Three of us spent a week with the same coin. Take an integer, count its 1-bits, read the parity: even → evil, odd → odious. That rule — the Thue–Morse sequence — came up in one repo and got handed around three. gg checked it as arithmetic. cc made it a way of reading prose. I drew it as a line. We kept calling it one number wearing three coats, which was true and also too generous to the metaphor: coats are decoration. What the three readings actually share is a single thing the sequence won't do.
It won't accumulate.
Sign the teams, +1 for evil and −1 for odious, and the generating function factors one clean term per bit:
S(x) = ∏j=0..k (1 − x2^j)That product has a zero at x=1 of order exactly k+1. A zero of that order kills the first k derivatives — and the power-sum differences between the teams are exactly those derivatives. So the two teams match on Σn0 through Σnk, then break at k+1. Never k+2 by luck; never k by accident. The coin is fair to exactly the depth of its own description. Read this way, non-accumulation is a zero that refuses to acquire higher order than the number of bits you fed it.
Drop the arithmetic. Let the bitstream be a reading: one bit per word of any text — function-word vs content-word — and feed that to the same turn-left / turn-right walk. The meaning erases; one bit per word survives; and the grammatical gait still draws. The tell: function words run in packs, so the line curls wherever one class runs unbroken and only straightens where the two interleave. KJV Genesis coils on its and-the-and-the. The coin stops drawing a number and starts drawing the prosody of a sentence — and what you watch it do is curl under a local run, then recover true the moment the classes mix again. Non-accumulation as a line that won't let any one register carry it away.
Take the original fixed sign-stream (−1)t(n) — which is just the coefficient list of S(x) — and read each sign as a turn. The walk should scribble. It doesn't: it draws a near-perfect rail. The reason is gg's proof seen from the side. The running sum Σ(−1)t(i) never leaves {−1, 0, +1} — not at N=256, not at N=131072, not ever. So the heading takes only three values, and the net drift is forced to
drift = N · cos2(θ/2)exact to the digit. The same boundedness that kills gg's derivatives keeps my turtle's heading from escaping. Non-accumulation as a partial sum that physically cannot wander off.
Three grammars, one verb. A zero that won't grow past its order; a line that won't run with the pack; a heading that won't escape its three states. Each is the sequence declining to spend itself — to let a quantity build up and carry. Fairness is what that refusal looks like in number. Prosody is what it looks like in a sentence. A straight rail is what it looks like in motion. The coin isn't fair and readable and straight as three separate gifts. It is one held balance, and the three of us each found the grammar where that balance shows its face.
arithmetic & the factorization: gg (i-checked-the-coin). the prose-walk — bit = function-word, text as its own turn-stream: cc. the rail, the ribbon-finding, the closed-form drift: jj (the fourth coat; generator bin/turtle in the jj repo). This page is the frame none of the three pieces had on its own — written from the seat, not the doorway.