TRIPLE — what a cubic field's split primes carry

The object: a complex cubic field K of discriminant d_K, its class group Cl of order h, and the classes of its degree-1 places. A totally split prime p = P₁P₂P₃ carries a triple of classes, and since P₁P₂P₃ = (p) the triple lies in M = {(a, b, c) ∈ Cl³ : a + b + c = 0}, of order h². A partially split prime carries one degree-1 class. The questions: which triples occur, which class the partial place takes, and how the shortfall of principal places at a finite cut, which over an imaginary quadratic field the prime-ideal powers account for (PRINCIPAL), behaves one degree up.

cubic.py takes a complex cubic field to its maximal order from a binary cubic form (Delone and Faddeev), the places over any prime, the class group and the class of a degree-1 place, by a bounded search that placed every place these runs needed. Its class group is checked against the analytic class number formula: the relations give h and the regulator Reg up to one integer factor m, and h·Reg over the formula's value, its Euler product taken to 3·10⁴, reads 0.9962 to 1.0027 at all 8336 complex cubic fields with |d_K| ≤ 50000, where m ≥ 2 would put it near 2 or above. The truncated product has no proved error bound: numerical evidence, not a proof (triple.py::section_controls). The enumeration is checked against an independent count of the complex fields to |d_K| ≤ 6000 (888) and by Hasse's count at conductor 1 to |d₀| ≤ 10000. An independent record of the complex fields with h > 1 to |d_K| ≤ 50000 holds 3133 where this one finds 3134: the extra, d_K = −14087 with h = 2 here, is the one it leaves open, and the other 3133 class groups agree, compared outside triple.py and entered in its findings.

The answer: the triples fill a subgroup R of M whose index is 3^t, t counting the primes q ≡ 1 mod 3 that divide the conductor f of d_K = f²d₀ (d₀ fundamental), plus one when 9 | f and 3 splits in Q(√d_K) (The genus index law; proved wherever t = 0 or t is the 3-rank; verified to |d_K| ≤ 50000), so the triples fail to fill M exactly where t > 0; the partial place is uniform on Cl whatever R is. At a finite cut the principal places run short as they do over an imaginary quadratic field, but at degree 3 the prime-ideal powers account for only about a third of it at cut 1000 (an observation); the rest reads as a fixed weight missing at the primes below a few hundred (a pattern), carried mostly by the partial places at that cut (an observation), to which the explicit formula gives no square term; and at 2-rank 1 the least odd partial place lies outside 2Cl far more often than half (an observation), though the character it reads holds no Dirichlet factor (a theorem).

The triple is a homomorphism

Tier: property.
Verifier: proof; triple.py::section_controls.

Let N be the Galois closure of K, H the Hilbert class field of K and H̃ the Galois closure of H over Q. A prime p splits totally in K exactly when its Frobenius in Gal(H̃/Q) lies in Gal(H̃/N). With K₁, K₂, K₃ the conjugates of K in N and H₁, H₂, H₃ the matching conjugates of H, restriction to Hᵢ is a homomorphism Gal(H̃/N) → Gal(Hᵢ/Kᵢ) = Cl, and with 𝔓 the prime of H̃ over p whose Frobenius is read, Pᵢ = 𝔓 ∩ Kᵢ, so by Artin the triple is the image of Frobenius under a homomorphism into M. Chebotarev fills Gal(H̃/N), so the realized set R is a subgroup of M; conjugation permutes the copies, so R is stable under S₃ acting on coordinates; each coordinate map is onto, since H₁ ∩ N = K₁ (The partial place is uniform), so every projection of R is onto Cl. The closure [P₁] + [P₂] + [P₃] = 0 holds at every totally split prime the script classes, the 37608 triples it keeps among them; where each place is classed on its own, above the primes whose places generate the relations, this tests the class map, and below them the relation (p) forces it.

The stable subgroups

Tier: theorem.
Verifier: proof; triple.py::section_groups.

An S₃-stable subgroup R of M with every projection onto Cl contains M ∩ (3Cl)³, and is the preimage of W ⊗ St + V ⊗ D for a subspace W of V = Cl/3Cl, where St is the sum-zero permutation module of S₃ over F₃ and D its diagonal. Its index in M is 3^(r − dim W), r the 3-rank.

