>[!warning] >This content has not been peer reviewed. Transition Sharpness determines how abruptly the [[Format]] switches between full [[Fidelity]] and throttled rendering. It is the exponent in the [[Resource Triangle]]: $W^n = \Omega^n + N^n$ A low $n$ (close to 1) means a gradual, smooth transition. The [[Format]] degrades slowly as the [[Signal-to-Noise Ratio]] drops. A high $n$ means a sharp, sudden switch — the [[Format]] holds full fidelity until a critical threshold and then drops rapidly. $n$ is not a universal constant. It evolves with the [[Noise Floor]]: $n(z) = n_0 + \ln\frac{H(z)}{H_0}$ At $z = 0$ (the present), $n_0 \approx 1.24$ — **derived** from backbone $d_B$ (Baseline 1.7); SPARC $1.25 \pm 0.05$ confirms (Lelli et al. 2016; [[SPARC Evaluation Verification]]). **Derived** (Baseline 1.7): $n_0$ is the backbone dimension $d_B \approx 1.22$ of a 3D critical percolation network (A1 + A5 + connectivity + percolation threshold). [[Backbone Dimension]]; [[RST Baseline 1.0]]. At $z = 2$, $n \approx 2.34$ (phase-hardening). --- ## RST Formalization **Symbol:** $n$ **Calibration:** SPARC gives $n_0 = 1.25$ ([[SPARC Evaluation Verification]]). **Derivation (Baseline 1.7):** $n_0 = d_B \approx 1.22$ from 3D critical backbone ([[Backbone Dimension]]). [[RST Baseline 1.0]]. **Evolution:** $n(\theta) = n_0 + \ln(\theta/\theta_0)$, where $\theta = 3H$ is the expansion scalar. --- ## In the Equation **"The Switch"** ($n$) $n$ appears as the exponent in both the [[Resource Triangle]] ($W^n = \Omega^n + N^n$) and the [[Fidelity]] function ($\mu = \eta/(1+\eta^n)^{1/n}$). It controls the shape of the transition between the classical and throttled regimes. For $n = 1$: gradual. For $n = 2$: the standard MOND interpolation. For $n \to \infty$: a hard step function. --- ## Special Cases | $n$ | Transition | Use | |:---|:---|:---| | 1 | Gradual, smooth | Simplest form; analytic solution | | 1.25 | Slightly sharper | **Calibrated local baseline** ($z = 0$) | | 2 | Standard MOND | Analytic solution; close to $z \approx 2$ prediction | | $\to \infty$ | Step function | Idealized limit (not physical) | See [[The Spectrum of Relational Topologies]] for the complete multi-sector dimensional ladder of $n$ values.