The image and limit

Writing S for an infinite set of primes disjoint from the support of s:

The last four nest strictly in that order; transparency sits outside the chain, being the only one whose single member is finite. Every image parameterizes the same way — a SUPPORT choice, then an EXPONENT choice on it — and the three fates are its extremes. Mortality pins both coordinates minimal and is the only true corner. Breadth pins the support maximal and leaves the exponents free; depth pins an exponent maximal and leaves the support free: each is a FACE, extremal in one coordinate. So two runs can both hold the depth fate and still differ, and a generic member of an image is interior and holds none of the three — a free policy holding no fate is the ordinary condition, not an anomaly. The two coordinates being separately free is also why fate purity fails over Z for free policies: 2^∞·∏(odd primes) carries breadth and depth at once and is reached under both demands that host both. Greedy independence lands where the two coordinates are extreme in opposite directions — every prime, every exponent 1 — which is the primorial limit.

Scope. The chain is a SQUAREFREE-SEED statement: from any other seed semisimplicity admits nothing, its image is the single point {s}, and it leaves the chain by the same route transparency never entered. Independence reaches its shape from any seed, the seed's own exponents frozen. Closure halves exhaustive over a state × move battery, reach halves constructed plus a stated continuation argument; the extremes are a synthesis over the five. Redrawn over F₂[x] — states monic polynomials, moves of degree ≥ 1 — the same five shapes, the same chain and the same extremes, 0 mismatches against the Z table.

verifiers: explore_fate_image.py, explore_fate_image_ff.py

Under the dynamics demand the greedy image takes three cardinalities across the rings run, and the space between them is empty. Over Z it is a single point. Over a quadratic number ring it is FINITE: the run LOCKS — past some move it never opens a new place again — so only finitely many openings ever survive. Over F₂[x] and over the coordinate ring of an elliptic curve over F₂ the openings recur forever and each choice is made permanent, so the image has the cardinality of the CONTINUUM. Nothing sits between: a surviving opening is a tie, hence carries at least 2 choices, and a run's openings are countable — finitely many surviving give a finite product, countably many give at least 2^ℵ₀ and at most ℵ₀^ℵ₀, which is the same cardinal. So no greedy image is countably infinite. All three come from one formula — the product, over the surviving openings, of the tie multiplicity at each — so what separates the rings is how many openings survive and not how deep their arithmetic runs. Over Z the product is empty: the run opens constantly and not one of its openings is a tie.

Scope. The dynamics demand, and the (demand, ring) pair rather than the ring: greedy independence over F₂[x] ties at its openings exactly as dynamics does and still reaches one point, because coprimality READMITS the declined sibling a move later and no opening survives. The number-ring half is a rule in range, censused seeds over two quadratic rings growing by ideals and by elements alike, 0 branch merges; the continuum half runs over F₂[x], the complete h = 1..5 elliptic ladder at q = 2, and a genus-2 ring of class number 15, where the places a declined choice can sit at — the passengers of the next paragraph — widen from the degree-1 places to the supports of the classes' least representatives, and their cap survives; the empty middle is a derivation from the formula and inherits its scope. The SIZE of a RING's finite image is not claimed here — the per-opening product is a lower bound on it, the correction being a within-degree multiplicity the product does not carry. Take the ring out and the size has a law, a different one per mechanism that makes an image finite: how big a finite image is.

verifiers: explore_greedy_image_nf.py, explore_greedy_image_ec.py, explore_greedy_image_g2.py

The multiplicity that formula multiplies is not a free parameter. A move's cost at a place is a power of the place's rational prime, so equally priced moves lie over ONE rational prime. In a quadratic ring that prime carries two places only when it splits, and the two are conjugate — exchanged by the ring's symmetry — so every ideal-world tie is a conjugate pair of multiplicity exactly 2, and the two branches leaving it run move-for-move identical in every cost thereafter: the pair's members are indistinguishable to the pricing, which is the proof and not only the census. At degree three both halves of that argument open, and both openings are measured on the cubic field of discriminant −283. One rational prime can carry places of two NORMS, and that ring's censused ties are all of one species: its norm-2 place moving two depths against its norm-4 place moving one, equal cost 4 over the one prime 2 — a pair no quadratic ring can present, and no conjugation behind it, since nothing in the ring's symmetry exchanges places of different norms. And a prime splitting completely carries THREE places — the first such prime there is 71 — so a state seating all three at depth 1, beside five further split places planted so that each one's contribution to λ covers a cheaper place's next climb, prices its three cheapest moves as a three-way tie, and the three forced branches lock three DISTINCT limits at 71 per move. So the base of the finite count is the tie's WIDTH: 2 is the quadratic width, not a law of number rings.

