How information gets lost—and why it matters for health.

01 / The finite forest

Let's begin with five states we can follow completely.

A forest can look very different across time while still maintaining the organization we are asking it to continue. Our small model gives that idea five named states.

In this declared system, B, P, R and A presently realize the forest organization. F does not. We can express that with a remarkably compact statement:

These five labels are exact categories in a mathematical benchmark, not a validated ecological classification of real forests.

02 / Which futures belong to the model?

Before measuring capacity, I have to decide which histories count.

Our model declares two environmental scenarios—mild and severe—and two horizons—now and long. It also declares six kinds of event: staying in a state, disturbance, regeneration, compositional adaptation, severe collapse, and canopy turnover.

The same event is not necessarily admissible in every context. Under mild conditions at the longer horizon, stressed P can regenerate or adapt. Under severe conditions at that horizon, the model also permits collapse.

A collapse may be lawful in the severe scenario while failing the continuation test. Lawfulness concerns whether it can occur under the model; viability concerns whether the required organization persists.

Paper IV compares two ways of encoding this process: one places contextual restrictions directly in the history types, while the other starts from permissive candidates and filters them explicitly. The formal equivalence proves they give the same downstream viable continuations and capacities when the corresponding structures are preserved.

03 / The most informative response representation

Let's keep the different kinds of stability distinct.

To make the census fully inspectable, I give the observer four possible response atoms. E means an exposed response. The three stable atoms remember how stability arose: S_B from baseline, S_R from recovery, and S_A from adaptation.

Here is the complete detailed-capacity census under one declared context: mild conditions at the short horizon.

Every state has a different detailed capacity in this census. In this mathematical context, the representation distinguishes all five forest states. No probability weights have been assigned to the attained response categories.

04 / Simplifying the observer

Now let's let the observer forget how stability arose.

Suppose the question only needs to know whether a stable response or an exposed response is present. We can replace the three detailed stable atoms with one common label S, while preserving E.

Let's apply this one rule to the same five states. In each row, the simplified capacity is just the projection of the detailed response atoms.

B, P, R, A, F

The five declared forest states: baseline, stressed, recovered, adapted and collapsed.

E and S

Exposed and stable response categories in the simplified capacity representation.

S_B, S_R, S_A

The three distinct stable-origin atoms retained by the detailed response representation.

π

The projection that sends each detailed stable atom to S and leaves E unchanged.

∅

The empty capacity: no viable response atoms are represented in the declared context.

We've lost exactly one distinction. Recovered and Adapted now both produce {S}. The projected representation cannot tell them apart. The five detailed states therefore form four capacity classes: B, P, {R,A}, and F.

This loss occurs in the representation itself. Once both R and A have been recorded simply as {S}, a deterministic operation using only that projected capacity has no information that identifies which original state produced it.

05 / Keep only the answers used by Health

A second grouping happens even when the simplified capacities differ.

Now we'll ask two declared adequacy questions. Does any viable response exist? And is a stable response among them? Those questions can be written directly on a projected capacity c.

Look at B and R. Their projected capacities are still different: {S,E} versus {S}. But the two adequacy questions return the same answers for both. That difference remains in the measurement representation and is simply invisible to these declared queries.

For full Health, I also check present realization. The Collapsed state fails it. Paper IV checks the full family of scenarios, horizons and two requirements and establishes the same three Health-profile groups: {B,R,A}, P, F. The displayed mild/now table illustrates their pattern; the complete-model classification is the stronger verified result.

06 / Locate the first loss

What disappears from the representation, and what remains unused?

Now let's put our new ladder to work. Suppose I ask you to distinguish the recovered state R from the adapted state A. Their detailed capacities contain S_R and S_A, but their projected capacities both become {S}. If all we retain is that projected value, we have lost the needed distinction.

Next suppose the distinction I want is between baseline B and recovered R. Their projected capacities remain different: {S,E} and {S}. What disappears is their distinction under the two declared health-adequacy questions, which both answer yes for B and R.

We can see both situations together and decide what a new scientific question would require.

07 / Preparing for a richer measurement

Can we add back information while preserving the old Health answers?

Suppose we have used the simplified capacity representation for some time, and its declared Health questions have established a collection of judgments. Later we want a richer observation that distinguishes baseline stability, recovered stability and adapted stability.

We can keep all those response categories in the detailed representation while retaining the ability to project it back to the earlier stable-or-exposed capacity exactly.

This sets up the next result, conservative enrichment. If the enriched observation can be projected back into the original one and the established adequacy questions continue to evaluate that projected representation, the previous judgments can remain intact while new distinctions become available for justified additional questions.

I'll develop the exact preservation and compatibility conditions in Lesson 13. For now, the point is that getting more informative does not necessarily require giving up the conclusions already supported by the earlier representation.