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 — a degree whose discount is gone being covered, so a fresh opening goes to the least uncovered degree. 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 — every coordinate but the deep one at exponent 1 — 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 — it is bounded by twice the least 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 of the clock's own dial. 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

Dial the covering rule and one thing decides whether the openings stop on their own: whether it can put degrees out of reach faster than the walk opens them. A rule keyed to the SEATED SET never can: what it covers reads only the degrees already opened, which no declined move changes, so the cheapest fresh opening holds a constant price that the doubling clock eventually undercuts. An opening covering its DIVISORS opens exactly the degrees the bare rule opens, the least uncovered degree only ever increasing; one covering its MULTIPLES opens the primes alone, and thins the climb into going HIGHER in the same walk, the clock firing more often for want of cheap openings. Such a rule is FED BY the openings it would have to get ahead of. A rule keyed to the CLOCK outruns them outright, the clock doubling where the openings step by one: at “covered while d ≤ c·T” the openings are dead at c = 1 on every branch whose deep item has degree at most 2 — a home at degree 3 climbs on at c = 1 and stops at c = 3/2 — and at c = 1/2 dead only on the branch whose deep item has degree 1 — the degree-2 branch pays 2·(T/2) = T for its own repeat and can still afford the least uncovered degree, so the threshold is the deep item's degree against c. And a stop is read as a PRICE and never as a silence: at the last state of each stopped walk the supply was still offering a fresh opening — at degree 1025 or 2049 — and the walk took a repeat costing 1024 instead.

Scope. A rule in range: 10 walks over 8 covering rules against a supply carrying items at every degree to 4096, each state asserted to sit far below that ceiling and to have a fresh opening left to DECLINE, so that the supply is never what stops anything. The two thresholds are exact by derivation and measured to a factor of two. The seated-set half holds for the whole kind, the frozen frontier by argument and the most generous member run: an opening covering every degree below 2^d puts the covered set past every bound in four openings — degrees 2, 4, 16, 65536 — and the walk still takes each next opening the moment the doubling repeat price passes its frozen price. Outrunning the DEGREES is not stopping the walk.

verifier: explore_ladder_stop.py, explore_seated_cover.py

Read a function field's admission test as a covering rule and it is DIVISIBILITY: base^d − 1 must divide the seated set's invariant, and an invariant is an exponent there, so what accumulates across the seated set is an LCM. At that reading nothing is ever covered for free, at base 2 or base 3. At the strictly more generous PRODUCT reading exactly one degree is freed — degree 6 at base 2, and nothing at base 3 — and it is the classical exception that frees it: 2^d − 1 carries a prime factor dividing no smaller member at every d > 1 except d = 6 (Bang, 1886), so an LCM of smaller terms can never supply what a fresh degree demands, while at d = 6 the demand is 2⁶ − 1 = 3²·7, whose factors a PRODUCT of smaller terms can supply and an LCM cannot. So a function field's sprawl is a fact about its covering rule and not only about its supply. The boundary is the load-bearing half: every reading run here has integer degrees and one base, which is the function field's shape, while a number ring's degree is a logarithm of a norm and its test runs over residue-field orders sharing no base, so nothing there plays Bang's part. That exonerates the covering rule where the openings are known not to stop; where they are known to stop, the exoneration comes from the clock instead, whose gaps never grow, and has nothing to do with covering.

Scope. A rule in range at four (reading, base) rows, run on the walks above, beside a control reproducing from the schedule side that the openable degrees are every degree. Bang's theorem is an external input and is not re-derived here.

verifier: explore_ladder_stop.py

Let the supply run out. The image is a SUM over the homes of a PRODUCT — the home's own width times the width of every other degree the walk can still open. For n items at every degree 1 to D that is n^D−1·(n+1), which at n = 2 and D = 8 reads 384 limits: the “+1” keeping it off a power of 2 is a SECOND HOME and nothing else. Break the tie between the two homes — gap the supply so the least fresh degree is no longer twice the least covered one, steepen the price, or lift the supply's least degree so the fresh side wins outright — and the sum has one term, a pure power of the width: under a gapped supply 128 = 2⁷ at width 2 and 243 = 3⁵ at width 3. The product runs over the degrees the walk COULD open rather than the ones present: park a degree above the home where the walk steps past it and it stands empty at every reachable state, 16 against the 32 a count of present degrees gives. Where the widths differ by degree the product shows itself as one, 48 = 2·3·2·4.

