Exact classification of multiplicative-ancestry shadows

Geometry

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Status: proved necessary-and-sufficient theorem

Date: 2026-08-15

Depends on: ES-SQUARE-STABILIZER-EXTENSION-SHADOW.md, ES-SQUARE-TRAP-SIGNED-BOX-IDENTITY.md, THEORY.md

Claim boundary: completely classifies direct shadow by j for square-completed descendants of the form k=jB along a modulus-ancestry edge. It does not classify ancestry edges for which j does not divide k and does not prove universal completed coverage.

1. Setup

Fix

\boxed{j\ge1}

and put

\boxed{m=4j-1.}

Let

R_j=\mathcal R_m(j)

be the completed signed divisor box of j reduced modulo its own modulus, and let

\boxed{H_j=\operatorname{Stab}(R_j).}

Take an integer

\boxed{B\ge1}

and define

\boxed{k=jB.}

Assume the modulus-ancestry condition

\boxed{m_j\mid m_k.}

Since j is a unit modulo m, this is equivalent to

\boxed{B\equiv1\pmod m.}

2. Exact product factorization

Write

j=\prod_p p^{E_p}, \qquad B=\prod_p p^{F_p}.

Then

k=\prod_p p^{E_p+F_p}.

The signed exponent interval satisfies

[-E_p-F_p,E_p+F_p] = [-E_p,E_p]+[-F_p,F_p].

Therefore reduction modulo m gives the exact product-set identity

\boxed{ \mathcal R_m(k) =R_j\,R_B,}

where

R_B=\mathcal R_m(B).

Because the zero exponent is always allowed,

\boxed{1\in R_B.}

Hence

\boxed{R_j\subseteq\mathcal R_m(k).}

This one inclusion turns the direct-shadow condition into an equality condition.

3. Direct shadow forces equality

The completed trap is the negative signed box:

S_t=-\mathcal R_{m_t}(t).

Along the ancestry edge, layer k is directly shadowed by j exactly when

S_k\bmod m\subseteq S_j.

Multiplying by -1, this is

\mathcal R_m(k)\subseteq R_j.

But the previous section already gave

R_j\subseteq\mathcal R_m(k).

Therefore:

Lemma — multiplicative shadow is exact reduction

For k=jB on an ancestry edge,

\boxed{ S_k\bmod m_j\subseteq S_j \iff S_k\bmod m_j=S_j.}

There is no proper completed containment in this multiplicative setting.

4. Necessity of stabilizer support

Assume direct shadow holds. Then

R_jR_B=R_j.

Let r be any prime divisor of B.

Because v_r(B)>=1, the signed box R_B contains the residue r: choose exponent +1 at r and zero at every other prime.

Thus

r\in R_B.

From

R_jR_B=R_j

we obtain

R_jr\subseteq R_j.

Multiplication by the unit r mod m is a bijection of the ambient unit group, so

|R_jr|=|R_j|.

A finite subset contained in another subset of the same size must be equal. Hence

R_jr=R_j.

Therefore

\boxed{r\in H_j.}

Since r was arbitrary:

\boxed{ \text{direct shadow} \Longrightarrow r\bmod m_j\in H_j \text{ for every prime }r\mid B.}

5. Sufficiency

Conversely, suppose every prime divisor of B lies in H_j modulo m.

Then every signed product formed from those primes also lies in the subgroup H_j, so

R_B\subseteq H_j.

Hence

R_jR_B \subseteq R_jH_j =R_j.

Since 1 in R_B, the reverse inclusion holds. Therefore

R_jR_B=R_j.

Thus

S_k\bmod m_j=S_j.

6. Exact iff theorem

Combining necessity and sufficiency:

Theorem — multiplicative-ancestry shadow classification

Let

\boxed{k=jB}

and assume

\boxed{4j-1\mid4k-1,}

equivalently

\boxed{B\equiv1\pmod{4j-1}.}

Then the following are equivalent:

the completed layer k is directly shadowed by j;
the reduction is exactly equal:

S_k\bmod(4j-1)=S_j;

every prime divisor r of B satisfies

\boxed{r\bmod(4j-1)\in H_j.}

Symbolically,

\boxed{ S_{jB}\bmod m_j\subseteq S_j \iff \operatorname{supp}(B)\bmod m_j\subseteq H_j.}

This is a complete classification of direct shadows on multiplicative ancestry edges.

7. Irreducible multiplicative descendants

The theorem identifies the exact obstruction to shadowing.

A multiplicative ancestry descendant k=jB can remain directly novel relative to j only if

\boxed{ \exists r\mid B: r\bmod m_j\notin H_j.}

Such a prime is a genuinely new projected direction in the ancestor quotient

G_j/H_j.

Therefore the natural multiplicative ancestry kernel of the extension is obtained by deleting all prime-power factors whose prime bases lie in H_j.

If nothing remains, the descendant is exactly redundant.

If something remains, Kneser theory should be applied only to those quotient-visible prime directions.

8. Prime extension as an immediate corollary

If B=r is prime, the ancestry condition itself gives

r\equiv1\pmod m.

Thus r is automatically in H_j.

Hence every prime multiplicative extension on an ancestry edge is completely shadowed:

\boxed{ k=jr,\quad m_j\mid m_k \Longrightarrow S_k\bmod m_j=S_j.}

This recovers ES-SQUARE-PRIME-EXTENSION-SHADOW.md without a separate hypothesis beyond ancestry.

9. Relation to Kneser defect quotients

The theorem turns cross-layer novelty into a quotient statement.

For a multiplicative descendant, every extension prime outside H_j has a nontrivial class in

G_j/H_j.

Those are exactly the directions capable of enlarging the projected completed box.

Thus:

\boxed{ \text{cross-layer novelty} = \text{nontrivial extension support in the ancestor stabilizer quotient}.}

This is the exact algebraic merger of modulus ancestry and internal stabilizer theory for the divisible-index case.

10. Next target

The remaining ancestry problem splits cleanly:

multiplicative ancestry j|k: solved by the theorem above;
nonmultiplicative ancestry j\nmid k: still open and governed by the affine relation

k=(4s+1)j-s.

The second class is now the genuinely new cross-layer frontier.

A useful next program is to determine whether those nonmultiplicative edges admit an analogous factorization after translating the layer index by the ancestry parameter s, or whether they require a different quotient object entirely.