The split triple

That quadratic cap is a fact about a quadratic field's units (the floor), so the first question one degree up is whether anything caps the reading here. Nothing does. Over the 1103 cubic fields of |d_K| ≤ 6000 — 888 complex, 215 totally real — read over the odd unramified primes to 1000, every stratum of signature and class number is covered 100.0 per cent at a sweep bound of 250 by a degree-1 place: a place whose residue field is the prime field, which at degree 2 is what the quadratic reading runs over. Coverage is a LOWER bound here, the principality test being a positive certificate, so a reading of 100.0 per cent is exact rather than an understatement. This matters because the rings the boundary question actually runs over are number rings in general and not quadratic ones of either sign.

What the lift separates is coverage from the WAIT. The mean least principal characteristic runs 5.6 at complex class number 1 to 66.0 at complex class number 7, and 6.1 to 47.3 across the totally real side, against the real quadratic side's 6.1 to 178.6 on the same statistic at the same prime cap. Two thirds of the primes carry a degree-1 place at degree 3 rather than one half, with three chances at a totally split one, and that is what buys the shorter wait — the twelve cyclic fields in the population read one third. But below the sweep bound of 250 the coverage grades with the class number, and sharply: complex class numbers 6 and 7 read 66.7 per cent at a bound of 100, and class number 4 reads 83.3 per cent at a bound of 50. So the absence of a floor is what makes the coverage full, and the class group is what decides how long it takes; the two are separate facts.

Scope. Observation, over the odd unramified primes to 1000 with coverage read at the bands 50, 100, 250 and 1000. The class number stratifying the fields is a RELATION reading, which the true one divides, certified only where it is 1. The two ranges compared are not the same — |D| ≤ 4000 on the quadratic side against |d_K| ≤ 6000 here — and the difference runs AGAINST the comparison, a wider discriminant range carrying larger class groups on average and so a longer wait; that is an argument from the class number formula rather than something measured here. The grading in the class number is not the instrument's either: at a fixed discriminant a larger class number goes with a smaller regulator, so a generator search reaches one SOONER at large class number, and any bias runs against the grading rather than producing it.

verifier: explore_cubic_principal.py

A model right about how MANY and still wrong about WHEN the first one arrives is a model wrong about spacing. A first arrival sits at 1/q when the hits fall independently at density q and at 1/(2q) when they are perfectly regular, so a first hit landing SOONER than a correctly priced model predicts — the residual the share reading a degree down is left with — is the signature of principal places spaced more evenly than chance. The statistic that reads it directly is the index of dispersion of a field's count. Under independence a field's number of principal-carrying characteristics is a sum of Bernoullis at its own per-prime densities, with a mean and a variance both fixed by those densities; the index is the squared deviations from those means, summed over the fields of one class number, over the summed variances. So it is 1 when the places fall independently, and below 1 when the counts sit closer to their means than independence allows — which is regularity, and is what an early first hit is made of.

Lifted to degree 3 the mechanism reproduces. Take the 236 cubic fields of |d_K| ≤ 6000 with class number above 1 and place all 36,929 of their degree-1 places below 1000, then read the COMPLEX ones, the totally real fields with class number above 1 being too few at this discriminant to fill a stratum. Against each prime bin's own MEASURED density, taken with the field itself left out so that the share deficit cannot enter the reference, the index reads 0.325, 1.583, 0.461, 0.291, 0.364 and 0.123 at class numbers 2 through 7. Five below 1, so the direction ports across the degree — but five below 1 is one direction and not five results, because how far below 1 a stratum sits says nothing without its field count. The index is a chi-square over that count, so its spread under independence is √(2/n): 0.15 at the 94 fields of class number 2, which puts 0.325 a full 4.6 spreads below 1, and 0.58 at the six fields each of class numbers 6 and 7, which leaves 0.364 and 0.123 only 1.1 and 1.5 spreads out however far below 1 they look. Only class number 2 clears on its own, with class number 5 marginal at 2.1. Nor does the index deepen with the class number; those five are a scatter with no direction in them. Nor does the SIZE port — at class number 2, the only one where a degree-2 figure on this same reference exists to compare against, degree 2 reads 0.142 against 0.325 here.

