Which rings walk

The model reads λ at depth a as (N−1)·p^{⌈(a−1)/gap⌉} — the prime-to-p part N−1 at every depth, the power of p climbing one factor per gap. Call a column standard where that holds. The model is derivable and not only fitted: λ at depth a is the exponent of the units modulo the place's a-th power — the filtration the door's premium is counted in — and wherever e < p − 1 the p-adic logarithm converges on the principal units and forces exactly that staircase. Below that line the standard column is forced, and the colour and the gap are the whole hand-over. Above it the model may miss, and every measured miss sits at e ≥ p − 1 with f = 1 — the licence's own line read as a boundary, not tameness and not p = 2: the model misses at both ramified places of the doubly ramified ring Z[x]/(x³ − 2), in which 2 and 3 each ramify totally, one of them tame in the p ∤ e sense and one the first miss at an odd p. Within that zone the shapes differ, and the two new ones are permanent and headless. Wild ramification — p dividing e — shifts the staircase's PHASE: the place over 3 there jumps one step early at every depth ≡ 1 mod 3 from depth 4 on, its first cubing landing at depth 3 rather than at 1 + e. A doubling zone at a tame place shifts its LEVEL: the place over 2 runs one 2-power ABOVE standard from depth 3 on, forever, the extra factor bought by the squarings at the depths below e/(p−1), where the principal units are still squaring rather than stepping. A residue field of size two drops the level instead. And a plateau wider than e — a head — needs the head criterion's three clauses to meet at once, torsion alone never sufficing: ±1 sits in every 2-adic field, and the level-shifted place carries no head. The residue field can also rescue the zone: the inert place over 2 of Z[x]/(x³ − x − 1) has e ≥ p − 1 and f = 3, and stays standard.

Scope. The licenced half is a derivation; the readings are rules in range. Over two quadratic rings — Z[√−5] and Z[ω] with ω² = ω − 6 — the model holds at 32 of the 35 places of norm ≤ 60, read to depth 14; across those and Z[i], every miss is a place over 2 with residue field F₂, while both tame ramified columns, over 5 and over 23, are reproduced. At Z[x]/(x³ − x − 1) it misses nothing — 551 places of norm ≤ 4000 to depth 14, 0 off, the range where the gap column is exercised at degree three, including the pair over 23 that shares a norm and parts at depth 3. At the doubly ramified ring the two shifted columns are hand-derived from the depth maps and confirmed by an independent brute-forcer — 21 ramified and 39 unramified readings, 0 off, after the same instrument reproduced 90 filed readings at Z[i] and the other cubic — and the head census reads width 0 at every place of norm ≤ 30 in both cubic rings, at depth 14.

verifiers: explore_element_schedule_nf.py, explore_cubic_ring.py, explore_wild_ring.py

A ring with no head anywhere has no walk to speak of. A place whose column is standard at gap 1 has door 1 at every depth — its λ gains a fresh factor of p at every further depth, which no invariant built from the walk's past can already carry — so it is priced at its bare norm forever, and the walk locks on the cheapest SUCH place, which need not be the ring's cheapest place. Both headless cubic rings were walked from the void. At Z[x]/(x³ − x − 1) the two coincide: the least-norm place is the cheapest move the ring ever offers, and the walk repeats it from the first move and touches nothing else. At the doubly ramified ring the two cheapest places are its ramified pair, and the walk can hold neither. The norm-2 place's first column entry is N − 1 = 1, which divides every invariant, so its door starts at 2 over the empty state itself and rises with slope e = 3 in the surplus: the cheapest norm in the ring is walk-INVISIBLE. The wild place over 3 wins the void opening instead, is seated twice and stranded at exponent 2, and the lock lands on the cheapest standard gap-1 column, norm 5. So the three quadratic rings lock late and elsewhere BECAUSE their cheapest place carries a head — Z[i]'s ramified place over 2 is the void's cheapest opening at 4 and is abandoned at exponent 3 when its plateau prices the next door at 32 — and which rings walk interestingly is a question about heads rather than about class groups, with the sharpening that a headless ring's ramified places are unreachable or strandable.

Scope. A rule in range: two headless cubic rings walked from the void through the same walker the quadratic rings ran, both walks locking well inside their runs. Z[x]/(x³ − x − 1) is also the first ring where the head criterion and the reading it replaced disagree and an engine decides: its place over 2 has e = 1, so p − 1 ≤ e predicts a head, while f = 3 fails the f = 1 clause and the ladder carries none — every place over 2 in the three quadratic rings has f = 1, which is why the separating clause was never exercised before. The doubly ramified ring exercises the e = (p−1)p^t clause instead: its places over 2 and 3 both have e = 3, neither a power of 2 nor twice a power of 3, and carry no head.

