A Worked Example
A Worked Example
The worked example gives an explicit finite instance of the basic construction. It uses the smallest case in which the profile construction and factor map are nontrivial: a four-state comparison world. No geometry, curvature, or three-sphere theorem is involved; only comparison profiles and rectangular completeness appear. The example shows rectangular completeness in the full product case and its failure in a deleted-cell structure that is not closed under the criterion.
The setup
A comparison world is a set of states together with comparison predicates — functions that take two states and return 0 or 1. Consider four states, represented for calculation by two binary internal features, a left feature and a right feature:
states: (0,0), (0,1), (1,0), (1,1)
The product interpretation is not supplied in advance. It is the structure the comparison data must recover. The initial data consist only of two predicates:
cL(u, v) returns the left feature of the first state, u.
cR(u, v) returns the right feature of the second state, v.
The example therefore starts from four states and two predicates, with no independent coordinate system or product structure among the primitive data. The remaining structure is extracted from these predicates.
Step one — profiles
For each state, its left profile records how it behaves as the first argument across all comparisons; its right profile records how it behaves as the second argument. Two states are identified by the left congruence (written α) when their left profiles are identical, and by the right congruence (β) when their right profiles match. (These are Definitions 2.3.1–2.3.2 of the monograph, applied by hand.)
Computing the left profiles: cL returns the first state’s left feature, so a state’s left profile is fixed entirely by its own left feature. States (0,0) and (0,1) share left feature 0, so they are α-equivalent; (1,0) and (1,1) share left feature 1. The left congruence sorts the four states into two classes:
α-classes: { (0,0), (0,1) } and { (1,0), (1,1) } — call them L=0 and L=1.
Symmetrically, cR returns the second state’s right feature, so a state’s right profile is fixed by its own right feature:
β-classes: { (0,0), (1,0) } and { (0,1), (1,1) } — call them R=0 and R=1.
The predicates determine exactly two left classes and two right classes without supplying a product decomposition.
Step two — the factor map
The factor map Θ sends each state to the pair (its α-class, its β-class) — that is, to (left class, right class). Laying the four states into a grid with α-classes as rows and β-classes as columns:
| R = 0 | R = 1 | |
|---|---|---|
| L = 0 | (0,0) | (0,1) |
| L = 1 | (1,0) | (1,1) |
Every cell is filled by exactly one state. Thus Θ is bijective: each (left class, right class) pair is realized once. The comparison data recover a product of a left factor and a right factor — the canonical decomposition the program calls rectangular completeness (Definition 2.5.1; equivalently, bijectivity of Θ; derived under the stated hypotheses in Theorem 2.6.4).
The mechanism appears here in its smallest nontrivial form: comparison data determine the product structure rather than presupposing it.
What it looks like when closure fails
The closed/open distinction appears when the same profile structure is present but one admissible pair is unrealized. Remove the state (1,1):
states: (0,0), (0,1), (1,0)
The profiles still produce two left classes and two right classes (the remaining states still show both feature values on each side). So the grid still has four cells — but only three are filled:
| R = 0 | R = 1 | |
|---|---|---|
| L = 0 | (0,0) | (0,1) |
| L = 1 | (1,0) | empty |
The cell (L=1, R=1) is admissible — the comparison structure contains a left class L=1 and a right class R=1, so the combination is a coherent possibility — but no state realizes it. The factor map is no longer onto; the resulting structure is not rectangularly complete. In the closure interpretation, this unrealized admissible cell indicates an omission requiring external selection: a possibility admitted by the comparison data, not realized by any state, and not internally detectable as an exclusion.
The rod — where calculation and interpretation part
The rod example separates the calculation from the interpretation. Suppose the two features are constrained to be equal, so that only states with left = right exist:
states: (0,0), (1,1)
The profiles still yield two left classes and two right classes, so the grid still has four cells — but only the two diagonal cells are filled:
| R = 0 | R = 1 | |
|---|---|---|
| L = 0 | (0,0) | empty |
| L = 1 | empty | (1,1) |
The calculation establishes the undisputed finite fact: two cells are admissible — the comparison structure contains both left classes and both right classes — and two are unrealized.
Closure adds an interpretation of that arithmetic: an admissible-but-unrealized cell is read as the signature of a constraint imposed from outside the two features themselves — here, a rigid coupling locking left to right, a relation the two features do not generate between them. On the closure reading, this world is open: the off-diagonal cells are excluded by a constraint not recovered from the two features alone.
One may instead treat the diagonal as the entire comparison world, a complete state space of two elements. That position is coherent, but it declines the closure criterion rather than refuting the calculation. It treats the exclusion of the off-diagonal cells as primitive rather than as evidence of an external constraint. Closure is the criterion that a self-contained comparison structure contains no admissible but unrealized profile pairs; accepting primitive exclusions amounts to adopting a different criterion of self-containment.
What this shows, and what it doesn’t
The calculation and the closure interpretation are distinct. The calculation is elementary: predicates determine congruences, congruences determine the factor map, and the factor map is bijective for the full product but non-surjective for the deleted world and the rod. The closure principle — that an admissible unrealized cell signals openness, an externally maintained absence — is the interpretive criterion applied to that calculation. The example illustrates the criterion; the theorem-level justification is supplied by the monograph.
The worked example covers only the first two constructions — profiles and the factor map — in the smallest world where they are visible. It does not address transport, curvature, or the three-sphere theorem, which require the full development. Its role is to display the basic rigidity mechanism: admissible combinations recognized by the comparison structure must be realized in a closed world. The later constructions apply the same no-unrealized-admissible-pair discipline to frames, transport, and relations among them, subject to the additional hypotheses stated in the papers.
Sources and dependencies: comparison worlds, left/right profiles, intrinsic congruences, and the factor map are defined in CSM Chapter 2 and summarized in Notation. The deleted-cell and rod readings are examples of the closed/open distinction discussed in Foundations, Overview, and Logical Status. The example is illustrative; the theorem-level statements are the monograph definitions and results it cites.
The full theory is in the monograph; the first-principles account is in Foundations; the result-by-result development is in Overview.