Counting the source-1 structural-gap semigroup

Theorem

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Status: proved analytic corollary of the exact source-1 shadow classification

Date: 2026-08-15

Project: Free Computation Foundation / CENTL

Claim boundary: the analytic machinery is classical. The Type-A/B-specific input is the exact theorem that every depth whose prime factors are all 1 mod 3 is completely shadowed by layer 1.

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1. The exact gap semigroup

Define

\mathcal S_3 = \left\{ n\ge1: p\mid n\Longrightarrow p\equiv1\pmod3 \right\}.

The exact saturated-base theorem gives

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

Every nontrivial member of this multiplicative semigroup is a global structural gap because its complete Type A/B layer is shadowed by depth 1.

Let

S_3(X)=\#\{n\le X:n\in\mathcal S_3\}.

Then the structural-gap counting function satisfies

\boxed{G(X)\ge S_3(X)-1.}

2. Dirichlet series

Let f_3(n) be the indicator of S_3.

Its Dirichlet series is

F_3(s) = \sum_{n\ge1}\frac{f_3(n)}{n^s} = \prod_{p\equiv1\pmod3} (1-p^{-s})^{-1}.

Let chi_3 be the nontrivial primitive Dirichlet character modulo 3.

Using the Euler products for zeta(s) and L(s,chi_3),

\zeta(s)L(s,\chi_3) = (1-3^{-s})^{-1} F_3(s)^2 \prod_{p\equiv2\pmod3}(1-p^{-2s})^{-1}.

Therefore

\boxed{ F_3(s)^2 = \zeta(s)L(s,\chi_3) (1-3^{-s}) \prod_{p\equiv2\pmod3}(1-p^{-2s}). }

Write

F_3(s)=\zeta(s)^{1/2}H_3(s),

where

H_3(s) = \left[ L(s,\chi_3) (1-3^{-s}) \prod_{p\equiv2\pmod3}(1-p^{-2s}) \right]^{1/2}.

The factor H_3(s) is analytic and nonzero in a neighborhood of s=1 suitable for the classical Selberg-Delange method.

3. Selberg-Delange asymptotic

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

\zeta(s)^z H(s)

with z=1/2 gives

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

where

\boxed{ C_3 = \frac{H_3(1)}{\Gamma(1/2)} = \frac1{\sqrt\pi} \left[ \frac23L(1,\chi_3) \prod_{p\equiv2\pmod3}(1-p^{-2}) \right]^{1/2}. }

Using the classical value

L(1,\chi_3)=\frac{\pi}{3\sqrt3},

this is an explicit positive Euler-product constant.

A numerical truncation gives approximately

C_3\approx0.3012,

but the exact product expression is the theorem-level object.

4. Strong structural-gap lower bound

Since every nontrivial member of S_3 is a structural gap,

G(X)\ge S_3(X)-1.

Hence

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

In particular,

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

This is substantially stronger than the earlier lower bounds

\sqrt X

from a single polynomial reciprocity tower and

X/\log X

from prime-depth residue classes.

5. Comparison with the realized backbone

The prime-modulus backbone gives

R(X)\gg X/\log X.

The source-1 gap semigroup gives

G(X)\gg X/\sqrt{\log X}.

Thus the presently proved structural-gap lower bound is asymptotically larger than the presently proved lower bound for infinitely realized depths:

\frac{X/\sqrt{\log X}}{X/\log X} =\sqrt{\log X}.

This does not imply that gaps outnumber realized depths in the full spectrum. These are only lower bounds generated by two specific theorem families.

6. Density consequence

Because

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

the source-1 semigroup itself has natural density zero among all positive integers.

So the theorem does not yet prove that structural gaps have positive density.

It does prove that the complement of the exact-depth spectrum is quantitatively much thicker than an isolated polynomial or prime sequence.

7. Why the exponent 1/2 appears

The allowed generating primes are precisely the primes

p\equiv1\pmod3,

which occupy one of the two reduced residue classes modulo 3 and hence have Dirichlet density 1/2 among primes not equal to 3.

The square-root logarithmic exponent is the analytic reflection of that half-density prime generator set.

In broad terms:

\boxed{ \text{half the prime classes generate the exact source-1 shadow semigroup} \Longrightarrow X/(\log X)^{1/2}\text{ semigroup size}. }

8. Potential extensions

The source-2 and source-4 shadow families are also multiplicative semigroup problems with additional target residue constraints.

A natural next analytic task is to count:

\left\{ K\le X: K\equiv2\pmod7, \ \ell\mid K\Rightarrow (\ell/7)=+1 \right\}

and

\left\{ K\le X: K\equiv4\pmod{15}, \ \ell\mid K\Rightarrow (\ell/15)=+1 \right\}.

Their generator primes also occupy character-positive subsets, so similar Selberg-Delange behavior is expected after the residue constraint is incorporated.

The union of the three saturated-base semigroups may yield a larger explicit constant in the structural-gap asymptotic lower bound.

9. Publication references for the analytic step

The analytic deduction is an application of the classical Selberg-Delange method. A modern reference is:

Régis de la Bretèche and Gérald Tenenbaum, Remarks on the Selberg--Delange method, arXiv:2010.12929.

No novelty is claimed for Selberg-Delange, Dirichlet characters, or the Euler-product manipulation.

10. Research significance

The exact shadow theory has now produced an analytic theorem about the size of the forbidden depth set.

The chain is:

\boxed{ \text{exact source-1 shadow classification} \to \text{multiplicative gap semigroup} \to \text{Euler product} \to \text{Selberg-Delange} \to G(X)\gg X/\sqrt{\log X}. }

That is a direct bridge from the finite-looking Type A/B shadow graph to asymptotic number theory.