verifiers: explore_cubic_ring.py, explore_wild_ring.py

The positive half of that question has now been asked at degree three. The first cubic field, ordered by |discriminant|, whose class group is nontrivial — d = −283, Q[x]/(x³ + 4x + 1), the group of order 2, every cubic field of smaller |d| certified trivial — carries its head at its CHEAPEST place, and by the head criterion's simplest shape: the place of norm 2 has e = f = 1, so its completion is Q₂ itself and its column is the exponent of (Z/2^a)^* — 1, 2, 2, 4, 8, …, a flat step, then doubling: tail gap 1, sup gap 2, exact at every depth by derivation rather than sweep. And the ring walks: over the seeds of norm ≤ 40 plus the void, five lock places — 2, 3, 4, 5 and 17 per move — where the headless cubic has a point. The head holds 11 of the 44 seeds at 2 per move, the bare norm-2 place at exponent 1 included: a door-2 jump from a shallow depth clears the flat step in one move and lands where the column doubles every depth, so cost 2 undercuts the universe forever — where Z[√−5]'s headed place loses its shallow grab and holds only when seeded past its flat window, and Z[i]'s is abandoned at exponent 3 above. So heads decide walks at residue degree 3 and nontrivial class group together, and not by quadratic accident.

Scope. A rule in range: 44 ideal seeds through the same walker the quadratic and headless cubic rings ran, all locking; the void locks the norm-3 place at 3 per move. The column model behind the walks is brute-forced at 44 cells over every place of norm ≤ 30, 0 off, the head column read to depth 13; the walks consume deeper cells under the closed forms, which at the two places over 2 are licensed by derivation — the head column as (Z/2^a)^*'s exponent, the norm-4 column's staircase bounded from both sides — with the brute as the check. The head and the class group carry separate certificates: the group's order sandwiched between a relation kernel and an exhausted-box witness, the kernel's bits equalling the certificate's at all 8 generator places.

verifiers: explore_cubic_field_shop.py, explore_headed_cubic_walk.py

A carrier supplies nothing until it is seated, and the price of seating it is its norm raised to its own door — so the norm is what sets its cost, and the residue degree is what sets the supply that cost buys. At a place of residue degree 3 over the rational prime q the prime-to-q part of λ is N − 1 = q³ − 1 = (q−1)(q²+q+1), so the place supplies the primes ℓ ≡ 1 mod 3 dividing q²+q+1 from a rational prime far below anything q−1 or q²−1 reaches: a supply of 7 costs norm 8, where residue degree 2 and below needs 29, and a supply of 13 costs 27 against 53. It also ATTAINS the floor every supplier has — a place carrying v_ℓ = k needs norm at least ℓ^k + 1 — where degree 2 and below is slack, 8 being neither a prime nor a prime square. What the cheap end costs the rest of the ring is fixed by the same arithmetic. A supply is cheap at degree 3 only when q is small, so the configuration wants 2 INERT; and in a cubic field e ≤ 3, so the head criterion's e = (p−1)p^t admits only (p, e) = (2,1), (2,2) and (3,2). Taking 2 inert kills both shapes over 2 — the one place there has f = 3 where the criterion needs f = 1 — and leaves the partially ramified place over 3 — e = 2, f = 1 — as the only seat a head has left, and then only where the completion holds the cube roots of unity, which is one of the two ramified quadratic extensions of Q₃ and not the other. So a head and a cheap carrier are compatible, at exactly one place each, and the configuration has a shopping spec rather than an obstruction.

Scope. Rules in range, with the arithmetic derived and the head clause inheriting the head criterion's own tier. The price half is a table: the least residue cardinality carrying each of the nine primes below 24, read at residue degree 3 against degree 2 and below. The compatibility half is a census of the cubic fields of |discriminant| ≤ 2000 — 3906 polynomials reduced to 331 fields, 1529 places brute-read to depth 3 with 0 excess off the three predicted shapes, and 623 further places left to the criterion, every one of them f ≥ 2 or p ≥ 5 and so headless without a brute. Of the 331, 88 have 2 inert and 9 of those carry a head, all 9 at the predicted place and all 9 with the head one depth wide.

verifiers: explore_cubic_undercut.py, explore_cubic_carrier.py

The shopped ring is d = 321, Q[x]/(x³ − x² − 6x − 3), totally real: the first field by |d| holding 2 inert (one place, norm 8, with N − 1 = 7), a head over 3, and a degree-1 place over 7 for the supply to be read by. Its undercut is charged on a menu rather than read off a table — the cheapest place of residue degree 2 or below in that same ring supplying v₇ ≥ 1 is the degree-1 place over 29, so the two prices standing in one universe are 8 and 29. And a walk pays it: over a belt of 35 seeds of norm ≤ 64, three reach a state seating the norm-8 place beside a consumer over 7, and at the first of them the consumer's door reads 2 where the same state with the carrier deleted reads 1 — the consumer being the norm-49 place, a price of 2401 against 49, its own norm squared where it had been paying that norm. The head does not forbid this: 3 per move is what the head charges once LOCKED, and from a state that leaves its own door above 1 it charges 27 or 243, so a norm-8 place is reachable on a menu the head is standing in. A walk also locks ON the norm-8 place once, at 8 per move.

