IndisputableMonolith.ILG.ISWSign
The ILG.ISWSign module defines the ISW driver B(a,k) = -1 + f + dlnw/dlna for the Infra-Luminous Gravity framework. It assembles the driver from the kernel w(k,a) and the growth prefactor B supplied by GrowthODE. Cosmologists modeling modified gravity corrections to large-scale structure and the integrated Sachs-Wolfe effect would cite this module. The module consists of direct definitions and positivity lemmas built on the two imported modules.
claim$B(a,k) = -1 + f + dlnw/dln a$, where $w(k,a) = 1 + C (a/(k τ_0))^α$ is the ILG kernel and $f$ is the growth rate obtained from the first-order correction $D = a(1 + B a^α)$ in an EdS background.
background
The ILG kernel module supplies the scale-dependent weight $w(k,a) = 1 + C (a/(k τ_0))^α$ that modifies gravitational clustering at infra-luminous scales. The GrowthODE module derives the prefactor B by substituting the ansatz $D = a(1 + B a^α)$ into the linear growth equation for an EdS background. The present module combines these objects to form the ISW driver, which enters the time derivative of the gravitational potential relevant for CMB temperature anisotropies.
proof idea
This is a definition module, no proofs. It introduces the ISW driver together with auxiliary statements (isw_driver, f_growth_gt_one, dlnw_pos, isw_driver_positive) by direct substitution of the kernel and growth prefactor imported from Kernel and GrowthODE.
why it matters in Recognition Science
The module supplies the ISW driver required for computing the integrated Sachs-Wolfe signal in ILG cosmology. It closes the chain from the kernel definition through the growth correction to an observable large-scale structure signature. No downstream theorems are listed, indicating the driver is intended as an interface for subsequent ILG calculations of CMB and matter power spectra.
scope and limits
- Does not derive the kernel form from first principles.
- Does not solve the growth ODE numerically.
- Does not extend beyond the EdS background assumption.
- Does not include radiation or dark-energy components.