The cascade boundary

The demand on an escaping trajectory really is a conjunction over characteristics, and not a disjunction it gets to satisfy at one characteristic of its choosing. That distinction is the whole difference between a sweep over primes and a statement about rings, so it is worth having as a derivation rather than a premise. An escaping trajectory's move costs must diverge: a norm-finite ring has finitely many places under any ceiling, so a stretch of moves that stayed under one would pick some place forever at bounded cost, and such a place absorbs the tail — a lock. Now every rank-1 characteristic's door is a λ-growing move available whatever the state has done, at cost p^v+2; so each of them, on its own and at every stage, CAPS what the escape can be paying, and every one of those exponents must therefore run away at once. The escape does not get to pick which characteristic to leave unsettled. It has to outrun all of them together, and it has to do so through places lying over OTHER characteristics — deepening p's own column would seat it there at a constant price — which is exactly why a carrier's norm is one more than a multiple of p^v+1: at a place away from p, nothing but that norm minus one can carry a factor of p. Comparing the carrier against the door is also, on its own, what confines the multiplier to m ≤ p − 1.

Scope. The derivation above reads moves as ideals, and an element move is not confined to a single place — but every step of it survives that. The divergence argument does not rest on moves being single-place, only on there being finitely many candidates under a cost ceiling, and a number field has finitely many IDEALS under one just as it has finitely many places: so one ideal recurs as the multiplier forever, one place inside it is what moves λ, and a power of that recurring BUNDLE is the always-available move at a fixed price. The door is charged on a generator, at the factor it costs to make the place's power principal — 1 exactly under unique factorization — and the multiplier's range widens by that factor. So what follows is a statement about the ideal dynamics of any characteristic-zero ring AND about its element dynamics.

verifiers: explore_module_law.py, explore_element_cascade.py

A standing move is a recipe — state ↦ a move admissible whatever the state has done — and never a fixed element: over Z the door at a fixed q is 5 at N = 71 and 25 at N = 355, and it is standing all the same. Give a greedy walk a family of standing moves, each priced by ONE coordinate of the state — an integer read off the state, through a nondecreasing function — and the walk pays at most the least of those prices at every step, since greedy is the minimum over a set containing them. Under norm-finiteness (finitely many moves under any cost ceiling, a move recurring forever absorbing the tail) either some coordinate stays bounded along the walk, and then a move of bounded cost recurs forever and the walk locks, or every coordinate runs away AT ONCE. A member priced constant in the state locks the walk outright; escape is a conjunction over the whole family. The two branches are two results of this corpus one dial apart: over Z the dynamics-greed walk's cheapest standing price goes constant and the trajectory locks, and in a characteristic-zero ring the door's price grows with the state, so an escape has to outrun every index at once — the conjunction above. Nothing but whether the standing price is constant or grows separates the two arguments.

Over Z the door menu is such a family, one recipe per prime, standing because λ(q^d) carries an unbounded power of q, so some power of q always grows λ; its coordinate at q is v_q(λ(N)) − v_q(N), the rung V read at q less the depth of q's own column, and V itself at a prime the state has never seated. Write x for it. The door at an odd q dividing N is exactly q^x+2, and q^x+2 bounds the door at every state, every strict case an opening at cost q, where q − 1 carries a factor λ lacks and one seat of q already grows it. And the coordinate has a floor: at odd q it never falls below −1, because q^e dividing N makes λ(q^e) divide λ(N), and λ(q^e) carries q^e−1. The lock-prime law's recurrence invariant — the lock prime's λ-exponent sitting one below its column depth — IS x = −1: the coordinate at its floor. So Z's lock is not a second mechanism beside the cap; it is the cap pinned at the bottom of the coordinate that prices it, and the budget inequality below bounds the same coordinate's rise from above, at one per move at odd p, where the invariant pins it from below. At q = 2 the floor and the bound both shift by one, as the door does, the bound failing at the virgin state alone.

