IndisputableMonolith.Gravity.ILGSpatialKernel
Defines the ILG spatial-kernel amplitude C = φ⁻² together with the companion penalty J_φ and half-rung budget identities used in galactic rotation-curve modeling. Gravity and ILG workers cite it for the numerical band on C and the exact link C + J(φ) = 1. Most statements are short algebraic rewrites from the golden-ratio minimal polynomial and the J-cost definition.
claimThe spatial-kernel amplitude is the constant $C = \varphi^{-2}$. Companion objects are the kernel weight $\alpha$, the penalty $J_\varphi := J(\varphi)$, and the half-rung budget. The module records $C = 2 - \varphi$, $0 < C < 1/2$, the exact identity $C + J(\varphi) = 1$, and $J(\varphi) = \varphi - 3/2$.
background
Information-Limited Gravity (ILG) replaces a dark-matter halo with a recognition kernel whose spatial amplitude is fixed by the golden ratio $\varphi$, the unique positive solution of $\varphi^2 = \varphi + 1$. In RS-native units the cost functional is $J(x) = (x + x^{-1})/2 - 1$ (equivalently $\cosh(\log x) - 1$), forced unique by the T5 step of the unified forcing chain.
This module sits downstream of Constants (which supplies $\varphi$ and the tick $\tau_0$) and Cost (which supplies $J$). It specialises those primitives to the radial kernel that multiplies the baryonic acceleration in the ILG force law. The sibling definitions introduce $C_{\mathrm{kernel}} = \varphi^{-2}$, an auxiliary $\alpha_{\mathrm{kernel}}$, the penalty $J_\varphi$, and the half-rung budget that appears when the eight-tick octave is halved.
proof idea
Definition module with a thin layer of algebraic lemmas. $C = \varphi^{-2}$ is unfolded via $\varphi^2 = \varphi + 1$ to obtain the closed form $C = 2 - \varphi$. Positivity and the strict bound $C < 1/2$ are immediate from $1 < \varphi < 2$. The penalty identities rewrite $J(\varphi)$ by direct substitution into the cost formula, yielding $J(\varphi) = \varphi - 3/2$. The complement relation $C + J(\varphi) = 1$ is then a one-line arithmetic check. Half-rung budget statements double or halve the same constants; no analytic estimates or external analysis are required.
why it matters in Recognition Science
Supplies the numerical kernel amplitude that every ILG rotation-curve prediction multiplies against baryonic surface density. The exact band $0 < C < 1/2$ and the complement identity $C = 1 - J(\varphi)$ let downstream gravity theorems replace floating parameters by pure $\varphi$-expressions, keeping the theory inside the RS forcing chain (T5 J-uniqueness, T6 $\varphi$ fixed point). Although the module currently has no recorded reverse edges, its siblings are the natural inputs to any theorem that assembles the full ILG acceleration law or compares the kernel against SPARC-style data. The construction is closed: every identity is proved from the golden-ratio equation and the definition of $J$, with no remaining sorry.
scope and limits
- Does not derive the ILG force law or fit rotation curves.
- Does not prove uniqueness of C outside the φ-ladder ansatz.
- Does not address temporal or spin kernels, only the spatial amplitude.
- Does not import observational data or error budgets.
- Does not claim C equals any measured dark-matter fraction.
depends on (3)
declarations in this module (23)
-
def
C_kernel -
def
alpha_kernel -
def
Jphi_penalty -
theorem
C_kernel_eq_two_minus_phi -
theorem
C_kernel_pos -
theorem
C_kernel_lt_half -
theorem
C_kernel_band -
theorem
Jphi_penalty_eq_phi_minus_three_halves -
theorem
Jphi_penalty_eq_Jcost_phi -
theorem
half_rung_budget -
theorem
half_rung_budget_doubled -
theorem
C_is_complement_of_Jphi -
theorem
half_rung_components_band -
def
C_kernel_competing -
theorem
C_kernel_competing_pos -
theorem
C_competing_gt_C_kernel -
theorem
C_competing_violates_budget -
def
channel_weight -
theorem
channel_weight_eq -
theorem
three_channel_factorization -
structure
ILGSpatialKernelCert -
def
ilgSpatialKernelCert -
theorem
ilg_spatial_kernel_one_statement