The clock

Replace the clock's factor b by the SET of exponents the tick may stand at, the tick advancing to the least member at or above the exponent just landed on. That set is exactly {a : v(a+1) > v(a)} for v the valuation λ carries at the place, and the whole of what it gives the dynamics is its gap: the distance from one member to the next, which is the price of a repeat move divided by the degree. A ring whose gaps are BOUNDED stops opening new places, at about the deep place's degree times the largest gap; a ring whose gaps are UNBOUNDED opens forever. Which one a ring has is settled by the principal-unit filtration (O/P^a)^* = k^* × (1+P)/(1+P^a): the p-th power map sends level i to p·i in equal characteristic — a multiplicative set, gaps unbounded, SPRAWL — and to min(i+e, p·i) in mixed characteristic, which is i+e past a leading stretch — constant gap, LOCK, with the gap the RAMIFICATION INDEX e. A set whose gap grows without being multiplicative is realized by no Dedekind domain with finite residue fields, which is where λ is defined at all. So the dichotomy is the characteristic — equal against mixed — and what varies within the locking side is ramification and nothing else. That is the same local invariant the module law states from the other end: its eventually constant price q^e = p^ef is this gap's cost, and its rank-∞ case is this unbounded gap.

Scope. The arithmetic half is a derivation from the filtration, checked at the three worlds run. The measured half is a rule in range: every place of norm ≤ 200 in two quadratic number rings — 90 in all, 82 split, 5 inert, 3 ramified — read at depths 1..24, the tail gap equalling the ramification index at 90 of 90, 1 at every split and inert place and 2 at every ramified one, 0 off; beside three function-field degrees whose gap is already 16 by depth 24. Three of the 90 carry a LEADING stretch of wider gaps, and they are exactly the three with p − 1 ≤ e — the depths below e/(p−1), where the principal units are still squaring rather than stepping. The stop location is measured at 2, 2, 3, 5 against products 1, 2, 3, 5 at gaps 1, 2, 3 and 5, exact at three and floored at the fourth — measured in the abstract schedule, where a gap can be dialled, and not in a ring.

The stop law is a COMPARISON of two curves — a bounded recurrent price against an opening cost that grows in the degree — and each half can fail. Where openings never get dearer a bounded gap stops nothing, which is the SECOND curve going flat and is the degree-blind dial, the one setting of the schedule that cannot tell a cheap item from a dear one. The FIRST can fail once the recurrent price READS THE STATE, and does wherever that reading GROWS with the seated set: every opening then makes the recurrent move dearer before any item can take its first repeat. The cheapest recurrent move reads 248, 494 and 740 against a cheapest opening of 124 — the recurrent curve standing ABOVE the opening curve — at gaps 1, 2 and 3 alike, so this is not the bounded/unbounded axis in another coat, and 122 items are seated with none above exponent 1 and no runaway at all: a limit shape the family has no name for. So a bounded gap stops a walk only where the recurrent price cannot read the state. The rings above are not disturbed — a walked state raises what one seated place must climb to move at all from 5 to 7, the widening priced below, where a planted companion drives the same one to 17 — but that is a measurement at three rings and not a bound, so the ring half of the stop law now rests on it.

What would settle it is a race between two rates, and both are properties already in hand. A place's door is the least climb there that raises λ against the whole state at once, and what it reads of that state is one number: the surplus, the power of p standing in the state's invariant above the exponent the place has reached, for p the rational prime the place sits over. Raising the surplus to k is bought from a carrier — a place over some OTHER rational prime whose own λ supplies that power — and a carrier needs norm at least p^k + 1, so a supply costs exponentially in what it supplies, at base p. The deep place answers with its recurrent price, its own norm raised to a door growing linearly in the surplus — base p^f, for f its residue degree. Both are exponential in the surplus and their exponents stand in ratio f, so the two rates part at f = 1, with the recurrent side dearer above it — the shape of the recurrent curve standing over the opening one. That is what the race would DECIDE, and it is not itself a verdict about walks: the comparison runs as the surplus grows, a ring walk's surplus stays small, and at f = 1 the rates agree and the constant in that linear door decides. Against it stands the self-correction — a seated place that does move lands past the surplus it outran, which resets the race rather than losing it. Neither this comparison nor the self-correction is read off the three rings, whose records carry no such reading, and the carrier's own residue degree enters only through the constant in that floor — degree 3 attains it where degree 2 and below are slack — so it moves the race by a constant and never by an exponent.

verifiers: explore_tick_pump.py, explore_lock_budget.py, explore_bare_cost.py, explore_populated_door.py, explore_cubic_undercut.py

