IndisputableMonolith.ILG.ISWSign
The ISWSign module defines the ISW driver B(a,k) = -1 + f + dlnw/dlna by combining the growth rate with the logarithmic derivative of the ILG kernel. Cosmologists working on modified gravity corrections to large-scale structure and CMB anisotropies would cite it for first-order ILG calculations. It is a definition module that imports the Kernel and GrowthODE modules and supplies supporting lemmas on sign and positivity.
claim$B(a,k) = -1 + f + \frac{d\ln w}{d\ln a}$, where $w(k,a)$ is the ILG kernel $w(k,a)=1+C\cdot(a/(k\tau_0))^\alpha$.
background
The module sits inside the Infra-Luminous Gravity (ILG) formalization. The upstream Kernel module introduces the kernel $w(k,a)=1+C\cdot(a/(k\tau_0))^\alpha$. The upstream GrowthODE module supplies the prefactor for the first-order growth correction in an EdS background, obtained by substituting the ansatz $D=a(1+B a^\alpha)$ into the growth ODE.
This module assembles those pieces into the ISW driver expression and records auxiliary facts about its sign and the growth rate f.
proof idea
this is a definition module, no proofs
why it matters in Recognition Science
The module supplies the ISW driver definition that supports the sibling results isw_driver, isw_driver_positive, f_growth_gt_one and dlnw_pos. It thereby connects the ILG kernel and growth ODE to the sign of the driver that enters Integrated Sachs-Wolfe calculations in the Recognition Science framework.
scope and limits
- Does not derive numerical values for C, alpha or tau0.
- Does not address backgrounds other than EdS.
- Does not include higher-order corrections beyond the linear B term.
- Does not perform numerical integration of the growth ODE.