GROWTH — a modulus grown by the cheapest move a demand admits

The object: a state Z/M and a structural demand on extensions, grown by the greedy move, from Z/M to Z/Mm with m ≥ 2 the least multiplier the demand admits. The demands never mention primes: the independence demand (Z/Mm ≅ Z/M × Z/m), new idempotents, the dynamics demand (λ, the unit group's exponent, grows), new orders of units, the transparency demand (M grows, λ fixed), and a rate, the m maximizing the idempotent doublings bought per bit. This page asks what each demand grows, where it stops if it stops, and what decides where it goes, and then asks it again over the ideals of a number ring and over F₂[x], where the price of one tick at one place decides the walk. Its verifiers are growth.py, standing.py, sizeless.py, module_law.py and sprawl.py, with bridges.py for the lock by a single multiplication, ghosts.py for the ghosts a seed picks, cascade.py for the full price over Z and lid.py for the lock over a number ring.

A trajectory runs from a state, its seed, and its limit is a supernatural number, a product over the primes (over a Dedekind ring O, its places P) with exponents in {0, 1, …, ∞}, each the depth it reaches down its prime's or place's column, the chain of quotients O/P^b (Z/q^b over Z), and the three fates are properties of that limit, not a partition: BREADTH, every prime seated, at a positive exponent; DEPTH, some prime at infinite exponent; MORTALITY, the limit a finite integer. Breadth is seating, not "every exponent finite", which every integer satisfies. That each greedy law over Z below realizes exactly one fate is a fact about greed: a policy free to take any move the independence demand admits can seat every prime but one, by never taking 2, which is none of the three.

The three fates

Tier: theorem.
Verifier: proof; growth.py::section_a.

The least m ≥ 2 coprime to M is the least prime q not dividing M, since every smaller m is built from primes M carries. So greedy independence never picks a composite: every new factor Z/q is a field and a squarefree seed stays squarefree, with neither demanded. From any seed its picks are the primes the seed lacks, in increasing order, so from 1 the trajectory is the primorial tower (checked at every M ≤ 10⁵ and on 100 seeds for 25 steps). New idempotents picks the same q, since the count of idempotents, 2^ω(M) with ω(M) the number of distinct primes dividing M, rises only with a new prime (checked at every M ≤ 10⁵). The rate from p_k#, the product of the first k primes, is the next prime: every new prime of m is at least the least absent prime q, so m buys at most 1/log₂ q doublings per bit, with equality only at m = q (the unique maximizer printed for 1 ≤ k ≤ 6 over m ≤ 1000). Three demands, one attractor, BREADTH.

A finite abelian group realizes exactly the divisors of its exponent as orders, since the exponent is some element's order and a cyclic group realizes every divisor of its order (checked at every 3 ≤ M ≤ 1200), so new orders is the dynamics demand, and dynamics runs down one prime's column: DEPTH, the lock-prime law below.

Transparency's fate is MORTALITY. λ(a) | λ(b) when a | b and λ(lcm) = lcm(λ), so {M : λ(M) | L} is closed under lcm, and it is finite; its maximum is the wall

W(L) = 2^(v₂(L)+2) · ∏ p^(v_p(L)+1)   over odd p with (p − 1) | L,

at even L, and W(odd L) = 2. A transparent walk stays among the divisors of W(λ(seed)) and cannot stop below it, since while M < W the least prime factor of W/M is admissible. So it dies at exactly W(λ(seed)): at all 99 seeds 2 to 100, seed 73 climbing to W(72) = 20,174,525,280 in 13 steps. Nothing here uses greed, since every state below W has an admissible prime: the states some policy reaches are the multiples of the seed dividing W, every run ends at W and only there, and the longest has Ω(W/seed) moves, Ω counting prime factors with multiplicity (searched over every move to a state up to 4W at the 80 seeds from 2 to 100 with W ≤ 10⁵).

So the primorial tower is not an attractor of free growth. It is the attractor of independence, the CRT axiom read as a demand, and of the two demands that pick what independence picks. The scope is the least-move laws on Z/M and the rate's best ratio; additive moves and non-cyclic ambients are not charted, and the least move softened into a weight, growth at an inverse temperature, is THERMAL's.

