linearZ_pos

This module treats cosmic Z-history as the remaining degree of freedom in the dark-energy shape problem (U5). Under the BIT mechanism the equation-of-state deviation is $w(z) = -1 + \delta w \cdot Z(z)/Z_{\mathrm{today}}$, so the normalized deviation equals the normalized Z-history. Deriving the shape is therefore exactly deriving $Z(z)$.

The linear history is the candidate $Z(z) = Z_{\mathrm{today}} \cdot a(z) = Z_{\mathrm{today}}/(1+z)$, i.e. accumulation linear in the scale factor. It is the unique monotone positive history that forces the canonical kernel $\delta w(z) = \delta w_0/(1+z)$ once the linear-accumulation premise is granted. Positivity of that history on $z \ge 0$ is the elementary domain fact needed before ratios and monotonicity statements are used.

Unfold the definition to $Z_t/(1+z)$. From $z \ge 0$ obtain $1+z > 0$ by linear arithmetic, then invoke the real-positivity tactic on the quotient of a positive numerator by a positive denominator. No external lemmas beyond the definition are required.

Inside the honest reduction of U5, the linear-in-$a$ history is the single remaining hypothesis that turns the BIT kernel into the canonical $\delta w(z) = \delta w_0/(1+z)$ shape. Positivity guarantees that the normalized ratio $Z(z)/Z_{\mathrm{today}}$ is well-defined and positive for all observational redshifts, matching the boundary conditions $\delta w(0) = \delta w_0$ and $\delta w \to 0$ in the early universe. Sibling facts (value today, antitonicity) complete the elementary calculus package around this history; the open residue of U5 is why accumulation should be linear in $a$ rather than in cosmic time or $a^p$.