ELEMENT — the walk that must stay principal

The object: a greedy growth walk under the dynamics demand in the element world, where a move multiplies by an element and so seats the place power it aims at together with the minimal divisor cancelling its class, over the ring of a curve over F₂ and over an imaginary quadratic ring. This page asks what the ring hands the walk, what the element reading hides of a gcd and of the count of ideals by norm, which colouring of its places is the coarsest that prices every move, what that divisor costs and does, whether the chain of clock holders survives it, what the walk converges to, and which of the limit's coordinates are unbounded. Its verifiers are element.py, element_ring.py, rider.py, sensor.py and module_law.py, with limit.py the trivial group's reference walk.

The curve rings are the functions on a curve over F₂ regular away from one rational point at infinity, in the imaginary model y² + H(x)y = F(x) with deg F = 2g + 1: F₂[x]; four elliptic curves h2 to h5, named by their class numbers; and y² + y = x⁵ + x of genus 2, class number 15, named g2. The class group is Pic⁰ of the curve and a place P of degree d has class [P − d·∞]; a place's colour is its (degree, class), and σ(d, c) counts the places of a colour, the table of these counts over degree and class being the ring's supply matrix. λ(M) is the exponent of the unit group of the ring mod M, and L its value at the state. The door, the notch and the chain's words are LIMIT's, degree 1 born covered: a move at an item of exponent a has door 1 when it opens a degree other than 1 that no seated degree is a multiple of, and T + 1 − a otherwise, T the least power of 2 at or above every exponent. A move is a core, the place power at its door, plus a rider, the least-degree effective divisor of the class that makes the product principal, MINREP of that class; m(c) is MINREP(c)'s degree and R the largest m. A state's menu is its least moves and their common price. The number rings are the imaginary quadratic ones, Z[√−5] and the ring of integers of Q(√−23) first, of class numbers 2 and 3, where a price is a norm and m a least norm.

The answer, in one line: the ring hands the walk its class group, over a curve a supply matrix whose colours no coarser colouring can replace, over a number ring a ladder per place, and over each of these curves every class summons exactly one minimal rider. Over a curve, past a finite transient (the walk to its last opening that pays a rider, Greed pays no avoidable rider) and a finite notch the riders arrive too slowly to break the chain at every branch run (the argument lacks one step, The eventual chain), so the walk keeps exactly one runaway, with a few rider-fed coordinates beside it that a recursion on (Z/n)², n the order of the runaway's class, sorts into unbounded and stopped, the unbounded ones growing at most like the logarithm of the runaway's exponent. Over a number ring the ideal walk's lid (CLOCK: A lidded block locks the walk for good) has a twin in the element world, priced with riders, so a walk reaches a lidded block exactly when its prices are bounded infinitely often (The element lid); a lock repeating one vehicle (The bare door) carries its rider at the core's own rate (The lock carries its rider, a rule at two rings, Z[√−5]'s lock riderless), and over an imaginary quadratic ring that rider lies over the core's own prime.

The ring is a supply matrix

Tier: rule (verified at six rings: every colour to degree 9 against a brute count of places; the counts to degree 400 computed by an exact recursion, whole at every degree); theorem for the finiteness of the principal-free degrees.
Verifier: element.py::cells, element.py::section_c, element.py::section_g.

In the group ring Z[Pic⁰], Φ(z) = ∏_P (1 − [cls P] z^(deg P))⁻¹ over the projective curve counts effective divisors by class, and Riemann–Roch gives every class 2^(n + 1 − g) − 1 effective divisors of degree n ≥ 2g − 1. So E = Φ·(1 − z)(1 − 2z) is a polynomial of degree 2g, read off the places of degree ≤ 2g − 2 (none below genus 2), and the logarithmic derivative gives Σ_(d | n) d·A_d^(n/d) = w_n + (1 + 2ⁿ)[0], with A_d the sum of the classes of the degree-d places, A^(k) pushed forward by multiplication by k, and w the coefficients of zE′/E. Division by n comes out whole at every degree, and with ∞ dropped from degree 1 the counts equal a count by root finding over F₂[x]/(ϖ), ϖ running over the irreducibles of F₂[x], an inert place counted at class 0. The class numbers are 1, 2, 3, 4, 5 and 15 by Cantor's enumeration, each the zeta numerator at 1.

