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 — and those fifty cells are re-read at two further pricings (the strand law below), one rising item at every settled cell there too. 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 a 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

Call the ramp's members its rungs and its last member the seat — for the ladders arithmetic supplies, the powers of the residue characteristic — and the head's width the sup gap's excess over the tail. WHERE: on a headed ladder every strand rests exactly one above a rung at or below the seat, and the fill is top-down — cutting the items into finer blocks never vacates a resting depth and never opens one deeper than a depth already held, so the deepest reachable rung fills first and the shallower ones fill in behind it. A strand CLIMBS to its rest — 63 of the 67 in range moved, one to four moves each — stalling one above the rung whose next door outruns the budget; a stalled climber and an untouched opening leave the identical state, and only the move count separates them. HOW MANY is the admission census, a count asked of the schedule with no walk in it: an item strands exactly when its entry and its every climb door price strictly under what the runaway pays to cross its own splice — degree 1 paying the sup gap once — and it rests at the first rung whose exit door prices at or above that. Every per-item cell prints exactly its census; every coarser cut sits at or under it, stranding at most one item per block that lost; and the recurrent budget is the WRONG threshold to count seats against — that count is blind to the width while the census runs 5 to 39 across the widths swept. So the seat grades the DEPTHS and the width prices the ADMISSIONS: a head is two coordinates doing two different jobs, and no single “how headed” number could have found either.

Scope. A rule in range — and the law is the RAMP'S rather than one pricing's, so what it is a rule UNDER is a list of properties and not a count of prices swept. There are three. TWO are the price's: nondecreasing in the DEGREE; nondecreasing in the DOOR. The THIRD is the walk's: a tie at the runaway's crossing price goes to the splice. A fourth was filed and retired — the born degree's opening not undercut by a fresh discount: the schedule hands each degree one discount off its first opening, and the least degree, the born one, starts with its discount already spent (The pricing schedule), so it opens at its own door where a discounted degree opens at a door of one level, and that condition grades two DIFFERENT doors against each other, which degree-monotonicity — one door, two degrees — cannot reach. Delete it and the single late riser reads degree 2 rather than degree 1 at four cells, one at each of the four prices out of nine swept whose discount undercuts and at no cell of the other five — but every one of those cells is on the ladder whose every door is one level, and that is the only ladder such a cell can be on. Wherever the ladder has a door two or more levels wide at all, a discounted rival pays at least the born opening to pass it, so it cannot pass before degree 1 opens; degree 1 then climbs the same one-level rungs at the least price on any menu, draws level, and is cheaper or first at every rung the two share — a rival can catch it and never pass it, and degree 1 is the unique runaway under the three alone. On the all-ones ladder the census is empty, so the retired condition names the runaway where there is nothing to count: twenty-five cells of the kind it was feared to reach — an undercutting price, a non-empty census, a crossing door above the tail — sit inside the sweep, each printing its census with degree 1 rising, and taking degree 1 out of the born set leaves every strand set unchanged. And doors never fall as the tick advances, a property of the landing rule rather than a hypothesis on anything: the tick goes to the least ladder member at or above the depth just landed on, a tick is already a member, and that rule is monotone — so no ladder whatever can send a tick backwards, and none does at any cell walked here. Each of the three is NECESSARY and a witness is the proof: a price monotone in the degree while charging LESS for a deeper door seats items the census excludes at 7 of 10 cells, every failure that way round and none the reverse — a standing item's door grows with the tick, and a growing door costs more only if the price says so. Delete either of the other two and there is a witness for that too: a price falling in the degree leaves the rising item something other than the runaway at every cell, or nothing rising at all, and sends the census's own climb past the splice, where it is undefined rather than wrong; the tie order reversed breaks 8 of the 9 cells whose runaway has crossed. What is NOT on the list is unboundedness in the door — a price that stops growing there reproduces the census everywhere. What is NOT proved is that the list SUFFICES: eight prices no closed form in the PRICE family generates reproduce the census exactly, count and depth set, at 78 of 80 cells, the other two named unreadable rather than counted — their censuses stand above what the move budget can seat and their runaways are not across at 1200 moves. That is evidence at eight prices and not a closed derivation. What the argument does close is three of its five STEPS outright under a per-item clock, each with an observable at zero over 113 settled cells: a standing item's door is frozen while the item does not move; the affordability window has SHUT at the crossing step, nothing on the menu priced under what the runaway pays there, so everything the census admits was bought before it; and nothing but the runaway moves after. The cap above — every coarser cut at or under the per-item census — is the DOOR side's second consumer and follows rather than being measured: a shared tick dominates each member's own, so it only ever raises a door and only ever shrinks the admitted set, a CONTAINMENT and not only a count, read both ways at 144 settled cells, where deleting the door side puts 18 of 40 coarser cells above their per-item census. The remaining step names the runaway — exactly one item ever rises past the crossing, and it is degree 1 — and it is the overtaking argument above. And no step reads the ladder's SHAPE at all, only that a tail exists and that the crossing is priced at the last door wider than it — which on a ramp arithmetic supplies is the sup gap, its one wider step. The identity holds at every settled cell of four ladders no arithmetic ramp produces, whose own censuses run 0 to 676. The two closed forms the price family does generate are instances, and they show what a price moves: the ADMITTED SET alone — squaring the degree per level shrinks it, since entries pay their degree's power against a crossing price that degree 1 holds fixed, and pricing degree plus levels widens it — while the resting places, the top-down fill and the budget never move. Read at ten ladders by five block structures at the standing price and at each of those two, by ten ladders and three at each of the eight, the census swept nine widths on the seat-4, tail-4 ladder under ten cuts and certified first against every recorded cell of the standing pricing, and the born set dialled at nine prices by fourteen ladders; every cell reading uncrossed or off its census re-read at 400 moves and again at 1200.

verifiers: explore_headed_block.py, explore_splice_cap.py, explore_rung_schedule.py, explore_price_hypotheses.py, explore_census_theorem.py, explore_born_set.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 last-clock-mover reading 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; sufficiency needs the branching test below. 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 last-clock-mover reading 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 measured walks: 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 never lowers it — 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 swappable-pair 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, how many items strand — take a ladder's sup and nothing else (The clock); where the strands rest already reads the ramp (the strand law above), and this reading takes the whole continuation.

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