Type I cannot rescue a hard-prime `q=7` Type-II miss

Corridor

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Status: proved exact companion to the q=7 Type-II filter

Date: 2026-08-15

Depends on: STRONG-ES-Q7-EXACT-FILTER.md, ES-TWO-TARGET-SIGNED-BOX-EQUIVALENCE.md

Claim boundary: identifies the first corridor position at which the two exact targets coincide in obstruction. It does not prove Erdős--Straus.

1. The two targets at shift `7`

Let p be Mordell-hard and put

C=\frac{p+7}{4}.

The unit group modulo 7 is cyclic of order six. Its quadratic-residue subgroup is

Q=\{1,2,4\}.

The Type-II target is

-1\equiv6\pmod7,

a nonresidue. The Type-I target is

-p^{-1}\pmod7.

2. Forced generator of `Q`

Hard primes satisfy p≡1\pmod8, so 2\mid C. The residue

2\pmod7

generates Q:

2,\qquad 2^2\equiv4,\qquad 2^3\equiv1.

Already the local signed set of a single factor 2 is

\{2^{-1},1,2\}=\{4,1,2\}=Q.

Thus for every Mordell-hard prime the signed box contains the full quadratic-residue subgroup.

3. Every hard class is a residue modulo `7`

The six Mordell-hard classes satisfy

p\bmod7\in\{1,2,4\}=Q.

Hence

\boxed{\Bigl(\frac p7\Bigr)=+1.}

Because 7≡3\pmod4, the Type-I target is then a nonresidue:

\Bigl(\frac{-p^{-1}}{7}\Bigr) = \Bigl(\frac{-1}{7}\Bigr) \Bigl(\frac p7\Bigr)^{-1} = (-1)\cdot(+1) = -1.

Concretely:

\begin{array}{c|c} p\bmod7 & -p^{-1}\bmod7\\ \hline 1 & 6\\ 2 & 3\\ 4 & 5 \end{array}

In the first row the two targets coincide. In the other two rows the Type-I target is a primitive order-six class.

4. Theorem

For a Mordell-hard prime, the following are equivalent:

-1\notin\mathcal R_7(C);
-p^{-1}\notin\mathcal R_7(C);
every prime factor of C is a quadratic residue modulo 7.

Proof

The existing q=7 theorem gives (1)\Leftrightarrow(3). Under (3) the signed box lies in Q, while both targets are nonresidues, so (3)\Rightarrow(2). Conversely, if some prime factor is a nonresidue then the Type-II theorem already produces -1 in the box, and inversion symmetry is not needed. If one prefers a direct Type-I check: a nonresidue factor together with the already-full subgroup Q fills the whole group, which contains both targets. QED.

5. Consequence

At the second corridor position, Type I contributes no additional hard-prime coverage beyond Type II. Combined Erdős--Straus failure at q=7 is exactly the already-classified Type-II miss.

The first corridor position q=3 is the same phenomenon: for hard p one has p≡1\pmod3, so -p^{-1}\equiv-1\pmod3 and the two targets coincide.

The next prime shift q=11 is different. There Type I can rescue a Type-II miss; see Q11-TYPE-I-COMPANION.md.

1. The two targets at shift 7