THERMAL — the growth walks at finite temperature

The object: the demands of GROWTH with the least move softened into a Boltzmann choice, an admissible m picked with probability proportional to m^−β at inverse temperature β > 1, the greedy walk being β = ∞. This page asks which parts of each fate survive heat, what each demand grows at finite β, what changes over F₂[x], where a tick sits on a log clock, and where the column locked on one place turns critical. Its verifiers are thermal.py, floors.py, spectrum.py, fieldtop.py, quadtop.py, cubtop.py, cubrace.py, realrace.py and lossdensity.py.

From a state M the thermal law picks m in the admissible set A(M) with probability m^−β / Ψ_M(β), where Ψ_M(β) is the sum of m^−β over A(M); for β > 1 it is a law at every state with A(M) nonempty, since ζ(β) bounds Ψ_M(β), and a finite nonempty A(M), as under transparency, makes it one at every β; β → ∞ recovers the least move. σ_β(n) is the sum of d^−β over the divisors d of n, and ξ(M) = W(λ(M))/M is the headroom under GROWTH's wall W(L), the largest modulus whose λ divides L. The answer, in one line: heat keeps what a demand's admissible set forces and melts what only the argmin enforced, so independence grows a random limit, its prime depths independent and geometric, dynamics grows all of Ẑ at every finite β > 1, transparency dies at the same wall, and over F₂[x] a walk with its moves restricted in degree always dies where one over Z can live.

The zeta measure

Tier: theorem.
Verifier: proof; thermal.py::section_i.

Under independence A(M) is the integers m ≥ 2 coprime to M, so once p divides M no later pick touches it: each prime enters once, at the depth of that pick. At a pick with p | m write m = p^k·u with u coprime to Mp: the weight p^−kβ·u^−β factors and the range of u does not depend on k, so

P(v_p(m) = k | p divides the pick) = p^−β(k−1)·(1 − p^−β)

exactly, at every state and every time, and the same factoring over the primes of one pick makes the depths independent. The move m = p is admissible while p ∤ M, and Ψ_M ≤ ζ(β) − 1, so p divides the pick with probability at least p^−β/(ζ(β) − 1) at every state p does not divide; by Lévy's extension of Borel–Cantelli every prime enters almost surely. From seed 1 the limit is the zeta measure, the random supernatural number ∏ p^(G_p) with G_p independent and geometric of ratio p^−β, the exponent law of a ζ(β)-random integer given p | n; from a squarefree seed s the primes of s stay at depth 1, the zeta measure from s. It is squarefree, the crystal the greedy walk grows surely, with probability ∏ (1 − p^−β) = 1/ζ(β), the reciprocal of the zeta function, and 1/ζ(2) = 0.6079 is the squarefree density read as the fraction of worlds that grow the primorial profile. As β → 1+, P(G_p ≥ k) → p^−(k−1), the law of the p-adic valuation of a Haar-random p-adic integer given that p divides it, place by place: β = 1 is at once ζ's pole and the edge of the law.

The hot limit is Ẑ

Tier: theorem.
Verifier: proof; thermal.py::section_z.

Under dynamics A(M) = {m ≥ 2 : λ(Mm) > λ(M)}. Since a | b gives λ(a) | λ(b), the set is upward-closed under divisibility: m admissible and m | m′ make m′ admissible. Its complement is {m ≥ 2 : Mm divides W(λ(M))}, the divisors of ξ(M), finite, so Ψ_M(β) = ζ(β) − σ_β(ξ(M)). Multiplying by p maps A(M) into itself injectively and scales every weight by p^−β, so

P(p divides the pick) ≥ p^−β

at every state, the exact value being p^−β(ζ − σ_β(ξ/p))/(ζ − σ_β(ξ)), with σ_β(ξ/p) = 0 when p ∤ ξ. The floor ignores the state, so by Lévy's Borel–Cantelli every prime divides infinitely many picks almost surely, v_p(M_t) → ∞ at every p, and the limit ring lim Z/M_t is Ẑ, the whole profinite completion, from every seed at every β > 1. The greedy walk's lock onto one prime (GROWTH: The lock-prime law) is an artifact of β = ∞: from seed 2 the greedy walk sits at 2^42 after 40 moves, while at β = 2 with picks m ≤ 500 all of 100 runs have opened a second prime by then. When v_p(λ) runs ahead of v_p(M), p's own admissible powers are high and weigh little, yet p rides into the state on p times any admissible move.

The proof used two facts, A(M) nonempty and upward-closed and the weight positive, summable and multiplicative, so any thermal growth on the divisibility order with those two properties reaches Ẑ almost surely. At a tombstone, M = W(λ(M)), transparency is dead, ξ = 1 and Ψ_M = ζ(β) − 1: dynamics there is the free zeta law on every m ≥ 2, at 24, 240 and 504.

The closure trichotomy

Tier: theorem.
Verifier: proof; thermal.py::section_z.

