Prime-depth backbone projection — Free Computation Foundation

Ancestry

Read with PRIME-DEPTH-DICHOTOMY.md and PRIME-MODULUS-BACKBONE.md.

Status: proved corollary of the prime-depth dichotomy

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this classifies the direct-shadow source available for a prime depth with composite target modulus. It does not prove universal López Type A/B coverage or the Erdős-Straus conjecture.

Read with PRIME-DEPTH-DICHOTOMY.md and PRIME-MODULUS-BACKBONE.md.

Theorem

Let k be a prime depth and put

m_k=4k-1.

If m_k is composite, then there exists a prime

q\mid m_k, \qquad q\equiv3\pmod4, \qquad q<m_k.

Set

j=(q+1)/4.

Then

m_j=q

is itself prime, so j is a prime-modulus backbone depth, and

\boxed{T_k\bmod q\subseteq T_j.}

Thus every structurally impossible prime depth is directly shadowed by a prime-modulus backbone layer.

Proof

Because

m_k\equiv3\pmod4,

its prime factorization contains at least one prime factor q=3 mod 4. Since m_k is composite, choose such a factor with q<m_k.

Write

m_k=Aq.

Because both m_k and q are 3 mod 4,

A\equiv1\pmod4.

Write

A=4s+1, \qquad q=4j-1.

Then

4k-1=(4s+1)(4j-1),

k=s(4j-1)+j

and

4k=(4s+1)(4j-1)+1.

Therefore

k\equiv j\pmod q, \qquad 4k\equiv1\pmod q.

Since k is prime,

T_k=\{-1,-4,-k,-4k\}.

Reducing modulo q=m_j gives

\{-1,-4,-j,-1\}\subseteq T_j,

because 1|j and j|j.

QED.

Graph interpretation

On the subsequence of prime depth values, every node has one of two forms:

\boxed{ \text{prime depth }k \to \begin{cases} \text{backbone node}, & 4k-1\text{ prime},\\ \text{direct edge to a backbone node}, & 4k-1\text{ composite}. \end{cases}}

So the prime-depth part of the shadow graph has depth at most one above the prime-modulus backbone.

This is substantially stronger than merely knowing that composite 4k-1 produces some earlier shadow source.

Multiple backbone parents

Every distinct prime divisor

q\equiv3\pmod4

of 4k-1 supplies a backbone shadow parent

j=(q+1)/4.

Thus the set of prime-modulus parents of an impossible prime depth is read directly from the 3 mod 4 prime factors of its target modulus.

Examples:

k=19:  4k-1=75=3*5^2
       q=3  -> j=1

k=29:  4k-1=115=5*23
       q=23 -> j=6

k=79:  4k-1=315=3^2*5*7
       q=3  -> j=1
       q=7  -> j=2

Consequence

The genuinely difficult part of the depth spectrum is therefore even more concentrated on composite depth values.

Prime depth values do not create long new shadow ancestry chains. They are either independent backbone layers themselves or are removed immediately by a backbone parent.