The sixth stratum reads the other way, and sharply. At class number 3 the index reads 1.583 on 83 fields, 3.8 spreads ABOVE 1 rather than below — so not thinness — and above 1 on the two coarser references as well: against the NOMINAL density it reads 2.927, and against the nominal density with the stratum's own level divided out — the factor by which its measured share sits under nominal — 1.624. So neither the level nor the share deficit accounts for it. Those fields disagree with each other about their own principal density by more than independence allows, and it is a question about class number 3 rather than about degree 3. The triple block and the two-populations block say what the structure is: the triple of classes a totally split prime carries is not uniformly distributed at this class number, and the stratum is not one population.

Read at the positions where the first hit is decided rather than over the whole range, the spacing derives the first hit here as it does a degree down. Against each stratum's own density, taken with the field left out, the first principal characteristic arrives early at every complex class number with 18 fields or more and clears its noise only at the degenerate half of class number 3 — the 38 fields whose split primes carry one class three times (the two-populations block) — at 0.837 of the independent model, 2.8 spreads out; there the count's survival function (the chance it is still zero at each position, summed) fed the index at every position returns 1.009 ± 0.07, and at class number 2 it returns 0.956 ± 0.045 against 0.929. The class-number-3 mixture, over-dispersed over the whole range, reads 0.90, 0.90, 0.56 and 0.43 at the first four positions — its two populations disagree about how MANY and not about WHEN — so the early hit is the degenerate half's, on an index curve that falls as the quadratic class-number-2 curve does, to the same 0.46 by the fifth position.

Scope. Observation, on 225 complex cubic fields across the six strata. The class number stratifying them is a RELATION reading, which the true one divides, so a field where the two differ is filed one stratum off. What the six strata carry JOINTLY, which is what the prediction was about, is a combined standardized deviation of −2.9 by Stouffer's method, the class-number-3 stratum pulling the other way inside it. The statistic counts the characteristics whose places are NOT principal, which no search exhibiting a generator can decide — a search that finds nothing is silent, not negative. It is read instead off a MAP from a place to its class, written in a fixed generating set and reduced against the relations known for it: triviality PROVES principality, and non-triviality proves the reverse only once that relation lattice is saturated. The gap is priced three ways rather than argued — at 62 fields whose class number 1 is certified by exhibited generators the map calls nothing non-principal, 40 of its negative verdicts are confirmed by an exhaustive certificate run against them and none refuted, and 69,323 accepted elements pass the norm accounting with no mismatch.

verifier: explore_cubic_class_map.py, explore_cubic_first_hit.py

The uniformity that produces the nominal density is an assumption about a TRIPLE, and at class number 3 it has exactly two alternatives. Let p be odd, unramified and totally split in a cubic field K — (p) = P₁P₂P₃ with all three places of residue degree 1. Their ideal classes sum to the class of (p), which is trivial, and the class group imposes nothing else; write M for the group of triples in Cl(K)³ summing to zero, of order h². The model behind the nominal density is that the triple is UNIFORM on M, under which a split prime carries a principal place with probability q_split = 1 − (h−1)(h−2)/h², or 7/9 at h = 3.

What the triple actually is, is a Frobenius, and the step from that to a SUBGROUP is the one input this rests on. Let N be the Galois closure of K and H̃ the Galois closure of its Hilbert class field, the maximal unramified abelian extension, whose group over K is Cl(K). A prime is totally split in K exactly when its Frobenius is trivial in Gal(N/Q), so these primes are exactly the ones whose Frobenius in H̃/Q lies in Gal(H̃/N) — and on that subgroup, sending an element to its triple of classes is a group HOMOMORPHISM into M, each coordinate being the restriction to one conjugate copy of the class field. Chebotarev fills Gal(H̃/N), so the realized triples are its image: a subgroup of M, stable under permuting the coordinates because conjugating permutes the copies.