Each demand's admissible set has a shape, and the shape decides how much of the fate it forces alone. Independence avoids the support of M, and that alone forces entry once. Dynamics is upward-closed and cofinite, and that shape under the thermal law at finite β forces Ẑ; the greedy walk, β = ∞, has the same admissible sets and locks onto one prime. Transparency is the divisors ≥ 2 of ξ(M), downward-closed and finite, and that shape alone does not force death: A(M) = {2} at every M is downward-closed, finite, never empty, and grows 2^∞. Transparency dies because each move consumes at least one prime factor of ξ(M), a budget: every trajectory, at every β and under any policy at all, ends at W(λ(seed)) within Ω(W/seed) moves, Ω counting prime factors with multiplicity. So seeds 5, 7 and 73 end at 240, 504 and 20,174,525,280, the walls the verifier computes; that every trajectory ends there is the proof's. One fate's mark, entry once, is authored by shape alone, one fate by shape and the finite-β weight; the third needs shape and budget.

What each fate needs of the ground

Tier: theorem.
Verifier: proof; floors.py.

Take the integers away one property at a time. Death needs only order: a demand confining the walk to a finite region absorbs under every policy within the region's height, since every move ascends strictly. Where it dies is the demand's. Under the interval demand, every element of the region above the state, the graves are the reachable elements with no successor, and there is one grave iff the reachable part has a maximum; on {1 < 2, 1 < 3} the grave is the policy's, greedy dying at the cheaper element and the thermal law splitting the two. Transparency is the interval case with a maximum, its fibre above a state s being the multiples of s dividing W(λ(s)), which is why it dies at one wall.

Reaching everything needs only multiplication: on a countable cancellative commutative monoid with a positive summable multiplicative weight, under an up-closed nonempty demand every element almost surely divides the state from some time on, since m ↦ am maps the admissible set into its multiples of a injectively and gives P(a divides the pick) ≥ w(a) at every state, for every element a and not only the atoms. Being arithmetic rests on unique factorization: entry-once, geometric depths and the crystal's probability ∏ (1 − w(g)) = 1/Σ w, the product over the atoms g and Σ w the weight summed over every element, hold on a free monoid word for word, and the last is the Euler product equalling the Dirichlet sum.

Freeness cannot be dropped. In the numerical monoid {x^n : n = 0 or n ≥ 2}, cancellative and not free since (x²)³ = (x³)², both atoms divide every x^n with n ≥ 5, so every independence walk from the identity, its moves sharing no atom with the state, dies within two moves, while under up-closed demands every element of the same monoid eventually divides the state: two fates part inside one monoid. Under the weight w(x^n) = t^n, 0 < t < 1, its Euler product 1/((1 − t²)(1 − t³)) leaves its Dirichlet sum 1 + t²/(1 − t) at t⁶, one element with two factorizations, 1024/945 against 13/12 at t = 1/4. And independence consumes: a move once taken is never admissible again, so the demand owns no fixed standing move, and a standing family for it is made of recipes (GROWTH: The standing-move dichotomy).

The grading and the radical

Tier: theorem.
Verifier: proof; thermal.py::section_z, thermal.py::section_i.

Heat grades each property of a fate by how much of the selection it needs. Mortality needs none: it holds on every trajectory. Independence's destination, every prime seated, needs only a floor, independent of the state, on the probability that an unseated prime divides the pick (p^−β/(ζ(β) − 1) under independence, The zeta measure), and holds almost surely but never surely, since picking odd prime squares forever is admissible. The squarefree profile, the increasing route (GROWTH: The route is the deleted place's) and the lock needed the argmin itself and melt at every finite β: from seed 1 the first independence pick is 2 with probability 2^−β/(ζ(β) − 1) and a power of 2 with probability 1/((2^β − 1)(ζ(β) − 1)), 0.3876 and 0.5168 at β = 2.

The limits, for a squarefree seed s:

demand          β = ∞                    β < ∞
transparency    W(λ(seed))               W(λ(seed))
independence    ∏ p, the crystal         the zeta measure from s
dynamics        one column               Ẑ

Ẑ/J = ∏ F_p, with J, the elements of Ẑ that every prime divides, the Jacobson radical of Ẑ, so cold independence grows the quotient by the radical and hot dynamics grows Ẑ itself: what hot dynamics holds beyond cold independence is exactly J, while hot independence adds finite depths and never J, entry being once. Cold independence and hot dynamics are the two cells of the table that forget the seed: cold dynamics locks onto a prime the seed sets, 2 from seed 2 and 7 from seed 11 (GROWTH: The lock-prime law).

The melt over F₂[x]

Tier: theorem.
Verifier: proof; thermal.py::section_f.

