Quantitative counting in the Type A/B exact-depth spectrum

Theorem

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Status: proved corollaries from the prime-modulus backbone, prime-depth dichotomy, and shadow-gap theorems

Date: 2026-08-15

Project: Free Computation Foundation / CENTL

Claim boundary: this does not classify the full exact-depth spectrum and does not prove universal Type A/B coverage or Erdős-Straus. It gives rigorous lower bounds for both infinitely realized depths and structurally forbidden depths.

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1. Counting functions

Let

R(X) = \#\{k\le X:k\in\mathcal D_\infty\},

where D_infty is the set of depths realized by infinitely many primes.

Let

G(X) = \#\{k\le X:k\notin\mathcal D_\exists\},

where D_exists is the set of depths realized by at least one prime.

The full functions are unknown. The theorem families now give order-X/log X lower bounds for both sides.

2. Prime-modulus backbone count

Define the prime-modulus backbone

\mathcal B = \{k:4k-1\text{ is prime and }>7\}.

The prime-modulus backbone theorem gives

\boxed{ \mathcal B\subseteq\mathcal D_\infty. }

The map

k\mapsto q=4k-1

is a bijection between backbone depths k<=X and primes

q\le4X-1, \qquad q\equiv3\pmod4, \qquad q>7.

Thus

\boxed{ \#(\mathcal B\cap[1,X]) = \pi_{3\bmod4}(4X-1)-2. }

Consequently

\boxed{ R(X) \ge \pi_{3\bmod4}(4X-1)-2. }

The prime number theorem in arithmetic progressions gives

\pi_{3\bmod4}(x) \sim \frac{x}{2\log x}.

Therefore

\boxed{ R(X) \ge \left(2+o(1)\right) \frac{X}{\log(4X)} }

and in particular

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

This is only a lower bound. Composite target moduli contribute many additional realized depths.

3. Prime-child gaps give an X/log X structural-gap lower bound

The quotient-5 prime-child theorem says:

\boxed{ k\text{ prime},\quad k\equiv4\pmod5 \Longrightarrow k\notin\mathcal D_\exists.}

Indeed every such prime depth can be written

k=5j-1

and is fully shadowed by the earlier layer j.

Therefore

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

By the prime number theorem in arithmetic progressions,

\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.}

This supersedes the earlier square-root lower bound as the strongest currently recorded unconditional gap count from one explicit shadow family.

4. Prime-depth dichotomy gives the exact prime slice

For every prime-valued depth k, PRIME-DEPTH-DICHOTOMY.md proves

\boxed{ k\in\mathcal D_\infty \iff 4k-1\text{ is prime},}

and if 4k-1 is composite then

\boxed{k\notin\mathcal D_\exists.}

Thus the prime-depth subsequence contains no intermediate finitely-realized case.

The quotient-5 count above uses only one easy subfamily of the composite-modulus side. The complete dichotomy says every prime k with composite 4k-1 is actually a structural gap.

Any sharper analytic estimate for prime pairs

k,\quad4k-1

would immediately sharpen the structural-gap count among prime depth indices, but no such extra estimate is required for the present X/log X theorem.

5. A separate square-root gap family remains explicit

The first reciprocity-gap sequence is

A_n=3n^2+3n+1, \qquad n\ge1.

Every A_n is a structural gap, and

A_n\le X \iff n\le \frac{\sqrt{12X-3}-3}{6}.

Hence, independently of prime-depth arguments,

\boxed{ G(X) \ge \left\lfloor \frac{\sqrt{12X-3}-3}{6} \right\rfloor }

for X>=7.

This family is no longer the strongest counting bound, but remains valuable because it consists of explicit polynomial depth gaps rather than a prime residue class.

6. Quantitative two-sided spectrum theorem

The current exact theorem package gives simultaneously

\boxed{ R(X)\gg\frac{X}{\log X} }

and

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

More explicitly, the proved mechanisms give

R(X) \ge \pi_{3\bmod4}(4X-1)-2 \sim \frac{2X}{\log(4X)},

while

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

These are lower bounds on opposite sides of the same minimal-depth spectrum.

Thus both infinite-arrival nodes and permanent structural gaps occur at least at prime-scale frequency among depth indices.

7. Interpretation

The C_AB depth line is not a nearly full set with a few exotic holes, nor a thin list of isolated arrivals.

Already-proved arithmetic mechanisms force substantial populations on both sides:

\boxed{ \text{many infinitely realized depths} + \text{many impossible depths}. }

The remaining problem is to classify the composite-depth region not already decided by ancestry, quotient, or shadow theorems.

8. Paper-level use

The quantitative spectrum can now be summarized as:

the realized spectrum is infinite and grows at least like X/log X;
the structural-gap set is infinite and also grows at least like X/log X;
prime depths are completely classified by primality of 4k-1;
explicit polynomial and Mersenne shadow families give additional deterministic gaps;
universal DSC-P would classify every remaining directly novel composite depth as infinitely prime-realizable.

9. Novelty boundary

The prime number theorem in arithmetic progressions and related analytic counting are classical. The candidate contribution is their application to the Type-A/B minimal-depth spectrum after the prime-child and prime-depth shadow theorems identify explicit realized and forbidden depth families.

No claim is made that the analytic counting theorems themselves are new.