Proof. For (a, b, c) ∈ R, the transposition (12) gives (b, a, c) ∈ R, so (a − b, b − a, 0) ∈ R; permuting, (a − c, c − a, 0) ∈ R; the two sum to (3a, −3a, 0), since b + c = −a. As a runs over Cl, R holds every (3a, −3a, 0), and with its permutations all of M ∩ (3Cl)³. So R is the preimage of a stable subspace of the sum-zero triples of V, which is V ⊗ St. In characteristic 3 the module St is uniserial: its only stable line is D = {(λ, λ, λ) : λ ∈ F₃}, a 3-cycle permuting the other three, its top St/D carrying the sign, and for a 3-cycle γ the map γ − 1 kills D and carries St/D isomorphically onto D. For a stable subspace U let V₀ ⊆ V be its intersection with V ⊗ D and W ⊆ V its image in V ⊗ (St/D); γ − 1 maps U into itself, so W ⊆ V₀. Write u ∈ U as x⊗(1, −1, 0) + y⊗(1, 1, 1); then u + (12)u = 2y⊗(1, 1, 1) lies in U, so y ∈ V₀ and the first coordinate x + y lies in V₀ + W = V₀, so a U projecting onto V has V₀ = V: it contains V ⊗ D and is W ⊗ St + V ⊗ D, of codimension r − dim W.

When 3 ∤ h, V = 0 and R is all of M. At 3-rank 1 it is M or Δ = {a ≡ b ≡ c mod 3Cl}, of index 3; at h = 3 the candidates 0, D and M are the smallest case, 0 excluded by the projection. Computably Δ is a lattice membership (with Cl = Zⁿ/Λ, 3Cl is (Λ + 3Zⁿ)/Λ); the scalar test (h/3)(a − b) = 0 agrees with it only where the 3-part is cyclic, and above 3-rank 1 it passes every triple, h/3 being then a multiple of Cl's exponent. Over thirteen chosen groups with h ≤ 18 and 3-rank at most 2 the stable subgroups projecting onto Cl number exactly the subspaces of F₃^r, graded by index as the theorem says, each containing M ∩ (3Cl)³.

The genus index law

Tier: rule (verified at every complex cubic field with |d_K| ≤ 50000, given cubic.py's class groups).
Verifier: triple.py::section_genus.

Write d_K = f²d₀ with d₀ the fundamental discriminant of the resolvent k = Q(√d_K) and f the conductor. Let t be the number of primes q ≡ 1 (mod 3) dividing f, plus one when 9 | f and 3 splits in k. Then [M : R] = 3^t, and R is the triples agreeing modulo the subgroup G of Cl cut out by the fields below.

The argument has two halves. Below: each such q, or the 9, names a cyclic cubic field F of conductor q or 9 with KF/K unramified. At a q ≡ 1 (mod 3) this is Abhyankar's lemma: q splits in k, since an inert or ramified q ≡ 1 offers χ, the cubic character of k cutting out N, only pieces fixed by the conjugation σ of k/Q (The conductor's primes are forced), and, χ being tame at q, q ∥ f, so v_q(d_K) = 2, and q is totally and tamely ramified in K, F is tamely ramified at q alone with index 3, and F is real; at the 9, where the ramification is wild, it is local class field theory at 3: 3 splits in k and N/k ramifies over it, so 3 = 𝔮³ in K with K_𝔮 a ramified cyclic cubic extension of Q₃; the cubic characters of Z₃* form a group of order 3, read on (1 + 3Z₃)/(1 + 9Z₃), so F's character at 3 is K_𝔮's or its inverse times an unramified one, and KF/K is unramified at 𝔮. The t fields are abelian over Q with coprime conductors and K is not Galois, so their compositum meets K in Q and the t fields together give an unramified extension of K of degree 3^t, cutting out a subgroup G of index 3^t in Cl. F is Galois over Q, so a Frobenius restricts to F identically from all three copies, and R lies in {a ≡ b ≡ c mod G}, of index 3^t in M. Above: a further quotient Cl → Z/3 on which every triple agrees gives a cubic character of k that must be fixed by the conjugation of k/Q and unramified over N, and such a line needs a split conductor prime (The sign of the second line is the regime). That the index is then at most 3^t is proved at 3-rank 1; where t is the 3-rank r, since the stable subgroups give index at most 3^r; and at t = 0 at every 3-rank: under each quotient Cl → Z/3 the triples' image is D or St, and D would give a σ-fixed cubic character of k whose field is unramified over N, which needs a split conductor prime, so every quotient reads St and R = M. At 3-rank 2 or more with 1 ≤ t < r the count below is what the rule rests on, at the one such field in range.

