bandwidthKernel
The bandwidth kernel is the ILG time kernel defined as the ratio of demanded recognition rate to available bandwidth. Researchers modeling saturation-driven gravity modifications cite it to locate the Newtonian-to-ILG transition. The definition is a direct division of the two input rates with no further computation.
claimThe ILG time kernel satisfies $w_t(R_d,R_b)=R_d/R_b$, where $R_d$ is the recognition event rate demanded by Newtonian dynamics of mass $M$ at dynamical time $T_{dyn}$ and $R_b$ is the holographic bandwidth limit $A/(4ell_P^2 k_R 8tau_0)$ for boundary area $A$.
background
Recognition bandwidth is the maximum recognition events per unit time allowed by the holographic bound: $R_{max}(A)=A/(4ell_P^2 k_R 8tau_0)$, with each event costing $k_R=ln(phi)$ bits and processing limited to one cycle per $8tau_0$. Demanded rate is the Newtonian requirement $R_{demand}=M/T_{dyn}$ (in units where Planck mass is 1). The module BandwidthSaturation treats ILG gravity as the batching compensation that occurs when demand exceeds this limit, forcing multiple dynamical times into each 8-tick cycle.
proof idea
One-line definition that returns the quotient of demandedRate by availableBandwidth.
why it matters in Recognition Science
This definition supplies the ratio used by the three saturation theorems in the same module (kernel_unity_at_saturation, kernel_gt_one_when_saturated, kernel_lt_one_when_sub). It realizes the ILG time kernel $w_t$ that amplifies gravity below the critical acceleration, directly implementing the eight-tick octave (T7) and the holographic bound that forces modified dynamics. It closes the path from recognition throughput limits to observable ILG parameters C_lag and alpha.
scope and limits
- Does not compute numerical values for specific masses or areas.
- Does not enforce positivity constraints on the input rates.
- Does not derive the saturation acceleration from first principles.
- Does not select among BIT kernel families.
formal statement (Lean)
151noncomputable def bandwidthKernel (demandedRate availableBandwidth : ℝ) : ℝ :=
proof body
Definition body.
152 demandedRate / availableBandwidth
153
154/-- The bandwidth kernel equals 1 at saturation (transition point). -/