Growth

Greedy growth laws trichotomize by demand. Breadth: demanding independence — the extension splits, Z/Nm ≅ Z/N × Z/m — forces the pick to be the least prime not dividing N (the least-new lemma): every new channel is a field and squarefree-ness never breaks, with neither demanded. From seed 1 the trajectory is the primorial tower; from any seed the picks are exactly the missing primes in increasing order (the healing rule — healing adds missing windows, never removes: a seed's excess prime powers persist). Depth: demanding new dynamics — λ must grow — collapses onto one prime's column (the lock-prime law, below). Mortality: demanding transparent growth — capacity with λ frozen — halts: λ is monotone, the rings with λ(n) | L are finitely many, and the greedy dies at exactly W(λ(seed)), the seed's wall: W(L) is the largest modulus whose λ divides L, and the identity W(L) = denom(B_L/2L) at even L gives 24, 240, 504, 480, 264, 65520 at L = 2..12 — the image-of-J orders of stable homotopy. One level down, every trajectory's limit is a supernatural number, and the fates are three PROPERTIES of it — every prime seated, some prime at infinite depth, a finite integer. Not a partition, and not "all primes at finite depth", which every finite integer satisfies: the greedy laws realize exactly one fate each, but that purity belongs to greed rather than to the demands (The image and limit). And all three are instances of one dichotomy on greedy walks, stated on The cascade boundary.

Scope. Greedy laws over extensions of Z/N, extended to F_q[x] and quadratic number rings by the module law and the class-group split (Growth over other fields); the wall identity is self-verified against from-scratch Bernoulli numbers at every even L ≤ 112. Additive moves and non-cyclic ambients are open.

verifiers: explore_growth_laws.py, explore_headroom.py

Under dynamics-greed the minimal λ-growing move is always a prime power, priced by a per-prime door formula, and the first non-ghost pick locks the trajectory into that prime's column forever — no rival door ever falls. Ghost openings (a prime already dividing λ with no window of its own; the factors each adds to λ close cheaper destinies behind it) strictly increase inside the odd prime factors of λ(seed), so the lock is certain within ω_odd(λ(seed)) + 1 moves: the basin map is decidable, decided at the first non-ghost move. Every prime is a reachable destiny (Linnik blocker seeds). Dynamics-greed marks up exactly what independence loves first: from every state N ≥ 3 opening the 2-window costs at least 16 and the 3-window at least 9, and those two are the only primes never opening at their own cost (every prime p ≥ 5 opens at cost p from some state); from the void (seed 1) greedy dynamics grows the 3-adic column, never opening the 2-window at all — that invisibility at birth is GREED'S and not the demand's, since a free policy under the same demand reaches the 2-column from the same void.

Scope. A mixed-characteristic privilege: in equal characteristic the same demand sprawls (the local module law).

