[I]s there any attempted framework for “proof” outside a mathematical context, determining the meaning and veracity of real-world propositions with linguistic concepts, in a systematic, structured and rule-based way, like “the train is late”?
'No' is the currently received view on this question. This is because the notion of facts being free of normativity is largely rejected, even if one attempts to defend "objective" observations as being intersubjective. Subjective bias, theory-laden definition, and bundles of hypothesis seem to make this impossible and have raised questions regarding the incommensurability of scientific theories.
This is an excellent question. 'Is it possible to use math and logic to make science more rigorous is" the right impulse, one that motivated operationists, logical positivists, and logical empiricists to search for ways to squeeze certainty using mathematics and logic. It's a conversation that lasted almost five decades until incommensurability (SEP) especially by Feyerabend and Kuhn and others in the spirit of Duhem became considered endemic to scientific theory. This whole discussion is part of the larger dichotomy between scientific realists and instrumentalists (SEP).
Given the long conversation, let's grab some ideas from Hempel's A Logical Appraisal of Operationism which is contained in Aspects of Scientific Explanation. In it, Hempel attempts to compare the methods of operationists like Bridgman with ideas that Carnap, Ramsey, and others done ending in commentary involving Quine's Two Dogmas.
After examining the operational definition, Hempel comes to the conclusion that such efforts "would evidently be overly restrictive" and that it would "need to be construed more broadly" (pgs.125,126). A measured object isn't measured continuously and its characterization is thus dispositional (SEP). More importantly, he goes on to say that "observational vocabulary of science is essential to the idea of operational definition". This notion of splitting up statements that comprise a scientific theory as exclusively observational or theoretic and ruling out meaningless metaphysics was at the core of the program of the logical empiricists.
The general idea is that it should be possible to have claims in natural language become irrefutably logical and represent elementary intersubjective facts about the natural world. Doing so would form a foundation of a scientific theory and make that theory rigorous and irrefutable once verified, a doctrine called verificationism. Confirmationism was then argued to save verification. That was found to be lacking by Popper who advocated for falsificationism. Hempel:
[T]he original requirement of full verifiability or full falsifiability by experiental data has to give way to the more liberal demand for confirmatbility... [t]his demand can be shown to be properly applicable to entire theoretical systems rather than to individual hypotheses - a point emphasized, in effect, already by Pierre Duhem.
He then goes on to recognize Quine's attack on the analytic-synthetic divide.
What does this all mean? It means, to sum up, that there is no clear demarcation between the psychological and empirical, and the logical and theoretical. Or more precisely, analytic statements which are essentially objective facts cannot be cleanly differentiated from synthetic statements which are essentially semantic in nature. Observational claims are necessarily tinged by theories. This is known as the theory-ladenness of scientific theories and a pillar in arguing the incommensurability of scientific theories.
So, to recap, the drive to show it is possible to make facts irrefutable collapsed by the 1970's into discussion about is it even possible to have two scientists understand each other's theories:
From Blackwell's A Companion to the Philosophy of Science, the article Incommensurability, p. 172:
One common and natural interpretation of the idea that there is no common measure among rival scientific theories is that they cannot be phrased in a common set of linguistic terms, or, to put it more simply, that they cannot both be translated into a single language.
Quite the reversal! In prior decades, the ambition was to craft a technique for mechanizing observations and their statements, and now science found or rather finds itself struggling to explain how competing theories even are understood by opposing camps. And there are other problems raised by the Duhem-Quine Problem and related concepts like confirmation holism.