The Type A/B minimal-depth spectrum is infinite and co-infinite

Theorem

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Status: proved synthesis theorem

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this theorem concerns the minimal depth spectrum inside López Type A/B congruence solutions. It does not prove that every prime has finite C_AB, and it does not prove the Erdős-Straus conjecture. Literature priority for this spectrum formulation remains under review.

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1. Spectrum

Define the prime minimal-depth spectrum

\mathcal D_{\mathbb P} = \{k\ge1:\exists\text{ prime }p\text{ with }C_{AB}(p)=k\}.

Define also the hard-class infinite-realization spectrum

\mathcal D_H = \{k:\text{infinitely many primes }p\equiv h\pmod{840} \text{ for some }h\in H\text{ satisfy }C_{AB}(p)=k\}.

The prime-modulus backbone and the Mersenne shadow lattice give opposite infinite families.

2. Infinite realized side

Let

q=4k-1

be prime with q>7.

The prime-modulus backbone theorem gives a target Type A/B candidate at depth k whose target modulus is a new CRT coordinate relative to all previous layers. Finite congruence avoidance plus Dirichlet then gives infinitely many primes, even in the hard class

p\equiv1\pmod{840},

with

\boxed{C_{AB}(p)=k.}

There are infinitely many primes

q\equiv3\pmod4,

so there are infinitely many such depths

k=(q+1)/4.

Therefore

\boxed{\mathcal D_H\text{ is infinite}.}

In particular

\boxed{\mathcal D_{\mathbb P}\text{ is infinite}.}

3. Infinite impossible side

The Mersenne shadow theorem gives, for a,b>=1,

a+2\mid b+2, \quad a<b \Longrightarrow T_{2^b}\text{ is completely shadowed by }T_{2^a}.

Hence

\boxed{C_{AB}(n)\ne2^b}

for every integer n whenever such a proper divisor relation exists.

In particular, for every r>=2, take

b=3r-2.

Then

3\mid b+2,

so depth

\boxed{2^{3r-2}}

is completely shadowed by depth 2.

Thus there are infinitely many positive integers k that are not a minimal Type A/B depth for any integer, and therefore certainly not for any prime.

Hence

\boxed{\mathbb N\setminus\mathcal D_{\mathbb P}\text{ is infinite}.}

The same global gaps are absent from the hard-class spectrum, so

\boxed{\mathbb N\setminus\mathcal D_H\text{ is infinite}.}

4. Spectrum theorem

Combining the two sides:

Theorem

The Type A/B minimal-depth spectrum is infinite and co-infinite:

\boxed{

|\mathcal D_{\mathbb P}|=\infty, \qquad

|\mathbb N\setminus\mathcal D_{\mathbb P}|=\infty. }</div>

Moreover the hard-class infinite-realization spectrum is also infinite and co-infinite:

\boxed{

|\mathcal D_H|=\infty, \qquad

|\mathbb N\setminus\mathcal D_H|=\infty. }</div>

5. Stronger binary subsequence statement

Inside the subsequence of depths

k=2^b,

the Mersenne theorem proves that every b>=3 with composite b+2 is structurally impossible.

Only b+2 prime can escape that particular obstruction.

Therefore, if

E(B)=\#\{b\le B:2^b\text{ survives the Mersenne obstruction}\},

then

E(B)\le\pi(B+2)+O(1),

and hence

\boxed{E(B)/B\to0.}

So the complement of the spectrum occupies a density-one set of exponents inside this explicit sparse subsequence.

6. Conceptual meaning

The minimal Type A/B depth parameter is therefore not simply an unbounded complexity statistic whose every sufficiently large value eventually appears.

Its spectrum has two permanent arithmetic forces:

arrival: independent prime-modulus coordinates create infinitely many realizable exact depths;
deletion: modulus ancestry and exact trap containment create infinitely many structurally forbidden depths.

Thus the object is naturally a genuine arithmetic spectrum with both infinite support and infinite holes.

7. Paper-level consequence

The depth program can now state a clean unconditional theorem package:

\boxed{ \begin{array}{c} C_{AB}\text{ has unbounded finite values};\\ \text{infinitely many depths are infinitely prime-realizable};\\ \text{infinitely many depths are structurally impossible};\\ \text{therefore the minimal-depth spectrum is infinite and co-infinite.} \end{array} }

This is independent of whether López's universal Type A/B coverage conjecture is ultimately true.

Even if every prime has finite C_AB, the set of values taken by C_AB still has an infinite deterministic gap structure.

8. Next questions

What is the density of the spectrum among all positive depths?
Can additional infinite shadow lattices enlarge the known complement?
Can every non-shadowed depth be shown to lie in the spectrum under universal DSC-P?
What is the asymptotic counting function

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

How does the spectrum decompose into prime-modulus backbone values, composite-rescue values, and structural gaps?