Weight a monic m by |m|^−β = 2^(−β·deg m). There are 2^d monics of degree d, so the free partition function over degree ≥ 1 is r/(1 − r) with r = 2^(1−β), rational in 2^−β, and 1 at β = 2. The zeta measure transfers word for word, each place g's depth geometric of ratio |g|^−β, and the crystal probability is 1/ζ_(F₂[x])(β) = 1 − 2^(1−β), exactly 1/2 at β = 2 against Z's 0.6079: every closed form here is rational in 2^−β. λ is divisibility-monotone here too, so dynamics' admissible set is upward-closed, and never empty since the wall is finite (GROWTH: The wall over F₂[x]), and |gm| = |g|·|m| gives the floor P(g divides the pick) ≥ |g|^−β: hot dynamics reaches the full profinite completion of F₂[x] at every β > 1. The sprawl (GROWTH: The sprawl over F₂[x]) is an artifact of β = ∞ as Z's lock is: x + 1, starved forever by the greedy walk, opens almost surely. Hot dynamics' limit is the same object on the whole range 1 < β < ∞, so the limit's only transition sits at β = ∞, and the thermal law needs no tie-break, where the greedy walk over F₂[x] needed one.

The two tests

Tier: criterion (proved; checked on 20 states against every monic of degree ≤ 9, the fresh law on every Δ ⊆ {1, …, 12} at d ≤ 14).
Verifier: proof; thermal.py::section_f.

Over F₂[x], λ(g^a) = (2^d − 1)·2^⌈log₂ a⌉ at g of degree d (GROWTH: The wall over F₂[x]), so with c = v₂(λ(M)) a move m is admissible iff it passes the clock test, some place of Mm at depth above 2^c, or the odd test, some place of m absent from M whose 2^(deg g) − 1 does not divide λ's odd part: the 1-units are 2-groups and the Mersenne factors are odd, so nothing else moves λ. The fresh law decides the odd test: for d ≥ 2 and a set Δ of degrees, 2^d − 1 divides the lcm of 2^d′ − 1 over d′ in Δ iff d divides some d′ in Δ. One way is direct; the other takes a primitive prime of 2^d − 1, whose order of 2 is d (Zsigmondy's theorem), and at d = 6, where none exists, 9, whose order of 2 is 6. So the odd test passes at most once per degree, never at degree 1, and opening degree d′ closes it at every divisor of d′. The two tests are the module law's split (GROWTH: The module law) read as a list of moves.

The mortality split

Tier: theorem.
Verifier: proof; thermal.py::section_f, thermal.py::section_b.

Restrict the moves to degree at most D. Every admissible move raises c or passes the odd test at a degree from 2 to D not passed before, so a trajectory of T moves has T ≤ (D − 1) + (c_T − c₀), and a clock at c needs a depth above 2^(c−1), so 2^(c_T − 1) < deg M₀ + D·T. Together T < D + log₂(deg M₀ + D·T) − c₀, false for large T, so T is bounded: every restricted trajectory over F₂[x] halts, at every β and under any policy. Over Z it need not: from 2^a with a ≥ 3 and a ≥ 2 + max v₂(p − 1) over the odd primes p ≤ B, the move 2 stays admissible under the restriction m ≤ B forever, since the odd part of the state can lift λ's 2-part no higher. So every infinite trajectory over F₂[x] picks moves of unbounded degree, since a tail inside degree D would halt. This is the module law's rank split read as mortality: its pump prices a tick, a move raising the clock c, at a constant, which a restricted walk pays forever, and the log clock at a degree that outgrows every restriction, at every temperature. The split has a price. Keep a halted world alive by multiplying in, at each halt, the least-degree monic after which a move of degree ≤ D is admissible: that bill is the least d·(2^c + 1 − ⌊D/d⌋ − v_g(M)) over irreducibles g of degree d ≤ D, since no injection reopens an odd test. Once a degree-1 place g leads a revival with D < 2^c, its one admissible move is g^D, which raises the clock by one and halts the world again, while any other place of degree d, standing at most 2^c − ⌊D/d⌋ since the world halted at clock c, costs at least d(2^c + 1) at that next halt: g leads forever, each revival buys exactly one move, and the bill at every later halt, of clock c′, is 2^(c′−1) − D, its power-of-2 term doubling with every move bought. Over Z the world from 2^a above needs no revival. Heat melts what selection authored and leaves what the module authored.

The place spectrum

Tier: theorem; observation (the points, their order, the least below 1000).
Verifier: proof; spectrum.py::section_q.

Lock dynamics on one prime q and deepen it: the column q^a. Its headroom is set by q's own arithmetic. At odd q, λ(q^a) = (q − 1)·q^(a−1), and W(λ) carries q to the exponent a, exactly the state, so q never divides the headroom: ξ(q^a) is 2^(v₂(q−1)+2) times every odd prime p ≠ q with p − 1 dividing (q − 1)·q^(a−1), each at the exponent v_p(q − 1) + 1. The exclusion p ≠ q is part of the formula: q itself has q − 1 dividing λ, but its exponent a is spent by the state; dropped, the formula is wrong at every odd column checked (q ≤ 47, a ≤ 6). At q = 2, λ(2^a) = 2^(a−2) for a ≥ 3, the 2-part cancels, and the Fermat primes are left. As a grows the headroom rises to a limit C_q, the same 2-part and the same exponents over every odd prime p ≠ q of the form d·q^i + 1 with d dividing q − 1. C_q is a supernatural number, a formal product that may hold infinitely many primes (whether 2·3^i + 1 is prime infinitely often is open), and its divisor sum σ_β(C_q) is the convergent product of the factors' sums. The column's normalizer falls to Ψ_col(q) = ζ(β) − σ_β(C_q).

