The win-cone: how many victories stay reachable
A new lab instrument, measured over 4,000 replayed games from the campaign corpus.
For any position, the win-cone asks: for each of the five colors, what is the minimum number of placements that could still produce a winning chain, granting perfect draws? It is an optimistic distance — real play can only be slower — so a color outside the cone is provably unreachable, and the cone's width (how many colors keep a feasible route) is how open the game still is. Merges are solved exactly (minimum Steiner trees over the board's empty cells), and every entry point respects the real printed dots.
This is stage 0a of the Alpha Mosaic program: the companion to the entropy curve, a future evaluation feature for the search and learning agents — and a measure of when a game of Mosaic is truly decided (later than you might think).
The cone barely narrows — Mosaic stays open
Mean cone width (colors with a feasible route to 7) by move. A cold deal starts at 5.0 of 5 with the Prism and 2.9 without; the striking part is how little of that closes before the end.
With the Prism
Open start
FindingGames end by someone arriving, not by the space closing. The cone stays near-maximal deep into every configuration — colors are almost never eliminated outright; they lose the race. Mosaic's endings are positive events (a chain completes), not exhaustions of possibility.
The inversionThe Prism's real gift is instant optionality. Anchored games open at the full width of 5.0 — the wild centre admits every color from move one. Open games start at just 2.9 of 5 (the founding tile only offers doors in its own colors) and spend the early game widening back to full. The blocks' reports showed the anchor hands an edge to whoever moves first; the cone shows what that edge is made of.
The decision curve: watching the winner pull away
Mean cone distance of the eventual winner's goal vs the field's goals, aligned to the end of the game (position 0 = the winning move). Five players, both rules; sole winners only.
Five players · with the Prism
Five players · open start
How early does the cone name the winner?
Share of games where the eventual winner's goal is strictly the closest in the cone, k moves before the end. Chance level is one over the number of real players.
With the Prism
Open start
FindingMosaic stays undecided until the final stretch. Ten moves out — half a five-player game — the cone's pick is barely better than chance (20% anchored, 22% open, vs 20% by luck at five players). Then it sharpens fast: 46% / 54% five moves out, 61% / 67% at three. Even perfect geometric knowledge cannot call the winner early — the game is decided in the endgame, not accumulated from the opening. (The strictly-closest bar counts ties against the cone, so these are conservative reads.)
For the programThis calibrates the agents to come: the cone is the strongest purely geometric signal available, and its early-game blindness is measured here before Apex or any network exists. Whatever a trained evaluation later adds over 20% at ten moves out is, by construction, knowledge the geometry alone doesn't carry.
The full table
Appendix
| Metric | 5p prism | 4p prism | 3p prism | 2p prism | 5p open | 4p open | 3p open | 2p open |
|---|---|---|---|---|---|---|---|---|
| Games replayed | 500 | 500 | 500 | 500 | 500 | 500 | 500 | 500 |
| Mean cone width, move 1 | 5.00 | 5.00 | 5.00 | 5.00 | 2.90 | 2.89 | 2.95 | 2.92 |
| Mean cone width, move 15 | 4.98 | 4.98 | 4.97 | 4.97 | 4.95 | 4.94 | 4.97 | 4.93 |
| Winner-vs-field gap, 10 from end | 0.54 | 0.33 | 0.51 | 0.16 | 0.66 | 0.53 | 0.40 | 0.22 |
| Cone names winner, 5 from end | 46% | 60% | 52% | 70% | 54% | 62% | 54% | 76% |
| Cone names winner, 10 from end | 20% | 19% | 34% | 41% | 22% | 26% | 32% | 46% |
| … chance level | 20% | 25% | 33% | 50% | 20% | 25% | 33% | 50% |
| Shared wins excluded | 1 | 4 | 3 | 3 | 7 | 15 | 10 | 8 |
Method: deterministic re-simulation of the first 500 seeds of every study's corpus (game 0 of each hash-verified against the recorded corpus line), omniscient tile supply, cone computed after every move. Distances use 8 as the "no route" sentinel. Shared exhaustion wins are excluded from winner-aligned curves.