Multiplicative reciprocity defect quotient for Type A/B square lifts

Geometry

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Status: proved universal theorem and classification

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this theorem is at multiplicative-coset resolution. It does not imply exact residue shadowing, universal Direct-Shadow Completeness, López Type A/B coverage, or the Erdős-Straus conjecture. The group theory, CRT, Dirichlet theorem, and quadratic reciprocity used in the proof are classical; the Type-A/B-specific organization is the research contribution under review.

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1. Ancestor multiplicative groups

Let

d=4a-1

be squarefree and put

G_a=(\mathbb Z/d\mathbb Z)^\times.

Let

D_a = \langle \ell\bmod d:\ell\mid a,\ \ell\text{ prime}\rangle \le G_a.

The multiplicative trap theorem gives

T_a\subseteq-D_a.

Every generator of D_a has Jacobi symbol +1 modulo d, so

D_a\subseteq K_a,

where

\boxed{ K_a = \ker\left( (\mathbb Z/d\mathbb Z)^\times \xrightarrow{(\cdot/d)} \{\pm1\} \right) }

is the Jacobi-positive subgroup.

Since d=3 mod 4, the Jacobi character is nontrivial and

[G_a:K_a]=2.

2. The multiplicative defect quotient

Define

\boxed{ \mathcal M_a = K_a/D_a. }

This is the multiplicative reciprocity defect quotient of the ancestor.

Its order is

|\mathcal M_a| =[K_a:D_a] = \frac{[G_a:D_a]}{[G_a:K_a]} = \boxed{\frac{\iota(a)}2},

where

\iota(a)=[G_a:D_a]

is the multiplicative trap index.

Using the quotient factorization

\iota(a)=2^{\kappa(a)}\Theta(a),

we obtain

\boxed{

|\mathcal M_a| =2^{\kappa(a)-1}\Theta(a). }</div>

Thus M_a contains both:

the quadratic reciprocity-defect information of dimension kappa(a)-1;
all higher-order multiplicative information measured by Theta(a).

3. Square-lift prime defects

For positive odd s, define

j_s=\frac{1+d s^2}{4}, \qquad 4j_s-1=d s^2.

Square-lift reciprocity proves that every prime

q\mid j_s

satisfies

\left(\frac qd\right)=+1.

Therefore

q\bmod d\in K_a

and has a well-defined multiplicative defect class

\boxed{ \delta_a^{\rm mult}(q) =[q]\in\mathcal M_a. }

The class is trivial exactly when

q\bmod d\in D_a.

4. Full multiplicative conservation law

Factor

j_s=\prod_q q^{e_q}.

Because

4j_s=1+d s^2,

we have

j_s\equiv4^{-1}\pmod d.

But

4\in D_a

because 4a=1 mod d and a in D_a. Hence

[4^{-1}]=1 \quad\text{in }\mathcal M_a.

Therefore

[j_s]=1 \quad\text{in }\mathcal M_a.

By multiplicativity:

Theorem

Every square lift satisfies the exact finite-abelian conservation law

\boxed{ \prod_{q\mid j_s} \left(\delta_a^{\rm mult}(q)\right)^{e_q} =1 \quad\text{in }\mathcal M_a. }

Unlike the earlier F_2 conservation law, the exponents are retained in the full finite abelian group. Odd-order information is not discarded.

5. Multiplicative signature defect of a lift

Define

D_{j_s\to a} = \langle q\bmod d:q\mid j_s\rangle \le K_a.

Its image in the defect quotient is

\boxed{ \Delta^{\rm mult}_{j_s\to a} = D_{j_s\to a}D_a/D_a \le\mathcal M_a. }

Equivalently,

\Delta^{\rm mult}_{j_s\to a} = \langle\delta_a^{\rm mult}(q):q\mid j_s\rangle.

The projected Type A/B trap set of the lift lies in

-D_{j_s\to a},

while the ancestor trap lies in

-D_a.

Thus the lift is ancestor-shadowed at multiplicative-coset resolution exactly when

\boxed{ \Delta^{\rm mult}_{j_s\to a}=1. }

This is stronger than quadratic-signature shadowing.

6. Universal multiplicative-shadow classification

Theorem

The following are equivalent:

\(\iota(a)=2\);
\(D_a=K_a\);
\(\mathcal M_a=1\);
every positive odd square lift has

\Delta^{\rm mult}_{j_s\to a}=1;

every positive odd square lift is ancestor-shadowed at multiplicative-coset resolution.

Proof

The equivalence of 1, 2, and 3 follows from

[G_a:K_a]=2

and

D_a\subseteq K_a.

If D_a=K_a, square-lift reciprocity puts every prime divisor of every lift in K_a=D_a, proving 4 and 5.