Each finite place q has its critical point β_q, the root of Ψ_col(q) = 1. Given the depth a thermal dynamics walk reaches along a column, conditioned on taking only moves q^r, the normalizers left whole, the set of lower depths it passed through holds each depth a independently with probability 1/(1 + Ψ_(q^a)), Ψ_(q^a) the normalizer there (MEMORY: The unbounded-memory law at q = 3; the argument holds wherever every move q^r, r ≥ 1, raises λ, at every depth of an odd column and from depth 3 on at q = 2), so β_q is where that coin turns fair in the deep limit. There is exactly one, and it lies in (1, β*), β* = 1.72865 the root of ζ = 2: Ψ_col(q) is a sum of m^−β over a fixed set of m, so strictly decreasing; the sum of 1/p over the primes d·q^i + 1 converges, so σ_β(C_q) stays bounded as β → 1 while ζ does not; and C_q has a divisor above 1, which puts Ψ_col(q) below ζ − 1, and that is 1 at β*. A column whose headroom has the larger divisor sum at every β crosses at the smaller β. The points are β_2 = 1.60449, β_3 = 1.49595, β_5 = 1.42057, β_7 = 1.42344 and β_11 = 1.43666, so they are not monotone in q; over the 168 primes below 1000 the least is β_541 = 1.28150.

The 2-adic place is the top

Tier: theorem.
Verifier: proof; spectrum.py::section_q.

β_2 > β_3 > β_q for every prime q ≥ 5. Each Ψ_col(q) decreases, so β_q < β_r follows once σ_β(C_q) > σ_β(C_r) at every β > 1; write σ for σ_β and b for β.

(a) At odd q ≠ 3 the headroom holds 8 and 3, so σ(C_q) ≥ σ(8)·(1 + 3^−b). C_2 is 3 times the Fermat primes F ≥ 5, each of the form 2^(2^n) + 1 with n ≥ 1, known or not, and the sum of 2/F over all n ≥ 1 is below 0.5255. With Y the sum of F^−b, Y ≤ 0.5255·2^−b ≤ 0.263 at b ≥ 1, so ∏ (1 + F^−b) − 1 ≤ Y·e^Y ≤ 0.684·2^−b, which is below σ(8) − 1. So β_q < β_2.

(b) At q = 3, σ(C_3) ≥ 1 + 2^−b + 4^−b + 7^−b, and the same bound taken against 5, the sum of 5/F being below 1.3137, gives σ(C_2) ≤ 1 + 3^−b + 1.7085·(5^−b + 15^−b). Now 7^−b ≥ 1.7085·15^−b since 15/7 > 1.7085; 2^−b − 3^−b ≥ 2^−b/3; and 2^−b/3 + 4^−b > 1.7085·5^−b, since (5/2)^b/3 + (5/4)^b ≥ 2.083 at b ≥ 1. So β_3 < β_2.

(c) At q ≥ 5, σ(C_3) = σ(8)·∏ (1 + p^−b) over the primes p = 2·3^i + 1 with i ≥ 1, and the sum of (2·3^i)^−b is at most 0.75·3^−b ≤ 0.25, so the product is at most 1 + 0.9631·3^−b, below the factor 1 + 3^−b in σ(C_q). So β_q < β_3.

The top place is the one whose headroom is thinnest: the 2-column cancels its own 2-part and sees only the Fermat primes. The verifier recomputes the three constants (0.68335, 1.70837, 0.96302) and checks each inequality on a grid of b from 1 to 12.

The algebraic clock over F₂[x]

Tier: theorem; observation (the points at degrees 1 to 6 and their order).
Verifier: proof; spectrum.py::section_f.

Over F₂[x] the column is g^a at an irreducible g of degree d, and λ(g^a) = (2^d − 1)·2^⌈log₂ a⌉ (The two tests). So f^j, with f ≠ g of degree e, divides the headroom of g^a iff e divides d and j ≤ 2^⌈log₂ a⌉: the headroom holds every place but g whose residue field embeds in g's, F_(2^d), each at a depth that grows without bound, and g itself to depth 2^⌈log₂ a⌉ − a. At the states a = 2^μ the headroom holds no power of g, and the limit is the divisor sum S_d(z) = ∏ (1 − z^e)^−(N_e − δ) over e dividing d, with z = 2^−β, N_e the number of irreducibles of degree e, and δ = 1 at e = d and 0 otherwise, since g itself is absent. The zeta function is 1/(1 − 2z), so the critical equation ζ − S_d = 1 clears to a polynomial in z, and every critical point is minus log₂ of an algebraic number. Write γ_d for the critical point of a place of degree d, read along the states a = 2^μ. At d = 1 the polynomial is 2z² − 4z + 1: 2^−γ₁ = 1 − 1/√2 and γ₁ = log₂(2 + √2) = 1.77155, below 2, where z = 1/4 and this ring's zeta is 2.

