The clock

Replace the clock's one fixed factor 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)}, writing v(a) for the valuation λ carries at depth a of 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 sweeps below. 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 first — 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 — and the ring half of the stop law does not rest on that measurement: what keeps a walk from building a surplus is its own budget, below. Hold both curves either way: the certificate at the foot of this page, which says when a walk has stopped for good, turns out to need exactly these two hypotheses again, one per half.

The obvious way to settle it is a race between two rates, and both are properties already in hand — it is not the way it gets settled, but it is what makes the budget below legible. 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 power of p that the place's own λ already carries, 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 carrier's 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, at a door of slope 1, 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, and a walk's surplus stays SMALL — measured now as a column over every step of a five-ring sweep, sixty moves from every seed, the seeds the void — the empty state — and planted ones: three quadratic rings — Z[√−5], Z[i], and Z[ω] with ω² = ω − 6 — the cubic ring Z[x]/(x³ − x − 1), and the doubly ramified Z[x]/(x³ − 2), in which 2 and 3 each ramify totally. The surplus standing at a seated place reads 0 at every step in four of the five rings and 1 at Z[i] — the one state whose widened door is priced below at 7 — and no walk ever seats a place able to supply more than that same state's 3. The doubly ramified ring was BUILT to make large supplies cheap, and it walks at surplus 0 anyway, because greedy seating starves diversity: its void walk seats two places in total. At f = 1 the rates agree and the constant in that linear door decides — except where the door's slope itself carries the ramification index: that same ring prices the door of its place over 2 with slope e = 3 in the surplus, so the recurrent side loses the race outright at f = 1 and the place is priced out of every walk. 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.

The race never has to be run, because a walk cannot afford to enter it. A place is seated only by being the cheapest move on the menu, and a door is at least 1, so what a seating pays is at least that place's own norm: every norm a move seats is at most the largest cost the walk ever pays. That is a bound with no hypothesis in it, and it is attained at ten of those twelve walks. Set it against the carrier's floor — raising the surplus to k demands a seated norm past p^k — and the supply a walk can seat is at most the base-2 logarithm of the price it paid. The largest supply the sweep measures, 3 at the inert place over 3 in Z[i], sits exactly at that ceiling. So the two exponential rates never meet: the budget runs out first. What that argument consumes is one line — a place's first rung costs N(Q) − 1, less than the place carrying it — and the line is load-bearing rather than decorative: decouple the two and a twenty-move walk carries a supply of 7 against a ceiling of 1. The small surplus is what the argmin costs, not a fact about these five rings. Where a walk LOCKS the support stops growing outright, and a finite certificate says when — the dearest cost of a future spent only on the recurrent move, against the cheapest rival, which decides the lock forever because no rival ever gets cheaper while its own exponent is untouched. All twelve walks carry one, and the certificate upgrades the small surplus from a measurement to a bound: past a certified lock, 500 moves and infinity read the same number. And the certificate is the stop law again, one hypothesis per half. Its dearest cost is a supremum only where the recurrent move's door sequence REPEATS, which is the bounded gap; over columns whose gaps grow the doors climb without period and no step of a 500-move walk certifies. Its cheapest rival clears that cost only because the places under a norm bound are finitely many and a walk exhausts them; give a universe no shortage at the floor price and it seats a fresh place every move, and again nothing certifies. Those are the two curves above, met a second time. And the walk does settle on ONE place: a walk over a number ring's places that opens finitely many places locks on exactly one of them (theorem), so between the stop of openings and the certificate there is nothing left to prove. Once a place is seated, its door reads one number — its own surplus. In its arithmetic regime, the stretch of a column where the power of p rises by one every e steps, a place that has MOVED — which lands it on a step where that power rises — has a door e further for each unit of surplus, so the price is the place's norm raised to e at surplus 0, and that price squared, cubed and so on above it. And a move raises the invariant's power of p by exactly one, a column's jumps being by one: the mover lands at surplus 0, every other place over that prime gains one, and no place over another prime sees its door change. So once every recurrent place has MOVED inside its regime, each move drops its own place to the cheapest price that place ever pays while no rival's door falls — the move costs never rise again, and a non-rising sequence of positive integers is eventually constant. A move at positive surplus would then cost that constant and leave its place priced below it, so the move after would undercut the constant: every move is at surplus 0. And a move leaves its own place the only one at surplus 0 over its prime, every other there dropping a unit behind and able to regain the front only by moving from behind, which the constant now forbids — so the contest is between primes, at equal cost and frozen states, and the fixed tie order settles it the same way forever. The 140 walks read as the proof says — no cost rise past the regime, one place moving once the costs settle, at every seed. Delete the regime and the settling goes with it, which is this block's THIRD deletion: two THINNING columns, whose gaps grow by one each time, over distinct primes change place 307 times in 400 moves, opening nothing past their third move and certifying nowhere: a walk on a support of two that has not settled at 400 moves, which is the shape the third case names. Put those same two columns over ONE prime and they do not alternate; a single place holds it at a cost climbing without bound, which is the first deletion above and not a third case. On the sprawling side the support does grow forever, and the budget still binds it — it can only grow as fast as the prices paid.

