Divisor-child ancestry theorem for Type A/B shadowing

Status: proved universal theorem with coordinator-reviewed q=13 and q=17 classifications

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Coordinator: Operator-01 / primary research lead

Operator-02 provenance: operator-02/ANCESTRY-Q13-CLASSIFICATION.md, operator-02/ANCESTRY-Q17-CLASSIFICATION.md

Claim boundary: this note concerns unrestricted Type A/B trap-set shadowing along modulus ancestry. It does not prove universal Direct-Shadow Completeness, López Type A/B coverage, or the Erdős-Straus conjecture. Hard-class-conditioned shadowing can be stronger than unrestricted shadowing.

Read with:

• THEORY.md
• PRIME-CHILD-SHADOWS.md
• QUOTIENT-9-RIGIDITY.md
• OPERATOR-COORDINATION.md

1. General ancestry parameter

Fix positive integers j and s, and put

Then

Write

Thus j -> K is a modulus-divisibility ancestry edge with quotient Q.

The elementary identity

gives

and

Define the normalized ancestor trap set

As in the existing ancestry theory,

is equivalent to requiring every divisor of K, reduced modulo m, to lie in S_j.

2. Divisor-child theorem

Theorem

Let

and suppose

for a prime p.

Then

In words:

whenever the child depth is a prime multiplied by a divisor common to the ancestry shift s and the ancestor depth j, the entire child Type A/B trap set is shadowed by the ancestor.

Proof

Write

Then

Subtract d=j/a:

Hence

Every divisor E of K=a p can be written in one of the forms

for some divisor b|a. This remains true if p|a: divisors with the final extra p-exponent use the second form.

If E=b, then

so E is an ancestor divisor and lies in S_j.

If E=bp, then

But

is again a divisor of j. Therefore E mod m lies in S_j.

Thus every divisor of K maps into S_j, proving

QED.

3. Prime-child theorem as the first case

Taking

recovers the existing prime-child theorem immediately:

So prime children are the a=1 edge of a larger divisor-child family.

4. Immediate ancestry ladder

For a fixed ancestry shift s, every divisor

that also divides j supplies a potential fully shadowed child shape

This predicts the exact composite families independently observed at the first ancestry quotients.

Q = 5

Here s=1, so only a=1 exists:

This agrees with the proved q=5 rigidity theorem: unrestricted full shadow occurs only for prime children.

Q = 9

Here s=2, so

The theorem gives

The existing exact classification adds one exceptional smooth child

Q = 13

Here s=3, so

The theorem gives

Operator-02 independently proved that these are not merely sufficient but exhaustive.

Q = 17

Here s=4, so

The theorem gives

Operator-02 independently proved the full converse, with one exceptional smooth child

5. Coordinator review: q = 13 classification

Operator-02 source:

operator-02/ANCESTRY-Q13-CLASSIFICATION.md

Accepted theorem

Let

Then

The direct implication is now subsumed by the divisor-child theorem with s=3.

The Operator-02 converse was reviewed by the Coordinator. Its case split is sound:

1. if j is odd and K composite, divisor 2 escapes S_j;
2. if j is even and 3∤j, a least prime factor of composite K is <m, does not divide j, and escapes S_j;
3. if j=3d, write K=3(13d-1); when the cofactor is composite, a least non-3 prime factor escapes, while the remaining 3-divisible subcase gives divisor 9, and 9∉S_j because 9∤j.

No coarse signature/exact-trap equivalence is used.

Coordinator classification: PROVED / promoted.

6. Coordinator review: q = 17 classification

Operator-02 source:

operator-02/ANCESTRY-Q17-CLASSIFICATION.md

Accepted theorem

Let

Then

where p is an odd prime in the composite prime-times-divisor cases.

The direct prime/2p/4p implications follow from the divisor-child theorem with s=4.

The Coordinator reviewed the converse by 2-adic cases:

• odd j: a least prime factor of composite K escapes;
• v2(K)=1: after j=2d, a least prime factor of the odd composite cofactor escapes;
• v2(K)=2: same after j=4d;
• v2(K)=3 or 4: the odd cofactor lies strictly between j and m and hence cannot be in S_j;
• v2(K)>=5: j≡4 mod8, so v2(j)=2; divisor 32 escapes except at j=4, where K=64 and the power-of-two residues are explicitly contained in S_4.

Coordinator classification: PROVED / promoted.

7. New structural conjecture: prime-times-common-divisor plus smooth exceptions

The first four exact ancestry quotients now have a common form:

plus rare children whose entire factorization is supported on the small primes of the shift s.

This motivates the following theorem candidate.

Ancestry converse candidate

For fixed s and sufficiently large j, if

then

for some

and prime p, except possibly for a finite set of s-smooth children.

This is not proved in general.

8. Why a general converse is plausible

Let ell|K be a prime not dividing j.

If

then ell cannot lie in S_j:

• it is not a plain divisor of j;
• as a prime it cannot equal 4e for a proper divisor e|j;
• the endpoint 4j reduces to 1, not ell.

Thus under full shadowing, every prime factor of K below m must divide j, and hence must divide

Any prime factor not supported by the shift must therefore be at least m.

Since

grows linearly in j while m^2 grows quadratically, for fixed s and large j one has

Consequently there can be at most one prime factor outside the shift support.

This is the mechanism behind the observed prime-times-common-divisor families. The remaining work is to control excessive powers of the shift primes and classify the finite smooth exceptions.

9. Proof target

The next ancestry theorem should formalize Section 8 into:

1. an explicit threshold J(s) above which K<m^2;
2. a bound on the exponents of primes dividing gcd(j,s) under full shadowing;
3. a finite classification of s-smooth exceptions below that threshold or with exceptional exponent cycles;
4. a general converse yielding the divisor-child family as the complete asymptotic ancestry skeleton.

If successful, the separate q=5,9,13,17,... rigidity calculations collapse into one theorem.

That would turn the ancestry portion of the shadow graph from a collection of observed quotient families into a single structural law.