IMAGE — the set of limits a greedy walk reaches

The object: a greedy growth walk under the dynamics demand, every tie followed, over Z, over an imaginary quadratic ring and over the ring of a curve over F₂, in the ideal and the element world. Its greedy image is the set of limits of all its branches. This page asks how big that set is, what the walk must be run for, which of a ring's features reach the count, whether a lock carrying its rider adds a limit or removes one, what the image becomes when every policy is free, and what a policy between greed and freedom keeps of greed's fate. Its verifiers are image.py and free_image.py, with race.py for the widest tie a cubic ring stages, cgreedy.py for the walk between, dam.py for the dam past linear slack, slack.py for a slack polynomial in the tide, track.py for the anchor, envelope.py for the depth dam and the record run's price, choosy.py for the runs that seat less, dambar.py, fwide.py, pstar.py and gapbar.py for the dam's own bar, anchor.py for the run led by its anchors, pointwise.py for the bar as the exact price at each tide, and single.py, crosswide.py and crossall.py for the least multiple of it one run keeps.

A place's colour is what the dynamics reads of it: over a curve its degree in the ideal world and its (degree, class) in the element world, as in ELEMENT; over a number ring its norm and local kind, the (e, f) and width that fix its ladder (CLOCK), with its class in the element world. The supply σ(ω) counts the places of colour ω, LIMIT's count of items per degree read by colour. A limit gives every place an exponent or ∞; a walk that locks repeats one vehicle forever, sending that vehicle's support to ∞ and freezing the rest. The shape of a state or a limit forgets which place of a colour carries which exponent, keeping per colour the multiset of exponents. The rider reach is the union of the supports of the least divisors of every class, the minimal riders MINREP(c), c a class.

The answer, in one line: the image is a sum over the limit shapes a walk reaches of products of multinomials over the supply, so a walk is run only to find its shapes; over Z that is one point, over an imaginary quadratic ring, on every seed run, where the walk locks within 60 moves, a finite number, 2^t in the ideal world, t the ties a trajectory meets, and over a curve with a rational place besides the one removed, at LIMIT's corner, in the ideal world, the continuum; and the lock that carries its rider removes a limit, the one choice it can reach, at the two seeds of Q(√−23) whose tie lies in its support. Free every policy in the ideal world and the ring drops out: the image is one of five forms fixed by the demand and the seed, and the ring reaches it only through the wall. Between the two, over Z, a walk free to take any move within a factor c of the least one is dammed, below a ratio set by the largest least move its states can hold: its support stays on finitely many primes, so its fate is greed's, depth and never breadth, at every c from every seed: that move is at most 2's price, set by the largest 2-adic depth of p − 1 over the state's odd primes p, and at infinitely many e every j·2^e + 1 with j below a given J, which would deepen it, is composite. The same e dam a slack a little past linear, any move up to max(c·m₀, κ·m₀·ln(ln m₀)), m₀ the least move, at every c and κ < 1/4, from every seed. Below c = 11/5 the supports of the factor-c walk's limits from the void (M = 1), with the primes they send to ∞, are computed exactly. A slack polynomial in the least move escapes from the void: once the walk has read the primes up to P, seating each that raises λ, that move is at least the least element order unborn at P, at least a fixed power of P by Linnik's theorem, so a nondecreasing slack at least p(t) on the prime powers, p(t) the least prime ≡ 1 mod t, seats infinitely many primes, and one at least p(t)² reaches breadth. No slack escapes from every seed: an odd prime q held at exponent a with no held prime ≡ 1 mod q^a holds the least move at most q, so f(q) < p(q^a) dams the walk, as from 3^N whenever 3^N ≥ f(3), and an escaping run climbs an infinite chain of primes, each ≡ 1 mod the one before and at most f of it. From the void every escape pays the prime after P from a least move at most the largest a state on the primes up to P can hold, and so p(r^e) from one at most r^(e+1), 2^(e+2) at 2. Up to each least move t some escape moves no further than the first of these prices at t, though none does so at every t: each pays 11251/3373 times it at some t up to 100,000. The walk reading the primes in order is f-greedy exactly when the slack f also pays p(r^e) at r^e for each prime r seated.

The colour count

Tier: theorem.
Verifier: proof; image.py::section_o.

Every permutation π of the places that preserves colour and fixes the seed maps branches to branches. λ of a state, the exponent of its unit group (GROWTH, the dynamics demand), is an lcm of place contributions read off (degree, exponent), or (norm, kind, exponent) over a number ring; a door is the least r with λ_P(a + r) not dividing L, where a is P's exponent, λ_P(b) is λ(P^b) and L is the state's λ; a price is d·r over a curve, d the place's degree, or N^r over a number ring, N the place's norm, joined in the element world by the least rider of class −r·c, c the place's class, its degree m(−r·c) added over a curve and its norm multiplied over a number ring; and a rider is a least divisor of class −r·c, which π, keeping degree or norm and class, sends to a least divisor of the same class. With every least vehicle on the menu, π carries menus to menus (over the curves one least divisor per class, ELEMENT: The minimal rider is unique, and at the two quadratic rings walked; over Q(√−15) two, which π may swap), and the limit map commutes with π, and the image Im(s) of the seed s is a union of orbits. The orbit of a limit X is counted by its shape and the supply:

|Im(s)| = Σ over the limit shapes reached of
          ∏ over colours ω of σ(ω)! / ∏_k n(ω, k)!,

a colour refined by the seed's exponent, σ(ω) then counting the places of the refined colour, and n(ω, k) the places of ω at exponent k in X, the places at 0 included. The walk is run to find the shapes, and the supply prices each. The product of tie multiplicities over a trajectory's openings is the case of one shape whose opened colours carry one place at exponent 1 each, and a tie between two colours separates limits wherever its branches reach different shapes. A count of trajectories is not a count of limits: a limit forgets the order of seating and the finite exponents at the places it sends to ∞.

The verifier checks the count on states, not limits, by brute force from the void, to a finite depth. An engine over every place of degree ≤ 9, λ read as an integer, reaches exactly the shapes of the walker that reads colours alone (limit.py's corner, element.py's element world) at every depth, and every shape's labelled states number exactly its product: F₂[x] and ELEMENT's curves h2 and g2 in the ideal world to depth 9, 9 and 8 (6,480, 12,096 and 4,608 states at the last), h2, h3 and g2 in the element world to depth 11. In the element world the (degree, class) colouring is the coarsest that prices every menu (ELEMENT: The coarsest colouring), and the one coarsening run on the count fails it: grouped by degree alone the element world's states sit off the degree multinomial at every depth of all three rings. A price reading one place's label breaks the count at every depth of F₂[x].

A quadratic ring's ideal image is 2^t

Tier: rule (the tie shape and the conjugate never seated proved; the count verified at Z[√−5] and Q(√−23) over the void and every ideal of norm ≤ 40; the bound on a tie's width attained at one cubic ring).
Verifier: image.py::section_q, race.py::section_width.

A least move P^r costs N(P)^r, a power of P's rational prime, so two of equal cost lie over one prime, and a prime carries two places only when it splits: every tie is a conjugate pair, one colour with σ = 2. Conjugates share a ladder, so if K is the least exponent with λ_P(K) not dividing L, a seated P at exponent a ≥ 1 has door K − a and its unseated conjugate door K: the conjugate is dearer by a factor of at least N(P)^a at every later state, its starvation, and is never seated; a conjugate seated shallower, at b < a, stands dearer by a factor N(P)^(a−b) and never moves. Every seed run reaches a lock, read as 20 repeated tie-free moves within 60, the lock of GROWTH: The lock over a number ring taken as final there, and before it a trajectory makes finitely many ties t, the same t on every branch by the colour count, and the two members of each stand at different exponents in every limit. The image is one shape, of 2^t limits. Its symmetry is a swap at each split prime separately, so it is a single Galois orbit only when t ≤ 1, and then only at a Galois-stable seed. Over a field of degree n every tie still lies over one rational prime, which carries at most n places, so a tie is at most n wide, and the bound is reached: at x³ − x + 3, discriminant −239, 3 splits completely, its three places each complete to Q₃ and share a ladder, and they tie at the void; the winner runs at price 3, and once it is seated λ = 2, which each partner's N − 1 = 2 divides, so the partners' doors rise to 2, price 9 and more, and they are never seated, one shape of 3!/(1!·2!) = 3 limits.

