A chosen paired readout gives affine ranks 1,1,2,2,3,3. The seventh role publishes the six-role archive; it does not add a coordinate or another counter digit.
Try the demonstration
Role 7: 6 inner roles included; affine readout rank 3; 64 addresses in this radix-2 block. The seventh role publishes the completed six-role archive.
Exact rank and address counts for a supplied six-digit paired counter. The widget does not derive a physical spatial frame, clock duration, or necessity of six roles.
What the controls change
The role slider selects one of six inner dependency stages or the seventh, outer completion. The radix chooses how many values each counter digit can take. At an inner stage a, the display counts the first qa addresses and reports the affine rank of their paired readout. At role seven it keeps all six digits and marks publication.
The nested boxes express dependence: higher roles use organized lower-level work. They are not seven independent time axes. The controls inspect successive parts of one declared construction; they do not simulate six independently adjustable laboratory clocks.
One update drives every digit
Write an address using six radix-q digits, with d₁ changing fastest. An elementary increment advances the whole counter and performs any required carries. A higher digit changes only after the lower digits complete a cycle.
Thus the first a digits range through qa addresses before returning together. The full six-digit archive contains q⁶ addresses. The source state retains the digits even when a later spatial image forgets some of their distinctions.
Why the rank rises only every other role
Supply three independent directions e₁,e₂,e₃ and pair adjacent digits with opposite signs. The readout is Q=ℓ[(d₁−d₂)e₁+(d₃−d₄)e₂+(d₅−d₆)e₃]. The first digit opens one direction; its partner extends values along that same direction. Digits three and five open the other two directions.
This proves the displayed sequence for the supplied pairing. The rank is computed from the full selected address block, not from the position of one point or the number of visible boxes.
Try the binary and ternary examples
With radix two, role four includes sixteen addresses and has rank two. Role six includes sixty-four addresses and has rank three. Moving to role seven changes publication status while preserving those numbers. With radix three, the corresponding address counts are eighty-one and seven hundred twenty-nine, with the same ranks.
These are exact finite counts. They do not assign physical seconds or metres. Nor is the private carry sequence a particle trajectory: projecting a carry can create a jump that does not respect an intended spatial neighbourhood.
What the model contributes to SMM/DTT
The example realizes an important conceptual separation: one generating process can have nested internal work and a later interface that makes its result available. It also locates the additional assumptions. Three independent directions, the paired signs, and the publication rule were supplied rather than forced by the counter alone.
A stronger source model would derive those choices, retain enough predictive memory, and explain its physical clock and local transport. The demonstration is a transparent finite starting point for that research. Reset restores the completed binary example, and the initial diagram and equations remain readable when interactive scripting is unavailable.
Continue with the articles
- One action, two temporal roles
Inner generation and outer record order can be roles of one process. A completed return becomes an observable advance only when its action respects the retained records and their tests.
- Six inner stages and one publication interface
A six-digit nested counter with paired signed readout realizes ranks 1, 1, 2, 2, 3, 3 before one completion interface. Its exact counts reveal sampling bias and the restricted assumptions behind the six-channel bound.
- Generating a neighborhood and preserving local transport
A generated graph predicts local influence only when later operations respect its edges and retained ports. Guarded transport supplies a finite-update support bound while exposing runtime and controller-memory requirements.
- What spatial readout forgets
Different inner states can share one spatial readout and still have different successors. A concrete counterexample and a kernel-preservation criterion identify the memory needed for autonomous completed evolution.