verifier: explore_lock_prime.py

The prime 2 pays three separate-looking premiums — the cold door, the lock-prime law's floor of 16 on opening the 2-window, where 3's floor is 9 and every prime from 5 up opens at face value from some state; the double step, λ(4) = λ(8) = 2, so deepening 4 to 8 raises nothing and a 2-locked trajectory jumps its 2-part from 4 straight to 16, one cost-4 move in a column whose every other deepening costs 2; and the budget inequality's excess unit: the 2-door carries one more exponent than an odd prime's, and one move can raise the exponent of 2 in λ by two where a move at an odd prime raises its exponent by exactly one. They are two facts wearing three names. The splitting: for e ≥ 3, (Z/2^e)^× = ⟨−1⟩ × ⟨5⟩ with 5 of order 2^e−2, so λ(2^e) = 2^e−2, where every odd prime power's unit group is cyclic. The evenness: −1 has order 2 modulo every odd prime, so λ(N) is even for every N ≥ 3 — on a pure 2-power state the witness is −1 ≢ 1 mod 4, the splitting's own root, the corner the two facts share. A counterfeit 2-channel — the 2-column re-priced with the odd pattern λ*(2^e) = 2^e−1, everything else untouched — switches the splitting off, and the prices sort themselves: the double step and the excess rise die at once (the moves from seed 4 run 2, 2, 2, … in place of 4, 2, 2, …, and every move raises the exponent of 2 in λ by exactly one), so the 2-column becomes behaviorally odd, while 3's floor stands at 9 on the evenness alone. The cold door is their conjunction rather than a third fact: from a state with the 2-window closed the 2-door is the least 2^r with λ(2^r) not dividing λ(N), which is 2^v+3 at v the exponent of 2 in λ(N) — the evenness forces v ≥ 1 and the splitting carries the door's extra exponent — and the two markups compound rather than add: at a hypothetical v = 0, λ(4) = 2 already fails to divide an odd λ(N) in both tables, so neither formula engages and both worlds agree at door 4; the splitting's exponent bites exactly on the states the evenness has already floored. Floor 16 real, 8 counterfeit, 3's unmoved at 9. The two facts are the two exponents of −1 − 1 = −2: zero at every odd prime — −1 ≠ 1 there, the evenness — and exactly one at 2 — −1 sits one step deep and no deeper, the splitting. Two invariants of the one unit of finite order that Z owns; the 2-column is where −1 lives.

Scope. The splitting and the evenness are proved; the price recomputation in both worlds is exhaustive over odd N ≤ 20,000, with the cyclic control at every odd p ≤ 13 up to exponent 5 and the double step run from seed 4.

verifier: explore_two_adic_prices.py

Channel-count capabilities are exactly the demands where greedy growth is optimal: k distinct prime powers multiply to at least p_k#, so “grow the cheapest ECC-bearing tower” picks 210 — growth equals choice on precisely the demands the fates already grow. Single-part capabilities never gap: the first progress step is the jump.

Scope. The class is proved, from the prime-power product bound; the single-part no-gap statement is a rule over the swept demands.

verifier: explore_growth_capability.py

Greedy “stay cyclic” growth walks 1 → 2 → 4 and dies — no 4m with m ≥ 2 has cyclic units — though the 2·3^e column is immortal, and one step of lookahead finds that column forever: the depth fate, forced by a preservation demand.

Scope. Proved — the death from the cyclic-unit classification, and with it the immortal column and the one-step lookahead that finds it.

verifier: explore_growth_capability.py

Under a cost strictly monotone under strict divisibility, a prime p is guaranteed to open its window under EVERY tie-break iff its sublevel set {q prime : c(q) ≤ c(p)} is finite. While p is admissible every pick costs at most c(p) and so comes from that set, and each pick consumes one member permanently — a finite set therefore leaves p the eventual unique minimum, while an infinite one feeds a tie-break that starves p forever. Ω is the specimen on the failing side: it gives every prime the same cost, so no sublevel set is finite, and an odd-preferring tie-break keeps 2 admissible and unpicked indefinitely. So healing — every missing prime eventually seated — holds under every tie-break exactly when the cost is proper, every prime's sublevel set finite. Of the deleted archimedean place, “least m” uses properness and one choice of linear extension, and nothing else about size is load-bearing for where a breadth trajectory ends up. What size does author is the route, the order the primes arrive in: any enumeration of the primes whatever is the route of some proper divisibility-monotone cost, so entry order — with it the rung identities, the Linnik ordering, the plateaus — is the deleted place's own contribution, no cost of this class singling one route out.

