Genus-theoretic identification of the square-lift defect quotient

Geometry

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Status: exact identification using classical genus theory

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: genus theory, genus characters, the principal genus theorem, and the identification of principal genus classes with squares in quadratic class groups are classical. This note does not claim those results as new. The Type-A/B-specific object is the distinguished subgroup generated by the prime ideals in the norm factorization of (1+sqrt(-d))/2, and its role in minimal-depth square-lift shadowing.

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Classical background references include:

F. Lemmermeyer, The development of the principal genus theorem, arXiv:math/0207306;
T. Ibukiyama, Genus character L-functions of quadratic orders in an adelic way and maximal orders of matrix algebras, arXiv:2303.14983;
P. Stevenhagen, Rédei reciprocity, governing fields, and negative Pell, arXiv:1806.06250.

1. Imaginary quadratic field of an ancestor

Let

d=4a-1

be squarefree, and put

K=\mathbb Q(\sqrt{-d}).

Since d=3 mod 4, the fundamental discriminant of K is

\boxed{D=-d.}

Write

d=\prod_{i=1}^r p_i.

For each odd prime factor define the associated prime discriminant

\boxed{ p_i^*=(-1)^{(p_i-1)/2}p_i.}

Because an odd number of the p_i are 3 mod 4,

\boxed{D=\prod_i p_i^*.}

2. Genus-signature vector of a split prime

Let ell be a rational prime not dividing d that splits in K, and let mathfrak p be either prime ideal above ell.

Classical genus characters attached to the prime-discriminant factors evaluate on such a prime ideal by quadratic symbols

\left(\frac{p_i^*}{\ell}\right).

For odd ell, quadratic reciprocity gives the exact identity

\boxed{ \left(\frac{p_i^*}{\ell}\right) = \left(\frac{\ell}{p_i}\right). }

Indeed:

if p_i=1 mod 4, then p_i^*=p_i and reciprocity introduces no sign;
if p_i=3 mod 4, then p_i^*=-p_i; the (-1/ell) factor cancels exactly the reciprocity sign.

For ell=2, the corresponding Kronecker-symbol identity agrees with the supplementary law for (2/p_i) because (p_i^*)^2=p_i^2 mod 8.

Thus the genus-character vector of a split prime is exactly our local Legendre-sign vector

\boxed{ \lambda_d(\ell) = \left( \left(\frac\ell{p_1}\right),\ldots, \left(\frac\ell{p_r}\right) \right), }

up to the standard conversion +1 -> 0, -1 -> 1.

3. The one relation is the splitting/Jacobi relation

Since ell splits in K,

\left(\frac{D}{\ell}\right)=+1.

Multiplying the individual prime-discriminant genus characters gives

\prod_i \left(\frac{p_i^*}{\ell}\right)=+1.

In binary coordinates this is exactly

\boxed{J_d(\lambda_d(\ell))=0.}

Hence split-prime genus signatures lie in

\ker J_d,

a vector space of dimension r-1.

The realization lemma in SQUARE-LIFT-SIGNATURE-CLASSIFICATION.md shows conversely that every vector in ker J_d is attained by infinitely many split rational primes.

So ker J_d is exactly the full split-prime genus-signature space.

4. Principal genus theorem

For an imaginary quadratic field, narrow and ordinary ideal class groups agree. Classical genus theory gives a homomorphism from the ideal class group to its genus-character signature space whose kernel is the principal genus.

The principal genus theorem identifies the principal genus with the subgroup of square classes.

Therefore the genus-signature map induces an isomorphism of F_2 vector spaces

\boxed{ \operatorname{Cl}(K)/\operatorname{Cl}(K)^2 \cong \ker J_d. }

This is the classical object underneath the local-signature space used in the Type A/B square-lift analysis.

5. The distinguished ancestor norm factorization

Define

\alpha_1=\frac{1+\sqrt{-d}}2.

Then

N(\alpha_1)=a.

Factor

a=\prod_{\ell\mid a}\ell^{e_\ell}.

Every rational prime ell|a splits in K. The element alpha_1 selects one prime ideal mathfrak p_{ell,1} above each such ell, and

\boxed{ (\alpha_1) = \prod_{\ell\mid a} \mathfrak p_{\ell,1}^{e_\ell}. }

Let

S_a \subseteq \operatorname{Cl}(K)/\operatorname{Cl}(K)^2

be the F_2 span of the classes

[\mathfrak p_{\ell,1}] \bmod\operatorname{Cl}(K)^2 \qquad(\ell\mid a).

Under the genus-signature isomorphism, each selected prime ideal maps to

\lambda_d(\ell).

Therefore the image of S_a is exactly the ancestor divisor-signature space

\boxed{V_a.}

6. Exact identification of the reciprocity defect quotient

Recall

\mathcal R_a = \ker J_d/V_a.

