Asymptotic skeleton of Type A/B ancestry shadows

Status: proved universal structural theorem

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this theorem classifies the prime-factor skeleton of unrestricted full-shadow ancestry children. It does not classify all smooth exceptional children, prove universal Direct-Shadow Completeness, López Type A/B coverage, or the Erdős-Straus conjecture.

Read with:

• ANCESTRY-DIVISOR-CHILD-THEOREM.md
• PRIME-CHILD-SHADOWS.md
• QUOTIENT-9-RIGIDITY.md

1. Setup

Fix positive integers s and j, set

Then

and

Hence

and

Let

Full unrestricted shadowing is equivalent to

Assume throughout the main theorem that

This simple range is not claimed optimal; it is chosen because it makes the size separation uniform.

2. Size separation

Lemma 1

For j>=s+1,

and

Proof

The first inequality is immediate:

For the second,

At j=s+1 this equals

The quadratic is increasing for j>=s+1 because

Thus K<m^2. QED.

3. Small new prime factors cannot occur

Lemma 2

Assume full shadowing. If a prime

satisfies

then

and therefore

Proof

Because ell is prime and lies strictly between 1 and m, membership in S_j can occur only as a plain divisor of j.

Indeed:

• if ell is odd, it cannot equal 4e for an integer e;
• if ell=2, it also cannot equal 4e;
• for a proper divisor e<j, 4e<m and is divisible by 4;
• the endpoint e=j gives 4j mod m = 1.

Thus full shadowing forces ell|j.

But ell|K as well, so

and hence ell|s. QED.

4. At most one prime outside the shift support

Call a prime factor of K external if it does not divide s.

Lemma 3

Under full shadowing and j>=s+1, there is at most one external prime factor of K, counted with multiplicity.

Proof

By Lemma 2, every external prime factor is at least m.

If two external prime factors occurred, including two copies of the same prime, then

contradicting Lemma 1. QED.

Therefore either:

1. K is s-smooth; or
2. there is a unique external prime p, appearing to exponent one, and

where every prime factor of B divides s.

Since p>=m,

Hence

This is a strong finite bound on the entire shift-supported cofactor.

5. Odd prime exponents in the cofactor

Assume the nonsmooth case

Lemma 4

For every odd prime r|s,

Proof

Suppose

Then

is a divisor of K.

Because x<=B<=s<m, full shadowing requires x in S_j as an ordinary integer residue.

From

the common r-valuation is at most a. Therefore

But x is odd, so it cannot equal 4e for a divisor e|j. Hence x notin S_j, contradiction. QED.

Thus no odd shift prime can occur in B to an exponent larger than its exponent in s.

6. The only possible exponent defect is dyadic

Let

If s is odd, there is no dyadic issue and Lemma 4 already gives

If s is even, suppose

The divisor

satisfies

By the gcd identity, x does not divide j. Therefore its only possible membership in S_j is through

But again the gcd identity gives

so

Hence

Combining with Lemma 4:

Theorem 1: general cofactor bound

In the nonsmooth case under full shadowing and j>=s+1,

with p prime, p>=m, and

Every prime divisor of B also divides j.

Thus the only way B can fail to divide s is by carrying one or two additional powers of 2 beyond the dyadic exponent already present in s.

7. Odd-shift converse skeleton

When s is odd, the dyadic defect disappears.

Then

Since every prime divisor of B divides j, and the exponent bound from Lemma 4 is already no larger than in s, we still need to ensure the full exponent occurs in j.

But B|K and B|s, so reducing

modulo B gives

Therefore

Hence

We obtain:

Theorem 2: odd-shift asymptotic skeleton

Let s be odd and j>=s+1. If

then exactly one of the following holds:

1. smooth case: every prime factor of K divides s;
2. divisor-child case:

where <div class="math" role="math">\boxed{a\mid\gcd(j,s)}</div>

and p is prime.

The second case is exactly the family proved sufficient in ANCESTRY-DIVISOR-CHILD-THEOREM.md.

Thus for odd ancestry shifts, above the explicit elementary threshold j>=s+1, the divisor-child family is the only nonsmooth full-shadow mechanism.

8. Consequences for the quotient ladder

s = 1, Q = 5

The shift is odd and has no nontrivial smooth primes. The theorem leaves only

recovering the q=5 prime-only rigidity for all j>=2. Tiny j is checked separately in the existing exact theorem.

s = 3, Q = 13

For j>=4, full shadowing implies either:

or

with the 3p case requiring 3|j.

Operator-02's exact q=13 classification proves that no smooth power exception survives, completing the finite small/smooth residue of the general theorem.

s = 5, Q = 21

Without any dedicated q=21 computation, the theorem already gives for j>=6:

This is a new theorem-level prediction for the next odd-shift ancestry quotient.

s = 7, Q = 29

For j>=8:

Again no quotient-specific search is needed to obtain the skeleton.

9. Even shifts

For even s, Theorem 1 gives

in every nonsmooth full-shadow case.

The observed q=9 and q=17 classifications show that the extra dyadic possibilities are usually eliminated, leaving B|s, with rare smooth power-of-two exceptions.

The remaining general even-shift theorem target is therefore sharply finite:

classify whether a cofactor B<=s with B|4s but B not|s can ever occur in a nonsmooth full-shadow child, and classify the purely s-smooth children.

For fixed s, this is a finite set of cofactor shapes.

10. Why this matters

The ancestry shadow graph originally appeared to require a separate classification at each quotient

The divisor-child theorem provided a universal sufficient family.

The present theorem supplies the converse skeleton:

For odd shifts, that tiny cofactor is exactly a divisor of gcd(j,s).

So an infinite collection of quotient-by-quotient shadow problems has collapsed into:

1. one universal prime-times-divisor law;
2. one finite smooth-exception problem for each shift;
3. a small dyadic correction for even shifts.

This is a structural theorem about the ancestry skeleton of the Type A/B shadow graph.