Exact q=7 filter for Mordell-hard primes in the strong/Type-II corridor

Corridor

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Status: proved exact factorization criterion

Date: 2026-08-15

Depends on: STRONG-ES-FINITE-SHIFT-CORRIDOR.md, FAB-HARD-FIRST-FILTERS.md, STRONG-ES-MIZONY-THEPAULT-PROVENANCE.md

Claim boundary: classifies the fixed Type-II shift q=7 for Mordell-hard primes. It does not prove the strong conjecture or Erdős--Straus.

1. Fixed shift q = 7

Let p be Mordell-hard. Then

p\equiv1\pmod8.

Put

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

The exact fixed-shift Type-II criterion is

\boxed{-1\in\mathcal R_7(C).}

Because p!=7, one has

\gcd(C,7)=1.

The unit group

(\mathbb Z/7\mathbb Z)^\times

is cyclic of order six.

2. Residue classes by order

The nonzero classes modulo 7 split as follows:

\boxed{ \begin{array}{c|c|c} \text{residues} & \text{order} & \text{quadratic character}\\ \hline 1 & 1 & +\\ 6 & 2 & -\\ 2,4 & 3 & +\\ 3,5 & 6 & - \end{array}}

The Type-II target is

\boxed{-1\equiv6\pmod7,}

the unique order-two element.

3. Primitive-order-six valuation

Let

E_6(C) = \sum_{ r^e\parallel C, r\equiv3,5\ (7) } e

be the total valuation carried by primitive order-six prime factors.

The product of their local signed exponent intervals is, in additive C_6 notation,

\boxed{ P_E=\{-E,-E+1,\ldots,E\}\pmod6,}

where E=E_6(C).

Hence:

if E>=3, the primitive factors alone hit class 3, i.e. -1;
if E<=2, they alone miss -1.

4. An order-three factor plus a primitive factor forces a hit

Any prime factor in residue class

2\text{ or }4\pmod7

has order three.

Its signed local set already fills the order-three subgroup

\boxed{\{0,2,4\}\subset C_6.}

If even one primitive order-six factor is also present, its local set contains

\{0,1,5\}.

The sum is the whole group:

\boxed{ \{0,2,4\}+\{0,1,5\}=C_6.}

Therefore the target class 3 is hit.

Likewise any prime factor congruent to 6 mod7 hits -1 directly.

5. General q=7 miss classification

For an arbitrary integer C coprime to 7, the target -1 is missed exactly in one of the following two situations.

Quadratic-residue branch

Every prime factor of C belongs to

\boxed{1,2,4\pmod7.}

Then the entire signed box lies inside the quadratic-residue subgroup of order three and cannot contain -1.

Primitive sparse branch

Every prime factor outside class 1 belongs to

\boxed{3,5\pmod7,}

there are no factors in classes 2,4,6, and

\boxed{E_6(C)\le2.}

These are the only misses.

6. Mordell-hard parity kills the primitive sparse branch

Now use the hard-prime condition

p\equiv1\pmod8.

Then

p+7\equiv0\pmod8,

\boxed{2\mid C.}

But

2\pmod7

has order three.

Therefore C always contains a nontrivial order-three factor.

If any primitive order-six factor were also present, Section 4 would force a Type-II hit.

A factor 6 mod7 also forces a hit directly.

Hence the primitive sparse branch is impossible for a Mordell-hard miss.

We obtain the exact theorem.

7. Exact hard-prime q=7 theorem

Theorem

For a Mordell-hard prime p, put

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

Then

\boxed{ -1\notin\mathcal R_7(C) \iff \text{every prime factor of }C \text{ is a quadratic residue modulo }7.}

Equivalently,

\boxed{ q=7\text{ fails} \iff \operatorname{supp}(C) \subseteq \{\ell:\ell\equiv1,2,4\pmod7\}.}

Thus any prime factor

\ell\equiv3,5,6\pmod7

immediately yields a Type-II solution at shift 7.

8. Hard residue classes modulo 840

The six Mordell-hard classes have

p\pmod7\in\{1,2,4\}.

Since

C\equiv2p\pmod7,

the three pairs of hard classes give:

\boxed{ \begin{array}{c|c|c} p\bmod840 & p\bmod7 & C\bmod7\\ \hline 1,169 & 1 & 2\\ 121,289 & 2 & 4\\ 361,529 & 4 & 1 \end{array}}

These product residues are automatically compatible with the quadratic-residue-only miss branch.

9. Consecutive-neighbor form

Let

A=\frac{p+3}{4}.

Then

\boxed{C=A+1.}

The previous exact q=3 filter says a hard-prime miss at q=3 is equivalent to

\boxed{ \text{every prime factor of }A \text{ being }1\pmod3.}

The present theorem says a miss at the next corridor position q=7 is equivalent to

\boxed{ \text{every prime factor of }A+1 \text{ being a quadratic residue modulo }7.}

Thus a hypothetical strong/Type-II counterexample must satisfy both factorization restrictions on consecutive integers.

10. Strategic consequence

The first two corridor defects are now exact splitting conditions rather than approximate sieves:

\boxed{ \begin{array}{c|c|c} h & C_h & \text{required miss condition}\\ \hline 0 & A & \operatorname{supp}(A)\subseteq\{1\pmod3\}\\ 1 & A+1 & \operatorname{supp}(A+1)\subseteq\mathrm{QR}(7) \end{array}}

The next useful targets are q=11 and q=19, where hard congruences force small prime factors into the shifted integers and may similarly collapse the general Kneser defect to a pure higher-residue splitting condition.