At Z[√−5] t is 0, 1 and 2 at 35, 15 and 2 of 52 seeds, and at Q(√−23) 0 and 1 at 65 and 14 of 79: every seed's branches share one t, its limits number 2^t in one shape, and the starvation held at every state visited, 4,028 checks of strict dearness.

The lock's rider erases a choice

Tier: rule (verified at the void and every principal ideal of norm ≤ 40: 26 seeds at Z[√−5] and 27 at Q(√−23)).
Verifier: image.py::section_q, element_ring.py::section_walks.

In the element world conjugates carry inverse classes, so a conjugate pair is one colour where its class is its own inverse and two colours where it is not. At Z[√−5], of class number h = 2, the ten seeds with a tie reach two limits each in one shape. At Q(√−23), h = 3, four seeds reach two limits in two shapes, and two reach two lock-time states and one limit. Those two are the seeds (2) and (2)², and their two states differ at the two places over 2 and nowhere else. The lock vehicle is (2) = P₂P₂′, a core at door 1 and its rider (ELEMENT: The lock carries its rider), which sends both places to ∞: which of the two stands deeper is a finite gap the state keeps forever and the limit forgets. So the rider adds no limit and erases one, and on the seeds run the count is 2^(t − ε), ε the ties whose two members both lie in a rider-carrying lock's support. Z[√−5]'s lock (2) is the ramified place squared with no rider, and Q(√−23)'s inert lock (5), at 3 of its 27 seeds on the single walk, carries one place: neither erases anything. Each of the 1,599 element menus the branched walks read (image.py, every branch followed, a count apart from the 3,180 states of the single walks of ELEMENT: The number rings price in (min, ×)) equals the ring's own least raising elements.

A curve's image is the continuum

Tier: rule (proved from LIMIT's ideal limit at the corner and ELEMENT's element world, both rules; the element half at every ring whose principal places number two or more at all but finitely many degrees, which six rings meet at every degree above 2 opened past the transient in 300 moves, the hypothesis read only at the degrees the run opens).
Verifier: image.py::section_c, image.py::section_e.

In the ideal world at the corner, over a curve holding a rational place besides the one removed (all six run), every branch's limit is ∞ at one runaway C of degree 1 or 2 and one place at exponent 1 at each other opened degree, every supplied degree above 1 opening, and none rational when C has degree 2 (LIMIT: The ideal limit). So the shapes are two at most, and

|Im| = σ(1) ∏_(d ≥ 2) σ(d)  +  σ(2) ∏_(d ≥ 3) σ(d),

each product over the supplied degrees, σ(d) > 0, the second present exactly when σ(2) > 0, as the void menu then ties. Each product has infinitely many factors of 2 or more, so each shape's orbit is the continuum and its points are choice functions on the degrees: the runaway, then one place per degree. The runaway degrees over every branch followed, every tie branched over the first 10 moves, are {1, 2} at the five rings with a degree-2 place and {1} at h5, which has none; the runaway and the first ten openings of the degree-1 shape already offer 10^11.6 at F₂[x] and 10^11.6 to 10^17.0 over the curves. No lock is needed for a coordinate to be decided: every coordinate of this limit is written down.

In the element world, past the transient, the walk to its last opening that pays a rider, an opening at degree d takes a principal core and pays no rider (ELEMENT: Greed pays no avoidable rider), so it chooses one of σ(d, 0) places. A declined one is never seated later: as a core it is raised by a clock move (LIMIT), T the notch, at d(T + 1) against its seated sibling's d·T at most, neither paying a rider, and as a rider it cannot come, a rider's place being alone in its colour. The recurrent price doubles while an opening costs its degree, so the openings never stop, and every limit carries each opened colour outside the rider reach as one place at a positive exponent and the rest at 0: a factor σ(d, 0) per degree opened past the transient. Over every branch followed at the six rings, ties branched over the first 8 moves and each branch walked to 300, every opening past the transient chose from two or more principal places but F₂[x]'s one of degree 2, and every opened colour outside the reach ended with one seated place. The reach is finite, since the class group is and each class has finitely many least divisors: h − 1 rational places over an elliptic curve and six places over g2.

The gap

Tier: rule (the point proved, the continuum as the curve section's rule, the finite count verified at the two quadratic rings on the seeds run).
Verifier: growth.py::section_c, image.py::section_q.

Over Z least moves at distinct primes are powers of distinct primes and never tie (GROWTH: The lock-prime law), so every branch is the one walk and the image is a point. Over an imaginary quadratic ring it is a finite number wherever the walk locks, as every seed run does within 60 moves, and over a curve with a rational place besides the one removed, at LIMIT's corner, the continuum; neither is countably infinite. Over the curve, however many choices the walk loses to erasure, a limit forgetting the order of seating and the finite history of a place it sends to ∞, one shape's orbit is already the continuum, so the count cannot fall to the countable.

The free image has five forms

Tier: theorem.
Verifier: proof; free_image.py::section_k1, free_image.py::section_k2, free_image.py::section_k3.

Let a policy take any move its demand admits, the demand one of the five named below: transparency, independence, new idempotents and dynamics as GROWTH defines them, and semisimplicity, which asks that the new modulus be squarefree. The free image of a demand and a seed s is the set of limits of every maximal run from s. In the ideal world, a move multiplying the state by any proper ideal, several places at once allowed, over every ring this page walks and any Dedekind ring with finite residue fields, finitely many places of each norm and infinitely many places (the rings of integers away from finitely many places of any global field), it takes one of five forms, W(L) the wall, for any L the maximum of the states whose λ divides L (shown below). A limit is finite when its support is finite and every exponent finite, that is when it is a state, and X ≥ s means every exponent of X is at least the seed's:

transparency     the one finite point W(λ(s));
semisimplicity   s·∏_{P∈S} P, S infinite and off supp(s), at a
                 squarefree seed; the point s at any other;
independence     s·∏_{P∈S} P^(a_P), S infinite and off supp(s),
                 every a_P finite;
new idempotents  every X ≥ s of infinite support;
dynamics         every X ≥ s that is not finite.

Closure: a coprime move carries no seated place, so the seed's and each seated exponent are final, and a coprime move always exists, the places being infinitely many, so S is infinite; a squarefree product admits no second copy; a new idempotent needs a new place, so the support is infinite; and a dynamics run never halts, a place P off the current state M with λ_P(1) > λ(M) always existing since finitely many places have bounded size, while infinitely many non-unit moves cannot converge to a finite limit. λ(lcm) = lcm(λ) and λ is monotone under divisibility, so {M : λ(M) | L} is closed under lcm, and finite, since finitely many places have N(P) − 1 dividing L and each ladder is unbounded (GROWTH: The module law); its maximum is W(L) (GROWTH: The three fates over Z, GROWTH: The wall over F₂[x]); a transparent run from s stays among the multiples of s dividing W(λ(s)), and any smaller state admits a place of the quotient, so every run ends at W. Reach, the places listed by norm and ties in a fixed order: seat the least unseated support place each step at its target exponent (1 where the target is ∞), bring one seated place short of its finite target up to it (seed places included), raise every seated place bound for ∞ by one, several places making one move, and under dynamics add a pad when λ has not risen: an unseated support place Q with λ_Q(1) not dividing λ, seated at its target exponent (1 where the target is ∞), which exists when the support is infinite; or else the least place bound for ∞, seated or not, pushed up its unbounded ladder until λ rises, which exists since a target of finite support in the image is not finite. The least unseated place rises strictly, so every support place is seated at a finite step, and from then rises at every step or reaches its finite target within finitely many. At a squarefree seed the last four nest strictly, witnessed by exponents 1, 2, 3 on an infinite support, one infinite exponent on an infinite support whose other exponents are 1, and s·P^∞, one infinite exponent on a finite support; the transparency point lies in none.

