The pricing schedule

Take the ring out and four ingredients remain: items carrying an integer DEGREE; one GLOBAL CLOCK multiplying the tick by b whenever any item is deepened; a PRICE f(degree, staleness), staleness being how far the tick has moved since that item last did; and a fresh-opening DISCOUNT spendable m times at each degree, the least degree born with its discount already spent. The ring enters only as a SUPPLY — how many items each degree holds. Dial each as far as the move model admits — b at 3/2, 2, 3 and 4, the degree's exponent at 0, 1 and 2 and an additive form, one to three discounts, every born-spent set — and ONE INFINITE COORDINATE survives every dial but one. The dial that kills it is the degree-blind price, which cannot tell a cheap item from a dear one — and only where the clock's steps are wider than 1. At step 1 an opening and a repeat cost the same, the tie deepens rather than opens, and one item runs away regardless.

A FLAT SUPPORT is the weaker half and does not survive as far, so the two are separate claims. At b ≥ 3 the ceiling on the deep item's degree has already fallen to the least degree while the clock still seats an item above it, so that item is seated, clocked, then undercut and STRANDED above exponent 1 forever — exactly one such item on every branch that hands the clock on, and none on the branch whose first repeat was also its last. The doubling clock of the ideal limit is precisely where that cannot happen.

Scope. Proved for the abstract schedule, and a rule in range: 60 walks over ten schedules and six supplies, with the abstract walker certified equal to the exact one in cost, in every move type's multiplicity and in the covered set at 120 states over each of six supplies. The dial survey is crossed with the clock's landing rule rather than read beside it — 7 landing rules × 8 dials × 2 clocks — and the single infinite coordinate holds at all 49 crossed cells whose price can see a degree UNDER THE GLOBAL CLOCK. Give each item its own tick instead and the count standing above exponent 1 moves with the dial and with the clock's step width together, which is the mixed run below. Strands arise on both sides by different mechanisms — there the ceiling falls under a seated item, in a mixed run the item's own step widens under it.

verifiers: explore_price_schedule.py, explore_tick_pump.py

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 above. 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.

verifiers: explore_tick_pump.py, explore_lock_budget.py, explore_bare_cost.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 vehicle past the splice — and e forever after, which is what the vehicle 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. A ring reads a place's door — the least climb there that raises λ — against the whole state's invariant, 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. The carrier is cross-prime — the INERT place over 3 has 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. Write p for the rational prime a place sits over. The units modulo a power of that 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 power of p in the state's invariant, less the exponent the place stands at, 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

Strip the ring again, and what the ideal world read as a supply by degree the element world reads as a supply matrix — how many places carry each (degree, class) pair, over the ideal classes as a finite group with its addition table. Hand a walker that, plus the clock's reading at each state. Recompute the rider and its cost from the matrix alone, as a shortest path over that group with each class weighted by its cheapest place, and the walker's menu — the cheapest admissible moves at a state, one entry where the greedy move is unique and several where it ties — is the ring's own, entry for entry, at every state. It holds over six function-field rings, where a place's degree is an integer, and over two quadratic number rings, where a place's degree is the logarithm of its norm, so that costs which added as degrees now multiply as norms. The least cost stops being a least total degree and becomes a least norm PRODUCT, its weights no longer integers, and nothing breaks: the two arguments that use it — the shortest path, and the bound saying a core bought past its door is never cheaper — need only that it is a minimum over an ordered monoid with a monotone operation, and the logarithm carries one such monoid to the other.

The matrix does not give everything. A function field's places share one clock, the doubling tick; a number ring gives each place its own ladder — the set of depths at which that place's own λ moves, its TAIL gap the ramification index above — so no single tick supplies the doors, and the matrix, being a COUNT of how many places carry each pair, holds no per-place ladder to read one from. The ring hands a LADDER COLUMN over beside it, and that is the whole price of the crossing. Norm and gap together come close:

for a place of norm N at depth a with gap g, rad(N) being the rational prime beneath it. That is the ring's own λ at every place except the ones carrying a head, where the principal units are still squaring rather than stepping — over these two rings exactly the places with p ≤ e + 1, which are the places over 2. So the ring enters as a supply matrix and a ladder column — one entry per place, and over a function field the same entry throughout — and the ladder is what hands the walker its doors wherever the state does not widen one, which over these two rings is everywhere.

The bundle's form also says why an element lock's recurrent price is FLAT, which the ideal world's own argument could not reach: that one is about a single place's valuation, and an element move seats several places at once. It does not have to reach several. The recurrent move is the core at a fixed power together with its rider, so the price FACTORS — the first factor flat because deepening a place carries the state's valuation to just below the new depth, leaving the next door at the constant gap, and the second a lookup on a finite group, fixed once the door is. The element world's flat price is the ideal mechanism composed with the rider being a price.

Scope. The cost monoid's carry-over is a derivation; the menu equality is a rule in range on both sides, the walker handed the clock's reading at each state — the tick over a function field, the door itself over a number ring — so what is under test is the bundle and the cost, never the ladder. Six function-field rings, menus compared entry for entry over the whole branch tree out to four moves and nine moves of the greedy line each, none differing; and two quadratic number rings, 675 states — every element seed of norm ≤ 40 walked twelve moves, plus the whole branch tree out to four moves — again none differing, and of the 344 and 347 menu entries, 103 and 307 seat a bundle rather than a single place. The ladder column is read at the 35 places of norm ≤ 60 over depths 1..14, as a biconditional against p − 1 ≤ e so that an agreement at one of the head-carrying places would count against it too: 32 agree, 3 part from depth 3 on, 35 of 35 on the criterion. Each class's cheapest rider is unique at both number rings, by an exhaustive enumeration, and at every function-field ring run, so the reading is never tested where a class has two. The recurrent tails are read at 53 seeds with 40 moves past each lock, 0 non-flat.

verifiers: explore_class_schedule.py, explore_element_schedule_nf.py