The degenerate regime

Which regime a field falls in is arithmetic, and it is read off the discriminant. A complex cubic field's discriminant factors as d_K = f²d₀, where d₀ is the discriminant of a quadratic field: that field k = Q(√d₀) is the resolvent named above, and the factor left over, f, is the conductor, which as an ideal of k is (f). The field is degenerate exactly when some prime of k dividing (f) is SPLIT over Q and carries a nontrivial local 3-part — a group's 3-part being its subgroup of elements of 3-power order, the group here the units modulo that prime's full power in (f). Read on the integer f alone that is a congruence: when a rational prime q ≡ 1 (mod 3) divides f, or 9 divides f and 3 splits in k. Neither form sorts by f by itself — f = 18 carries the degenerate regime at d₀ = −11 and the uniform one at d₀ = −19. And the sorter is the first case of a count. Write t for the number of such primes — one per rational prime q ≡ 1 (mod 3) dividing f, one more when 9 divides f and 3 splits in k — the lines the conductor carries, a line being a subgroup of order 3, here of that local 3-part — and R for the subgroup of M the realized triples fill. Then [M : R] = 3^t at every field, and R is the set of triples whose three classes agree modulo one subgroup of index 3^t in Cl(K), the principal genus defined below — the genus index law. A field whose conductor carries no line fills M; one line at 3-rank 1 is the subgroup Δ of the degeneracy block; and at 3-rank r — the number of cyclic factors of the class group whose order 3 divides — the word DEGENERATE covers the r values t ≥ 1.

Both directions are proved, and the uniform one is the mechanism itself. Write σ for the nontrivial automorphism of k, conjugation. The degenerate regime asks for a field E, abelian of degree 9 over k, stable under σ, containing the Galois closure N of K and unramified over it. Gal(E/k) has order 9, and it carries two pieces of order 3 to speak of: the subgroup Gal(E/N), a line, and the quotient Gal(N/k), on which σ acts by INVERSION, N being cut out by a character σ inverts. The line Gal(E/N) is FIXED by σ, and the reason is what E is built from rather than any computation: E is the compositum of N with one conjugate copy of K's Hilbert class field H, so restriction identifies Gal(E/N) with Gal(H/K) — and an automorphism fixing that copy of K pointwise restricts to σ on k while lying IN that abelian group, so it acts on that group trivially. With 2 invertible on a group of order 9 the two pieces then split apart, and an order count forces both to have order 3: Gal(E/k) is (Z/3)², one line inverted and one fixed. So the second line E needs is the FIXED one, necessarily.

What the unramifiedness then costs is the sorter itself. A second character must restrict, at every prime of the conductor, to a POWER of what the first one does there. Where σ FIXES such a prime the first character is locally inverted, so its powers hold no nontrivial fixed character and the second has to be unramified there — while a fixed character must ramify somewhere over f: over an imaginary quadratic field the characters unramified everywhere are all inverted, an ideal times its conjugate being principal. Where σ SWAPS two primes, being fixed relates the two components and constrains neither alone, and — the local 3-part being cyclic with the first character generating it — the condition is automatic. So a fixed line is usable — able to serve as E's second line — exactly at a split prime of the conductor carrying a nontrivial local 3-part, and a field whose conductor carries no such prime is UNIFORM — with nothing granted and no measurement entering. A field of conductor 1 has no prime there to carry anything, so it is uniform outright.

The converse is what the genus field carries. A fixed line is a cubic character of k that σ fixes, and the field it cuts out is therefore abelian over Q: it is kF for a cyclic cubic field F over Q of conductor q, or 9. The local condition that made the line usable makes KF unramified over K, so the compositum of K with its t fields F is an unramified extension of K with group (Z/3)^t — the genus field — and the subgroup of Cl(K) it corresponds to under class field theory, the principal genus, has index 3^t. Each F is Galois over Q, so the Frobenius of a totally split prime restricts to F the same way from all three places: the three classes agree modulo the principal genus, and [M : R] ≥ 3^t. One line already makes R proper, and at 3-rank 1 a proper R is Δ (below), which is the converse; and the bound is why t never exceeds the 3-rank r, so a field of 3-rank 1 with two lines does not exist.