Scope. The conjugate-pair half is a rule — the identical-pricing argument, with 618 tie encounters across the two quadratic rings' censuses, all of multiplicity 2, as its check. The cross-norm species is an observation at the one cubic ring run — three ties in its 453 censused move readings, plus one constructed seed, all the same pair. Width 3 is an observation at a CONSTRUCTED state: no censused seed reached one, so the claim is that the ring presents the tie and its branches diverge; what seeded runs reach stays geography.

verifiers: explore_greedy_image_nf.py, explore_headed_cubic_walk.py

Where the tick doubles, a greedy run's limit is

one place C carrying an unbounded exponent, one further place P_d at exponent 1 at each degree d the run ever opens, and every seated place other than C standing at exponent 1 FOREVER — a single deep coordinate over a flat support that never stops growing. C is itself one of the opened degrees' places exactly when its degree is 2. Four facts make it. The tick doubles at a repeat move and a fresh opening leaves it alone, so every exponent ever written is 0, 1, or the tick at its own repeat plus 1. No place's price is ever below d·T/2. A deepened place of degree 1 is therefore permanent, every rival's price rising by at least as much as its own at every later repeat, and it settles at its FIRST repeat with no transient at any ring walked. And a fresh opening at degree d multiplies λ by 2^d−1, so degree 1 is never eligible — 2¹−1 = 1 divides everything — while the fresh ladder opens degrees 2, 3, 4, … one place at a time forever, since 2^d−1 carries a prime no smaller one does, by Bang's theorem, whose only exceptions are d = 1 and d = 6, where a square of 3 does the same work.

Scope. Proved by induction over the move model, calling on no computation, for runs growing by IDEALS in a world whose tick doubles; and a rule in range, 11 branches at six function-field rings walked 300 moves each with every move's tick and exponent read. The move model is base 2 throughout, and the leg that makes the support grow FOREVER leans on it: Bang's theorem is the base-2 Zsigmondy, so the endless ladder is a claim about F₂ and not about doubling ticks in general. The deep place's degree is bounded by TWICE the least degree the ring's supply carries, which is 1 at all six — hence 1 or 2, and 0 of 140 censused repeat moves at degree 3 or above. That bound is the doubling clock's and the supply's jointly, so the limit's SHAPE and its deep coordinate's bounded DEGREE are two statements and not one — the schedule is where they come apart, the bound moving with the dial while the deep coordinate stands. That single deep coordinate is a CORNER and not the general law: priced at the clock a ring actually carries, the runaways — the coordinates carried up without bound — number 1 where the cost of a repeat is flat and one per residue characteristic where it climbs. A second runaway needs BOTH a climbing cost and more than one characteristic, and no ring supplies both — a function field climbs and has one, a number ring has many and goes flat (the corner).

verifier: explore_greedy_limit.py

Grow by ELEMENTS rather than by ideals — every state and every move generated by one element — and a move seats a whole bundle of places at once. A bundle can raise an exponent with no repeat move at all, breaking the exponent ceiling the ideal limit rests on, but it cannot do so often: an opening costs the bare degree when the place it seats is PRINCIPAL — generated by a single element — and strictly more otherwise, and past a handful of degrees there are principal places to spare, so a greedy run stops paying for bundles within its first few moves. After that the only source left is the run's own repeat move, one per ERA — the stretch between two repeats — against an era whose length doubles. The limit is then ∞·C, plus at most the places of one orbit of the ideal classes, fixed by C's own class and gaining about one unit an era, plus one place at exponent 1 at each opened degree. It collapses to the ideal shape exactly when that orbit is empty or is C itself. So the two ways of growing differ by a logarithmically deep coordinate and never by the headline.