The wall is Adams' number

Tier: property (the closed form is Adams' theorem; checked at every L ≤ 200).
Verifier: growth.py::section_b_z.

"λ(M) | L" and "z^L ≡ 1 (mod M) for every unit z" are one condition, so W(L) is the number J. F. Adams defined (On the groups J(X) II, Topology 3, 1965), the gcd over the integers y of y^s(y^L − 1) for large s, and his theorem gives its closed form at even L, W(L) = denominator(B_L / 2L):

L      2    4    6    8    10     12
W(L)  24  240  504  480   264  65520

the orders of the image of the J-homomorphism in degree 2L − 1 (Adams, On the groups J(X) IV, Topology 5, 1966, with the Adams conjecture proved by Quillen and by Sullivan), by the same definition and not by coincidence; nothing here uses homotopy. The gcd, taken over y up to a cutoff, equals W(L) at two settings of s and the cutoff at every L ≤ 200, and the Bernoulli form does at every even L ≤ 200, the Bernoulli numbers computed from scratch and checked against von Staudt–Clausen first. The squarefree kernel of W(L) is the von Staudt–Clausen product ∏ p over (p − 1) | L.

The wall over F₂[x]

Tier: theorem.
Verifier: proof; growth.py::section_b_f2.

At an irreducible g of degree d the unit group of F₂[x]/(g^a) has exponent (2^d − 1)·2^⌈log₂ a⌉: the units are F_(2^d)^× times the 1-units 1 + g·F₂[x]/(g^a), and (1 + u)^(2^j) = 1 + u^(2^j) is 1 at every such u iff 2^j ≥ a (checked element by element at every g^a of degree ≤ 9 with d ≤ 4 and a ≤ 5, 26 prime powers). So λ(g^a) | L iff (2^d − 1) | L and a ≤ 2^v₂(L), and

W(L) = ∏ (∏_{deg g = d} g)^(2^v₂(L))   over d with (2^d − 1) | L.

At L = (2ⁿ − 1)·2^v the admissible degrees are the divisors of n and W(L) = (x^(2ⁿ) + x)^(2^v): at v = 0 the wall ring is a product of finite fields, one F_(2^d) for each irreducible of each degree d dividing n, and at v > 0 that polynomial's power, no longer a product of fields. The wall exists in both rings for the same reason; its value is the ring's, elementary here and Bernoulli over Z.

The lock-prime law

Tier: theorem (the door, the lock and its bound, every prime a lock, the lock by one multiplication); rule (the condition for a full-price first move against the cascade's dead level, compared at every level of every odd q < 1000 where the two can differ: one level, q = 11).
Verifier: proof; growth.py::section_c, standing.py::section_e, bridges.py::section_r, cascade.py::section_w.

The least λ-raising move is always a prime power q^r, the door lemma: λ(Mm) is the lcm of λ at the prime powers exactly dividing Mm, so m's prime-power part at a prime q where λ(q^(v_q(Mm))) does not divide λ(M) is a legal move no larger. At each prime q the least raising power q^r has a closed-form exponent r, q's door. With a = v_q(M) and v = v_q(λ): at odd q with a ≥ 1 the door is v − a + 2; an opening, the first move at a prime M lacks (a = 0), has door 1 when (q − 1) does not divide λ and v + 2 when it does; at q = 2 the same scan runs over λ(2^j) = 1, 2, 2, 4, … (the formula matches the least raising power found by search at every q ≤ 47, over every odd M ≤ 2,000 and the states eleven greedy walks visit in their first six steps, standing.py::section_e). Least moves at distinct primes are powers of distinct primes, so they never tie: greedy dynamics over Z needs no tie-break.