The verifier runs the closure over Z and F₂[x] on a battery of 14 states and 12 states against every move to a cap, 8,634 state-move pairs: the zeros of the three demands that do not read λ, independence, semisimplicity and new idempotents, are their definitions read back, and its content is that every battery state has a dynamics move. It runs the transparent reach from four seeds per ring, 2, 6, 12, 30 and 1, x, x², x² + x + 1, exact at all eight; and the reach construction from the void and a squarefree seed: 56 constructions clean for 12 steps, every step admissible (the construction builds each below its target and rising), every place off the seed at ∞ and, from the squarefree seed, its places raised to 3 among them, 3,124 dynamics targets of finite support, each with an infinite exponent, run 12 steps, every place bound for ∞ rising at every step after its seating, and each of eight out-of-image targets stalling, its step and reason printed.

The free image forgets the ring

Tier: theorem.
Verifier: proof; free_image.py::section_k4.

Every image but transparency's is described by conditions on the seed's places and, off them, on the multiset of exponents alone, every place taken as one colour. So each is a union of orbits of G_s, the permutations of all places fixing the seed's exponents: the colour count's symmetry with the coarsest colouring there is, every place alike, whose supply is infinite, so the orbit is no number to count; and each of these four holds a continuum orbit, but semisimplicity's at a seed that is not squarefree, the point s. An image can have more symmetry than its demand. Dynamics reads λ_P, so a permutation breaking the ring's colouring changes which moves it admits, at 161 of the 5,586 (state, move) pairs tested over Z and 96 of the 3,048 over F₂[x], and two degree-preserving ones over F₂[x] at none; yet every dynamics target with such a pair exchanged is reached, dynamics' image being a union of orbits of G_s, and the verifier runs the construction toward six such targets from two seeds per ring, one exchanged pair each, three of the six moved by the exchange, every construction completing 12 steps. The one image that reads the ring is transparency's point, the wall, where the ring's arithmetic sits: Adams' number over Z (GROWTH: The wall is Adams' number), a product of whole degree classes of irreducibles over F₂[x] (GROWTH: The wall over F₂[x]). The ring reaches the greedy image through its colours and the free image through the wall, and nowhere else.

The fates are the image's extremes

Tier: property.
Verifier: proof; free_image.py::section_k5.

A limit in a free image has two coordinates, the support and the exponents. BREADTH is the support every place and DEPTH an exponent infinite, each a face, extreme in one coordinate with the other free; MORTALITY, a finite limit, is where both bottom out. Every move at least doubles the size, so a run from a seed dies exactly when its limit is finite. Breadth and depth hold together at 2^∞ times every odd prime; a member missing infinitely many places with every exponent finite holds no fate; and new idempotents and dynamics hold the same fates as different images, so fates do not determine an image. A finite reachable set puts a demand at MORTALITY, since no run can repeat a state; a ban on depth does not: fresh dynamics, which admits only a new place lifting λ, never deepens and never dies, such a place always existing. The verifier runs it 20 steps from four seeds, every place it seats at exponent 1 by definition, and over F₂[x] no two of them of one degree, which holds by construction: two places of one degree share λ_P(1) = 2^d − 1.

Greed's point

Tier: rule (resting on GROWTH's lock rules, this page's curve rule and the five forms; the places at ∞ read at the 27 principal seeds of norm ≤ 40 of Q(√−23)).
Verifier: free_image.py::section_c, free_image.py::section_k3, image.py::section_c, image.py::section_q, element_ring.py::section_walks.

Where a greedy walk locks, over Z and over an imaginary quadratic ring on the seeds run (GROWTH: The lock-prime law, GROWTH: The lock over a number ring), its limit has a finite support and its lock vehicle's places at infinity, one place over Z, and at Q(√−23) in the element world both places over 2 at 24 of 27 seeds on the single walk with one tie-break, an element-world run being an ideal-world run whose moves are principal: a member of the dynamics image, finite support with an infinite exponent, that no smaller image holds. Over Z greedy dynamics runs from the void to 3^∞, while a free policy reaches 2^∞ by refusing the least move, 4 lifting λ where 2 does not: that the walk never sends 2 to ∞ is greed's choice, not the demand's. Where the walk sprawls, over a curve over F₂ at LIMIT's corner, its limit keeps the infinite exponent and seats one place per opened colour outside a finite rider reach (the curve section above), a support infinite and missing infinitely many places, interior in its coordinate. Both hold the depth fate alone; only the support separates them. Over Z the walk is deterministic only because least moves at distinct primes never tie; a tie-break is needed wherever moves share a cost, and the greedy image prices it, nothing over Z, a factor of two per surviving tie over the two quadratic rings walked, the continuum over F₂[x].

The c-greedy walk is dammed

Tier: theorem (the dam, the striking rule for the tide ceiling, the feeder formula and the empty seating's bound on P*, τ(P) ≤ 2^(W(P)+3), every c dammed from every seed, every slack at most max(ct, κ·t·ln(ln t)), κ < 1/4, dammed from every seed); rule (τ(P) and the dam ratio verified at every prime P ≤ 10,000; the limit patterns of the image from the void verified exact for every c in [1, 11/5); the least k with k·2^e + 1 prime, and the ratio it proves, at every e ≤ 600; the lower dam's steps verified at J ≤ 40 and x ≤ 2000); observation (the record tides' form, at four tides).
Verifier: proof; cgreedy.py::section_tau, cgreedy.py::section_dams, cgreedy.py::section_census, cgreedy.py::section_two, dam.py::section_dam, pstar.py::main.

Over Z, a class of policies between greed and freedom: the c-greedy walk, c ≥ 1, takes at a state M any m ≥ 2 that raises λ with m ≤ c·m₀(M), where the tide m₀(M) is the least such m. Composite moves are allowed, so one move may seat several primes. c = 1 is the greedy walk, and the image I_c(s) of every run from s grows with c, inside the free dynamics image. Wider still, the f-greedy walk, f nondecreasing with f(t) ≥ t, takes any λ-raising m ≤ f(m₀(M)), and the c-greedy walk is f(t) = ct. The tide is a prime power, the least of the prices q^door(q) over the primes q (the door lemma, GROWTH: The lock-prime law). Write P⁺ for the prime after P and τ(P), the tide ceiling, for the largest tide of a state whose primes are all at most P; τ(P) ≤ P from P = 5 on, since a state's largest odd prime p has price p, no smaller prime being 1 mod p, and a power of 2 has tide at most 4.

The dam. If f(τ(P)) < P⁺, every f-greedy run from a seed whose primes are at most P keeps them at most P: at each state the tide is at most τ(P), so every move is at most f(τ(P)), below P⁺, and brings in no prime past P, and the claim follows by induction along the run. The support stays finite while M grows without bound, so some exponent goes to ∞: the limit holds DEPTH and never BREADTH. A walk restricted to prime-power moves runs a subset of these runs and is dammed with them. With the dam ratio ρ(P) = P⁺/τ(P), a seed is dammed at every c < ρ(P) for any prime P at or past its largest prime.

