My understanding of Kant's synthetic a priori is that it refers to a kind of knowledge that can be deduced, but without reference to experience. This would mean there are two conditions limiting what may validly be considered to be synthetic a priori knowledge: (1) it must be able to be deduced from other principles or information and (2) it must be able to be deduced without reference to experience. Is this correct, in your view?
Your way of interpreting synthetic a priori knowledge is inadequate since it cannot be distinguished from the rationalist view. It is the rationalists (e.g. Descartes, Spinoza and Leibniz) who believed that certain concepts and knowledge (specifically, metaphysical knowledge) can be gained independently of sense experience by virtue of reason alone. (cf https://plato.stanford.edu/entries/rationalism-empiricism/) so your (2) clearly is the rationalist position. The rationalists also believed that once we settle with the first principle (e.g., “Cogito ergo sum”), all other metaphysical principles can be deduced from it, so they held your (1).
To discriminate Kant’s synthetic a priori knowledge from rationalists’ innate knowledge, you have to appeal to Kant’s theory of transcendental deduction (https://plato.stanford.edu/entries/kant-transcendental/). A transcendental deduction is involved with finding out the preconditions that allow we humans to perceive the world the way we perceive. For example, when I see furriness, snout nose, floppy-ear and an adorable smile, why do I immediately attach objecthood (dog) which sustains all these properties? Or, when I see smoke after fire, why do I think that the fire is the cause of the smoke? Kant wanted to find out the reasons for our thinking these ways, and argued that the reason that we humans perceive that way (e.g., causality and objecthood) must be due to the fact that we humans are wired as such. That is, it is impossible for us not to perceive the world the way we perceive. Stated equivalently, we humans can perceive the world only through the way we are wired. This is why we cannot perceive ultraviolet wavelengths, unlike bees. Had we been wired differently, the world would have looked differently to us humans. Kant argued that we humans are wired by 12 a priori categories. This reasoning of Kant is called the transcendental deduction.
Synthetic a priori knowledge (such as math, laws of physics, and metaphysics, according to Kant) must be understood under this transcendental view of human knowledge. Knowledge of this type is necessarily true neither because the truth of the predicate can be deduced from the subject (Hume’s analytic truth), nor because it is known by the rationalist intuition (Leibniz’s dogmatic assertions). To Kant, synthetic a priori knowledge is necessarily true because it can be purely deduced from the nature of the presuppositions, that is, the categories. Knowledge of this type is synthetic since the truth of the predicate is not embedded in the subject; it is a priori since the truth is knowable without resorting to sense perception.
Kant’s transcendental epistemology is valuable for its synthesizing effect. Hume rejected a priori metaphysical knowledge; Leibniz rejected synthetic metaphysical knowledge. Kant argued that metaphysical knowledge must be synthetic a priori.
Your general question (let's call it the question suggested by your question) requires a long answer. I hope this helps.
I would study this first:
I realize that links can break, and websites can disappear, but it would be difficult to properly answer your question with one finger on my little screen!
Once that is done (and completed) you may want to do a search: Pitt.edu Einstein on Kant. There is some very interesting material on these pages prepared by Prof. John D. Norton so you may want to prowl around a bit. For instance, after you read Einstein on Kant, go back toward the top and hit "main text" too.
Before answering, here's a suggestion for reading an abridged version of the "Critique of Pure Reason" written by, wait for it... yes, Immanuel Kant. It's called the "Prolegomena to any Future Metaphysics". It's roughly 120 pages long. Kant explains in the introduction that he wrote it for those who do not have the stamina to last through hundreds of pages of the 'Critique' which at one point just becomes a constant repetition of the same analysis under the heading of a 'new' category.
Jefferysbrother and Nanhee did a splendid job on the technical description which they offered above. This one will be slightly more down home. They mentioned the phrase "wired that way" which is where this comment will direct your attention. Kant in the Prolegomena asked two questions; how are pure Mathematics and pure Science possible? That is, how can it be determined that these two systems involve objective truth or reality without some attribution to any human inclusion in these type of judgements. I do not feel the need to expand on the nature of the problem here. I'm going to assume that those who read this clearly understand the conundrum.
Kant points out that human perambulation involves some very complicated computational dynamics to happen. For example it has been noted that if we had to use a deliberative consciousness to climb a flight of stairs then we would not be able to do it. So Kant creates a term to complement the 'synthetic a priori', he termed it the 'subjective essential'. Think of it as a built in gyroscopic sentience, something akin to its mental co-relative, projicience. This sensory motor allows us to navigate our way through doorways and stairs and ladders and hills, etc. All without any conscious input on our part. Adios. CS