Call a carrier load-bearing at a consumer when deleting it from the state lowers that consumer's door, which is the reading the deletion above makes. The supply is load-bearing only while the consumer is shallow, and that is also why it keeps biting. A consumer seated over ℓ at depth a, with its own ramification index e, already supplies v_ℓ from its own column, so a carrier supplying k is load-bearing exactly to the deepest a whose self-supply is still under k. On the standard column that supply is ⌈(a−1)/e⌉ and the window is a ≤ e(k−1) + 1 — the consumer's residue degree never entering, and its ramification cancelling at k = 1, where the bound reads a ≤ 1 for every e. That window is the headless one, and measured against columns computed place by place it is exact at every consumer with no head, at every k read. The consumer over 7 here is one of them: its window is the single depth 1, 7 dividing 7 exactly once.

A headed consumer — one whose column holds a flat step its ramification does not account for — departs from that window, and the interest is that it departs in BOTH directions. From k = 2 the unramified place over 2 runs one depth WIDER, and the ramified place of Z[i] runs one depth NARROWER at k = 2 — a window of 2 where the headless form says 3 — before running two wider at k = 3. So a head can COST a carrier depth as easily as buy it, and there is no single correction to apply: the window is read off the column. What makes the direction go either way is the head's transient, the column climbing FAST at the bottom before it flattens, so a shallow window can sit inside the fast stretch and a deeper one inside the flat. At k = 1 nothing departs at all — the window is the single depth 1 at every place measured, headed or not, and it cannot be otherwise, since that window needs a self-supply of zero and every column starts at zero and reaches one by depth 2.

Both ways of widening that window are open at the smallest prime there is. A supply of v_ℓ = k needs a carrier of norm q ≡ 1 mod ℓ^k, so its floor ℓ^k + 1 is set by how small ℓ is and not by residue degree — which is why the degree-3 undercut above is a k = 1 effect, the primes it reaches being those ≡ 1 mod 3 and never 2. At ℓ = 2 the floor is ATTAINED four times over, at norms 3, 5, 9 and 17 for k = 1, 2, 3, 4, all four inside the belt these walks charge; k = 5 is the first to leave it, at 97 against a floor of 33. At ℓ = 3 the belt holds k ≤ 2, and at ℓ ≥ 5 it holds k = 1 alone where it holds anything — at ℓ = 17 and ℓ = 19 it holds nothing, their cheapest supply of any k costing 103 and 191. And a place over 2 with residue degree 1 is headed nearly for free, so a cheap carrier and a departing consumer turn up in the same place: a norm-5 carrier supplying k = 2 is load-bearing to depth 3 at an unramified consumer over 2 against the headless 2, and to depth 2 at the ramified place of Z[i] against a headless 3. The control is the inert place over 2, which has residue degree 2 and therefore no head: it sits at the headless window at every k.

The window then holds itself: the raised door prices the consumer out of the menu, a door of 2 costing its norm squared, so the walk stops moving that place and it stays at the depth where the carrier can still be read. And held there the carrier moves the LIMIT and not only a door — at a planted pair, the norm-8 place and the degree-1 place over 7 both at depth 1, the walk locks at 11 per move with the carrier and at 7 per move without it.

Scope. An observation for the census, a rule in range for the window arithmetic, and an observation at one pair for the limit. The ring is found by the same enumeration of the cubic fields of |d| ≤ 2000. The walks: 35 seeds of norm ≤ 64 through the same walker the other rings ran, 397 states, of which 33 seat a carrier beside a consumer, all 33 load-bearing with the consumer at depth 1 at every one, each read against the same state with the carrier deleted; the consumer is moved no further times at any of the 33. The limit reading is one planted pair, whose two menus are both free of ties, so the difference is not a tie-break's. A narrower belt of norm ≤ 40 read no carrier state at all, no seed of that size holding both places — 7 · 8 exceeds 40 — which is why the belt is the wider one. The columns and the supply floor are separate readings: the columns are the unit-group exponents of eleven places over six rational primes, of which the eight over 2, 3 and 7 are the ones any window above is scored at, each computed by brute force over the residue ring rather than from a formula — every window above is read straight off one of those columns as the deepest surviving depth, never from a closed form — and the floor table is the least prime power in each class ℓ^k | q − 1 for the nine primes below 24 and k ≤ 6, scanned to two million. The places read have e ≤ 2 and residue degree ≤ 2, so a wilder consumer is outside what was measured, and the windows run to k = 4, a window deeper than its own column not being scored at all. Nothing above rests on the columns' asymptotic shape, which is reported separately: a window is a shallow-depth reading, and the fast stretch a head puts at the bottom of a column is exactly what an asymptotic reading discards.

verifiers: explore_cubic_carrier.py, explore_carrier_window.py