The striking rule. τ(P) is the largest prime power T > 4 that some state on the primes ≤ P holds as a tide, every price at least T, or 4, the tide of the state 4, when there is none, and it is found without a search, and by a formula below. Fix T, let k_ℓ be least with ℓ^(k_ℓ) ≥ T, and v_ℓ the ℓ-adic valuation. A state with every price at least T stays one, with λ's part below T only larger, when: every prime of [T, P] is seated once, its own price being at least T in any case and its only part of λ below T being p − 1; 2 is unseated, since seated at a its price is at most 2^(v + 3 − a), v = v₂(λ), and its own part of λ tops the rest of λ's 2-part only where that price is at most 4, so unseating it keeps λ and raises its price to 2^(v + 3); and each odd ℓ < T sits at exponent 0 or 1, since seated at a its price ℓ^(v − a + 2), v = v_ℓ(λ), reaches T only when v ≥ a + k_ℓ − 2 > a − 1, so the other primes' p − 1 supply λ's ℓ-part and a = 1 keeps λ at a higher price. Each change keeps λ's part at the primes below T or enlarges it, the only part a price below T reads, and a larger such part only raises every price of a prime below T, a prime at or past T pricing at least T in any case, so none loses the tide. With S the seated odd primes below T and N the lcm of p − 1 over S and the primes of [T, P], λ = N, and every price is at least T exactly when

ℓ in S             v_ℓ(N) ≥ k_ℓ − 1        (price ℓ^(v+1));
odd ℓ < T, not S   (ℓ − 1) | N and          (price ℓ^(v+2),
                   v_ℓ(N) ≥ k_ℓ − 2          ℓ if (ℓ − 1) ∤ N);
2                  v₂(N) ≥ max(1, k₂ − 3)   (price 2^(v+3), 4 at
                                             odd N).

Every bar is monotone in N and N(S ∪ S′) = lcm(N(S), N(S′)), so the feasible sets are closed under union and the greatest, G, exists when any does. Striking finds it: start from every odd prime below T and at most P, strike each member whose seated bar fails under the current N, and repeat to a fixpoint X. G is never struck, N(G) dividing the current N at every stage, so G ⊆ X, and X meets every seated bar. If X meets the unseated bars and 2's, X = G; if it fails one, so does every subset of X, a prime outside X being unseated there too while N only shrinks, and no state on the primes ≤ P has tide T.

The rule gives τ(2) = τ(3) = 4, τ(5) = τ(7) = 5, τ(11) = 7 and τ(13) = 11, each equal to the largest tide over a box of states on the same primes (exponents to 6, 4 and 3 at 2, 3 and 5, to 2 elsewhere), and at every prime P ≤ 10,000 a state built from G, at the prime where τ last rose, has tide exactly τ(P) by the door formulas, so the ceiling is attained. τ(P)/P trends down, 0.636 at 11, 0.465 at 101, 0.225 at 1009, 0.183 at 9973, and ρ's last record at each of the tides 5, 17, 107, 227 and 839 climbs, 11/5 = 2.200 at P = 7, 97/17 = 5.706 at 89, 643/107 = 6.009 at 641, 1583/227 = 6.974 at 1579 and 5867/839 = 6.993 at 5861, with no record after it to 10,000, and the least dam is at P = 7 for c = 2, 47 for 3, 67 for 4, 83 for 5 and 641 for 6. Observed at the four record tides d past 5, d − 1 is 2⁴ or 2r with r prime, and d is passed first at a prime 1 mod 16, d or r: 17 at 97 = 6·16 + 1, 107 at 643 = 6·107 + 1, 227 at 1583 = 14·113 + 1 and 839 at 5867 = 14·419 + 1.

The feeder formula. The fixpoint unrolls into one round. Fix the prime power T > 4, write p_T(s) for the least prime ≡ 1 mod s at least T, and for s = r^j < T call the primes below T that are ≡ 1 mod s the feeders of s; an odd prime p < T is unfed when p^(k_p − 1) has none, so that only a prime of [T, P] meets its seated bar. The least P at which some state on the primes ≤ P has every price at least T is P*(T), the largest of p_T(s) over the forced s without feeders, s = 2^max(1, k₂ − 3) and ℓ^(k_ℓ − 2) at odd ℓ with k_ℓ ≥ 3, and of min(p_T(s), p_T(p^(k_p − 1)) over the feeders p of s) over the s that have feeders, all unfed. So τ(P) is the largest prime power T > 4 with P*(T) ≤ P, or 4 when there is none. Proof: put ν(s) = min(p_T(s), ν(p^(k_p − 1)) over the feeders p of s), well founded as a feeder p exceeds s. A prime of [T, P] or a seated feeder of r^j is what puts r^j in N, and every bar reads v_r(N) only up to k_r − 1, where r^(k_r − 1) < T. Striking at P keeps exactly the p with P ≥ ν(p^(k_p − 1)): downward from the largest, p's seated bar is met by a prime of [T, P] or by a kept feeder of p^(k_p − 1), larger than p, and the set so kept meets every seated bar, so lies in X, while X lies in it by the same induction. So v_r(N) ≥ j exactly when P ≥ ν(r^j). An odd ℓ < T meets its unseated bar exactly when P ≥ ν(s) at each prime power s exactly dividing ℓ − 1 and at s = ℓ^(k_ℓ − 2) when k_ℓ ≥ 3, and a kept ℓ meets these already, each being at most ν(ℓ^(k_ℓ − 1)); 2's bar asks P ≥ ν(2^max(1, k₂ − 3)). ν only grows along divisibility, and a feeder ℓ of s puts s under the power exactly dividing ℓ − 1, so P*(T) is the largest ν(s) over the s with feeders and the forced s without. At a feeder p that is not unfed, ν(s) ≤ ν(p^(k_p − 1)), whose feeders are larger than p; following them ends at an s whose feeders are all unfed, where ν(s) is the minimum above outright. One case is a Sophie Germain pair ℓ = 2r + 1, r odd, with (T + 1)/2 < ℓ < T: r's only feeder is ℓ, as 2rm + 1 ≥ 4r + 1 > T at m ≥ 2, and ℓ is unfed, so P*(T) ≥ min(p_T(ℓ), p_T(r)). Computed by striking and a search over the p_T values, P* equals the formula at the 1278 prime powers T that follow some t in 4..10,000 and at 36 drawn between 5639 and 1.62·10^6, seven of them among those; one-member feeder sets alone fall short at T = 2243..2663, where 373's feeders are 1493 and 2239, both unfed, and P*(2243) = p_T(373) = 15,667. The empty seating, no odd prime below T seated, bounds P*: its bars ask only for a prime of [T, P] ≡ 1 mod each prime power s exactly dividing some ℓ − 1, ℓ an odd prime below T, mod ℓ^(k_ℓ − 2) at k_ℓ ≥ 3 and mod 2^max(1, k₂ − 3), so P*(T) is at most the largest p_T(s) over these s, the moduli it reads. Each has φ(s) < T/2, φ Euler's totient: an odd s exactly dividing ℓ − 1 = s·m has m even, φ(2^j) = 2^(j − 1) with 2^j < T, and ℓ^(k_ℓ − 2) < T/ℓ.

