A structural-gap family of order X / sqrt(log X)

Theorem

Read with:

Status: proved analytic consequence of the depth-1 semigroup shadow theorem

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this is a counting theorem for one explicit family of Type A/B structural gaps. It does not determine the density of the full depth-spectrum complement and does not prove the Erdős-Straus conjecture. The analytic input is the classical Selberg-Delange theorem for multiplicative Dirichlet series.

Read with:

1. Structural semigroup

Define

\mathcal S_3 = \{n\ge1:\ p\mid n\Rightarrow p\equiv1\pmod3\}.

The element 1 is included for multiplicative convenience.

The depth-1 semigroup shadow theorem proves that every

n\in\mathcal S_3, \qquad n>1,

is globally impossible as a minimal Type A/B depth:

\boxed{C_{AB}(x)\ne n}

for every integer x for which C_AB(x) is defined.

Thus

\boxed{ \mathcal S_3\setminus\{1\} \subseteq \mathbb N\setminus\mathcal D_{\mathbb P}. }

2. Dirichlet series

Let

F(s) = \sum_{n\in\mathcal S_3}\frac1{n^s} = \prod_{p\equiv1\ (3)}(1-p^{-s})^{-1}, \qquad \Re s>1.

Let chi_{-3} be the nontrivial real Dirichlet character modulo 3.

The Euler products for zeta(s) and L(s,chi_{-3}) give the exact identity

\boxed{ F(s)^2 = \zeta(s)L(s,\chi_{-3}) (1-3^{-s}) \prod_{p\equiv2\ (3)}(1-p^{-2s}). }

Indeed:

at p=1 mod 3, the product zeta L contributes (1-p^{-s})^{-2};
at p=2 mod 3, it contributes (1-p^{-2s})^{-1}, cancelled by the displayed correction factor;
at p=3, the zeta factor is cancelled by 1-3^{-s}.

Therefore

F(s)=\zeta(s)^{1/2}G(s),

where, in a neighborhood of s=1,

G(s) = \left[ L(s,\chi_{-3}) (1-3^{-s}) \prod_{p\equiv2\ (3)}(1-p^{-2s}) \right]^{1/2}

is analytic and nonzero after choosing the positive real branch at s=1.

3. Selberg-Delange asymptotic

The classical Selberg-Delange theorem for a Dirichlet series of the form

\zeta(s)^zG(s)

with z=1/2 gives

\boxed{ S_3(X) :=

|\mathcal S_3\cap[1,X]| \sim \frac{G(1)}{\Gamma(1/2)} \frac{X}{\sqrt{\log X}}. }</div>

Since

\Gamma(1/2)=\sqrt\pi

and

L(1,\chi_{-3})=\frac{\pi}{3\sqrt3},

we obtain the explicit constant

\boxed{ C_3 = \sqrt{ \frac{2}{9\sqrt3} \prod_{p\equiv2\ (3)} \left(1-\frac1{p^2}\right) }. }

Thus

\boxed{ S_3(X) \sim C_3\frac{X}{\sqrt{\log X}}. }

The Euler product converges absolutely to a positive number, so

C_3>0.

4. Structural-gap lower bound

Let

G_{AB}(X) =

|[1,X]\setminus\mathcal D_{\mathbb P}|.</div>

Because every member of S_3 except 1 is a global structural gap,

G_{AB}(X) \ge S_3(X)-1.

Therefore

\boxed{ G_{AB}(X) \ge (C_3+o(1)) \frac{X}{\sqrt{\log X}}. }

In particular,

\boxed{ G_{AB}(X) \gg \frac{X}{\sqrt{\log X}}. }

The same bound applies to the complement of the hard-class infinite-realization spectrum because these depths are impossible for every integer, not only for hard-class primes.

5. Comparison with previous gap families

The known structural-gap lower bounds have now progressed through three scales:

Mersenne power-of-two lattice

G_{AB}(X)\gg\log X.

Prime-depth ancestry

G_{AB}(X)\gg X/\log X.

Depth-1 multiplicative semigroup

\boxed{ G_{AB}(X)\gg X/\sqrt{\log X}. }

Because

\frac{X}{\sqrt{\log X}} \gg \frac{X}{\log X},

the multiplicative structural semigroup is asymptotically the strongest explicit deletion family currently proved in the project.

6. What this does not establish

Since

\frac{1}{\sqrt{\log X}}\to0,

this family still has natural density zero among all positive depths.

Therefore the theorem does not imply that the full depth spectrum has density less than one, nor does it determine whether the total complement has positive density.

It proves something different and concrete: the complement contains an explicit multiplicatively defined family much larger than any previously isolated project family.

7. Research interpretation

The result shows that structural gaps are not rare isolated curiosities created only by Mersenne divisibility or prime target depths.

They form a substantial multiplicative population:

\boxed{ \text{prime-factor restriction} \Longrightarrow \text{exact source-coset containment} \Longrightarrow \text{global impossible depth}. }

The next natural question is whether the union of semigroup-shadow families from the other coset-saturated sources has a larger asymptotic order than X/sqrt(log X), or even positive density.