# LIMIT — where one greedy walk goes

The object: a greedy growth walk over F₂[x] or the ring of a curve over
F₂, under the dynamics demand, read as an instance of a **pricing
schedule** — items of integer degree, a notch on a ladder, a price of
degree and door, ν openings per degree at door 1, and a
rule saying which degrees are covered. This page asks what the walk
converges to, which features of that limit belong to the ring and which
to the schedule, what bounds the runaway's degree, and what can stop the
support from widening. Its verifier is limit.py.

Items carry a positive integer degree d, the **supply** σ(d) counting
the items of degree d, finitely many per degree. A state gives
every item an exponent a ≥ 0, its **support** being the items at a
positive one, and carries a **notch** T on a ladder S, an infinite set
of positive integers containing 1. A degree is **covered** when it is
born covered, when ν openings have been made at it, or when a **covering
rule** says so. A move on an item is an **opening** when its exponent is
0 and its degree uncovered (door 1, landing at 1, the notch unmoved),
and a **clock move** otherwise (door r = T + 1 − a, landing at T + 1,
the notch moving to the least member of S at or above T + 1). It costs
the price κ(d, r), r being its **door**, the rise in its item's
exponent. A greedy walk starts from the void, every exponent 0 and the
notch at 1, and at each state takes a least move on its menu, the moves
the state offers. A move's type is its kind, opening or clock move, with
its item's degree and exponent; a tie is a menu whose least price
several types share, and a branch is one way of breaking every tie. The
verifier branches every tie of a walk's first few moves, breaks every
later one canonically (a clock move first, then the lower degree, then
the lower exponent) and counts branches reaching one state once; the
counts of branches below are of those. The item a clock move moves is its
**holder**; the item holding every clock move from some point on is the
**runaway**, and an earlier holder is a **strand**. Every exponent
stays at or below the notch, so every door is at least 1. The
**corner** is S the powers of 2, κ(d, r) = d·r, ν = 1 and degree 1
born covered.

The answer, in one line: under a price that rises strictly with the door
at no lower degree, the holders of clock moves form a chain of strictly
falling degree, so a walk that clocks forever has one runaway, its
strands are the chain's earlier members and everything else stands at
exponent 0 or 1; once a runaway holds the clock, with no degree covered
but by birth or its openings, the support widens forever when infinitely
many supplied degrees are not born covered, the ladder's gaps are
unbounded and the price is unbounded in the door, and stays finite when
the gaps are bounded and only finitely many openings are cheap; the
runaway's degree is at most 2^(1/α) times the least supplied degree at
the price d^α·r, α > 0, on every ladder, and a geometric ladder of ratio
b > 2 lowers that; and on unbounded gaps, at a price unbounded in the door,
over infinitely many supplied degrees not born covered, a covering rule
read off the born and opened degrees alone, finitely many at each
state, never stops the widening, while one keyed to the clock can.

## The ring is a supply
Tier: rule (the door proved; the walk verified at 720 states over six
ring supplies).
Verifier: limit.py::section_d, limit.py::section_x.

For P irreducible of degree d over F₂, (F₂[x]/P^a)^× is F_(2^d)^× times
(1 + P)/(1 + P^a), and (1 + y)^(2^j) = 1 + y^(2^j) makes the second
factor a 2-group of exponent 2^⌈log₂ a⌉. So λ(P^a), the unit group's
exponent, is lcm(2^d − 1, 2^⌈log₂ a⌉), checked by brute force at 47
pairs (P, a) with deg P ≤ 4 and deg P^a ≤ 12. A move raises λ at P
either by seating P (exponent 0 to 1) while 2^d − 1 does not divide λ's
odd part, an opening at door 1, or by carrying P to T + 1, the notch T
here being the 2-part of λ, which the move doubles. A place, or an item,
is seated at a positive exponent. The index [P^a : P^(a+r)] is 2^(dr),
and its log d·r is the move's additive index. A curve over F₂ has the same
completions, so the ideal world over either ring
(GROWTH.md#the-lock-over-a-number-ring), a move seating one place power,
is the corner schedule with the supply its place count per degree,
provided the covered supplied degrees are the opened ones and 1. The
next claim provides that, the corner opening degrees in increasing
order at the rising price d, and a walker covering by exact divisibility
of λ reads the corner's covered sets and notch (both walkers type and
price moves alike) at every one of 720 states, 120 along one canonical
walk over each of F₂[x] and five curves given by their zeta numerators,
1 − t + 2t², 1 + 2t², 1 + t + 2t², 1 + 2t + 2t² and
1 + 2t + 4t² + 4t³ + 4t⁴, each with one rational place removed and its
supply cut at degree 40. As a control, the corner schedule with a notch
that triples, the ladder of powers of 3 in place of 2, parts from that
walker at the second state.

