Measurement Across Environments

Conceptual map

1. I-07.01Objects and environments
2. I-07.02Scalar transport
3. I-07.03Composition and rankings
4. I-07.04Interaction dimension
5. I-07.05Normalization and covariance

1. Measurement is indexed by environment

A source measurement can be transported to another environment only when source-equivalent objects remain equivalent for the target measurement.

Let \(X\) be an object space and \(m:X\times \mathcal{E}\to Y\) an environment-indexed measurement. For source environment \(e_{0}\) and target environment \(e_{1}\), write \(m_{0}(x)=m(x,e_{0})\) and \(m_{1}(x)=m(x,e_{1})\).

2. A valid scalar revaluation

Let \(X=\mathbb{R}\) and define \(m_{0}(x)=x^{2}\) and \(m_{1}(x)=2x^{2}+1\). Source fibers pair \(x\) and \(-x\), and both receive the same target value. The transport is \(\phi _{10}(y)=2y+1\) on [0,\(\infty\)).

The formula is defined on the attained source range. Extending it to negative source reports is irrelevant to the model and would be an extrapolation.

3. An environment interaction creates a missing direction

Let the object be \(x=(x_{1},x_{2})\in \mathbb{R}^{2}\) and measurement

Objects (0,0) and (0,1) have the same source measurement zero and target measurements zero and one. Criterion (1) fails. Differentially, \(Dm_{0}\)=[1,0] hides direction (0,1), while \(Dm_{1}\)=[1,1] loads on it. The mixed derivative

identifies the object direction whose value is activated by the environment change.

4. Differential necessity and its limit

If the maps are differentiable and \(m_{1}=\phi _{10}\circ m_{0}\) with differentiable \(\phi _{10}\), then \(Dm_{1}=D\phi_{10}Dm_{0}\) and \(\operatorname{ker}(Dm_{0})\subseteq \operatorname{ker}(Dm_{1})\). This is a necessary local test. Passing it at one object does not establish global transport across source fibers.

When environment is continuous, a first-order expansion \(\partial _{e}m\) separates uniform revaluation from object-specific interaction. Integrating that derivative across a finite environment change needs smoothness and path information.

5. Implementation, exercises, and sources

Group or solve for source fibers, compare target measurements within each fiber, and estimate mixed derivatives only after fixing object and environment coordinates. Report attained ranges and transport residuals.

Exercises

1. Verify the transport and its domain in Section 2.
2. For (2), characterize every pair of environments with a deterministic scalar transport on all of \(\mathbb{R}^{2}\).
3. Construct a nonlinear example that passes the derivative kernel test at one object and fails global transport.

1. David Blackwell (1951), “Comparison of Experiments,” Proceedings of the Second Berkeley Symposium, 93–102.Information comparison through post-processing.
2. John M. Lee, Introduction to Smooth Manifolds, sections on differentials and constant-rank maps.Local kernel tests.
3. I-03 · The Boundary Between Exact and Local Sufficiency.Local versus global factorization.
4. Chae-Yeon Xon (2026), “It’s a Price! It’s a Value!: Economic Measurement Across Environments.”Related cross-environment measurement application.