A Constructive Fragment of Physical Propositions

We develop a proof-theoretic analysis of the Operational Standard of Matsas, Pleitez, Saa, Vanzella (2024) showing that admissible measurement in Minkowski Spacetime yields only finite observational sequences and thereby restricts the class of physically meaningful propositions to those admitting terminating extraction procedures or uniform stability conditions. These correspond exactly to the arithmetical fragment ${\mathrm{\Sigma }}_1^0\cup {\mathrm{\Pi }}_2^0$, and the induced realizability structure interprets ${\mathrm{\Delta }}_0$ Heyting Arithmetic on the code of observational data. A diagonal argument then establishes an operational form of incompleteness: there exist true arithmetical propositions about admissible extraction that no sound, recursively axiomatizable theory of spacetime can decide. The result is structurally analogous to classical incompleteness but arises from the evidential limits of measurement rather than from ontological assumptions.