How can we carry information about health from one context to another?

01 / From one context into another

Transport is about information, not physical movement.

Suppose I've observed a forest in one situation. I know something about its present state and continuation capacity. Under a later stage or a different set of conditions, what can that original information tell me?

I use the word transport for carrying a mathematical description or an answer across a declared relationship between source and target states. The relationship may express developmental change, environmental change, or a correspondence between descriptions.

One subtlety matters immediately: the relationship may connect a single starting state to several targets. A relation is allowed to do that. An ordinary function, by contrast, assigns one output for each input in its domain.

In the exact finite example from Paper II, I use the perturbed state P as the source and relate it to two possible targets: baseline B and recovered R.

There is no unique target state for P under this raw relation. Yet we haven't asked the central question of the lesson: does the distinction between the targets matter for the information we need?

02 / The information carried by each target

Different capacities can answer the same health questions.

The two target states have different represented capacities. I use S for a stable viable response and E for an exposed viable response:

Recall the two adult-stage requirements from our previous lesson. One asks whether any viable response exists. The other asks whether a stable viable response is available.

Both presently realize the declared organization. Under the fixed adult context and those licensed requirements, the targets give the same adequacy answers and the same corresponding full Health judgments.

That means I can't infer a single raw target capacity from the source P under this relation, but the two target capacities belong to the same health-visible class. Their difference in E isn't visible to the particular licensed questions we're asking.

03 / When exact transport exists

A deterministic rule needs an answer that the source can determine.

Now let's express the criterion precisely. Call the source capacity representation C₀ and the target representation C₁. A relation R connects source states x to target states y. I can transport the target capacity deterministically from the source capacity only if identical source capacity information never leads, along admitted relation pairs, to different target capacities.

R(x,y)

The source state x is related to a target state y.

x′ and y′

Another related source and target pair. The prime mark distinguishes the second pair.

C₀

The capacity information available at the source.

C₁

The capacity representation demanded at the target.

⟹

“Implies.” If the hypotheses hold, the conclusion must hold as well.

Our finite forest relation supplies the counterexample. The same P is related to B and R. Its source capacity is necessarily the same in both pairs, because the starting state is the same. Yet the two target capacities are {S, E} and {S}. A single-valued rule cannot assign both outputs to the same input.

This is an information limitation. It isn't repaired by writing a more elaborate formula: the source representation has not specified which raw target capacity to choose.

04 / Transport only the distinctions we need

The health-visible class can still be determined.

Lesson 3 gave us a way to group different capacities whenever every declared adequacy question returns the same answer. Paper II uses the context-licensed version of that idea.

Let q_A represent the quotient-class map for the licensed adult questions. Since {S,E} and {S} both pass the any-response and stable-response questions, they have the same class:

That leaves us with two different transport tasks. One asks for every detail of the raw target capacity. The other asks only for the equivalence class determined by the licensed health questions.

05 / The information required by a new context

Sometimes even the health-visible answer cannot be transported.

Let's bring back the developmental-stage example from Lesson 5. At the Newborn stage, only the any-response requirement is licensed. The capacities of P and R—{E} and {S}—both pass it. Their newborn health-visible classes agree.

At the Adult stage, we also license the stable-response requirement. P fails that requirement; R passes it. Their adult health-visible classes differ.

Imagine keeping only the newborn quotient-class label and discarding the original difference between {E} and {S}. When the adult question arrives, the missing distinction is exactly what I need. The single source label cannot determine which adult class to assign.

That makes exact deterministic transport from the newborn health-visible quotient to the adult quotient impossible for this example. I call it an obstruction: a proved impossibility under the information and relation we've specified.

Nothing about the actual capacities has to change in this counterexample. The obstruction comes entirely from changing which requirements are licensed. The same source information is now being asked to answer a question it was never designed to distinguish.

06 / Three claims to keep separate

Determining the target is not the same as preserving Health.

There is one more distinction I want us to keep with us. Suppose a map really does determine the requested target information. That tells me the target answer is well-defined from the source. It does not, by itself, say that the answer at the target is equal to the answer at the source.

The second task uses a coarser target than the first. If full raw capacity is determined, its target quotient class is also determined once the target requirements are fixed. But even then the source and target Health truths need not agree. Determinacy and preservation are different properties.

Paper II gives another useful illustration in its external-support discussion. Withdraw a declared external support and the resulting represented capacity can be determined exactly to be empty. Determining that loss is not the same as preserving Health; the present organization may still exist while the continuation capacity required for the original health judgment disappears.

Preservation requires more: present realization and adequacy must be compatible across the related states, and the relevant requirements must be coherently interpreted or licensed in both contexts. Those are explicit mathematical obligations, not automatic consequences of having a transport map.