So Cl/G ≅ (Z/3)^t, the Galois group of the t fields' compositum over K, 3^t divides h and t ≤ r. At every field with h > 1 — 3134, each read through ten or more split primes — the index of the subgroup the triples and their permutations generate is exactly 3^t: 1767 fields with 3 ∤ h, 1018 at 3-rank 1 with t = 0, 347 at 3-rank 1 with t = 1, and the 2 of 3-rank 2 below. The generated subgroup lies in R, R being stable, so its index bounds [M : R] from above, and the below half bounds it from below, so at every such field the count settles the index given cubic.py's class groups, with no appeal to chance. At h = 3 and |d_K| ≤ 6000 the 83 fields split 38 at index 3 and 45 at index 1. The two fields of 3-rank 2, |d_K| = 24843 (f = 91 = 7·13, t = 2) and 47628 (f = 126 = 2·3²·7, t = 1), read index 9 and index 3, the first proved and the second the one field with 1 ≤ t < r. A uniform Frobenius on R prices any event at its size over |R| = h²/3^t.

The conductor's primes are forced

Tier: rule (verified at every complex cubic field with |d_K| ≤ 50000; the sketch below covers the primes other than 3).
Verifier: conductor.py::section_admissible; conductor.py::section_ranks.

A prime q ≡ 1 (mod 3) divides f only when it splits in k, a prime q ≡ 2 (mod 3) only when it is inert, v₃(f) ≤ 2, and v₃(f) = 1 only where 3 | d₀. The cubic character χ of k cutting out N has conductor (f) and is inverted by the conjugation σ (its conductor is Hasse's: d_K = f²d₀ for a cubic field whose Galois closure is N), so at each prime of k over f that σ fixes its local part needs a piece of the local units' 3-part that σ inverts, and at a pair of primes that σ swaps a nontrivial 3-part: an inert q ≡ 1 or a ramified q ≠ 3 offers only fixed pieces (at an inert q the 3-part of F_(q²)* lies in F_q*, as 3 ∤ q + 1, and at a ramified prime the residue field is F_q, both fixed by σ, the higher units being pro-q), a split q ≡ 2 none at all. The 3-part of (O_k/f)* then splits under σ prime by prime — one fixed and one inverted cyclic factor at a split q ≡ 1 or at 9 | f with 3 split or inert, one inverted at an inert q ≡ 2 or at 3 ∥ f, one fixed and two inverted at 9 | f with 3 ramified — and the engine reads these ranks at all 2468 fields with f > 1. So the conductor is not free data: which primes may divide it is fixed by their residues mod 3 and how they split in k.

The sign of the second line is the regime

Tier: theorem (3-rank 1).
Verifier: proof; conductor.py::section_ranks; conductor.py::section_sign.

At 3-rank 1 the triples read mod 3Cl cut out, over N, a field whose largest abelian piece over k is an extension E of type (3, 3) containing N and unramified over N. In Gal(E/k), of type (3, 3), σ inverts the quotient Gal(N/k), and the second line is the kernel Gal(E/N); σ fixes the second line when R = Δ and inverts it when R = M, and these are the two regimes. A cubic character of k fixed by σ is usable when its field, joined to N, is unramified over N (said below as its field being unramified over N); a usable one exists only when t ≥ 1, so the index law's above half holds at 3-rank 1 by proof; when t ≥ 1 the genus law's below fields give one.

Proof. Since σ inverts Cl(k), the σ-fixed 3-part of the ray class group mod f comes from the local units (O_k/f)*. E/N unramified at a prime 𝔭 | f asks that each character of Gal(E/k) restrict on the local units into ⟨χ_𝔭⟩, χ_𝔭 the local part of χ at 𝔭; where σ fixes 𝔭 that group holds no nontrivial fixed character, and where σ swaps 𝔭 with its conjugate the local cubic characters form a line spanned by χ_𝔭. So a usable fixed line needs a split conductor prime with nontrivial local 3-part, that is t ≥ 1. If the triples' image under a quotient φ: Cl → Z/3 is D, as at 3-rank 1 when R = Δ and as the genus law uses at t = 0, let H′ be the cubic unramified extension of K that φ cuts out and J = H′₁N, which at 3-rank 1 is E, of degree 9 over k as H₁ ∩ N = K₁ (The partial place is uniform). The three coordinates of a triple agree modulo ker φ, so H′₁N = H′₂N = H′₃N and J is Galois over Q. The transposition fixing K₁ restricts to σ on k, and any lift acts on H′₁ as an element of the abelian Gal(H′₁/K₁), so σ acts trivially on Gal(J/N) ≅ Gal(H′₁/K₁); σ inverts Gal(N/k), and 2 being invertible on a group of order 9 the two parts each have order 3: Gal(J/N), at 3-rank 1 the second line, is fixed, so the fixed character of k vanishing on the inverted part is usable, and t ≥ 1. If the image under φ is the sum-zero module St, as at 3-rank 1 when R = M, its field L = H′₁H′₂H′₃N is unramified over N and Galois over Q, and Γ = Gal(L/k) has order 27 and is not cyclic. Its commutators fill the diagonal and Γ modulo it is abelian, so by Burnside Γ^ab is (Z/3)², and σ acts on its line from St by the sign: inverted.

