Every step of the reduced map has a shape, whatever the actual numbers: the letter (m, r) — how the step entered its block (m, the run of trailing 1-bits of the door y) and how it exited (r, the halvings at the exit). The stratum word is the orbit's itinerary written in that fixed alphabet: the sequence of reduced states with all size information thrown away, shapes only. Companion to The 3-adic mirror of the reduced Collatz dynamics, DOI 10.5281/zenodo.21303918 (reverse.md §14.14–14.15).
Two proved facts sit in deliberate tension here. The word is free: any finite letter sequence — any past, any future — is realized by an actual positive integer; the dynamics impose no grammar at any finite length (the full-shift theorem, §14.15.2, and the bicylinder corollary, §14.15.4). But each letter has a price: every prescribed letter multiplies the smallest integer realizing the window by a fixed factor (2m+r·3m), and a running orbit consumes the very bits that encoded each letter, never to recover them. Freedom at every finite length, escape to infinity at unbounded length — the gap between the two is exactly the program's open object (bridge.md §16).
Two runs, two numbers — both visible above. m is the blue run: the trailing 1s of the door y itself (its length is read straight off the bits, nobody chooses it). The block then runs its course and exits at the even number A = 3m·q − 1 (where q is what's left of y+1 after the entry) — and r is the green run: A's trailing 0s, the clean halvings before the next odd door. So r is a visible run too, just in the derived number, not in y. And q — the middle column — is the bridge between the two numbers: it is what remains of y+1 after the entry, and you can read it straight off y's bits — delete the blue run and flip the 0 that ended it to a 1 (the dotted bit); that's q, always odd. It is the payload the block carries into the tripling: A = 3m·q − 1. Since A's low bits come from q's, which come from y's bits just above the blue run, the whole letter (m,r) is pinned by y's last m+r+1 bits alone (the shaded window). The step then shifts that window away — the next door's low bits are fresh digits pulled down from higher up, and nothing regenerates the supply. That consumption is the digit budget (stage4.md 11.8.7.7): a number of a given size can only ever pay for boundedly many letters of its own itinerary. What exactly happens to the window's m+r+1 bits: they are fully spent buying the letter, in two stages. The blue run plus its boundary 0 become, in y+1, the factor 2m — divided out at entry, gone. The window's remaining bits pass through the tripling and come out as A's low bits: the r green zeros — divided out at exit, gone — plus one guaranteed 1 above them, which says only "the next door is odd," true of every door, so zero information. Nothing of the window reaches the next door; its bits descend from y's higher purple bits, shifted down and scrambled by the tripling's carries — and you can read that directly in the exit column: drop A's green zeros and the bits that remain are the next door. The window's anatomy, decomposed (the author's observation): the blue run never enters the tripling — it is stripped at entry, functioning purely as the unary spelling of m (its length is the information; its bits are all forced to 1). The 0 that ends it is forced too (a run ends by definition at a 0 — it flips into q's dotted low 1). So the window's only free payload is the r green bits of y just above the boundary — exactly the bits that, pushed through the tripling, become A's r green zeros. Consumed: m+r+1 bits; payload: r bits; the other m+1 are the self-delimiting address that locates and prices them. In every letter chip, m is blue like the 1-run it counts and r is green like the 0-run it counts; in the state column, d is shown blue when it equals m (the usual case — the block had no pre-converted 3-rises), and plain when d > m. A dead door (3 | y) reads forward like any other but no predecessor can reach it.
Families. Every state (ω, d) is a family of exactly d doors ya = 2d−a·3a·ω − 1, all funnelling through the single even value A = 3d·ω − 1 — click any row's state to see them. The whole future word from the next door on is family property; only the first letter's m component is the member's own. And the r green zeros are the family boundary — consumed, they end the family, and the next door opens a new one (reverse.md 14.1.1, 14.14.1).