · MODE OF BEING
E
what kind of thing
Entity kind
What kind of thing the system is. Substances have independent existence; emergents do not. Active substances initiate; reactive substances respond. Persistent substances are static. The entity kind constrains how a synthesizer can be deployed and held accountable.
persistent reactive active emergent
03 · LANDAUER FLOOR
T
cost class
Thermodynamic class
The cost floor of the system's operation, inherited from the Periodic Stack of Computation. L₀ is free (look-up, cache hit). L₁ is measurement (deterministic state read). L₂ is Landauer (every erasure pays). L₂ₘₐₓ is the wide-mismatch regime where the algorithm asks for far more than the silicon natively offers.
L₀ L₁ L₂ L₂ₘₐₓ
04 · ⊂ STACK258
P
primitive subset
Composition primitives
Which operations from the 258-primitive compute stack the system actually uses. A projection key — set-valued, not a single label. The mismatch between P(c) and the silicon's native ops is exactly the impedance the synthesizer pays for.
P ⊆ {1…258} families composition edges
05 · WHERE THE LOOP CLOSES
I
output channel
Interface
Through what channel outputs reach the world. Bodies are stakes-bearing. Tokens, signals, and "none" are not. The interface determines what failure modes the synthesizer can produce, and how fast feedback can close the loop.
none tokens signals bodies
06 · CHECKABILITY
V
how it can be checked
Verifiability
Can the output be checked against ground truth? At decision time (full), against a retrieved source (citation), after consequences play out (delayed), in aggregate (statistical), or not at all (none). Verifiability is structural, not optional — design for it.
full citation delayed statistical none
07 · WHAT THE WEIGHTS HOLD
R
representation
Encoding regime
What the system's parameters actually encode. The mathground inversion: encoding navigation, not facts. Encoding facts in weights is L₂-expensive and statistically verifiable at best; encoding how-to-navigate-an-external-substrate is L₁ with citation.
facts navigation world model grammar none
The Cell
A synthesizer is its coordinate.
Five axes are finite-valued. One is a projection (P). One (T) inherits from the compute stack below. The discrete subspace has exactly 8,000 cells — the addressable map of synthesizer space.
|Z| · |E| · |T| · |I| · |V| · |R| = 5 · 4 · 4 · 4 · 5 · 5 = 8,000
Discrete cell count · the addressable map of synthesizer space
8,000
discrete cells
258
compute primitives
7
orthogonal axes
Continue
See the table light up.
The seven axes project down. Two dimensions become a table you can scan; three become a tensor you can rotate.