IndisputableMonolith.Cosmology.DarkEnergyScaleAffinityDerivation
Derives the lower admissibility condition for the scale-affine cosmic-Z law: with only the early endpoint a=0 and today a=1 fixed, a normalized ledger fraction cannot pick a nonlinear scale coordinate without extra structure, so endpoint interpolation is forced. Cosmologists working the dark-energy residual cite it when reducing free shape in δw(z). The argument is a chain of no-hidden-coordinate lemmas into a certificate.
claimIf a normalized ledger fraction on the scale factor has no hidden scale coordinate beyond the endpoints $a=0$ and $a=1$, then its induced map is scale-affine: it preserves endpoint interpolation and yields a linear relation in the cosmic $Z$-coordinate, so the dark-energy residual takes the form $\delta w(z)=\delta w_0\cdot Z(z)/Z_{\mathrm{today}}$.
background
The parent setting is the cosmic $Z$ scale law: under the BIT kernel, the dark-energy equation-of-state residual is forced into the shape $\delta w(z)=\delta w_0\cdot Z(z)/Z_{\mathrm{today}}$. That module still leaves a last shape residue: which coordinates on the expansion history are admissible once only the early and late endpoints are fixed.
This module isolates the lower admissibility condition behind that scale-affine $Z$ law. A "no hidden scale coordinate" hypothesis means the normalized ledger fraction is not allowed an extra nonlinear reparameterization of the scale factor beyond the two endpoints $a=0$ (early) and $a=1$ (today). Without that extra structure, the only remaining freedom is endpoint interpolation.
Sibling objects package the hypothesis, the forced identity and linear-$Z$ maps, the canonical kernel and deviation forms, and a derivation certificate that records the implication chain.
proof idea
Not a single theorem: a small derivation stack. The core hypothesis is the no-hidden-scale-coordinate condition. From it the module proves successive strengthenings: the map is scale-affine, then the identity on the normalized interval, then linearity in the cosmic $Z$ coordinate, then the canonical deviation and kernel forms. A canonical instance of the hypothesis is shown to land on the canonical affine map. The stack closes with a certificate object that packages the derivation for downstream cosmology proofs.
why it matters in Recognition Science
Closes the lower half of the dark-energy plan's last shape residue. Upstream, CosmicZScaleLaw already forces $\delta w(z)\propto Z(z)/Z_{\mathrm{today}}$ under the BIT kernel; this module justifies why the coordinate on that residual must stay scale-affine once only $a=0$ and $a=1$ are given. No downstream edges are recorded yet, so the immediate consumers are the certificate and the scale-affine $Z$-law statements in the same cosmology layer. In Recognition terms it is bookkeeping on admissible ledger fractions, not a new forcing step in T0–T8, but it keeps the dark-energy residual free of smuggled nonlinear gauges.
scope and limits
- Does not derive the BIT kernel or the proportionality $\delta w\propto Z/Z_{\mathrm{today}}$ itself.
- Does not fix numerical values of $\delta w_0$, $Z_{\mathrm{today}}$, or cosmological parameters.
- Does not rule out nonlinear coordinates if extra structure beyond endpoints is allowed.
- Does not address spatial curvature, matter sector, or non-BIT kernels.
- Does not claim observational uniqueness of the dark-energy residual.
depends on (1)
declarations in this module (10)
-
structure
NoHiddenScaleCoordinate -
def
noHidden_to_scaleAffine -
theorem
noHidden_forces_identity -
theorem
noHidden_forces_linearZ -
theorem
noHidden_forces_canonical_deviation -
theorem
noHidden_forces_canonical_kernel -
def
canonicalNoHiddenScaleCoordinate -
theorem
canonicalNoHidden_maps_to_canonical -
structure
ScaleAffinityDerivationCert -
def
scaleAffinityDerivationCert