Degrees 1 to 6 give 1.77155, 1.50553, 1.47652, 1.39505, 1.48768 and 1.33584. Two places of one degree share their point exactly, and degree 5 sits above degree 3. Degree 1 is the top: for d ≥ 2 the headroom holds both places of degree 1 at unbounded depth, so S_d ≥ (1 − z)^−2 coefficient by coefficient, strictly above S_1 = (1 − z)^−1 from z¹ on.

Between the states 2^μ the column breathes: at a = 2^μ + 1 the headroom holds g itself to depth 2^μ − 1, the limit along those states is S_d/(1 − z^d), and at d = 1 that is (1 − z)^−2 = S_2, so the degree-1 column just after a tick sits at the degree-2 point.

The class split

Tier: theorem (the headroom reads the splitting law, and the ratio of the normalizers tends to the class number h); observation (the headroom's entrants and depths at P3, and the points at four columns of Z[√−5]).
Verifier: proof; spectrum.py::section_k.

At a prime P of a number ring, of norm N over p, λ(P^a) = (N − 1)·p^(ν(a)), with ν(a) growing without bound as a grows, since the 1-units of the completion contain a copy of the p-adic integers. A prime R ≠ P, over the rational prime r, enters the limit headroom up to the largest j with λ(R^j) dividing (N − 1)·p^∞, which depends on N, p and R's own chain and never on ν. So a column over a number field reads the splitting law, which primes lie over which r and with which norms, and at the primes R ≠ P never its own chain. In Z[√−5], at the prime P3 over 3, the headroom holds the prime P2 over 2 to depth 2, the conjugate P3′ without bound, paid by the column's own 3-power, and both primes over each split prime r = 2·3^i + 1: 7, 163, 487 and 39367 below 10⁶, the range spectrum.py reads, and 2·3¹⁶ + 1 next. The primes 19 and 1459 have r − 1 dividing 2·3^∞ but are inert, of norm r², and r² − 1 is divisible by 8, and never enter. Every entrant is nonprincipal: P2 and P3′ hold no element of norm 2 or 3, and each split r = 2·3^i + 1 is 3 mod 4, so no x² + 5y².

The element world, whose moves must be principal, splits the point. Its normalizer is the weight N(·)^−β summed over every principal ideal, (1/h) times the sum of L(β, χ) over the characters χ of the class group (ELEMENT: The class gap between principal and all ideals), less the same weight summed over the headroom's principal divisors, (1/h) times the sum over χ of the headroom's divisor sum twisted by χ, as Ψ_col(q) is ζ(β) less σ_β(C_q). At P3, where h = 2, the element point is 1.35270 against the ideal point 1.54922. The class number is read at the pole: only ζ_K has a pole at β = 1, while the headroom's sums, plain and twisted, stay bounded there: its entrants' norms are d·p^i + 1 with d dividing N − 1, geometric in i, so Σ N(R)^−1 converges, and each entrant's own factor is geometric too; so the ratio of the ideal to the element normalizer tends to h, and at P3 it falls through 2.0317, 2.0056 and 2.0010 at β = 1.2, 1.05 and 1.01. Four ideal points, each read at the states whose headroom holds no power of the column's own prime, order as P2 1.70395 > P3 1.54922 > 1.45633 at a prime over 7 > 1.31983 at the inert prime over 13: the place with residue field F₂ is again on top.

The F₂ place need not top

Tier: theorem (the comparison and the wild chain); observation (the points, at 45 fields).
Verifier: proof; fieldtop.py::section_q, fieldtop.py::section_a, fieldtop.py::section_o, fieldtop.py::wild_control, cubtop.py::control_chain.

Over a number field the place with residue field F₂ can lose the top of the spectrum; the least degree where it does is 3 (The race over cubic fields). Two facts set the race. Every column shares ζ_K and each Ψ decreases, so an odd column's point lies below the F₂ place's iff its headroom's divisor sum is the larger at the F₂ point. And at a place Q over 2 with residue degree 1 and ramification index e a power of 2, λ(Q^j), the exponent of U¹/U^j, is 2^⌈log₂ j⌉ for j ≤ 2e: since v(2) = e, a 1-unit 1 + y with v(y) = t squares to valuation min(2t, e + t) where the two differ and to at least 2e at t = e, so 1 + π reaches valuation exactly 2^k after k squarings while 2^k ≤ e, and every 1-unit reaches 2e one squaring later. So an odd column of norm N holds Q to depth 2^v, v = v₂(N − 1), while 2^v < 2e. At any e the same squaring bounds the depth below: 1 + y with v(y) ≥ t squares to valuation at least φ_e(t) = t + min(e, t), and φ_e increases, so the column holds Q to depth at least φ_e^v(1) (2, 4, 7, 10, … at e = 3, 3v − 2 from v = 2).

