Why more information can sometimes make prediction harder.

01 / The problem we'll solve

Imagine trying to determine a forest's future capacity.

Let's return to the stressed forest state P that we met in Paper II. Our declared mathematical relation connects P to two target states: the baseline state B and the recovered state R.

Those targets have different capacities. B has both a stable and an exposed response, while R has only a stable response. This relation tells us neither that one target is more probable nor which will occur in the real world; it is a finite formal construction designed to reveal what a source representation determines.

So I can't assign one exact raw target capacity to that single source representation. Yet both B and R answer our adult-stage health requirements identically: a viable response exists, and a stable response exists.

For these declared questions, both target capacities fall into the same health-visible equivalence class. The complete future capacity is not determined, while the information the questions require is determined.

02 / Add distinctions to what we know

A richer source representation cannot destroy an exact answer we already had.

Let's call a relatively coarse source description S₀ and a more detailed one S₁. Imagine that S₀ records a forest's tree count, while S₁ records tree count, composition, and juvenile regeneration. S₁ retains every distinction S₀ can make and may make additional ones.

In our mathematical language, this is the refinement relation S₁ ⪰ S₀. The richer description can recover the coarser one on the values that actually occur.

Why is that true? Whenever two states are indistinguishable to S₁, they must also be indistinguishable to S₀. If S₀ already determines the target, S₁ has retained everything needed for the same conclusion.

This is an exact logical result about deterministic information. Collecting additional noisy observations in an empirical study introduces separate statistical questions about error and estimation.

03 / Ask for finer information at the target

More detail in the answer creates a stronger requirement.

Now let's hold our source representation S fixed and change what we demand to know about the target. Let T₁ retain more detailed information than T₀. If we can exactly determine T₁, we can determine T₀ as well, because the coarser answer can be recovered from the finer one.

The reverse is not guaranteed. Knowing that a stable response is available doesn't necessarily tell me whether an exposed response is also available.

For me, the two theorems reveal a helpful asymmetry. Giving the source additional distinguishing information cannot remove exact determinacy already established. Asking for a more detailed target adds distinctions that the source must uniquely determine.

04 / The least demanding adequate answer

What is the coarsest future description that still answers our questions?

In Lesson 8 I introduced the canonical query-visible quotient q_A. It groups target states according to their answers to every question in a declared family A. It is the coarsest representation that preserves exactly those answer distinctions.

Now I ask whether a representation can satisfy two conditions at once: it must be sufficient for our target questions, and it must be exactly determined from the available source information along the declared relation.

The left-hand side asks whether our source can determine the target's query-visible class. The right-hand side asks whether any target representation exists that is both sufficient for every declared question and exactly determinate from the source. “Not equal to ∅” simply means that at least one such representation exists.

R

The declared relationship between source and target states.

S

The information representation available at the source. Here S denotes an encoding, not the stable-response symbol used in the earlier forest model.

A

The family of target questions whose answers we want to preserve.

q_A

The canonical quotient that groups target states by all their declared query answers.

Determinate

The same source representation always determines the same requested target value on all related pairs.

SuffDet

The collection of target representations that are both sufficient for A and determinate from the fixed source through R.

Here is the central meaning: there exists an exactly determinable representation sufficient for all our target questions if and only if the canonical query-visible quotient is itself exactly determinable.

If even that coarsest sufficient answer cannot be determined, adding target detail cannot solve the exact problem. If it can be determined, we already have a sufficient, determinate target: the quotient itself.

05 / Why the conclusion follows

The proof follows the ideas we've already learned.

I'll walk through both directions. First, suppose our source exactly determines q_A. By definition, q_A retains every distinction that the declared questions can see. It is therefore a sufficient target representation. So we have at least one target that is both sufficient and determinate.

Now suppose instead that some other target representation T is both sufficient and determinate. From the principal theorem in Lesson 8, its sufficiency means T refines q_A: it preserves every query-visible distinction and perhaps additional ones.

From today's target-coarsening theorem, determining T also determines its coarser quotient q_A. So the existence of any sufficient, determinate target forces the quotient itself to be determinate.

Those two steps close the argument. Representation sufficiency and target coarsening work together to identify a precise information boundary, rather than a complicated numerical prediction rule.

06 / Return to the forest

One source, two raw capacities—and one health-visible answer.

Let's return to the exact relation inherited from Paper II. Source P is related to B and R. The baseline target B has {stable, exposed}; recovered target R has {stable}. Their raw capacities differ, so no one raw-capacity value is determined by the fixed source representation along this relation.

But both targets satisfy the two licensed adult adequacy questions: some viable response exists and a stable response exists. Their canonical query-visible quotient classes therefore agree.

That's a concrete reason to ask for precisely what the health question needs. Insisting on the complete target capacity requires us to resolve a distinction our declared questions don't consume.