I like this description, Such clarity!
In his Sat Sandarba, Srila Jiva Goswami defines adhokshaja like this: The Godhead,Krishna,is He who has reserved the absolute right not to expose Himself to mundane human senses. Adhokshaja refers to that which cannot be measured by one's material senses. When that transcendental knowledge reveals itself to our service-inclined senses, only then will we realize it; otherwise, even the best students in the world cannot even partially know the science of Krishna. That science cannot be learned through sadhana, by the best forms of intelligence, or the best powers of deliberation.
Despite this, many people rush to discuss it with an enjoying spirit, considering it a mundane philosophy or text. The Absolute Truth is never revealed to such people. We cannot measure objects of the fourth or fifth dimension with our present material sense devises. Being unable to do so, we often heave a sigh of relief with phrases like "To the infinity." Vaikuntha, or the transcendental object, is of the fourth dimension. Because we are unable to perceive Him, we want to turn Him into something "impersonal." But the transcendental Absolute Truth is never impersonal.
- by His Divine Grace Srila Bhaktisiddhanta Saraswati Takur Prabhupada, Sarasvati Samlapa, page 77