A Rational Coordinate Space for the Exponential Period - Formal construction, hidden-coordinate extension, compactification, algebraic adjunction, and Abel linearization

A finite-dimensional rational coordinate space is constructed for the principal exponential period. The construction begins with the principal logarithmic value

g := Log(−e) = 1 + iπ,

and studies the ℚ-vector space

V_exp := span_ℚ{1, i, g} ⊂ ℂ.

It is proved that {1, i, g} is a ℚ-basis, that

V_exp = {x + i(y + kπ) : x, y, k ∈ ℚ},

that the coordinates are explicit and unique, and that every principal logarithm of a root of unity belongs to V_exp. The same data are then recast as an intrinsic coordinate model

E_Q := ℚ³,  ε_Q(x, y, k) = x + i(y + πk),

whose realification fits into the exact sequence

0 → K → E_R --ε_R→ ℂ → 0,  K = ℝ(0, −π, 1).

It is proved that E_Q ∩ K = {0}, so the model is genuinely larger than ℂ while remaining arithmetically rigid on its rational lattice. An explicit splitting

E_R ≅ ℂ × ℝ

is constructed, and this yields a fiberwise compactification

Ĉ × ℝP¹

which preserves the projection to the visible complex value. The algebraic part of the construction is separated from the exponential part by proving

V_exp ∩ Q̄ = ℚ(i).

This yields a canonical adjunction theorem for algebraic numbers outside ℚ(i). It is further proved that V_exp is not multiplicatively closed, so the space is an exact additive coordinate model for branch-logarithmic quantities, not a period algebra. Finally, an abstract Abel-linearization theorem is stated and proved: whenever a map admits an Abel coordinate L with

L(E(z)) = L(z) + 1,

every forward and backward iterate is affine in that coordinate.