A least move is a deepening (q | M), a fresh opening (q prime to λ), a ghost (q | λ, (q − 1) ∤ λ: a prime whose period already runs in λ though q does not divide M), or an opening at full price q^(v + 2) (2^(v + 3) at 2) with q | λ and (q − 1) | λ. At odd q the last is least from some state exactly when no prime ≡ 1 (mod q^(v + 1)) lies below its price, which agrees with the cascade's dead level at every level of every odd q < 1000 where the two can differ, the one level (11, 1) (CASCADE: The full-price opening at odd p is a dead level over Z). No state M ≤ 20,000 has one (10,967 deepenings, 6,782 fresh openings, 2,251 ghosts), but the prime seed 961440481, whose λ carries every ℓ − 1 for the primes 5 ≤ ℓ < 81 and 3² exactly, opens 3 at 81: 28 and 55 are composite. Every pick but a ghost leaves v_q(λ) = a_q − 1 (a_q − 2 at q = 2 once a_q ≥ 3), which holds q's door at 1, price q, from then on (at 2 once a₂ ≥ 3; the one pick landing at a₂ = 2 is from the seed 2, where 2's next least move costs 4 and every rival at least 5), while every other least move only grows and never costs q. So the first pick that is not a ghost locks the trajectory onto q: every later pick deepens q, which is what a lock onto a place means over every ring below. A ghost at a prime q′ is legal only while (q′ − 1) ∤ λ, which only gets harder. When q′ is picked, every unseated prime q″ < q′ already has (q″ − 1) | λ, since otherwise its opening would cost q″ < q′, ghost or fresh, and be picked first; λ only grows, so no such q″ is a ghost later: ghosts come in increasing order, and since a ghost adds to λ only primes below it, every one divides λ(seed): the lock falls within ω_odd(λ(seed)) + 1 picks, ω_odd counting the odd primes dividing λ(seed), and the basin map seed ↦ lock is computable. A ghost's dowry, the factors of q′ − 1 it adds to λ, shuts cheaper locks behind it: seed 11 ghosts 5 and locks 7; seed 71 ghosts 5, 7 and locks 17; the prime seed 20231, with 20230 = 2·5·7·17², ghosts 5, 7, 17 and locks 19, the bound attained. From the void, M = 1, the walk's states are 3, 9, 27, …: 2's first door is 2, price 4.

Every prime is some seed's lock. For odd q let B be the lcm of {p − 1 : p ≤ q} and of ℓ^(c_ℓ) over the primes ℓ < q, c_ℓ least with ℓ^(c_ℓ + 2) > q, and take a prime R ≠ q, the blocker, with R ≡ 1 (mod B) and R ≢ 1 (mod q), which Dirichlet's theorem provides since q ∤ B. The seed qR has λ = R − 1, q's door is 1, price q, every opening below q is shut at a price above q, and no ghost fires: the first pick deepens q. At q ≤ 47 the least blockers are R = 5, 13, 13, 61, 61, 241, 2161, 23761, then 55441 for q = 29 to 43, and 1275121; the seed 2 locks at 2 with none.

The same construction steers a state already grown: from any M, a single multiplication of M by the number built below locks the walk onto any prime q. At 2 the multiplier raises 2 to depth max(3, v₂(λ) + 2), where 2's door is 1, price 2, the least move there is. At odd q it raises q to depth v_q(λ) + 1, so q's door is 1, times a prime R′ ≡ 1 (mod B′), R′ ≢ 1 (mod q), not dividing M, where B′ is the lcm over the primes ℓ < q of ℓ − 1 and ℓ^(a_ℓ + c_ℓ + 2), a_ℓ the depth of ℓ in M and c_ℓ as above (the run takes the exponent a_ℓ + c_ℓ + 4, a multiple the argument allows): every least move below q then costs more than q, R′'s own at least R′, and deepening q moves no rival's price down. The basin map is computable, and from any state every value is one multiplication away: 200 tested states each lock onto 2, and five states, 3¹⁵ and 17⁶ among them, each lock onto 3, 5, 7, 13 and 17, each run's 30 picks watched after the multiplication deepening the target at price q.

The ghosts are the witnesses' records

Tier: theorem; known (primes p ≤ x with a prime factor of p − 1 above x^(1/2) number at least (1/2 − o(1)) x/log x).
Source: M. Goldfeld, On the number of primes p for which p + a has a large prime factor, Mathematika 16 (1969), 23–27, Theorem 1.
Verifier: proof; ghosts.py::section_family, ghosts.py::section_corner, ghosts.py::section_sets.

