Principal places

Write L₁(K) for a field's least PRINCIPAL rank-1 characteristic — the quantity that decides which rings the unwidened walk closes in the element world. It runs over the ODD characteristics, as the walk does, 2 being settled by the budget inequality. Over an imaginary quadratic field it is settled by arithmetic alone. An odd prime is rank-1 when the discriminant is a nonzero square modulo it, and the places above such a prime are principal exactly when the prime is represented by the PRINCIPAL form, which after completing the square reads 4p = u² + |D|y².

That test carries a floor, and the floor is most of the story. y = 0 would make p a square, so y ≠ 0 and 4p ≥ |D|: no principal rank-1 characteristic sits below |D|/4, whatever the class group does. The bound is attained — D = −19995 has L₁ = 4999 on the nose — and attained precisely at odd discriminants where (|D|+1)/4 is itself an odd prime, that being the principal form's value at (0, 1); never at even ones, where |D|/4 divides the discriminant and so ramifies. So the per-place reading reaches FURTHER than the factor and stops just as surely: at sweep bound P no field with |D| > 4P is closed by it at all. At P = 1000 that is 856 of the 1217 fundamental discriminants down to −4000 and none of the rest, against at most 36 of that same 1217 for the factor-4 route — at most, because a factor of 4 or less is necessary there and not sufficient, so the rival is counted in its own favour. Wider by at least 23 times, and both finite.

Scope. The floor is proved; the census is verified over the 6079 fundamental discriminants down to −20,000. Inside |D| ≤ 1000 the reading looks like near-total coverage — 296 of 305 fields, 97.0%, mean 291.2 and median 181 — and that is the RANGE's number rather than the world's, since |D| ≤ 1000 lies wholly under the cap. The crossover — coverage falling from that near-total to nothing across the |D| bands to 20,000 — belongs to the floor and not to the class group: a first-hit estimate on the correct density of principal places puts it near 10⁴ and is out at every one of the 34 class numbers in range, by 2.9 to 8.6 times. The density is not what fails far out — measured, it sits between 0.887 and 0.980 of its predicted value over 100,000 primes — and the starting point is part of what does, there being nothing to hit below the floor. Only part: the companion field below carries no floor at all and the estimate is still out there, so a density verified in the tail is simply not the density at the bottom of the range where a LEAST member is read. The multiple by which L₁ exceeds the floor is a function of neither the discriminant nor the class number, falling across one at fixed other and rising the other way; what it does is stay modest, averaging between 1.3 and 3.7 across the sweep, which is what makes the cap the right order of magnitude and not merely an upper bound. The restriction to imaginary quadratic fields is load-bearing and not a convenience: the floor exists because such a field's UNITS are finite, so the generators of a principal place form a finite orbit and the norm form is definite. Where units are infinite the generator moves through an indefinite form with no floor at all, so none of this transfers to a real quadratic or higher-degree field.

verifier: explore_principal_place.py

Run the same statistic where the units are infinite and the floor does not weaken — it is absent. Over a REAL quadratic field the same completion of the square reads u² − Dy² = ±4p, which constrains no size whatever, and principality is decided instead by a cycle of reduced forms: the place is principal exactly when the form built from ±p reduces into the PRINCIPAL cycle, both signs being needed because ideal classes match indefinite form classes only in the narrow sense.

D = 19996 carries L₁ = 3 where the floor would have demanded 4999, and across the 6081 real fundamental discriminants to 20,000 the coverage that fell away with |D| on the other side does not move: 100.0, 100.0, 99.9, 99.9 and 99.8 per cent over the same five bands. So what capped the per-place reading was the UNIT RANK and not the reading.

