📖 Irreducible Element (in a Ring) An element $p$ of a ring is irreducible if it is not a unit (not $\\pm 1$ in $\\mathbb{Z}$) and if the only factorizations $p = ab$ have one of $a$, $b$ as a unit. In $\\mathbb{Z}$, irreducible elements are the prime numbers and their negatives. From: gal-edwards Learn more:

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📖 Splitting Field A splitting field of $f \\in F[X]$ is an extension $E/F$ such that $f$ splits completely in $E[X]$ and $E$ is generated over $F$ by the roots of $f$. From: gal-weintraub Learn more:

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📐 Invertibility Characterization A linear map $T \\in \\mathcal{L}(V, W)$ is invertible if and only if $T$ is injective and surjective. From: linalg-axler Learn more:

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📖 Discriminant The **discriminant** of a polynomial $f(x) = \\prod (x - r_i)$ is $\\operatorname{disc}(f) = \\prod_{i < j} (r_i - r_j)^2$. For $K/F$ Galois, $\\operatorname{disc}(f) \\in F$ and $\\sqrt{\\operatorname{disc}(f)} \\in K$. From: gal-morandi Learn more:

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📖 Algebraic Closure An **algebraic closure** of $F$ is an algebraically closed field $\\overline{F}$ that is algebraic over $F$. It exists and is unique up to $F$-isomorphism. The **absolute Galois group** is $\\operatorname{Gal}(\\overline{F}/F)$. From: gal-morandi Learn more:

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📐 Genus and the Riemann-Roch Space For an algebraic function field $K/k$, the **genus** $g$ is defined such that for any divisor $D$, the Riemann-Roch space $L(D)$ is a finite-dimensional $k$-vector space with $\\dim L(D) \\geq \\deg D - g + 1$. From: gal-morandi Learn more:

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📖 Exponential Waiting Time If blocks arrive at rate $\\lambda$, the waiting time $T$ between blocks satisfies $P(T > t) = e^{-\\lambda t}$, with density $f(t) = \\lambda e^{-\\lambda t}$, mean $1/\\lambda$, median $\\ln 2 / \\lambda$, and variance $1/\\lambda^2$. From: mining-pools Learn more:

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🎮 Interactive: Non-Euclidean Lines Demo Explore straight lines in hyperbolic geometry. In the Poincare disk, geodesics appear as circular arcs perpendicular to the boundary. From: Four Pillars of Geometry Try it:

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The Four Pillars of Geometry (Stillwell) | Math Academy

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📐 Satoshi\ An attacker with share $q < p = 1-q$, starting $z$ blocks behind, succeeds with probability $P = 1 - \\sum_{k=0}^{z} \\frac{\\lambda^k e^{-\\lambda}}{k!}\\left(1 - (q/p)^{z-k}\\right)$ where $\\lambda = zq/p$. If $q \\geq p$, $P = 1$. Proof: Sketch: while honest miners produce $z$ blocks, the attacker\ From: mining-pools Learn more:

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📖 Mining Pool A mining pool aggregates many miners\ From: mining-pools Learn more:

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📖 PPS / FPPS Pay-per-share: the pool pays a fixed price for every share submitted, block or no block — the operator absorbs all luck variance. FPPS additionally pays the expected transaction-fee revenue per share. From: mining-pools Learn more:

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📐 Bonferroni Bound Testing $m$ cells at once, requiring per-test $p < \\alpha/m$ keeps the probability of even one false positive below $\\alpha$. For the era: $m = 1440$, $\\alpha = 0.05 \\Rightarrow p < 3.47 \\times 10^{-5}$. Proof: Union bound: $P(\\text{any false positive}) \\leq \\sum_{i=1}^{m} P(\\text{test } i \\text{ false positive}) = m \\cdot (\\alpha/m) = \\alpha$. From: mining-pools Learn more:

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📐 The Era Verdict Across all 1,440 pool-epoch cells of the 3.125 era (epochs 417–476), zero cells survive the Bonferroni correction: no pool defied its probabilistic outcome. Every pool met expectation within noise. From: mining-pools Learn more:

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📐 Memorylessness For an exponential waiting time, $P(T > s + t \\mid T > s) = P(T > t)$: no matter how long you have waited, the distribution of the remaining wait is unchanged. Proof: $P(T > s+t \\mid T > s) = \\frac{P(T > s+t)}{P(T > s)} = \\frac{e^{-\\lambda(s+t)}}{e^{-\\lambda s}} = e^{-\\lambda t} = P(T > t)$. From: mining-pools Learn more:

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📐 Luck Dispersion Over an epoch of 2016 blocks, a pool expecting $\\lambda$ blocks has luck $= 100 \\times k/\\lambda$ with standard deviation $\\sigma_{\\text{luck}} = 100/\\sqrt{\\lambda}$ percent. Proof: The count $k$ is Poisson($\\lambda$), so $\\sigma_k = \\sqrt{\\lambda}$. Luck is $k/\\lambda$ scaled by 100, hence $\\sigma = 100 \\cdot \\sqrt{\\lambda}/\\lambda = 100/\\sqrt{\\lambda}$. From: mining-pools Learn more:

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📖 Hashrate Share A miner controlling hashrate $h$ of network total $H$ wins any given block with probability $h/H$ and finds blocks at rate $\\lambda = 144 \\cdot h/H$ per day. From: mining-pools Learn more:

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📖 Stale Block A valid block that loses a race at the same height: when two blocks compete, the next block decides the winner; the loser\ From: mining-pools Learn more:

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📖 PPLNS Pay-per-last-N-shares: each found block\ From: mining-pools Learn more:

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📖 Expected Blocks Baselines Flat baseline: a pool\ From: mining-pools Learn more:

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📖 Exponential Waiting Time If blocks arrive at rate $\\lambda$, the waiting time $T$ between blocks satisfies $P(T > t) = e^{-\\lambda t}$, with density $f(t) = \\lambda e^{-\\lambda t}$, mean $1/\\lambda$, median $\\ln 2 / \\lambda$, and variance $1/\\lambda^2$. From: mining-pools Learn more:

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