We wish you could have defined physical reality (many different definitions are going round, but none satisfactory) and the scope of mathematics as a language. Language is the transposition of some information/command on the mind/CPU of another person/operating system. Mathematics tells us how much a system changes in the right hand side, when the parameters of the left hand side change. This information is universal and invariant in cognition. To that extent, mathematics is a language of physics. But it does not describe what, why, when, where, or how about the parameters or the system. It gives partial information. Generalizing such partial information misleads. Thus, it cannot be the only language of Nature.
Mathematics, explains the accumulation and reduction of numbers linearly or non-linearly of confined or discrete objects. Even analog fields are quantized. Accumulation or reduction is possible only in specific quantized ways and not in an arbitrary manner (even fractions or decimals are quantized). Proof is the concept, whose effect remain invariant under laboratory conditions. Logic is the special proof necessary for knowing the unknown aspects of something generally known. Thus, the validity of a mathematical statement rests with its logical consistency.
The validity of a physical statement rests with its correspondence to reality. What is the precise and scientific definition of space? Does the Hilbert and other spaces or sub-spaces have any physical significance and conform to the precise definition of space as a class?
Synchronization is the operation or activity of two or more things at the same time or rate. There is nothing strange about it. A vane is a broad blade attached to a rotating axis or wheel which pushes or is pushed by wind or water and forms part of a machine or device such as a windmill, propeller, or turbine. Your description: "In the vane, the normalized vector that represents the elementary module is eigenvector of a private operator that attaches a spatial location as the imaginary part of the eigenvalue to the elementary module" is presenting the same fact in a rather incomprehensible language for laymen. How can "Modules act as observers and all observers travel with the vane". Modules can be inert. Can there be inert observers?
You begin with: "Construct in a modular way". However, also non-discrete items exist. Universe contains continuums and these continuums appear to relate to the discrete objects. Modular design is a design approach that subdivides a system into smaller parts called modules or skids that can be independently created and then used in different systems. Continuum is a continuous sequence in which adjacent elements are not perceptibly different from each other, but the extremes are quite distinct. You cannot deny at least quarks and leptons that constitute every object in the universe. They are perceptible different from each other. What you appear to say by: continuum which "relate to the discrete objects", is, all objects are modular in the space-time continuum. But how do you justify your statement: "â€ŽFurther, we as intelligent observers of these facts, want to place everything into an appropriate model, such that we can comprehend our environment. This model appears to be capable to generate intelligent species"? Do we regulate creation? Or is the creation like this because we comprehend it like this? Kindly educate us on these issues.