Prime-modulus backbone and unbounded depth for the square-completed Type-II layers

Geometry

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Status: proved theorem

Date: 2026-08-15

Depends on: ES-SQUARE-COMPLETION-TRAP-GEOMETRY.md, ES-SQUARE-TRAP-SIGNED-BOX-IDENTITY.md, PRIME-MODULUS-BACKBONE.md

Imported classical tools: Chinese remainder theorem and Dirichlet's theorem on primes in reduced arithmetic progressions

Claim boundary: proves arbitrarily large exact finite first-hit depths for the square-completed Type-II congruence system. It does not imply failure of universal Type-II coverage and does not prove Erdős--Straus.

1. Completed trap layers

For

a\ge1

put

\boxed{m_a=4a-1}

and define the square-completed Type-II trap

\boxed{ S_a = \{-4D\pmod{m_a}:D\mid a^2\}.}

The ordinary López trap satisfies

T_a\subseteq S_a.

The first question is whether the much larger completed layers might accidentally contain the neutral residue 1, which would destroy the old CRT exact-depth construction.

They do not.

2. The neutral residue is never trapped

Theorem

For every

a\ge1,

one has

\boxed{1\notin S_a.}

Proof

Suppose instead that there exists

D\mid a^2

with

-4D\equiv1\pmod{4a-1}.

Then for some positive integer q,

\boxed{4D+1=q(4a-1).}

Because the left side is 1 mod 4 and 4a-1≡3 mod4,

\boxed{q\equiv3\pmod4.}

Write the canonical squarefree-root factorization

D=sb^2, \qquad a=sbc,

with s,b,c positive.

The assumed equation becomes

1+4sb^2=(4sbc-1)q.

Rearrange:

1+q =4sbcq-4sb^2 =4sb(cq-b).

Put

\boxed{t=cq-b.}

Since the left side is positive,

\boxed{t>0.}

Thus

\boxed{q=4sbt-1.}

But from the definition of t,

\boxed{b+t=cq\ge q.}

On the other hand, for positive integers s,b,t,

q=4sbt-1 \ge4bt-1 >b+t.

The last strict inequality holds for all b,t>=1.

This is a contradiction. Therefore

\boxed{1\notin S_a.}

QED.

3. The target residue -1 is always trapped

Take

D=a.

Since

a\mid a^2,

this is a valid square divisor.

Moreover

-4D=-4a\equiv-1\pmod{4a-1}.

Hence

\boxed{-1\in S_a\quad\text{for every }a.}

So every layer contains the familiar central Type-A/B spine while still excluding the neutral residue.

4. Exact-depth theorem at a prime target modulus

Let a be such that

\boxed{q=4a-1}

is a prime greater than 7.

Define

L_{a-1} = \operatorname{lcm} \left( 840, \{4j-1:1\le j<a\} \right).

Since q is prime and larger than every earlier modulus, it divides none of them. Since q>7, it also does not divide 840.

Therefore

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

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

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

This residue is reduced modulo qL_{a-1}.

Dirichlet's theorem therefore gives infinitely many primes p in this class.

For every earlier layer j<a,

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

By the neutral-residue theorem,

1\notin S_j,

so no earlier completed layer captures p.

At the target layer,

p\equiv-1\pmod q

and

-1\in S_a.

Thus the first square-completed Type-II congruence hit occurs exactly at a.

Finally, 840|L_{a-1}, so all these primes satisfy

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

Therefore:

Theorem — square-completed prime-modulus exact depth

Whenever

4a-1>7

is prime, there exist infinitely many Mordell-hard primes whose first square-completed Type-II layer is exactly a.

5. Unbounded completed depth

There are infinitely many primes

q\equiv3\pmod4.

Every such prime q>7 has the form

q=4a-1

for

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

Applying the exact-depth theorem gives arbitrarily large finite first-hit depths.

Hence:

Corollary

\boxed{ \text{the square-completed Type-II first-hit depth is unbounded}.}

This remains true even when restricted to primes in the single Mordell-hard class

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

6. The completed prime-modulus backbone

Define

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

Then every depth in B_sq is realized infinitely often as an exact square-completed first-hit depth among Mordell-hard primes.

This is the same index set as the old López prime-modulus backbone, but it now belongs to the complete symmetric Type-II layer.

The completion can dramatically reduce the depth of individual primes, but it cannot produce a universal constant ceiling.

7. Finite census versus theorem

A finite search can therefore show substantial compression without suggesting boundedness.

The current exact finite census through

p\le50,000,000

finds all 93,457 Mordell-hard primes captured by square-completed layers no deeper than

\boxed{a=624.}

The unique deepest observed prime is

\boxed{p=2,031,121.}

Its completed witness is

\boxed{ a=624, \qquad D=576, \qquad m_a=2495.}

Indeed

576\mid624^2

and

2,031,121+4\cdot576 =2,033,425 =2495\cdot815.

The divisor 576 is mixed relative to

624=2^4\cdot3\cdot13:

576=2^6\cdot3^2

lies above the midpoint in the 2 and 3 coordinates and below it in the 13 coordinate.

For comparison, the first López Type-A/B hit for the same prime is

\boxed{1403.}

Thus square completion more than halves the observed depth of this finite record prime.

These are finite computational facts, not a bound theorem. The exact-depth result above proves that later primes must eventually exceed every fixed depth.

8. Strategic consequence

The completed system has two simultaneous features:

strong finite compression: mixed-sign square divisors can solve hard primes much earlier than the López boundary orthants;
provable unbounded latency: prime-modulus CRT coordinates force arbitrarily large exact first-hit depths.

Therefore the all-prime proof cannot be a bounded-depth theorem.

The right target is structural coverage:

\boxed{ \text{prove every prime has some completed layer, while accepting that the first such layer is unbounded}.}

This matches the lesson already learned from the López depth spectrum, now in the complete square-completed Type-II geometry.