kernel_gt_one_when_saturated
plain-language theorem explainer
When recognition demand Rd exceeds available bandwidth Rb (with Rb positive), the bandwidth kernel exceeds one, marking the saturated regime where ILG batching activates. Researchers deriving modified gravity from holographic throughput limits would cite this to separate Newtonian and ILG domains. The proof is a direct algebraic reduction: the kernel is the ratio Rd/Rb, so the inequality follows immediately from the hypothesis via the division property.
Claim. Let $R_d, R_b$ be positive real numbers with $R_b < R_d$. Then the bandwidth kernel $K(R_d, R_b)$ satisfies $1 < K(R_d, R_b)$, where $K$ is the ratio of demanded recognition rate to available bandwidth.
background
The BandwidthSaturation module shows how ILG gravity arises when Newtonian dynamics demand more recognition events per unit time than the holographic bound allows. The system compensates by batching multiple dynamical times into each 8-tick cycle, producing a time kernel greater than one. The bandwidth kernel is the ratio of demand rate to bandwidth capacity; saturation occurs precisely when this ratio exceeds unity. Upstream results include the ILG kernel definition $w(k,a) = 1 + C (a/(k τ_0))^α$ from ILG.Kernel and the general bandwidth function from RecognitionBandwidth.
proof idea
Term-mode proof. Unfold the definition of the bandwidth kernel to expose the ratio Rd/Rb. Rewrite the target inequality using the algebraic fact that 1 < x/y holds exactly when y < x for y positive. Discharge the goal directly with the hypothesis Rb < Rd.
why it matters
This theorem identifies the boundary between Newtonian and ILG regimes inside the Recognition framework. It supports the module's listed results on saturation acceleration and kernel compensation, which in turn feed into the emergence of modified dynamics below a_sat. It instantiates the eight-tick octave (T7) and holographic bound constraints that force batching when throughput is exceeded, consistent with the overall forcing chain from T0 to T8.
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