A place's ladder is the orbit of one recursion — ψ(i) = min(p·i, i+e), started at level 1, the multiplying branch below the Kummer seat e/(p−1) and the stepping branch above it. The junction between the two is the ladder's splice. The multiplying segment REACHES the seat only when the seat is a power of p, and where it does and the residue layer dies there, the two gears cancel and the step overshoots — which is the splice made visible in the ladder itself, as a head: a leading stretch of wider gaps of width w before the constant e settles in. Such a place hands the dynamics TWO numbers, the tail gap e and the sup gap e+w, and they decide different things. The sup decides where the walk STOPS and how many coordinates stand above exponent 1; the tail decides what the runaway PAYS forever. Both are gaps the item LANDS on, which is why each is a price and not a statistic: reachable depths climb the ramp to exactly p^t+1, whose next member is the landing, so the item pays e+w once — the barrier it must clear to get its recurrent move past the splice — and e forever after, which is what that move then costs. So Z[i]'s place over 2 (e = 2, w = 3) stops where gap 5 stops and is priced where gap 2 is priced, and Z[2^1/8]'s stops at 12 and is priced at 8. Nothing else about the head enters: given the two numbers, the schedule's laws are the constant family's unchanged, and the ramp's internal shape, the splice's position and the number of splices add nothing.

Scope. A rule in range on generated ladders, and the arithmetic behind them a derivation. Seven ladders ψ produces, each run at two clocks against the constant ladder of its own sup: agreement on all four aggregate columns at 14 of 14 — least uncovered degree, runaway count, strand count, and which fate the run holds — and disagreement on the recurrent budget at 14 of 14, which reads the tail where the control reads the sup. The stop reading holds at four ladders ψ cannot produce at all (a splice-free ladder over a signature where a splice is guaranteed, an arithmetic rather than geometric ramp, a splice three steps above the seat, and a ladder with two overshoots), which is what makes it a fact about walks rather than about the recursion. WHICH places carry a head is read at 24 places over 21 local fields: the head needs e = (p−1)p^t AND the whole residue layer to die, at residue degree f = 1 with μ_p in the completion. The older reading "exactly the places with p−1 ≤ e" holds at all 90 places of the two quadratic rings measured here and is false as a criterion — wrong at 9 of the 24, Z[√3] against Z[√−3] carrying the same p and e with a head at one and none at the other.

BOTH NUMBERS ARE A LONE PLACE'S, which is what a per-item clock is. The state's invariant a door is read against is an LCM over every seated place, so a populated door can only be WIDER than the lone-place one and never narrower. At 472 readings over the SEATED places of those two quadratic rings it is never wider either — off that column all three rings widen. At a third ring it is: Z[i]'s ramified place at exponent 3 has a lone-place door of 5 and a door of 7 in the locked state. Its carrier is the INERT place over 3: residue field F₉, so its λ carries that field's 9 − 1 = 8, and that 2-part already covers the depths the place over 2 would have escaped at. And the widening obeys a law that is structural rather than observed. The units modulo a power of a place split as a cyclic group of order prime to p, times a group of p-power order — so the only part of λ that moves with depth is the power of p inside it, and a seated door is just the least climb whose λ carries a higher power of p than the whole state's invariant does. That is ONE number, and it is the whole of what a seated place ever sees of the state around it. Whether it has a CLOSED FORM is a second question, and ramification does not scope the answer. Where a place's λ gains exactly one factor of p per depth — the shape a cyclic local unit group gives — the door is read off the surplus directly: the surplus plus 2, against a lone-place door of 1. That holds at all 413 readings on a column of that shape and breaks on every one of the five columns that lack it, at 237 of their 304 readings — and Z[i]'s place above is one of the five, which is why its door reads 7 where the form would give 2: that column plateaus rather than climbing, so it takes several depths to gain the factor the form spends one on. And what takes a column off the shape is not ramification: λ is the unit group's EXPONENT rather than its order, and the two part exactly where that group stops being cyclic, which happens at an UNRAMIFIED place whenever 2 splits — as it does in the ring of integers of Q(√−23), whose places over 2 climb 1, 2, 2, 4, 8, … So the boundary is cyclic against not, and unramified is no defence. What no ladder here can express is the SOURCE rather than the widening: an item's price in this family is a function of its own exponent, and this one is a function of another place's residue cardinality. Nor is the excess bounded — a single planted companion drives that same door from 5 to 17.

verifiers: explore_headed_ladder.py, explore_populated_door.py, explore_head_width.py

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, as above — 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 the two rules above — 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 — one ladder at one block structure being a CELL, each 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, at a block-split seed as well as at 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