Three binary differences give 27 sites from 64 addresses. Uniform address sampling gives multiplicities 8 at the centre, 4 at face centres, 2 at edge centres, and 1 at corners.
Try the demonstration
Address 0, digits d₁…d₆ = 0 0 0 0 0 0 → (0, 0, 0). This site has 8 of the 64 addresses: probability 12.5% under uniform address sampling. Total: 64 addresses, 27 distinct sites.
| Site class | Sites | Addresses each |
|---|---|---|
| Centre | 1 | 8 |
| Face centres | 6 | 4 |
| Edge centres | 12 | 2 |
| Corners | 8 | 1 |
Exact binary enumeration and multiplicities for the stated paired map. Probabilities assume uniform address sampling, and the counter scan is not identified with physical transport.
Choose an address and read the highlighted site
Move the address slider from zero to sixty-three. The live result displays its six digits in the order d₁ through d₆, with d₁ the least significant digit. The three plots are slices at z=−1,0,+1. Every circle contains the number of source addresses mapping to that site; the selected site has a highlighted border.
The plots show the complete finite archive, not a growing trajectory. Circle size reinforces multiplicity, while the printed numbers and summary provide the exact information without relying on colour.
Calculate the paired map
Each digit is either zero or one. Pair adjacent digits and take their difference:
A single pair has four possible addresses. Its differences are zero for (0,0) and (1,1), positive one for (1,0), and negative one for (0,1). Therefore one coordinate has three possible values with multiplicities one, two, and one. Three independent pairs give 3³=27 sites while retaining 4³=64 source addresses.
The image does not inherit a uniform site measure
Multiplicities multiply across the three coordinate pairs. A site with three zero coordinates has 2³=8 representatives. With two zeros it has four, with one zero it has two, and with no zero it has one. Counting the corresponding classes gives:
Under uniform address sampling the centre has probability 8/64=1/8 and each corner has probability 1/64. A uniform distribution on sites would instead give every site 1/27. Such uniformity would require an additional weighting or sampling rule.
Two equal present sites can have different futures
Address zero has digits (0,0,0,0,0,0), while address three has (1,1,0,0,0,0). Both map to the central site. Their immediate successors are one and four: the first maps to (1,0,0), and the second maps to (0,1,0).
Thus the current spatial site does not determine the next projected counter value. The full digit state contains predictive information that the image loses. This does not contradict the exact twenty-seven-site count. A present display and a sufficient state for future prediction solve different problems.
A finite lesson for generated geometry
In SMM/DTT, inner generation can support more distinctions than an outer spatial record exposes. This example gives an explicit map and a complete count of the information compressed by it. It also shows why geometry, sampling measure, and predictive memory must be specified separately.
The construction supplies coordinate directions and a readout rule. It does not prove that physical space is a twenty-seven-site cube or that matter follows the counter scan. A physical extension would derive locality and a clock law for completed records. Reset selects address zero; the static default already contains all twenty-seven multiplicities for inspection.
Continue with the articles
- 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.
- 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.
- Retained observations versus predictive states
A readable record need not contain enough information to predict its next update. An exact leakage criterion identifies when a linear completion is autonomous and measures the error when hidden information returns.