All three fates (Growth) are instances of the dichotomy, and the branch is decided by whether some pricing coordinate stays bounded. Depth — dynamics-greed's fate — locks with its coordinate pinned at the floor. Mortality — λ frozen, the walk halting at the seed's wall, the largest modulus whose λ divides the seed's — is the bounded branch with the coordinate driven to its bottom: from every live state the jump straight to the wall is a standing move, priced by the wall's ratio to the state alone, which descends 48, 24, 12, 6, 3, 1 off seed 5. Breadth — demanding that the extension split, which picks the least prime not dividing N — is the growing branch over Z: independence CONSUMES each move on use, every move of cost at most 30 permanently inadmissible by step 10 from seed 1, so no standing recipe of bounded price exists and the walk cannot lock. But availability is a recipe's, and one recipe per reduced residue class mod m — the least unused prime in the class, priced by how many primes of that class the walk has picked, through the class's own primes — is a standing family whose every price grows, whose cap equals the greedy cost at every step but the one, if any, that picks a prime dividing m, and whose every coordinate diverges. That is the escape's conjunction with real conjuncts: Dirichlet's theorem, read per class, is what makes each recipe standing, and the walk's escape is the statement that every class is picked without end. A family reads nothing unless a coordinate can REPEAT a value along the walk — the greedy move itself, priced by N, is a standing move meeting every hypothesis on all eleven locked depth walks and reads each as "the coordinate diverges" — so the content sits in choosing coordinates that are projections of the state, bounded along a lock: over Z x is held at its floor while the class counts are driven up without end, and in a number ring the door's price grows with the state.

Scope. The dichotomy is proved for any greedy walk under the three hypotheses, and the floor at −1 for every odd prime; the door bound is censused over twelve seeded walks — 402 exact and 1326 strict over 1728 odd-prime/state pairs, the floor reached at 113, and along seed 71's locked tail x₁₇ = −1 out to N = 59,981,858,965. The class family is a rule IN RANGE: m ∈ {3, 4, 5, 8}, eleven seeds, twelve steps, every recipe found, every class coordinate repeating and ending above its start, and strict at exactly one step at each seed the prime dividing m does not divide and at none where it does; the class recipes are standing by Dirichlet's theorem, an import checked only inside a sieve to 200,000, and their divergence is unbounded only through that same theorem. The wall jump is measured at 13 states off three seeds. What the dichotomy does not recover is WHICH prime a locking walk locks: that is the lock-prime law's, and lies outside it.

verifiers: explore_standing_move.py, explore_standing_recipe.py

A door cheap enough to be affordable is too cheap to buy the exponent that would keep the ladder climbing. Write V = v_p(λ); a rung is a value of it. V rises only at an opening, for the reason the conjunction above turns on, and an opening is paid at its face value N(Q), which the ladder premise holds below the virgin door. Since p^v_p(X−1) divides X − 1 it is smaller than X, so v_p(N(Q) − 1) < log_p N(Q) < log_p of the door. At odd p the virgin door is p^V+2, so V rises by at most ONE per move, the ladder's cap is pinned at p forever, and the excess — how far V outruns the carrier's own exponent, which is the only thing that could widen that cap — is unbuyable outright. At p = 2 the unit group's extra Z/2 makes the door 2^V+3 and the rise at most TWO: exactly one unit of excess, and banking it needs a place of norm 2^V+2 + 1 with the multiplier pinned to 1. The numbers 2^k + 1 that are prime powers are exactly the Fermat primes together with 9 = 2³ + 1 — a prime 2^k + 1 forces k a power of 2, and Mihailescu leaves 9 the only proper power — so the entire excess supply is the five known Fermat primes together with 9, and the open finiteness question is the Fermat primes' own. The inequality reads a door's SIZE and never its shape, so no residue-degree or lifting-the-exponent route escapes it. Two corrections travel with it. The carrier condition is on a place's NORM, so the supply is the prime POWERS and not the primes: the p = 2 rung at V = 1 is carried by 9 = 3², an inert place over 3 in a quadratic field, a rung a census over primes certifies dead. And over that widened supply the ladder at the pinned cap dies anyway, unconditionally, every candidate printed with its factorization — p = 2 climbs V = 1, 3, 4, 5, 6, 8 through the norms 9, 17, 97, 193, 257 and then misses under the door 2048 (513 = 3³·19, 1025 = 5²·41, 1537 = 29·53); p = 3 dies at V = 2 and p = 5 at V = 3. The death is terminal rather than a stall: a stuck V is a stuck door, so every later move sits under one fixed ceiling forever, and a norm-finite ring has finitely many places under a ceiling, so some place is picked infinitely often at bounded cost and absorbs the tail — which is the lock. So a characteristic-zero trajectory with a rank-1 place over 2, 3 or 5 LOCKS.

