Square-completed trap complement: global inversion and exact mixed-parameter count

Geometry

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Status: proved theorem

Date: 2026-08-15

Depends on: ES-SQUARE-COMPLETION-TRAP-GEOMETRY.md, ES-TYPEII-SQUARE-COMPLETION-LOPEZ-A.md

Claim boundary: this is an exact structural theorem for the square-completed Type-II layer. It counts divisor parameters, not necessarily distinct residue classes after modular collisions.

1. Setup

Fix

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

The square-completed Type-II layer is

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

The ordinary López Type-A/B boundary is represented by

\boxed{ D\mid a \quad\text{or}\quad a\mid D\mid a^2.}

The remaining divisors are the mixed cross-orthant parameters.

2. Divisor complement is residue inversion

For every divisor

D\mid a^2

define its complement

\boxed{D^*=\frac{a^2}{D}.}

The corresponding square-trap residues satisfy

\begin{aligned} (-4D)(-4D^*) &=16DD^*\\ &=16a^2\\ &=(4a)^2\\ &\equiv1\pmod{4a-1}. \end{aligned}

Therefore:

Theorem — global complement/inversion law

\boxed{ -4D^* \equiv (-4D)^{-1} \pmod{4a-1}.}

Thus the complete square-divisor layer is invariant under multiplicative inversion:

\boxed{S_a^{-1}=S_a.}

This is the divisor-lattice origin of the inverse symmetry already visible in the signed-box formulation.

3. López A/B inverse pairing is the boundary restriction

Take a lower-box divisor

e\mid a.

Its complement is

e^* =\frac{a^2}{e} =a\frac ae.

This belongs to the upper box.

The lower residue is the Type-A residue

\boxed{-4e.}

The complementary upper residue is

-4e^* =-4a\frac ae \equiv-\frac ae \pmod{4a-1},

which is a Type-B residue.

Hence:

Corollary — López mutual inversion is inherited from divisor complement

For every e|a,

\boxed{ (-4e)^{-1} \equiv -\frac ae \pmod{4a-1}.}

So the familiar Type-A/Type-B inverse relationship is not an isolated two-family phenomenon. It is exactly the restriction of the global involution

D\leftrightarrow a^2/D

on the complete Type-II square divisor lattice.

4. Exact number of boundary parameters

Let

\operatorname{Div}(a^2)

be the positive divisor set of a^2.

The lower region

L(a)=\{D:D\mid a\}

has cardinality

|L(a)|=\tau(a).

The upper region

U(a)=\{D:a\mid D\mid a^2\}

is the image of Div(a) under multiplication by a, so

|U(a)|=\tau(a).

Their intersection is exactly

L(a)\cap U(a)=\{a\}.

Therefore the López boundary parameter count is

\boxed{|L(a)\cup U(a)|=2\tau(a)-1.}

5. Exact mixed-parameter count

The complete square divisor lattice has size

\boxed{|\operatorname{Div}(a^2)|=\tau(a^2).}

Hence the number of mixed square-divisor parameters is exactly

\boxed{ M(a) = \tau(a^2)-2\tau(a)+1.}

This counts the square-completed Type-II divisor parameters that lie in neither López boundary orthant.

Again, this is a parameter count. Distinct mixed divisors may in principle collide modulo 4a-1.

6. Vanishing criterion

a=\ell^E

is a prime power, then

\tau(a)=E+1, \qquad \tau(a^2)=2E+1.

Thus

M(a) =(2E+1)-2(E+1)+1 =0.

Conversely, if a has at least two distinct prime factors, the exponent box has at least two coordinates. Choosing one coordinate strictly below its midpoint and another strictly above its midpoint produces a mixed divisor.

Therefore:

Theorem — exact mixed-region criterion

\boxed{ M(a)=0 \iff a\text{ is a prime power}.}

Equivalently,

\boxed{ M(a)>0 \iff \omega(a)\ge2.}

So multi-prime layers are exactly the layers where the complete Type-II divisor geometry contains parameter directions invisible to López A/B.

7. Mixed parameters occur in complement pairs

The complement involution preserves mixedness.

Indeed, write

a=\prod_i\ell_i^{E_i}, \qquad D=\prod_i\ell_i^{U_i}.

Then

D^* =\prod_i\ell_i^{2E_i-U_i}.

If D is mixed, some U_i<E_i and some U_j>E_j. Under complement those inequalities reverse, so D^* is again mixed.

The only fixed point of complement in the full divisor lattice is

D=a.

But a lies on both López boundaries and is not mixed.

Therefore complement acts without fixed points on the mixed region.

Corollary

\boxed{M(a)\text{ is even}.}

The genuinely new Type-II parameters arrive in inverse residue pairs.

8. Example a = 12

For

a=12=2^2\cdot3,

we have

\tau(12)=6, \qquad \tau(144)=15.

Hence

\boxed{M(12)=15-12+1=4.}

The mixed divisors are

\boxed{8,9,16,18.}

They pair under complement as

8\leftrightarrow18, \qquad 9\leftrightarrow16.

At modulus

4a-1=47,

the p=2521 certificate uses D=16, while its inverse mixed residue is represented by D=9.

This makes the square-only rescue a literal cross-orthant inverse pair.

9. Strategic consequence

The complete Type-II layer now has a clean three-part decomposition:

\boxed{ \text{lower López orthant} \ \cup\ \text{mixed inverse pairs} \ \cup\ \text{upper López orthant}.}

The old Type-A/B inverse theorem is the boundary shadow of a global complement symmetry, and the exact amount of missing divisor geometry is measured by

M(a)=\tau(a^2)-2\tau(a)+1.

This supplies natural quantitative inputs for the next shadow program:

measure how many mixed parameters survive hard-class admissibility;
classify modular collisions among the mixed inverse pairs;
determine whether mixed-pair arrival is concentrated on the old Type-A/B composite-rescue core;
extend ancestry/shadow reductions using complement symmetry to halve the new parameter search.