Interface Realizability Problem

Existing page: Interface Realizability Problem
Recommended public status wording: Preliminary research question — formal scope and canonical source not yet fixed.
Suggested display title: When Can Two Structured Descriptions Be Connected by an Admissible Interface?

Catalog summary

Determine when two specified structured descriptions admit a comparison that satisfies explicitly stated preservation and compatibility requirements. This is a preliminary research question: the domains, allowed interface type, and admissibility conditions still need to be fixed.

What the problem concerns

Two mathematical descriptions may concern related phenomena while carrying different kinds of information. An interface is a proposed way to connect them. Depending on the setting, it could be a map, a functor, a correspondence, or a more structured comparison.

The realizability question asks whether a comparison of the required kind actually exists and satisfies the conditions needed for the intended application. Those conditions must be stated for that setting. Calling an interface “admissible” does not supply them.

What is already established

The public archive identifies this as a line of inquiry about passage between structured domains. It has not yet attached a canonical manuscript, fixed the two domains, specified a class of interfaces, or supplied a general realizability theorem. The linked general Admissibility entry also remains a placeholder awaiting exact source wording.

Accordingly, this page records a problem under formulation. It does not yet state a single fully specified conjecture.

What remains open

For two explicitly specified structured domains, determine whether there exists an interface satisfying a declared set of preservation and compatibility conditions.

To turn this into a concrete mathematical problem, specify:

1. The two domains. What are their objects, comparisons, and relevant structures?
2. The allowed interface. Is the candidate a map, functor, correspondence, or another stated kind of object, and in which direction does it act?
3. The admissibility conditions. Which structures must be preserved exactly, up to a declared equivalence, or within an explicitly allowed defect? Which information may be discarded?
4. The compatibility conditions. How must the interface behave under composition, restriction, refinement, or change of presentation, where these operations are part of the chosen problem?

These are specifications still needed for this entry. They are not a proposed universal definition of admissibility.

Why the question requires more than matching outputs

Two descriptions can agree on one observable while disagreeing on distinctions the intended interface must preserve. Agreement of outputs is evidence for an interface only to the extent that the chosen admissibility conditions concern those outputs. Stronger requirements require additional evidence.

Example

As an illustration, suppose one description distinguishes states a and b, while a second description records only a common output c. The map sending both a and b to c exists.

If the declared interface is allowed to forget that distinction, this map may be acceptable. If the interface must permit recovery of whether the original state was a or b, the same map fails: no recovery map can send the single output c back to both distinct states.

The example shows why realizability depends on the stated requirements. It does not identify state recovery as a requirement of every physical interface or as the settled content of this research problem.

What would count as a resolution

First, fix a concrete setting and attach its canonical formulation. A resolution in that setting would then construct an interface and verify every declared requirement, or prove that no interface of the allowed kind can meet them.

A broader classification would characterize which pairs of domains admit such interfaces under fixed rules. Until those rules are supplied, “the Interface Realizability Problem is solved” would have no determinate mathematical meaning.

Source and present limitation

Interface Realizability Problem, Interface Problems in Mathematical Physics, and Admissibility, reviewed September 8, 2026. These public records motivate the explanation above but do not supply a canonical formal definition or theorem. The proposed wording makes that limitation visible rather than filling it with an invented claim.