Once experiments are designed to generate structured variation, a familiar anchor disappears.
There is no longer a guarantee that results will:
collapse to a single value
And without that collapse, a practical question becomes unavoidable:
how do we compare results that are not expected to agree?
This is where the shift becomes technically demanding.
Because comparison, as traditionally practiced, has been tied to convergence.
The inherited model of comparison
In standard experimental logic, comparison is straightforward.
Multiple measurements:
- target the same quantity
- differ due to error
- are refined until they agree
Comparison serves one purpose:
determine how close each result is to the true value
Disagreement is temporary.
Convergence is the goal.
What breaks when convergence is not assumed
If results are configuration-dependent, this model no longer holds.
Measurements may be:
- individually precise
- internally consistent
- reproducible within their setup
And yet:
systematically different across setups
At this point, comparison cannot mean:
which one is correct?
Because the premise of a single target is no longer secure.
The shift: from identity to relation
Comparison must be redefined.
Not as:
reduction to sameness
but as:
identification of structured relations between outcomes
This is the key move.
Results are not compared by collapsing them.
They are compared by:
- mapping how they differ
- identifying patterns in those differences
- determining how one result transforms into another under changes in configuration
What replaces convergence
Instead of convergence, we seek:
- consistency of difference
- stability of transformation
- coherence across configurations
A set of measurements is well-understood when:
the relations between them are stable, reproducible, and analysable
Agreement is one special case of this.
But it is no longer the only one.
A simple illustration
Suppose two experimental configurations produce slightly different values.
Under standard logic:
- one is closer to the truth
- the other reflects error
Under relational comparison:
- the difference itself is the object of interest
We ask:
- does the difference persist across repetitions?
- does it vary predictably with changes in configuration?
- can it be systematically related to identifiable constraints?
If so, the difference is not noise.
It is:
a stable relation between two measurement regimes
Transformation as comparison
The most powerful form of comparison becomes:
the ability to transform results from one configuration into another
This may take the form of:
- correction functions
- mapping relations
- regime-dependent adjustments
But the key is:
these transformations are themselves empirical objects
They can be:
- tested
- refined
- compared across experiments
Comparison becomes:
the study of how outcomes relate under controlled variation
Families of equivalence
In some cases, different configurations will produce results that can be grouped.
Not because they are identical, but because they are:
equivalent under a defined transformation
This introduces the idea of equivalence classes:
- sets of outcomes that belong together
- defined by their relational structure
- not by strict numerical identity
A “constant” may then appear as:
an invariant within such a class—not across all possible configurations
Returning to gravitational measurement
Different experimental approaches to gravitational interaction:
- torsion balances
- atom interferometers
- free-fall systems
produce slightly different values.
Instead of forcing convergence, we can ask:
- what transformations relate these results?
- which configurations cluster together?
- where do systematic differences emerge?
- how do these differences depend on experimental structure?
The goal is not to eliminate discrepancy.
It is to:
map the relational space in which these discrepancies occur
Statistical practice, reoriented
Even statistical tools take on a different role.
Instead of:
- averaging toward a single value
we can:
- model structured variation
- identify correlations with configuration parameters
- quantify stability of differences
Statistics shifts from:
collapsing variation
to:
describing its organisation
Why this is harder
Comparison without convergence is more demanding.
Because:
- it requires tracking more variables
- it resists simple summary
- it produces richer, but more complex, outputs
There is no single number to report.
There is:
a structured set of relations
This is harder to communicate.
But it is also more informative.
What counts as success
A set of measurements is successful when:
- differences are reproducible
- transformations between results are stable
- equivalence structures can be identified
- dependencies are clearly mapped
Success is not:
agreement
It is:
coherent relational structure
What this enables
Once comparison is redefined, new possibilities open:
- linking experimental regimes that were previously treated as incompatible
- identifying hidden parameters through systematic divergence
- refining models to account for structured dependence
- expanding the domain of what can be meaningfully compared
The field of inquiry becomes richer.
Not because it is less constrained, but because:
constraints are now part of what is being compared
Closing
Comparison has long been tied to convergence.
But convergence is only one way for results to relate.
When experiments are designed under visible conditions, comparison must follow suit.
It becomes:
the study of how stable outcomes relate across structured variation
Not:
which result is correct
But:
how different results belong to a coherent system of relations
The next step pushes this further:
if comparison is relational, what does it mean to model a system—not as a set of objects with properties—but as a structure of relations across configurations?
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