The number of ghosts a walk over Z picks before its lock is at most ω_odd(λ(seed)) (The lock-prime law), and it is unbounded over seeds. Write g(w) and h(w) for the least primes ≡ 1 (mod w) and ≡ 1 (mod w²). Fix x and a set W of odd primes, each w with w³ > x, g(w) ≤ x and g(w) < w², none dividing another's w′ − 1, and let the seed C(x, W) be a product of distinct primes r > x, one for each prime ℓ ≤ x outside W, with r ≡ 1 (mod ℓ^(e_ℓ)), ℓ^(e_ℓ) the largest power of ℓ at most x, and no w dividing r − 1 (Dirichlet). Its ghosts up to x are exactly the w-adic records R = G ∪ H, G = {g(w)} and H = {h(w) ≤ x}: the primes at which some v_w(p − 1) first reaches 1 or 2 (as p − 1 < w³ it reaches no more), in increasing order, each at price p; the next pick costs more than x. Proof: the seated ghosts are always the records below the next pick, and v_w(λ) is the largest v_w(q − 1) over them, since the seed puts no w in λ, a ghost's dowry adds q − 1, and every other prime power of a p − 1 with p ≤ x is in λ from the start. So the least unseated record p*, outside W as no witness divides another's w′ − 1, has p* | λ and (p* − 1) ∤ λ, and any such prime below it would be a record below it, all seated: p* is picked, at door 1, price p*. Every other price exceeds p*: w, with (w − 1) | λ since no witness divides w − 1, opens at w² > g(w) while g(w) is unseated and at w³ > x after; 2 at 2^(v₂ + 3), a seated ghost g at g^(v + 1) ≥ g^(e_g + 1), a prime p ≤ x with (p − 1) | λ at p^(v + 2), a seed prime or a prime above x at more than x. Once R is seated no ghost at most x is left. By Goldfeld's theorem at least (1/2 − o(1)) x/log x primes p ≤ x have a prime factor w of p − 1 above x^(1/2); such a w has w³ > x and g(w) ≤ p ≤ x < w², and divides p − 1 for fewer than x^(1/2) primes p ≤ x, so at least (1/2 − o(1)) x^(1/2)/log x distinct w arise, and one dyadic window (A, 2A] holds at least a 1/log₂ x share of them, none dividing another's w′ − 1. Distinct w there have distinct g(w), as g − 1 < x holds at most one prime above x^(1/2), so that C(x, W) picks at least c·x^(1/2)/log² x ghosts, c > 0 fixed and x large. A brute walk, with its own λ and no door formula, and growth.py's walk pick 3, 8, 15 and 26 ghosts at x = 100, 400, 1,000 and 2,500, each W the window (A, 2A] giving the most, R = G there, the last seed 341 primes; the brute walk's ghosts up to x are R at 33,131 windows (x = 30..400) and at 6,832 pairs (x, W) built where w² < x, x ≤ 2,500, H nonempty at 1,330 of them (ghosts.py::section_sets). Over x = 30..5,000 no two witnesses share a record; h(w) never can, as h(w) − 1 = w²m with another witness dividing m exceeds 2x.

The two-adic prices

Tier: theorem.
Verifier: proof; growth.py::section_c, growth.py::section_d.

The cold openings: from every M ≥ 3 that lacks the prime, the opening at 2 costs at least 16 and the opening at 3 at least 9, both attained, while every prime from 5 to 47 opens at its face value from λ = 2. They are two facts about −1. The splitting, Z₂^× ≅ {±1} × (1 + 4Z₂) where every odd Z_p^× is procyclic, whose finite shadow is λ(2^a) = 2^(a−2) at a ≥ 3; and the evenness, −1 of order 2 in every odd residue field, so λ is even wherever an odd prime divides M. A counterfeit column at 2 wearing the odd pattern λ*(2^a) = 2^(a−1) turns the splitting off and leaves the evenness. On odd M > 1 the real 2-door is v + 3 at v = v₂(λ) and the counterfeit v + 2: the "+3" is the splitting's and the floor v = 1 the evenness's, so the cold opening at 2 is their conjunction, floor 16 real and 8 counterfeit, while at M ≥ 3 prime to 3, 3's door v₃(λ) + 2 is the evenness alone and does not move. The splitting also prices the walk from 4, which pays 4 then 2, 2, … (λ(4) = λ(8)) where the counterfeit pays 2 throughout, and it prices, at a rank-1 characteristic p (a prime with an unramified place of norm p over it; over Z every prime is one), the carriers (CASCADE), the places whose opening lifts v_p(λ): a carrier's norm h·p^(v_p(λ)+1) + 1 cheaper than p's opening at p^(v+2) has h ≤ p − 1 at odd p and raises v_p(λ) by one, but at 2 that price 2^(v+3) admits h ∈ {1, 2, 3} and h = 2 raises it by two; the counterfeit admits h = 1 only. That norm, raising v₂(λ) by two, is 2^(v+2) + 1, and the prime powers of that form are the Fermat primes from 17 on and 9, Mihăilescu's theorem leaving 9 = 2³ + 1 the only proper power (9, 17, 257 and 65537, the exponents 3 to 20 being the part of v ≥ 1 searched). The splitting and the evenness are the two valuations of −1 − 1 = −2: its 2-adic depth 1 and its unit at every odd place.

