01A conclusion needs its own evidential basis
In the previous chapters, we distinguished a mechanical trace from its interpretation, examined declared conditions, and saw why recognising a transition requires a specific criterion. The next step is to connect each conclusion to the evidence supporting it.
The Manifest distinguishes compatibility from structural attribution, and internal validation from agreement with physical reality. It also requires each extension or interpretation to have its own argument, evidence and scope.
A correct calculation shows what follows from the declared relations. To support a description of a real mechanism, it additionally needs a connection to the observation method, conditions and relevant data. That connection is the subject of this chapter.
Connection to the corpus: Corpus Manifest v1.0, §10, p. 6 · §§12–13, p. 7 ↗
02A record needs an accompanying description
Before comparing results, we need to know which quantity was observed, how its value was obtained and under which conditions. A photograph, a trajectory and a force measurement provide different kinds of information.
Measurement uncertainty characterises the spread of values attributed to the quantity being measured, using the available information. Measurement error is a measured value minus a reference value. Repeatability concerns agreement among repeated measurements under specified common conditions. Little scatter between repeats is not a complete uncertainty assessment, which may also include contributions from systematic effects.
In the following example, we use an interval as a declared comparison criterion. Its numbers are chosen for the exercise. They do not come from measurements and are not an uncertainty estimate or a confidence interval.
Metrological terminology: VIM3, 2.26 · uncertainty ↗ · 2.16 · error ↗ · 2.21 · repeatability ↗ · 2.20 · repeatability conditions ↗ · 2.15 · precision ↗
03Two descriptions and an explicit criterion
Return to the dimensionless geometric descriptions in the chapter on traces. In A, y = 0; in B, y = x2, with 0 ≤ x ≤ 1. We now choose to examine y at the exactly specified position x* = 1/2.
yA = 0 · yB = 1/4 = 0.25
Declare the following rule: a description satisfies the criterion when its value lies within the selected closed interval I. We compare three different criteria, keeping the descriptions and the position fixed.
| Comparison interval I | A: y = 0 | B: y = 0.25 |
|---|---|---|
| [0.22, 0.28] | Outside | Inside |
| [−0.02, 0.28] | Inside | Inside |
| [0.10, 0.15] | Outside | Outside |
In the first row, only B satisfies the criterion among A and B. In the second, this criterion does not distinguish them. In the third, neither satisfies it. The last result concerns these particular descriptions and this particular comparison; it does not establish physical impossibility.
The interval in the second row is wider. Comparing the first two rows shows how the width of the declared criterion affects discrimination. This exercise cannot be presented as improving an actual measurement: no measurement was made here.
The mathematical form of the comparison
Let F = {A, B} be the declared family of descriptions and vi the value of y given by description i at x*. Define locally:
KF(I) = { i ∈ F : vi ∈ I }
The three rows of the table give {B}, {A, B} and ∅, respectively. If I ⊆ J, then KF(I) ⊆ KF(J): widening the interval does not exclude a description that already satisfies the criterion.
The position x* is exact by assumption. No position uncertainty, parameter estimation or probabilities are included. The calculation checks only membership of the two values in the declared interval.
Original dimensionless example with invented comparison intervals and local notation. No measured data, confidence intervals or FBM validation criteria are presented.
04A unique choice depends on the candidate list
Add a third description C, with y = x/2 on the same domain, 0 ≤ x ≤ 1. At x* = 1/2, it also gives y = 1/4. It therefore satisfies the first criterion as well.
B and C are different curves, although they agree at this particular position. The initial uniqueness of B concerned the list {A, B}. With the list {A, B, C}, the same criterion retains B and C.
The statement “only this description agrees” needs to specify which alternatives were examined and under which criterion.
The example gives a concrete way to read the distinction between compatibility and attribution. Success in a limited comparison does not establish a general claim of unique structure. The next question is which additional observation could distinguish the remaining alternatives.
C is an original construction. Connection to the compatibility / attribution distinction: Corpus Manifest v1.0, §10, p. 6 ↗
05A compact case-reading record
To make a case understandable and inspectable by another reader, we can organise its presentation around the following information. The table is filled with our example so that the basis of the result is explicit.
| Field | In this example |
|---|---|
| Question | Which of A and B give a value of y within I at x* = 1/2? |
| Declared description | A: y = 0 and B: y = x², with dimensionless coordinates and 0 ≤ x ≤ 1. |
| Origin of the values | Exact calculation from the equations. I = [0.22, 0.28] was chosen for the exercise. |
| Criterion and result | Membership of the closed interval I. With F = {A, B}, KF(I) = {B}. |
| Limit of the conclusion | B is the only description satisfying this criterion within this list. C shows that the result does not establish general uniqueness. |
In an empirical case, the origin of the values would need to be connected to the actual records, method, settings and uncertainty assessment available in the study. References should lead to the evidence from which the conclusion is drawn.
If an item is missing, we state that in its place. This lets the reader distinguish an open question from a supported result. The record is a presentation aid, not a universal validation protocol.
Connection to the corpus: Corpus Manifest v1.0, §§12–13, p. 7 ↗
06Preparing to read the Atlas
Theory specifies the structural question we ask. Mathematics lets us state the relations, comparison and scope of the result precisely. Empirical study needs to show how these connect to a particular system.
The Manifest presents the Atlas as the corpus’s empirical foundation in articulators and includes traces, contact continuity, friction, dominance distribution and validation within its scope. When reading such a case, we look for the connection between the declared setup, observation, interpretation and supporting evidence.
This chapter prepares that reading. The numerical example remains an educational construction; an actual Atlas case needs its own source and data.
Connection to the corpus: Corpus Manifest v1.0, §3.3, p. 3 · §10, p. 6 · §§12–13, p. 7 ↗
07Sources and further study
This chapter is a new introductory synthesis. Section and page references for FBM point to the public Corpus Manifest v1.0. VIM3 is used for metrological terminology. The numerical example, intervals and case-reading record were created for this page and are not experimental results. The full definitions and development of FBM remain in the corresponding monographs.