The partially split primes' principal share runs short of one in the class number at the bottom of the range and climbs toward it (the two-populations block), and the deficit does not close at the top of the range. Read over the 18,689 partially split primes of the 227 complex cubic fields of discriminant to 6000 whose class number exceeds 1, against each field's own profile of class ORDERS, the principal cell runs — as a standardized deviation z of its count from its nominal — −12.1, −6.1, −4.1, −2.8 across the four prime bins — the odd primes below 1000 cut at 30, 100 and 300: shrinking at every step, and a measured residual at the top rather than a vanishing, since those binomial z are floors on their own magnitude. The index of dispersion — a count's spread over the spread independence gives — taken on that same principal cell over the 227 field cells of the top bin rather than over the fields of one class number, reads 0.526 there, which carries −2.8 to −3.9. None of it touches the derivation of that one in the class number (the triple block), which is a Chebotarev asymptotic no count below 1000 reaches. And the orders grade the shortfall's shape, the low ones short and the highest long — where half of that is bookkeeping and the rest is not. A stratum's order cells sum to its place count, so a short trivial class has to inflate the others; what says the order grades anything is that the MIDDLE orders are short as well, leaving the surplus on the top order instead of spread across the rest by size. The same shape stands at degree 2, where the prime-power term behind that shortfall is read over a whole field rather than one place at a time — and the two tables have been read against each other, in one denomination: the level, whose nominal is 1 by derivation at both degrees. The cells where the order equals the class number — the classes that generate — sit at one constant — the ceiling — across the strata at both degrees over the shared range: 1.0935 here against a degree-2 mean of 1.0932, each inside the other's spread, on two populations sharing no field, two splitting types, an ordinary class group against a narrow one. That number is a point on a curve rather than a constant of either arithmetic: with the prime cut made a parameter, both ceilings decay toward 1 — 1.25 to 1.10 at degree 3 and 1.19 to 1.09 at degree 2 over cuts 250 to 1000 on fixed field sets, replicating out of sample on a disjoint cubic population four times the size — and at degree 2, whose primes reach 10,000, the surplus falls from 0.193 to 0.024, a factor of 8 where a 1/log decay allows 1.7, so the tail is neither a constant nor logarithmic. Nor is the ceiling bookkeeping: a short trivial cell forces the others above 1 on average, but the excess over that forced mean decays the same way at both degrees. And the top of the shared range decides against one curve: with the cubic population widened to 1,274 fields at discriminants to 24,000, the band of primes from 630 to 1000 reads 0.978 ± 0.012 at degree 3 against 1.043 ± 0.010 at degree 2 — a separation of four joint spreads, holding at three restricted to the original strata and at four on the disjoint new fields alone. Degree 2 sits four of its own spreads above 1 exactly where degree 3 is consistent with 1: the cubic surplus exhausts inside the shared range while degree 2's persists past it, so the two ceilings are not one curve — what survives is the decay's consistency along the lower cuts.
What carries the persistence is the prime-power term of the Chebotarev explicit formula, and a population carries its leading piece — the prime SQUARES — exactly when its Frobenius class is a square in its Galois group. The formula counts prime powers at weight 1/k at the k-th power; a count of primes omits them and so runs short of the main term in the classes the powers land in — one half for each prime below √x whose Frobenius squares into the class, a third per cube, the ramified ideals at one half each on the classes of order at most 2. Put that weight back per field and prime window and the quadratic surplus is removed exactly: the imaginary top band from 1.0426 to 1.0001 ± 0.0097, the imaginary tail from 1.0240 to 1.0000 ± 0.0011 below 10,000, the real top band from 1.0613 to 1.0151 ± 0.0066 — and the degree-2 table beneath the ceiling flattens with it, the trivial class's fall being the inert primes' squares — an inert prime being one whose ideal stays prime in the field — all of them principal. The population degree 3 reads is the PARTIALLY split primes' degree-1 place, whose Frobenius is a transposition, and a transposition is never a square in the symmetric group S3, the Galois group of the closure: that population carries no prime-square term at all — its levels move by at most 0.0017 under the cube term — which is why the complex cubic and the totally real cubic (top band 1.0167 ± 0.0100, the combined primes 630 to 10,000 at 1.0042 ± 0.0025) exhaust