Popper's falsifiability criterion doesn't make much sense if the theories in question are not DEDUCTIVE:
Karl Popper: "According to the view that will be put forward here, the method of critically testing theories, and selecting them according to the results of tests, always proceeds on the following lines. From a new idea put up tentatively, and not yet justified in any way - an anticipation, a hypothesis, a theoretical system, or what you will - conclusions are drawn by means of LOGICAL DEDUCTION. These conclusions are then compared with one another and with other relevant statements, so as to find what logical relations (such as equivalence, derivability, compatibility, or incompatibility) exist between them. We may if we like distinguish four different lines along which the testing of a theory could be carried out. First there is the logical comparison of the conclusions among themselves, by which the internal consistency of the system is tested. Secondly, there is the investigation of the logical form of the theory, with the object of determining whether it has the character of an empirical or scientific theory, or whether it is, for example, tautological. Thirdly, there is the comparison with other theories, chiefly with the aim of determining whether the theory would constitute a scientific advance should it survive our various tests, and finally, there is the testing of the theory by way of empirical applications of the conclusions which can be derived from it." The Logic of Scientific Discovery, p. 9 https://www.amazon.com/Logic-Scientific-Discovery-Routledge-Classics/dp/0415278449
Except for special relativity, models and theories in modern physics are empirical, not deductive. The method by which they are obtained is "guessing the equation", not "deducing the equation":
Richard Feynman: "Dirac discovered the correct laws for relativity quantum mechanics simply by guessing the equation. The method of guessing the equation seems to be a pretty effective way of guessing new laws." http://dillydust.com/The%20Character%20of%20Physical%20Law~tqw~_darksiderg.pdf
Below Einstein defines two types of theory - empirical and deductive - and it is unquestionable that the equations of a deductive theory do matter. The problem is: Do the equations of an EMPIRICAL theory matter in physics?
Albert Einstein: "From a systematic theoretical point of view, we may imagine the process of evolution of an empirical science to be a continuous process of induction. Theories are evolved and are expressed in short compass as statements of a large number of individual observations in the form of empirical laws, from which the general laws can be ascertained by comparison. Regarded in this way, the development of a science bears some resemblance to the compilation of a classified catalogue. It is, as it were, a purely empirical enterprise. But this point of view by no means embraces the whole of the actual process ; for it slurs over the important part played by intuition and deductive thought in the development of an exact science. As soon as a science has emerged from its initial stages, theoretical advances are no longer achieved merely by a process of arrangement. Guided by empirical data, the investigator rather develops a system of thought which, in general, is built up logically from a small number of fundamental assumptions, the so-called axioms." https://www.marxists.org/reference/archive/einstein/works/1910s/relative/ap03.htm
The equations of an empirical theory do (should) not matter in physics. Any such equation, together with its implications, forms a local cluster that has no logical connections with anything else in theoretical physics. Popper's "four different lines along which the testing of a theory could be carried out" are obviously meaningless in the case of an empirical theory.
