General odd-prime-shift ancestry rigidity — corrected scope

Ancestry

The previous version claimed that for an odd prime s, with

Status: REVISE / PREVIOUS ALL-j CLAIM RETRACTED

Date: 2026-08-15

Project: Free Computation Foundation / CENTL

Claim boundary: the former statement in this file claimed an exact classification for every j>=1. That statement is false. The correct universal theorem holds in the asymptotic range j>=s+1; small j can contain additional full-shadow children. See ODD-PRIME-SHIFT-ASYMPTOTIC-RIGIDITY.md.

1. Retraction of the former all-j statement

The previous version claimed that for an odd prime s, with

Q=4s+1, \qquad K=Qj-s, \qquad m=4j-1,

full unrestricted shadowing always satisfied

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

This is false for small j.

A concrete counterexample is

\boxed{s=17,\quad j=2,\quad K=121=11^2,\quad m=7.}

Here

S_2=-T_2=\{1,2,4\}\pmod7.

The divisors of 121 are

1,11,121,

which reduce modulo 7 to

1,4,2,

respectively. All lie in S_2, so

T_{121}\bmod7\subseteq T_2.

But 121 is neither prime nor 17p.

Therefore the all-j theorem is disproved.

2. Why the former proof failed

The previous write-up itself exposed the problem: it attempted to use

K<m^2

for all small j, but this inequality fails when s is large relative to j.

For example, at s=17, j=2,

K=121>49=m^2.

The least-prime-factor escape argument therefore does not apply.

A proof cannot replace this missing range with an unspecified “finite check per s” while still claiming a uniform all-s, all-j theorem.

3. What remains valid

The following results remain valid:

the prime-child theorem;
the divisor-child theorem:

a\mid\gcd(j,s),\ K=ap,\ p\text{ prime} \Longrightarrow T_K\bmod m\subseteq T_j;

the asymptotic ancestry skeleton for j>=s+1;
the exact quotient-specific classifications already proved independently for Q=13, 17, 21, and 29;
the corrected odd-prime-shift theorem in ODD-PRIME-SHIFT-ASYMPTOTIC-RIGIDITY.md.

4. Small-j exception family is a real object

The counterexample above is not isolated.

Finite scouting finds additional small-j full-shadow children for prime shifts, for example:

s=19, j=4:   K=289=17^2
s=53, j=4:   K=799=17*47
s=71, j=8:   K=2209=47^2
s=71, j=16:  K=4489=67^2
s=83, j=2:   K=583=11*53
s=89, j=10:  K=3481=59^2

These are not counterexamples to the asymptotic skeleton because every one lies in the small range

j<s+1.

They define a separate small-ancestor exception problem governed by the exact multiplicative geometry of S_j.

In particular, dyadic ancestors j=2^a are already understood through the Mersenne trap lattice:

S_{2^a}=\langle2\rangle \pmod{2^{a+2}-1}.

That multiplicative closure naturally permits composite child divisors to remain inside the ancestor trap image.

5. Correct research program

The odd-prime-shift program now splits cleanly:

Large-j theorem

For

j\ge s+1,

the exact nonsmooth classification is prime or sp, and smooth children are eliminated in the corrected theorem note.

Small-j exception census

For

1\le j\le s,

classify full shadows by the finite group / trap geometry of S_j.

This small range is not noise. It intersects the dyadic trap lattice and other multiplicative-saturation phenomena and may possess infinite families as s varies.

6. Scientific record

The invalid all-j proof remains available in Git history for provenance but must not be cited as a theorem.

The canonical status is now:

all-j odd-prime-shift rigidity:      FALSE
asymptotic j>=s+1 rigidity:          PROVED
small-j exception classification:    OPEN