Limits of eight construction routes

A degree-12-over-quadratic tower cannot reach a group with no block of size 12

Why

This inverted tower has a quadratic subfield and relative degree 12, so its action preserves two blocks of twelve roots. A group with no block of size 12 is excluded. Checked as an exact finite computation with AllBlocks over TransitiveGroup(24, t).

Scope

Only the inverted tower, and only groups genuinely lacking a size-12 block. Of forty groups tested this way, twelve were excluded and the other twenty-eight do have a size-12 block system and are not touched by this. Nothing here says the tower fails on those.

A positive-norm radicand cannot reach a real-root count of 2 modulo 4

Why

The sign of the degree-12 norm is the product of the embedding signs, so a positive norm forces an even number of negative embeddings. The quadratic step then gives

and with that count even, r is divisible by four. So every real-root count congruent to 2 modulo 4 is out of reach whenever the prescribed rational norm square class is positive.

Scope

A quadratic step on a totally real degree-12 base with a prescribed positive rational norm class. It is a statement about that constraint and not about the base: choosing a negative norm class, or a base with complex places, changes the arithmetic. The general form is r = 24 − 4s − 2w, of which this is the parity corollary.

Unramified quadratic extensions added no groups in the measured bank

Measured

846 class field builds over 12 bases returned 36 distinct groups and not one group or real-root count outside what was already covered. The discriminant ratio held at exactly 2.0000 on all 846.

Scope, and what it is good for

Across these 12 bases, the 846 builds added no group or real-root count beyond the existing bank. The method remains useful for small-discriminant constructions. The computation does not establish the same limit for other bases or stronger constructions.

In the measured twists, the real-root count changed and the group did not

Measured

376 builds across 60 targets: every one hit its predicted real-root count, none reached a different group, and a full census found that all 24 distinct cells they landed on were already covered.

Scope

For the quadratic-over-degree-12 tower and the tested twists drawn from the base, the construction controlled the real-root count while retaining the group. Changing the group requires a different construction or additional evidence about twists outside this sample.

The degree-8 tower over mixed-signature bases reaches few groups per base

Measured

950 builds produced 39 cells not previously covered, all of them heavily covered by other teams, at a rate of 0.41 groups reached per base and with a large discriminant gap against the best known.

Scope

These builds gave little improvement in group coverage or discriminant size. They did reach low real-root counts in cells this construction had not previously covered, a capability worth retaining when that signature is the target.

Eleven degree-12 groups carry no quadratic character at all

Measured

Exact Galois computation on totally real representatives, two per group where available: every quadratic-character slot came back trivial.

Scope

A supply fact about those eleven groups as bases for a quadratic-character construction. It says nothing about reaching the degree-24 groups above them by another route.

Uncorrelated draws cover the already-realized ratio-one groups

Measured

Among the 963 groups whose finest block system is six blocks of four, the 32 groups at correlation ratio one, covering 320 cells, are all already realized. Drawing a quartic over a sextic with no relation among its six conjugates returns nothing new here.

Scope

Ratio one only. The one-relation and higher strata are where the unrealized groups in this population sit, and those are a different draw with an explicit condition on it.

Klein-freeness selects most of the measured population

Measured

Full Klein rank is the ordinary value on the decomposed population: 651 of 1,162 groups, and none of the 296 that carry a field of degree 8 or 12. A test that selects for Klein-freeness therefore selects for the default and separates almost nothing.

Scope

Selecting for Klein-freeness leaves most of this population in play. The reverse implication provides a useful filter: a field carrying a subfield of degree 8 or 12 cannot land on any of those 651 groups, so it can be excluded before forming a resolvent.