Unlike special relativity, Einstein's general relativity is not a deductive theory. It is a not-even-wrong empirical concoction - a malleable combination of ad hoc equations and fudge factors allowing Einsteinians to predict anything they want. Its creation marked the transition from deductivism to empiricism in physics, or from "deducing the equation" to "guessing the equation". Einstein and his mathematical friends spent years tirelessly "guessing the equation" until "excellent agreement with observation" was reached:
Michel Janssen: "But - as we know from a letter to his friend Conrad Habicht of December 24, 1907 - one of the goals that Einstein set himself early on, was to use his new theory of gravity, whatever it might turn out to be, to explain the discrepancy between the observed motion of the perihelion of the planet Mercury and the motion predicted on the basis of Newtonian gravitational theory. [...] The Einstein-Grossmann theory - also known as the "Entwurf" ("outline") theory after the title of Einstein and Grossmann's paper - is, in fact, already very close to the version of general relativity published in November 1915 and constitutes an enormous advance over Einstein's first attempt at a generalized theory of relativity and theory of gravitation published in 1912. The crucial breakthrough had been that Einstein had recognized that the gravitational field - or, as we would now say, the inertio-gravitational field - should not be described by a variable speed of light as he had attempted in 1912, but by the so-called metric tensor field. The metric tensor is a mathematical object of 16 components, 10 of which independent, that characterizes the geometry of space and time. In this way, gravity is no longer a force in space and time, but part of the fabric of space and time itself: gravity is part of the inertio-gravitational field. Einstein had turned to Grossmann for help with the difficult and unfamiliar mathematics needed to formulate a theory along these lines. [...] Einstein did not give up the Einstein-Grossmann theory once he had established that it could not fully explain the Mercury anomaly. He continued to work on the theory and never even mentioned the disappointing result of his work with Besso in print. So Einstein did not do what the influential philosopher Sir Karl Popper claimed all good scientists do: once they have found an empirical refutation of their theory, they abandon that theory and go back to the drawing board. [...] On November 4, 1915, he presented a paper to the Berlin Academy officially retracting the Einstein-Grossmann equations and replacing them with new ones. On November 11, a short addendum to this paper followed, once again changing his field equations. A week later, on November 18, Einstein presented the paper containing his celebrated explanation of the perihelion motion of Mercury on the basis of this new theory. Another week later he changed the field equations once more. These are the equations still used today. This last change did not affect the result for the perihelion of Mercury. Besso is not acknowledged in Einstein's paper on the perihelion problem. Apparently, Besso's help with this technical problem had not been as valuable to Einstein as his role as sounding board that had earned Besso the famous acknowledgment in the special relativity paper of 1905. Still, an acknowledgment would have been appropriate. After all, what Einstein had done that week in November, was simply to redo the calculation he had done with Besso in June 1913, using his new field equations instead of the Einstein-Grossmann equations. It is not hard to imagine Einstein's excitement when he inserted the numbers for Mercury into the new expression he found and the result was 43", in excellent agreement with observation." https://netfiles.umn.edu/users/janss011/home%20page/EBms.pdf
"Guessing the equation" is naturally followed by "guessing the fudge factor". In the video below, at 0:57, a fudge factor is added to an equation in an empirical model (Einstein's general relativity), then at 2:16 the fudge factor is removed:
SPACE'S DEEPEST SECRETS Einstein's "Biggest Blunder"
"A fudge factor is an ad hoc quantity introduced into a calculation, formula or model in order to make it fit observations or expectations. Examples include Einstein's Cosmological Constant..." https://en.wikipedia.org/wiki/Fudge_factor
Can one add a fudge factor analogous to the cosmological constant to the Lorentz transformation equations? One cannot, and the reason is simple: Special relativity is deductive (even though a false postulate and an invalid argument spoiled it from the very beginning) and fudging is impossible by definition - one has no right to add anything that is not deducible from the postulates.
Nowadays, except for special relativity, theories and models in physics are empirical, non-deductive - they cannot be presented as a set of valid arguments built up logically from a small number of simple axioms (postulates). This makes them unfalsifiable a priori. Only deductive theories (models) can be falsified, either logically or experimentally. That is:
1. Arguments can be checked for validity.
2. The reductio-ad-absurdum procedure can be applied.
3. Showing, experimentally, that a postulate or a deduced consequence is false makes sense - the deductive structure allows one to interpret the falsehood in terms of the whole theory. (In the absence of a deductive structure any detected falsehood or absurdity remains insignificant - one can ignore it or "fix" it in some way, e.g. by introducing a fudge factor.)
The only alternative to deductivism is empiricism - theories are essentially equivalent to the "empirical models" defined here:
"The objective of curve fitting is to theoretically describe experimental data with a model (function or equation) and to find the parameters associated with this model. Models of primary importance to us are mechanistic models. Mechanistic models are specifically formulated to provide insight into a chemical, biological, or physical process that is thought to govern the phenomenon under study. Parameters derived from mechanistic models are quantitative estimates of real system properties (rate constants, dissociation constants, catalytic velocities etc.). It is important to distinguish mechanistic models from empirical models that are mathematical functions formulated to fit a particular curve but whose parameters do not necessarily correspond to a biological, chemical or physical property." http://collum.chem.cornell.edu/documents/Intro_Curve_Fitting.pdf