Combining the preceding sections yields:

Theorem

There is a natural genus-theoretic identification

\boxed{ \mathcal R_a \cong \frac{\operatorname{Cl}(K)/\operatorname{Cl}(K)^2}{S_a}. }

Equivalently,

\boxed{ \dim\mathcal R_a = \operatorname{codim}_{\operatorname{Cl}(K)/\operatorname{Cl}(K)^2}S_a. }

Since

\dim\mathcal R_a=\kappa(a)-1,

we obtain

\boxed{ \kappa(a)-1 = \operatorname{codim} \left\langle [\mathfrak p_{\ell,1}]:\ell\mid a \right\rangle \text{ in } \operatorname{Cl}(K)/\operatorname{Cl}(K)^2. }

7. Meaning of kappa(a)=1

The automatic square-lift signature theorem now has a classical interpretation.

We have

\boxed{ \kappa(a)=1 \iff S_a = \operatorname{Cl}(K)/\operatorname{Cl}(K)^2. }

In words:

kappa(a)=1 exactly when the split prime ideals occurring in the distinguished principal factorization of alpha_1 already generate the entire genus class group.

When this happens, every later square-lift prime has a genus signature already generated by the ancestor norm factors, which is the class-group explanation of universal ancestor shadowing at local quadratic-signature resolution.

When kappa(a)>1, the ancestor norm factorization misses one or more genus directions. The realization theorem then produces infinitely many split primes in those missing genus classes, and those primes can be forced to divide later square-lift norms.

8. Full class-group conservation for every lift

For odd s, define

\alpha_s=\frac{1+s\sqrt{-d}}2, \qquad N(\alpha_s)=j_s.

j_s=\prod_\ell\ell^{e_\ell},

then

(\alpha_s) = \prod_\ell \mathfrak p_{\ell,s}^{e_\ell}

for the uniquely selected split prime ideals above the rational factors of j_s.

Hence

\boxed{ \sum_\ell e_\ell[\mathfrak p_{\ell,s}] =0 \quad\text{in }\operatorname{Cl}(K). }

Reducing modulo squares and then modulo S_a gives exactly the defect conservation law

\boxed{ \sum_\ell(e_\ell\bmod2)\delta_a(\ell)=0 \quad\text{in }\mathcal R_a. }

Thus our binary conservation law is the genus-level projection of a stronger principal-ideal identity in the full class group.

9. Relation to Rédei matrices

Classical Rédei matrices also encode quadratic residue symbols over F_2, but they are built from the prime-discriminant factors of a quadratic field discriminant and are used, in particular, to study 4-ranks of quadratic class groups.

The Type A/B reciprocity matrix is not automatically a Rédei matrix:

its rows are primes dividing d=4a-1;
its columns are primes dividing the norm a=(d+1)/4;
it records genus signatures of a distinguished family of split prime ideals selected by alpha_1.

The exact identification above shows that the correct classical interpretation is a genus-class evaluation matrix for the ancestor norm factors.

Whether this matrix can be obtained as a natural submatrix, presentation, or derived map from a classical Rédei construction is a separate question and should not be asserted without proof.

10. What remains Type-A/B-specific

The genus-theory identification does not trivialize the research program. It sharpens the novelty boundary.

Classical theory supplies:

the ambient genus class group;
its quadratic characters;
the principal genus theorem;
the interpretation of Legendre-sign vectors.

The Type-A/B-specific structure supplies:

the distinguished norm element

\alpha_1=(1+\sqrt{-d})/2

tied to a=(d+1)/4;

the subgroup S_a generated by the prime ideals in that specific norm factorization;
the later norm family

\alpha_s=(1+s\sqrt{-d})/2;

its role as the square-lift component of minimal López Type A/B witness depth;
direct shadowing, projection excess, and the exact residue obstruction beyond genus information.

The candidate novelty is therefore not “we discovered genus theory.” It is the way this distinguished norm/genus structure controls the Type A/B shadow hierarchy.

11. Next theorem targets

The class-group interpretation suggests a stronger sequence of questions:

characterize the subgroup S_a directly from the ideal class of the factors of alpha_1;
determine whether kappa(a) can be read from standard genus/class-group invariants plus the factorization of alpha_1;
classify the possible full ideal-class configurations of the primes dividing alpha_s after quotienting by the subgroup generated at s=1;
identify the ray-class refinement that remembers actual residues modulo d, not only genus classes;
express squarefree projection excess E_j as a failure of triviality in that ray-class quotient;
test whether exact direct shadowing becomes a subgroup/coset inclusion theorem in the corresponding ray class group.

The fourth through sixth questions are now a particularly promising bridge from the classical genus layer back to the exact Type A/B residue problem.

12. Prior-art boundary

The genus-class isomorphism and principal genus theorem are classical. The research record should cite them explicitly and must not advertise the F_2 character space itself as novel.

The potentially novel content remains the minimal-depth Type A/B shadow architecture and the distinguished norm-factor subgroup S_a inside it, subject to broader literature and independent review.