Every c is dammed. A tide is at most 2's price, and 2's price is bounded through one valuation, v = v₂(λ): unseated at odd M > 1 it is 2^(v + 3), the 2-door v + 3 of GROWTH: The two-adic prices, and seated at a it is at most 2^(v + 3 − a), as in the striking rule. With w the largest v₂(p − 1) over a state's odd primes, v is w unless 2's own part tops it, at v = a − 2 with a ≥ 3 and a price at most 2, so 2's price is at most 2^(w + 3), equal to it at odd M > 1 and an inequality where 2 is seated, and at most 4 on a power of 2. Hence τ(P) ≤ 2^(W(P)+3) for P ≥ 3, W(P) the largest v₂(p − 1) over the odd primes p ≤ P. If j·2^e + 1 is composite for every j ≤ J, let P be the largest prime ≤ J·2^e: a prime p ≤ P with 2^e | p − 1 would be j·2^e + 1 with j < J, so W(P) ≤ e − 1, τ(P) ≤ 2^(e+2) and P⁺ > J·2^e, and ρ(P) > J/4. Such e exist for every J, infinitely many: with L_J the lcm of the orders of 2 modulo 2j + 1 over j ≤ J, every e = 1 + t·L_J, t ≥ 1, makes 2j + 1 a proper divisor of j·2^e + 1 = 2j·2^(e−1) + 1. So ρ is unbounded with no prime-pattern hypothesis, and every c-greedy walk over Z, from every seed and at every c, is dammed: its fate is greed's, depth and never breadth. The construction is astronomical, L₂₉ = 240,360,120 placing ρ > 7.25 at a P near 10^72,355,607.6. The Proth route does better: the least k with k·2^e + 1 prime gives ρ > k/4 at the prime before it, by the same count, and its records over e ≤ 600, each prime proved (Miller–Rabin deterministic below 3·10^24, Proth's theorem above), give ρ > 8.75 at a P near 10^10.9 (k = 35 at e = 31), the first past 5867/839, and ρ > 224 near 10^130.9 (k = 896 at e = 425). 2's bar is not what holds the tides of the table: dropped from the striking rule, it changes τ at no prime P ≤ 10,000, the bound 2^(W(P)+3) standing at least 1.377 times above τ(P) there (first at P = 7547).

Past every linear slack. The same covering dams a slack a little past every linear one. The order of 2 modulo n = 2j + 1 divides Carmichael's λ(n), which is even with λ(n)/2 ≤ j, so L_J divides 2·lcm(1..J); and lcm(1..x) ≤ 4^(x−1) at every x ≥ 1, since lcm(1..x)/lcm(1..⌈x/2⌉) is the product of the primes with a power in (⌈x/2⌉, x], each dividing C(x, ⌊x/2⌋) ≤ 4^⌊x/2⌋ (its Legendre term at that power is 1), and the steps ⌊x/2⌋ telescope to x − 1. At e = 1 + L_J the dam holds when f(2^(e+2)) ≤ J·2^e, which g(t) = max(ct, κ·t·ln(ln t)) meets when 4c ≤ J and κ·ln((L_J + 3)·ln 2) ≤ J/4; as (L_J + 3)·ln 2 < 4^J, κ = 1/(8 ln 2) meets it at every J ≥ 4. So every f-greedy walk with f ≤ g past some t, at any c and κ ≤ 1/(8 ln 2), is dammed from every seed, at the largest prime below J·2^(1+L_J) for every large J, and Every c is dammed is its case κ = 0. With ψ(J) = ln lcm(1..J), ln((L_J + 3)·ln 2) < ln 3 + ψ(J) at J ≥ 3, and ψ(J) ~ J by the prime number theorem (known: Hadamard and de la Vallée Poussin, 1896), so every κ < 1/4 is dammed the same way at every large J. The steps are checked at J ≤ 40 and x ≤ 2000. The largest κ the covering admits at one J, J/(4·ln((L_J + 3)·ln 2)), runs from 0.331 (J = 5) to 0.558 (J = 22) over 4 ≤ J ≤ 40, but each J dams only the seeds below its one P: a κ at or past 1/4 is dammed from every seed this way only if ln((L_J + 3)·ln 2) ≤ J/(4κ) at infinitely many J, which is not proved, and ln L_J/J lies in 0.51 to 0.82 at the twelve J the run tables.

The image from the void. Below c = 11/5 the dam at 7 keeps every state on 2, 3, 5 and 7. There the dynamics reads an exponent a only up to a cap B_p = ⌊log_p 1000⌋ + 3: past it, p's part of λ, a − 1 (a − 2 at 2), exceeds v_p(r − 1) for every prime r ≤ 1000, so p's price stays p and a move raises λ at p exactly when p divides it. A state and its capped state share tide and moves, and the capped states form a finite graph. Every move raises a capped exponent or leaves the capped state fixed, so a run climbs to one capped state and then takes its self-loops, moves on its capped primes alone, forever: its limit sends a nonempty union of self-loop supports to ∞, keeps the other capped primes at a finite exponent at or past the cap, and the rest as they stand, and every such union is reached by cycling. The graph changes with c only where c·m₀ crosses an integer, at c = m/m₀ with m₀ a tide of a state on 2, 3, 5, 7, so in {2, 3, 4, 5}, and the census runs at every such point in [1, 11/5). A limit pattern is a limit's support with the primes it sends to ∞; the patterns there are exact, and they arrive at eight values of c:

c      patterns arriving (support | at ∞)
1      3 | 3
4/3    2 | 2;  2,5 | 5
7/5    2,5,7 | 5;  2,5,7 | 5,7;  2,5,7 | 7
8/5    2,5 | 2;  2,5,7 | 2
5/3    3,5 | 3;  3,5 | 3,5;  3,5 | 5;  5 | 5;  5,7 | 5;
       5,7 | 5,7;  5,7 | 7
7/4    2,7 | 2;  2,7 | 7
9/5    2,3,5 | 3;  2,3,5 | 3,5;  2,3,5 | 5
2      2,3 | 2;  2,3 | 2,3;  2,3 | 3;  2,3,5 | 2;  2,3,5 | 2,3

At c = 4/3 the image is exactly 3^∞, 2^∞ and 2²·5^∞: the c-greedy walk reaches 2^∞, the limit greed never reaches from the void (Greed's point), from c = 4/3 on. No pattern below 11/5 holds both 3 and 7. The dam at 7 is sharp at its own ratio: at c = 11/5 a state on 2, 3, 5, 7 with tide 5 admits 11, and the census seats primes to 23, completing under the next dam, ρ(37) = 41/17. The census merges a pattern's finite exponents past the cap, so how many limits a pattern carries is not computed; only the patterns and their thresholds are law.

Polynomial slack reaches breadth from the void

Tier: theorem (the floor τ(P) ≥ β(P), the escape, breadth, the depth dam, the anchor, its chain, the slack below p(q) dammed from every seed, the least slack past the dam dammed from the void at 23, the record run's price max(S, B) ≤ E ≤ max(S, B, t), every p(s) a record seat, B ≤ F ≤ U, the anchor bar, the pointwise price E_min = F, the crossing and the held crossing); rule (the inequalities the floor, the escape and breadth lean on, verified at every prime 3 ≤ P ≤ 10,000 and at every move of the riding run whose seat is at most 10,000; the anchor's m₀ ≤ q and its edges over the capped walks stated below; E = S, max(S, B) ≥ t at 3 ≤ t ≤ 10,000, the readings of E, S, B and U and the two runs that seat less, verified at every t ≤ 10,000; F's readings against B, at every 4 ≤ t ≤ 1822, and against the runs and p(q), and the pointwise price's construction, at every 4 ≤ t ≤ 10,000; c₀ ≥ 11251/3373, by the crossing test over its 1922 crossings, and the crossing-led run within 11251/3373 F, to tide 100,000, those 1922 passing at the bound at the horizon 10^6, and at the horizon 10,000 the bound 739/233 by its 309 crossings and the run within 739/233 F to tide 10,000); observation (min{p(q) : q ≥ t} at t ≤ 10,000).
Verifier: proof; slack.py::section_record, slack.py::section_riding, track.py::section_anchor, track.py::main, envelope.py::main, choosy.py::main, dambar.py::main, fwide.py::main, anchor.py::main, pointwise.py::main, single.py::main, crosswide.py::main, crossall.py::main.

