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Type-I companion to the exact `q=11` filter

Corridor · hosted from the CENTL repository

Research library · Corridor

Corridor

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Source in the repository

Status: proved exact companion

Date: 2026-08-15

Depends on: STRONG-ES-Q11-EXACT-FILTER.md, ES-TWO-TARGET-SIGNED-BOX-EQUIVALENCE.md, HARD-Q7-TYPE-I-NO-RESCUE.md

Claim boundary: classifies when Type I rescues a hard-prime Type-II miss at shift 11. It does not prove that one of the two targets always hits, and therefore does not prove Erdős--Straus.

1. Forced factor and the two boxes

Let p be Mordell-hard and

C=\frac{p+11}{4}.

Then 3\mid C and C is odd. Write Q for the quadratic-residue subgroup modulo 11:

Q=\{1,3,4,5,9\}.

The forced prime 3 generates Q. Its simple local set is

\{3^{-1},1,3\}=\{4,1,3\}.

Call this the thin QR box

Q_{\mathrm{thin}}=\{1,3,4\}.

The complementary QR classes are 5 and 9. The existing Type-II theorem says that if the total nontrivial QR valuation of C is at least two, then the signed box contains all of Q.

2. Location of the Type-I target

Because 11≡3\pmod4 and p≡1\pmod4,

\Bigl(\frac{-p^{-1}}{11}\Bigr) = -\Bigl(\frac p{11}\Bigr).

Thus the Type-I target is a quadratic residue if and only if p is a nonresidue modulo 11. The ten nonzero classes give:

\begin{array}{c|c|c} p\bmod11 & -p^{-1}\bmod11 & \text{side}\\ \hline 1 & 10 & \mathrm{NR}\\ 2 & 5 & \mathrm{QR}\\ 3 & 7 & \mathrm{NR}\\ 4 & 8 & \mathrm{NR}\\ 5 & 2 & \mathrm{NR}\\ 6 & 9 & \mathrm{QR}\\ 7 & 3 & \mathrm{QR}\\ 8 & 4 & \mathrm{QR}\\ 9 & 6 & \mathrm{NR}\\ 10 & 1 & \mathrm{QR} \end{array}

The Type-II target is always 10.

3. Full-QR Type-II misses

Theorem — full-QR rescue

Suppose every prime factor of C is a quadratic residue modulo 11, and the nontrivial QR valuation is at least two. Then the signed box equals Q, Type II misses, and Type I hits if and only if

\boxed{\Bigl(\frac p{11}\Bigr)=-1,}

equivalently p\equiv2,6,7,8,10\pmod{11}.

Proof

Valuation at least two fills Q by the existing q=11 lemma. The Type-II target 10 lies outside Q. The Type-I target lies in Q precisely on the nonresidue classes listed above. QED.

4. Thin QR Type-II misses

Theorem — thin-QR rescue

Suppose v_3(C)=1 and every other prime factor of C is 1\bmod{11}. Then the signed box equals the thin QR box {1,3,4}, Type II misses, and Type I hits if and only if

\boxed{p\equiv7,8,10\pmod{11}.}

Proof

The local set of 3 is exactly {1,3,4}, and primes 1\bmod{11} do not enlarge it. The Type-I target lies in that three-element set precisely for the three rows p\equiv7,8,10 of the table. QED.

The two remaining nonresidue classes p\equiv2,6 have Type-I targets 5 and 9, which lie in Q\setminus Q_{\mathrm{thin}}. Those classes are rescued only when extra QR valuation fills the whole subgroup.

5. Combined q=11 miss

Combining the Type-II classification with the two rescue theorems:

Theorem — combined hard-prime miss at 11

A Mordell-hard prime misses both exact targets at shift 11 if and only if it is a Type-II miss of one of the following kinds:

  1. full-QR miss with residue side: every prime factor of C is QR modulo 11, the QR box is full, and p is itself a quadratic residue modulo 11;
  2. thin-QR miss outside {7,8,10}: v_3(C)=1, every other QR prime is 1\bmod{11}, there is no primitive nonresidue packet, and p\not\equiv7,8,10\pmod{11};
  3. thin primitive Branch B: the existing Type-II Branch B holds, and the Type-I target additionally avoids the resulting signed box.

In particular Type I does contribute new hard-prime coverage at q=11, unlike q=3 and q=7.

6. Finite signal

Through 2{,}000{,}000 there are 4519 Mordell-hard primes. After the exact q=3 and q=7 combined misses, the Type-II q=11 filter solves 1057 further primes, and the Type-I companion solves an additional 13 primes that Type II missed. The remaining combined 3,7,11 core has size 711.

Those 13 Type-I-only rescues are the first concrete corridor primes at which the second target is essential for original Erdős--Straus rather than for the strong/Type-II conjecture.