Prime-modulus backbone of the Type A/B depth spectrum

Ancestry

Let

Status: theorem note

Date: 2026-08-14

Claim boundary: this note applies classical CRT, Dirichlet's theorem on primes in arithmetic progressions, and the Type A/B congruence characterization. It does not claim the Erdős-Straus conjecture is solved, and it does not make a literature-priority claim without a separate prior-art review.

1. Setup

Let

m_k=4k-1, \qquad T_k=\{-e,-4e\pmod{m_k}:e\mid k\},

and for a prime p define

C_{AB}(p)=\min\{k\ge1:p\bmod m_k\in T_k\},

with C_AB(p)=infinity if no such k exists.

The elementary control lemma used below is

1\notin T_j\qquad\text{for every }j\ge1.

2. Prime-modulus exact-depth theorem

Theorem

Let k be such that

q=4k-1

is a prime greater than 7. Then there exist infinitely many primes p satisfying

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

Moreover all of these primes may be chosen in the Mordell hard class

\boxed{p\equiv1\pmod{840}}.

Proof

Define

L_{k-1}=\operatorname{lcm}\bigl(840,\{4j-1:1\le j<k\}\bigr).

Because q=4k-1 is prime and

q>4j-1

for every j<k, the prime q divides none of the earlier moduli 4j-1. Since q>7, it also divides neither 840. Therefore

\gcd(q,L_{k-1})=1.

By CRT there is a unique residue class a mod qL_{k-1} satisfying

a\equiv1\pmod{L_{k-1}}, \qquad a\equiv-1\pmod q.

This residue is reduced:

\gcd(a,qL_{k-1})=1,

because it is 1 modulo every prime divisor of L_{k-1} and -1 modulo q.

Dirichlet's theorem now gives infinitely many primes

p\equiv a\pmod{qL_{k-1}}.

For every earlier layer j<k,

p\equiv1\pmod{4j-1}.

Since 1 is never a Type A/B trap residue,

p\bmod(4j-1)\notin T_j.

At the target layer,

p\equiv-1\pmod{4k-1}.

Because 1|k, the residue -1 belongs to T_k. Therefore the first Type A/B hit is exactly k:

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

Finally, 840|L_{k-1}, so all these primes satisfy p=1 mod 840. QED.

3. Consequence: finite C_AB values are unbounded

There are infinitely many primes

q\equiv3\pmod4.

Every such prime q>7 can be written uniquely as

q=4k-1

with

k=\frac{q+1}{4}.

Applying the theorem to each such k produces infinitely many distinct exact finite values of C_AB.

Hence

\boxed{C_{AB}(p)\text{ is unbounded even when restricted to finite values.}}

This is stronger than the earlier observation that, for every K, infinitely many primes can be forced to have C_AB(p)>K or C_AB(p)=infinity. The prime-modulus construction exhibits arbitrarily large layers that are definitely reached, and reached by infinitely many primes.

4. Prime-modulus backbone

Let D_H denote the hard-class minimal-depth spectrum:

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

Define

\mathcal B = \left\{\frac{q+1}{4}:q>7\text{ prime},\ q\equiv3\pmod4\right\}.

Then

\boxed{\mathcal B\subseteq\mathcal D_H.}

In fact every depth in B is realized infinitely often already inside the single hard class 1 mod 840.

We call B the prime-modulus backbone of the Type A/B minimal-depth spectrum.

5. Quantitative lower bound for the spectrum

The prime number theorem for arithmetic progressions gives

\pi(x;4,3)\sim\frac12\operatorname{Li}(x).

Therefore the number of prime-modulus backbone depths up to K satisfies

|\mathcal B\cap[1,K]| = \pi(4K-1;4,3)+O(1) \sim \frac{2K}{\log(4K)}.

Consequently

\boxed{

|\mathcal D_H\cap[1,K]| \ge (1+o(1))\frac{2K}{\log(4K)}. }</div>

This is only a lower bound. Composite target moduli 4k-1 contribute many additional realized depths in the computational spectrum.

6. Explicit Type B family on every backbone depth

The target residue in the proof is

p\equiv-1\pmod{4k-1}.

This is a Type B congruence with

d=k, \qquad n=1.

Write

p=(4k-1)t-1.

Then the standard Type B construction gives

x=kt, \qquad y=ktp, \qquad z=kp.

Indeed,

\frac1{kt}+\frac1{ktp}+\frac1{kp} = \frac{p+1+t}{ktp} = \frac{(4k-1)t+t}{ktp} = \frac4p.

Thus every prime produced by the theorem comes with an explicit exact Erdős-Straus decomposition.

7. Why this matters for the shadow program

The prime-modulus backbone gives an infinite family of layers that cannot be structural gaps.

For these k, the earlier shadow closure is guaranteed to have an uncovered reduced class. Therefore any proposed global theory of the shadow graph must contain an infinite irreducible backbone indexed by primes 4k-1.

This creates a sharper theorem target:

understand the guaranteed prime-modulus backbone;
characterize which composite 4k-1 layers join it;
characterize which composite layers are structural gaps because of admissibility or shadow closure;
study the arrival function on both the backbone and the additional composite-modulus spectrum.

The hard-class minimal-depth spectrum is therefore provably infinite before any conjectural Type A/B coverage assumption is used.