Scope. The boundary is NOT closed in general: it closes at the three smallest characteristics, and the rings whose rank-1 characteristics all avoid 2, 3 and 5 stay open. The door values are Z's and transfer for the reason the reduction restricts to rank-1 places at all — such a completion is Q_p, whose unit filtration is Z_p's verbatim. Component tiers: the inequality and the three deaths are proved (the deaths with a printed factorization of every candidate, so no primality claim is load-bearing on the kill side), checked against 5,656 greedy moves × 46 unseated primes and a brute scan over the 26,152 prime powers below 300,000; the supply correction is an observation on one witness, which is all a widening needs. The excess here is a valuation gap and NOT the ghost count the lock-prime law bounds — the two are different quantities, and the ghost mechanism was the proposal this inequality refutes.

verifiers: explore_ghost_wander.py, explore_module_law.py

Under the conjunction the inequality above also runs on, and not a second dependency on top of it, every characteristic-zero ring with a rank-1 characteristic below 1000 has its cascade boundary closed: the odd characteristics here, 2 by the inequality above. At the pinned cap the affordable carriers at rung V are exactly the p − 1 numbers m·p^V+1 + 1 with 1 ≤ m ≤ p − 1 — the door gives that range and nothing else does, since m·p^V+1 + 1 < p^V+2 bounds m and every such m clears the door in return — so each characteristic is a direct O(p)-per-rung walk rather than a scan of everything under the door, which is what carries the ladder past the three the inequality above kills. All 167 odd primes below 1000 reach a death rung, one where every one of their candidates is a certified non-prime-power. That rung is ERRATIC and not increasing: D(19) = D(23) = 1 against D(719) = 62, and D(17) = 2 sits beside D(29) = 8. Two structural facts came out with it. The odd multipliers contribute NOTHING: for odd p they make the carrier even, so only a power of 2 could qualify, and across every rung of all 167 ladders none does — the effective supply is the ⌊(p−1)/2⌋ even multipliers exactly, half of what the cap appears to grant. And what the sweep buys is the RING statement rather than a statement about primes: the reduction demands a carrier at every characteristic carrying a rank-1 place, so closing ONE of a ring's characteristics closes the ring, and an initial segment of the characteristics closes every ring that meets it.

Scope. Verified p ≤ 1000 at a rung ceiling of 120. Three positive controls carry it: the directly generated supply equals the earlier range scan at every rung tested, the rebuilt prime-power test agrees with the earlier one below 20,000 and departs from it only in the safe direction on a 121-digit specimen, and the published deaths at 3 and 5 reproduce unchanged. That test is primality plus exact integer roots and never factorization, which is load-bearing rather than incidental: a test that factors reports NOT-A-PRIME-POWER when it merely fails to certify, and at these sizes that manufactures false deaths — a false death reports the boundary closed where it is open. What stays open is the rings whose rank-1 characteristics ALL exceed 1000.

verifier: explore_cascade_chars.py

The ladder is DELAYED by the widening and not saved by it. Re-walked with the multiplier allowed up to τ·p, every one of the 167 odd characteristics below 1000 still reaches a death rung at τ = 2 — the ideal walk's close, bound for bound — as do all 124 below 700 at τ = 3 and all 94 below 500 at τ = 4. The death rung deepens by about the factor itself — on average 1.97, 2.82 and 3.61 times the unwidened one, each figure taken over its own factor's range of characteristics, so they summarise three walks rather than track one series — and it always arrives: the deepest moves from 62 at p = 719 to 116 at p = 859. The parity half of the supply stays empty throughout — across all four walks one odd-multiplier carrier is ever accepted, 64 = 7·3² + 1 at p = 3, V = 1, admitted at τ = 3 and again at τ = 4, and it is the power of 2 the parity argument demands.