The bound is attained, and the fixed characters are what attain it. Any subgroup of M stable under permuting the places and projecting onto Cl(K) contains every triple of cubes, so R is decided by its image in (Cl/3Cl)³, and the stable subgroups there projecting onto Cl/3Cl are indexed by the subspaces of Cl/3Cl, the one answering to a subspace W having index 3^{r − dim W} — at 3-rank 1 nothing but M and Δ, and at a class number prime to 3 nothing but M, prime or composite. Write W for the subspace R answers to. Read on the side of the fields, the largest subfield abelian over k of the compositum of the three copies of the 3-elementary class field — the subfield of the Hilbert class field whose group over K is Cl/3Cl — carries exactly r − dim W independent cubic characters FIXED by σ, each a ray class character mod f meeting the local condition above. A fixed character unramified everywhere factors through the class group of k, whose 3-part σ inverts, so it is trivial; hence the fixed usable characters inject into the fixed lines at the split primes of the conductor, of which there are t. So r − dim W ≤ t, and with the genus field's bound the index is 3^t exactly.

And the fixedness carries that whole distance, which is why granting it was never cheap. Drop it and what is left is the bare EXISTENCE of a σ-stable E unramified over N — and that holds at every field of the population: all 123 uniform fields carry a usable second line that is INVERTED, as do 15 of the 70 degenerate ones — so 138 of the 193 fields get their E from an inverted line and the remaining 55 from the fixed one. An inverted line is supplied two independent ways, by an unramified cubic character of k or by the conductor alone, and over this population is usable as soon as it exists at every configuration but one — a ramified 3 with 9 dividing f, which one field occupies. So existence is a CONSTANT here and sorts nothing: without the fixedness the mechanism excludes no field from the degenerate regime, so it cannot predict the uniform one at all — which is a weaker position than predicting the wrong one, not a stronger one.

Scope. The biconditional is a CRITERION, proved in both directions: at 3-rank at most 1 a field is degenerate exactly when its conductor carries a line, and in general the realized subgroup is proper exactly when t ≥ 1 and is Δ exactly when t = r. The genus index law is a RULE, proved above and exact at all 1367 complex cubic fields of |d_K| ≤ 50,000 with 3 dividing the class number and ten or more totally split primes read — the generated order times 3^t equal to h² at every one, none over or under — with t ≤ r at all 3133 fields there of class number above 1; the biconditional reproduces at every class-number-3 field of every band, 489 uniform and 137 degenerate to 50,000, and at every 3-rank 1 field of class number 6, 9, 12, 15 and 18. The law was first read as a pattern on the 193 fields of relation class number 3 with |d_K| ≤ 13,000 — 70 degenerate and 123 uniform, read off the first 83 and holding over two bands it had never seen, 55 fields at 6000 < |d_K| ≤ 10,000 and 55 above, with the dichotomy of the two-populations block replicating in both, no field of either band landing in the window [0.45, 0.95] — and the regime label at class number 3 is that block's measurement, inheriting its reading of the class map. The argument never consults the signature; the totally real side is unread. The two congruence forms are one condition because the conductor's primes are not free data: a cubic character of k inverted by σ can ramify at q ≡ 1 (mod 3) only where q splits in k and at q ≡ 2 (mod 3) only where q is inert, with v₃(f) ≤ 2 throughout (rule, verified on all 193 conductors). One inverted line is asserted from a rank rather than checked — d_K = −11907, the population's one field with 9 | f and 3 ramified in k, which is that one configuration. It is degenerate and carries a usable fixed line, so the one reading above that it enters is the count of 15.

verifier: explore_genus_index.py, explore_cubic_regime_sorter.py, explore_ray_class_lines.py, explore_ray_class_inverted.py

The diagonal D is the class-number-3 face of a condition that reads the same at every field, and reading it there costs the word EQUAL. Write 3Cl for the subgroup of classes that are cubes, {3x : x in Cl} in the additive notation the triples already use, and Δ for the triples in M whose three classes agree modulo it — a − b and b − c both in 3Cl. At h = 3 the cubes are trivial and Δ is D exactly; in general Δ asks for less than equality, and it is the degenerate subgroup. Its index in M is set by one invariant: the 3-rank r, equivalently the dimension of Cl/3Cl over the field of three elements. Then [M : Δ] = 3^r. The map sending (a, b, c) to (a − b, b − c) in (Cl/3Cl)² has Δ for its kernel; it lands in the diagonal of that square, since (a − b) − (b − c) = (a + b + c) − 3b vanishes on M; and it hits every element of that diagonal, (u, 0, −u) lying in M.