## No supplied degree is covered before it opens
Tier: theorem (the lcm reading, from Bang's theorem); rule (the product
reading and base 3, verified 1 < d ≤ 300).
Verifier: proof; limit.py::section_d.

Bang's theorem (1886), at base 2: for d > 1 and d ≠ 6, 2^d − 1 has a
prime factor p dividing no smaller 2^j − 1. The order of 2 mod p is then
d, so p | 2^j − 1 iff d | j; at d = 6, 63 = 3²·7 and v₃(2^j − 1) ≥ 2 iff
6 | j. So 2^d − 1 divides the lcm of the 2^j − 1 over a set J iff J
holds a multiple of d, and a walk that opens degrees in increasing order
never covers a supplied degree before opening it. Read as the product of
the 2^j − 1 the test is more generous: it covers d = 6 before 6 opens,
63 dividing 3·7·15 over {2, 3, 4}, and no other degree below 301; at
base 3, the 3^j − 1 of a ring over F₃, neither reading covers a degree
early. This is the ideal world's
own statement of GROWTH.md#the-sibling-shadow's frontier.

## The chain
Tier: theorem.
Verifier: proof; limit.py::section_c.

Let the price satisfy κ(d′, r′) > κ(d, r) whenever d′ ≥ d and r′ > r,
at any ladder under one notch shared by every item. Take a clock move of Y
whose previous clock move had another holder Z. Z landed one above the
notch u it met, and since only openings came between, every other
exponent is at most max(1, u); so Y's door exceeds the door of Z's own
re-clock at that state. Z could have been moved, so
κ(d_Y, r_Y) ≤ κ(d_Z, r_Z), which the hypothesis forbids unless
d_Y < d_Z. So the holders' degrees strictly fall at every change
of holder, and no holder returns.

A walk with infinitely many clock moves therefore has exactly one
runaway. Its strands are the chain's earlier members, at most as many as
the supplied degrees below the first holder, each at a finite exponent
at least 2, and every item never clocked stands at 0 or 1 forever. The
proof reads no ring, no ladder, no ν and no degree's value. The power
prices d^α·r, α ≥ 0, and the sum price d + r meet the hypothesis. A
price that also reads whether the item is seated needs the hypothesis
with either price on the left and a seated one on the right. One is
the multiplicative index, the factor by which a move multiplies the
order of the unit group: over a ring whose places of degree d have norm
q^d, q^(dr) − q^(d(r−1)) for an unseated place at door r and q^(dr) for
a seated one. The multiplicative index meets the hypothesis except at
q = 2 and degree 1, where an unseated item at a door one larger can tie,
a case the proof never meets: at the notch T′ that Z's move set, an
unseated Y's door T′ + 1 exceeds the holder's T′ − u by u + 1 ≥ 2. Over
the schedule walks the verifier runs, 17 changes of holder were read, a
change counted each time a walk steps through it, and every one fell.

The one step that consults an order is the last: over degrees that are
only partially ordered, κ(d_Y, r_Y) ≤ κ(d_Z, r_Z) with r_Y > r_Z gives
only that d_Y is not at or above d_Z, and two incomparable items could
trade the clock forever. Two items A and B trading it in turn, at gaps
g_k, need
κ(d_B, g_(k−1) + g_k) ≤ κ(d_A, g_k) < κ(d_A, g_k + g_(k+1)) ≤ κ(d_B, g_(k+1)),
d_A and d_B their degrees, so g_(k+1) > g_(k−1) + g_k: at bounded gaps
two items cannot, by an order on prices and not on degrees; more than
two is not derived here. A deep coordinate is an exponent that grows
without bound, and under the chain's total order a walk that clocks
forever has exactly one, its runaway's, on every ladder; without that
order, at bounded gaps a pair cannot trade the clock in turn forever,
by greed's order on prices, and at unbounded gaps, the corner's,
nothing here excludes it; neither is the ring's doing.

