linearZ

In the BIT (Bosonic Identity Theorem) picture, the dark-energy equation of state takes the form $w(z) = -1 + \delta w \cdot Z(z)/Z_{\mathrm{today}}$, where $Z(z)$ is the integrated cosmic Z-complexity at redshift $z$ and $Z_{\mathrm{today}} = Z(0)$. The module reduces the dark-energy shape problem (U5) to deriving that history: under the BIT kernel the normalized deviation equals the normalized Z-history exactly (shape reduction), with boundary conditions $\delta w(0) = \delta w_0$ today and $\delta w \to 0$ in the early universe.

The remaining freedom is which monotone positive history $Z(z)$ is realized. The linear-in-$a$ choice $Z(z) = Z_{\mathrm{today}} \cdot a(z)$ is the member that yields the observationally familiar $1/(1+z)$ kernel. Scale factor and redshift are related by $a(z) = 1/(1+z)$ in the usual FLRW convention used here.

Definitional. The body is the single expression $Z_t/(1+z)$; there is no tactic proof or lemma application. Downstream lemmas unfold this definition and rewrite with field arithmetic.

This is the concrete accumulation law that turns the conditional half of U5 into a theorem. Feeding it into the BIT deviation formula yields exactly $\delta w(z) = \delta w_0/(1+z)$ (linear_accumulation_forces_canonical_kernel) and the equation of state $w(z) = -1 + \delta w_0/(1+z)$ (linear_accumulation_kernel). The certificate structure CosmicZShapeCert packages shape reduction plus this forcing step, stating that the only residual freedom is the accumulation law itself.

Sibling facts record the elementary analytic properties needed for that package: value today equals $Z_t$, positivity for $z \ge 0$ when $Z_t > 0$, and antitone behavior on $z \ge 0$. The same history is reused in the scale-law module (CosmicZScaleLawCert, scaleAffine_forces_canonical_deviation). What remains open in the framework is why accumulation should be linear in $a$ rather than in cosmic time or $a^p$; that single premise is the honest residue of U5.