Two SCALAR tests have stood in for that condition, and neither is it in general. Asking that the three classes have equal 3-parts — a class's being its component in the class group's — by multiplying the differences by h stripped of ALL its factors of 3, is the reading the shortfall block below was built on, and at h = 9 that multiplier is 1: the test then demands the three classes be equal outright, which at Cl = Z/9 no field obliges, so a whole stratum — the fields at one class number — reads uniform. The right scalar is h/3, and it is the subgroup condition exactly when the 3-part of the class group is CYCLIC. Write Cl = A ⊕ T with T that 3-part and h = 3^eu for u prime to 3: multiplication by h/3 kills A and acts on T as 3^e−1 times a unit, whose kernel at cyclic T is A ⊕ 3T, which is 3Cl. Cyclicity enters at exactly one step, and above it the scalar form does not weaken — it INVERTS. At r ≥ 2 every cyclic factor of T has exponent at most e − 1, so 3^e−1 annihilates T outright and the test reads DEGENERATE at every such field whatever its split primes do.

Two complex cubic fields of |d_K| ≤ 50,000 have class group Z/3 × Z/3, and they part. At |d_K| = 24843 with h = 9, over 20 of its totally split primes, the realized triples generate a subgroup of order 9 and index 9 in M: it IS Δ. At |d_K| = 47628 with h = 9, over 31, they generate order 27 and index 3 — proper in M, and containing Δ strictly. That containment is measured, and it is also forced: the realized subgroup contains every triple of cubes and so contains Δ (the sorter block), where an arbitrary subgroup of order 27 could have met Δ in anything. So generating a proper subgroup of M is not the same as generating Δ, the single word DEGENERATE covers two regimes whose model shares differ by a factor of 3, and the two fields are two values of one count, the conductor's lines: 91 = 7·13 carries two, and 126 = 2·3²·7 carries one, its 9 buying nothing over a resolvent in which 3 ramifies — the index 9 and the index 3 the genus index law demands. A discriminant does not name a field: cubic fields share discriminants and |d_K| = 47628 carries four, the one meant being the one with h = 9, which is unique there.

What that moves is the class-number-9 stratum of the shortfall block, whose thirty fields were all read as uniform by the stripped test. Nine of them generate Δ. Against the share those nine are owed, 1/27, they read 0.805 at z = −0.57 — against the 2.414 at z = +2.39 that the share 1/81 was giving them — and the twenty-one that fill M read 0.727 at z = −0.74. Both halves sit below 1 and both are consistent with it inside their own spread, at expected corrected counts of 8.6 and 7.3. So the 1.200 the stratum pooled is the average of a correct cell and a cell whose denominator was three times too small, and it is not a corrected level.

Scope. The index [M : Δ] = 3^r, and the agreement of the h/3 scalar with the subgroup condition where the 3-part is cyclic, are PROPERTIES, derived above; the agreement is measured too, at zero disagreements over the 1365 fields with 3 dividing h, 3-rank at most 1 and ten or more totally split primes read. That the two fields of 3-rank 2 part is an OBSERVATION, on both such fields among the 3133 complex cubic fields of |d_K| ≤ 50,000 with class number above 1, and there are no others there: a class number read from relations is a multiple of the true one, so a quotient's 3-rank never exceeds its group's and no field of 3-rank 2 hides; their parting is the genus index law read at t = 2 and t = 1. What a field GENERATES is a lower bound on the subgroup its triples fill — filling M settles it, and a proper subgroup leaves open that the subgroup is M and every prime read landed inside, at an expected 0.09 fields misread that way across the 1283 fields of the box below with class number above 1 and at least ten split primes read. The class-number-9 split is an OBSERVATION, nine fields against twenty-one.

verifier: explore_noncyclic_level.py, explore_rank2_hunt.py

Outside class number 3 the all-equal triple is short too, and the shortfall GROWS. Over the |d_K| ≤ 6000 box the count of totally split primes below 1000 carrying one class at all three places reads 498 against the uniform model's 580.0 at h = 2 (0.859, z = −3.4), 12 against 27.5 at h = 4 (0.436) and 6 against 19.2 at h = 5 (0.312), and the 45 uniform fields at h = 3 pool 245 against 377.0 (0.650, z = −6.8). At h coprime to 3 the only all-equal triple is the ALL-PRINCIPAL one, so this is one event across every stratum, and no deformation of the per-place share reaches it: the smallest one that moves that share and nothing else, fitted per prime bin, leaves 0.66, 0.46, 0.35 standing at h = 3, 4, 5 — a depressed share gives a flat factor, and this one deepens.