The F₂ column's headroom holds every place of norm 2^k + 1 (3, 5, 9, 17, 257, 65537, and any Fermat prime past them) at depth 1, every other place of norm 2 without bound, and nothing else. Where 2 is unramified, an odd column holds each place of norm 2 to depth v + 2 ≥ 3 by the chain of (Z/2^j)^*, a factor at least 1 + x + x² + x³ with x = 2^−β against each conjugate's 1/(1 − x). Where 2 is totally ramified there is one F₂ place, and an odd column with v = 1 holds it only to depth 2, while the F₂ column holds every place of Fermat norm. The cyclic fields read here are cut out by a character even on the 2-part of its conductor; each has a twin, odd there, with the same splitting at 2, at the conductor's odd prime and at 5, not read. In the cyclic octic field of conductor 544, where 2 and 17 are totally ramified and 5 splits completely, the F₂ column holds the eight places of norm 5 and the one over 17, the column at a place of norm 47 holds the place over 2 to depth 2 and its own seven conjugates, and that column tops the F₂ place by 0.083 below 10⁶ even with the place over 2 held without bound, so the loss leans on no wild chain there, the tail past 10⁶ certified on the exact arm, wild chain included; the octics of conductor 2336 and 3104 lose by 0.048 and 0.080 the same way. Of the six cyclic quartics read with one F₂ place and 5 split completely and m < 100 (conductor 16m, m = 13, 17, 37, 53, 73, 97), the one of conductor 272, the only one of the six where 3 has residue degree 2, loses by 0.0059 at the wild chain's exact depth: its F₂ column holds two places of norm 9 beside four of norm 5. The F₂ place tops at the five other quartics; at the twelve quadratic fields read (−7, 17, −8, 161, and the eight fundamental discriminants |D| < 400 where 2 and 3 ramify and 5 splits), as it must (The F₂ place tops every quadratic field), least gap 0.030 at D = 264, the runner-up the column over 3; and at the 24 fields of degree 4, 6, 8 and 10 inside Q(ζ_m), m < 5000 prime, where 2 and 5 split completely, least gap 0.076. The menus stop at 10⁶. A missed place can only lower its own column's point. At each winner the most they could hide, every place of norm d·p^i + 1 past 10⁶ with d dividing N − 1, at full count and unbounded depth, leaves it above the F₂ point; the F₂ column misses only places over Fermat primes past 65537, at least 2^32 + 1, a mass below 2^−27 at degree at most 16, which moves the F₂ point by far less than every gap where it tops.

The F₂ place tops every quadratic field

Tier: theorem (a finite reduction; its one computation brackets every configuration over the whole range where the F₂ point can lie).
Verifier: proof; quadtop.py::bracket, quadtop.py::control, quadtop.py::control_quadratic.

Over every quadratic field with a place of residue field F₂, that place's point is the largest of the spectrum. Every place over 2 is then an F₂ place, and the two over a split 2 are conjugate, so only the odd columns race. By the comparison, an odd column P of norm N sits below the F₂ place iff its headroom sum is the larger at the F₂ point. Write x = 2^−β and v = v₂(N − 1). The F₂ column's sum is T₂ = (1 − x)^−(g−1), g the number of F₂ places, times 1 + N(R)^−β at every place R of Fermat norm 2^k + 1: a norm in a quadratic field is p or p², and p² − 1 is a power of 2 only at p = 3, so these are 3, 5, 9, 17, 257, 65537 and the Fermat primes past 65537, at most two places each. P's headroom holds each F₂ place to depth v + 2 where 2 is unramified, and where 2 ramifies, to depth 2 at v = 1 and at least 4 at v ≥ 2, by the wild chain at e = 2; every place of Fermat norm 2^k + 1 with k ≤ v other than P, at depth at least 1, since 2^k divides N − 1; and where N is a Fermat norm, P's conjugate without bound. Every factor is at least 1, so dropping every other place, P's sum over the F₂ column's is at least A·X/(T₂·W): A the F₂ places' factors 1 + x + … + x^depth, X = 1/(1 − N^−2β) for the conjugate, W P's own factor 1 + N^−β where N is a Fermat norm, times the factors of the other Fermat-norm places with k > v, times a bound on the Fermat primes past 65537, which are at least 2^32 + 1. This reads the field only through the splitting of 2 (split or ramified) and of 3, 5, 17, 257 and 65537: 486 configurations. It reads the column only through v and whether N is a Fermat norm, and W moves with v only where v crosses k in {1, 2, 3, 4, 8, 16} while A grows with v, so six generic classes of v and the columns of Fermat norm bound every column; a column over a Fermat prime past 65537 is bounded by the class v = 16, whose bound counts its own place and its conjugate.