At h = 3 that pins it to three possibilities. M is 2-dimensional over F₃, the permutation module is NOT semisimple there since 3 divides the order of the permuting group, and the diagonal D = {(t, t, t)} is exactly the fixed space of the 3-cycle; any stable subgroup of order 3 lies in it, by nilpotence of (σ − 1). So the realized set is M, D or 0 and nothing else. M is the model at q_split = 7/9. D is a DEGENERATE regime in which every split prime carries ONE class three times over, collapsing q_split to 1/3. And 0 is class number 1, a misfiling of the stratifier rather than a regime. A partially split prime — one carrying a single degree-1 place — needs its own argument, and it gives 1/h at every cubic field. There (p) = PQ with Q of degree 2, and the Frobenius is a TRANSPOSITION: it lies in no subgroup on which the map to the classes is a homomorphism, so listing the subgroups of the class group is not the right menu — what [P] realizes is a COSET. It does lie in the larger subgroup fixing one copy of H, and on that subgroup restriction to H is a homomorphism, the degree-1 place being the Frobenius's fixed point. The coset is a coset of Gal(H/(H ∩ N)), whose index in the class group is [H ∩ N : K] and so 1 or 2 — and 2 would mean such a place is NEVER principal. It is 1 at every cubic field. Where K is cyclic that is immediate, N being K itself; otherwise it holds because N/K is ramified. Some prime ramifies in the quadratic field sitting inside N — the resolvent, off which the sorter reads a field's regime (The degenerate regime) — and the inertia subgroup there, the part of the Galois group that prime's ramification generates, is then a transposition or the whole group. Each of those leaves a place of K whose ramification index doubles in N. So [P] is uniform on the whole class group, q_partial = 1/3 here, and the two regimes differ through q_split alone. And D is nontrivial exactly when 3 divides h, which is what makes this an answer to why class number 3 and no other rather than a correction that would apply anywhere.

Scope. Proved. The subgroup statement holds for an arbitrary cubic field, as does the coset statement for a prime with one degree-1 place and its consequence that the density is 1/h; the trichotomy into M, D and 0 is its class-number-3 case. The two facts imported are Chebotarev's density theorem and the unramified-abelian description of the Hilbert class field, and nothing here computes in H̃, whose degree over Q is 54 at a generic field. Where K is CYCLIC the degeneracy is forced without any of this: Gal(K/Q) permutes the three places and the automorphism group of Z/3, of order 2, leaves that action trivial. What the derivation above adds is that the same regime is reached with no Galois action on K at all, which is what puts it at a whole population rather than at the cyclic fields.

verifier: explore_cubic_split_triple.py, explore_cubic_transposition.py

Both regimes occur, and a field's membership in one is total rather than a tendency measured noisily. Across the 83 complex cubic fields of |d_K| ≤ 6000 whose relation class number is 3, over the 2023 totally split primes they carry between them, THIRTY-EIGHT carry one class at all three places over EVERY split prime they have — an equal-class fraction of exactly 1.000, on 17 to 27 primes each. The other 45 read 0.000 to 0.375, mean 0.2143 against the uniform model's 1/3. No field lies between: the window [0.45, 0.95] holds none of the 83 and the sorted column jumps from 0.375 straight to 1.000. The map's saturation gap cannot manufacture that, and the DIRECTION is what says so rather than a price — two places read as one class only when their difference is PROVED to lie in the relation lattice, so an unsaturated lattice moves a field toward the uniform regime and never toward the degenerate one, making the 38 a positive certificate.