Over Z, widen the slack of the f-greedy walk (The c-greedy walk is dammed) further. Write p(t) for the least prime ≡ 1 mod t, and β(P) for the least prime power dividing no p − 1 with p ≤ P prime: the least element order not yet born at P's rung (TOWER: The birthday formula), so p(β(P)) > P.

The floor. The record run reads the odd primes in increasing order from the void and seats, once, each p whose p − 1 does not divide λ; a skipped prime's p − 1 divides λ already. Once every prime ≤ P is read, its state M_P has λ = lcm(p − 1 : p ≤ P), and every price there is a prime power q^e with e > v = v_q(λ): 2^(v+3) at 2, unseated; q^(v+1) at a seated odd q and q^(v+2) at a skipped one; at least the prime itself past P, a prime dividing no p − 1 with p ≤ P. None divides λ, so the tide is at least β(P), and τ(P) ≥ β(P), a floor under the dam's ceiling 2^(W(P)+3). Linnik's theorem (Mat. Sb. 15 (1944) 139–178) gives p(m) ≤ C·m^L at every m ≥ 2 for constants C and L; as p(β(P)) > P, β(P) > (P/C)^(1/L), so the tide ceiling grows at least as a fixed power of P and ρ(P) ≤ P⁺/β(P) ≪ P^(1 − 1/L), past the trivial ρ(P) ≤ P⁺/4.

The escape. If f(t) ≥ p(t) at every prime power t, the record run is an f-greedy run. Its next seat R(P) is the least prime past P whose R − 1 does not divide λ, and p(β(P)) is one, so R(P) ≤ p(β) ≤ f(β) ≤ f(m₀(M_P)). It seats infinitely many primes, each once: its support is infinite and no exponent grows, so it escapes the dam, holding neither depth nor breadth, 2 and the skipped primes staying out. f(t) = C·t^L qualifies, and under the generalized Riemann hypothesis, where p(m) ≪ (m log m)² (Bach and Sorenson, Math. Comp. 65 (1996) 1717–1735), a slack of that order does.

Every place. If f(t) ≥ p(t)² at every prime power t, an f-greedy walk from the void reaches breadth. The riding run moves 4 first, λ rising from 1 to 2 at 4 ≤ f(3); it then reads the primes as the record run does from λ = 2, 3 skipped, moving each prime R it seats with the oldest skipped prime q unseated, m = R·q, or R alone when none waits. A rider leaves λ unchanged, q − 1 dividing λ, so λ stays lcm(p − 1 : p ≤ P) and the move raises it through R. 2 at exponent 2 prices 2^(v+1) and a seated rider q^(v+1): every price is still a prime power not dividing λ, the tide at least β, and m < R² ≤ p(β)² ≤ f(m₀). Its seats never end and each carries the oldest waiting prime, if any, so every skipped prime rides at a finite step: the limit gives 2 the exponent 2 and every odd prime the exponent 1, breadth without depth. By Linnik f(t) = C²·t^(2L) qualifies. So a policy strictly between greed and freedom changes greed's fate: from the void a slack polynomial in the tide reaches the free image's breadth, which every c-greedy walk misses.

At every prime 3 ≤ P ≤ 10,000 the computed tide at M_P and τ(P) are at least β(P), and R(P) ≤ p(β(P)); τ(P)/β(P) is 5.05 at P = 9973, where β = 361 = 19². Along the riding run's 302 moves, the move 4 and the seats 5 to 9887, every move raises λ, its rider is below its seat with q − 1 | λ, the tide before it is at least β, and m ≤ p(β)²; at the time every prime ≤ 10,000 is read, 631 skipped primes wait, the least 2897. The record run's slack R/m₀ reaches 27.7 at 9973, close to P/β by construction: it does not locate the least slack that escapes.

The depth dam. Write u_r for the largest v_r(p − 1) over the held primes p (odd p at r = 2). The prices bound the tide by them: m₀ is at most r^(u_r+2) at an unheld odd r, r^max(1, u_r−a+2) at r held at exponent a, and 2^(u₂+3) at 2 (4 before an odd prime). A tide that stays at most T keeps every move at most f(T) and the support finite, so an escaping run's tide is unbounded, and with it every u_r. Each e past the seed's u_r is then crossed by one move, which brings a prime ≡ 1 mod r^e, at least p(r^e), from a state with u_r ≤ e − 1, whose tide is at most r^(e+1) at odd r (r^max(1, e−a+1) if r is held at exponent a) and 2^(e+2) at 2. So from the void every escaping run has f(r^(e+1)) ≥ p(r^e) at every odd prime r and f(2^(e+2)) ≥ p(2^e), at every e ≥ 1, and one pair failing dams every run from the void. Write B(t) for the largest p(r^e) whose point, r^(e+1) or 2^(e+2), is at most t: every slack under which a run escapes from the void is at least B. It is carried by small r at depth: B(t)/t is largest at B(2048) = p(2⁹) = 7681, 3.7505, over t ≤ 10,000, and B(10,000) = 17,497. The depth dam is the dam read at the prime powers, the anchor its held case at e = a. Write the dam's bar F(t) = max{P⁺ : τ(P) ≤ t} at t ≥ 4, 5 below 4: by the dam every slack under which a run escapes from the void is at least F there, and F is the least nondecreasing slack with f(τ(P)) ≥ P⁺ at every P. So F(t) = P*(T⁺(t)), which the feeder formula computes, T⁺(t) the least prime power past t, the largest P with τ(P) ≤ t being the prime before it, as P* only grows with T. Take P the prime below p(r^e): a state on the primes ≤ P holds no prime ≡ 1 mod r^e, so u_r ≤ e − 1 and the prices hold its tide at most the point, so τ(P) ≤ r^(e+1) (2^(e+2) at 2) and F there is at least P⁺ = p(r^e). So F ≥ B at every t. At every prime power s with p(s) ≤ 9973, τ at the prime below p(s) is at most s's point, and on 100 ≤ t ≤ 1822, where τ's table to 10,000 fixes F, F > B at every t, F/B largest 15.3249 at t = 503 (F = 3019, B = 197).

The anchor. Neither the escape nor breadth holds from every seed, and the held case says why. Call a held prime ≡ 1 mod q^a an anchor of q at depth a. If a state holds an odd prime q at exponent exactly a and no anchor of q at depth a, q's own part, a − 1, is the whole of v_q(λ), the other primes giving v_q(p − 1) < a; so the move q raises λ and m₀ ≤ q. Every move is then at most f(q), and an anchor is at least p(q^a), so if f(q) < p(q^a) none ever enters: q's exponent may grow, but an anchor at the new depth would be one at depth a, m₀ stays at most q, and the run's primes stay at most the larger of the state's and f(q). The run is dammed. At the seed 3^N, q = 3 and a = N, and 3^N ≥ f(3) puts f(3) below p(3^N) > 3^N: every f-greedy walk from such a seed is dammed, whatever f.

