In mathematics, axioms and postulates, are not based on observations. That would be self-contradictory, since they are not physically observable phenomenon. One might observe physical approximations to such abstract entities, but the abstractions themselves cannot be observed. "The conclusions of mathematical operations (correctly performed) are always logically consistent", as you said, but that does not prove that the starting premises are true, it only proves that the conclusion follows from them. The conclusions of Euclidian Geometry follow from its axioms. But the conclusions of non-Euclidean geometry do not, instead, they follow from a different set of axioms.
The definition you appear to be using for the word "information", is not that used in Information Theory. You stated that "Information theory tries to make the concepts opaque to the less privileged." It did not try, that was merely a side-effect. What it tried to do is give the less privileged, and everyone else, better communications, such as our modern cell phones and high definition television. Most people believe it succeeded.
Shannon did deal with electromagnetic waveforms. But he did not deal with electromagnetic theory. It was not necessary for him to do so, since he demonstrated that ALL waveforms, electromagnetic or otherwise, are subject to the same laws of information theory - they are in essence, mathematical laws, rather than physical laws. Hence, if you use math to describe the physics, the resulting physical laws, whatever they may be, are going to be subject to the math laws.
"Indices are results of past measurement, which are fairly repetitive." Measurements of many phenomenon are not repetitive at all. Physics merely restricts its domain, to those that are. That restriction, is the ultimate reason that math is so effective, when applied to physics.
"How can you be sure about "ALL", when the machine acts only on command programmed by the designer with limited capabilities?" By postulate.
Searle's Chinese Room has nothing to do with computers or perception per se. Searle's point is that there is not enough "information" in any communication or description, to ever enable one to distinguish between an intelligent being and an entity with no intelligence, but complete, a priori knowledge, in cases in which auxiliary channels of information (such as being able to directly observe the entities) are not available.