Quotient-13 shadow rigidity

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

Date: 2026-08-15

Project: Free Computation Foundation / CENTL

Provenance: proved in Operator-02 lane (operator-02/ANCESTRY-Q13-CLASSIFICATION.md); promoted to parent theorem document

Claim boundary: unrestricted Type A/B trap-set shadowing only. Does not prove universal Direct-Shadow Completeness, López Type A/B coverage, or the Erdős-Straus conjecture.

Read with:

• PRIME-CHILD-SHADOWS.md
• QUOTIENT-9-RIGIDITY.md
• QUOTIENT-17-RIGIDITY.md
• QUOTIENT-21-RIGIDITY.md

1. Quotient-13 ancestry

Fix source depth j >= 1 and set

Then

so this is ancestry of quotient q = 13 = 4·3 + 1.

2. Classification theorem

Theorem

Let

Then

In the second alternative necessarily 3 | j and p = (13j-3)/3.

3. Normalized criterion

Define

The set S_j is inverse-closed. Full unrestricted shadowing holds if and only if every divisor of K has residue in S_j.

4. Direct implications

Prime child

Immediate from the prime-child ancestry theorem.

Thrice-prime child

Suppose K = 3p with p prime. Then 3 | K, so 3 | j. Write j = 3d. Then

and p ≡ d = j/3 mod m. Divisors of K are {1, 3, p, K}, reducing to {1, 3, j/3, j}, all of which divide j. Full shadowing holds.

5. Converse

Assume full shadowing.

Odd j

If j is odd then K is even. If K is composite then 2 | K, so it suffices that 2 ∉ S_j. For odd j: 2 ∤ j; unwrapped 4e ≥ 4; wrapped endpoint 4j ≡ 1 mod m. Hence 2 ∉ S_j and shadowing fails. Thus odd j forces K prime.

Even j with 3 ∤ j

Then gcd(j,K) = 1. If K is composite, its least prime factor ℓ satisfies ℓ ≤ √K < m for j ≥ 2 (because K < m²). Also K is odd, so ℓ is odd, ℓ ∤ j, and therefore ℓ ∉ S_j.

Even j with 3 | j

Write j = 3d and N = 13d - 1, so K = 3N. If N is prime we are done. Assume N composite and let ℓ be its least prime factor.

• If ℓ ≠ 3: then ℓ ∤ j (using gcd(N,d) = 1), so ℓ ∉ S_j.
• If ℓ = 3: then 5 ∤ wait — 3 | N forces d ≡ 1 mod 3, hence 3 ∤ d and 9 ∤ j. The divisor 9 of K satisfies 9 ∉ S_j (not a divisor of j; not of the form 4e with e | j). Shadowing fails.

(Note: 3 | d and 3 | N cannot occur simultaneously, since N = 13d - 1 ≡ -1 mod 3 when 3 | d.)

6. Infinite families

Dirichlet gives infinitely many primes K ≡ 10 mod 13 and infinitely many primes p ≡ 12 mod 13 yielding K = 3p shadows.

7. Novelty boundary

Elementary arithmetic and the prime-child theorem are prior within this program. The contribution is the exact unrestricted quotient-13 composite-child classification.