## When the clock runs forever
Tier: theorem.
Verifier: proof; limit.py::section_b.

Between clock moves the notch stands still, so every clock move's price
does, while each opening is spent once. If the opening prices κ(d, 1)
tend to infinity over the supply, only finitely many items opening below
any bound, or the supply is finite, the walk clocks infinitely often.
Unbounded is not enough: at κ(d, 1) = 1 at every even d and
κ(d, r) = d·r otherwise, every degree supplied, on the doubling ladder,
every clock move once the notch reaches 4 has door at least 2 and costs
at least 2, and the walk opens forever. At the degree-blind price κ = r
every opening costs 1 and no change of holder is possible. A clocked
item lands at T + 1 and its next door is the ladder's gap there; once
that gap exceeds 1 every clock move's door is at least 2, since every
other exponent is at most 1 and the notch at least 2. So on any ladder
but the exact one, S = {1, 2, 3, …} of gap 1, a degree-blind walk over
infinitely many supplied degrees that its born set and covering rule
leave uncovered until they open, clocks a bounded number of times and
then opens forever: no runaway. On the doubling ladder that is at most
two clock moves, final notch at most 4, and whether they happen at all
is the tie-break's: over 300 moves, the second and third moves when its
ties go to the clock, none when they go to the opening.

## The support
Tier: theorem.
Verifier: proof; limit.py::section_s.

Take the chain's price and the default covering, no degree covered but
by birth or by its ν openings; a covering rule is
LIMIT.md#what-stops-the-widening's.
Once the runaway X holds every clock move, its door at each re-clock is
the ladder's gap T_k − T_(k−1), T_1 < T_2 < … the members of S, and the
**recurrent price** is κ(d_X, gap). If the gaps are unbounded and κ is
unbounded in the door, then for any opening a large enough gap prices
the re-clock above it, and since openings leave the notch where it is,
every cheaper opening is taken before that re-clock: over infinitely
many supplied degrees not born covered the support widens forever, with
min(ν, σ(d)) openings at every degree not born covered, each opened
item outside the chain at exponent 1. If the gaps are bounded by G, no
opening priced above
κ(d_X, G) is taken once X holds every later clock move, the re-clock
being always on the
menu; when only finitely many openings cost κ(d_X, G) or less, the walk
locks with a finite support. The degree-blind price κ = r on the exact
ladder has infinitely many, and a branch alternating its ties clocks and
opens forever.

At the corner price over two items per degree, 240 moves, on the branch
taking the clock at every tie, the void's κ(1, 2) = 2 = κ(2, 1) first:
the exact ladder and the gap-2 ladder never open, gap 3 opens to degree
2 and gap 5 to degree 4, all at or below d_X·G; the squares, the
triangular numbers and the ladders of ratio 3/2, 2, 3 and 4 are still
opening in their last quarter.

## The ideal limit
Tier: rule (proved at the corner; verified over six ring supplies, 10
moves branched and 290 more on every branch).
Verifier: limit.py::section_l.

At the corner, over a supply holding a place of degree 1 or 2 (every
ring the verifier runs holds one), the limit of every branch is

    ∞·C + one place at exponent 1 at each opened degree but C's,

with C of degree 1 or 2 and no strand: the ideal world's single deep
place over a flat support, every other seated place at exponent 1,
that never
stops widening. The chain gives
the shape, and the support theorem the widening, since the doubling
ladder's gaps are unbounded. The first holder keeps the clock: degree 1
being born covered, every rational rival is unseated and bids T + 1,
while a holder of degree d₁ ≤ 2, the void menu's winner (the
attainment section below), re-clocks at d₁·T/2 ≤ T. A degree-2 C
is the opened degree-2 place itself, so no rational place is seated at
all on that branch. Every supplied degree
above 1 opens, one place each; over 300 moves that is 290
degrees on every branch with C rational and 291, C's own opening among
them, on every branch with C of degree 2. F₂[x] and the four curves
with a degree-2 place have three branches each, the curve with no place
of degree 2 or 3 one, with C rational.