The F₂ point has a floor and a ceiling. ζ_K is at least the configuration's own local factors times (1 − p^−2β)^−1 at every other prime, the least local factor, and every β where that, less the bound on the F₂ column's sum, is still at least 1 lies at or below the point. And Ψ ≤ ζ_K − 1 ≤ ζ² − 1 puts the point below 2.3353. A·X and T₂·W both decrease in β, so where A·X at each point of a grid of step 10^−4 exceeds T₂·W at the point before, no β between can make the column win: a bracket that cannot miss a narrow dip. Over every class of every configuration, from its floor past the ceiling, none fails, each ratio far above double rounding. The race is closest at the column over 3 where 2 and 3 ramify and 5, 17, 257 and 65537 split: there the bound counts nothing but the place over 2 to depth 2, against the F₂ column's two places each of norm 5, 17, 257 and 65537 and the one of norm 3, and the least ratio is 1.0058. With 2 split that column's is 1.0626, and every class but it and the generic v = 1 stays above 1.2.

The race over cubic fields

Tier: theorem (the bound decides every column but the column of norm 3 at 675 configurations; the F₂ place tops every cyclic cubic field with one); rule (whether the F₂ place tops, verified at every cubic field with an F₂ place and |d_K| ≤ 3000, both signatures).
Verifier: proof; cubtop.py::sweep, cubtop.py::control_chain, cubtop.py::control_quad_sweep, cubrace.py::Race, cubrace.py::hunt, cubrace.py::control, realrace.py::hunt, realrace.py::control_real, realrace.py::float_check.

Over a cubic field the F₂ place can lose the top, while over Q (The 2-adic place is the top) and every quadratic field with one (The F₂ place tops every quadratic field) it tops, so 3 is the least degree where it can lose. Over the complex cubic field of discriminant −1420, cut out by 3x³ + 7x² + 7x + 5, where 2 is totally ramified, the column of a place of norm 3 has its point in [1.59889, 1.59897] and the F₂ place in [1.59661, 1.59670]. It tops over every cubic field with an F₂ place and smaller |d_K|, of either signature, and over every cyclic one; it loses at about 1 in 200 cubic fields (The F₂ place loses at about 1 in 200 cubic fields).

The quadratic reduction carries to degree 3. A norm is now p, p² or p³, and p³ − 1 is never a power of 2, so the Fermat norms are the same, at most three places each. Where 2 = Q₁Q₂ with Q₂ of norm 4, Q₂'s column is even: its headroom holds Q₁ without bound and each place of Fermat norm 2^k + 1 at depth at least 1, since 2^k divides 3·2^∞, while the F₂ column holds no place over 2, since 3 does not divide 1. So the F₂ column's headroom lies inside Q₂'s, Q₂'s point is the lower at every β, and only odd columns race, and there a column over 3 also holds Q₂ to depth v + 1, since λ(Q₂^j) divides 3·2^(j−1). Where 2 ramifies, the chain at any e (The F₂ place need not top) holds the ramified F₂ place in an odd column to depth at least 2, 4, 6, 8 at v = 1, 2, 3, 4 for e = 2 (where an unramified F₂ place beside it is held to v + 2) and 2, 4, 7, 10, … for e = 3 (2 totally ramified). The floor, as in the quadratic bound, reads the local factors at 2, 3, 5, 17, 257 and 65537 and takes the inert factor (1 − p^−3β)^−1 at every other prime. The bound reads the field through the splitting of 2 (four types with an F₂ place) and of 3, 5, 17, 257 and 65537 (five types each): 12500 configurations, the F₂ point below 2.7511, since ζ³ < 2 there.

Over every class of every configuration, from its floor past that ceiling, the bracket decides every column but one: the column of a place of norm 3, in 675 configurations, 250 with 2 = Q₁²Q₂ and 425 with 2 totally ramified, every one with two or three places of norm 5 in the F₂ column and 3 of type 12, 1³ or 1²1, never 111 or 3. There the bound credits that column only the F₂ places to depth 2 (3 at an unramified one) and its own conjugates, and its least ratio is 0.9015. A cyclic cubic field is Galois and ramified only at 3 and at primes ≡ 1 mod 3, so with an F₂ place 2 splits completely, and 5, 17, 257 and 65537, each ≡ 2 mod 3, split completely or stay inert: 48 configurations, none failing, the least ratio 1.0254 at the column of norm 3 where 3 ramifies.

Where the bound fails, the race is read at a field. Of the cubic fields with |d_K| ≤ 3000 (Hunter's theorem; the 96 totally real ones match LMFDB's list), twelve lie in a failing configuration, ten complex and two totally real, each failing at the column of norm 3 alone. There both points are read exactly but for the tail: the column of norm 3 holds a ramified place over 2 to depth exactly 2, since λ(Q³) = 4, an unramified F₂ place to depth 3, its own conjugates of norm 3 without bound, and at depth 1 each place over an odd prime whose norm M has M − 1 dividing 2·3^∞ (7, 19, 163, 487, 1459, …). ζ_K is the Euler product below 10⁶ times a tail between 1 and the cube of ζ's own tail there, since the places over each prime have Σ e f = 3, and each point is read at both ends, the column of norm 3's lower end carrying the most its places past 10⁶ could add. The F₂ column drops places of Fermat norm past 65537, a factor below exp(6·2^(−32β)) in its σ, which its lower end carries. Each end is bisected to 10⁻⁹; at −1420 and 1940, at 30 digits and midway between the brackets, the F₂ upper end's bound on its column's Ψ − 1, Ψ = ζ_K − σ, is negative and the norm-3 lower end's positive, so both losses are certified, each Ψ decreasing. The F₂ place loses at −1420, 1940, −2764 and 2804, and tops at the other eight, by 0.0043 at its closest (−2284), and at both fields below 1420 (−460, −940). At −1420 the F₂ column holds places of norm 3, 5, 5, 9, 17, 17, 17 and 65537, and the column of norm 3, beyond the place over 2, one place of norm 1459; 1940 has the same F₂ column, its brackets 0.0022 apart, as at −1420. The bound's ratio does not order the race: −1420 loses at 0.9608 and −2284 tops at 0.9206.