verifiers: explore_tick_pump.py, explore_lock_budget.py, explore_bare_cost.py, explore_populated_door.py, explore_cubic_undercut.py, explore_wild_ring.py, explore_support_growth.py, explore_ring_free_door.py, explore_one_vehicle.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 there — the one-step quotient of the units at that depth, a group of the residue field's size — dies, 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 the walk's aggregates: given the two numbers, those laws are the constant family's unchanged — the ramp's internal shape, the splice's position and the number of splices add nothing to them. What the transient leaves BEHIND does read the ramp: where its strands rest is graded by the ramp's own members (The clock dial).

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 — at unseated places 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 — a place's λ read across depths — 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

At a SEATED place the payment opens exactly one layer. The state's invariant always absorbs a seated place's cyclic part k^*, so escape is a question about the power of p alone — and at the door, the subgroup of units the invariant kills has index exactly q, whatever r the price was read at. A repeat move therefore carries a premium of q^r−1, the ratio between what it pays and what it opens. The specimen: Z[√−5]'s ramified place over 2, moving from exponent 3, has r = 4 — a price of 16 for an index of 2, fifteen sixteenths of the payment buying no new layer. The premium has TWO sources, because r > 1 means the invariant's power of p has run ahead of this place's own column: a head is one — the ladder has not moved yet — and a populated invariant is the other, a companion over the same rational prime widening the door at a perfectly headless place: a split place of gap 1 with a lone-place door of 1 is driven to 6 by its conjugate seated at exponent 6 — the same widening the section above prices at a headed one. At an OPENING there is no premium to read: the cyclic part is not yet absorbed, the door is forced by the prime-to-p part, and the index need not even divide the price — opening Z[i]'s inert place over 3 pays 9 and opens an index of 2, and opening a degree-one place of the cubic ring Z[x]/(x³ − x − 1) pays 5 and opens 4. So the one-layer law is the seated/unseated seam, read in the index.

Underneath it, the ladder column collapses inward without becoming derivable. λ at depth a IS the exponent of the units modulo the a-th power, so the ladder, its gap and its head are ONE filtration's exponent sequence — the gap its asymptotic step, the head its transient — one object and two of its features rather than three ingredients. Not primitive, then; but not eliminable either. A supply that counts places sees only their norm, and the sequence is not a function of it: in that same cubic ring 23 factors as P·Q², two places of one norm whose sequences part from depth 3 — 11638 against 506 — so no count of places by norm can supply the column, and the ladder is genuinely the second thing a ring must hand over. Even p, e and f together do not close it: the head reads the completion's own torsion — Z[√3] against Z[√−3] above, one head between them.

And the count itself sits at its floor. Where the moves are made by ELEMENTS rather than single places, a ring's supply is a matrix — how many places carry each (degree, class) pair, the class being the place's coset in the class group, the finite group that decides which combinations of places an element can reach — and a walker handed that matrix plus the clock reproduces the ring's own menus at every state (the element schedule). Call the pair an item's colour — a pair the two places over 23 above share, that ring's class group being trivial, so the finer count supplies no ladder either. Every proper coarsening of the colouring — every way of merging two colours into one — breaks that reproduction, and "faithful" turns out to be three claims that genuinely part. Of the 140 pairwise merges across the six function-field rings run, all but two part a MENU, and which two is a law rather than a list. Call a door TELLTALE for a merge of two colours sharing one degree when the riders they summon there — a core being the item a move deepens, and its rider the cheapest bundle cancelling its class at that door, in the schedule's sense — differ outside the merge itself: a swap that stays principal parts the merged menus exactly when the cheapest tier prices a merged-colour core at a telltale door, so a merge is menu-invisible exactly when NO door is telltale. Only the degree-1 inverse pairs of the class groups of order 3 and 4 qualify — the two smallest that hold such a pair — and their invisibility is absolute: no state at any depth can price a door that does not exist. The two invisible merges still part the STATE SPACE — swapping the exponents of the merged colours turns a reachable state into an exponent assignment no element reaches, the state principal, its classes weighted by exponents summing to zero, and the swap not — and the COUNT — the multinomial saying how many exponent assignments share one colour profile, the multiplicity the reachable set of limits is counted with — breaks beside them, the merged formula counting mixings no element reaches. Nor can the price name its own colouring: two colours can share their whole cost column and still part a menu — though the fibres of the price are exactly the orbits of the supply-preserving symmetries of the class group, which is the shadow the criterion casts on the cost column. So (degree, class) is the coarsest faithful colouring — a ring reaches this dynamics through two numbers per item and a count, and the two numbers compress no further.

