scaleAffine_forces_canonical_kernel
plain-language theorem explainer
Scale-affine ledger admissibility forces the BIT dark-energy equation of state onto the canonical kernel w(z)=-1+δw₀/(1+z). Cosmologists closing the RS dark-energy shape residue cite this under the no-extra-coordinate condition. The proof is a short rearrangement of the companion deviation identity via the BIT-kernel definition and linear arithmetic.
Claim. For any amplitude $\delta w_0$, any nonzero today-value $Z_{\mathrm{today}}$, and any scale-affine Z-law, at every redshift $z$ with $1+z\neq 0$, the BIT kernel on the Z-history generated by that law equals $-1+\delta w_0/(1+z)$.
background
The module Cosmic Z Scale Law tightens the last shape residue in the dark-energy plan. CosmicZHistory already shows that under the BIT kernel the equation-of-state deviation satisfies $\delta w(z)=\delta w_0\cdot Z(z)/Z_{\mathrm{today}}$, so the dark-energy shape is exactly the normalized cosmic-Z history. The open question is why that normalized history should be the scale factor $a(z)=1/(1+z)$.
A scale-affine Z-law packages a normalized fraction $Z_{\mathrm{frac}}$ with endpoints $Z_{\mathrm{frac}}(0)=0$ and $Z_{\mathrm{frac}}(1)=1$, plus ledger-uniform interpolation: $Z_{\mathrm{frac}}$ preserves convex combinations between the early zero-complexity endpoint $a=0$ and today $a=1$. That is the no-extra-coordinate condition (equal scale-factor fractions carry equal recognition-ledger fractions). The generated history is $Z(z)=Z_{\mathrm{today}}\cdot Z_{\mathrm{frac}}(a(z))$.
The BIT kernel is $w(z)=-1+\delta w_0\cdot Z(z)/Z_{\mathrm{today}}$; its deviation is $w(z)+1$. The sibling result already forces that deviation to $\delta w_0/(1+z)$.
proof idea
Short rearrangement, not a fresh derivation. Apply the sibling theorem that scale-affinity forces the canonical BIT deviation $\delta w(z)=\delta w_0/(1+z)$ on the Z-history generated by the law. Unfold the definition that the deviation equals the BIT kernel plus one. Finish by linear arithmetic to recover $w(z)=-1+\delta w_0/(1+z)$.
why it matters
This is the kernel form of the forced dark-energy equation of state under scale-affinity. It fills the canonical_kernel_forced field of the cosmic Z scale-law certificate, which packages identity, redshift-history, deviation, and kernel forcing as one inhabited record.
Downstream, the no-hidden-scale-coordinate derivation converts a NoHiddenScaleCoordinate hypothesis into a scale-affine law and applies this theorem, so the same canonical $w(z)$ follows from the no-hidden-coordinate condition alone.
In the RS cosmology stack this closes the shape residue named in the module doc: once the ledger is scale-affine from $a=0$ to $a=1$, the BIT kernel is forced to $w(z)=-1+\delta w_0/(1+z)$ with no free functional freedom left in the redshift dependence. Status is theorem conditional on the named admissibility law, zero sorry.
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