Scope. Proved for the steady state; a rule in range over six rings walked 300 moves on every branch of the tie sweep, whose branch sets are complete — three extra coordinates at one ring on all twenty of its branches, three to six at another over all 120 — though a COUNT like that is part mechanism and part opening stretch, which the next claim separates. The element world has no reordering lemma — a declined minimal move is still minimal at the successor at only 466 of 719 types at one ring — so its shapes are read over every branch and never off one continuation.

verifier: explore_element_limit.py

Those extra coordinates gain about one unit an era, so asking whether any of them grows FOREVER looks like a question about longer runs — and no run can answer it. The era doubles, so five more of them cost about 32× the moves and buy five more units. The answer is arithmetic instead. A repeat move at the deep place lands it exactly one above the tick and turns the tick over to twice itself, so the next increment is the old tick less whatever the place was handed in between; and that increment, scaled by the place's own ideal class, is what fixes where the next bundle lands. Reduce both the tick and the increment modulo the number of ideal classes and the whole future is a walk on a FINITE set of states. So the classes the bundles visit are eventually periodic, and the dichotomy is exact: a place the CYCLE reaches is fed forever and its coordinate is unbounded, a place reached only by the RUN-IN — the stretch before the states start repeating — stops at an exponent one can name. Over the six rings the verdict is the deep place alone at four of them and exactly THREE unbounded coordinates at each of the other two, on every branch. At one ring the income does stop — and stops harder than a run could show, the state where nothing is handed over mapping to itself, so the income cannot restart rather than merely having halted.

Which also settles what the branch counts above were counting. The cycle depends only on the deep place's class, so the steady state is the same on every branch, and the ring that reads three to six deep places decomposes exactly: three unbounded, at most one bounded, and up to three left behind by the opening stretch before the deep place settled — three ranges read separately, which at any one branch sum to that branch's own count and so cannot be added at their largest. The MINIMUM of a spread like that is the mechanism and the rest is history.

Scope. Proved for the steady state — which is what makes the repeat move the only source of bundles, and so makes the recursion exhaustive — and a rule in range at every branch of all six rings' complete branch sets. The recursion is initialized once per branch and has no fitted parameter after that: at all 964 later repeat moves the increment is exactly the one that lands the place one above the tick, and a longer climb is on offer at each of them. Against the runs themselves it reproduces the bundle ledger era for era, 964 eras with no mismatch. Across the six rings every run-in is 0 or 1 era and every cycle 1, 2 or 4 — lengths of the state as the computation carried it, and a range fact rather than a law. The one law is the other direction, that the period of the tick's own doubling divides the cycle, and the recursion swept off the ring below reaches a cycle of 20 and a run-in of 12 at one of the orders these rings themselves carry. Both verdicts assume no late opening pays for a bundle again — read off place counts exact to degree 1200 against runs reaching degrees 291 to 293 — but a late bundle only ADDS units, so it could restart a bounded coordinate and can never stop an unbounded one.

verifier: explore_rider_recursion.py

Between the unbounded coordinates and the ones the opening stretch left behind sits a third kind, and one ring of the six has it: a place that goes on deepening PAST the opening stretch — so it is no strand — and then stops forever, because the places its class reaches lie in the run-in and never in the cycle. It stands at six such places, stopping at exponents 2 to 6, and two of them are of degree 2, which no other ring's bundles reach at all. The obvious reading of “why only there” is arithmetic on the ideal classes: that ring has 15 of them and 15 factors, while the other ring carrying bundles has 5, a prime, with no proper subgroup for the deep place's class to sit in.

The reading is wrong twice over, and settling it needs no ring at all. Write d for the ORDER of the deep place's class — the least number of copies of it summing to the trivial class — and the whole recursion is a map on pairs (tick, increment) read modulo d: the tick doubles, the next increment is the tick less the units handed back, and the units handed back are a function of the class the increment summons. Sending an increment to that class is a BIJECTION onto the classes the deep place's own class generates, so the class summoned IS the increment's residue mod d, and the question is the residue track of one orbit of one map on d² states. The number of ideal classes never enters it. First death: at the ring in question the deep place's class GENERATES the whole group, its order being the full 15, so the proper subgroup the reading appealed to is not where that class sits. Second: the factorization decides nothing anyway. Sweep the map over every order, every pattern of returned units and every seed a clock can stand at, and at order 5 — prime, and the exact order carried by the ring with NO such coordinate — 46.4% of orbits ESCAPE, meaning they carry on the run-in a non-zero residue the cycle never returns to; over the patterns a ring could actually hold, 48.3%. The rate tracks d upward and never how d factors, and the sweep's only structural bar is a bijection: an odd order whose pattern is injective makes the whole map a bijection of a finite set, so no orbit has a run-in and there is nothing to escape from.

So whether a coordinate stops is settled inside those d² states, from the order, the pattern and the seed, with no ring, no run and no limit in it — and the seed is the one thing no algebra hands over. It is the state at the first repeat move past the opening stretch, and the opening stretch is exactly what a run is for. What separates the two rings sits precisely there: one seeds at residues its own cycle also carries, so 8 of its 20 branches take a run-in and none of them escapes; the other seeds, on 89 of its 120 branches, at residues its cycle never returns to, which is what a stopping coordinate needs. Which kind a ring's coordinates are therefore cannot be read off the ring before walking it — which is the finding rather than a shortfall against it, and the branch counts are what make it one: 77 of those 120 branches carry a stopped place and the rest do not, all of them over ONE ring with ONE class group, so the answer is not a function of the algebra at all.

