Almost all prime depth values are structural gaps

Ancestry

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Status: proved analytic corollary of the prime-depth dichotomy

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this is a density statement about prime values of the depth parameter k. It does not claim that almost all input primes fail Type A/B, and it does not prove the Erdős-Straus conjecture. The analytic ingredient is the classical Brun/Selberg upper-bound sieve for two linear forms.

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1. Prime-depth realization is a prime-pair condition

The prime-depth dichotomy proves that for prime k,

\boxed{ k\text{ is a realizable minimal Type A/B depth} \iff 4k-1\text{ is prime}.}

Thus the number of realizable prime depth values up to X is

R_{\rm prime}(X) = \#\{k\le X:k\text{ prime and }4k-1\text{ prime}\}.

This is a two-linear-form prime problem.

2. Upper-bound sieve

Consider

F(n)=n(4n-1).

For every odd prime ell, the congruence

F(n)\equiv0\pmod\ell

has exactly two residue classes:

n\equiv0\pmod\ell

n\equiv4^{-1}\pmod\ell.

They are distinct. The prime 2 contributes one harmless local condition.

This is therefore an admissible dimension-two sieve problem. The classical Brun/Selberg upper-bound sieve gives

\boxed{ R_{\rm prime}(X) \ll \frac{X}{(\log X)^2}. }

No conjecture about the infinitude of the prime pairs k,4k-1 is used.

3. Relative density among prime depths

The total number of prime depth values up to X is

\pi(X) \sim \frac{X}{\log X}.

Therefore

\frac{R_{\rm prime}(X)}{\pi(X)} \ll \frac1{\log X} \to0.

Hence:

Theorem

\boxed{ \text{Among prime depth values }k, \text{ a relative density-one set are structural gaps.} }

Equivalently,

\boxed{ \#\{k\le X:k\text{ prime and }k\notin\mathcal D_{\mathbb P}\} = (1-o(1))\pi(X). }

Using the prime number theorem,

\boxed{ \#\{k\le X:k\text{ prime and structurally impossible}\} = (1-o(1))\frac{X}{\log X}. }

4. Stronger global gap lower bound

Every structurally impossible prime depth is also a gap in the full depth spectrum. Therefore, if

G(X)=|[1,X]\setminus\mathcal D_{\mathbb P}|,

then

G(X) \ge \pi(X)-R_{\rm prime}(X).

The sieve estimate gives

\boxed{ G(X) \ge (1-o(1))\frac{X}{\log X}. }

This strengthens the explicit single-residue-family bound

G(X)\ge\left(\frac14+o(1)\right)\frac{X}{\log X}

coming from prime depths k=4 mod 5.

Because these prime-depth gaps are impossible for every integer, the same lower bound applies to the complement of the hard-class spectrum.

5. Graph interpretation

The prime-depth backbone projection shows that every impossible prime depth is directly shadowed by at least one prime-modulus backbone parent.

Thus the relative-density theorem may be read graphically:

almost every prime-valued node in the depth graph is not a new spectrum vertex at all; it is a one-edge satellite of an earlier backbone vertex.

Only the sparse prime-pair nodes

k,\quad4k-1\text{ both prime}

remain as genuine prime-depth spectrum vertices.

6. What remains unknown

The sieve theorem is an upper bound. It does not prove that infinitely many prime depths are realizable.

Infinitude of prime k for which 4k-1 is also prime is a separate prime-pair problem not resolved here.

This does not affect the global spectrum's infinitude because the prime-modulus backbone produces infinitely many realized depths k=(q+1)/4 as q ranges over primes 3 mod 4, and those depth values need not themselves be prime.

7. Structural message

The depth spectrum is now known to have a remarkable asymmetry on the prime subsequence:

\boxed{ \begin{array}{c} \text{all prime depths}\\ \downarrow\\ \text{almost all: composite target modulus}\\ \downarrow\\ \text{directly shadowed by backbone}\\ \downarrow\\ \text{structural gaps} \end{array}}

The genuinely difficult spectrum classification is therefore overwhelmingly a problem about composite depth values.