The clock dial

Run the schedule at a block clock and the limit's shape splits by the recurrent price. Where the gaps are BOUNDED that price is flat and exactly ONE item runs away, however finely the items are cut into blocks. Where they GROW the price climbs and the runaways number exactly the blocks. The mechanism is the door's RESET: a recurrent mover lands on the next member of its own ladder, so its next door is its own gap and a bounded ladder returns it to the same price forever, one global minimum holding; a growing ladder lifts it past that price instead, the minimum ROTATES, and every block takes a runaway of its own. So a single deep coordinate over a flat support is the FLAT-COST CORNER of a law that otherwise reads one runaway per block, and what collapses the product to one place is the total order on prices rather than anything p-adic. Two things move with the cut and one does not. The budget does not: the flat minimum is the runaway's own price read at its own gap — which is the staleness the reset leaves it at — however the blocks are drawn. The count standing above exponent 1 does, against a runaway count fixed at 1, so every coordinate a finer cut adds there is a STRAND. And the ceiling on the deep item's degree is per BLOCK, so a state's deepest degree is a MAX over blocks and rises with the cut, from 1 at a single block to as far as 26 at singletons.

THE SECOND REGIME IS ARITHMETICALLY UNREACHABLE, which is why the corner was never in danger and why it is safe for the wrong reason. Rotation needs a climbing price AND more than one block. A function field's gaps grow, and it has one residue characteristic and so one block. A number ring has many blocks, and its ladders are eventually constant-gap at the ramification index — a gap set that grows without being multiplicative being realized by no Dedekind domain with finite residue fields (The clock) — so its price is eventually flat. The two realizable worlds each miss one hypothesis and they miss different ones, so the finiteness of the residue field is the whole of what stands between this law and its second regime, and no ring either world supplies would ever have shown that the law has a second regime at all.

A HEAD DOES NOT BUY IT BACK, which is where that “eventually” is paid. A head climbs, so a headed place is in the rotating regime for its own length — but every such ladder flattens at its splice, and past it each block's price is a constant, so the global minimum picks one and holds it. Crossing both dials at once — a block tick over headed ladders, which is the ring's actual shape run with the ring's actual ladders — exactly one item is still rising in every run, and it stands past the depth where its own ladder's gaps have settled to the tail, so the budget it sets is read at the TAIL gap and the head moves nothing in it. A head buys a TRANSIENT and never a second infinite coordinate.

Scope. A rule in range at one schedule, and the unreachability half a derivation from two rules on The clock — the characteristic dichotomy and what gap sets a ring can realize — which it inherits the scope of. A runaway is counted as an item still rising over a trailing stretch of moves. Six ladders × five block structures — each cell walked 120 moves — give one rising item at all 20 bounded-gap cells, and rising items equalling the distinct blocks they occupy at the 6 growing cells that can discriminate, the per-item rows being consistent and VACUOUS for that reading since every item is its own block there. The crossed cell adds ten headed ladders × five, one rising item at all 50, read against each cell's deepest exponent so that a walk stopping short of a splice shows as vacuous rather than passing quietly. Both dial ends are certified against the family's own two clocks — one block against the global, singletons against the per-item — move for move, menu for menu and seat for seat, as the control. Two cells built to break it hold: blocks whose two supplies are isomorphic still give one runaway, by tie-break alone, from a block-split starting state as well as from the skewed one; and a shared tick with UNSHARED ladders, which is what two places over one rational prime are, puts the runaway on the NARROW item and never carries the wide one above exponent 1. What a block clock strands it strands ACROSS blocks and never inside one.

verifiers: explore_block_clock.py, explore_headed_block.py

The reading identifies its deep coordinate as the last item to move the clock. At a global clock that is sound for a reason the identification does not carry: a clock rises only where the landing exponent exceeds the tick, so a clock mover always sits above exponent 1, and under a global clock exactly one coordinate is recurrent — 557 of 557 states — so the identification is a corollary of the ideal limit theorem, which is a global-clock law. Under a per-item clock the last clock mover is the recurrent item at 2089 of 45709 states, 4.6%. Rebuilt to forget what the walk actually carries to infinity, the reading reproduces the imported one at every global-clock state — and its two extreme continuations, the walk continued with every tie broken toward the first candidate against every tie broken toward the last, agree at only 5810 of the 45709. That agreement is necessary and not sufficient; the sufficient test is the next block's. A state whose reading moves with a tie still to be broken has no limit for any reading function to return. So what fails at a per-item clock is the object and not the instrument.

