Infinite squarefree factor-lift families at every ancestry quotient

Ancestry

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Status: proved infinite exact construction

Date: 2026-08-15

Depends on: ES-SQUARE-SQUAREFREE-FACTOR-LIFT.md, THEORY.md

Imported classical tool: Dirichlet's theorem on primes in reduced arithmetic progressions

Claim boundary: constructs infinite exact completed-shadow families for every allowed modulus-ancestry quotient. It does not classify all shadows at a fixed quotient or prove universal completed coverage.

1. Allowed ancestry quotients

Every modulus-ancestry quotient between layers has the form

\boxed{Q=4s+1}

with

\boxed{s\ge1.}

The corresponding ancestry relation is

\boxed{k=Qj-s}

or equivalently

\boxed{k=j+s(4j-1).}

We now construct infinitely many exact squarefree factor-lift shadows for every such Q.

2. Choose a squarefree divisor of the ancestry parameter

Fix

\boxed{Q=4s+1.}

Choose any squarefree positive divisor

\boxed{A\mid s.}

Put

\boxed{t=s/A.}

Because

Q=4s+1,

we have

\boxed{\gcd(t,Q)=1.}

Indeed every common divisor of t|s and 4s+1 divides 1.

3. Choose the lifted prime

Let r be a prime satisfying

\boxed{r\equiv-t\pmod Q.}

Then

\boxed{B=\frac{r+t}{Q}}

is a positive integer.

Define

\boxed{j=AB}

and

\boxed{k=Ar.}

For all sufficiently large choices of r, one may also assume

\gcd(r,A)=1,

so k is squarefree because A was chosen squarefree.

4. Exact ancestry identity

Using

r=QB-t,

we obtain

\begin{aligned} k &=Ar\\ &=A(QB-t)\\ &=QAB-At\\ &=Qj-s. \end{aligned}

Thus j,k satisfy the exact ancestry relation.

Hence

\boxed{4k-1=Q(4j-1).}

So the modulus quotient is exactly the prescribed value Q.

5. The new prime lifts the missing ancestor factor

Let

m=4j-1.

Since

j=AB,

we have

m=4AB-1.

Now

\begin{aligned} r-B &=QB-t-B\\ &=(Q-1)B-t\\ &=4sB-t\\ &=4AtB-t\\ &=t(4AB-1)\\ &=tm. \end{aligned}

Therefore

\boxed{r\equiv B\pmod m.}

This is the exact factor-lift congruence.

6. Factorization match

Write the squarefree divisor A as

\boxed{A=p_1p_2\cdots p_h}

with distinct primes p_i.

Then the later squarefree index is

\boxed{k=p_1p_2\cdots p_h r.}

The earlier index has the factorization

\boxed{j=p_1p_2\cdots p_h B.}

Modulo m:

each shared prime factor p_i maps to itself;
the new prime r maps to the remaining ancestor factor B.

Thus the prime factors of k lift a complete factorization of j.

ES-SQUARE-SQUAREFREE-FACTOR-LIFT.md applies and gives

\boxed{ S_k\bmod m \subseteq S_j.}

Therefore k is structurally impossible as a completed minimal depth.

7. Infinite family by Dirichlet

The target prime residue class is

\boxed{-t\pmod Q.}

Because

\gcd(t,Q)=1,

this is a reduced class.

Dirichlet's theorem therefore supplies infinitely many primes

\boxed{r\equiv-t\pmod Q.}

Discarding finitely many primes that divide A, infinitely many remain with

\gcd(r,A)=1.

Hence:

Theorem — all-quotient factor-lift family

For every ancestry quotient

\boxed{Q=4s+1}

and every squarefree divisor

\boxed{A\mid s,}

there exist infinitely many ancestry pairs j<k satisfying

\boxed{4k-1=Q(4j-1)}

for which the later index k is squarefree and

\boxed{S_k\bmod(4j-1)\subseteq S_j.}

Thus every allowed ancestry quotient supports infinitely many exact completed structural gaps.

8. Quotient nine recovered

Take

Q=9, \qquad s=2.

Choose

A=2, \qquad t=1.

Then

r\equiv-1\equiv8\pmod9,

B=\frac{r+1}{9},

j=2B=\frac{2(r+1)}9,

and

k=2r.

This is exactly the quotient-nine semiprime family already isolated.

9. Quotient thirteen

Take

Q=13, \qquad s=3, \qquad A=3, \qquad t=1.

Every prime

\boxed{r\equiv12\pmod{13}}

gives

B=\frac{r+1}{13}, \qquad j=3B, \qquad k=3r,

with

\boxed{4k-1=13(4j-1)}

and complete direct shadow by j.

For example

r=103

gives

\boxed{j=24, \qquad k=309.}

10. Multiple families at one quotient

When s has several squarefree divisors, one quotient supports several distinct factor-lift templates.

For

Q=25, \qquad s=6,

one may choose

A=2,\ 3,\ 6

with corresponding

t=3,\ 2,\ 1.

This yields prime progressions

\boxed{ \begin{array}{c|c} A & r\pmod{25}\\ \hline 2 & 22\\ 3 & 23\\ 6 & 24 \end{array}}

and three infinite shadow families at the same ancestry quotient.

So the factorization of the ancestry parameter s controls the multiplicity of explicit shadow templates.

11. The rank-one case A = 1

The divisor

A=1

is squarefree and always allowed.

Then

t=s, \qquad j=B, \qquad k=r

is prime.

The construction reduces to the prime-index ancestry absorption theorem.

Thus the all-quotient theorem continuously interpolates from:

rank-one prime target shadows (A=1);
semiprime target shadows (A prime);
higher squarefree target shadows (omega(A)>1).

12. Structural interpretation

For fixed quotient

Q=4s+1,

the ancestry parameter s is not just an additive offset.

Its squarefree divisors specify how much of the earlier layer factorization can be carried unchanged into the later index.

The remaining ancestor factor B is then replaced by one new prime r in a Dirichlet progression chosen so that

r\equiv B\pmod{m_j}.

Therefore the fixed-quotient shadow geometry has an explicit arithmetic template:

\boxed{ \text{shared squarefree factor} +\text{one lifted prime} \Longrightarrow \text{complete shadow}.}

13. Consequence for the nonmultiplicative frontier

The nonmultiplicative ancestry graph is not a collection of isolated coincidences.

It contains infinite exact shadow families at every allowed quotient and often several independent families at the same quotient.

The remaining classification problem is therefore:

determine which shadows are generated by these one-prime factor lifts;
allow several new primes lifting several factors of j;
characterize the residual ancestry edges not decomposable into stabilizer extensions or factor lifts.

A successful theorem would turn nonmultiplicative shadowing into a finite factorization-matching problem attached to s=(Q-1)/4.