Multiplicative-coset and Jacobi shields survive square completion

Geometry

---

Status: proved exact containment theorem

Date: 2026-08-15

Depends on: ES-SQUARE-TRAP-SIGNED-BOX-IDENTITY.md, MULTIPLICATIVE-TRAP-COSET.md, QUADRATIC-TRAP-SIGNATURE.md

Claim boundary: shows that the existing coarse multiplicative/character shields remain valid for the full square-completed Type-II layer. It does not prove exact completed coverage or Erdős--Straus.

1. Setup

Fix

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

and let

G=(\mathbb Z/m\mathbb Z)^\times.

Define the divisor-generated subgroup

\boxed{ H_a = \langle \ell\bmod m: \ell\text{ prime},\ \ell\mid a\rangle \le G.}

The full square-completed Type-II layer is

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

By the signed-box identity,

\boxed{ S_a =-\mathcal R_m(a),}

where

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

2. Every completed signed divisor lies in the divisor-generated subgroup

Every prime base occurring in the signed product belongs to H_a.

Because H_a is a group, it contains all positive and negative powers of those bases.

Therefore

\boxed{ \mathcal R_m(a)\subseteq H_a.}

Multiplying by -1 gives:

Theorem — completed trap-coset containment

\boxed{ S_a\subseteq-H_a.}

Thus square completion enlarges the exact trap only inside the same multiplicative coset envelope that already contained López Type A/B.

3. The Jacobi sign remains fixed negative

Let

\chi_a(x) = \left(\frac{x}{m}\right)

be the Jacobi character modulo m=4a-1.

The divisor-prime calculation from the earlier trap-signature work gives

\boxed{ \chi_a(\ell)=+1 \qquad(\ell\mid a).}

Hence chi_a is identically +1 on the subgroup generated by those primes:

\boxed{ \chi_a(h)=+1 \qquad(h\in H_a).}

Because

m\equiv3\pmod4,

one has

\boxed{ \chi_a(-1)=-1.}

Therefore every completed trap residue satisfies

\boxed{ \chi_a(x)=-1 \qquad(x\in S_a).}

So the complete Type-II layer remains entirely in the Jacobi-negative half of the unit group.

4. Exact hierarchy of safe-region tests

The complete square layer now fits into the same containment hierarchy as the old López layer:

\boxed{ T_a \subseteq S_a \subseteq -H_a \subseteq \{x\in G:\chi_a(x)=-1\}.}

Thus there are still three increasingly coarse resolutions:

exact López boundary trap T_a;
exact complete Type-II trap S_a;
multiplicative coset envelope -H_a;
Jacobi-negative half.

The important correction is that the exact layer in the middle is now the complete square-divisor Type-II layer rather than only the López boundary orthants.

5. Old coset shields remain sound

Any residue class outside

-H_a

is automatically safe from every square-completed Type-II certificate at layer a.

Likewise every Jacobi-positive residue is automatically safe.

Therefore all arguments that use only the outer multiplicative-coset or character obstruction remain valid after square completion.

What changes is only the exact occupancy inside the negative coset.

This means the pre-existing character quotient and multiplicative quotient machinery does not need to be discarded or re-proved from scratch.

6. What square completion changes

The old López trap occupies two monotone orthants of the signed exponent box.

The completion fills every mixed-sign exponent vector as well.

Hence square completion increases the density inside -H_a while preserving:

the same divisor-generated ambient subgroup;
the same Jacobi signature;
the same quotient-character safe regions.

Symbolically,

\boxed{ \text{completion changes exact occupancy, not the outer shield}.}

This is useful for the all-prime program because every coarse elimination theorem can be retained while the exact residual core is recomputed with the larger layer.

7. Power-of-two saturation

For

a=2^b,

the old Mersenne theorem gives

T_a=-H_a.

Since prime-power layers have

S_a=T_a,

one also has

\boxed{ S_{2^b}=-H_{2^b}.}

Thus the binary layers continue to saturate the complete multiplicative coset.

For general layers the inclusions may be strict.

8. Research consequence

The unified completed-layer proof architecture should now be

\boxed{ \text{Jacobi shield} \supseteq \text{multiplicative coset shield} \supseteq \text{exact completed signed box} \supseteq \text{López boundary orthants}.}

The proof search can therefore proceed coarse-to-fine:

discard candidates by Jacobi sign;
discard more by quotient characters of G/H_a;
apply Kneser/stabilizer structure to the exact symmetric box S_a;
use cross-layer ancestry only on the small exact residual.

This preserves the strongest parts of the pre-DSC obstruction theory while upgrading the exact layer to standard Type II.