A supplied degree with no place of class 0 is principal-free. To degree 400 they are none, {1, 4}, {1, 3}, {1, 2, 4}, {1, 5} and {1, 2, 4, 5, 7}, and every supplied degree above them to 400 holds at least two principal places (at least one at F₂[x], where every place is principal). They are finitely many on every such curve: the places of class 0 are those splitting completely in the Hilbert class field, a curve over F₂ in which ∞ splits, and the Weil bound on that field counts about 2^d/(d·h) of them at degree d, h the ring's class number, so every large degree holds one. At the trivial group the walker is LIMIT's: 603 menus and states equal to limit.py's corner walk over F₂[x].

The element reading sees only classes

Tier: criterion (proved in any Dedekind domain with finite class group; brute force at three rings).
Verifier: proof; sensor.py::section_c; sensor.py::section_s.

Two nonzero non-units α and θ are element-coprime when no non-unit element divides both, ideal-coprime when (α) + (θ) is the whole ring. They are element-coprime iff the classes of the primes of G = (α) + (θ), taken with multiplicity, form a zero-sum-free sequence in the class group: no nonempty part of it sums to 0. So the two tests agree on every pair iff h = 1, h the class number, and a hidden gcd, one the element test cannot see, carries at most D(Cl) − 1 primes, D the Davenport constant and Cl the class group.

Proof. A common non-unit divisor δ gives a principal ideal (δ) dividing G, and a principal divisor (δ) of G gives δ dividing both; a divisor of G is a part of its prime sequence, principal iff its classes sum to 0. If h = 1 every nonempty sequence sums to 0. If h > 1, take a nonprincipal prime P and integral I₁, I₂ in the class of P⁻¹ with I₁, I₂ and P pairwise coprime, which every class allows; then PI₁ = (α) and PI₂ = (θ) meet in G = P, one nonzero class. A zero-sum-free sequence is at most D(Cl) − 1 long by the definition of D, and for a cyclic class group D(Z/h) = h.

So the element reading of a gcd is a reading of its class sequence and nothing finer: it sees a common prime only through a part of the sequence that closes up to a principal ideal. Over Z[√−5] (h = 2), taking the non-unit elements of norm at most 120 up to units, 904 of their 3240 unordered pairs are hidden, each gcd one nonprincipal prime, the norm-least being 2 against 1 ± √−5. Over the ring of Q(√−23) (h = 3), hidden gcds of one prime and of two occur and none longer, so the bound D(Z/3) − 1 = 2 is attained; over Z[i] none is hidden. The script checks a brute search for common divisors against the criterion at every pair of non-unit elements of norm at most 120, up to units, at each ring, after checking the class map against every element of norm at most 3000.

The class gap between principal and all ideals

Tier: theorem (the identity, by character orthogonality); rule (the genus form at Z[√−5], verified n ≤ 10⁴).
Verifier: proof; sensor.py::section_g.

Counting elements up to units counts principal ideals, which is the count of all ideals over h plus the class group's nontrivial L-functions over h. A principal ideal's indicator is (1/h) Σ_χ χ(I) over the characters of Cl, so

Σ_(I principal) N(I)^(−s)  =  (1/h) Σ_χ L(s, χ),

the trivial character giving the ring's Dedekind zeta function. At Z[√−5] genus theory names the one nontrivial L-function as L(s, χ₋₄)L(s, χ₅), with χ₋₄, χ₅ and χ₋₂₀ the Kronecker symbols (−4/·), (5/·) and (−20/·): the principal ideals of norm n number (φ_n + ψ_n)/2, with φ_n = Σ_(d | n) χ₋₂₀(d) and ψ_n = Σ_(d | n) χ₋₄(d)χ₅(n/d), at every n ≤ 10⁴, and with ψ_n dropped the count fails.

