Quotient-29 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

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

Then

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

In the second alternative necessarily 7|j.

Direct implication

This is the divisor-child theorem with shift s=7.

The only divisors of 7 are 1 and 7, giving prime children and 7p children when 7|j.

Converse for j >= 8

Because s=7 is odd and j>=s+1, the odd-shift asymptotic skeleton applies.

Hence every full shadow is either:

7-smooth, so K=7^u; or
prime or 7p.

If K=7^u>7, then 49|K. Whenever 49<m, the odd divisor 49 lies below m, while

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

shows 49∤j. Thus 49∉S_j, impossible.

Therefore a smooth exception requires

m\le49.

With j>=8, this leaves only

8\le j\le12.

Exact finite 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, shadowed.
j=5:  K=138, m=19; divisor 2 escapes.
j=6:  K=167;        prime, 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, shadowed.
j=11: K=312, m=43; divisor 2 escapes.
j=12: K=341, m=47; divisor 11 escapes.

Thus no smooth or other exceptional child survives the finite window. For j>=13, the skeleton leaves only prime or 7p.

This completes the classification. QED.

Regression note

Earlier drafts referred to a bounded computational gate in the small range. The canonical proof no longer needs that gate: all twelve small values are displayed with explicit escaping divisors or primality.