The Hidden-space Bridge System: A Three-axiom Foundation Connecting the Real and Imaginary Domains

Abstract

We present a generative-layer proposal called HIBS (Hidden-space Bridge System): a conjecture that the complex plane C is not a foundational object but rather the observable slice of a hidden generative layer S under a projection pi. We introduce an abstract space S whose elements are called hidden numbers h — neither real nor imaginary. "Hidden number" is an intermediate notation for the generative layer; it is not a new kind of number. Under certain arithmetic operations, a hidden number is mapped by the flow operator onto the real line R or onto the imaginary line iR; under other operations it remains inside S. Three axioms are postulated: (1) the existence of two maps f: S -> R and g: S -> iR that are non-injective (no left-inverses); (2) closure of S under addition and subtraction, together with the fact that multiplication and division necessarily map onto R; and (3) the square-root operation maps onto iR, with the positive half-axis mapping to iR^- and the negative half-axis to iR^+. The main mathematical results of the paper are: (i) C is the projection image of S under pi (pi ∘ iota = id_C); (ii) iota is not a multiplicative monomorphism — the "rotation" semantics of C multiplication cannot pass through iota into S. These two results together give an axiomatic-level account of the structural asymmetry between the real and imaginary parts of C.

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