The module law

Tier: theorem; observation (the pump's start at the ramified rings run).
Verifier: proof; module_law.py::section_m, module_law.py::section_control.

At a place P of a Dedekind ring O, with ramification index e over p and a residue field of N(P) = p^f elements, (O/P^a)^× splits as (O/P)^× × U₁/U_a, U_b being the units ≡ 1 mod P^b, so λ(P^a) = (N(P) − 1)·E(a) with E(a) the exponent of U₁/U_a, a power of p. A tick, one more factor p in λ bought by steps down P's column, is a jump of E, and E has two shapes.

In equal characteristic O_P = (O/P)[[t]] and (1 + u)^(p^j) = 1 + u^(p^j), so E(a) = p^⌈log_p a⌉ exactly, whatever f is: a log clock ticking at depths p^j + 1 (2, 3, 5, 9, 17, 33 in characteristic 2), whose tick from depth p^j + 1 costs N(P)^(p^j(p − 1)), unbounded along any column. In mixed characteristic the p-th power of a 1-unit z adds exactly e to v_P(z − 1), v_P the valuation at P, once v_P(z − 1) > e/(p − 1), and U₁ has no torsion past that bound, so E(a + e) = p·E(a) from a threshold on: a pump, one tick per e of depth, priced N(P)^e = p^(ef), where ef is the Z_p-rank of U₁. So in mixed characteristic a tick costs p^(ef), p to the Z_p-rank of U₁, a constant; in equal characteristic U₁ has no finite rank and the tick price grows without bound down the column. The threshold is not read off e and p alone: Q₂(√−2), Q₂(√−5) and Q₂(i) share e = 2 and f = 1 and start their pumps at depths 4, 5 and 6, a pump's start being the least depth from which E(a + e) = p·E(a) holds at every later a. The pump runs from depth 1 at every unramified odd place, where e/(p − 1) < 1, from 3 at Z₂, and from 6 at Q₂(i), the latest of fourteen local rings run to depth 40.

E is computed from the elements 1 + w·π^b, π a uniformizer and w over a basis of the residue field, which span U₁/U_a since each U_b/U_(b+1) is the residue field under addition. E from these elements equals E read off every element of U₁/U_a at 140 (ring, depth) in mixed characteristic and 38 in equal; (N(P) − 1)·E(a) equals the exponent of the whole unit group (O/P^a)^× at 126 (ring, depth); and the local rings reproduce the exponent of (O/p^b)^× of Q(√−23) and Z[√−5] at 38 (field, p, b).

The lock over a number ring

Tier: theorem (the ideal world's lock iff its costs are bounded infinitely often, at p^(ef) a move; the element world's two rides); observation (every ideal walk from 134 seeds over two fields deepens one place at each of its last 25 of 40 moves).
Verifier: proof; module_law.py::section_n, module_law.py::section_e, lid.py::section_lid.