The coarsest colouring

Tier: theorem (the witnesses built at six rings for every pair of colours of degree ≤ 6).
Verifier: proof; element.py::section_k.

No coarser colouring than (degree, class) tells every state, an assignment of exponents that is principal, from a non-principal one and prices every menu. A coarsening merging colours (d₁, c₁) and (d₂, c₂) identifies two assignments that exchange the exponents of an item X₁ of the first and an item X₂ of the second. A witness is such an exchange where one side is principal and the other is not, or where both are principal and their least costs differ. Neither reads a colour, so a witness stands against every coarsening that merges its pair, and every proper coarsening merges one: pairs suffice. Every class holds an affine effective divisor avoiding X₁ and X₂, a pad: of its 2^(n + 1 − g) − 1 effective divisors of a large degree n on the projective curve, at most 2^(n − d₁ + 1 − g) − 1 contain X₁ and 2^(n − d₂ + 1 − g) − 1 contain X₂, and one containing neither is a pad once its multiple of ∞ is dropped. If the classes differ, X₁ one above X₂ beside a pad making the assignment principal gives an exchange moving the class sum by c₂ − c₁ ≠ 0. If they agree, at c, with d₁ < d₂, fix a pad for every class and let k be their largest exponent; take T a power of 2 above d₂ + R + k + 1, seat the pad of class −Tc, X₁ alone or X₂ alone at T, and principal places outside the pair and the pad, at exponent 1, covering every supplied degree to d₂ + R (the context; a principal-free degree is covered through a multiple holding a principal place, which the finiteness of the principal-free degrees provides). The notch is T on both sides and the core at T has door 1; every other seated item's clock costs at least T + 1 − k, an unseated item of a covered degree more, and an opening more than d₂ + R. So the least costs are d₁ + m(−c) against d₂ + m(−c). The verifier builds the witnesses with MINREP(−Tc) as the pad, doubling T until it avoids the pair: over the six rings 531 cross-class and 118 same-class pairs carry them, and without the context 28 of the 118 price alike. So a ring reaches the dynamics through two numbers per place and a count, with its class group's addition, which m and MINREP read.

The minimal rider is unique

Tier: theorem.
Verifier: proof; element.py::section_m.

Every class of Pic⁰ of an imaginary hyperelliptic curve has exactly one reduced representative (Mumford's representation; Cantor, Math. Comp. 48, 1987, for y² = F(x) in odd characteristic, and Koblitz, J. Cryptology 1, 1989, for y² + H(x)y = F(x) in every characteristic): a pair (u, v) with u of degree ≤ g dividing v² + Hv + F, standing for an effective affine divisor of degree deg u. Let Δ attain m(c). Its degree is at most g, the reduced representative's; it holds no inert place, no place with its conjugate beside it and no ramified place twice, since each is the divisor of a polynomial in x and removing it lowers the degree. So Δ is reduced, and Δ is the reduced representative: every class has one minimal rider, m(c) = deg u ≤ g. A place in MINREP(c) is the only place of its colour, since a second would give a second minimal rider; and m is a shortest path in the class group, m(c) = min over colours (d, c′) of d + m(c − c′) at c ≠ 0, with m(0) = 0, read off σ and the group's addition, with no further reference to the curve. At all six rings the shortest path meets the reduced divisor in degree and in support at every class, and R = g: 0, 1, 1, 1, 1, 2.

The bare door

Tier: theorem.
Verifier: proof; element.py::menu.

A principal effective divisor V raising λ carries some place Q of colour (d, c) at least to Q's door r, say to v ≥ r. V − vQ is effective of class −vc, so deg V ≥ d·v + m(−vc), and MINREP(−vc) plus v − r copies of Q is effective of class −rc, so that is at least d·r + m(−rc). At equality V − vQ is MINREP(−vc) and (v − r)Q + MINREP(−vc) is MINREP(−rc), the minimal rider being unique (The minimal rider is unique). So the least moves are exactly r·Q + MINREP(−rc), costing d·r + m(−rc) at the door r, and a longer core never wins, and the shortest path makes it arithmetic, m(−rc) ≤ j·d + m(−(r + j)c) for a core j steps past its door: 13.7 million longer cores offered, up to g + 1 past the door, none cheaper. The cost is a price of the core's colour and door; the effect is not a price: the move raises places outside its core, which no one-item move does. Two cores can summon one divisor, so the menu is read by vehicle, the whole divisor a move multiplies in, core plus rider.