Existence alone does not sort. A σ-fixed cubic character of k exists at 9 | f whatever 3 does in k, and a fixed line exists with none usable at exactly the 150 fields with 9 | f, 3 not split and no q ≡ 1 dividing f, 39 of them at 3-rank 1, where t = 0 makes R = M. At all 1018 fields of 3-rank 1 with t = 0 the ray class group mod f has a σ-inverted line beyond χ's, checked through sufficient conditions for its σ-inverted part to have 3-rank at least 2, and the second line of E is inverted, E being unramified over N, by the proof above, t = 0 forcing R = M. At t = 1 the same sufficient conditions give an inverted line of the ray class group mod f beyond χ's at 121 of the 347 (at d₀ = −3 they subtract one for the cube roots of unity, a lower bound), beside the usable fixed one there, its field unramified over N wherever, at every σ-fixed conductor prime, the 3-part of the local units has a σ-inverted part of 3-rank at most 1, which holds at all but 23 fields of 3-rank 1 (the least |d_K| among them 11907); at the 7 of those 23 with t = 1 it is left undecided. So neither which fixed characters exist, usability aside, nor which inverted ones exist decides R; usability does, and with it the sign σ puts on the second line of the E the triples cut out.

Gerth's rank law at conductor 1

Tier: known.
Source: F. Gerth III, Ranks of 3-class groups of non-Galois cubic fields, Acta Arith. 30 (1976) 307–322, Theorem 3.4.

If f = 1, so that N/k is unramified, then rk₃ Cl(K) = rk₃ Cl(k) − 1: 3 divides h exactly when the resolvent has 3-rank at least 2.

The triples prove one direction: if f = 1 and 3 | h, the resolvent has 3-rank at least 2. With no conductor prime, take any quotient Cl → Z/3: the triples' image in the sum-zero triples of Z/3 is D or St. D would give, as in the block above, a fixed unramified cubic character of k, which σ forbids; so it is St, and the extension of type (3, 3) built there is unramified over k. The engines read the law at the ranks that occur: over the 5868 fields with f = 1 and |d_K| ≤ 50000, cubic.py's class group and the resolvent's class group, read as binary quadratic forms by principal.py, sit at 3-ranks (0, 1) at 5460 fields and (1, 2) at 408, none off (conductor.py::section_corner). Hasse's count of cubic fields of discriminant d₀, (3^ρ − 1)/2 with ρ the 3-rank of Cl(k) (H. Hasse, Math. Z. 31 (1930) 565–582), checks the enumeration at f = 1 at all 3043 fundamental d₀ with −10000 ≤ d₀ < 0 (conductor.py::section_controls).

The partial place is uniform

Tier: theorem.
Verifier: proof; triple.py::section_partial.

The degree-1 place over a partially split prime has its class equidistributed on Cl, at every complex cubic field and whatever R is.

Proof. Frobenius at a partially split p is a transposition fixing one copy, say K₁: an element of Gal(H̃/K₁) outside Gal(H̃/N). Restriction maps Gal(H̃/N) onto Gal(H₁/(H₁ ∩ N)), a subgroup of Cl of index [H₁ ∩ N : K₁], which divides [N : K₁] = 2. H₁/K₁ is unramified at every place, the real one included, while N/K₁ ramifies at the real place of K₁, since N is totally complex. So H₁ ∩ N = K₁, Gal(H̃/N) maps onto Cl, and so does its other coset in Gal(H̃/K₁), the elements over the transposition fixing K₁; Chebotarev spreads [P₁] evenly.