Scope. Both statistics run over odd characteristics. It does not follow that the reading never caps: eight fields in range are uncovered at this bound. And removing the floor exposes that the floor was never the whole explanation of the estimate it was blamed for — with none left, the first-hit estimate is still out, by 1.15 at class number 1 rising to 3.23 by class number 6, and the shortfall is graded a second time by the sign of the fundamental unit's norm where the density is identical either way. How large that shortfall is runs in the share block, and what it is in the block on the prime-power term.

verifier: explore_real_principal.py

The quantity to watch is not the first hit but the SHARE: what fraction of the rank-1 characteristics below a bound are principal, against the nominal one in the narrow class number — the count of form classes taken up to PROPER equivalence, which is h over an imaginary field and doubles it over a real one whenever the fundamental unit has norm +1. Pooled over primes to 1000 that share is short, and shorter as that number grows. Read as a FUNCTION of the prime it CLIMBS — at narrow class number 8 it runs 0.662, 0.855 and 0.964 of nominal over the prime ranges 1–1000, 1000–3000 and 3000–10,000, and by the top range the deficit is under 6 per cent at every narrow class number through 8. So the shortfall is a bottom-of-range effect and nothing standing, and it behaves the same over both signs of the discriminant once the floor is out of the imaginary side — the opposite of the floor above, which was a phenomenon of the unit rank alone.

Two rival explanations die on the way. It is not the size of the discriminant, and cutting both ways at once separates them: hold the class number and let the discriminant grow and the ratio does not move; hold the discriminant and let the class number grow and it falls. And it is not the genus congruences, which is what it was blamed on — the share factors exactly into a principal-genus share and a within-genus share, both are short by the same shape, and at the fields carrying only ONE genus, where the genus factor is identically 1, the share is still short by an ordinary amount — 0.885 of nominal at narrow class number 3, on twenty fields. The sign of the fundamental unit's norm adds nothing once the NARROW class number is fixed: it grades the first hit because it doubles that number, and for no other reason found here.

Scope. Observation, over the fundamental discriminants of both signs to 4000 and the odd primes to 10,000; the climb is a pattern across class numbers and the two refutations are read on cuts whose field counts are printed beside every entry. The least principal characteristic lands at or below 100 at 94.2 per cent of these fields, so that decade is where the first hit is read; restricted to it the share falls to 0.332 of nominal at narrow class number 8. Pairing each field's own local share against its own first hit — the share taken from that field's stratum with the field itself left out, read per prime rather than pooled, and integrated to a predicted first hit — turns the discrepancy over rather than closing it. Measured over predicted runs 0.714 to 0.924 across the wide class numbers 2 to 8, so the priced model now arrives LATE, and what is left is FLAT in the class number where the discrepancy it replaced climbed. That flat residual is the spacing itself, in size as well as sign: the count's index of dispersion — its variance over the variance independent draws would give — read at the positions where the first hit is decided rather than over the whole range, and carried through the count's survival function (the chance it is still zero at each position, summed), returns the measured first hit within noise at every class number, 1.017 ± 0.028 at class number 2; and the same law run a degree up returns it at the one cubic stratum whose early hit is significant, 1.009 ± 0.07 against 0.837 uncorrected (the spacing block there). The one part that was never arithmetic is the value at class number 1, where every RANK-1 characteristic is principal and the old model was out by a factor of 1.15 on the shape of its own draw.

verifier: explore_principal_share.py, explore_paired_division.py, explore_first_hit_survival.py

The share above is ONE class's. The two places over a rank-1 characteristic carry INVERSE ideal classes, so the characteristics below a bound sort, up to that inversion, into the classes of the narrow class group — the group whose size is the narrow class number — and the principal ones are those landing in the TRIVIAL class, which is its own inverse. An even sorting would give each class one in that number; write the level of a class for its observed count over that nominal. Read per class, the levels form a ladder in the class's ORDER — the least power of the class that is trivial, which a class and its inverse share, so the ambiguity in the sorting does not reach it. Paired within a field against that same field's classes of order above 2, the classes of order 2 run 0.151 below them over real fields and 0.216 below over imaginary ones, orders 3 and 4 run short by less, and the classes of order 5 or more run LONG, by 0.112 and 0.040. Priced absolutely within each narrow class number instead, the table carries two gradients: at a fixed order the level falls as the class number grows — the trivial class from 0.833 to 0.334 of nominal by class number 20 over primes below 1000 — while the classes that GENERATE the whole group read 1.09 across ten strata, a spread of 0.026 with no trend. A fixed ceiling at the generators, and a shortfall deepening beneath it.

