Prime-extension shadow family for square-completed Type-II layers

Geometry

---

Status: proved infinite exact shadow theorem

Date: 2026-08-15

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

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

Claim boundary: proves an infinite family of structural gaps inside the square-completed Type-II depth system. It does not prove universal completed coverage or Erdős--Straus.

1. Setup

For every positive layer index k, put

m_k=4k-1

and define the completed trap

\boxed{ S_k =\{-4D\pmod{m_k}:D\mid k^2\}.}

Equivalently,

\boxed{S_k=-\mathcal R_{m_k}(k)}

where R is the centered symmetric signed divisor box.

Fix an earlier layer

\boxed{j\ge1.}

Let r be a prime satisfying

\boxed{r\equiv1\pmod{m_j}.}

Define the later layer

\boxed{k=jr.}

2. Modulus ancestry is automatic

Modulo

m_j=4j-1,

we have

4j\equiv1.

Therefore

4jr-1 \equiv r-1 \pmod{m_j}.

Since

r\equiv1\pmod{m_j},

it follows that

\boxed{m_j\mid m_k.}

Thus j is a genuine modulus ancestor of k.

Equivalently, the standard ancestry relation gives

k\equiv j\pmod{m_j}.

3. The new prime factor is invisible in the ancestor group

Factor

j=\prod_i\ell_i^{E_i}.

Then

k=jr

has the same prime factors as j, together with one additional copy of r if r is new, or one extra exponent at r if r already divides j.

But modulo m_j,

\boxed{r\equiv1.}

Therefore every signed power of r is the identity:

\boxed{r^z\equiv1\pmod{m_j}\quad\text{for every integer }z.}

So the extra signed exponent direction created by multiplying the layer index by r contributes no new residue at all in the ancestor unit group.

4. Exact signed-box reduction

The completed signed box of k=jr reduces modulo m_j to the signed box generated by the prime powers already present in j.

Hence

\boxed{ \mathcal R_{m_k}(k)\bmod m_j = \mathcal R_{m_j}(j).}

Multiplying by -1 gives the completed trap identity:

Theorem — exact prime-extension shadow

\boxed{k=jr, \qquad r\text{ prime}, \qquad r\equiv1\pmod{4j-1},}

then

\boxed{ S_k\bmod(4j-1) =S_j.}

Thus every congruence hit at layer k is already a hit at the earlier layer j.

The later layer is completely directly shadowed.

5. Structural-gap consequence

For every integer n, if

n\bmod m_k\in S_k,

then reduction along

m_j\mid m_k

gives

n\bmod m_j\in S_j.

Therefore k=jr cannot be the first completed hit of any integer.

Hence:

Corollary — prime-extension structural gap

\boxed{ r\equiv1\pmod{4j-1} \Longrightarrow jr\text{ is impossible as a square-completed minimal depth}.}

This holds without any hard-prime restriction.

6. Infinite family from every layer

Because

\gcd(1,4j-1)=1,

Dirichlet's theorem gives infinitely many primes

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

Therefore:

Theorem — every layer has infinitely many dead prime extensions

For every fixed

\boxed{j\ge1,}

there are infinitely many composite layer indices

\boxed{k=jr}

that are completely directly shadowed by j in the square-completed Type-II system.

So the completed minimal-depth spectrum has infinitely many structural gaps above every base layer.

7. Example: j = 2

Take

j=2, \qquad m_j=7.

Every prime

r\equiv1\pmod7

produces a dead descendant

k=2r.

For example

r=29

gives

k=58, \qquad m_k=231=33\cdot7.

Modulo 7, the new prime is

29\equiv1,

\boxed{S_{58}\bmod7=S_2.}

Thus layer 58 carries no completed first-hit information beyond layer 2.

8. Relation to the prime-index theorem

ES-SQUARE-PRIME-INDEX-SPECTRUM.md showed that a prime-index layer a is structurally dead whenever 4a-1 is composite.

The present theorem is complementary:

the prime-index theorem kills layers because the target modulus has an earlier 3 mod 4 divisor;
the prime-extension theorem kills composite indices because a new prime factor of the layer index becomes the identity in the ancestor unit group.

Both mechanisms become transparent in the completed signed-box coordinate.

9. Natural generalization

The proof uses only one property of the extension factor: every new signed exponent direction must be trivial modulo the ancestor modulus.

Thus a broader sufficient condition is immediate.

Suppose

k=jB

and every prime divisor r of B satisfies

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

Then every signed power contributed by B is trivial modulo m_j, and therefore

\boxed{S_k\bmod m_j=S_j.}

The prime-extension theorem is the irreducible one-prime version of this more general identity-extension shadow.

10. Research consequence

The composite completed spectrum now has at least two explicit infinite deletion mechanisms:

prime-power/Mersenne ancestry gaps, where the old López layer is already complete;
identity-extension gaps, where new prime directions collapse to the identity in an ancestor group.

The remaining composite-index problem should therefore factor out all prime divisors of the layer index that are 1 modulo an ancestry modulus before treating the residual signed box.

This suggests an ancestry reduction kernel for completed layers: remove identity directions first, then apply Kneser only to genuinely nontrivial projected prime factors.