IndisputableMonolith.ILG.XiBins
The ILG.XiBins module supplies the analytic radial shape factor n(r) = 1 + A (1 - exp(-(r/r0)^p)) for use in Recognition Science ILG constructions. Researchers discretizing radial profiles into bins cite these definitions when building xi mappings. The module consists of direct definitions for n_of_r together with monotonicity statements for n and the auxiliary xi functions. No complex proofs appear; the content is built from the exponential expression and basic calculus.
claim$n(r) = 1 + A(1 - e^{-(r/r_0)^p})$ for parameters $A, r_0, p > 0$, with auxiliary maps $u(r)$ and bin index functions $xi(u)$ and $xi(bin)$ that preserve monotonicity.
background
The module sits inside the ILG domain and introduces an explicit closed-form radial profile to replace numerical integration in global shape calculations. n_of_r encodes the given exponential expression. n_of_r_mono_A_of_nonneg_p records that the profile is non-decreasing in the amplitude A whenever A is nonnegative. xi_of_u converts the radial coordinate to an auxiliary variable u, while xi_of_bin and xi_of_bin_mono handle the subsequent mapping onto discrete bins while preserving order.
proof idea
This is a definition module, no proofs. The sibling declarations supply the core expression for n(r) and the two monotonicity statements that follow immediately from differentiation of the exponential term.
why it matters in Recognition Science
ILG.XiBins supplies the radial profile required by higher-level binning and discretization steps in the ILG layer of the Recognition Science framework. It supports the transition from continuous radial functions to the eight-tick octave and phi-ladder constructions that appear in downstream mass and forcing-chain results.
scope and limits
- Does not derive the exponential form from the J-cost equation or RCL.
- Does not fix numerical values for A, r0 or p.
- Does not address three-dimensional volume elements or angular dependence.
- Does not connect the shape factor to the Berry threshold or Z_cf.