Supersignum Algebra: a formal presentation of the circular–hyperbolic two-branch structure

The purpose of this manuscript is to present a complete formalization of a proposed “supersignum” structure whose informal clauses are the following: a symbol 𝑔 is intended to encode simultaneously the circular imaginary unit 𝑖 satisfying 𝑖² = −1 and the hyperbolic unit 𝑗 satisfying 𝑗² = 1; the exponential 𝑒^{𝑥𝑔} is intended to unify the circular functions (cos 𝑥, sin 𝑥) and the hyperbolic functions (cosh 𝑥, sinh 𝑥); the expression 𝑔(−𝑔) is intended to be sign-blind at the level of the square; the nilpotent unit 𝜀 is intended to be excluded; and an additional logarithmic clause is intended to compare the phases associated with 𝑖 and 𝑗. The present text writes these data as a sequence of definitions, propositions, theorems, and proofs. The resulting ordinary commutative real algebra is

𝒜 := ℝ[𝑠, 𝑔]/(𝑠² − 1, 𝑔² − 𝑠) ≅ ℝ[𝑔]/(𝑔⁴ − 1) ≅ ℂ × 𝕊,

with two canonical idempotent sectors. The exact universal trigonometric functions are obtained from the power series

𝐶_𝑠(𝑥) = ∑{𝑛≥0} (𝑠^{𝑛}𝑥^{2𝑛})/(2𝑛)!,  𝑆_𝑠(𝑥) = ∑{𝑛≥0} (𝑠^{𝑛}𝑥^{2𝑛+1})/(2𝑛+1)!,

which satisfy 𝑒^{𝑥𝑔} = 𝐶_𝑠(𝑥) + 𝑔𝑆_𝑠(𝑥) together with the full differential and addition identities. The literal set notation 𝑔 = {𝑖, 𝑗} is shown not to define an ordinary algebra under ordinary set multiplication; a tagged-set realization is then constructed and proved isomorphic to ℂ × 𝕊. The sign-blind identification 𝑥 ∼ −𝑥 is shown not to be compatible with single-valued addition, and the corresponding multivalued quotient is written explicitly as a commutative hyperring. The exclusion of 𝜀 is rewritten as the requirement that the algebra be reduced. The logarithm clause is shown to fail in the split-complex branch and to acquire a precise realization after passage to the classical tessarine algebra.