Scope. A rule in range: 46571 final states — every state a walk of each measured length ends at — over eleven ladders, headed and constant-gap, at the two supplies and both clocks, the rebuilt reading checked against the imported one at every global-clock state.

verifier: explore_headed_image.py

The sufficient test BRANCHES the continuation over every minimal-cost tie: a final state is determined when every branch locks on one reading, undetermined when two branches lock on different readings — both being genuine least-cost continuations — and unresolved when a cap or the 48-move horizon fires, never counted as determined. At twelve of the fourteen headed cells — seven headed ladders at the two supplies — every final state is determined: 142 states, the enumeration exhaustive, no caps. The constant ladder of the same sup — the headed cell's sup control — determines NOTHING: the few percent of control states the extreme continuations had passed all part readings under the full branching, 2491 states across every control of gap 2 or wider, both supplies, to the last. The one constant-gap survivor is the exact ladder, gap 1, which keeps its 25. The two cells outside the twelve are the widest ladder's, sup 12, whose stop sits past the 24-move walk: carried to 84 moves every state locks — 5712 of 5712 and 10168 of 10168 — and the enumeration reads 4432 and 9202 determined against ZERO undetermined, the remainder unresolved at the instrument's own caps. No headed state anywhere in range is a tie.

WHY is a derivation on the door law rather than a shape. On a constant-gap ladder every landed item's door is the gap forever — depth buys no discount — so two landed items of one degree are priced identically at every step of every continuation, and a continuation running one away has an equal-cost mirror running the other, the two readings parting exactly where the pair's kept exponents do; a ramp prices position and closes the fork. Checked both ways: no determined state anywhere carries such a swappable pair — landed, distinct exponents, equal door future, exactly one of the two forgotten by the reading — and among the control states that FELL under branching, h5's at gaps 2 through 8 carry the witness almost to the last, 458 of 462 at the largest cell, while F₂[x]'s falls and the gap-12 cells' carry a second fork the census does not name. So a head does not give the per-item image a different SIZE — it gives it a DOMAIN, and the constant family has none to give.

At a GLOBAL clock no tie appears under branching — 0 of 747 states undetermined, a horizon band uncertified — and headed ladders part from their controls on size and members instead. A cell settles at the move count past which its reading set stops changing; the settled set agrees with the control's in size at 10 of 14 cells and as a set at 8 of 14. The sharpest form: the tail-2 ladders of head widths 2 and 3 over F₂[x] settle at four readings exactly as their gap-4 and gap-5 controls do, and the members that separate the sets carry a coordinate at exponent 3 where the control's carry 2 — one MORE degree seated in the same walk, the ramp's cheap door spent on depth and on openings at once. What does not part is settling: 11 of 14 headed cells settle by move 12 against 13 of 14 controls, the misses the sup-6 and sup-12 ladders, whose stop sits at the deep degree times the sup — at the widest, the lock arriving between 60 and 84 moves. The walk's aggregate readings — where it stops, what runs away, what strands — take a ladder's sup and nothing else (The clock); this register is the first that does not.

Scope. Rules in range, with the door-law half a derivation. Every verdict is at a 48-move horizon, so determined means determined at the horizon; interchangeable-item ties are collapsed and the collapse audited against the unpruned enumeration where it terminates. The derivation's premise — door equal to the gap at every landed item — is checked at 33293 final states, 0 off, and its consequence both ways, with a 47-state residual at observation: fallen states carrying no swappable pair, the second fork unnamed. The global count's 231-state band is unresolved at the horizon — 102 whose two extreme continuations return no reading either, 129 passing that two-extremes agreement while holding a branch that does not lock, uncertified rather than refuted. The exact ladder's survival is an observation at its 25 states.

verifiers: explore_headed_image.py, explore_image_domain.py