Counting bounds for the Type A/B minimal-depth spectrum

Theorem

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Status: proved quantitative synthesis theorem

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: these bounds concern the López Type A/B minimal-depth spectrum. They do not prove universal Type A/B coverage or the Erdős-Straus conjecture. The analytic inputs are classical prime-number theorems in arithmetic progressions; the project-specific inputs are the prime-modulus backbone, prime-depth dichotomy, and Mersenne shadow lattice.

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1. Counting functions

Let

D_H(X)=|\mathcal D_H\cap[1,X]|,

where D_H is the hard-class spectrum of depths realized by infinitely many primes.

Let

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

and

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

Every depth impossible for all integers is absent from every prime spectrum, so a global structural-gap family also lower-bounds the complement of D_H.

2. Lower bound for realized depths

q=4k-1

is prime with q>7, the prime-modulus backbone gives infinitely many hard-class primes of exact depth k.

Therefore every prime

q\equiv3\pmod4, \qquad q\le4X-1,

apart from finitely many initial cases, contributes a distinct depth

k=(q+1)/4\le X.

Hence

\boxed{ D_H(X) \ge \pi(4X-1;4,3)-O(1). }

By the prime number theorem for arithmetic progressions,

\pi(Y;4,3) \sim \frac12\frac{Y}{\log Y},

\boxed{ D_H(X) \ge (2+o(1))\frac{X}{\log(4X)} \gg \frac{X}{\log X}. }

The same lower bound applies to D_P(X).

3. Macroscopic lower bound for structural gaps

The prime-depth dichotomy gives a much denser deletion family than the original Mersenne construction.

Every prime depth

k\equiv4\pmod5

satisfies

5\mid4k-1.

Writing

k=5j-1

gives

4k-1=5(4j-1),

and because k is prime the complete target trap layer reduces into the earlier layer j. Thus every such prime k is a global structural gap:

\boxed{k\notin\mathcal D_{\mathbb P}.}

Therefore

\boxed{ G(X) \ge \pi(X;5,4)-O(1). }

The prime number theorem in arithmetic progressions gives

\pi(X;5,4) \sim \frac14\frac{X}{\log X},

hence

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

In particular,

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

Because these depths are impossible for every integer, the same bound applies to the complement of the hard-class spectrum:

\boxed{

|[1,X]\setminus\mathcal D_H| \gg X/\log X. }</div>

4. General ancestry residue families

The mod 5 family is only the first member of an infinite collection.

Fix s>=1 and put

A=4s+1.

Every prime depth satisfying

\boxed{k\equiv-s\pmod A}

has

A\mid4k-1

and is directly shadowed by

j=(k+s)/A<k.

Since

\gcd(s,A)=1,

Dirichlet gives infinitely many prime depths in each such shadow family.

The families overlap, so their densities cannot simply be added. The single s=1 family is enough for the unconditional X/log X lower bound above.

5. Mersenne gaps remain structurally distinct

The power-of-two Mersenne lattice still supplies a different kind of deletion:

2^b\notin\mathcal D_{\mathbb P}

whenever b>=3 and b+2 is composite.

This gives

G(X) \ge (1-o(1))\log_2 X

from the binary subsequence alone.

That bound is now numerically weaker than the prime-depth X/log X bound, but it remains conceptually important because it comes from exact multiplicative-coset saturation rather than prime-target divisor sparsity.

6. Quantitative spectrum theorem

We now have unconditional lower bounds of the same broad order on both known sides:

\boxed{ D_H(X)\gg\frac{X}{\log X}, \qquad

|[1,X]\setminus\mathcal D_H| \gg\frac{X}{\log X}. }</div>

Likewise

\boxed{ D_{\mathbb P}(X)\gg\frac{X}{\log X}, \qquad G(X)\gg\frac{X}{\log X}. }

More explicitly, the two current constructions give

D_H(X) \ge \pi(4X-1;4,3)-O(1)

and

G(X) \ge \pi(X;5,4)-O(1).

7. What these bounds do not say

They do not determine the natural density of the full spectrum or its complement.

Both X/log X lower bounds are compatible with density zero.

The exact prime-depth dichotomy also shows that understanding the spectrum on prime values of k is equivalent to understanding prime pairs

k,\ 4k-1,

on the realizable side. No infinitude of such prime pairs is assumed or needed for the global spectrum lower bound, because the prime-modulus backbone allows composite depths k as well.

The central counting problem remains

\boxed{ D(X)=|\mathcal D\cap[1,X]|, \qquad X-D(X). }

8. Research significance

The spectrum now has four proven quantitative structures:

exact individual Dirichlet realization certificates;
at least X/log X realized depths from the prime-modulus backbone;
at least X/log X global structural gaps from one prime-depth ancestry family alone;
an independent infinite Mersenne deletion lattice dominating the power-of-two exponent subsequence.

The next counting objective is to understand the union of the prime-depth ancestry families and to find comparable infinite deletion mechanisms on composite depths.