Quotient-17 shadow rigidity — Free Computation Foundation

Ancestry

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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-Q17-CLASSIFICATION.md); promoted to parent theorem document

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

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Theorem

Let K = 17j - 4 and m = 4j - 1. Then

\boxed{ T_K \bmod m \subseteq T_j \iff \begin{cases} K \text{ is prime},\quad\text{or}\\ K = 2p \text{ with } p \text{ an odd prime},\quad\text{or}\\ K = 4p \text{ with } p \text{ an odd prime},\quad\text{or}\\ (j,K) = (4, 64). \end{cases} }

Direct implications

Prime child: parent prime-child theorem.
K = 2p: j = 2d, p = 17d - 2 ≡ j/2 mod m; divisors map to {1,2,j/2,j}.
K = 4p: j = 4d, p = 17d - 1 ≡ j/4 mod m; divisors map into {1,2,4,j/4,j/2,j}.
(4,64): explicit check; S_4 = {1,2,4,8} mod 15 absorbs all powers of two.

Converse outline

Odd j: gcd(j,K) = 1, K odd; least prime factor of composite K escapes S_j (√K < m for j ≥ 2).
v₂(K) = 1: write K = 2N; if N composite its least odd prime factor escapes S_j.
v₂(K) = 2: write K = 4N; same if N composite.
v₂(K) ≥ 3: forces j ≡ 4 mod 8 and v₂(j) = 2 exactly.

- v₂ ≥ 5, j > 4: divisor 32 ∉ S_j. - v₂ = 3 or 4: odd part N satisfies j < N < m and is odd, hence N ∉ S_j. - Only remaining shadow: (4,64).

Full case-by-case write-up: operator-02/ANCESTRY-Q17-CLASSIFICATION.md.

Structural remark

Unlike quotients with odd prime s = (q-1)/4, the case s = 4 = 2² produces a short 2-adic family rather than a single odd factor s·p. This is the first fully classified quotient whose composite-child list is governed by 2-adic valuation rather than a single odd prime factor.