Square-completed López layer equals a symmetric signed divisor box

Geometry

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Status: proved exact identity

Date: 2026-08-15

Depends on: ES-SQUARE-COMPLETION-TRAP-GEOMETRY.md, ES-SQUARE-TRAP-COMPLEMENT.md, FAB-KNESER-FULL-STABILIZER-DEFECT.md

Claim boundary: this is an exact finite-group identity at each layer. It does not by itself prove that every prime is hit by some layer.

1. Square-completed layer

Fix

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

The square-completed standard Type-II trap layer is

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

Write the prime factorization

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

Every divisor of a^2 has the form

D=\prod_i\ell_i^{U_i}, \qquad 0\le U_i\le2E_i.

2. Center the divisor exponents

Define

\boxed{z_i=U_i-E_i.}

Then

\boxed{-E_i\le z_i\le E_i.}

Moreover

D =a\prod_i\ell_i^{z_i}

as an identity in the multiplicative group of rational numbers, and modulo m every prime factor of a is a unit because

\gcd(a,4a-1)=1.

Thus D/a is a well-defined unit modulo m.

Since

4a\equiv1\pmod m,

we obtain

\begin{aligned} -4D &=-4a\frac Da\\ &\equiv-\frac Da\\ &=-\prod_i\ell_i^{z_i} \pmod m. \end{aligned}

3. Exact signed-box identity

Define the symmetric signed divisor box of a modulo m by

\boxed{ \mathcal R_m(a) = \left\{ \prod_i\ell_i^{z_i}\pmod m: -E_i\le z_i\le E_i \right\}.}

The centered exponent map above is a bijection between divisors D|a^2 and signed exponent vectors.

Therefore:

Theorem — square trap = negative signed box

\boxed{ S_a =-\mathcal R_{4a-1}(a).}

Equivalently,

\boxed{ p\bmod(4a-1)\in S_a \iff -p\in\mathcal R_{4a-1}(a).}

Thus the complete square-completed Type-II congruence layer is exactly a symmetric multiplicative product box.

4. López Type A and B are the two monotone orthants

Under

z_i=U_i-E_i,

the lower divisor condition

D\mid a

becomes

\boxed{z_i\le0\quad\text{for every }i.}

This is the Type-A orthant.

The upper divisor condition

a\mid D\mid a^2

becomes

\boxed{z_i\ge0\quad\text{for every }i.}

This is the Type-B orthant.

Hence the López Type-A/B trap is the restriction of the signed box to

\boxed{ [-E_1,0]\times\cdots\times[-E_s,0] \quad\cup\quad [0,E_1]\times\cdots\times[0,E_s].}

The complete Type-II square layer allows the entire centered box

\boxed{[-E_1,E_1]\times\cdots\times[-E_s,E_s].}

The omitted certificates are precisely the mixed-sign exponent vectors.

5. Inversion becomes z -> -z

Divisor complement

D\longmapsto\frac{a^2}{D}

sends

U_i\longmapsto2E_i-U_i,

so the centered coordinates transform as

\boxed{z_i\longmapsto-z_i.}

Therefore the complement/inversion theorem becomes the obvious central symmetry of the signed box:

\boxed{ \mathcal R_m(a)^{-1}=\mathcal R_m(a).}

Likewise Type A and Type B are exchanged by

z\longmapsto-z.

This recovers the López mutual-inverse theorem as orthant reflection.

6. The Kneser machinery applies directly

The box

\mathcal R_m(a) = \prod_i \{\ell_i^{-E_i},\ldots,1,\ldots,\ell_i^{E_i}\}

is exactly the kind of finite multiplicative product set treated by the repository's Kneser defect theory.

Let

G_m=(\mathbb Z/m\mathbb Z)^\times

and

H_a=\operatorname{Stab}(\mathcal R_m(a)).

Passing to

G_m/H_a

therefore imports the full-stabilizer order gap:

for every prime-power factor

\ell_i^{E_i}\parallel a

whose residue lies outside H_a,

\boxed{ \operatorname{ord}_{G_m/H_a}(\ell_iH_a)>2E_i+1.}

And any target miss can be studied by Kneser expansion exactly as in the fixed-shift FAB box.

This is the direct algebraic merger of the old shadow layer and the newer signed-box defect theory.

7. Prime-power layers revisited

a=\ell^E,

the signed box is one-dimensional:

\mathcal R_m(a) = \{\ell^z:-E\le z\le E\}.

Every exponent is either nonpositive or nonnegative. Therefore the two López orthants already fill the entire box.

This re-proves immediately that

\boxed{S_a=T_a}

for prime-power layers.

The mixed Type-II geometry is therefore exactly the higher-dimensional phenomenon

\boxed{\omega(a)\ge2.}

8. Example a = 12

For

a=12=2^2\cdot3, \qquad m=47,

the completed signed exponent box is

\boxed{(z_2,z_3)\in[-2,2]\times[-1,1].}

The López families use only

(z_2,z_3)\le(0,0)

coordinatewise or

(z_2,z_3)\ge(0,0)

coordinatewise.

The p=2521 mixed divisor D=16 has

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

\boxed{(z_2,z_3)=(2,-1).}

This is visibly a mixed-sign point.

Its signed-box value is

2^2\,3^{-1}\equiv17\pmod{47},

and therefore the trap residue is

-17\equiv30\pmod{47},

exactly the square-only residue that captures 2521.

9. Strategic consequence

The two major research languages can now be unified:

Shadow language

Layer a, modulus 4a-1, congruence trap containment, ancestry, collective cover.

Kneser language

Symmetric signed product box, stabilizer quotient, expansion defect, high-order exceptional atoms.

The identity

\boxed{S_a=-\mathcal R_{4a-1}(a)}

shows that these are the same completed Type-II layer.

The next proof program should therefore work on one object and use both toolkits:

shadow/CRT geometry across different layers;
Kneser expansion inside each completed layer;
complement/inversion to pair mixed candidates;
the exact mixed-parameter count to measure how much new mass is added beyond López A/B;
finite tests of whether the old Type-A/B zero-density survivor core disappears rapidly under the completed symmetric layers.

This is the cleanest current synthesis between the pre-DSC Type-A/B research and the post-DSC exact Type-I/II route.