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zeekay
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The seven engine specifications (lux-{photon,wave,nova,nebula,prism,ray,field}-protocol)
were fragmented short notes documenting layers of the same Quasar consensus stack.
This commit merges them into a single canonical reference, lux-consensus-engines.tex,
that uses one preamble, one set of theorems, one bibliography, and one composition
chapter showing how the engines layer into the LP-020 finality pipeline.
The seven originals are reduced to single-page breadcrumb stubs pointing at the
consolidated paper; their sections/ content is preserved on disk for git history.
- new: lux-consensus-engines/ (50 pages, 646KB PDF, ~2900 lines tex)
- breadcrumbs: each original keeps cover + title + stub paragraph naming its chapter
- composition: chapter 9 records the layered contract Photon < Wave < Nova < ...
- bibliography: consolidated from the seven original thebibliography blocks
58 lines
2.4 KiB
TeX
58 lines
2.4 KiB
TeX
\section{Estimator Convergence}
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\label{sec:ray-estimator}
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\begin{theorem}[Adversary Estimate Convergence]
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\label{thm:ray-estimator}
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The Bayesian adversary estimator $\hat{\gamma}_r$ converges to the true adversarial
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fraction $\gamma$ with rate:
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\[
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\mathbb{E}[|\hat{\gamma}_r - \gamma|] \leq (1 - \eta)^r |\hat{\gamma}_0 - \gamma| + \frac{\eta \sigma_k}{\sqrt{2 - \eta}}
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\]
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where $\sigma_k = \sqrt{p(1-p)/k}$ is the sampling standard deviation and $\eta \in (0, 1)$
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is the learning rate.
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\end{theorem}
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\begin{proof}
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The estimator follows the recursion $\hat{\gamma}_r = \eta X_r + (1 - \eta)\hat{\gamma}_{r-1}$
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where $X_r = 1 - n_c/k$ is the observed adversarial fraction. Each $X_r$ has mean $\gamma$
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and variance $\sigma_k^2 = p(1-p)/k$.
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The bias decays as $(1-\eta)^r$:
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\[
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\mathbb{E}[\hat{\gamma}_r] - \gamma = (1-\eta)^r(\hat{\gamma}_0 - \gamma)
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\]
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The variance satisfies:
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\[
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\text{Var}[\hat{\gamma}_r] = \eta^2 \sigma_k^2 \sum_{i=0}^{r-1} (1-\eta)^{2i}
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\leq \frac{\eta^2 \sigma_k^2}{1 - (1-\eta)^2} = \frac{\eta \sigma_k^2}{2 - \eta}
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\]
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Combining bias and standard deviation by Jensen's inequality gives the stated bound.
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For $\eta = 0.1$, $k = 20$, and $r \geq 30$: the bias term is negligible
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($(0.9)^{30} < 0.04$) and the variance term contributes $\sqrt{0.1 \cdot 0.25/20 / 1.9} \approx 0.026$,
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giving an estimate accurate to within 3\% of the true $\gamma$.
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\end{proof}
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\begin{lemma}[Conservative Threshold]
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\label{lem:ray-conservative}
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If the estimator uses an upper confidence bound $\hat{\gamma}^+ = \hat{\gamma}_r + z_{\delta} \sqrt{\text{Var}[\hat{\gamma}_r]}$
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with $z_\delta$ chosen for confidence level $1 - \delta$, then the derived threshold
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$\beta^*$ is sufficient for safety with probability $\geq 1 - \delta$.
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\end{lemma}
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\begin{proof}
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The upper confidence bound ensures $\hat{\gamma}^+ \geq \gamma$ with probability $1 - \delta$.
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When $\hat{\gamma}^+ \geq \gamma$, the derived threshold satisfies:
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\[
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\beta^* = \frac{128 \ln 2}{\ln(q_1(\hat{\gamma}^+) / q_0(\hat{\gamma}^+))} \geq \frac{128 \ln 2}{\ln(q_1(\gamma) / q_0(\gamma))}
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\]
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since $q_1/q_0$ is decreasing in $\gamma$ (more adversaries make the ratio closer to 1).
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Thus $\beta^*$ overestimates the required threshold, maintaining safety.
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\end{proof}
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%% -----------------------------------------------------------------------
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%% 5. SAFETY ANALYSIS
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%% -----------------------------------------------------------------------
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