## The ceiling
Tier: theorem.
Verifier: proof; limit.py::section_c, limit.py::section_s.

Openings leave the notch where it is, so the first clock move is made at
T = 1, where an item of the least supplied degree d_min offers a move at
price at most κ(d_min, 2): its opening, its clock at door 1, or, its
degree covered, its clock at door 2. The first holder, of degree d₁,
therefore has κ(d₁, 1) ≤ κ(d_min, 2), and by the chain the runaway's
degree D is at most d₁. At κ = d^α·r, α > 0, that is D ≤ 2^(1/α)·d_min,
and at the sum price D ≤ d_min + 1, on every ladder. The verifier
runs a sweep of 14 schedules, varying the ladder, the price, ν and the
born-covered degrees, over three ring supplies (F₂[x], the curve
1 + 2t + 2t² and the genus-2 curve) at every schedule and a designed one
at twelve, 54 rows, every tie branched over the first 10 moves; its 71
branches at those prices and the ladders run in the support section find
no runaway above it, the runaway read as the item at the top exponent
when the run ends, the last three clock moves sharing its degree.

The ladder can lower this. Let D > d_min. Strands sit above D, so every
item of degree d_min stands at 0 or 1, and at each re-clock after the
last change of holder

    κ(D, T_k − T_(k−1)) ≤ κ(d_min, T_k + 1).

At κ = d^α·r, α > 0, that is D ≤ d_min·R_k^(1/α) with R_k = (T_k + 1)/
(T_k − T_(k−1)). On the ladder T_k = ⌈b·T_(k−1)⌉, R_k tends to
b/(b − 1), so

    D ≤ d_min · (b/(b − 1))^(1/α),

the ceiling, 2·d_min at the corner, the first-clock bound again. It is sharper
exactly when b > 2 (1.5·d_min at b = 3 and α = 1) and weaker at b < 2,
where the first-clock bound governs. Since
R_k/(R_k − 1) = (T_k + 1)/(T_(k−1) + 1) exactly, the bound reads the
liminf of R_k, finite as soon as T + 1 grows by a fixed ratio infinitely
often. A steeper degree penalty lowers both bounds. No runaway crossed
either where the sweep computes it.

## Attainment is the void menu's
Tier: rule (verified at 54 rows: 14 schedules over three ring
supplies and 12 over a designed one, every tie branched over 10
moves).
Verifier: limit.py::section_c.

Until the first clock move the notch is 1, and there the least
born-covered degree d_c bids κ(d_c, 2), the least fresh degree d_f, the
least supplied degree not born covered, bids κ(d_f, 1), and an item
opened meanwhile at that degree bids κ(d_f, 1) again by a clock at door
1, so at a price strictly increasing in the degree, as every one in the
sweep is, the first holder is a winner of that one comparison. The
degrees that hold a clock move in the run, over every branch of a row, its
**census**, are exactly the comparison's winners at all 54 rows, but not
in general: at b = 4 with degree 2 born covered and every degree from 2
supplied, the winner 3 sits above the ceiling ⌊2·4/3⌋ = 2 and a degree-2
item takes the clock from it, the census reading {2, 3}. At the corner
every ring supply with a degree-2 place ties it, κ(1, 2) = 2 = κ(2, 1),
the source of every degree-2 branch; the curve with no place of
degree 2 or 3 bids 4 against 2 and has one branch. A row whose supply
has nothing at 1 or 3, with degree 2 born covered and 4 the least fresh
degree, ties at κ(2, 2) = 4 = κ(4, 1) and its census reads {2, 4}:
degree 4, the ceiling itself, holds a clock move on one of its branches.