alike, on 1 in every band, where both quadratic populations hold, the real quadratic at 1.061 ± 0.007 in the top band. And the cubic's TOTALLY split places, whose Frobenius is the identity, carry the term as the formula says: on the 1,274 fields their cumulative ladder 1.115, 1.086, 1.063, 1.048 over cuts 250 to 1000 corrects to 1.045, 1.022, 1.010, 1.004 ± 0.005, and their top band 1.019 ± 0.014 raw reads 0.991 ± 0.013 corrected. So the fracture runs between square and non-square Frobenius classes — not between quadratic and cubic places, not between the two signs of the discriminant, whose ceilings agree at the tail and whose narrow and ideal class groups read one verdict, and not by the unit rank — the number of independent units of infinite order, 0 and 1 over the two quadratic signs, 1 and 2 over the two cubic ones — which decides nothing at any measured value. The two cubic populations are not resolved from each other — 2.4 joint spreads apart, on opposite sides of 1, the complex side unmeasured past 1000 — whether they share one asymptote is open. What the partially split primes DO carry is a second object, and read as a COUNT rather than a level it is not a ladder: the generating class's excess is a fixed number of primes per field — 3.0 ± 0.3, 3.5 ± 0.4, 4.3 ± 0.6 and 3.8 ± 0.7 at cuts 250, 400, 630 and 1000, a ratio of 1.26 ± 0.26 across the range where the prime-square term's shape would read 11/6 = 1.83, 2.2 spreads off — in place by cut 250 and flat after it, not repaid within the range; the level's 1.246 to 1.096 over those cuts is that count over a growing denominator, and the top band's 1 is the same count over a window that holds none of it. No prime-power term touches it, and the real quadratic residual it was once paired with has resolved into one narrow class and a classical prime race (what the shortfall is), so it stands alone. And it has a SEAT, and the seat is a statement about a fourth field. Sort a field's partially split places by the prime beneath them — the places are read over the odd primes, so the least place is the one over the smallest odd prime that splits partially — and call a place's position its index in that order, 1 at the least. A class is a square when it is the square of some class and a non-square otherwise; the squares form a subgroup, of index 2 exactly when the group holds one class of order 2, and the fields read here are those — every one with a class group cyclic of even order, 686 of them at discriminants to 24,000 and 840 over the disjoint range from 24,000 to 48,000. Such a field has one unramified quadratic extension, and the Galois closure of that extension is also the closure of a quartic field, with Galois group S4, whose cubic resolvent — the cubic field inside the closure whose three conjugates answer to the three ways of pairing four letters — is the cubic field itself. A partially split prime's degree-1 place is then a square exactly when the prime's Frobenius in S4 is a transposition, the prime carrying two degree-1 places in the quartic field, and a non-square when it is a 4-cycle, the prime inert there; the two classes are of one size, neither is a square in S4 nor a power of any element outside its own class, so no prime-power term of any explicit formula separates them and the even sorting gives each one half. Against that half, the place at position 1 is a non-square 84 and 80 times in 100 on the two populations — +0.338 ± 0.014 and +0.300 ± 0.014 over one half — and the same at every class number a bar can read: at class number 2, where a non-square is simply a non-principal place, +0.341 ± 0.017 and +0.307 ± 0.017, beside +0.322 ± 0.038 and +0.322 ± 0.035 at class number 4, where the non-squares are the classes that generate; at class number 6 the non-squares are the two generating classes and the class of order 2 together, and they read +0.392 ± 0.039 on the first population where the generating classes alone read +0.251 ± 0.062, so the cell the seat lives on is the quotient by the squares and not the generators. The excess decays over the positions — +0.246 ± 0.009 and +0.200 ± 0.008 pooled over positions 1 to 3, +0.136 ± 0.010 and +0.108 ± 0.009 over 4 to 6, +0.109 ± 0.009 and +0.094 ± 0.008 over 7 to 10 — and it is the position that grades it and not the prime: on the table of prime against position the non-square share runs about 0.85 and 0.80 at position 1 at every prime from 3 to 11, 0.75 and 0.67 at position 2, 0.70 and 0.63 at position 3, 0.62 and 0.60 from position 4 on, flat across the primes to 250 (position 1 against the rest at a fixed prime +0.088 ± 0.034 and +0.159 ± 0.029; the slope of the position-1 share on the prime −0.005 ± 0.007 and −0.003 ± 0.005). Fixing the prime beneath the least place fixes how every prime below it splits, so the seat is not the arithmetic of a small norm. What the discriminant moves is the LEVEL of that profile and not its shape: the position-1 share falls by 0.034 ± 0.013 per