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 door 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] and the complete h = 1..5 elliptic ladder at q = 2; the empty middle is a derivation from the formula and inherits its scope. The SIZE of a 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.

verifiers: explore_greedy_image_nf.py, explore_greedy_image_ec.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 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 Zsygmondy, so the endless ladder is a claim about F₂ and not about doubling ticks in general. The deep place's degree is 1 or 2 — 0 of 140 censused repeat moves at degree 3 or above — and that ceiling is a consequence of the doubling, not of the shape.

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 a capped tie sweep enumerated — three extra coordinates at one ring on every branch carried, three to five at another, those two being the rings whose sweep hit its cap. Whether the extra coordinates are UNBOUNDED is OPEN: at one ring the bundle income stopped dead once a parity locked. The element world has no reordering lemma — a declined minimal move is still minimal at the successor at only 46 of 85 types at one ring — so its shapes are read over every branch and never off one continuation.

verifier: explore_element_limit.py

Take the ring out and four ingredients remain: items carrying an integer DEGREE; one GLOBAL CLOCK multiplying the tick by b whenever any item is deepened; a PRICE f(degree, staleness), staleness being how far the tick has moved since that item last did; and a fresh-opening DISCOUNT spendable m times at each degree, the least degree born with its discount already spent. The ring enters only as a SUPPLY — how many items each degree holds. Dial each as far as the move model admits — b at 3/2, 2, 3 and 4, the degree's exponent at 0, 1 and 2 and an additive form, one to three discounts, every born-spent set — and ONE INFINITE COORDINATE survives every dial but one. The dial that kills it is the degree-blind price, which cannot tell a cheap item from a dear one — and only where the clock's steps are wider than 1. At step 1 an opening and a repeat cost the same, the tie deepens rather than opens, and one item runs away regardless.

A FLAT SUPPORT is the weaker half and does not survive as far, so the two are separate claims. At b ≥ 3 the ceiling has already fallen to the least degree while the clock still seats an item above it, so that item is seated, clocked, then undercut and STRANDED above exponent 1 forever — exactly one such item on every branch that hands the clock on, and none on the branch whose first repeat was also its last. The doubling clock of the theorem above is precisely where that cannot happen.

Scope. Proved for the abstract schedule, and a rule in range: 60 walks over ten schedules and six supplies, with the abstract walker certified equal to the exact one in cost, in every move type's multiplicity and in the covered set at 120 states over each of six supplies. The dial survey is crossed with the clock's landing rule rather than read beside it — 7 landing rules × 8 dials × 2 clocks — and the single infinite coordinate holds at all 49 crossed cells whose price can see a degree UNDER THE GLOBAL CLOCK. Give each item its own tick instead and the count standing above exponent 1 moves with the dial and with the clock's step width together, which is the mixed run below. Strands arise on both sides by different mechanisms — there the ceiling falls under a seated item, in a mixed run the item's own step widens under it.

verifiers: explore_price_schedule.py, explore_tick_pump.py

Replace the clock's factor b by the SET of exponents the tick may stand at, the tick advancing to the least member at or above the exponent just landed on. That set is exactly {a : v(a+1) > v(a)} for v the valuation λ carries at the place, and the whole of what it gives the dynamics is its gap: the distance from one member to the next, which is the price of a repeat move divided by the degree. A ring whose gaps are BOUNDED stops opening new places, at about the deep place's degree times the largest gap; a ring whose gaps are UNBOUNDED opens forever. Which one a ring has is settled by the principal-unit filtration (O/P^a)^* = k^* × (1+P)/(1+P^a): the p-th power map sends level i to p·i in equal characteristic — a multiplicative set, gaps unbounded, SPRAWL — and to min(i+e, p·i) in mixed characteristic, which is i+e past a leading stretch — constant gap, LOCK, with the gap the RAMIFICATION INDEX e. A set whose gap grows without being multiplicative is realized by no Dedekind domain with finite residue fields, which is where λ is defined at all. So the dichotomy is the characteristic — equal against mixed — and what varies within the locking side is ramification and nothing else. That is the same local invariant the module law states from the other end: its eventually constant price q^e = p^ef is this gap's cost, and its rank-∞ case is this unbounded gap.

Scope. The arithmetic half is a derivation from the filtration, checked at the three worlds run. The measured half is a rule in range: every place of norm ≤ 200 in two quadratic number rings — 90 in all, 82 split, 5 inert, 3 ramified — read at depths 1..24, the tail gap equalling the ramification index at 90 of 90, 1 at every split and inert place and 2 at every ramified one, 0 off; beside three function-field degrees whose gap is already 16 by depth 24. Three of the 90 carry a LEADING stretch of wider gaps, and they are exactly the three with p − 1 ≤ e — the depths below e/(p−1), where the principal units are still squaring rather than stepping. The stop location is measured at 2, 2, 3, 5 against products 1, 2, 3, 5 at gaps 1, 2, 3 and 5, exact at three and floored at the fourth — measured in the abstract schedule, where a gap can be dialled, and not in a ring. The stop law is a COMPARISON of two curves — a bounded recurrent price against an opening cost that grows in the degree — and where openings never get dearer a bounded gap stops nothing.

verifiers: explore_tick_pump.py, explore_lock_budget.py