Quotient-9 shadow rigidity

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

Date: 2026-08-15

Project: Free Computation Foundation / CENTL

Claim boundary: this does not prove universal Direct-Shadow Completeness, López Type A/B coverage, or the Erdős-Straus conjecture. It completely classifies unrestricted full direct shadows along the ancestry quotient q=9.

Read with:

• PRIME-CHILD-SHADOWS.md
• SQUARE-LIFT-TOWERS.md
• RECIPROCITY-TOWER-SHADOWS.md
• DYADIC-TRAP-LATTICE.md

1. Quotient-9 ancestry

Fix a source depth j>=1, put

and define its quotient-9 ancestry child

Then

Thus this is exactly the odd-square lift with square multiplier

Also

and

2. Normalized trap sets

It is convenient to negate the trap sets. Define

The set S_n is inverse-closed. Indeed, if e|n and f=n/e, then

so

and conversely.

For the ancestry pair (j,K), if e f=K, then modulo m

Therefore, once every divisor residue e mod m belongs to S_j, all companion residues 4e mod m belong as well by inverse closure.

Hence:

Lemma

For K=9j-2,

This reduces quotient-9 full shadowing to the image of the target divisor lattice.

3. Classification theorem

Theorem

Let

Then the complete unrestricted target layer is shadowed by j if and only if exactly one of the following occurs:

1. K is prime;
2. K=2p with p prime;
3. (j,K)=(2,16).

Equivalently,

In the second alternative j is automatically even because 9j-2 is even.

4. Prime child direction

If K is prime, the general prime-child ancestry theorem applies immediately:

5. Twice-prime child direction

Suppose

with p prime.

Then K is even, so j is even. Put

Since

we have

Subtract d=j/2:

Therefore

The divisors of K=2p are

Modulo m these become

and every one is a divisor of j.

Thus every target divisor residue lies in S_j, so by the normalized-divisor lemma

6. Exceptional child K = 16

For

we get

The normalized base trap set is

Every divisor of 16 is a power of two, and its residue modulo 7 lies in

Therefore

This is the unique composite child in the quotient-9 classification that is neither prime nor twice a prime.

7. Converse when j is odd

Assume j is odd. Then K=9j-2 is odd.

Suppose K is composite. Let ell be its smallest prime factor. Then

For every composite odd child beyond the tiny initial range,

and the few initial values are checked directly. In fact the inequality already holds for every relevant odd j>=1 with composite K.

Also

so

Because ell is an odd integer with

it cannot be a member of the normalized base set S_j:

• it is not a divisor of j;
• every unwrapped value 4d with d<j is divisible by 4, hence cannot equal odd ell;
• the only wrapped endpoint 4j reduces to 1.

Thus

But ell|K, so the normalized-divisor lemma says full shadowing fails.

Therefore, when j is odd,

8. Converse when j is even

Now let

Then

where

Also

because

Hence

Suppose N has an odd prime factor ell.

Choose the smallest such factor. Since ell|N and

we have

Also ell<m except for the prime case N=ell itself. If N is composite with an odd prime factor, choosing a proper prime factor gives ell<=sqrt N<m.

As above, an odd ell with 1<ell<m and ell not|j cannot lie in S_j.

Therefore full shadowing forces either:

1. N is prime, giving the twice-prime family K=2N; or
2. N has no odd prime factor, so

We must classify the second possibility.

9. The power-of-two residual is unique

Suppose

Then

The powers of 2 mod 9 have period 6, and -1 mod 9 occurs exactly when

For r=3,

which is the exceptional shadow already found.

Now suppose r>=9. Then d=(2^r+1)/9 is odd, so

has exact 2-adic valuation 1.

The target depth is

so 16|K and therefore 16 is a target divisor.

For r>=9,

But 16 cannot lie in S_j:

• 16 is not a divisor of j, because v_2(j)=1;
• if 16=4e in the unwrapped second family, then e=4, but 4 not|j;
• the wrapped endpoint is only 1.

Hence the target divisor 16 escapes the base normalized trap set.

Therefore no r>=9 works.

So the unique power-of-two residual is

Combining all cases proves the classification theorem. QED.

10. Structural interpretation

The quotient-5 theorem said:

The new quotient-9 theorem says:

This is the first exact evidence that composite-child shadowing is controlled by the factorization shape of the child depth itself.

The square multiplier 9=3^2 permits one extra divisor coordinate, but only in an extremely rigid way.

11. Infinite shadow subfamilies

The theorem gives several infinite families.

Prime children

Prime children satisfy

Dirichlet gives infinitely many primes in this class, hence infinitely many quotient-9 prime-child shadows.

Twice-prime children

Write

Then

Every prime

gives

and hence a quotient-9 full shadow.

Dirichlet gives infinitely many such primes.

Therefore the quotient-9 graph has at least two infinite arithmetic components:

and

12. Counting

For child depths K<=X, the prime component has asymptotic count

The twice-prime component corresponds to primes

and therefore has asymptotic count

Ignoring their disjoint tiny exception, quotient 9 therefore contributes