Now stop the ladder instead. The image is the PRODUCT over the homes alone of (width + 1), less one for the state with no deep item yet, and the widths ABOVE the homes never enter: two supplies agreeing at widths 2 and 3 on the homes and differing at every degree above both read 11 = 3·4 − 1. So a stop does not merely make an image finite, it makes it SMALL — at most (n+1)² − 1 for n the larger home's width, whatever the supply, where an exhausted ladder multiplies in a width for every degree it could ever reach. And a stopped image stands above a bare count of its homes' widths for two reasons — the second home, and the one degree a stop leaves OPTIONALLY open — which a tie-break removes together, they being the same degree. At the linear price — the degree's exponent 1 — a walk stops at its first clock or never, so every opening a stopped walk makes happens at the first tick, and a degree opened there is exactly a degree that won the comparison at the empty state, which is what a home is.

The power-of-2 reading does not travel between the two mechanisms. Under a stop the tied case reads 8 = 2³ at width 2 and the untied one 2, both powers; what separates them there is width 3, 15 = 3·5 against 3.

Scope. The exhausted law is DERIVED, from the homes being the winners of that comparison and an exhausted state having one shape per home; that every shape it allows is actually REACHED is measured and not derived. A rule in range over nine families — seven untied across three dials of the supply and the price, beside two tied controls — with the decomposition asserted at every reachable state rather than read off the count, and the covering-rule rig's own tied rows reproduced through this rig's enumerator as a positive control, read first. The stopped law is PROVED at the linear price, where the stop-at-the-first-clock step is derived rather than measured, and a rule in range over eleven families with none off the closed form — at ONE discount per degree and a doubling clock, both load-bearing: more discounts break the step that leaves one flat item, and a deep move's undercut threshold is (b−1)/b of the tick, so the threshold is the clock's as much as the degree's. And it is a closed form only where the covering rule stops EVERY branch: the covered home's threshold is half the fresh home's, so a rule set between them stops the covered branch and leaves the image infinite on the fresh one, whose factor is then not a count. Nothing here is a claim about a RING's finite image, which the first mechanism cannot reach at all: a degree there is the logarithm of a norm, so the supply never runs out and the ladder stops on a flat recurrent price instead.

verifiers: explore_void_untie.py, explore_stopped_untie.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 fixed by that place's ramification, how many times the rational prime beneath it repeats there (The clock) — 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 whose ladder carries a leading stretch of wider gaps (The clock), where the principal units are still squaring rather than stepping — over these two rings 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 the criterion for a leading stretch (The clock), so that an agreement at one of the places carrying one 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 — proved by an enumeration clearing the Minkowski bound — and at every function-field ring run; where a class does have several, the width decides: offering them all keeps the orbit count, and taking a canonical one keeps it at width 2 and breaks it at width 3. 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

A rider is not priced only at its own degree. It raises the exponent at every place it seats, and a place's door falls by exactly what its exponent gains, with no repeat at all — so a FRESH element move can drop the menu's minimum, which is an effect the ideal world has no move for. A tie is therefore breakable from outside itself: the cheapest move at a later state need be neither member of it, and the loser is priced out rather than beaten on its own terms. Every fall in a menu minimum along these walks sits at a place the move that caused it seats — its core, or its rider — and at none anywhere else. Over six function-field rings the element walks read 18 falls in 1,370 moves, 14 at a core and 4 at a rider; the ideal walks read 37 in 1,632, all at a core, there being no riders there to read.

Scope. A rule in range: every fall in the menu minimum attributed along ideal and element walks over six rings, 0 outside the causing move's own support. Two families of declined moves that a comparison of door prices leaves uncovered — the reading behind the separation lemma — sit at a fallen minimum at 74 of 74 and 32 of 32, which is what a discounted door accounts for and a comparison of prices at one state cannot. That a rider never advances the tick is read across 2,351 moves and 44,828 place readings, 0 moving it.

verifier: explore_undercut.py