The tie at d_f = 2d_c is the additive index's; at the sum price d + r it
falls at d_f = d_c + 1. Priced by the
multiplicative index, with d = d_c and d_f ≥ 2d, the void bids are
q^(2d) − q^d < q^(2d) − 1 ≤ q^(d_f) − 1 and the born-covered bid wins:
every multiplicative row, at q = 2 on the ladders of ratio 2, 3 and 4
and at q = 3 on the doubling one, reads one branch and no strand.

## The strand
Tier: rule (verified at the 54 rows).
Verifier: limit.py::section_c.

At every b > 2 the void tie still seats an item of degree 2·d_min, the
tie being a notch-1 fact, while the ceiling has fallen below 2·d_min: to
1 over a ring supply with a degree-2 place, and to 3 at b = 3 and 2 at
b = 4 over the designed supply with degree 2 born covered, whose row in
the preceding section has census {2, 4}. On that branch the item is
clocked and then undercut: at b = 3 it re-clocks at 2·d_min·(2T/3)
against the T + 1 door of an unseated rival of degree d_min, priced
d_min·(T + 1), and loses once T > 3. By the chain the rival is the
runaway and the seated item its one strand, at a finite exponent
above 1. Every branch that hands off strands exactly one item: at
b = 3, 2 of 3 branches over F₂[x], 3 of 4 over the genus-2 curve and 2
of 3 over the designed supply; at b = 4, 1 of 2 over each of the three;
none on the ladders of ratio 2 or 3/2. By the chain one deep
coordinate survives every dial the sweep turns; a flat support does
not, since a strand stands above exponent 1.

## What stops the widening
Tier: theorem (the opened-set half); rule (the clock-keyed half, proved
at the corner over a supply holding every degree; checked at 22 branches
over six values of θ, and the opened-set half read at the covering
below 2^d).
Verifier: proof; limit.py::section_k.

An **opened-set** rule, whose extra covered set is a function of the
born and opened degrees holding finitely many degrees at each state,
covers nothing new on a clock move. On a ladder with unbounded gaps and
the chain's price unbounded in the door, where one runaway holds the
clock, over infinitely many supplied degrees not born covered, between
openings the horizon, the least uncovered supplied degree with an item
left to open, keeps a constant price while the recurrent price is
unbounded with the gaps, so the horizon opening undercuts some later
re-clock, and after any step some opening is made: no opened-set rule
stops the widening, however fast it covers. Under the rule that an
opening at d covers every degree below 2^d, the canonical branch,
breaking every tie to a clock move, then the lower degree, then the
lower exponent, opens 2, 4, 16 and 65536 at notches 8, 16, 64 and
262144, where the verifier's supply ends; the next horizon is 2^65536.

A clock-keyed rule can. Under "d is covered while d ≤ θ·T", at the
corner over a supply holding every degree, where the horizon costs at
least ⌊θ·T⌋ + 1, the first clock's holder keeps the clock, so its degree
is the runaway's, D. Before the first clock the notch is 1 and a degree-1
item, born covered and so never opened, bids 2 by a clock at door 2, so
the first holder's degree d₁ is 1 or 2. The first clock lands the holder
at the notch, and each later clock at T + 1 while the notch doubles, so
the holder bids d₁·T/2 at every notch T. A change of holder needs a
lower degree (LIMIT.md#the-chain), and at d₁ = 2 a degree-1 item bids
T + 1 against the holder's T, so D = d₁. The notch is a power of 2 at
least 2 after the first clock, so D·T/2 is an integer and the recurrent
price D·T/2 undercuts the horizon's ⌊θ·T⌋ + 1 iff D·T/2 ≤ θ·T iff
θ ≥ D/2. Below D/2, an opening is taken only at or below the standing
clock price, D·T/4 through the notch T/2, so no degree above D·T/4 has
been opened by the move to T once D·T/4 ≥ 2, and just after the move to
T the horizon is at most max(⌊θ·T⌋, ⌊D·T/4⌋) + 1, under D·T/2 at a
large notch, so an opening follows every late clock move. The widening
stops at the first clock or never, with no T in the condition: at
θ = 1/4 every branch keeps opening, at θ = 1/2 and 3/4 the degree-1
branches stop and the degree-2 branches keep opening, and from θ = 1
every branch stops.