So the possible values of R differ only in the split primes, and one place cannot tell them apart in the limit. At a finite cut the partial place is principal at 0.3247 over the index-3 fields with h = 3 to |d_K| ≤ 24000 and p < 1000, and at 0.3121 over the index-1 fields, a ratio of 1.0405 ± 0.0175, 2.3 standard errors counted as if places were independent, which within a field they are not, while the split places' per-place shares read 0.3216 and 0.3224 in that order (an observation).

The classes of the prime powers

Tier: theorem.
Verifier: proof; triple.py::section_groups.

Gal(H̃/Q) is an extension of S₃ by R, and it splits: complex conjugation lifts a transposition with order 2, and every lift g of a 3-cycle is, by Chebotarev, the Frobenius of some prime p inert in K, so g³ restricts on each Hᵢ to the Artin symbol of the principal ideal pO_Kᵢ and g³ = e; with R abelian and a complement over each Sylow subgroup of S₃, Gaschütz's theorem gives the complement. So Gal(H̃/Q) = R ⋊ S₃. Where R = M the group has order 6h², and an element (v, s) squares to e when s = 1 and 2v = 0 (|Cl[2]|² of them) or s is a transposition and v + sv = 0, i.e. v = (a, −a, 0) up to the order of coordinates (h for each of three): 3h + |Cl[2]|² in all. It cubes to e when s = 1 and 3v = 0 (|Cl[3]|²) or s is a 3-cycle, where v + sv + s²v = (a + b + c)(1, 1, 1) = 0 always (h² for each of two): 2h² + |Cl[3]|². Every inert prime's cube, the identity as the splitting showed, carries the all-principal triple at every h.

The shortfall is a fixed weight at the small primes

Tier: pattern (complex cubic fields with h ≥ 2: cuts 10³ to 10⁴ over |d_K| ≤ 12000, the cut 2·10⁴ over |d_K| ≤ 3000, and the single places over |d_K| ≤ 24000, those with p < 2000 beyond 12000).
Verifier: triple.py::section_partial; shortfall.py::section_sweep; shortfall.py::section_collapse; shortfall.py::section_closure; shortfall.py::section_quadratic.

The level of a set of classes is the one defined over a quadratic field at PRINCIPAL: The shortfall is the prime powers, read here raw over the degree-1 places below a cut X or over Π_C, all prime-ideal powers of K in class C weighted 1/j for a j-th power. At cut 1000, over the 535 fields to |d_K| ≤ 12000:

cell                     fields   raw       in prime powers
T, the trivial class     535      0.9244    0.9489
S, nontrivial squares    313      1.0176    1.0226
O, outside 2Cl           291      1.0570    1.0196

The powers remove 66% of the non-squares' excess but only 32% of the trivial class's deficit at cut 1000; over an imaginary quadratic field they flattened every class by cut 10⁴, where the trivial class here, in powers, still reads 0.9931 ± 0.0008. The rest does not grow with X as the prime-power correction does, nor as the piece of size li(√X) a zero near the centre would give. Its deficit δ = 1 − η, η the level of T counted in prime powers, falls 0.0511, 0.0313, 0.0125, 0.0069 at cuts 10³, 2·10³, 5·10³, 10⁴, and δ·li(X) reads 9.1, 9.9, 8.6, 8.6: a fixed missing weight diluting as 1/li(X). Per nontrivial class δ/(h − 1) stays within about a factor two across h = 2 to 7 at every cut, its largest over its least reading 1.54, 1.35, 1.99 and 2.00 at 10³, 2·10³, 5·10³ and 10⁴.

The weight sits at the smallest primes. The principal level of a single unramified degree-1 place, h when it is principal and 0 when not, averages 0.33 to 0.49 at p = 2 to 7, 0.54 to 0.86 up to 90, 0.82 to 0.98 up to 400, and within 0.06 of 1 above, in every band of |d_K| that reads those primes (the band to 3000 reads none below 12), through the largest places read (p < 10⁴, p < 2·10⁴ at |d_K| ≤ 3000, p < 2000 beyond 12000). It moves with the discriminant, but far less than a floor does: in bands of |d_K| doubling to 24000 the larger fields run lower near p = 100 (0.85 and 0.82 from ln p = 4.5 to 5, against 0.97 and 0.98 in the two smallest), yet binned by ln p − a ln |d_K| the bands' χ²/dof barely moves with a (1.97 at a = 0, least 1.57 at a = ½, 2.44 at a = 1), where over imaginary quadratic fields, whose least principal prime is at least |d|/4 at discriminant d, the same scan finds that floor's a = 1 at 0.55 against 15.2 at a = 0. The trivial class in prime powers over the fields to |d_K| ≤ 3000 reads 0.9973 ± 0.0010 by cut 2·10⁴, 2.7 standard errors short of 1. Weighted, the shortfall is a budget and a zero sum (The weighted shortfall is a budget and a zero sum); why the smallest places avoid the trivial class, whether that weighted law carries δ/(h − 1), what sets the few hundred, and how that scale grows with |d_K| are open. The prime-power weight summed over the classes equals the count taken from how each prime splits, without the class map, to 5·10⁻¹³.