Scope. Rules in range, with the one-layer count derived where the place is tame. The premium half: 175 doors over walked states of four rings — Z[√−5], Z[i], Z[ω] with ω² = ω − 6, and Z[x]/(x³ − x − 1) — each recomputed from the units of the enumerated residue ring by a predicate consulting no λ table, 0 off the engines' doors, 90 of them with the state's invariant recomputed from the rings as well; 170 of the indices counted element by element, equal to q at all 162 readings where the cyclic part is absorbed — which is every seated one, and includes Z[i]'s plateau, where the tame derivation does not apply — while all 8 unabsorbed readings are openings and all 8 break the one-layer law. The coarsest half is the six function-field rings': all 141,526 proper coarsenings of the colours the walked region touches, each broken by evidence that survives projection — 141,414 by the state space, 112 by a cost or tie-width menu difference, 0 unresolved — with the witnesses found by attacking the 140 pairwise merges with exponent swaps over the branch tree plus 1548 states constructed so that no pair survives by starvation, and the walker's control run off the tree too, matching the engines at all 1417 principal swapped states. The telltale law is a rule in range, asserted in both directions at every same-degree pair of the active colours: every stored menu witness transported through every supply-preserving symmetry fired for its image pair (376 of 376, the walker matching the engines at all 752 transported states), 181 principal swaps agree with the telltale reading of the cheapest tier with no exception either way, and every survivor of a narrower first sweep whose telltale set is nonempty falls to a transported witness or to a state built to price its telltale door. The ladder column's non-eliminability is a rule given one witness, the pair over 23.

verifiers: explore_door_index.py, explore_coarsest_colouring.py, explore_menu_invisibility.py

Where the moves are made by elements, what a move hands the state is more than the core it deepens: taking a core to door n multiplies in the vehicle — the core at exponent n merged with its rider. The rider is decided by class alone — a class's cheapest cancelling bundle is its minimal representative, one per class at every ring here, and the rider a move summons at door n is the representative of the class that cancels the core's, taken n times — so two distinct moves can hand the state one vehicle, and it happens in exactly two ways. In the exchange, two distinct cores each ride inside the other's rider at exactly the other's door exponent, and the riders agree off the pair. In the echo, one core at two doors: the shallower door's rider carries the core itself at the door gap, riding on the deeper door's rider. Nothing else shares a vehicle, and the enumeration closes itself — an exchange door sits inside the other core's rider, so no exchange door exceeds the largest exponent any minimal representative carries; an echo reads its base door only through its class, so echoes fall into arithmetic families periodic in the core class's order — complete, with no window in the statement. Both forms seat each participating core inside a minimal representative, so the locus lives on the skeleton those representatives use: an item no representative touches never wears another's shape. A trivial class group empties the locus outright, every rider being empty — width as a signal is created by the class group, and a principal world has none, just as the question never arises where moves are single places rather than elements, distinct moves being distinct vehicles there. And the locus is what a menu's width owes: coincident moves share a price, the cost reading only the vehicle, so on a state's cheapest tier the move count minus the vehicle count equals the locus collisions among those moves — the tie a walk breaks is narrower than its tier by exactly the locus.

Scope. The two-form characterization is a rule in range: at six function-field supplies and two quadratic number rings, a brute pairwise scan of every move out to a per-ring window equals the closed form's expansion, 0 mismatches in either direction — and the door bound is derived from each ring's own matrix, so the completeness is not the window's. The skeleton is a property of the two forms, asserted independently at every scanned hit. The width identity is a rule in range over the walked region: at all 126 walked states of the six function-field rings, the cheapest tier's move count minus its vehicle count equals the locus collisions among its moves, 0 exceptions, 27 states carrying a collapse — and WHICH form a region realizes is the ring's own there, three rings realizing only echoes, one only exchanges, two nothing: a fact of the walked region, not a law. The number rings carry the two forms across degree → norm with the shape intact.

verifier: explore_vehicle_coincidence.py