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 columns read depth by depth and confirmed by an independent brute-forcer — 21 ramified and 39 unramified readings, 0 off, after the same instrument reproduced the 90 recorded 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 is answered at field 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 — the local field it sits in — 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 column's leading flat stretch, and Z[i]'s is abandoned at exponent 3 above. So heads decide walks at field degree three 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

Which places a walk from the void ever seats is set by the kind of its first move — at a void tie, by the break that decides it. Call the place the void's cheapest opening seats the winner. Where the winner is UNRAMIFIED — split or inert, any residue degree — the walk seats exactly one place EVER, up to a void-menu tie at the opening price (where the tie crosses kinds at two norms, below, one branch seats the loser too); an unseated price never falls as the invariant grows, and the prices the walk pays never climb past the rivals. What separates the unramified winners is whether the winner's column ever pauses, and among unramified columns that happens exactly at residue field F₂: squaring a unit 1 + 2u lands 1 + 4(u + u²), and u ↦ u + u² over the residue field is the Artin–Schreier map, which vanishes identically over F₂ and no other finite field — so there every unit congruent to 1 modulo the place squares to 1 modulo its cube, and the norm-2 column pauses once: 1, 2, 2, 4, then doubling. At every other unramified winner the column climbs strictly, at every residue degree f — at odd norm by the licence's staircase, at norm 2^f with f ≥ 2 because the map is nonzero somewhere on F_2^f, so depth 3 already holds a unit of order 4 and every level above climbs by one — its door stays 1, and the walk pays its bare norm, CONSTANT from the first move, every rival having lost the void at or above that price. At norm 2 the winner opens at door 2 — its first column entry is N − 1 = 1, which divides every invariant — for 4, the flat step holds its door at 2 for one more 4, and from the third move the column doubles at every depth, pricing the recurrent move at 2, the least price the family has: the paid sequence FALLS to that floor and never rises again. The tie caveat is structural exactly there — at the quadratic shape a norm-2 winner never arrives alone, since a split 2 hands the void both its places, the loser (the winner's conjugate) tied at exactly 4 — and benign: the conjugate's column is the winner's own, so its price jumps to 16 on the walk's second move and widens with the invariant.

Beyond the quadratic shape the tie CROSSES KINDS and stops being benign. A cubic ring whose 2 factors into two places at residue degrees 1 and 2, holding no place of norm 3 to undercut them, hands the void a norm-2 winner beside a norm-4 rival, both at exactly 4 — the norm-4 place at door 1, the norm-2 place at door 2, nothing conjugating one into the other — and those two are all the places over 2, so the tie is exactly binary. The tie recurs on the second move, and the tree of breaks ends three ways. Seat the norm-2 place twice and the paid falls 4, 4, 2, 2, 2, …, the norm-4 place never seated. Mix the first two moves — the two orders land in one state, so they merge — and both sit seated, paid 4, 4, 4, 2, 2, …, the norm-4 place held at exponent 1 by prices it can never meet again. Seat the norm-4 place twice and the paid never leaves 4, the norm-2 opening priced out without bound. In each ending exactly one coordinate runs away — the norm-2 place in the two that fall, the norm-4 place in the constant one — so the same ring, from the same void, under the same law, runs to two different runaway places: the greedy limit is MULTIVALUED at such a void, and the tie-break chooses the world.

Where the winner is RAMIFIED, its column's flat stretches make the prices the walk pays RISE through the transient — they may touch the floor on the way (Z[i] pays 2 twice) but climb off it — and the climb can hand the menu to a second cheap ramified place. Z[i]'s winner is ramified and it owns no second; Z[√−30] — ramified at 2, 3 and 5, least split norm 11 — pays 3, 3, 5, 5, 17, seating the place over 3 as its opening and the one over 5 on its third move: a ramified place seated from the void that was not the cheapest opening. Seeds free of ramified places tell the same story at scale: the prices climb past a stalled place's and seat it mid-walk, even at exponent 2 in one move.