So a run escapes only if f(q) ≥ p(q^a) at every state, for every odd q it holds at exponent a with no anchor at depth a. It then climbs an infinite anchor chain. Take q₀ past the seed's primes, the largest of the support right after the move that brings it in (an escaping run has infinitely many such), and q_(i+1) the first anchor of q_i at depth 1 to enter. Each exists, or q_i is never anchored and the run is dammed at f(q_i). Every q_i with i ≥ 1 exceeds q₀, so it enters after q₀, the largest when it entered. At the move bringing q_(i+1), take the least j with q_(j+1) not yet held: q_j is held (q₀ by the remark, the rest by the choice of j) and has no anchor yet, q_(j+1) being its first, so m₀ ≤ q_j ≤ q_i and q_(i+1) ≤ m ≤ f(q_i). Two classes follow. Every f with f(q) < p(q) at every large prime q, for f nondecreasing the same as f(t) < min{p(q) : q prime ≥ t} at large t, is dammed from every seed, at any q₀ past that point, as p(q) ≤ p(q^a). This adds nothing to the covering if that minimum is at most ct at large t: it lies between 2t and 3t at 9990 of the 9999 t in 2..10,000, and whether it stays below ct asks for primes k·q + 1 with k bounded, open. So the anchor's force is per run. The dam's bar F is dammed from the void at 23: on the primes ≤ 11 the tide is at most τ(11) = 7 and every move at most F(7) = 13, so the first prime past 11 is 13, which has no anchor until p(13) = 53 is held, while every move is at most F(13) = 23, as τ(19) = 11 and τ(23) = 17. The bar f(τ(P)) ≥ P⁺ does not suffice from the void either.

m₀ ≤ q holds at every odd q held with no anchor at depth 1 over the 687 states of capped walks (exponents to 3 at odd primes, 5 at 2): F's from the void, and those under f = min{p(q) : q ≥ t} − 1 from the void, 3, 5, 7 and 3·5·7, which hold no prime past 7; along the latter's edges no move passes f(q) or brings an anchor of such a q. Under F the capped walk from the void reaches 591 states, all on the primes ≤ 23.

The record run's price. For nondecreasing f the record run is f-greedy exactly when f ≥ E, E(t) = max{R : m₀(M) ≤ t} over its seat states M, R the seat taken from M. Write S(t) = max{p(r^e) : r seated, r^e ≤ t}. Then max(S, B) ≤ E ≤ max(S, B, t). Below: a seat r is past every held prime, so it enters with u_r = 0 at exponent 1 and each e is crossed from a tide at most r^e by a seat at least p(r^e); and the run escapes under E, so E ≥ B. Above: the next seat is the least unread prime R with R − 1 ∤ λ, and every read prime p has p − 1 | λ, a skipped one by the rule and a seated one by its seat. The tide m₀ is a price at some r. At a seated r it is s = r^(u_r+1), which does not divide λ, so p(s) is unread and R ≤ p(s) ≤ S(m₀); at an unheld r with r − 1 ∤ λ, r is unread and R ≤ r = m₀; at an unheld r with r − 1 | λ, R ≤ p(m₀/r), and at 2, R ≤ p(m₀/4), each a power not dividing λ and B's term at the point m₀. Every f of the class has f(t) ≥ t, so the record run is f-greedy exactly when f ≥ max(S, B): every escape from the void pays F ≥ B, and the record run pays its seated primes' powers besides. At every 3 ≤ t ≤ 10,000, max(S, B) ≥ t, and B never binds: E = S, with E(10,000) = 359,747. Every p(s), s a prime power, is a record seat: no prime below it is ≡ 1 mod s, so s divides no p − 1 read before it, nor, being a prime power, their lcm λ. The tide-led run seats at each move the least prime ≡ 1 mod the least power of the tide's prime not dividing λ, p(m₀) at a seated r, p(m₀/r) at a skipped one and p(m₀/4) at 2, and an open tide prime r (unheld, r − 1 ∤ λ) itself. Run until its tide passes 10,000, every seat it takes is a record seat, its held primes up to 10,000 are the record run's 302, and its envelope is E at every t ≤ 10,000. Skipping each open r instead, by seating the least prime P > r ≡ 1 mod the part of r − 1 not dividing λ, prices below E at 6816 of the 9999 t in 2..10,000 and above it at 484; it never falls below 0.2538·E on 3..10,000, and is 324,469, 0.9019·E, at 10,000. Both runs escape, so both envelopes are at least F. With τ tabled to P = 153,757, where it first passes 10,000, F is fixed at every 4 ≤ t ≤ 10,000, and there E/F is at most 7.6829, at t = 1997 (median 3.8921 over t ≥ 100, 2.3397 at 10,000): each run holds 1997 with no anchor until it seats p(1997) = 87,869. The skipping run meets F at t = 227 and 228, where it is 1583, and nowhere else on 100..10,000; below 100 it meets F at 4..12 and 17..24, the record run at 5 and 6. The hull U(t) = max{p(s) : s a prime power ≤ t} of the escape above bounds E, and F ≤ U: τ(P) ≤ t puts β(P) ≤ t by the floor, and P⁺ ≤ p(β(P)) ≤ U(t). E = U at only 2640 of the 9999 t in 2..10,000, those where U's largest p(s) has s a power of a seated prime; at the rest s is a power of a skipped prime or of 2, whose p(s) the bounds force only at B's point.

The anchor bar. An endless run seating one odd prime per move at exponent 1, 2 never seated, with E_run its envelope as E is the record run's, pays p(P) for each seat P: E_run(P) ≥ p(P). The tide never falls: λ only grows, which lowers no price, and the seat's own price becomes at least P ≥ m₀. The seat P raised λ, so it was priced at most P and taken from a tide at most P; if no held prime is ≡ 1 mod P, P's price is P and the tide stays at most P until one enters, at least p(P) (if none ever does, the run's endless new primes are all seated from a tide at most P), and if one is held, it entered from a tide at most P. So seating 1997 pays p(1997) = 7.6829 F(1997), the record run's peak. Each move's class is the primes ≡ 1 mod the least power of the tide's prime not dividing λ, or at an open r ≡ 1 mod m*, the part of r − 1 not dividing λ, whose least member is r. The chain-led run seats the class prime with least max(P/F(m₀), V(P)), V the chain value V(q) = min over primes a ≡ 1 mod q of max(a/F(q), V(a)), cut to 0 past 10,000: to 10,000 it stays within 6.7315 F, never seats 1997, and is below E at 9317 t and above it at 28. Continued past its tide 10,000 by the record rule, seating the least open prime r from a tide at most r, such a run seats a new prime at every move and escapes, the least open prime rising so that every v_r(λ) and so the tide grow without bound, and a move from a tide past t no longer counts at t.

The pointwise price. Write E_min(t) for the least E_run(t) over the runs escaping from the void. Then E_min = F at every t ≥ 4. Every escape pays F, so E_min ≥ F. Above, take Q = F(t): some state on the primes ≤ Q has tide past t. There each prime r ≤ t prices past t, which forces v_r(λ) at least r's exponent a, a price r^max(1, v−a+2) > r at odd r and 2^max(1, v+3−a) > 4 at 2; so λ's part at the primes ≤ t is that of lcm(q − 1) over the held odd primes q, and the squarefree state on them has tide past t: λ's part there is kept, a held exponent falling to 1 and 2 unseated only raise a price, an unheld r ≤ t keeps its kind, r − 1's primes lying below t, and every larger prime is priced at least itself. Cut it to a set D that is tide-minimal, each prime needed for a tide past t. D is irredundant: a p whose p − 1 has its part at the primes ≤ t dividing the others' lcm leaves with the tide still past t, its own price, if p ≤ t, rising from p^(v+1) to p^(v+2). So D seated from the void one prime at a time, in any order, raises λ at every move, each move at most Q. A new odd prime at exponent 1 that raises λ lowers no price and prices itself at least the tide, so the tide never falls; continued from D by the tide-led run, it rises without bound and the run escapes, its moves from tides at most t being D's. So E_min(t) ≤ Q. And Q lies in D and in the state of every run attaining E_min(t) when its tide first passes t, the primes below Q holding the tide at most t. At all 28 steps of F on 4..10,000 the construction holds: D is 11, 23, 29, 47, 59, 73, 97 at 17..46 (tide 47), and 460 primes at 8543..10,000 (F = 153,757, tide 10,007), 1997 and p(1997) among them. No run attains E_min at every t: one would be F-greedy, and F is dammed from the void at 23.