Scope. The criterion is proved in both directions; the route sentence is a rule, its realizability checked at three scrambled costs. The divisibility-monotone hypothesis is load-bearing, and the demand throughout is independence. Verified in range: fields for free across 3 tie-break policies × 6 seeds × 20 steps, every pick prime; the starvation specimen at 30 steps; three scrambled proper costs entering all 46 primes in range, each at its own rank and all three at the same destination.

verifier: explore_size_crystallization.py

To DIE a law needs only order; to REACH EVERYTHING it needs multiplication; to BE ARITHMETIC — independent windows, squarefree, the crystal — it needs unique factorization. Floor 0, pure order. Any demand confining the trajectory to a finite region absorbs, under every policy, within that region's height — no weight, no monoid, no ring. For the interval demand (admissible = everything above the state in the region, which is mortality's own shape) the grave is a no-successor element, and it is unique exactly when the reachable region has a maximum. Mortality is therefore a poset phenomenon, and the tower dies at ONE wall because the states above its seed with λ frozen — the multiples of the seed dividing W(λ(seed)) — have a maximum; a region with several maximal elements leaves the grave to the policy — greedy dies at the cheapest, thermal (the argmin softened to picking with probability proportional to |m|^−β) splits between them (the multi-tombstone freedom, 30 of 240 censused regions, minimal specimen one element below two incomparable ones). That is a degree of freedom the tower never shows. Floor 1, a cancellative commutative monoid carrying a summable multiplicative weight. Up-closed nonempty demands reach the top of any such monoid: the injection m ↦ am is cancellativity, so the admissible mass surviving inside the multiples of a is bounded below by w(a) times the whole — uniformly in the state, and for EVERY element rather than only for atoms. Cofinality therefore needs no atomicity, and countability comes free from summability. The machinery that reads a world's genesis from inside it (The inside view) runs at this floor already — it is monoid-generic. Floor 2, freeness. Each atom enters the state once and never again, its entry depth geometrically distributed, so the limit is the crystal with probability ∏_g(1 − w(g)) = 1/ζ_M by the Euler product: the partition function of thermal breadth is the zeta function OF THE MONOID, Riemann for the integers and rational for F₂[x]. And freeness is necessary, not decorative (the coprimality collapse). In the numerical monoid ⟨x², x³⟩ — cancellative, not free — every xⁿ with n ≥ 5 carries both atoms, so every independence trajectory dies by its second move, while the same world's up-closed demands still reach its top: without unique factorization the breadth fate collapses into mortality, and breadth's immortality is revealed as a unique-factorization privilege. The identity breaks with it — the Dirichlet sum 13/12 against the Euler product 1024/945 at t = 1/4, first diverging at x⁶ = x²x²x² = x³x³, one element and two multisets. Two statements, not one: the collapse is what freeness buys, and P(crystal) = 1/ζ_M is what it measures.

Scope. Each floor's law is proved at its own structure and its necessity witnessed one floor down; the multi-tombstone freedom is a rule over a 240-region census (divisor lattices of 60, 72, 210 at every seed, plus 200 random sub-regions of the divisors of 360); the coprimality collapse is proved and its 2-move census exhaustive. Instance worlds: the integers, monic F₂[x], and the numerical monoid. Cancellativity is the floor-1 hypothesis rather than a proved necessity — what is proved on that boundary is that an absorbing element admits no nontrivial weight at all, which is an observation over a scanned specimen.