All of it is one term of the Chebotarev explicit formula. That formula counts prime POWERS, weighted 1/k at the k-th power; a count of primes omits them, and so runs short of the formula's main term in exactly the classes the powers land in: one half for every prime below √x whose place class, DOUBLED, is the class in question — and every INERT prime, one whose ideal stays prime in the field, squares to a principal ideal, so the trivial class takes all of those — a third per cube, and one half for each ramified ideal, whose class has order at most 2. Put those weights back, field by field and prime window by prime window, and the imaginary table flattens WHOLE: the generating classes from 1.0875 to 1.0011 ± 0.0033 below 1000 and from 1.0240 to 1.0000 ± 0.0011 below 10,000; the trivial class from 0.52–0.79 to 0.96–1.02 below 1000 over class numbers 6 to 14; every cell of the table within 0.03 of 1 below 10,000. So the trivial class is short by (h + |Cl[2]|)/2 · π(√x)/π(x), where h is the order of the group the places are sorted into — the narrow class number here — |Cl[2]| its number of classes of order at most 2 and π(√x)/π(x) the fraction of the primes below x that lie below √x — the inert primes' squares, h/2 of them per class-count, and the squares of the split primes whose class has order at most 2, |Cl[2]|/2 of them, the only split squares that are principal — which is the grading by the class number and by the number of independent classes of order 2 at once, the ramified ideals adding their half-weights on those same classes; the decay along the prime range is π(√x)/π(x) itself, the factor of 8 from cut 250 to 10,000 that ruled out a constant and a logarithmic law; and the ceiling sits at the classes that receive nothing, the NON-SQUARES, which in a cyclic group of order 4, 8 or 16 are the generators alone — order and squareness parting wherever the class number has an odd factor, and at 6, 10, 12 and 20, where the odd-order classes are squares, the order ladder inverts exactly there. Over a real field the same weights remove three quarters or more of the surplus — the generating classes from 1.1041 to 1.0190 ± 0.0025 below 1000 and from 1.0246 to 1.0008 below 10,000 — and what they leave, read as COUNTS of primes per field rather than as levels, is ONE class: the narrow class of a principal ideal with a generator of NEGATIVE norm — present exactly when the fundamental unit has norm +1, and of order 2 — which sits 2.10 ± 0.14 primes per field below its nontrivial siblings below 1000, pooled over narrow class numbers 4 to 10 with every stratum between 1.6 and 2.5, and 1.5 ± 0.3 below them at 10,000, while the other classes of order 2 read flat at +0.18 ± 0.16. At narrow class number 2 that seat is an identity: the discriminant is a product D₁D₂ of two NEGATIVE prime discriminants, and a split prime's place lies in that class exactly when the Kronecker symbol (D₁/p) is −1 — zero mismatches over 1133 fields. So at narrow class number 2 that class's deficit is the prime race between two ODD quadratic characters — ones taking −1 at −1, which a negative prime discriminant gives — and above it the race of the narrow characters that take −1 on the class, odd in the same sense; read at primes comparable to the discriminant, before the bias the prime squares drive has developed: over the fundamental d < 0 to 16,000 the sum over p < 1000 of the Kronecker symbol (d/p) averages +1.2 where the squares alone would put it at −5.5, and reaches that value only where |d| is far below the cut (−12.9 against −12.5 at |d| < 100 and p < 10,000); the class's shrinking by 10,000, from −1.07 ± 0.07 to −0.84 ± 0.18 primes per field, is the bias catching up. The bias against principal primes is a theorem in its log-weighted form — Aoki and Koyama, 2023: a prime race against the identity in the Hilbert class field, the extension whose Galois group is the class group itself, under the Deep Riemann Hypothesis and the non-vanishing at ½ of the class-group L-functions, with coefficient (|Cl/Cl²| − 1)/2; the inert primes' squares are what the unweighted count at a finite cut shows on top of it.

