Growth over other fields

At a place with residue field F_q, q = p^f, ramification e, the 1-unit exponent E(a) = exp(U₁/U_a) obeys: in mixed characteristic U₁ has Z_p-rank ef and the p-power map is an isomorphism U_b → U_b+e past the wild threshold, so E(a+e) = p·E(a) eventually — a linear pump, one λ-tick per e of depth at the eventually constant price q^e = p^ef = p^rank. In equal characteristic (1+u)^pm = 1 + u^pm gives E(a) = p^{⌈log_p a⌉} exactly, with rank infinite — a log clock whose tick prices are unbounded on any recurring column. The lock is the finite-rank case; the sprawl is its rank-∞ limit.

Scope. The two cases are proved (Cohen structure / direct computation); instances brute-verified at Z, Q(√−23), and F₂/F₃/F₉[[t]].

verifier: explore_module_law.py

At class number h > 1 the depth fate runs once per currency. Growing by ideals, all three censused fields reproduce Z's geography: every seed locks a window at the module-law price, each quadratic tie a Galois orbit broken by fiat (only Q's greedy needs no tie-break clause), and the cubic field of discriminant −283 locks five distinct places across its census where the headless cubic — one with no head, no place whose column opens with a stretch of wider gaps — has a one-point basin map (the headed walk). Growing by elements — principal states and moves — the world pays its class group in price, and the basin map is a norm race among the principal vehicles the seed's dowry — the windows and the λ it starts with — leaves alive. A nonprincipal window's cheapest vehicle is the free-rider door, the window bundled flat with a partner at the product norm — in Z[√−5] a norm-6 move deepens the 3-window while carrying the 2-window along — and every ride lifts the passenger one depth until (2) wakes and absorbs the trajectory: Z[√−5]'s element census is a (2)-monobasin at 4 per move. The monobasin is field geography, not an h > 1 law. Q(√−23) (h = 3) has two basins: 38 censused seeds ride to (2) at 4 per move, three lock the inert 5 at 25 per move — a pure principal column at the untaxed ideal price, whose very opening carries period 24 and swallows the novelty of every cheaper move; the passenger lift is real but conditional on riding. The cubic (h = 2) has three: 17 seeds to the square of its cheapest place, 3 to a principal norm-17 place tax-free, and 1 to the full rational (2) at 8 = 2³ — the first tail to pay its ring's degree, because that seed's dowry supplies the power of 2 that covers the cheap square's every step, so the square never moves λ and the degree-priced vehicle is the cheapest LEFT: a dowry can edit the vehicle menu. THE ELEMENT PRICE: a tail pays its cheapest 1-move principal vehicle still ticking — still moving λ against the state — the power P^m (m the window's class order, norm p^fm, f the residue degree) against the rational (p) (norm pⁿ, n the ring's degree) — so p^{min(fm, n)} per move wherever both tick. The class-group tax exists only at fm > n, and it is pure myopia — in Q(√−23) the tax-free cube P₂³ (P₂ a prime above 2) sits at norm 8 forever, three depths per purchase at the untaxed 2 per tick, and a per-move greedy never picks it. And the tax need not buy depth at all: the cubic's taxed square pays 4 = 2² per move while the exponent E above climbs two doubling steps per move — 2 per step, the ideal world's own price at that place — so where the vehicle squares a place whose E-column carries a flat step, the tax buys granularity, two steps a move, and not step price.

Scope. Censused wall to wall at Z[√−5] (h = 2) and Q(√−23) (h = 3) and over the norm ≤ 40 belt at the cubic field of discriminant −283 (h = 2), both currencies; the basin counts are each field's census. The element price is the pattern the block is badged at: three fields, seven censused tail species, no proof; the still-ticking clause is load-bearing, the cubic's dowry edit being its counterexample otherwise. The pricing it rides is the module law above.

verifiers: explore_number_field_lock.py, explore_module_law.py, explore_headed_cubic_walk.py

The partition function is rational: ζ_F2[x](β) − 1 = r/(1−r) with r = 2^1−β, and the crystal probability is exactly 1 − 2^1−β — 1/2 at β = 2, against Z's 1/ζ(2) ≈ 0.608: no transcendental content survives. The hot-limit theorem transfers, so thermal dynamics reaches the full profinite completion almost surely at every β > 1; the sprawl's geography is a β = ∞ artifact exactly as Z's lock is. Over Z temperature's gift to the dynamics demand was depth (the radical); over F₂[x], where cold growth already runs one column's radical while starving breadth, the gift is breadth — the shadowed siblings open.

Scope. The closed forms are exact — partition function, crystal probability, degree law. The hot-limit transfer is the theorem's own argument run here — any thermal growth whose admissible sets are never empty and closed upward in divisibility reaches the full completion almost surely: λ is divisibility-monotone over F₂[x] too, so admissibility is closed upward, and the chance that a pick deepens a given irreducible g by at least k is at least |g|^−βk whatever the state.

verifier: explore_function_field_melt.py

What temperature never touches is the module law's diverging costs. Bound the alphabet — moves of degree ≤ D only — and every F₂[x] thermal trajectory halts, the death authored by the clock c — the exponent of the dyadic frontier the trajectory climbs — concentrating from a light seed (one starting well below the mode) at c*(D) = ⌊log₂ D⌋ + 2; halt times obey T ≤ (D−1) + (c_final − c₀). Z, by contrast, has an immortal bounded-alphabet witness: along the pure 2-column, m = 2 rides the linear pump forever. The dichotomy is the module law thermally dressed: selection-authored structure (geography, route, tie-breaks, the lock) melts at β = ∞ exactly; module-authored structure (the cost divergence) is temperature-invariant.

Scope. Bounded-alphabet model over F₂[x], light seed. The mode c*(D) is β-free; the tail distribution is not (priced by the dyadic phase of D).

verifiers: explore_halt_clock.py, explore_function_field_melt.py