The two groups' principal shares land on their own derived values, 1/3 and 4/9, at 0.3232 and 0.4318 — a factor of 4/3 apart, which is not the two q_split's ratio of 7/3 but what that ratio becomes once the partially split primes are averaged in, both regimes pricing those at 1/3 and those being three quarters of the primes carrying a degree-1 place at all. Each sits about 3 per cent under, and at the 45 that single mild lowness is two effects of opposite sign pooled over the SPLITTING TYPE. Among the 1131 totally split primes of the 45 non-degenerate fields, 973 carry a principal place: 0.8603 against the uniform model's 7/9, above it by a factor 1.106, and above — not short of — the 0.8556 that their own all-equal deficit predicts acting alone, those same primes carrying 245 all-equal triples, a pooled 0.2166 against the model's 1/3, which is the rate that prediction is computed at rather than the field-by-field mean above. So a principality shortfall among those primes is bounded and not merely unmeasured, and the comparison telescopes to a single count. Every unequal triple carries a principal place whatever the suppression does, so the two readings differ only INSIDE the all-equal class, where the question is whether one triple in three is the all-principal one: 87 of the 245 are, a fraction 0.3551 at z = +0.72, above it and not short. In q_split that is a deficit estimate of −0.0047 at a spread of 0.0065, so anything above one part in ninety of 7/9 would have shown at two spreads — and the sign is wrong for a shortfall at all. At the 38 degenerate fields the split side is exact, 297 of their 892 split primes carrying a principal class against 297.3 expected — a split prime there carries one class three times over and so is a SINGLE draw, so the degeneracy is a correlation among a triple's coordinates and nothing a single place can see. The deficit in both groups is the partially split primes', whose one degree-1 place lies under no triple constraint at all: their share runs 0.1911, 0.2594, 0.2455, 0.3311 across the prime bins against the 1/3 derived above, short at the bottom and at that value by the top.

The two sides of the agreement at 1/3 and 4/9 are not equally strong. At the degenerate fields both densities are 1/3, so the share is 1/3 whatever else the triple distribution does, which makes its landing at 0.3232 a confirmation. At the other 45 the share runs THROUGH q_split, which their own all-equal rate says is not 7/9, so their 3 per cent is a net of the two departures rather than a matching one; what those fields establish without qualification is that they are not degenerate, since at that regime the share would read 1/3. What that partial shortfall is, read against the order of the class and against the same reading one degree down, and where its excess sits — at the field's least places, read in the quartic field whose cubic resolvent the field is — is The generator ceiling.

And the excess dissolves. On the measured, field-left-out reference of the spacing block the index of dispersion reads 1.583 over the whole stratum, 0.220 on the 38 and 0.252 on the 45 — so the entire excess was the GAP BETWEEN TWO GROUPS' MEANS, and inside each the count is under-dispersed like every other class number, at 0.220 and 0.252 against the 0.123 to 0.461 the five unexceptional strata span. Class number 3 was never an exception to the regularity; it was two populations under one label, and the label is arithmetic rather than statistical.

Scope. Observation, on the 83 complex fields of relation class number 3 and their 2023 totally split primes, the reading by splitting type on those same primes. The class reading and its priced saturation gap are the spacing block's. The groups are read off the same primes the count is, which lowers a within-group spread mechanically, and what answers that is the size and the shape rather than the argument: removing a mixture carries an index down to 1 and no further, so 0.220 and 0.252 are under-dispersion and not bookkeeping, and a degenerate field's fraction is 1.000 with no freedom in it, so the label is not a fitted cut through a continuum but the answer to whether the field has any unequal triple at all. A separate control: a split prime carrying no principal place must have all three classes equal, three nonzero elements of Z/3 summing to zero being forced equal, and this holds at all 753 such primes. What SORTS a field into its regime is not answered by this measurement; the sorter block reads it off the discriminant instead.

verifier: explore_cubic_split_triple.py, explore_cubic_zero_tilt.py