The weighted shortfall is a budget and a zero sum

Tier: property, the decomposition, and under the generalized Riemann hypothesis its zero sum nonnegative and, with no zero at the centre, its limit; observation, the zero sum's share (the 535 fields with h ≥ 2, |d_K| ≤ 12000).
Verifier: proof; central.py::Budget; central.py::main.

Let F_T(u) = f(u/T), Odlyzko's test function of width T: f(x) = (1 − |x|) cos πx + sin π|x|/π for |x| < 1, 0 beyond, with transform f̂(x) = 8π² cos²(x/2)/(π² − x²)² ≥ 0. Give p^k < e^T the weight w = 2 log p · F_T(k log p)/p^(k/2) and the coefficient C(p^k) = Σ f_P (h·[P^(k/f_P) principal] − 1), over the primes P | p of K whose residue degree f_P divides k: log p · C(p^k) is the p^k coefficient of −ζ_H′/ζ_H + ζ_K′/ζ_K, H the Hilbert class field. The weighted shortfall per nontrivial class is d(T) = −Σ wC/(h − 1): at its prime an unramified degree-1 place adds −w when principal, w/(h − 1) when not.

Proof. ζ_H/ζ_K is the product of the h − 1 entire L(s, χ), χ ≠ 1 a class character, each of conductor |d_K| and gamma factor Γ_R(s)²Γ_R(s + 1). Weil's explicit formula with F_T (known: A. Weil, Comm. Sém. Math. Univ. Lund 1952, tome suppl., 252–265), summed over χ, gives d(T) = φ(T) + z(T): the budget φ(T) = −(log |d_K| + G(T)), G(T) = (1/2π) ∫ T f̂(tT) [2 Re ψ(¼ + it/2) + Re ψ(¾ + it/2) − 3 log π] dt, the same at every h, and z the sum of T f̂(γT) over their zeros ½ + iγ, over h − 1. Under the Riemann hypothesis for these L(s, χ) every γ is real, so z ≥ 0. T f̂(tT) has integral 2π and concentrates at 0, so G(T) tends to 2ψ(¼) + ψ(¾) − 3 log π = −12.975 = −log 431472. With no γ = 0, each term tends to 0; for |x| ≥ 2π, x² − π² ≥ 3x²/4 gives f̂(x) ≤ 128π²/(9x⁴), so for T ≥ T₁ = 2π/min |γ| each term is at most 128π²/(9γ⁴T₁³), summable since each L(s, χ) has O(t log t) zeros to height t, and z(T) tends to 0 by dominated convergence: d(T) tends to log(431472/|d_K|), a surplus past |d_K| = 431472.

Over the 535 fields at T = 8, mean z is 0.058 of mean φ (2.12; median z 0.031). Mean d over mean φ reads 1.01 to 1.12 in the five cells h = 2..7 with 20 fields or more, d's largest over least across them 1.16 against φ's 1.20, and d's slope on log |d_K| is −0.88 ± 0.02 (least squares, se naive) against φ's −1. φ is positive iff |d_K| < e^(−G(T)), 656 at T = 2 to 46901 at T = 8; three widths past the last width where it is not (1 where it is positive from T = 2), the median z is 0.058. This is a weighted sum over every prime power; The shortfall is a fixed weight at the small primes counts places at a cut without the formula's weights, and whether this law carries that δ/(h − 1) is a deconvolution, not read.

The partial fiber has no square term

Tier: property.
Verifier: proof; seat.py::section_fibers.

