Evidence Adds Up to a Fact

P meets M. That first touch is the same meeting whether M is #18's marker or a particle of a detector: a promise, two lines, nothing written.

The difference comes after. An isolated marker keeps the trace to itself. A detector particle passes it to its neighbours [environment fragments]. Each meeting extends the shared wave; no line is selected here.

Before detecting P, we reverse its joint meeting with M [inverse unitary operation]. This undo reaches P and M, but not the crowd. With no escaped trace, subsequent P readings recover the full fringe. #18 used a different move: turn M, read it, and sort already recorded P results [conditional interference]. That leaves the unsorted results unchanged.

A detector is a marker whose trace spreads. The escaped correlations suppress interference in P's readings alone [decoherence], even after we undo the P–M meeting.

Nothing here picks the line. Room 2 supplies a separate toy rule for choosing an answer; this room tests how much interference a limited undo can recover.

P has met D: a promise, nothing written (room 1). Now each of them shakes one more hand, and the two newcomers meet each other.

A handshake with a newcomer is a copy. The newcomer holds the promise too. Nothing is written, and the step can still be undone.

A handshake between two holders is a chance. Each makes one light touch. Usually it lands nowhere. Sometimes it lands on a line, with a chance set by that line's length.

The first time both touches land on the same line, that line is written for every holder at once. Length times length: the square. Nobody proposes, compares or refuses. A miss leaves nothing behind, and the number of misses before the write is the same on every table, so even someone counting them would learn nothing.

A line is written at the first handshake where two holders' touches land on the same line. The crowd makes the chances, not the choice: its wiring decides when and where, never the odds.

Other crowds (a lone thread that never writes, a ring, a room of friends) and other ways of putting the two touches together are under Variations below the picture. They change when and where the write comes, never the odds.

Once a line is written, witnesses meet the settled particle. What can their meeting do?

Copy. A witness that meets a particle whose line is already written sees a table with one line left, so it can only copy. The odds stay the square, however many witnesses come.

Re-sign. If every witness rolled its own lottery on the whole table, it would contradict the books already written, about one link in three.

Vote. If all books had to agree before anything counts, the longer line would win almost always. That is majority rule, and the square is gone by the third book. It is also how a folk tale spreads: the catchiest version wins.

There is one write, at the first match. Every later meeting copies. Spread buys permanence, not odds.

To undo a fact, you must undo every meeting that copied it, last one first. That means reaching every book that holds it: P's, M's and each witness's.

Give the undo a chance to reach each book (the "reach" dial). It undoes the fact only if it reaches them all.

The chance of an undo falls geometrically with the books that hold the fact. That is the shape of Nakamoto's catch-up odds: every confirmation multiplies the attacker's chance by the same factor.

Nothing forbids the undo. It just stops being reachable.

Simple form first. Each copy takes with some chance ("copy takes"). The fact can be erased only in the branch where nobody knows it.

A crowd: M tells three friends directly. Everyone fails only if all three fail: at 3 in 4, one in 64. A chain: M tells W1, W1 tells W2, W2 tells W3. If W1 failed, everyone after is silent: one in 4, however long the chain.

The chart shows the corresponding wave calculation: a witness's look turns it a little, and the fringe contrast recoverable before detection is set by the looks nobody undid. With cos(look) equal to the chance a copy fails in the counting analogy, the numbers are the same; the wave calculation retains both alternatives.

With a reach. Now let the undo reach each witness with some chance ("undo reaches") and undo, last first, every look it reaches. From a crowd, what comes back still falls by the same factor per witness: geometric, room 4's shape. From a chain it levels off after a few links. Missing a link costs only what the shallowest missed link costs; deeper misses cost nothing more.

A factor per unreached direct witness, and the factors multiply. A witness of a witness adds a listener, not a look. Evidence adds up; hearsay doesn't.

This is why a Bitcoin confirmation counts: every block re-hashes the history itself, a fresh direct look, not a forwarded opinion. Gossip that only forwards is spread without finality.

Two threads. After P meets D, P's later meetings and D's later meetings are two sub-trees of the fact. Each sub-tree costs the undo its own factor, and inside each, only the shallowest missed link counts.

Proved on the toy. The shallowest-link rule, the crowd's product and both averages are exact identities for chains and crowds of any length (the message box prints the formulas). Each unreached direct witness is worth the same amount of finality: half a bit for a 45° look.

The shallowest-link rule and the crowd's product are a machine-checked proof (Lean 4, Mathlib) for any number of witnesses. One honest clause: it is proved for a line-by-line copy of the toy's code in Lean, which reproduces every number the toy ever stored; for looks between 0° and 90°.

P has met M, and a witness W has copied M. Now someone turns P and reads it. Two ways to predict the reading:

The whole table. Nothing is ever cut; just run the turn on the joint table of P, M and W.

The narrowed table. Pretend a match already wrote one line, then turn that line alone, and average.

As long as the move misses at least one book that holds the fact, the two predictions agree exactly. Narrowing is conditioning on the books beyond reach. The table never narrows.

So nobody who misses a witness can tell where, or whether, the line was written. Only a move that reaches every book sees the difference: the fringe comes back.

A witness copies P's line and goes away. Later somebody reads P, straight or after a turn. Does the copy matter?

Compare the reading with the copy and without it. After a straight reading the two agree exactly: the copy told the world nothing the reading would not have said anyway. After a turn they differ: the copy has removed the fringe the turn would have shown.

A copy is a fact only for a reading that could contradict it. The same entry is a fact under one later turn and a dead letter under another, and the late choice signals nothing.

On the toy, more than half of what the vacuum writes is a fact for nobody in this sense (55.6% on the meter).

Inside a sealed box: an atom A, a detector D, the cat C. Outside: you, O. Step through the meetings.

The atom meets the detector, the detector meets the cat: copies. Inside the box the holders keep meeting, and the first match writes the line, somewhere in the box (room 2). The cat's book agrees with the detector's completely. Across the box wall there is nothing: your book knows nothing about the cat.

Opening the box is your first link: your book copies the cat's.

Decided inside the box, unreceived until your first link. "Alive and dead" is the outside's table, not the cat. Opening the box does not create the fact; it delivers it.