In the ideal world of a ring of integers, states are ideals M and a move multiplies by an ideal priced by its norm. The door lemma survives: if m raises λ, some place P of m raises it alone at no greater norm, so every least move is a prime power P^r, and ties sit among places over one prime: of one norm when those places share a residue degree, as in the quadratic fields run, while a cubic ring where 2 splits with residue degrees 1 and 2 ties 4 against 4 (CLOCK: A lidded block locks the walk for good). A door off the moved place never falls, since λ only grows under divisibility, but the moved place's own door can: (7) over Q(√−23) pays 23, 23, 27, then 3. So the lock is argued through boundedness. If infinitely many moves cost at most C, places of norm at most C are moved infinitely often, so some block, the places over one rational prime sharing one count (CLOCK: A block clock keeps at most one runaway per block), is clocked forever, and its count, the p-adic valuation of λ that its clock moves raise (CLOCK: A ring's door is one valuation and its clock a count), passes every threshold. That block is then lidded at one of its clock moves (CLOCK: A lidded block locks the walk for good), by the place that made it, whose door is exactly e, at the constant price p^(ef); an opening that could jump the block's count costs more than that, and every opening bought once the block is lidded costs at least its norm and at most the lid, so finitely many follow; a rival in the block is clocked only when its own constant price, p^(ef) at its own e and f, is below the lid, and the block is then lidded by the rival in turn, at a strictly lower lid; and two constant prices at distinct characteristics are powers of distinct primes. So a trajectory locks iff its costs are bounded infinitely often, and a lock pays p^(ef) per move. Whether costs can diverge instead is CASCADE's question, open there and not settled here; over Z the lock-prime law settles it, and over F₂[x] the sprawl settles it the other way.

At Q(√−23) (class number 3, 2 split) over 80 seeds, the void, every ideal of norm 2 to 40 and (17), and at Z[√−5] over 54, the void, every ideal of norm 2 to 40, a place over 29 and the product of places over 43 and 89, every trajectory deepens one place at each of its last 25 of 40 moves, and every such tail pays p^(ef), among them 2 at a split place over 2, 25 at the inert 5 and at Z[√−5]'s ramified 5, 4 at its wild ramified 2. Every move matches a scan of every ideal up to its cost, and every tie met in the two fields is a conjugate pair.

The element world, states and moves principal, breaks the door lemma. Over Q(√−23) no element has norm 2 or 3, so from the void the least raising element has norm 6: a place over 3 carrying a place over 2 at which λ does not rise, a compound with a flat passenger. The walk pays 6, 6, 6 and then rides (2) = P₂P₂′ at 4, one part flat at every move: the flat place over 2 of the first three moves, now at depth 3, ticks at every ride, and the other place becomes the passenger, lifted beside it (depths 40 and 37 after 40 moves). It rides forever: 4 is the least norm of any element but a unit, and (2) always ticks its deeper place. The lift is conditional: from (5) the walk pays 23, 23, then 25 on the principal prime power (5), which ticks at every depth, and no cheaper element ever raises λ again, so the places over 2 are never touched. An element of norm below 25 is a product of places over 2, 3 and 23 (13 splits, but its places are not principal and 26 is past 25): it puts depth at most 3 at a place over 2 and at most 2 at one over 3, whose unit groups then have exponent dividing 4 and 6, so dividing the 24 that (5)'s residue field puts in λ, and takes the place over 23 from depth 2 to 3, where its 1-units keep exponent 23. None is taken, so the depths stay where they are and the argument repeats at every move.

The sprawl over F₂[x]

Tier: theorem; observation (at least 53 of 60 moves open a place at exponent 1, from 63 seeds under two tie-breaks).
Verifier: proof; sprawl.py::section_s, sprawl.py::section_control.

Price a move by its degree and take a least raising m, a tie at the least degree broken by the least or the greatest integer encoding (x ↦ 2, so x² + x + 1 is 7). Every pick is again a prime power g^r, and at g of degree d and depth a, with c = v₂(λ), the pick is g^r with r = 2^c + 1 − a for a deepening, r = 1 for an opening at exponent 1 when (2^d − 1) ∤ λ, and r = 2^c + 1 for a clocked opening otherwise, one where (2^d − 1) already divides λ and only c can rise; each costs d·r. Once c ≥ 1 the openings at exponent 1 are the ghosts here, read as over a number ring with the residue characteristic 2 in q's place: 2 divides λ and 2^d − 1 does not, and the place lands at depth 1, where its door is 2^c, so it seats no lock. A deepening or a clocked opening raises c by exactly one and an opening at exponent 1 leaves it, so a trajectory's other moves number c_final − c_seed. An opening at exponent 1 always exists, at a degree at most one past the bit length of λ's odd part, while once the clock passes the seed's depths every other move has degree at least 2^(c − 1). If the openings stopped, the opening at exponent 1 would stand still while the others doubled, so openings recur forever under any tie-break and no trajectory locks or dies: the module law's log clock at every place at once. Over 63 seeds, the void and every polynomial of degree 1 to 5, ties broken to the least and to the greatest encoding, at least 53 of 60 moves are openings at exponent 1.