verifier: explore_selection_frame.py

Thermal growth sees the missing floor three ways, the dynamical one in two worlds. Statics. The ideal world's partition function is the Dedekind zeta ζ_K, the sum of N(I)^−β over ideals, giving crystal probability 1/ζ_K(β) = 0.5389 at β = 2 — so the family is three deep: Riemann for Z, rational for F₂[x], Dedekind for O_K. The element world's is (ζ_K + L(χ₋₄)L(χ₅))/2, matched coefficient by coefficient to n ≤ 10⁴. The gap between the two partition functions is therefore a class-group L-function. Dynamics. Element-coprimality is CLASS-BLIND: two elements are coprime iff the class sequence of their ideal gcd is zero-sum-free — no non-empty sub-sequence of classes summing to the trivial class — and the sensor is exact, element-coprime matching ideal-coprime, exactly when h = 1 (the sensor criterion). What it misses is budgeted: a hidden common divisor carries at most D(Cl) − 1 primes, the Davenport constant of the class group, which is one prime for C₂ — the minimal specimens are 2 and 1±√−5, silently sharing the prime above 2. So the element crystal is DEAD rather than leaky (the absorption profile law): the pure-square move stays admissible until depth 2, so an in-universe run can absorb only with every nonprincipal prime at depth ≥ 2 and every principal prime entered, and the squarefree fraction is 0. Dynamics in the infinite world. With no atom set to exhaust, the question is not whether a run absorbs but how fast each prime's exponent in the state — its column — grows, and the class group splits the columns in two. The SHALLOW half is a property, no limit taken: once a principal prime has entered the state, any later move divisible by it puts the trivial class into the gcd's class sequence, whose one-term sub-sequence sums to zero, so a principal column is written by at most one move and frozen thereafter, over any class group — floor 2's entry-once, read where the class group rather than freeness blocks the second write. What is bounded is the number of writes and not the depth: a fresh prime is coprime to the state, so a move carrying its square writes depth 2 in one stroke. At h = 1 this covers every prime and no column is bottomless — the sensor criterion read dynamically. The BOTTOMLESS half is a rate. Past depth 1 every move raising a nonprincipal P must carry a fresh nonprincipal factor, the leading such move being P·Q for a fresh nonprincipal Q; so with a and b the prime-zeta tails — the sums of N(P)^−β over the fresh principal and the fresh nonprincipal primes, those not yet in the state — and K the nonprincipal mass already spent, the one-step raising probability is N(P)^−β·b/(a + Kb), measured against the trajectory's own tails to within 4.3% at the widest norm cap the rig runs and closing as that cap widens (the bottomless rate). Each nonprincipal column therefore grows LINEARLY in the move count, at N(P)^−β/(ρ_β + C_β) with C_β the same sum over ALL the nonprincipal primes and ρ_β the trajectory's own tail ratio a/b: 0.2152 predicted against 0.2148 measured at the prime above 2, β = 2. The static tails are balanced, each class holding half the primes (Chebotarev; a/b = 1.0004 at 10⁶), but the trajectory's ratio runs 0.64 to 0.93: a nonprincipal prime is spent only with a partner, so its fresh tail stays the fatter, and putting the balanced 1 + C_β in the denominator gives a lower bound on the rate rather than the rate. The split is about a class being trivial, not about parity: it survives intact at Cl = C₃, where a shared P² is still zero-sum-free — what a larger class group changes is the constant, not the dichotomy. Thermometer. ζ_princ/ζ_K runs 0.6743 at β = 2 and 0.5063 at 1.05, converging to 1/h at the pole β = 1: heat both worlds toward the pole and their mass ratio reads the class number off. So the missing floor is a graded one. To be arithmetic at the IDEAL level needs only the mezzanine; to be arithmetic at the ELEMENT level — the crystal, independent windows — needs class number 1, and the class group measures how far a world sits from its own arithmetic.

Scope. Z[√−5] (h = 2, Cl = C₂), with Z[i] as the h = 1 control and Q(√−23) (h = 3, Cl = C₃) as the leg beyond parity. The sensor criterion is a criterion: the zero-sum-free law is proved in any Dedekind domain, and the h > 1 direction is constructive in any ring of integers. The absorption profile law is proved in-universe. Of its infinite-world limit — the class group as a partition of places into shallow and bottomless columns — the shallow half is a property over any class group, and the bottomless half is a rule in range: the rate's identity is derived, and measured at move-norm caps of 2000, 8000 and 32000, β = 1.5 and 2, five seeds each, the agreement improving with the cap. What is measured and not derived is that the trajectory's tail ratio ρ_β stays bounded, which is what the rate being positive rests on. The two static identities are classical: the partition split is genus theory and the pole ratio is residue equidistribution, and what is claimed here is the growth reading of each, not the identity.

verifier: explore_class_gap.py, explore_bottomless_columns.py