What deepens is a term, and it is the identity element's. The all-principal triple is the IDENTITY of Gal(H̃/Q) — the group M ⋊ S₃, of order 6h² where the realized triples fill M: at every class number prime to 3, and wherever the conductor carries no line. Every event here is a union of conjugacy classes of that group, and Chebotarev's explicit formula shorts the count of primes whose Frobenius lies in such a set by 1/k for every prime q with q^k in the window and Frob_q^k in the set. On the identity those powers are: a totally split prime's square, when its three classes are 2-torsion; a partially split prime's square, which lands on (2a, −a, −a) for a the class of its degree-1 place, so when that place is principal; and EVERY inert prime's cube — a prime carrying no degree-1 place, whose Frobenius is a 3-cycle paired with some v in M — since the cube of such an element is (s, s, s) with s the sum of v's coordinates, which is zero because v sums to zero. That is the norm relation doing the work. Counted in the group, the squares landing on the identity number 3h + |Cl[2]|² and the cubes 2h² + |Cl[3]|²: quadratic in h against a main-term share of 1/h², where a per-place deficit is flat.

Put the term back per field and per prime, 2 included, over q^k < 1000 — the corrected count is the primes counted plus the weights landing on the identity, the corrected total the totally split primes plus the weights landing in M, and the corrected level the first over the second times 1/h² — across the complex fields of class number above 1 among the 4865 cubic fields of |d_K| ≤ 24,000:

A raw shortfall running from 0.86 to 0.12 and a corrected level that does not move with h across the seven strata, the inert cube's share of the correction rising, unevenly, from a sixth to two thirds. Class number 9 is absent from the table because its thirty fields are not one stratum: they split nine to twenty-one between the regimes, and the two halves read 0.805 and 0.727 where the pooled figure read 1.200 — the degeneracy block above. The control is the totally split count itself, which needs no class reading: over all 4865 fields and their 804,769 odd unramified primes below 1000 it reads 0.1495 against 1/6 at z = −41, and 0.1674 corrected. The degenerate regime enters exactly. It is the 3-part of M collapsed to Δ, read per field as every split prime's three classes agreeing modulo the cubes, and below 9 | h that is the same demand as their 3-parts being equal, which sorts the SAME fields the equal-class fraction of the two-populations block does — all 413 at h = 3 agree. At h = 6 it separates 15 fields whose realized triples form a group of order 12 from 50 whose triples fill M, and a stratum that read 1.49 unsorted (z = +3.5) becomes the table's 1.117 against 1/36 and 1.097 ± 0.171 against the degenerate share 1/12; the degenerate fields at h = 3 read 0.968 ± 0.029 against their share of 1/3.

And the term is not the whole shortfall at this precision. Class number 2 sits OVER at 1.058 ± 0.017, z = +3.5 — the excess in the top prime bin, 1.114 at z = +5.3 on [300, 1000) where the raw count already stands at 0.98 of the model, and in the fields above |d_K| = 6000, 1.064 ± 0.019 against the box's 1.030 ± 0.038. The uniform regime at class number 3 sits UNDER at 0.877 ± 0.033, z = −3.7, the box and the fields above it agreeing; its all-equal count reads 0.934 ± 0.019 with the non-principal equal triples at 0.963 ± 0.023, so the equal event's residual is the all-principal one's. The strata from 4 up all sit above 1, pooled at z = +1.6. Both residuals lie inside one bound the reading leaves unallocated: the powers of the RAMIFIED primes, which no class map here places — a weight landing on the identity raises a level and one landing in M off the identity lowers it, and 2 alone ramifies in 178 of the 283 uniform fields at class number 3 — so whether they are those powers, the remainder below the discriminant, or a third thing is open. Reading the ramified primes by their inertia cosets decides first; a window scaled to the discriminant is the second instrument.

Scope. The cube's landing on the identity is a PROPERTY of the sum-zero relation; the flat corrected level is a PATTERN across seven strata, h = 2 to 8, at a spread Poisson on each stratum's expected corrected count; the two residuals are an OBSERVATION, stated with their signs. The window q^k < 1000 sits under |d_K| at every field above the box, so the remainder carries geometry the average does not remove. Where the realized triples fill M is settled by the genus index law of the sorter block: the index is 3 to the conductor's line count, so it is 1 at every class number prime to 3 and at every uniform field — the measured corrections need no such assumption, the share 1/h² does. The closed-form expectation of the weights, taking every base q ≤ 31 at its asymptotic share, overshoots the measured ones by a third to a half, those bases being where the small-prime deficit of the two-populations block lives; the term is read per prime and never by its leading coefficient. The class number is the relation reading of the spacing block.

verifier: explore_triple_cube_term.py, explore_genus_index.py