The sibling shadow

Tier: theorem.
Verifier: proof; sprawl.py::section_s, sprawl.py::section_v.

Once a place of degree d is seated, (2^d − 1) | λ, so each of its siblings, the other places of degree d, is clocked at degree d(2^c + 1), c = v₂(λ) as in the sprawl, strictly above the seated place's own deepening. So the walk opens at most one place of each degree and none at a degree its seed seats, under any tie-break. Beyond its seed's places its support holds at most one of the roughly 2^d/d places of each degree, a set of density zero: the sprawl's infinite support is never BREADTH.

From the void, ties broken to the least encoding, the walk has a closed form: x², x, x², then the least irreducible of each degree 2, 3, 4, …, preceded at each degree 2^j ≥ 4 by x^(2^j), the least monic of that degree. With x at depth 2^j + 1 its least move has degree 2^j and dominates every other move but the opening at exponent 1, and the frontier, the least degree δ with 2^δ − 1 not dividing λ, advances by one per opening, because 2^n − 1 has a prime factor dividing no earlier 2^k − 1 at every n ≥ 2 but 6 (Zsigmondy's theorem) and 9 | 63 covers 6. x + 1, x's sibling, never enters. The first eleven picks under the least encoding are 4, 2, 4, 7, 11, 16, 19, 37, 67, 131, 256; after 60 moves x sits at depth 65 and one place of each degree 2 to 54 is open. The limit is DEPTH, x at infinite exponent, beside an infinite support of density zero.

The standing-move dichotomy

Tier: theorem (the dichotomy, the depth reading and the jump); observation (the breadth coordinates' repeats and rises at D = 3, 4, 5 and 8 over twelve picks from each of eleven seeds, and the greedy move's own family on those seeds).
Verifier: proof; standing.py.

The lock needs nothing about rings. Give a greedy walk a standing family: recipes, each sending a state to a move admissible there, the i-th priced by κ_i(x_i), a nondecreasing function of one integer coordinate x_i of the state. In a world with finitely many moves under any cost ceiling, where a move recurring infinitely often absorbs the tail, being from some step on the only move taken, the walk pays at most the price cap min_i κ_i(x_i) at every step, so either some price κ_i(x_i) is bounded infinitely often and the walk locks, one move absorbing the tail, or every price diverges at once. A constant-priced member locks the walk outright, and escape is a conjunction over the whole family. A recipe is not a fixed move: the least move at 5 costs 5 at M = 71 and 25 at M = 355.

The schema reads all three fates. DEPTH: at each odd prime q the move q^(x_q + 2) is priced by the excess x_q(M) = v_q(λ(M)) − v_q(M). The deepening door is x_q + 2 exactly and every door at odd q is at most that, strict only at an opening costing q: q^(x_q + 2) itself raises λ, and a smaller power of q raises it only through a new q − 1, an opening (the least raising power, found by search, is checked against q^(x_q + 2) at every odd q ≤ 47 over every odd M ≤ 2,000 and every state the greedy walks from the eleven seeds 1, 3, 5, 7, 9, 15, 71, 100, 121, 210 and 1001 visit in their first six steps, standing.py::section_e, and its least move against brute search at every M ≤ 20,000, growth.py::section_c); at odd q the excess is at least −1, since q^a | M makes λ carry q^(a−1), and the lock-prime law's recurrence invariant is x_q = −1. The lock is the price cap sitting on the floor of the coordinate that prices it, and along every tail locked at an odd prime the excess reads −1 from the first deepening. At q = 2, where λ(2^a) = 2^(a−2) from a = 3, the floor falls to −2 from a = 3, and 2's door is x₂ + 3 at every state but 1 and 2. MORTALITY, a walk that stops and so sits beside the dichotomy: the jump to the wall, m = ξ(M) = W(λ(M))/M, is admissible from every state below its wall and priced by the headroom ξ alone, which the walk drives down to 1, where it ends: 48, 24, 12, 6, 3, 1 off seed 5. BREADTH: fix D ≥ 3 and give each reduced class mod D the recipe "the least unused prime in the class", priced by the walk's count in the class. Dirichlet's theorem makes every recipe admissible at every state; the greedy pick being the least unused prime, the price cap equals it except at a pick dividing D (at most one per seed when D is a prime power, as 3, 4, 5, 8, the moduli run, are); every coordinate repeats and every one rises. Breadth is the escape branch with every conjunct met, and that every one is met is Dirichlet's theorem in the schema's own terms.