unit of log|d| over the 1,526 fields together, each pooled band of positions of the second population sitting under the first's — by 0.046, 0.028 and 0.015 — and inside the second population it is level, 0.798 ± 0.025 and 0.802 ± 0.024 over its two halves, so whatever falls has fallen by 24,000. One confound has the shape of a seat at small norms and is priced rather than argued. Every ideal class holds an ideal of norm at most the Minkowski bound — 0.283 √|d| over a complex cubic field of discriminant d, 21.9 at |d| = 6000 and 43.8 at 24,000 — so the prime ideals under that bound generate the class group, and a field with few places under it carries a generator among them by construction. A null that draws every partially split place's class uniformly and keeps only the draws in which the places under the bound generate the group — a demand made of a subset of the true generating set, the ramified primes generating too and going undrawn, so it bounds the conditioning from above — the generation-conditioned null — reads 0.009 ± 0.006 at position 1 for the generating class's share on the 185 fields of class numbers 4, 6 and 8 among the first population, where that share stands 0.283 ± 0.032 above the uniform φ(h)/h — the fraction of the group's classes that generate it — and 0.037 ± 0.007 on the 24 fields at 6000, where it stands 0.375 ± 0.075 above: a tenth or less of either. And the correspondence between the class labels and the factoring of primes in the quartic field is not read off the labels it sorts: the quartic field is built from the order alone — a generator of the square of an ideal of order 2, a unit, a sign and a congruence modulo 4 — and the way each prime factors in it reproduces the type the labels predict at 101,592 primes across 616 fields of the first population and 99,146 across 602 of the second without one disagreement, which certifies the 2-part of every class label in those fields — the label modulo the squares — independently of the lattice of relations that assigned it. What puts the 4-cycles first is not derived.
Scope. Observation, the prime-power term classical and its accounting verified in range. The order grading and the top-bin residual are read on the 18,689 partially split primes of the 227 complex cubic fields of class number above 1 at discriminants to 6000, and the dispersion index that rescales those z is measured on the principal cell it rescales and is carried to no other. The cross-degree curve is read on frozen field sets — at discriminants to 6000, the 24 fields of class numbers 4 and 6 among the 227 walked, those being the only strata where the generating cell is free (a stratum's cells sum to its place count, so at a prime class number that cell is the trivial one restated) — the count per field on those same 24 fields, an observation — and for the top-band separation the six composite strata from 4 to 12 among 1,274 cyclic-class-group fields at discriminants to 24,000 — against 224 imaginary quadratic fields over five strata, each cut's error bar measured at that cut. The real quadratic reading adds 2,086 real quadratic fields at discriminants to 16,000, 524 of them in readable strata, read in the narrow class group with the ordinary class group's view printed beside it; the totally real cubic read adds the 28,072 totally real cubic fields at discriminants to 530,000, 406 usable under the widened population's eligibility. The seat is an observation read by position and against the prime on two disjoint populations of complex cubic fields with class group cyclic of even order — 686 at discriminants to 24,000 and 840 over 24,000 to 48,000 — and the quartic correspondence is a rule on the 616 and 602 fields of them where the quartic field was built; the generation-conditioned null is drawn 200 times per field on the 24 fields at 6000 and 40 on the 185 at 24,000 with its spread taken over draws, read as a size against an effect ten to thirty times larger and never as a test. Every widened or added population runs through the same estimators as the one before it and reprints that one's frozen figures before its own number is read. The prime-power accounting re-reads those same frozen populations with the correction added per field and window, the raw column reprinting the earlier populations' figures inside 0.0015 first and the arithmetic progressions to 10⁷ over 77 prime moduli as the control (the non-residue lead 1.15 · π(√x)/π(x) at eight spreads, 0.9 of one spread corrected); the cubic populations omit the prime 2 and the ramified primes, printed as an unallocated bound, zero in every band above 300.
verifier:
explore_cubic_transposition.py,
explore_cubic_order_level.py,
explore_ceiling_curve.py,
explore_ceiling_topband.py,
explore_ceiling_realquad.py,
explore_ceiling_realcubic.py,
explore_ceiling_fourthcell.py,
explore_ceiling_squares.py,
explore_ceiling_constant.py,
explore_ceiling_early.py,
explore_quartic_seat.py,
explore_quartic_second.py