The factor itself can be measured rather than bounded, and for a quadratic field it is exact: the ideal classes are the reduced binary quadratic forms of the discriminant, and the least norm in a class is its form's leading coefficient, so the factor is the largest leading coefficient among them. At Q(√−23) the forms are (1,1,6) and (2,±1,3), giving class number 3 and a factor of exactly 2 — against a Minkowski bound of 3.05, which is where a bound rather than a value comes from. It is worth saying what that 2 is NOT. The element trajectories over that same field pay 4 per move where the ideal price is 2, and the ratio agrees by coincidence: that one is the per-move price an element trajectory settles at once it has locked (the class-group split), which is a lock price, where this is the padding on a door. They have no reason to agree at another field.

Scope. The sweep bound falls as τ rises because the walk costs about τ³, and each bound was fixed from the clock before the walk ran; no walk at any bound has ever produced a survivor, so there was never one for a bound to be drawn around. What a wider τ does NOT buy is more fields: over the 305 fundamental discriminants down to −1000 the factor averages 9.58, only 11.8% of fields have one of 4 or less, and it sits at 0.705 of its Minkowski bound on average — the bound tracks the value, so the factor genuinely grows. The better reading is per-place rather than per-field: the padding needed at a characteristic runs only over the powers of that place's own class, so the factor is 1 whenever the place is PRINCIPAL, and there the unwidened walk closes the element dynamics outright. Since closing one characteristic closes the ring, what that leaves to measure is which characteristics of a field are principal (Principal places).

verifier: explore_element_cascade.py

The residual has members at every bound, so no sweep empties it. Let d = 2·3·5·⋯·P and take K = Q(√d). Since d ≡ 2 mod 4 the discriminant is 4d, so every prime up to P divides it, ramifies, and is not rank-1: L(K) > P. The argument is elementary, needs no computation, and is available at every P — so raising the sweep's bound raises the bar and never clears it. The residual is a moving target and not a shrinking remainder, and a close covering every ring at once is a theorem's job rather than a longer computation's. The analytic machinery this question first appears to want cannot see the construction at all, and why not is the whole content of it. A degree-n field's rank-1 characteristics carry density — a definite proportion of all the primes — at least 1/n, since a point stabilizer has index n and every element of it fixes that point; that proves L finite for every field and says nothing about where it sits, a density being a statement about the tail which survives deleting any finite set of primes. The ramified primes ARE such a set, and they are exactly what L is at the mercy of.

Scope. The construction is proved for every P; the rig controls it at every odd P ≤ 97, printing the pair. Degree 2 throughout. It closes no characteristic and frees no ring: what it settles is that the survivors are inhabited, not that any of them escapes.

verifier: explore_cascade_residual.py

Retail, the residual is cheap. Sweeping every fundamental discriminant to 10⁸ — one per quadratic field — gives 15 champions, the least |D| attaining each record L — reaching L = 127 at D = 46,305,413, where log|D| = 17.65 against the primorial witness's 107.07 over the same primes. The existence proof above pays 6.07 times the exponent for the same guarantee, and the gap is an entire logarithm: a witness bought by ramification costs roughly P in the exponent where one bought by residue conditions costs P/ln P. Both baselines count the primes STRICTLY below L, L being the one prime a champion is not required to kill. The champion law is log|D| ≈ ln2·π(L) and is best left in that form: inverting it gives L ≈ 1.44·log|D|·ln L, which is implicit, and the explicit expansion in |D| alone carries a second-order term nearly as large as its leading one at this reach: the naive L ≈ 1.4·log|D|·loglog|D| reads 71 where the top champion has 127. What makes the retail case cheap is therefore not a bound but DECIDABILITY: L(K) is computed rather than estimated, by a walk up the primes testing each in turn — in a quadratic field, one residue condition each — and the ladder above is then run at that single characteristic. The residual is uncloseable wholesale and cheap for any ring anyone actually writes down. Exactly two champions buy their way in with no ramified prime below L at all, D = −67 and D = −163: that is Rabinowitsch's condition, so their polynomials are Euler's prime-generating x² + x + 17 and x² + x + 41 — reached by a rig carrying no class field theory and therefore a control on the sweep rather than a result of it.

