The shopped carrier

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 place of residue degree 1 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 residue-degree-1 place over 29, so the two prices standing in one universe are 8 and 29. And a walk pays it: of a belt of 35 seeds — every seed of norm ≤ 64, each walked to lock — 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 tame one: a column with no head is the standard staircase exactly when e ≤ p − 1, and at a wildly ramified place the p-th power map climbs faster than the staircase at the bottom even with no head — Z[2^1/3]'s place over 2 has v₂ running 0, 1, 2, 2, 3, so its window at k = 2 is 2 against a formula reading 4. Every headless consumer read here is tame, and columns computed place by place confirm the window at every k. 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 a place over 2 of residue degree 1 runs one depth WIDER and Z[i]'s ramified place one depth NARROWER, 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. And a place over 2 of residue degree 1 is headed nearly for free, so the cheap supply and the departing consumer turn up in the same ring. The carrier there is only a place over 5 of residue degree 1 — which 5 split and 5 ramified both give, and which supplies v₂ = 2 either way, since what a place hands over is its residue cardinality and never its own column — so nothing has to be engineered, and quadratic fields holding one beside a place over 2 are common in all three shapes 2 can take. That is the whole distance from the degree-3 undercut, whose cheap carrier forced 2 inert and left the head a single seat.

Seat such a carrier beside a consumer over 2, then delete it again, and the deepest depth at which the consumer's door changes is that same window read off the brute column, at every field swept: 3 at a split consumer, 2 at an inert one, 2 at a ramified one, against a filed e(k−1) + 1 of 2, 2 and 3 — right at the headless inert consumer alone. So the departure is what a greedy engine CHARGES, on a menu priced against every other place in the universe, and not only what a unit-exponent column reads.

The transient costs that depth and buys size, which is one mechanism seen from two sides. At Z[i]'s ramified place over 2 the column's v₂ runs 0, 1, 2, 2, 2, 2, 2, 3, 3: it climbs fast at the bottom, which is what closes the window a depth early, and then sits flat for five rungs — and inside the window that flat stretch is what the door has to cross. Seating the carrier at consumer depth 1 there moves the door from 1 to 7, a move costing 2 against 128, where the degree-3 carrier's whole visible effect was one step. A k = 1 carrier cannot reach past the first rung of a flat stretch, so k ≥ 2 is what makes the second half visible at all.

The two halves belong to different objects, though — the depth to the shape, the size to a residue — and a place over 2 does not determine its completion: Q₂ has six ramified quadratic extensions, so one field's reading wears a shape's name without earning it. Write the quadratic fields as Q(√m) with m squarefree. Over every shopped field of |m| ≤ 60 the window takes exactly one value per shape, and it could not do otherwise: the split and inert shapes fix the completion outright, and at every ramified one — writing π for a uniformizer, a generator of the place, of valuation 1 — the square of 1 + π is 1 + 2π + π², sitting at exact level 2 since 2π has valuation 3 — so v₂ reaches 2 at depth 3 and the window is 2 at all six. So the depth a head COSTS is an invariant of the shape, the sweep its control. The door is not, and the ramified shape is what shows it: the door the carrier opens onto is 4 at every split consumer and 3 at every inert one, but 5, 6 AND 7 at the ramified one, with v₂ breaking at depth 6 in Q(√−6) and at depth 8 in Q(i). What it IS a function of is m mod 8 and nothing else: m ≡ 2 and 6 give door 5, m ≡ 3 gives 6, m ≡ 7 gives 7, no exceptions over the 28 — and those three values are the only ones there are, which is the ramified door below and not something a sweep could have told us. And that residue is NOT the completion — it is coarser, which puts the door strictly between the two. Q₂(√m) is fixed by the class of m in Q₂^* modulo squares, whose invariant is v₂(m) mod 2 together with the odd part mod 8, and those 28 fields realize SIX such classes — every ramified quadratic extension there is — against four residues, m ≡ 2 carrying odd parts 1 and 5 and m ≡ 6 carrying 3 and 7. So the door is constant across completions the residue MERGES, and Q₂(√2) and Q₂(√10) print the same column. Z[i] is then the EXTREME of its shape and not its representative, sitting in the m ≡ 7 class — the class that runs longest, for a reason the next claim derives.

Two of the three consumers pay for the supply with a WALK and the third never does, and the third is the one with the biggest door. Taking split, inert and ramified in that order, one field per shape: 429, 148 and 99 walked states seat a norm-5 place beside a place over 2, of which 269, 15 and 66 are load-bearing under deletion — and of THOSE only 156, 13 and none were seated by the walk rather than handed to it in the seed. So the split and inert consumers get the supply charged on a menu, which is what the priced fact had never had; the ramified consumer does not, every load-bearing state it has coming from a seed that already held both places. Its belt supplies no one-line cause for that: those walks lock at seven different places, 17 of the 48 on the carrier itself at 5 per move, so the consumer is not uniformly priced out either. What is measured is the walk-seated count and not a story about why.

