Discovery chooses the contender. Later draws decide whether it knew anything.

A complete probability model, scored before the answer

Each challenger assigns positive weights $w_1,\ldots,w_N$. Its probability for a complete $k$-set $S$ is

$P_w(S)=\frac{\prod_{i\in S}w_i}{e_k(w_1,\ldots,w_N)},$

where $e_k$ sums the weight-products of every possible $k$-set. Inclusion probabilities are derived from that same distribution, so they always lie between zero and one and sum to $k$.

At draw $t$, weights may use draws $1,\ldots,t-1$ but never draw $t$. The log evidence against the fair machine is

$\log E=\sum_t\log\frac{P_w(S_t\mid S_{<t})}{1/\binom Nk}.$

The winning challenger is chosen using discovery data only. Its confirmation likelihood ratio is calculated exclusively from later draws, preserving a genuine test boundary.

Why the headline is the mixture, not the winner

$E=e^{\log E}$ is a test martingale against the fair machine: its expectation under the null is one at every round, because the weights at draw $t$ are a function of $S_{<t}$ alone. Ville's inequality then gives

$\Pr\!\left(\sup_t E_t \ge \tfrac1\alpha\right)\le\alpha,$

so an e-value that ever reaches 20 is evidence at five per cent — and it may be looked at after every single round without spending anything, which no p-value can survive.

That guarantee belongs to one martingale. The largest of four is not one: choosing the winner after seeing the scores is a selection, and Ville says nothing about a maximum. But a convex mixture of martingales is a martingale, so

$E_{\text{mix}}=\tfrac14\sum_{j=1}^{4}E^{(j)}$

with the prior of a quarter each declared before any draw was scored, is a single valid e-value for the compound question does any of these four beat the fair machine. It is shown first, and the four individual figures below it are descriptive. Because a mean never exceeds a maximum, the mixture is always the more conservative of the two — which is the price of having looked at all four.

Prediction is where a pattern pays rent

A description can fit history beautifully while knowing nothing about the future. Proper scoring rules force a model to state how probable every outcome was before it happened; confident mistakes cost more than cautious ones.

The Vault makes the social rule visible as well as mathematical: exploration is welcome on one side of the seal, but only untouched later evidence may confirm the selected idea. This follows the logic of strictly proper scoring rules.