Greed pays no avoidable rider

Tier: theorem.
Verifier: proof; element.py::analyse.

An opening at an uncovered degree d costs d with a principal core and d + m(−c) > d otherwise, and every item of an uncovered degree is unseated. So an opening pays a rider only at a principal-free degree, each opened at most once, and the transient — the walk up to the last opening that pays a rider — is finite whenever the principal-free degrees are. Past it, only clock moves summon riders, at most R units each. Over every walk at the five curves h2 to h5 and g2 the transient ends at step 4 to 9 of 300 (steps counted from 0), and over the six rings, F₂[x] with them, none of 43,503 openings paid an avoidable rider, an opening in the stretch counted once per branch.

The margin step

Tier: theorem.
Verifier: proof; element.py::chain_step.

LIMIT's chain step fails twice here: a rider lifts a rival with no door paid, and d·r + m(−rc) is not increasing in the door across classes. What survives is a margin, R below the holder's notch: a place standing that low, lifted as the move's core, takes over only by falling in degree. Let the holder Z's last clock move meet notch t, so Z stands at t + 1 or above, and let the next clock move lift a place Y ≠ Z, standing at a_Y, past the notch T′ as the move's core, at door T′ + 1 − a_Y. Z's door is at most T′ − t, so its cost is at most d_Z(T′ − t) + R. If d_Y ≥ d_Z, Y's cost is at least

d_Z(T′ + 1 − a_Y) = d_Z(T′ − t) + d_Z(t + 1 − a_Y),

which exceeds Z's once a_Y ≤ t − R, and greed would have moved Z. So a change of holder to a place standing at most t − R, lifted as the move's core, falls in degree. At the price d·r, LIMIT's chain is the case R = 0, where every rival stands at most t and the premise always holds.

The eventual chain

Tier: rule (verified over every branch of six rings, an 8-move stretch continued to 300 moves; the argument below lacks its induction's base).
Verifier: element.py::analyse, element.py::section_w.

Past the transient every clock move at least doubles the notch and brings at most R rider units. Fix a notch t₀ past the transient. An item no clock move has lifted past the notch since t₀ stands at its exponent then plus R per clock move, O(log t), below t − R at a large notch t. A strand made past that notch stands at most t + j, j = O(log t) its rider units, so its door is at least T′ − t + 1 − j; if it left the chain above the later holders' degrees, every change past that notch having fallen, its cost is at least (d_Z + 1)(T′ − t + 1 − j), above the holder Z's d_Z(T′ − t) + R (The margin step) once T′ − t > R + (d_Z + 1)(j − 1), which T′ ≥ 2t and t > R + (d_Z + 1)(j − 1) ensure. No rider lifts an item past T′ ≥ 2t then either, every non-holder standing below T′ + 1 − R. So past the transient and a finite notch every crossing is made by a move's core and every change of holder falls: the changes are finitely many. A walk clocks forever, an opening costing at least its degree and each degree opening once while a clock move's price stands still between clock moves, so it has exactly one runaway. The step not written is the induction's base: an induction over the changes past that notch takes the next holder to be an item below t − R, which the margin step sends down in degree, or such a strand, which this bound prices out, and leaves the items last lifted between t₀ and that notch, the holder there among them, unplaced. The walks agree at every branch run: no change of holder past step 9 or notch 8 (Where the chain changes hands). The rider-fed coordinates beside the runaway can still grow; the orbit law below sorts them.

Where the chain changes hands

Tier: rule (verified over every branch of six rings, an 8-move stretch continued to 300 moves: 2, 2, 2, 4, 20 and 120 branches).
Verifier: element.py::analyse, element.py::section_w, element.py::apply.