Scope. The third kind itself is an observation at one ring over its complete branch set. The census is a rule in range over 4,907,558 swept orbits — every pattern at every order to 6, sampled above that to 16 — and two of its statements are proved outright: the bijection bar, and the tick's period dividing every cycle. Escape at the CLASS level over-reports the places: a class the cycle never summons again can still have every place it reaches fed by another class that IS on the cycle, which happens at 12 of that ring's 120 branches — 89 escaping against 77 carrying a stopped place. What holds exactly is the bridge, and it is checked against the runs' own deep places rather than against the recursion's other output, which would be a tautology: the escaping classes' places, less the cycle's and less the deep place's own, ARE the stopped coordinates, at every one of the 150 branches, and every place the recursion summons is one the run itself deepens, so the identity needs no remainder term.

verifier: explore_increment_group.py

Take a place P and suppose some other place, of no greater degree, is cheaper than it by more than the largest norm a minimal representative of an ideal class can carry, that neither of the two can be seated fresh, and that no bundle move can carry P along. Then P is never a minimal move again, and the limit reads its exponent as finite. For runs growing by ideals the last two suppositions are vacuous — there are no bundles and no classes to represent — which is why the lemma is proved there and measured for runs growing by elements. The gap cannot close: a repeat move changes it by a nonnegative amount, and repeating the cheap witness drops that witness to the bottom of the range, which widens the gap rather than closing it — while a place that could still be seated fresh prices its first move at 1 and jumps to d·T the moment it is taken, so a witness that can jump is no witness. WHICH degree the supposition means is where the two characteristics part. Over F₂[x] it is the DEGREE. Over a number ring it is the local degree e·f — the ramification index times the residue degree f, which is the degree the place already reports — because a seated place reads λ through a single p-adic valuation. The tick used above is one COMPONENT of a vector carrying one entry per rational prime; over F₂[x] there is a single residue characteristic, so the vector has one entry and reads as the scalar it was written as, while over a number ring two places share a clock only when they share a residue characteristic. Where nothing ramifies the local degree IS the degree, which is why the polynomial statement reads as a statement about degrees and never had to say which it meant. And a lock is not a rival argument for the same conclusion: it is the second sufficient condition on this one invariant, the case where the repeated move a locked run makes forever never reaches the coordinate's characteristic at all, so neither price moves.

No quadratic ring can test it — two places over one rational prime are conjugate there, of equal norm and equal ramification, so the hypothesis holds with EQUALITY at every pair such a ring can present. Separating the two readings needs a rational prime carrying two places of different ramification, which takes degree three at the least, and the cubic field of least |discriminant| supplies one at 23 = P·Q²: the pair shares the norm 23 and carries local degrees 1 and 2. There the readings disagree — by the residue degree the lemma freezes P, by the local degree it refuses — and one fresh seating lifts the tick to level three and reverses the comparison, P falling to 23⁴ against Q at 23⁵. The degree supposition IS the mechanism rather than a condition bolted onto one: a jump of j levels moves the gap in log cost by j times the difference of the two local degrees, which is nonnegative exactly when that supposition holds.

Scope. The port is PROVED for runs growing by ideals, by an induction over the move model calling on no computation, and is a rule in range for runs growing by elements — the two identities it rests on, that a seated place reads λ through one valuation and that a repeated move's price climbs, late in a run, by the local degree per level of the tick, re-read as controls at 5,945 readings over the 90 places of norm at most 200, with none off. That a lock's cases are instances of the same invariant is a rule in range: 36 of 36 frozen records certified over two rings and both worlds, none refused, each certified on exactly one of the walk's two branches — a conjugate pair freezes one member per branch, the other being that branch's own witness — and the lemma firing before the walk locks at 34 of the 36. The local-degree half is an OBSERVATION — one ring, one prime, one configuration — so what stands is the port, not a census of the rings that can separate the two readings; and the gap identity is read in the tail, which is where a frozen coordinate sits. The ring that decides it prices its cheapest place at that place's bare norm forever, so its own greedy run repeats one place and nothing here is a claim about walks.

verifiers: explore_undercut_nf.py, explore_cubic_undercut.py