Stabilizer-extension shadow theorem for square-completed Type-II layers

Geometry

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Status: proved exact theorem

Date: 2026-08-15

Depends on: ES-SQUARE-TRAP-SIGNED-BOX-IDENTITY.md, FAB-KNESER-FULL-STABILIZER-DEFECT.md, THEORY.md

Imported classical tool: Dirichlet's theorem on primes in reduced arithmetic progressions

Claim boundary: proves a broad infinite family of exact cross-layer redundancies generated by the internal stabilizer of an earlier completed signed box. It does not prove universal completed coverage or Erdős--Straus.

1. Completed signed box at an earlier layer

Fix

\boxed{j\ge1}

and write

\boxed{m=4j-1.}

The square-completed Type-II layer is

S_j=-R_j,

where

\boxed{ R_j = \mathcal R_m(j) = \left\{ \prod_{p^E\parallel j}p^z\pmod m: -E\le z\le E \right\}.}

Let

\boxed{ H_j = \operatorname{Stab}_{(\mathbb Z/m\mathbb Z)^\times}(R_j).}

Thus

\boxed{H_jR_j=R_j.}

Because 1 in R_j, one also has

\boxed{H_j\subseteq R_j.}

2. Extend the layer index

Let

\boxed{B\ge1}

be an integer satisfying two conditions.

Ancestry condition

\boxed{B\equiv1\pmod m.}

Stabilizer-support condition

For every prime divisor r of B,

\boxed{r\bmod m\in H_j.}

Define the later layer

\boxed{k=jB.}

3. The ancestry condition gives modulus divisibility

Since

4j\equiv1\pmod m,

we have

4k-1 =4jB-1 \equiv B-1 \equiv0 \pmod m.

Therefore

\boxed{m_j\mid m_k.}

So j is an exact modulus ancestor of k.

4. Signed exponent intervals factor under multiplication of layer indices

Write

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

allowing F_p=0 and permitting shared prime factors.

Then

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

For each prime p, the exponent interval satisfies

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

Therefore, after reduction modulo m, the complete signed box factors exactly as a product set:

\boxed{ \mathcal R_m(k) = \mathcal R_m(j)\,\mathcal R_m(B).}

This remains true even when j and B share primes.

5. The extension box lies inside the stabilizer

Every prime divisor r of B lies in H_j by hypothesis.

Because H_j is a subgroup, every signed power of every such prime also lies in H_j.

Hence

\boxed{ \mathcal R_m(B)\subseteq H_j.}

Therefore

\mathcal R_m(k) =R_j\mathcal R_m(B) \subseteq R_jH_j =R_j.

The reverse inclusion also holds because

1\in\mathcal R_m(B).

Thus:

Theorem — exact stabilizer-extension reduction

\boxed{ \mathcal R_m(k)=\mathcal R_m(j).}

Multiplying by -1 gives

\boxed{ S_k\bmod m_j =S_j.}

So the later completed layer is completely directly shadowed by the earlier layer.

6. Structural-gap theorem

Every integer captured by the completed layer k=jB is already captured at layer j.

Therefore:

Corollary — stabilizer-extension gap

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

and every prime factor of B belongs modulo 4j-1 to the stabilizer H_j, then

\boxed{jB\text{ is impossible as a completed minimal depth}.}

This is a cross-layer shadow theorem generated entirely from the internal Kneser stabilizer of layer j.

7. Prime-extension theorem as the identity case

B=r

is prime and

r\equiv1\pmod m,

then

r\in H_j

because the identity belongs to every subgroup.

Thus ES-SQUARE-PRIME-EXTENSION-SHADOW.md is the special case

\boxed{B=r\equiv1.}

The present theorem reveals the correct larger principle: an extension direction need not be the identity, only a stabilizer direction.

8. Infinite nonidentity extensions from any nontrivial stabilizer

Suppose

\boxed{h\in H_j.}

Then also

h^{-1}\in H_j.

Both are reduced residue classes modulo m.

By Dirichlet's theorem, there are infinitely many primes

\boxed{r\equiv h\pmod m}

and infinitely many primes

\boxed{s\equiv h^{-1}\pmod m.}

Choose such primes and set

\boxed{B=rs.}

Then

B\equiv hh^{-1}\equiv1\pmod m,

and both prime factors lie in H_j.

Hence

\boxed{S_{jrs}\bmod m=S_j.}

If h!=1, neither new prime direction is individually trivial in the ancestor group, yet their entire extension remains invisible because both directions lie in the stabilizer.

Therefore:

Theorem — nontrivial stabilizer cones

Every nonidentity element of H_j generates infinitely many two-prime structural-gap descendants of j.

9. Arbitrary stabilizer words

More generally, take any finite sequence

h_1,\ldots,h_t\in H_j

with

\boxed{h_1h_2\cdots h_t=1.}

Choose distinct primes

r_i\equiv h_i\pmod m

by Dirichlet and put

B=\prod_i r_i.

Then

B\equiv1\pmod m

and every prime factor lies in H_j.

Thus

\boxed{ S_{j\prod_i r_i}\bmod m=S_j.}

The completed shadow graph therefore contains an infinite stabilizer-extension cone above every layer.

10. Kneser-shadow synthesis

The theorem gives a precise bridge between two previously separate pieces of the research program.

Internal product-set theory

Kneser analysis attaches a stabilizer H_j to the signed box of a single layer.

Cross-layer ancestry theory

Modulus ancestry decides when a later congruence progression projects to an earlier modulus.

Fusion

If a later layer is obtained by adjoining only prime directions from H_j, with total extension product 1 mod m_j, then the internal stabilizer turns into exact cross-layer redundancy.

Symbolically:

\boxed{ \text{internal stabilizer} +\text{ancestry congruence} \Longrightarrow \text{complete direct shadow}.}

This is the first general theorem in which the Kneser stabilizer itself manufactures an infinite family of shadow-graph edges.

11. Next targets

Compute or characterize H_j for important layer families.
Determine whether every completely shadowed multiplicative extension arises from a stabilizer-support condition after factoring off an irreducible kernel.
Define the ancestry reduction kernel of a completed layer by deleting all extension prime powers whose projected classes lie in an ancestor stabilizer.
Study the residual kernel with Kneser's full-stabilizer order gap.
Apply the construction to squarefree semiprime and few-prime-support layers, where the signed boxes have very low dimension.