A branch is a distinct state, notch and seating, that the ties reach in the 8-move stretch, walked on canonically to 300 moves; paths reaching one state keep the first, so the changes of holder below are read on one history per state. Eleven changes of holder stood under the margin step's premise, one at h2 and ten at g2, and all fell; seventy-eight more fell without it, all at g2. Fifty-two did not fall, all at g2 and within the transient, and every one is a hand-off to the conjugate colour of the holder P's, where its image ι(P) under the curve's involution (x, y) ↦ (x, y + H(x)) lies: same degree, negated class, and standing level with P, so the premise, a place at most t − R, fails. None of the 52 vehicles is P^r·ι(P)^r. No change of holder comes after step 9 or notch 8 at any branch of any ring, but 53 come after the transient (one at h2, 52 at g2, all falling), so the change range, the steps up to the last change of holder, is not the transient. No rider crosses the notch alone on any branch's history, and in all 28 crossings a rider joins, counted once per move applied (merged successors included, so not on one history per state), the core crosses and every crossing place lands at the core's exponent.

The orbit law and the rider recursion

Tier: theorem (past the change range; verified over every branch of six rings, an 8-move stretch continued to 300 moves).
Verifier: proof; element.py::analyse, element.py::verdict.

Past the transient and the last change of holder, the only rider source is the runaway C's own move, and an era runs from one of C's clock moves to the next. At notch T its door r = T + 1 − a_C summons MINREP(−rγ), γ the runaway's class, of which ρ units land on C; C stands at T + 1 + ρ ≤ 2T since ρ ≤ R < T, so

T′ = 2T,   r′ = T − ρ,   ρ read off r mod n, n = ord γ.

The steady state is the self-map (T, r) ↦ (2T, T − ρ(r)) of (Z/n)², eventually periodic within n² steps. A colour some MINREP(−rγ) uses at an r in the cycle gains units forever and is unbounded; one used only in the pre-period stops at a finite exponent. The verdict reads the state at the change range's end, which the walk provides. Every door and every rider of the 895 eras past the change range is the recursion's.

The element limit

Tier: rule (verified over every branch of six rings, an 8-move stretch continued to 300 moves; unbounded and stopped read by the orbit law's recursion from each branch's seed, a limit past the eventual chain's missing base).
Verifier: element.py::section_w.

The limit is ∞ at the runaway, the cycle's colours unbounded, the pre-period's and the transient's leftovers at finite exponents, each opening's place at exponent 1 unless a later move raised it, and at some branches places at exponent 1 at a degree never opened, a rider's: one at both of h3's branches, two at 2 of h4's 4, one at 4 of h5's 20 and at 35 of g2's 120. At F₂[x] it is the ideal limit (LIMIT: The ideal limit): γ = 0, the runaway of degree 1 or 2, nothing beside it. At h3 γ = 0 too, the runaway the principal degree-2 place, with one rational place beside it at exponent 1. At h2 and h4 γ has order 2 and the cycle feeds nothing beside the runaway: at h2 the cycle's doors are even and summon nothing, ρ = 0 absorbing, and at h4 they are odd and the one rider they summon is the runaway itself, ρ = 1 every era. Either way the runaway alone is unbounded, beside at most one other place above exponent 1. At h5 and g2 the runaway is a rational place of a nonzero class, γ of order 5 and 15, and a cycle of length 4 feeds two rational places forever: three unbounded coordinates at every branch, the two beside the runaway gaining at most R units an era against a notch that doubles. A pre-period of length 1 leaves one more colour at a finite exponent at 23 of g2's 120 branches, and with the transient's leftovers g2 ends with two to five places above exponent 1 beside the runaway.

The number rings price in (min, ×)

Tier: rule (verified at Z[√−5] and Q(√−23): the classes against principality over the places of norm ≤ 50, m against every ideal of norm ≤ 200, the menu at 3,180 states over 53 seeds).
Verifier: element_ring.py::section_classes, element_ring.py::section_rider, element_ring.py::section_walks.

