>[!warning] >This content has not been peer reviewed. Fidelity is the [[Format]]'s rendering quality. It measures how faithfully the [[Format]] can reproduce the [[Signal]] given the current [[Noise Floor]]. When resources are abundant, Fidelity is 1 — the [[Format]] renders perfectly. When the [[Signal]] drops into the noise, Fidelity degrades. The [[Format]] can no longer maintain full detail. It begins to [[Zoom Logic|coarse-grain]], to approximate, to buffer. This single quantity explains why Newton works on Earth but fails in galaxy outskirts. It is not that gravity changes — it is that the [[Format]]'s rendering quality drops when the gravitational [[Signal]] approaches the cosmic [[Noise Floor]]. --- ## RST Formalization **Symbol:** $\mu$ **Definition:** The ratio of [[Workload]] to [[Total Budget]]: $\mu = \frac{\Omega}{W} = \frac{\Omega}{(\Omega^n + N^n)^{1/n}}$ Equivalently, as a function of the [[Signal-to-Noise Ratio]] $\eta = \Omega/N$: $\mu(\eta, n) = \frac{\eta}{(1 + \eta^n)^{1/n}}$ **Constraint:** $0 < \mu \leq 1$. Fidelity is bounded — the [[Format]] never amplifies, only attenuates or preserves. --- ## In the Equation **"The Quality"** ($\mu$) In the [[Resource Allocation Equation]] $I = \Omega \cdot \mu$, Fidelity connects the [[Signal]] to the [[Workload]]. High Fidelity ($\mu \to 1$) means the output matches the request. Low Fidelity ($\mu \to \eta$) means the output is the geometric mean of request and noise: $\Omega = \sqrt{I \cdot N}$. --- ## Why This Shape The boundary conditions of Fidelity are forced by the axioms, not chosen: 1. **High SNR ($\eta \gg 1$):** If you have enough resources, you maintain everything. $\mu \to 1$. 2. **Low SNR ($\eta \ll 1$):** The [[Energy Floor|Landauer Principle]] (Landauer 1961) says the cost of maintaining a bit against noise $T$ is $k_B T \ln 2$ — **linear** in the noise. This forces $\mu \to \eta$ (not $\eta^2$, not $\sqrt{\eta}$). See [[Relational Resolution Theory (RRT)]] for the relational extension. These two boundary conditions, plus the power-exhaustion rule ($\mu^n + \nu^n = 1$, derived from Translation-step allocation and the scale-free response of the Format), determine the function uniquely up to one parameter $n$. The form $\mu(\eta,n) = \eta/(1+\eta^n)^{1/n}$ is a standard MOND interpolation family (Famaey & McGaugh 2012); RST derives it from axioms. **Full derivation:** [[Fidelity Derivation]] — $\mu(\eta,n)$, the budget identity, and the Resource Triangle $W^n = \Omega^n + N^n$ all follow from RRT axioms and Landauer (1961). The axioms constrain $n \geq 1$; the value $n_0 \approx 1.24$ is derived by the Pure Axiom Substrate (rst_axiom_pure.py); SPARC confirms (1.25). See [[expanded theory applied/Derivation Chain Overview]], [[Transition Sharpness]]. --- ## The Budget Identity Fidelity and its complement (the [[Noise Projection]] $\nu = N/W$) fully allocate the budget: $\mu^n + \nu^n = 1$ Every unit of budget goes to either [[Signal]] work or [[Noise Floor|noise]] work. Nothing is wasted, nothing is created. --- ## Applied: sector inversion (gravity ↔ materials) The **same** μ function governs both galactic dynamics and the elastic-to-plastic transition in metals, but with the argument **inverted**: gravity uses μ(signal/threshold), materials use μ(threshold/signal). This produces opposite physical effects — amplification in gravity, saturation in materials — from the same budget equation. See **[[Fidelity Inversion — Gravity and Materials]]** for the full derivation, evidence summary, and falsifiability conditions. --- ## Applied: state-dependent source and $\mu$ When the **source** $I$ is driven by an external energy $E(\mathrm{state})$ (e.g. in the [[Reality Engine (RST)]]), the same identity $\mu = \Omega/W$ holds: the engine sets $I_{\mathrm{eff}} = I_{\mathrm{base}} + \alpha(E_{\mathrm{ref}} - E)$, so low-$E$ states get higher $I$ and thus higher $\mu$. Fidelity is still the single interpolation between high- and low-SNR regimes; only the *input* $I$ becomes state-dependent. See [[Energy input (RST)]]. See [[The Spectrum of Relational Topologies]] for the complete dimensional ladder of $n$ values across all sectors.