The F₂ place loses at about 1 in 200 cubic fields

Tier: theorem (a finite reduction, the bound's sweep fixing the 675 configurations that race: ordered by |d_K|, the cubic fields of a signature where the F₂ place loses have a density, the same in both signatures, and the density where it tops is positive); rule (proved: the loss's density is positive, from the race certified at −1420); observation (its value, 0.00491 ± 0.00006, over 34000 drawn splittings); known (the local laws and the count; Lévy's theorem).
Source: M. Bhargava, A. Shankar and J. Tsimerman, On the Davenport–Heilbronn theorems and second order terms, Invent. Math. 193 (2013), 439–499, Lemma 18 and Theorems 7 and 8 (arXiv:1005.0672); P. Lévy, Sur les séries dont les termes sont des variables éventuelles indépendantes, Studia Math. 3 (1931), 119–153, as stated in N. Bouleau, The lent particle method, application to multiple Poisson integrals, arXiv:1004.2819, Theorem 12(a).
Verifier: proof; lossdensity.py::failing_configs, lossdensity.py::Draw, lossdensity.py::control_race, lossdensity.py::control_sampler, lossdensity.py::control_shortcut, realrace.py::float_check.

The race reads a field only through the splitting type of each prime below 10⁶, no real place among them. The local laws (Source, Lemma 18) give a prime p its five types 111, 12, 3, 1²1 and 1³ in proportion to 1/6, 1/2, 1/3, 1/p and 1/p², independently across primes, in either signature, and the cubic fields with finitely many types prescribed have the product of those weights as their density (Theorems 7 and 8). So every field split as −1420 is at every prime below 10⁶ loses, and the loss has positive density; a configuration the bound decides tops at every field in it, so the top has positive density too.

The density exists and is the probability of a loss under the product of the local laws. Cut the race at Y, reading ζ_K below Y with its tail between 1 and the cube of ζ's tail, and each σ's places past Y by a bound: the sequences it calls a loss, and those it leaves open, are finite unions of prescribed types, so the losing fields' lower and upper densities lie between the first's density and the sum of both. As Y grows every bound past Y tends to 0, so the race decides every sequence whose two points differ at every large Y, and by dominated convergence the open sets' measure tends to the ties'. A tie needs σ₂ = σ₃ at a point β₀ where ζ_K(β₀) = 1 + σ₂(β₀). Fix the types at the primes the σ's read (2, 3, the Fermat primes and the primes 2·3^a + 1): σ₂ − σ₃ is then a fixed Dirichlet series, nonzero (σ₂ holds a place of norm 5 at every failing configuration, σ₃ none; elsewhere the bound decides, so the two points differ), with countably many zeros β₀ > 1, and log ζ_K(β₀) is a fixed part plus a convergent sum of independent terms, one per other prime, each on five distinct values, the likeliest of probability below 1/2. Every term misses any constant with probability above 1/2, so the misses' sum diverges and Lévy's theorem makes that sum's law continuous, so each β₀ is a tie with probability 0, and the ties have measure 0.

Its value is read by drawing. Of the bound's 12500 configurations, each with an F₂ place, all but 675 top; those weigh 86570/2598141 = 0.03332 of all cubic fields, and within them a draw of a type at every other prime below 10⁶ loses at 0.1474 ± 0.0019 of 34000 draws, 18 left open. The density is 0.00491 ± 0.00006 of all cubic fields, about 1 in 200, and 0.00607 ± 0.00008, about 1 in 165, of the 17/21 with an F₂ place. Cutting each draw's ζ_K at 10³, the σ's read whole, decides 92% of them, and the cut never disagrees with the race at 10⁶ on 500 draws raced both ways, since each cut's ends bound the true columns. The two ramified types of 2 lose at 0.1486 and 0.1462, alike within the spread. The fields to 3000 lose at 4 of the 388 with an F₂ place, against 2.35 expected (P(X ≥ 4) = 0.21, Poisson), though the types at 2 and 3 of all 515 fields to 3000 sit far from the local laws' weights: 111 at under half its weight.

Open fronts

In degree 4, the twins of the cyclic quartics read, odd on the 2-part of the conductor, which share their splitting at 2, the odd conductor prime and 5.