A smaller effect sits beside it: a class of small order also STARTS later. The least rank-1 characteristic lying in a class of order at most 2 has a median above that field's median over its classes of higher order at 0.873 of real fields and 0.982 of imaginary ones, where a size-matched placebo reads 0.492 and 0.465, and compared at a matched least characteristic the order-2 gap keeps three quarters of its size over real fields and two fifths over imaginary ones — the asymmetry between the signs being the one the imaginary floor would give, no characteristic below |D|/4 being principal at all, which fixes where the principal count STARTS where the prime-power term fixes how short it RUNS. Whether the later start is the same term read at a first arrival is not computed; the real side's residual is no longer its candidate, being one narrow class and a classical race.

Scope. The term is classical and the accounting is verified in range: the fundamental discriminants of both signs to 4000 (narrow classes on the real side, to 16,000 for the real ceiling), the odd primes to 10,000, the control being the arithmetic progressions to 10⁷ over 77 prime moduli, where the non-residue classes lead by 1.15 · π(√x)/π(x) at eight spreads and by 0.9 of one spread once the same weights go back. The ladder and the table are observations, read per CLASS on the same discriminants, over the odd primes below 1000 unless stated — the bottom of the range, where the shortfall lives; a field with fewer than twenty rank-1 characteristics carries no share and is dropped. The paired figures are COMPARATIVE — a class against the other classes OF THE SAME FIELD — and assert nothing absolute; the order-2 arm rests on 189 real fields and 880 imaginary, and pooled across fields the column is not to be read, a short trivial class inflating whichever column a field's composition hands the surplus to. Rotating the order labels within each field beats the observed spread across a stratum's orders at 12 of 13 strata, so the ladder is structure. The generating classes exist only where the narrow class group is CYCLIC; one degree up every class group in the population is cyclic, and the partially split primes there carry no prime-square term at all — their generating class's excess over the same cuts is a fixed count of primes per field, which no prime-power term touches (The generator ceiling). The seat of the real residual and its size are observations, read as counts per field on the real sweep to 16,000; the identity at narrow class number 2 is exact over its 1133 fields. That one class is not the explicit formula's x-independent term either, a fixed number of primes per class: read as counts the real cells are flat from cut 400 to 1000 and shrink by 10,000, to 0.67 ± 0.19 and 0.18 ± 0.53 of their cut-1000 value, where a constant would hold. The imaginary side is not read as counts below its discriminants: its trivial class jumps from −0.83 ± 0.09 primes per field at cut 630 to −0.05 ± 0.09 at cut 1000, which reads as the floor's deficit being repaid above |D|/4 — a reading of a pooled jump, not a per-field measurement. The trivial class's rise with |D| at a fixed narrow class number, read as a count across a 512-fold range, is 0.05 to 0.10 primes per unit of log|D| over the bottom decade and 0.14 to 0.35 below 1000 — a grading that GROWS with the cut, where the explicit formula's one |D|-dependent constant shrinks with it, so it is not that constant. Nor is it bounded where an early start would put it: read in windows of the prime rather than at cuts, the count's reach GROWS with |D| — at narrow class number 8 the residual above p = 100 is −0.04 ± 0.21 primes per field at the bottom of that 512-fold range and +1.28 ± 0.14 at its top — and it does not end at √D/2, the increment above that line reading +0.48 ± 0.08 and +0.54 ± 0.06 at narrow class numbers 4 and 8, nor is √D its scale: measured in units of √D, the window below √D/4 still rises with log|D| at 0.16 to 0.22 primes per unit, against 0.02 to 0.09 over the three bounded windows above it. A count densest at the smallest primes and nowhere zero below 1000, whose reach grows with the field: not derived.

verifier: explore_class_share.py, explore_class_order.py, explore_class_level.py, explore_ceiling_squares.py, explore_ceiling_constant.py, explore_ceiling_early.py