Scope. Verified over every fundamental discriminant to 10⁸, degree 2. The champion law describes that measured extremal family and bounds nothing: for an arbitrary field what is available is conditional, effective Chebotarev under GRH giving L ≪ (log|disc|)². The degree-n direction — where a higher degree makes each prime cheaper to bar while the discriminant rises — is named in the rig and not measured. The Euler pair is not evidence that a ramification-free route closes at 41: the later champions have L far below |D|/4, where Rabinowitsch says nothing, and past L = 41 none of them is ramification-free. A prime fails to be rank-1 with probability (p+1)/2p, which is (p−1)/2p from the residue condition plus 1/p from ramification, so ramification is a bonus term on top of it and not a rival to it — which is what a minimizer free to use either would be expected to do, and 15 minimizers is not enough to say more than that.

verifier: explore_cascade_residual.py

THE EXPONENT IDENTITY: a carrier q^a at rung V has a equal to the multiplicative order of q mod p^V+1 — the least e with q^e ≡ 1 there — exactly: otherwise the order itself already clears the modulus, and the carrier, being at least its square, clears the door. Three consequences follow: a divides p − 1, the order dividing p^V(p − 1) and a factor of p in it putting the carrier past the door; q < p, since p^V+1 divides (q^a−1)/(q−1) at the exact order, which forces q − 1 to divide the multiplier m ≤ p − 1; and q^a > p^V+1 then forces a ≥ V + 2, which subsumes the square exclusion. So the proper prime powers are confined to an enumerable set, and across every rung of all 167 ladders there is exactly ONE of them: 3⁵ = 2·11² + 1 at p = 11, which saturates the divisibility that produced the confinement — (3⁵−1)/2 = 121 = 11² where the derivation asks only that 11² divide it. A death is therefore a finite check plus a single question, is there a PRIME congruent to 1 mod p^V+1 below p^V+2. AND THE RUNG IS INVISIBLE TO EVERY SMALL-PRIME INSTRUMENT: a prime r not dividing p divides m·p^V+1 + 1 for exactly ONE class of m mod r, at every rung alike, so the multipliers left after sifting by every prime below z number about (p−1)∏(1−1/r) over those r — a function of p and z with no V in it — while the carriers themselves thin out like 1/V. Covering congruences, a finite set of congruence classes demanded to exhaust the multipliers, act through exactly that sifted set; and averaging inherits the same defect, a bound that does not fall with the rung bounding a sum over R rungs by R times its first value, the R cancelling. So no argument built on small-prime divisibility separates the density-zero target from the universal one.

Scope. The exponent identity and the invisibility are proved for every odd p, every rung and every sifting bound. The measurements around them are verified p ≤ 1000: pooled over the 144 primes whose death rung is at least 10, the carrier density across rungs 1 to 10 falls by a factor 0.245 against the 0.25 a (V+2)⁻¹ heuristic gives, while the sifted count moves by 1.025 and holds the level Mertens' theorem predicts for sifting to z — the flat ratio alone would read the same under a miscalibrated prediction, so the absolute agreement is what says the sifted count is the Mertens count. Sifting by every prime below 10⁴ empties no death rung at any swept p ≥ 100, and no larger z helps: closing a rung needs the moduli to multiply past p − 1, which makes it an alignment against one interval rather than a covering system. Pooling on a death rung of at least 10 conditions every rung on carrying a carrier, which can only understate the fall.

verifier: explore_cascade_theorem.py