And the tie can cross kinds AT ONE NORM. A cubic ring staging 5 = PQ² — P unramified of residue degree 1, Q ramified, both of norm 5, both at door 1 — with 2 and 3 inert, so nothing undercuts the pair, ties the void at exactly 5 across the unramified/ramified seam, and the break there chooses between the dichotomy's own readings: constant against rising. Seat P and the paid holds constant at 5 forever, Q's opening priced out without bound. Seat Q and the paid RISES off 5 for good — but the runaway is the menu's to sell, not the winner's to keep. The tie itself forces 2 inert, and the inert place over 2 opens at 8 against every invariant the ramified column alone builds — its column starts at 7, while those invariants are all 4 times a power of 5 — so the risen branch is undercut at 8-or-cheaper in every ring of the family: a place of residue degree 1 over one of 7, 13, 17, 19 or 23 — the cheapest the ring holds — takes the runaway at its own norm, the forced inert-2 seat paying one move at 8 on the way wherever that place is dearer than 8, and the ramified winner reclaims the runaway, paid 5, 5, 8, 25, 25, …, only where no such place exists, the inert-2 seat stranded either way it fires. So in the generic risen branch the runaway is a place the tie rejected at 7-or-worse against 5, while both tie members end stranded or unseated — up to three places seated on one walk — and winning the void appoints the paid direction, never the runaway. This tie never recurs between the kinds and its branches never merge; later ties, where they fire, are same-norm ties among the undercutters, each sub-branch printing one signature. So the paid sequence's direction reads the winner's kind three ways — up to void ties, where the cross-kind mixed branch entered norm-4-first still falls, the norm-2 place taking over on the second move, while the one-norm tie's branches each obey their winner's kind: it falls to the floor and stays exactly at a norm-2 winner, holds constant at every other unramified one, and rises through a ramified winner's transient.

Which ramified places can be caught splits on the prime below them. λ at depth 1 is N − 1, even at every place of odd norm — and only places over 2 have even norm — so every seating of an odd-norm place feeds the power of 2 in the invariant, and the door of a place over 2 climbs in lockstep with the menu: the seed sweeps seated an unplanted ramified place 753 times, never one over 2. An odd-characteristic one can stall instead, its door widening only when its column's next value divides the invariant: Z[√−30]'s place over 5 is pinned at 5 exactly so — its column's every value is divisible by 4, and the invariant's power of 2 is still 2 at the catch. Past a lock no strict seating is possible at all — a flat floor of prices meets nondecreasing ones only at a tie — so every catch happens before the walk locks. And beside an unramified winner it never happens at all: a stalled ramified price is caught by a climbing minimum, and an unramified winner's minimum holds or falls — in the minted rings a ramified place of norm 5 sits beside one norm-2 winner and one norm-4 winner, prices 5 at the void, and stands unseated through both walks. Which places a walk ever holds is the arithmetic of the winner's kind, not a law about ramification.

Scope. The unramified half is a theorem at every ring of integers — the strict climb is the unit filtration's at e = 1, and a rival's price never falls because the invariant only accumulates divisors while that rival's own exponent is untouched — with its premises machine-checked as controls: the odd-norm climb read to depth 21 at both such rings; the norm-2 and norm-4 columns brute-forced on the residue rings (the flat step to depth 17, the strict doubling to depth 8, closed forms matched at every cell, and the completion at an inert 2 the same at all three rings that share it, checked to depth 21); the never-falling price at five rigs — 1,338,247 consecutive unseated-price pairs (0 falls) at one, 153,051 (0 falls) at another, 3,109 walks (0 falls, 0 lost ties) at the third, 122,176 pairs (0 falls) across the fourth's cubic sweep, and 954,324 (0 falls) across the fifth, its controls included. Eleven quadratic rings walked from the void in all, six of them minted for the over-2 cell. The cross-kind tie is a rule: the three endings are derived from the two columns for every cubic ring meeting the signature — 2 factoring at residue degrees 1 and 2 and no place of norm 3, which leaves nothing else priced under 5 — and walked at 92 defining polynomials, at least 17 distinct rings by discriminant, six discriminants positive (totally real fields included), every tie tree printing exactly the three endings and the mixed orders merging in one state. The one-norm tie is a rule: the branch endings are derived from the three columns for every cubic ring staging 5 = PQ² with 2 and 3 inert and no other prime below 31 dividing the discriminant, the ramified column brute-verified on both ramified quadratic extensions of Q₅; witnessed at 9,478 defining polynomials over at least 2,577 distinct rings by discriminant, 1,132 of them walked and asserted branch by branch — every walked void printing exactly the two-member tie, every branch its derived paid, seatings and strand exponents, and the class with no undercutter being 28 polynomials over exactly five discriminants. Not claimed: ties of three (2 totally split), seeded ties of either shape, or ramified ties at any other prime, at higher degree, or under wild ramification. The ramified half is a rule in range: five quadratic rings walked from the void plus the seed sweeps through one walker — 34 mid-walk seatings, and Z[i]'s place over 2 never below 256 across 1,080 seeded walks. The first two rigs are independent and agree where their sweeps overlap; the third rides the first's walker and factory with one corrected column formula, its columns checked against an independent brute-forcer; the fourth is self-contained, its own walker reprinting both published quadratic cells as controls; the fifth is likewise self-contained, its walker reprinting both quadratic cells and the fourth's cubic as controls.

verifiers: explore_late_seating.py, explore_opening_law.py, explore_even_winner.py, explore_cross_kind_tie.py, explore_ram_tie.py