A place's class is its reduced binary quadratic form. m(c), the least norm of an ideal in class c, is a shortest path in the class group with each class's edge weighted by the least norm of a place in it, taken in (min, ×): a least ideal keeps its class when a place is swapped for a least place of the same class, so the curve's (min, +) holds after logarithms. The bare door's proof holds over any Dedekind ring with finite class group, norms multiplying where degrees added: a principal V raising λ carries some place J of class c to a depth v at least J's door r; V·J^(−v) is integral of class −vc, so N(V) ≥ N(J)^v·m(−vc) ≥ N(J)^r·m(−rc), since J^(v−r) times a least ideal of class −vc lies in class −rc; and at equality V is J^r times a least ideal of class −rc, so the least moves are the cores at their doors times least ideals of class −rc, which need not be unique there. What the matrix cannot carry is the door itself, which reads each place's own ladder λ_P, λ_P(b) = λ(P^b), against L: over F₂ every place shares the notch, and a number ring hands the walker a ladder per place: at a quadratic ring places of one norm share a ladder, and in general they need not (CLOCK: A norm does not fix a column). At both rings each class's minimum is attained once, m is (1, 2) and (1, 2, 2), and the menu read from the matrix and the ladders equals the ring's least raising elements at every state, where 1,528 of its 3,196 vehicles are compound, their support more than one place. None of 36,078 longer cores, up to 3 past the door, is cheaper, which at these two rings is arithmetic: every least ideal has norm 1 or 2, so a core j ≥ 1 past its door r costs at least 2N(J)^r, no less than the door's N(J)^r·m(−rc).

The lock carries its rider

Tier: rule (verified at the 53 seeds, the void and every principal ideal of norm ≤ 40, the last 20 of 60 moves; each lock's permanence proved).
Verifier: element_ring.py::section_walks, module_law.py::section_e.

Every seed ends on one vehicle repeated over its last 20 moves: (2), the ramified place over 2 squared, at all 26 of Z[√−5]; (2) = P₂P₂′ at 24 of Q(√−23)'s 27, and the inert (5) at 3. Each ride is permanent: at both rings 4 is the least norm of a non-unit and 2 the only element of norm 4 up to sign, and (2) raises L at every move once the deeper place over 2 gives the whole 2-part of L and is past its head (CLOCK: The ladder is ψ's orbit, save a head at the bend), where its column count (CLOCK: A ring's door is one valuation and its clock a count) rises by one every e units of depth, e its ramification index, deepening the one place over 2 of Z[√−5] by its e = 2 and the deeper of Q(√−23)'s two by its e = 1; the (5) lock holds because (5) raises L at every depth and no element of smaller norm ever raises L again (GROWTH: The lock over a number ring). A lock's vehicle is P^r times a least ideal of class −rc at a constant door r, so every place X of it gains its valuation there in units a move and the rider grows at the core's rate: at Q(√−23) both places over 2 gain one a move with a constant gap, the deeper set by the seed; at Z[√−5] the door is 2 against a class of order 2, so −2c = 0 and the lock has no rider. The margin step does not port: over a curve a clock move brings at most R rider units while the door doubles, so the rider share of the runaway's growth falls to zero; at a number ring's constant door the share is fixed, and the limit's unbounded places are the lock vehicle's support.

The element lid

Tier: theorem (the lid and the four statements over any number ring; the lock's one prime over an imaginary quadratic field).
Verifier: proof; rider.py::section_lid, rider.py::section_seeds, rider.py::section_table, rider.py::section_bite.

A place Y's price at door k is π_Y(k) = N(Y)^k·m(−k·c_Y), nondecreasing in k by the bare door, and its tail price is Λ_Y = π_Y(e_Y), e_Y the ramification index of Y over the rational prime below it. The bare door bound holds for every place of a vehicle, not only its core: a principal ideal in which Y has valuation k has norm at least π_Y(k). A block holder over p is a seated place at count exactly V = v_p(L) (CLOCK: A ring's door is one valuation and its clock a count), V being in this section the block's count and no longer a move's divisor; past its tail index its door is at most e_Y, so its next move costs at most Λ_Y. A creeper over p is a place Y over p some least ideal holds at least once and fewer than e_Y times, so that a rider can move it fewer than e_Y depths, less than one tail gap (CLOCK: The two numbers). The block is lidded when it has a block holder, V is past the tail index of every place over p, and p^(V+1) ≥ B, the largest Λ over its block holders and the creepers over p; its lid is the largest Λ over its block holders. These are the element world's own terms, priced with riders, not CLOCK's lid for the ideal walk.