The single run. A run's envelope stays within cF at every t exactly when every move from a tide m₀ is at most cF(m₀); write c₀ for the infimum of the c at which such a run escapes from the void. The crossing of a prime power d = r^e has the threshold T(d) = min(r^(e+1), p(d)), 2^(e+2) in place of r^(e+1) at 2: while v_r(λ) ≤ e − 1 the tide is at most T(d), r's price being at most the depth dam's point and p(d), unheld, pricing itself. λ only grows, so until a prime ≡ 1 mod d enters every move is at most the crossing's allowance X(d) = cF(T(d)); raising r's own exponent to a instead holds the tide at most r (4 after the move 4) until an anchor at depth a enters, and then at most r^(e−a+1) ≤ T(d), r's price being r^max(1, v−a+2) with v ≤ e − 1. At a held crossing, q odd and held at exponent a with v_q(λ) ≤ e − 1, q's price q^max(1, v−a+2) holds the tide at most q^e. So an escaping run passes every crossing by a viable prime b: each crossing b^e of b is passed by a viable prime ≡ 1 mod b^e, held before b or brought by a move at most cF(b^e), raising b's own exponent instead holding the tide at most b, from exponent e on, until one enters (the anchor, above). The crossing test at c and a horizon H asks every crossing d with T(d) ≤ H, or a smaller bound named, for a prime ≡ 1 mod d within X(d), viable with every prime it needs first within X(d), viability read at the crossings b^e ≤ H and every prime past H counted viable, so the test reads only moves from tides ≤ H. It is monotone in c, its least passing c never falling as H grows: at H = 10,000 its 309 crossings first pass at c = 739/233, at H = 100,000 its 1922 at c = 11251/3373 ≈ 3.33561, failing just below the latter at the least crossing alone, 2: T(2) = 3, and X(2) = 5c admits 3, 5, 7, 11 and 13, none viable within 16. 5 and 11 meet a held crossing with no prime ≡ 1 mod it in reach: 5's, 625, needs 11,251 = 18·625 + 1 held first, past cF(625) below the bound, and 11's, 11⁴ = 14,641, read only past 10,000, needs 673,487, past cF(14,641); 3, 7 and 13 meet theirs, 3, 49 and 13, with primes in reach, none viable. So every run escaping from the void pays 11251/3373 F at some t in 4..100,000: c₀ ≥ 11251/3373. At the horizon 10^6, crossing bound 100,000, the 1922 still pass at 11251/3373, the crossing 2 by 5, whose crossing 5⁸ is passed by 9,375,001, past the horizon and so counted viable, as at 1560 of the 3387 crossings viability reads: a range for the test, never a bound. Each bound is met to its horizon: the crossing-led run seats at each move the least unheld prime of the tide's class within cF(m₀) viable with every prime it needs first within cF(m₀), and at c = 739/233 it reaches tide 10,007 in 503 seats and at c = 11251/3373 tide 100,043 in 3386 seats, within cF, the latter seating 11,251 from the tide 625; continued past the horizon by the record rule, not shown within cF there, each escapes. The held crossing prices the chain-led run's peak: an escaping run that holds 19 before any prime ≡ 1 mod 361 pays p(361) = 10,831, which is 6.7315 F(361), at t = 361.

Open fronts

Whether a Dedekind ring with an infinite class group, where the rider reach need not be finite, can lose endlessly many choices and land between the finite and the continuum, is open.

Whether ρ tends to ∞ along every P, that is whether τ(P)/P → 0, is open: ρ is unbounded, but the bound at 2 alone would not give it if primes k·2^w + 1 with k bounded occur at infinitely many w, as expected and unproved, and τ(P)/P is computed only to P = 153,757, ρ reaching 17.998 at 153,749; the feeder formula reads F at 36 drawn t to 1.62·10^6, F(t)/t reaching 43.62 at t = 1,333,521, so ρ ≥ 43.62 at the prime before 58,168,417. How fast ρ grows is open too: the Proth route proves ρ > k/4 at a P near k·2^e, its k averaging 1.12 times e·ln2/2 over 100 ≤ e ≤ 600, its records higher; the floor τ(P) ≥ β(P) proves ρ(P) ≪ P^(1 − 1/L), L a Linnik exponent. With M_D(T) the largest p_T(s) over the moduli the empty seating reads, P*(T) ≤ M_D(T): if M_D(T) ≤ T + (A/2)·T·ln² T at every large prime power T, as a gap bound A·φ(s)·ln² T for those progressions gives, then ρ(P) ≤ 2 + A·ln²(2P) at every large P (τ(P) → ∞ by the floor): T⁺(t) ≤ 2t gives F(t) ≤ 2t + A·t·ln²(2t), and ρ(P) ≤ F(τ(P))/τ(P), τ(P) ≤ P. The hypothesis is its conclusion on P* up to the factor M_D/P*, 1.143 to 2.711 over the 36 draws (gapbar.py), and gives the gap bound back only where the largest normalized gap sits at φ(s) near T/2. That gap, G_D(T), the largest (p_T(s) − T)/(φ(s)·ln² T), has medians 0.647, 0.648 and 0.722 over the draws near 10^4, 10^5 and 10^6, a ratio 1.116 against a bar of 1.3 set before the run between independent waits' 1.07 under a bounded G_D and 1.6 under growth like ln T, each median below its model's (the real waits share primes); it sits at the Sophie Germain r above at 652 of the 1278 T following t ≤ 10,000 and 22 of the 36 draws.

Where the fate changes along the slack is open: a slack at most max(ct, κ·t·ln(ln t)), κ < 1/4, or below p(q) at every large prime q dams every walk from every seed; whether one run stays within a constant of F is open, and a deeper horizon rereads the crossings of the candidates already read, so the bound can rise at any horizon, as it did past 10,000 at the crossing 2. Heuristically, in a random model of the primes ≡ 1 mod q in (q, cF(q)] deciding each candidate q's viability, a crossing at an odd prime r with p(r) = kr + 1 fails with probability about exp(−(1 + k)c·F(p)/(p ln p)), p = p(r), most often at a Sophie Germain r (k = 2); so c₀ is infinite if F(t)/t (ρ(P) at t = τ(P) when τ(P⁺) > τ(P)) grows like ln t, and at least 1/(3C) if it grows like C ln² t. Over twelve t near each of 10^4, 10^5 and 10^6, F(t)/(t ln² t) has the medians 0.144, 0.146 and 0.176, each putting 1/(3C), past which finitely many such crossings are expected to fail, below the bound, and F(t)/(t ln t)'s median rises 1.83-fold from the first to the third, against 1.0 for a pure ln t and 1.50 for C ln² t: F(t)/t outgrows ln t; whether it outgrows C ln² t the windows, whose draws share most least primes, do not settle. The model prices the newest crossings only, never the older ones a horizon rereads, so it bounds c₀ from below only. The record run is f-greedy exactly when f ≥ max(S, B), which is S to 10,000. Whether some run from the void escapes under a slack at most max(S, t) at every t and below S somewhere, seating primes whose powers cost less than the record run's, is open: the tide-led run pays E to 10,000, and the skipping and chain-led runs pass it at some t. A walk from a seed escapes only if f(τ(P)) ≥ P⁺ at every prime P past the seed's primes and f(q) ≥ p(q^a) at every odd q it holds at exponent a with no anchor at depth a (the anchor, above); the first bar alone does not suffice, F being dammed from the void at 23 and every f from 3^N with 3^N ≥ f(3). How the classes between greed and freedom order by their images is open past the c-greedy one: its images nest in c and are computed, as limit patterns, from the void below 11/5 only, and a class that is not a threshold on the tide is not computed at all.