Odd-prime-shift ancestry rigidity above the small-ancestor window

Status: proved universal theorem

Date: 2026-08-15

Project: Free Computation Foundation / CENTL

Claim boundary: unrestricted Type A/B trap-set shadowing only. The theorem applies for j>=s+1. Small j can contain additional full-shadow children and is a separate exception problem. This does not prove Direct-Shadow Completeness, López Type A/B coverage, or the Erdős-Straus conjecture.

Read with:

• ANCESTRY-ASYMPTOTIC-SKELETON.md
• ANCESTRY-DIVISOR-CHILD-THEOREM.md
• QUOTIENT-ODD-PRIME-S-RIGIDITY.md — corrected/retracted all-j claim

1. Setup

Let r>=3 be an odd prime and put

No primality assumption on Q is required.

Assume

2. The theorem

Theorem

For every odd prime r and every j>=r+1,

In the second alternative necessarily

Thus above the explicit small-ancestor window, the entire unrestricted full-shadow structure of an odd-prime shift is exactly the divisor-child family.

3. Direct implication

The divisor-child theorem applies with shift s=r.

Since r is prime, the only divisors of r are

• a=1 gives prime children;
• a=r gives K=rp when r|j.

Hence both stated shapes are fully shadowed.

4. Converse from the asymptotic skeleton

Because r is odd and j>=r+1, the odd-shift asymptotic skeleton applies.

Therefore a fully shadowed child satisfies exactly one of:

1. K is r-smooth;
2. K=a p with

and p prime.

Since r is prime, the second case is already

It remains only to eliminate the smooth branch.

5. Elimination of smooth children

If K is r-smooth, then

for some integer u>=1.

The ancestry equation gives

hence

u = 1

Then

impossible.

u = 2

Integrality would require

because gcd(r,4r+1)=1.

But

impossible.

u = 3

Integrality requires

Use the exact identity

Therefore

For odd prime r>=3,

The only positive divisors of 17 at least 13 are 17 itself, which would give

not an odd prime.

Thus u=3 is impossible.

u >= 4

The child modulus relation gives

For every r>=3 and u>=4,

Indeed it is enough to check u=4:

which is equivalent to

true for r>=3.

But r^2 is a divisor of K=r^u and satisfies

Also

so

Because r^2 is odd, it cannot be of the form 4e for a divisor e|j. Hence

This contradicts full shadowing.

Therefore no r-smooth full-shadow child exists.

The converse is complete. QED.

6. Why Q need not be prime

The proof uses only

for modulus ancestry and the arithmetic identity defining K.

It does not require Q itself to be prime.

Thus the theorem simultaneously contains:

• r=3, Q=13;
• r=5, Q=21;
• r=7, Q=29;
• r=11, Q=45;
• and every later odd-prime shift.

For the first few shifts, existing exact quotient-specific theorems additionally handle the finite small range j<=r.

7. Small-j exceptions really exist

The threshold j>=r+1 cannot simply be deleted.

For example,

Then

and the divisors 1,11,121 reduce to 1,4,2, so full shadowing holds even though K is neither prime nor 17p.

Other small-window examples include:

r=19, j=4:  K=289=17^2
r=53, j=4:  K=799=17*47
r=71, j=8:  K=2209=47^2
r=83, j=2:  K=583=11*53

These are governed by the exact multiplicative structure of the small ancestor S_j, not by the large-j factor-size mechanism.

8. Structural interpretation

The theorem separates odd-prime-shift ancestry into two regimes:

versus

So the apparent infinite quotient-by-quotient complexity is actually concentrated into a finite-width diagonal strip j<=r.

Outside that strip, every odd-prime shift has exactly the same rigidity law.

9. Next target

Classify the small-ancestor exception strip

as r varies over odd primes.

For fixed j, the normalized ancestor trap set S_j is fixed, while

varies linearly with r.

Thus the exception problem becomes:

classify prime parameters r for which every divisor of r(4j-1)+j lies inside the fixed finite residue set S_j modulo 4j-1.

At dyadic ancestors j=2^a, the earlier Mersenne theorem gives

turning the condition into a concrete multiplicative-subgroup factorization problem.

That is the correct next layer of the ancestry diamond.