IndisputableMonolith.Cosmology.CosmicZHistory
Defines the BIT dark-energy equation of state and the cosmic Z-history that drives it: w(z) = -1 + δw₀ · Z(z)/Z_today. Introduces the bit kernel, bit deviation, and a linear Z accumulator with positivity and antitonicity. Cosmologists citing the RS dark-energy plan use this as the shape layer before the scale law. The module is mostly definitions plus elementary identities and order facts on the linear accumulator.
claimUnder the BIT kernel, the dark-energy equation of state is $w(z) = -1 + \delta w_0 \cdot Z(z)/Z_{\mathrm{today}}$, where $Z$ is a positive, antitone linear accumulator of cosmic history, normalized so that the fractional deviation today is $\delta w_0$.
background
Recognition Science cosmology treats dark energy as a residual cost defect on the recognition ledger rather than a bare cosmological constant. The BIT (bit-kernel) model packages that residual into a redshift-dependent equation-of-state correction. The module sits in the Cosmology domain and imports RS constants and the J-cost infrastructure, so the cost language is available even when the present file only needs the shape of $Z$.
The central objects are a bit kernel (the early-universe seed of the residual), a bit deviation $\delta w$ built from that kernel, and a linear cosmic accumulator $Z(z)$ that is positive and antitone, with a distinguished today-value $Z_{\mathrm{today}}$. The claimed shape is that the deviation tracks the normalized accumulator: $\delta w(z) = \delta w_0 \cdot Z(z)/Z_{\mathrm{today}}$, equivalently $w(z) = -1 + \delta w_0 \cdot Z(z)/Z_{\mathrm{today}}$.
Downstream, CosmicZScaleLaw treats this module as having already fixed that shape residue under the BIT kernel, so only the overall scale remains to be tightened.
proof idea
Definition-heavy module with supporting lemmas, not a single top-level theorem. It introduces bitKernel, bitDeviation, and linearZ, then records elementary equalities (bitDeviation_eq, bitDeviation_today, linearZ_today) and order facts (linearZ_pos, linearZ_antitone). Early-kernel and shape-reduction lemmas connect the BIT seed to the normalized $Z$ ratio. A short forcing pair (linear_accumulation_kernel / linear_accumulation_forces_canonical_kernel) argues that linear accumulation selects the canonical kernel shape. No deep analytic machinery: algebraic identities and monotone/positive properties of the linear accumulator.
why it matters in Recognition Science
This is the shape layer of the RS dark-energy plan. CosmicZScaleLaw imports it explicitly and states that CosmicZHistory proves, under the BIT kernel, $\delta w(z) = \delta w_0 \cdot Z(z)/Z_{\mathrm{today}}$, thereby clearing the last shape residue before scale fixing. In framework terms it supplies the redshift skeleton on which later cosmology can hang RS-native constants and cost defects, without yet claiming a numerical $\delta w_0$ or a full Friedmann integration. Anyone auditing the dark-energy chain cites this module for the $w(z)$ ansatz and the linear $Z$ accumulator.
scope and limits
- Does not fix the numerical value of $\delta w_0$ or today's dark-energy density.
- Does not derive Friedmann dynamics or luminosity-distance predictions.
- Does not prove uniqueness of BIT outside the linear-accumulation hypotheses stated here.
- Does not connect $Z(z)$ to observational catalogs or likelihood fits.
- Does not discharge scale-law residues left to CosmicZScaleLaw.
used by (1)
depends on (2)
declarations in this module (15)
-
def
bitKernel -
def
bitDeviation -
theorem
bitDeviation_eq -
theorem
bitDeviation_today -
theorem
bitKernel_early -
theorem
shape_reduction -
def
linearZ -
theorem
linearZ_today -
theorem
linearZ_pos -
theorem
linearZ_antitone -
theorem
linear_accumulation_forces_canonical_kernel -
theorem
linear_accumulation_kernel -
theorem
reciprocal_history_kernel -
structure
CosmicZShapeCert -
def
cosmicZShapeCert