Conversely, if every lift is multiplicatively shadowed then no nontrivial class in K_a/D_a can ever be represented by a lift prime. Section 7 proves that every class of K_a/D_a is represented by infinitely many such primes, so the quotient must be trivial. QED.

Therefore

\boxed{ \iota(a)=2 \iff \text{universal square-lift multiplicative shadowing}. }

7. Every multiplicative defect class is realized infinitely often

Assume

\mathcal M_a\ne1.

Let

C\in\mathcal M_a

be any class, nontrivial or trivial. Choose a unit residue

r\in K_a

representing C.

Use CRT to choose a reduced residue class R mod 4d satisfying

R\equiv1\pmod4, \qquad R\equiv r\pmod d.

Dirichlet's theorem gives infinitely many primes

q\equiv R\pmod{4d}.

For each such prime,

[q]=C\in\mathcal M_a.

Because q=1 mod 4, quadratic reciprocity gives

\left(\frac{-d}{q}\right) = \left(\frac qd\right) =+1,

since r in K_a.

Therefore the congruence

d s^2\equiv-1\pmod q

has a solution. Choose one odd solution s_0, and then every

\boxed{ s=s_0+2q n, \qquad n\ge0, }

is positive odd and satisfies the same congruence.

Hence

q\mid j_s

for infinitely many distinct square lifts, and all such lifts contain the prescribed multiplicative defect class C.

Thus:

Realization theorem

\boxed{ \text{Every class of }\mathcal M_a \text{ is represented by infinitely many split primes}\ \text{and occurs in infinitely many square-lift depths.} }

In particular, if

\iota(a)>2,

then infinitely many square lifts fail ancestor shadowing at multiplicative-coset resolution.

8. Relation to the quadratic defect quotient

The local quadratic-sign map induces a natural surjection

\boxed{ \mathcal M_a \twoheadrightarrow \mathcal R_a, }

where

\mathcal R_a = \ker J_d/V_a

is the earlier F_2 reciprocity defect quotient.

Its kernel contains precisely the defect information invisible to all local Legendre symbols.

Numerically,

|\mathcal R_a|=2^{\kappa(a)-1},

while

|\mathcal M_a| =2^{\kappa(a)-1}\Theta(a).

Therefore the kernel has order

\boxed{\Theta(a).}

So the previously defined deep multiplicative factor Theta(a) is exactly the amount of square-lift defect information discarded by the complete local quadratic-signature system.

9. Ray-class interpretation

Let

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

and use the finite modulus

d\mathcal O_K.

For an ideal A coprime to d, its absolute ideal norm gives a unit modulo d:

N(A)\bmod d\in(\mathbb Z/d\mathbb Z)^\times.

If a principal ideal is generated by

\beta\equiv1\pmod{d\mathcal O_K},

then

N(\beta)\equiv1\pmod d.

Hence the norm-residue map factors through the ray class group modulo d\mathcal O_K.

For the canonical selected prime ideals

\mathfrak p_{q,s}\mid(\alpha_s), \qquad N\mathfrak p_{q,s}=q,

the ray norm-residue is exactly

q\bmod d.

Composing the ray norm-residue map with

G_a\to\Gamma_a=G_a/D_a

produces precisely the multiplicative defect classes above.

Thus M_a is the concrete residue quotient seen by the norm map from the square-lift ray-class configuration. This does not identify M_a with the full ray class group; it identifies it as a canonical quotient of ray norm-residue data.

10. Why this sharpens the proof search

The hierarchy is now exact:

\boxed{ \begin{array}{c} \text{principal norm identity}\\ \downarrow\\ \text{ray norm-residue classes}\\ \downarrow\\ \mathcal M_a=K_a/D_a\\ \downarrow\\ \mathcal R_a\text{ (quadratic quotient)}\\ \downarrow\\ \text{Jacobi bit} \end{array} }

Every square-lift prime-factor configuration obeys a conservation law already in the full finite abelian group M_a, before any reduction to quadratic characters.

This is a stronger constraint than the binary reciprocity-defect law and is a natural candidate mechanism for the strong overlap observed in the exact Type A/B covering systems.

11. Next theorem target

For an ancestor with

\iota(a)>2,

classify the finite zero-product configurations

\prod C_i^{e_i}=1 \quad(C_i\in\mathcal M_a)

that can arise from square-lift prime factors, and determine how those configurations constrain the exact two-box trap image.

The especially important case is when M_a is cyclic or has small Davenport constant. Zero-sum/zero-product theory in finite abelian groups may then force short cancelling subconfigurations that correspond to intermediate divisor depths or exact shadow ancestors.

That is now a concrete bridge from the multiplicative defect quotient back to Direct-Shadow Completeness.