Quotient-21 shadow rigidity

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

Date: 2026-08-15

Project: Free Computation Foundation / CENTL

Provenance: Operator-02 first isolated/proved the 5p pattern; the canonical proof below is the Coordinator's independent proof via the general ancestry skeleton.

Claim boundary: unrestricted Type A/B trap-set shadowing only. Does not prove universal DSC-P, López coverage, or Erdős-Straus.

Read with:

• ANCESTRY-DIVISOR-CHILD-THEOREM.md
• ANCESTRY-ASYMPTOTIC-SKELETON.md
• QUOTIENT-21-29-RIGIDITY.md

Theorem

Let

Then

In the second alternative necessarily 5|j.

Direct implication

This is the divisor-child theorem with shift s=5.

The only divisors of 5 are 1 and 5, giving respectively:

• prime children;
• K=5p when 5|j.

Both are fully shadowed.

Converse for j >= 6

Since the shift s=5 is odd and

the odd-shift asymptotic skeleton applies.

Thus every fully shadowed child is either:

1. 5-smooth, so K=5^u; or
2. a divisor-child K=ap with a|gcd(j,5), hence a=1 or 5.

The second case is exactly prime or 5p.

For the smooth case, K>5 forces 25|K. If 25<m, then 25 is an odd divisor of K below m. But

shows 25∤j, and an odd integer cannot be of the form 4e. Therefore 25∉S_j, contradicting full shadowing.

Hence a smooth full shadow would require

With j>=6, only j=6 remains. Then

and divisor 11 satisfies 1<11<m and 11∤j, so 11∉S_j.

Therefore no smooth exception occurs.

Exact small cases j = 1,...,5

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

This completes the exact classification. QED.

Correction note

An earlier promoted draft inherited the sentence

N=21d-1 is always odd.

That sentence is false when d is odd. The theorem itself is unaffected because the canonical proof above does not use that parity assertion.

The repository preserves the earlier version in Git history for provenance.