At the cubic ring the consumer stays inside its window: no later state in the belt's walks moves that place again. Held there a carrier moves the LIMIT and not only a door — at a planted pair, the norm-8 place and the residue-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. The norm-5 carrier moves the limit at all three consumers, but deleting it removes its supply and its availability as a cheap VEHICLE — the place a walk's recurrent move rides — in one move, and no odd-norm place supplies nothing — every odd norm q has q − 1 even. What parts the two is which place the carried walk locks on: at the split and inert consumers it is a third place — the ramified place over 3 at 9 per move, and the norm-19 place at 19 — so there the carrier moved the limit without being it. At the ramified one the walk locks ON the carrier, 5 per move against 9 per move without it — so the single row where the recurrent price goes DOWN is the single row where supply and vehicle cannot be told apart.

Scope. Rules in range for the window arithmetic, for the shop and for the swept readings; observations for the censuses and for the limits. The cubic ring is found by the same enumeration of the cubic fields of |d| ≤ 2000, and its belt is 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. Both limit readings are one planted pair each, and the cubic one's two menus are both free of ties, so that difference is not a tie-break's. A narrower belt of norm ≤ 40 read no carrier state at all, 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. Over 5 the sweep is the 41 quadratic fields of |m| ≤ 60 holding a norm-5 place beside a place over 2 — 5 with 2 split, 8 inert, 28 ramified — and it carries the k = 2 window, the deepest load-bearing depth and the depth-1 door; the door's m mod 8 sort is a rule over those 28 fields and not a statement of which extension gives which. The belts, the limits and the k ≥ 3 rows are ONE field per shape, so a shape's name on any of those is not earned; the belts are 106, 28 and 48 seeds of norm ≤ 64, every one of them locking. Each deletion removes ONE norm-5 place, so where 5 splits the second norm-5 place stays seated and supplies the same k = 2, which makes the walk-seated counts a lower bound on carriers and never an upper one.

verifiers: explore_cubic_carrier.py, explore_carrier_window.py, explore_norm5_carrier.py

Everything above reads doors off measured columns. The ramified one does not need measuring, and deriving it says something the columns cannot: how many values the door can take at all.

Fix a squarefree m with 2 ramified — that is m ≡ 2 or 3 mod 4 — and give the place over 2 a state carrying exactly 2². A ramified place over 2 has only two residues, so every unit there is 1 plus something divisible by the uniformizer, and the whole exponent column is a pure power of 2. The door is therefore set by the least number of 2s that u⁴ − 1 carries at that place, minimized over units u.

Taking norms makes that a question about ordinary integers. At a ramified place the number of 2s an element carries is the number the norm of it carries, and u⁴ − 1 factors as (u − 1)(u + 1)(u² + 1), whose three norms are N − T + 1, N + T + 1 and (N − 1)² + T², for N the norm of u and T its trace. The last of those is a SUM OF TWO SQUARES, and that is where the answer lives.

Split on how many 2s u − 1 carries. Two or more, and the total is at least 7: three or more makes the other two norms carry 2 each, and exactly two makes u + 1 carry a third, since a unit here is 1 plus something and so u + 1 is always even. Exactly one, and the two outer norms carry exactly one each, so the sum of two squares decides alone. It gives 3 at m ≡ 2 mod 4 and 4 or 5 at m ≡ 3 mod 4. So the door is 5, 6 or 7 at every ramified quadratic and never anything else: the three values the sweep found are the three that exist, and the table above is complete rather than a sample.

Which class runs longest, and why, falls out of the same sum. For m ≡ 3 mod 4 the door is 6 plus the number of 2s in s² + 1, where s = (m − 3)/4. An odd square is 1 mod 8 — so when s is odd, s² + 1 lands exactly 2 mod 8 and contributes one factor of 2, and when s is even it contributes none. s is odd exactly when m ≡ 7 mod 8. That parity is the whole reason the m ≡ 7 class runs one rung longer — not a depth, and nothing about how any column happens to be shaped. The two even classes agree for the mirror reason: there the derivation reads m/2 only as odd and never any finer.

And that settles what owns the door. No step of the derivation reads m beyond mod 8, which is why the door is constant across completions the residue merges — the coincidence noted above is forced, not observed. The completion still decides which of the six ramified extensions the place sits in; the door it does not decide, and the length of the flat stretch the column shows is a consequence of the residue rather than a cause of the door. Checked at 1619 fields to |m| ≤ 2000 with no exceptions, though the proof does not rest on the check.

Scope. A derivation, holding for every squarefree m with 2 ramified, and it needs one hypothesis the page supplies rather than the theorem: that the state the place is handed carries exactly 2², which is what a norm-5 supply gives it. The case analysis is finite and complete — three cases on how many 2s u − 1 carries, and two parities inside the last — so the range checked at 1619 fields to |m| ≤ 2000 is corroboration and not the scope. What is NOT claimed: anything about the split and inert doors, which stay column readings; and anything about whether a walk ever pays this door, which is the observation above and is untouched by deriving the price.

verifier: explore_ramified_door.py