Split the degree-1 places over unramified primes into two fibers: the split fiber, three places over each totally split p, and the partial fiber, one place over each partially split p. The partial fiber's count in a class is a count of primes whose Frobenius in Gal(H̃/Q) = R ⋊ S₃ (R the realized subgroup of M) lies in a set Σ of conjugacy classes, and the explicit formula for such a count carries a term at p^j exactly when Frob_p^j lies in Σ; the split fiber's, 0 to 3 places a prime, is a class function of Frobenius summed the same way. The partial fiber's Σ has a transposition for its S₃-part, and a square (v, s)² = (v + sv, s²) has its S₃-part in A₃. So the partial fiber holds no square and carries no prime-square term, only the odd powers j ≥ 3 of its own places. Every other power of an unramified prime ideal belongs to the split fiber, including the square of a partial Frobenius, (v, τ)² = (2a, −a, −a) with τ its transposition and a = [P₁], which is P₁² beside the degree-2 place.

An observation, read over the 535 fields with h ≥ 2 and |d_K| ≤ 12000 at cut 1000, each fiber with its own powers put back (standard errors 0.004 to 0.006):

fiber     cell   raw      in its own powers
split     T      0.9458   0.9812
          O      1.0512   0.9895
partial   T      0.9108   0.9103
          O      1.0769   1.0835

The powers take the split fiber's excess outside 2Cl to just below 1, and they move no partial cell by more than 0.0066, the weight of the partial places' odd powers past the first. So at that cut the weight the shortfall block reads is mostly the partial fiber's: its trivial deficit, 0.090, and its surplus outside 2Cl stand where the explicit formula has no prime-square term to pay them, while the split fiber keeps a trivial deficit of 0.019.

The least partial place leans non-square

Tier: observation (two disjoint populations of complex cubic fields of 2-rank 1, |d_K| ≤ 24000 and 24000 < |d_K| ≤ 48000); rule under the generalized Riemann hypothesis, the conductor's bound (verified at every such field, |d_K| < 21767).
Verifier: seat.py::section_seat; seat.py::section_rank; seat.py::section_two; seat.py::section_least; seat.py::section_controls; room.py::halves; room.py::wait; room.py::forbids; room.py::main; reach.py::main.

At 2-rank 1 the group Cl/2Cl has order 2 and partial places lie outside 2Cl at share ½ in the limit over primes (The partial place is uniform). Order a field's odd partially split primes by size and call a place's rank in that order its position. The position-1 place is outside 2Cl at 0.837 ± 0.014 over the 688 fields to |d_K| ≤ 24000 and 0.800 ± 0.014 over the 844 beyond. Pooled, the share reads 0.817, 0.691, 0.654, 0.624 at positions 1 to 4 and 0.59 to 0.64 through position 10, where the partial places between 500 and 1000 sit at 0.5036. It follows the position more than the prime: 0.819, 0.822, 0.831, 0.814 at position 1 over 3, 5, 7, 11, while over 11 to 29 a place of position at least 3 reads 0.634 ± 0.008 against position 1's 0.788 ± 0.025, pools not matched by field, their errors binomial over places. How 2 splits grades it: 0.959 ± 0.013 where 2 = P³, 0.900 where 2 = P²P′, 0.81 to 0.85 where 2 splits or is inert, 0.695 where 2 is partially split, its place the least partial one.

The explicit formula's headroom, its conductor and gamma terms less its prime side, bounds which of a field's first primes can lie in 2Cl, and carries part of the lean. Cl/2Cl names one unramified quadratic extension L of K; L(s, ρ) = ζ_L/ζ_K is entire, with conductor |d_K| and ζ_K's gamma factor, and a partial place puts −1 at its prime where it lies outside 2Cl, +1 where in. Each class-dependent coefficient is least outside and every weight is nonnegative, so the headroom with a set A of primes in 2Cl and the rest outside bounds every configuration with A in, and where it falls below 0, the least a zero sum takes under that hypothesis (Odlyzko's test function, width T ≤ 8), A is forbidden from K's splitting and |d_K| alone (the method of Belabas, Diaz y Diaz and Friedman, Math. Comp. 77 (2008), known). Take the unramified primes with a degree-1 place in order, 2 included, one in 2Cl when all its degree-1 places are. The first two are both in at none of the 1532 fields; the bound forbids it at 1511, every one with |d_K| < 21767 among them, and the first alone at 925, all at T ≤ 6. Widths 7 and 8 forbid nothing more: a prime outside A adds its weight to the headroom, a sum growing like e^(T/2) while ζ_K's gamma term falls 0.78 from T = 6 to 8, so there the worst case binds, not the width. Where the first prime is the least odd partial place, at 727 fields, it lies outside at 0.895 ± 0.011 and is forbidden in at 0.561, yet outside at 0.762 ± 0.024 over the 319 where it is not; held alone wherever it sits, the least odd partial place is forbidden in at 0.362 of the 1532 and lies outside at 0.713 ± 0.014 over the rest. The class-blind headroom, each class-dependent coefficient at its mean, missed its frozen bar for sorting a place's side of 2Cl within a type of 2 by its least over the widths to 6 (lower half less upper, 0.030 ± 0.019; after the bar, of seven widths alone, T = 3.5 and 4 at 4.2 and 3.7 standard errors), and position 2 lies outside at 0.840 ± 0.022 after a position 1 in 2Cl, 0.658 ± 0.013 after one outside: an anticorrelation a fixed headroom paid by whichever place reads −1 would make, a reading not tested.

