The Verification Venue · pointed at a formula everyone quotes
The Fidelity That Loses the Master
Every textbook prices the error threshold at q* = σ^(−1/ν), which for a genome of fifty sites and a master that replicates twice as fast as everything else is 0.9862327045. That number is an approximation standing in for an answer: it is what you get if you assume no mutant ever mutates back onto the master. This page computes the exact threshold from the full 51 by 51 mutation-selection eigenproblem, then runs Wright-Fisher populations of up to a million genomes, about ten billion counted operations in your browser, to measure where finite populations actually lose the master.
Drag the fidelity slider and watch a live population redistribute across Hamming classes while the exact deterministic curve waits behind it. Then press the run button and let every core you have spend a minute finding out what a population a thousand times larger than the textbook demonstration actually does at the threshold.
Live master fraction (from all-master start)
1.000
exact deterministic value here: ·
The two thresholds (ν = 50, σ = 2)
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computing…
Drag it across the threshold and watch the collapse: there is no gentle decline, there is a cliff.
Logarithmic, up to a hundred million. The class-chain sampler is exact at any N, so drift visibly shrinks as you drag right.
The model, stated so you can check every line of it: binary genomes of ν = 50 sites; the all-zero master replicates at rate σ = 2, every other genome at rate 1; each site is copied correctly with probability q and flipped with probability u = 1 − q; each generation, N offspring pick parents with probability proportional to fitness. The exact answer is the Perron eigenvector of the (ν+1)×(ν+1) matrix M[i][j] = A_j·P(j→i); the master persists while its leading eigenvalue exceeds 1. The textbook formula comes from the same model with back mutation forbidden, which is a different model.
Layer two: the formula versus its own question
Seven stops from 1.1 to 3.0. The curve is recomputed exactly for your stop, live.
Seven stops from 10 to 80. Longer genomes, bigger gap.
The full run
Operations performed (counted, not estimated)
0
idle
Workers and elapsed
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The check · every number recomputed in front of you
Before anything is displayed, the compiled engine and a slow, deliberately obvious JavaScript reference are run over a battery of small instances, in your browser. If they disagree on any row, nothing below may be believed and the run button stays dead.
| battery instance | engine fingerprint | reference fingerprint | agree |
|---|---|---|---|
| loading engine… | |||
press to watch the check go red
The byte-reproducibility check (rebuilding engine.c and demanding identical bytes) cannot run in a browser; it runs offline: node research/the-fidelity-that-loses-the-master/verify-the-fidelity-that-loses-the-master.mjs. Two decompositions agreeing rules out boundary and slicing errors and almost nothing else; what none of these checks rules out is a shared misunderstanding of the model itself, which is why the verifier also runs a literal packed-genome simulation and the σ = 1 known answer against the engine.
Fixed parameters this page is built from, each recomputed by the verifier against this shipped text:
- Textbook threshold q* = σ^(−1/ν) = 2^(−1/50) = 0.9862327045
- Genome length nu = 50; master fitness sigma = 2; all other genomes fitness 1.
- Full sweep: N = 100000, 200 generations, 41 fidelities with u from 0.005 to 0.045.
- Finite-N study: N = 10000, 100000, 1000000; 160 generations; 9 fidelities with u from 0.008 to 0.02.
- Full-size runs are initialised from the exact Perron eigenvector at their own fidelity, so each measured tail is the stationary distribution and not a burn-in transient; the planted control seeds a 1% master colony inside a class-25 population.
- Steady state estimated from the second half of each run's generations.
What the big run found
| N | apparent threshold (where master fraction crosses 1/2) | displacement from exact |
|---|---|---|
| run the full computation to fill this | ||
The direction and size of the finite-N displacement is measured here, not asserted: if a population's confidence interval brackets the exact threshold, the page says no resolvable displacement was found. Each run is initialised from the exact Perron eigenvector at its own fidelity, so the tail measures the stationary distribution rather than a burn-in transient. Above the exact threshold the master class, once lost by a finite population, is not regained on any feasible timescale, so there the runs are checked for the collapsed state, and the Perron comparison is made only where the master class is recurrent (leading eigenvalue above 1).
The cost, measured
Speed factor vs main-thread JavaScript
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extrapolated from a slice: ·
Wall time for the full run
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What's exactly true, what's a model, and what this page refuses to claim
Exactly true, given the stated rules. The mutation kernel arithmetic; the eigenvalue computation to its stated tolerance; the conservation of N in every generation; bit-identical trajectories under any repartitioning of the work; and the σ = 1 stationary solution Binomial(50, ½), which is a theorem about the rules, verified against the engine as a planted control.
A model, chosen for this page. The single-peak landscape, σ = 2, ν = 50, Wright-Fisher non-overlapping generations. Whether any real virus sits near an error threshold is genuinely contested and estimates vary widely; this page asserts nothing about any organism, and does not use the phrase error catastrophe as a fact about nature. Every parameter here is a stated choice, attributed to no one.
Measured, not remembered. The direction and size of the finite-N displacement is read off this page's own runs with uncertainties. If the measurement does not resolve a displacement, the page says so rather than borrowing a sign from theory.
Separation from Muller's ratchet. That layer is irreversible accumulation of damage over time in an asexual lineage. This page is a different question: a threshold in copying fidelity with a stationary population on both sides. Nothing here accumulates; at every fidelity the population settles, and the only question is what it settles to.