Square-completed Type-II trap geometry: López A/B are the two boundary orthants

Geometry

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Status: proved exact structural theorem

Date: 2026-08-15

Depends on: ES-TYPEII-SQUARE-COMPLETION-LOPEZ-A.md, ES-TWO-TARGET-DIVISOR-SQUARE.md, COMPOSITE-CORE.md

External background: Miguel Angel López, A Complete Congruence System for the Erdos-Straus Conjecture, arXiv:2404.01508

Claim boundary: this identifies the López Type-A/B layer inside the complete square-divisor Type-II layer. It does not prove that every prime is hit by a square-completed layer.

1. Three trap systems at one layer

Fix

a\ge1, \qquad m_a=4a-1.

The existing López Type-A/B trap set is

\boxed{ T_a = \{-e,-4e\pmod{m_a}:e\mid a\}.}

The exact square-completed standard Type-II theorem suggests the enlarged set

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

For a prime p, a residue hit in S_a is a standard Type-II certificate whenever the representing divisor D is not divisible by p.

We now identify the old two-family trap exactly inside this square-divisor box.

2. Type A is the lower divisor box

Every divisor

e\mid a

is also a divisor of a^2.

Taking

D=e

in the square-completed trap gives

-4D=-4e.

Therefore the López Type-A residues are exactly the square-divisor parameters

\boxed{D\mid a.}

This is the lower half of the divisor exponent box.

3. Type B is the upper divisor box

Again let

e\mid a.

Take

\boxed{D=ae.}

Then

D\mid a^2.

Since

4a\equiv1\pmod{4a-1},

we have

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

Thus every López Type-B residue is also a square-completed Type-II residue at the same layer.

The Type-B parameters are exactly

\boxed{D=ae,\qquad e\mid a,}

or equivalently

\boxed{a\mid D\mid a^2.}

This is the upper half of the divisor exponent box.

Therefore:

Theorem — exact layerwise containment

For every a,

\boxed{T_a\subseteq S_a.}

More precisely,

\boxed{ T_a = \{-4D:D\mid a\} \cup \{-4D:a\mid D\mid a^2\} \pmod{4a-1}.}

So López A and B are not separate from square-completed Type II. They are the two boundary divisor regions of the complete Type-II layer.

4. Prime-exponent geometry

Write

a=\prod_{i=1}^s \ell_i^{E_i}.

Every divisor of a^2 has the unique form

D=\prod_{i=1}^s\ell_i^{U_i}, \qquad 0\le U_i\le2E_i.

Thus the square-completed parameter space is the full integer box

\boxed{ \mathcal B(a) = \prod_{i=1}^s[0,2E_i]_{\mathbb Z}.}

Type-A orthant

The condition D|a is

\boxed{U_i\le E_i\quad\text{for every }i.}

Type-B orthant

The condition a|D|a^2 is

\boxed{U_i\ge E_i\quad\text{for every }i.}

Hence the López parameter region is

\boxed{ \mathcal B_-(a)\cup\mathcal B_+(a),}

where

\mathcal B_-(a)=\prod_i[0,E_i], \qquad \mathcal B_+(a)=\prod_i[E_i,2E_i].

The complete Type-II square box is all of B(a).

5. The genuinely new region is mixed

A square divisor D|a^2 lies outside both López families exactly when its exponent vector is mixed relative to the midpoint vector (E_i):

\boxed{ \exists i,j: U_i<E_i, \qquad U_j>E_j.}

These are the cross-orthant regions omitted by both Type A and Type B.

Therefore a standard Type-II certificate that is not represented by López A or B at the same layer must use a mixed square divisor.

This gives a precise geometric meaning to the extra Type-II room.

6. Prime-power layers are already complete

Suppose

a=\ell^E

is a prime power.

Then every exponent

0\le U\le2E

satisfies either

U\le E

U\ge E.

Hence there is no mixed region.

Therefore:

Theorem — prime-power layer equality

If a is a prime power, then

\boxed{S_a=T_a.}

So square completion adds no new Type-II trap residue at all on a prime-power layer.

Every genuinely new square-completed parameter requires a layer a with at least two distinct prime divisors.

7. Composite layers create the cross-regions

If a has at least two distinct prime factors, then mixed divisor parameters exist.

For example, if

a=u^Ev^F\cdots

with distinct primes u,v, then the exponent choice

U_u=0, \qquad U_v=2F

is below the midpoint in one coordinate and above it in another.

This proves that the parameter-space completion is nontrivial on every non-prime-power layer.

Residue collisions modulo 4a-1 can in principle identify some mixed parameters with boundary residues, so parameter-space nontriviality alone does not assert strict residue-set enlargement for every such a. But strict enlargement occurs already in small examples and is exactly what happens in the canonical 2521 rescue below.

8. The p = 2521 mixed-divisor rescue

Take

p=2521, \qquad a=12=2^2\cdot3.

Then

m_a=4a-1=47.

The square-only certificate from ES-TYPEII-SQUARE-COMPLETION-LOPEZ-A.md uses

\boxed{D=16=2^4.}

Relative to

a^2=2^4\cdot3^2,

the exponent vector of D is

(U_2,U_3)=(4,0),

while the midpoint vector for a is

(E_2,E_3)=(2,1).

Thus

4>2, \qquad 0<1,

so D is genuinely mixed.

Its residue is

-4D=-64\equiv30\pmod{47}.

The ordinary López layer has divisors

1,2,3,4,6,12

and trap residues

\{-e,-4e:e\mid12\}.

None is 30 mod 47.

Therefore

\boxed{30\in S_{12}\setminus T_{12}.}

And indeed

2521\equiv30\pmod{47}.

So the square-completed layer solves 2521 at the prime modulus 47 through a mixed divisor that neither López boundary orthant can see.

9. Exact relation to the López Type-B parametrization

The upper-box embedding can also be written directly in López parameters.

A Type-B witness has positive d,n and modulus

4dn-1.

Put

a=dn

and choose the square divisor

\boxed{D=dn^2=an.}

Then

D\mid a^2

because

\frac{a^2}{D}=d.

Moreover

4D=4dn^2 =n(4dn) \equiv n\pmod{4dn-1}.

Hence

\boxed{-4D\equiv-n\pmod{4dn-1},}

which is exactly the López Type-B residue.

Thus both López parametrizations embed explicitly into one square-divisor formula:

\boxed{ \begin{array}{c|c} \text{López family} & \text{square divisor }D\\ \hline \text{Type A} & d\\ \text{Type B} & dn^2 \end{array}}

at the common layer a=dn.

10. Strategic consequence: reuse the shadow machinery

The mature Type-A/B research should not be discarded when moving to the exact Type-I/II formulation.

Instead, define the square-completed layer

S_a=\{-4D:D\mid a^2\}\pmod{4a-1}

and regard the existing López layer T_a as its two boundary orthants.

This creates a concrete next program:

extend exact trap cardinality from Div(a) to the square divisor lattice Div(a^2);
classify collisions among mixed square-divisor residues;
extend direct-shadow and ancestry theorems from T_a to S_a;
determine whether the old zero-density Type-A/B survivor core is killed by mixed square-divisor traps;
exploit the theorem that prime-power layers need no extension, so all new geometry is localized to multi-prime layers;
compare mixed-divisor shadow quotients with the exact Kneser stabilizer defects of the signed-box formulation.

The conceptual picture is now precise:

\boxed{ \text{López A/B} = \text{two boundary orthants of the complete Type-II square-divisor box}.}

The missing certificates live in the cross-regions.