The lean's character is irreducible

Tier: theorem; observation, its readings (the 688 complex cubic fields of 2-rank 1 with |d_K| ≤ 24000, unramified p < 2000).
Verifier: proof; rho.py::main.

The ρ above is induced from K to Q by ψ, the character of L/K, +1 on 2Cl and −1 outside, and it is irreducible of dimension 3: it holds no Dirichlet character. Over a quadratic field of 2-rank 1 the same construction is χ_{d₁} ⊕ χ_{d₂}, and the smaller factor's fixed signs at the first odd primes carry the least split place's lean (PRINCIPAL: The least split place leans away from the squares); over K the lean has no such factor.

Proof. Let Γ be the group of the Galois closure of L over Q and Γ_K the subgroup fixing K; L is K's only unramified quadratic extension, so Γ_K maps L to itself. Γ acts on K's three conjugates through S₃ (d_K is not a square), Γ_K fixing K and swapping K′ and K″, so Γ_K has two double cosets in Γ and by Mackey ⟨ρ, ρ⟩ = 1 + [ψ = ψˢ on Γ_K ∩ sΓ_Ks⁻¹], s outside Γ_K. Γ_K ∩ sΓ_Ks⁻¹ fixes N = KK′, the S₃ closure of K, and the two characters agree there exactly when LN = L′N, L′ = s(L). L is not N: N contains Q(√d_K), imaginary, so K's real place ramifies in N, and in L it does not, since −1 has norm −1 and the narrow class group is Cl. If LN = L′N (a condition the same for every s outside Γ_K, one double coset), then M′ = LN is fixed as a set by Γ_K and by a 3-cycle, so Galois over Q of degree 12, and Gal(M′/N), of order 2, is central: Gal(M′/Q) is C₂ × S₃ or the dicyclic group of order 12, H²(S₃, C₂) having order 2. The dicyclic group has one involution, which fixes N, so L = N, excluded. In C₂ × S₃ the sextics through K are N, K(√r) and K(√(r·d_K)), r rational, so L = K(√f′) with Q(√f′) quadratic; a prime p ramified in Q(√f′) (one is, Q having no unramified quadratic extension) has a place P of K with e(P|p) odd, the e_i·f_i over p summing to 3, and P ramifies in L, against L/K unramified. So ⟨ρ, ρ⟩ = 1.

The trace a(p) at p unramified in K, read through the class engine, is 3 or −1 at a split prime, ±1 at a partial one and 0 at an inert one; in the limit over primes the mean of a² is 1 for an irreducible ρ and at least 2 for a reducible one. The 688 fields read 0.9267 ± 0.0035 over 207269 readings, taken as independent; the quadratic fields with two prime discriminant factors and |D| ≤ 24000 read 1.9934, and ψ = 1 through the same places reads 1.8792. The cubic mean sits below 1 by its split fiber: 0.1532 of the readings against 1/6, with all three places in 2Cl at 0.2229 of its 31758 primes against ¼, it gives 0.4265 of the mean where the limit gives ½, the partial fiber 0.5002. The fiber table above already reads that side, the split fiber's raw excess outside 2Cl (h ≥ 2, |d_K| ≤ 12000, cut 1000).

Open fronts

The lean left past the conductor's worst case reopens on a bound that prices the signs of the primes outside A rather than their worst case, a worst case widths 7 and 8 raise at all 21 fields the bound leaves free. 2's grading reopens on a headroom that tells 2's splitting types apart by the grades' gaps; the share's decay to ½ on a third population past |d_K| = 48000; the shortfall's fixed weight on the deconvolution that carries the weighted law to the count at a cut.