Prime-child ancestry shadows

Status: proved theorem family inside the Type A/B minimal-depth program

Date: 2026-08-15

Project: Free Computation Foundation / CENTL

Claim boundary: this does not prove universal Direct-Shadow Completeness, López Type A/B coverage, or the Erdős-Straus conjecture. It proves an infinite family of exact direct-shadow edges along every modulus-ancestry quotient and completely classifies the unrestricted quotient-5 case.

Read with:

• THEORY.md
• DIRECT-SHADOW-COMPLETENESS.md
• DYADIC-TRAP-LATTICE.md
• RECIPROCITY-TOWER-SHADOWS.md

1. Modulus ancestry

Let j<K and suppose

Write

where necessarily

Write

Then

A small but crucial identity is

so

2. Prime-child shadow theorem

Theorem

Suppose

and the later depth K is prime.

Then the entire later Type A/B trap set is directly shadowed by the earlier layer j:

Proof

Because K is prime, its positive divisors are only

Therefore

Reduce these residues modulo

The ancestry identity gives

Hence

and

Thus the projected target trap set is contained in

But 1|j and j|j, so all three residues belong to T_j.

Therefore

QED.

3. Infinite prime-child shadows for every ancestry quotient

Fix any

The ancestry children of quotient q have depths

Equivalently,

Since

Dirichlet's theorem gives infinitely many primes in the residue class

For every such prime K,

is a positive integer and

The prime-child theorem therefore gives an exact direct-shadow edge

Thus:

Corollary

For every ancestry quotient

there are infinitely many exact Type A/B direct-shadow edges of that quotient.

4. Counting prime-child edges for fixed q

Let

Every such prime gives one prime-child shadow edge.

By the prime number theorem in arithmetic progressions,

So the prime-child mechanism alone contributes on the order of

shadow edges of fixed ancestry quotient q up to child depth X.

This immediately explains why the smallest quotient q=5 is expected to be especially prominent in finite shadow maps.

5. The quotient-5 family

For

we have s=1 and

The prime-child theorem gives

In this special quotient, the converse also holds.

6. Quotient-5 rigidity theorem

Theorem

For every positive integer j, put

Then

This concerns the unrestricted exact trap sets. Hard-class-conditioned shadowing can be stronger because inadmissible target residues have been removed.

Proof of the converse

Assume K=5j-1 is composite. Put

We show some divisor of K produces a target trap residue outside T_j.

Case 1: j is odd

Then K is even. Since K>2,

Thus

We claim

Equivalently, 2 does not belong to the normalized base set

Because j is odd, 2 is not a divisor of j.

For d<j,

so 4d mod m is an even integer at least 4, never 2.

For d=j,

Hence 2 notin S_j, so -2 notin T_j.

Therefore full shadowing fails.

Case 2: j is even

Then K is odd and composite. Let ell be a prime factor of K with

We may choose the smallest prime factor, so

Also

so

If ell belonged to the normalized base trap set S_j, then because ell<m there are only three possibilities:

1. ell=d for some divisor d|j, impossible because ell does not divide j;
2. ell=4d with d<j, impossible because ell is odd;
3. the wrapped value from d=j, namely 4j mod m=1, impossible because ell>1.

Thus

so

But ell|K, hence -ell in T_K. Full shadowing fails.

Therefore a composite child can never give unrestricted quotient-5 full shadowing.

Combined with the prime-child theorem,

QED.

7. Infinite quotient-5 family

Primes

are infinite by Dirichlet.

For every such prime,

and the quotient-5 rigidity theorem gives a full exact direct shadow.

The first bases are

j = 4,  6,  12, 16, 18, 22, 28, 30, 36, 40, ...

corresponding to prime children

K = 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, ...

This matches the unrestricted trap-set computation.

8. Why the hard-class shadow map had additional q=5 edges

Earlier finite work on the Mordell-hard residue classes found q=5 shadows such as

j=3 -> K=14
j=8 -> K=39

whose child depths are composite.

There is no contradiction.

Those computations imposed prime compatibility and the six hard classes modulo 840 before asking whether the surviving target candidate classes were shadowed.

The quotient-5 rigidity theorem concerns the complete unrestricted target trap set.

Thus:

and conditioning can create additional complete shadows after incompatible target classes disappear.

This distinction is now important enough to keep explicit in future theorem statements.

9. New interpretation of the ancestry graph

The observed ancestry quotient groups are not arbitrary.

Every quotient

carries an infinite prime-child shadow family.

The direct-shadow graph therefore contains a universal arithmetic backbone indexed by:

The remaining composite-child shadows are the genuinely richer part of the ancestry problem.

10. Next theorem targets

1. for each fixed quotient q, classify the composite children that are also fully shadowed;
2. compare composite-child shadows with square-lift quotients q=c^2;
3. determine when full local quadratic-signature shadowing explains a composite child;
4. determine when multiplicative quotient containment explains it;
5. isolate the residual composite-child shadows requiring exact two-box divisor geometry.

For q=5, this program is already complete in the unrestricted system: there are no composite-child full shadows.

11. Novelty boundary

Dirichlet's theorem and the prime number theorem in arithmetic progressions are classical. The modulus-divisibility identity is elementary, and López Type A/B congruences are prior art.

The candidate contribution is the prime-child direct-shadow theorem across all Type A/B ancestry quotients and the exact quotient-5 rigidity classification inside the minimal-depth/shadow framework.

Targeted literature searches have not yet located this formulation. That is not proof of publication priority.