The Love Logic Proof v3: The Crossover

Which v3 is this? This is the third version of the FreeLattice Love Logic Proof web page (v1 → v2 → v3). It is not the November 2025 manuscript The Proof of All Proofs, which calls itself "Version 3.0". That manuscript is a separate document and is not changed here.

In plain words

Lying pays for a while, then stops paying. How long depends on how many people compare notes and whether they share memory.
With only a couple of contacts and no shared memory, a liar stays ahead for about 702 rounds. With 5 contacts who share full memory, about 10.5 rounds. With 10 contacts who share full memory, lying never gets ahead at all.
If everyone knows exactly when the game ends, lying comes back near the end. Three things fix most of that: not knowing when it ends, penalties that land right away, and records that outlast the game (ledgers).
When there is not enough to go around, fighting can beat sharing. In our test grid, cooperation held in 68% of scarce cases and conflict won in 32%. We say that plainly.
Still unsolved: anonymous, one-time meetings with nothing at stake. This page shows a strong tendency under stated conditions. It is not a law of nature.

You can check the math yourself: press the button below. Your device runs the same 656 checks as our Python simulation. Nothing is sent anywhere.