Everything else on this site states the program’s results. This page does one thing instead: it runs the machinery on the smallest case where anything interesting happens, so the mechanism can be checked by hand rather than taken on faith. No geometry, no curvature, no three-sphere — just the first two moves, comparison profiles and the factor map, on a world of four states. The point is to see rectangular completeness hold, and then watch it fail when the world is no longer self-contained.

The setup

A comparison world is a set of states together with comparison predicates — functions that take two states and return 0 or 1. Take four states, and think of each as carrying two internal features, a left feature and a right feature, each either 0 or 1:

states: (0,0), (0,1), (1,0), (1,1)

We are not told in advance that these are “pairs of features.” That structure is exactly what the comparison data will have to recover on its own. All the world actually has is two predicates:

c_L(u, v) returns the left feature of the first state, u.
c_R(u, v) returns the right feature of the second state, v.

That is the entire world: four states, two predicates. No coordinates announced, no product structure assumed. Everything below is extracted from these two 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: c_L 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, c_R 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.

Nothing about “two features” was assumed; the predicates produced exactly two left classes and two right classes on their own.

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. That is what it means for Θ to be a bijection: each (left class, right class) pair is realized once. The world has recovered, from two predicates alone, that it is a product of a left factor and a right factor — the canonical decomposition the program calls rectangular completeness (Theorem 2.6.4). Two classes by two classes, four cells, four states, no cell empty and none doubled.

This is the whole mechanism in miniature: comparison data in, product structure out, nothing assumed.

What it looks like when closure fails

The interest is in what breaks the picture, because that is where the closed/open distinction lives. Take the same two predicates, but delete one state — remove (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 recognizes a left class L=1 and a right class R=1, so the combination is a coherent possibility — but no state realizes it. This is a gap. The factor map is no longer onto; the world is not rectangularly complete. And the gap is exactly the “silent absence” the program forbids in a closed world: a possibility the system’s own comparisons recognize, that nothing realizes, and whose omission no comparison detects from inside.

The rod — where calculation and interpretation part

The rod makes the seam between calculation and interpretation sharp, which is why it is worth dwelling on. Suppose the two features are locked equal — only states where 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)

What the calculation says is not in dispute: two cells are admissible — the comparison structure recognizes both left classes and both right classes — and two are unrealized. That much is arithmetic.

What closure adds is an interpretation of that arithmetic: it reads an admissible-but-unrealized cell 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: something external is holding the off-diagonal cells empty.

A reader is free to reply: why not take the diagonal as the entire world, a complete and self-sufficient state space of two elements? That reply is coherent, and it is worth being exact about what it amounts to. It does not refute the closure analysis; it declines the closure premise — it chooses to treat an unexplained exclusion as a brute fact rather than as a mark of externality. That is precisely the choice the program asks a reader to make consciously rather than by default. Closure is the stance that a genuinely self-contained world owes no silent exclusions; a reader who accepts brute exclusions is describing a different kind of world, not correcting the analysis of this one.

What this shows, and what it doesn’t

Keep two things separate, because the page’s value depends on the distinction. The calculation is not in dispute: predicates determine congruences, congruences determine the factor map, and the factor map is bijective for the full product, non-surjective for the deleted world and the rod. Anyone can check it. The closure principle — that an admissible unrealized cell signals openness, an externally maintained absence — is the program’s interpretation laid on top of the calculation, not a consequence of it. The example illustrates the principle vividly; it does not prove it. What proves its worth is whether the program built on it derives more than it assumes, which is the business of the monograph, not this page.

This is also only the first two moves — profiles and the factor map — on the smallest world that shows them. No transport, no curvature, no three-sphere; those act on far richer worlds and need the full development. But the example exposes something the geometric results otherwise hide: where their rigidity comes from. The uniqueness theorems downstream are not a new and separate force. They are this same pressure — no admissible cell may stand silently empty — applied not to two features but to frames, transport, and the relations among them. Seen from here, the later rigidity is less surprising: it is the no-silent-absence discipline carried upward through the structure. Whether each upward step succeeds is a question for the papers. What this page makes visible is that the source of the rigidity is already fully present in a grid of four states.

The full theory is in the monograph; the plain-language account in The Idea; the result-by-result development in the Overview.