bitKernel
plain-language theorem explainer
The BIT dark-energy equation of state is the kernel w(z) = -1 + δw₀ · Z(z)/Z_today, with Z the integrated cosmic Z-complexity at redshift z. Anyone reducing the U5 dark-energy shape problem to a Z-history cites this as the exact link. It is a one-line algebraic definition encoding the Bosonic Identity Theorem hypothesis; no proof content.
Claim. For amplitude $\delta w_0$, today's cosmic-Z value $Z_t$, a history $Z:\mathbb{R}\to\mathbb{R}$, and redshift $z$, the BIT equation-of-state kernel is $w(z)=-1+\delta w_0\cdot Z(z)/Z_t$.
background
The module treats the dark-energy shape problem (U5) under the BIT mechanism. That mechanism supplies an equation of state of 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 local programme is an honest reduction: under this kernel the normalized equation-of-state deviation equals the normalized Z-history, so deriving the dark-energy shape is exactly deriving $Z(z)$. Boundary conditions are forced: $\delta w(0)=\delta w_0$ today, and $\delta w\to 0$ as $Z\to 0$ in the early universe (recovering $\Lambda$CDM).
The remaining freedom is the accumulation law for $Z(z)$. If $Z$ accumulates linearly in the scale factor, the canonical $1/(1+z)$ shape follows; that linear-accumulation premise is the sole residue of U5.
proof idea
Pure definition (no theorem content). The body is the one-line formula $-1 + \delta w_0\cdot Z(z)/Z_t$. Downstream lemmas unfold this definition and simplify by ring or rewriting.
why it matters
This kernel is the entry point for the entire CosmicZHistory reduction of U5. It is consumed by the deviation wrapper $\delta w=w+1$, by the identity that $\delta w(z)=\delta w_0\cdot Z(z)/Z_t$, and by the early-universe recovery $w=-1$ when $Z=0$.
Under linear scale-factor accumulation it yields the canonical kernel $w(z)=-1+\delta w_0/(1+z)$. The same object is re-used in CosmicZScaleLaw and DarkEnergyScaleAffinityDerivation: scale-affine ledger admissibility forces the normalized Z-history to be the scale factor and therefore forces this canonical shape. The definition thus localizes every subsequent certificate to a single, named functional form rather than an ad-hoc ansatz.
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