A lidded block stays lidded, and every later price is at most its lid, which never exceeds B and, with no creeper over p, never rises. A jump of the count (CLOCK: A ring's block keeps one runaway, in counts) needs a newly seated place of norm ≡ 1 mod p^(V+1), and any vehicle holding it costs more than B. A place a move leaves at the top count after k depths has Λ at most the price paid when k ≥ e_Y, by the bound read at it; with k < e_Y it is either the place of the move's core, a block holder already, having crossed from count V at its door, or a creeper riding. So the four statements of CLOCK: A lidded block locks the walk for good stay one when moves must be principal: a walk reaches a lidded block, clocks some block infinitely often, makes finitely many openings, and has its prices bounded infinitely often, together or not at all. Past a lidded block every price is at most B and an opening costs at least its core's norm, the riders being finitely many places, so finitely many openings follow. Finitely many openings leave finitely many seated places, one of them a core's place infinitely often; prices bounded infinitely often bring cores of bounded norm, each opened once. Either way some block's count rises by a core's crossing infinitely often, so V passes every tail index over p and p^(V+1) every Λ over p, and after such a rise the crossing core's place is a block holder: the block is lidded. What does not carry is one vehicle forever, whose ideal proof reads single places.

Over an imaginary quadratic field the places over one prime share Λ, a split pair being conjugate with inverse classes and m(−c) = m(c): p·m(c) at a split place of class c, p² at a ramified or inert one. So a block's lid is constant once lidded, and it exceeds the ideal lid N(Y)^(e_Y) only at a split non-principal place, by a factor, the leading coefficient of its class's reduced form: at Z[√−5] the pair over 3 lids at 6. And a lock's rider lies over its core's prime. Let W = S^r·U repeat forever, S over p and U its rider. S's conjugate, or S itself when ramified, gives an ideal of class −r·c_S at norm p^r, and an inert S is principal, so N(U) ≤ p^r. If U held a place over q ≠ p, W's places over q would grow without bound, and at each rise of the q-count one of them, Y, is a block holder, past the tails once the count is, pending at most Λ_Y ≤ q²; but N(W) ≥ p^r·N(Y) > q², since q ≤ N(U) ≤ p^r and p^r ≠ q. Then W is not a least move. The quadratic field enters twice: in the bound N(U) ≤ p^r, and in Λ_Y ≤ q², which in a field of degree n reads only q^n.

Over the 122 fields Q(√−f), f squarefree and at most 200, of class numbers 1 to 20, the void walked 40 moves with every tie branched gives 207 ends and 7762 prices after a block is lidded: no lid lost or risen, no price over it, no jump on a lidded block, every end with a lidded block by move 4, and 3567 prices over the ideal lid at 13 fields. The 1206 principal seeds of norm ≤ 40, walked one path each, keep the lid: no lid lost or risen, no price over it, no jump on a lidded block. Of 4062 seeds built on a split non-principal place of norm ≤ 60 whose least rider lies over another prime, 100 take that two-prime vehicle and every one leaves it; no lock read lies over two primes. At Q(√−15) the class of the places over 2 has order 2 and holds both, so it has two least ideals, and P₂² and P₂P₂′ tie at 4 at every move: a number ring's minimal rider need not be unique, and the walk repeats a price rather than a vehicle.

Open fronts

Whether a lock's rider can sit over another prime than its core in a field of higher degree; whether a walk with finitely many openings always settles on one vehicle up to ties; and the eventual chain's induction base, the items last lifted between a notch past the transient and the large notch, with an explicit notch bound.