Exact ancestry rigidity for quotients 21 and 29

Ancestry

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Status: proved exact unrestricted-shadow classifications

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: unrestricted Type A/B trap-set shadowing only. Hard-class-conditioned shadowing can be stronger. This does not prove Direct-Shadow Completeness, López Type A/B coverage, or the Erdős-Straus conjecture.

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1. Quotient 21

Set

K=21j-5, \qquad m=4j-1.

This is the ancestry shift s=5.

Theorem 1

\boxed{ T_K\bmod m\subseteq T_j \iff \left( K\text{ is prime} \quad\text{or}\quad K=5p\text{ with }p\text{ prime} \right). }

In the 5p case, necessarily 5|j.

Proof: direct implication

The divisor-child theorem with s=5 has

a\mid5.

Thus a=1 gives prime children and a=5 gives 5p whenever 5|j. Both are fully shadowed.

Proof: converse for j >= 6

The odd-shift asymptotic skeleton applies because

j\ge s+1=6.

Therefore full shadowing implies either:

K is 5-smooth, hence K=5^u; or
K=a p with

a\mid\gcd(j,5),

so a=1 or 5.

The second case is exactly the claimed prime/5p family.

It remains to remove the smooth branch.

For an odd shift, any smooth child larger than s has some prime exponent exceeding its exponent in s. Here that means divisor

25\mid K.

25<m,

then the odd integer 25 is a divisor of K smaller than m. It cannot lie in S_j unless 25|j. But

\gcd(j,K)=\gcd(j,5)

has 5-valuation at most one, so 25∤j. Thus smooth shadowing would require

m\le25.

For j>=6, this leaves only

j=6,\qquad m=23,\qquad K=121.

But divisor 11 satisfies

1<11<m, \qquad11\nmid j,

so 11∉S_j and shadowing fails.

Therefore no smooth exception occurs for j>=6.

Small j = 1,...,5

The remaining cases are exact and tiny:

j=1: K=16, m=3;  divisor 2 escapes S_1.
j=2: K=37;        prime, hence shadowed.
j=3: K=58, m=11; divisor 2 escapes S_3.
j=4: K=79;        prime, hence shadowed.
j=5: K=100,m=19; divisor 2 escapes S_5.

This completes the quotient-21 classification. QED.

2. Quotient 29

Set

K=29j-7, \qquad m=4j-1.

This is the ancestry shift s=7.

Theorem 2

\boxed{ T_K\bmod m\subseteq T_j \iff \left( K\text{ is prime} \quad\text{or}\quad K=7p\text{ with }p\text{ prime} \right). }

In the 7p case, necessarily 7|j.

Proof: direct implication

The divisor-child theorem with s=7 has only

a=1\quad\text{or}\quad a=7.

Thus prime children and 7p children with 7|j are fully shadowed.

Proof: converse for j >= 8

The odd-shift skeleton applies for

j\ge s+1=8.

Hence a full shadow is either prime/7p, or 7-smooth:

K=7^u.

A smooth child larger than 7 contains divisor

49.

49<m,

then 49<m is an odd divisor of K. It cannot be in S_j unless 49|j, but

\gcd(j,K)=\gcd(j,7)

has 7-valuation at most one. Therefore a smooth full shadow requires

m\le49.

For j>=8, this leaves only

8\le j\le12.

These five cases are checked explicitly below.

Exact small window j = 1,...,12

j=1:  K=22,  m=3;  divisor 2 escapes.
j=2:  K=51,  m=7;  divisor 3 escapes.
j=3:  K=80,  m=11; divisor 2 escapes.
j=4:  K=109;        prime, hence shadowed.
j=5:  K=138, m=19; divisor 2 escapes.
j=6:  K=167;        prime, hence shadowed.
j=7:  K=196, m=27; divisor 2 escapes.
j=8:  K=225, m=31; divisor 3 escapes.
j=9:  K=254, m=35; divisor 2 escapes.
j=10: K=283;        prime, hence shadowed.
j=11: K=312, m=43; divisor 2 escapes.
j=12: K=341, m=47; divisor 11 escapes.

Every non-prime case in the smooth-exception window has an explicit divisor outside S_j.

For j>=13, the smooth branch is impossible by the 49<m argument, so the asymptotic skeleton leaves only prime or 7p.

This completes the quotient-29 classification. QED.

3. Ancestry ladder after these theorems

The exact unrestricted classifications now begin:

Quotient `Q`	Shift `s`	Full-shadow child shapes
5	1	prime only
9	2	prime, `2p`, plus `(j,K)=(2,16)`
13	3	prime, `3p`
17	4	prime, `2p`, `4p`, plus `(j,K)=(4,64)`
21	5	prime, `5p`
29	7	prime, `7p`

The quotient-21 and quotient-29 proofs are no longer separate ad hoc divisor searches. They are short corollaries of the universal odd-shift skeleton plus finite smooth-window elimination.

4. General prime-shift corollary template

Let r be an odd prime and set

Q=4r+1, \qquad K=Qj-r.

For

j\ge r+1,

the odd-shift skeleton gives

\boxed{ \text{full shadow} \Longrightarrow K=r^u \text{ or } K=p \text{ or } K=rp\ (r|j). }

Any smooth case K=r^u with u>=2 contains divisor r^2. Once

r^2<m=4j-1,

that divisor cannot lie in S_j, because r^2∤j follows from

\gcd(j,K)=\gcd(j,r).

Thus every odd-prime shift has only a finite explicit smooth window

4j-1\le r^2

to check.

This turns exact classification for all prime shifts into a finite arithmetic problem after the universal theorem.

5. Next target

Automate the finite smooth-window check for odd prime shifts r and determine whether any prime shift admits a genuine smooth exception.

If none do, then for every odd prime r we obtain the uniform exact theorem

\boxed{ T_{(4r+1)j-r}\bmod(4j-1)\subseteq T_j \iff K\text{ prime or }K=rp\text{ with }r|j. }

That is now a sharply bounded theorem program rather than an open-ended search.