A family says something only where its coordinates can repeat. Take the greedy move itself as a one-member family, its coordinate M and its price the walk's own cost at M: on all eleven locked depth walks it meets the hypotheses (the costs never fall in six steps), and its coordinate diverges while its price stays bounded. So on those eleven walks it meets the lock branch, and there the family explains nothing; the content lies in choosing coordinates that are projections of the state and stay bounded along a lock.

The properness criterion

Tier: criterion (proved both ways).
Verifier: proof; sizeless.py::section_h, sizeless.py::section_i, sizeless.py::section_p.

Take size out of the cost. A unital map Z/a → Z/b sends 1 to 1, so it exists iff b | a (and is checked so at every 2 ≤ a, b ≤ 60): window maps (SIZE) order the candidate moves by divisibility, and distinct primes are an antichain. With no cost at all, independence admits every coprime m: seed 2 may pick 9, losing squarefree-ness, and seed 3 may run 5, 7, 11, … with 2 never entering. Fields and squarefree-ness come back under any integer-valued cost γ, bounded below and strictly monotone under proper divisibility, Ω for one, since a coprime composite has a coprime prime factor that is cheaper: every pick is prime under every tie-break (Ω's specimen, sizeless.py::section_f).

Under such a cost a prime p enters under every tie-break iff its sublevel set {q prime : γ(q) ≤ γ(p)} is finite. While p is admissible each pick costs at most γ(p), so picks come from that set and each consumes a member for good. Conversely, if the sublevel set is infinite, either infinitely many primes cost less than p, and p never enters, or infinitely many tie with it, and a tie-break preferring them starves p: under Ω every prime costs 1, and preferring odd primes starves 2 for as long as the walk runs. So every prime arrives under every tie-break iff the cost is proper, every sublevel set finite. Size enters "the least m" through properness and one choice of order, and nothing else about size bears on the destination.

The route is the deleted place's

Tier: theorem (the route, and the arrival rule's shape); observation (its linear extensions counted over the first 20 primes).
Verifier: proof; sizeless.py::section_p, sizeless.py::section_r.

Rank the primes by any bijection ρ with the positive integers and price m by the sum of a·ρ(p) over p^a ∥ m: the cost is proper and strictly monotone, and from 1 the walk takes the primes in the order ρ, as three scrambled rankings of the first 30 primes print, one destination and three routes. Every proper cost of the criterion's kind, integer-valued, bounded below and strictly monotone, buys the destination, and the ranking inside it, up to the tie-break among primes of equal cost, buys the route: the ascending order of arrival, and with it every feature that makes the tower a sequence, is the ranking by size, the deleted archimedean place.

One arrival rule reads only the rings Z/q: admit q only once every prime factor of q − 1, the order of the unit group of Z/q, already divides the modulus. That is the Pratt certificate order. It forces 2 first, and while 2 is the only prime seated the admissible primes are exactly the Fermat primes; the increasing order is one of its linear extensions. Over the first 20 primes it has 8,506,240,142,400 of them against 20! ≈ 2.4·10¹⁸: the arrival rule provides 18.1 of the 61.1 bits a route carries, and the choice of the ascending order provides the other 43.

Open fronts

The standing-move dichotomy assumes that a move recurring infinitely often absorbs the tail; a walk in which some move recurs infinitely often without absorbing the tail would fall outside the dichotomy, and none is known. The wall over a number ring is not computed here. Over Z it is Serre's w_L(Q), the largest M with Gal(Q(μ_M)/Q) ≅ (Z/M)^× of exponent dividing L; over a number field K, w_L(K) reads K's cyclotomic extensions while the wall over O_K is an ideal read off (O_K/M)^×, and how the two are related is not worked out here.