Prime-shift Kneser dichotomy for the classical strong/Type-II corridor

Corridor

---

Status: proved application of Kneser's theorem

Date: 2026-08-15

Depends on: STRONG-ES-FINITE-SHIFT-CORRIDOR.md, FAB-KNESER-FULL-STABILIZER-DEFECT.md

Claim boundary: organizes fixed prime-shift Type-II misses into a pure quadratic-splitting branch and higher even-index defects. It does not prove universal Type-II coverage.

1. Setup

Let

p\equiv1\pmod4

be prime and let

\boxed{q\equiv3\pmod4}

be a prime shift with

q\ne p.

Put

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

Then

\gcd(C,q)=1.

Let

G=(\mathbb Z/q\mathbb Z)^\times,

and define the symmetric signed divisor box

\boxed{ R=\mathcal R_q(C).}

The standard Type-II target is

\boxed{-1\in G.}

Let

\boxed{H=\operatorname{Stab}_G(R)}

and

\boxed{n=[G:H].}

Because

1\in R,

one has

\boxed{H\subseteq R.}

2. A Type-II miss forces odd stabilizer order

Assume

\boxed{-1\notin R.}

Since

q\equiv3\pmod4,

the group G has order

q-1=2m

with m odd.

The cyclic group G has a unique element of order two, namely -1.

Every even-order subgroup of a cyclic group contains that unique order-two element.

Therefore if H had even order, then

-1\in H\subseteq R,

contradicting the miss.

Hence:

Theorem — even defect index

Every prime-shift Type-II miss satisfies

\boxed{|H|\text{ odd}}

and therefore

\boxed{n=[G:H]\text{ even}.}

3. Index two is exactly the pure quadratic branch

Suppose

\boxed{n=2.}

Then H is the unique index-two subgroup of G, namely the quadratic residues

\boxed{Q=G^2.}

Because R is H-periodic and contains 1,

H\subseteq R.

If R contained any quadratic nonresidue, then H-periodicity would force the complete nonresidue coset into R.

That coset contains -1, contradiction.

Therefore

R=H=Q.

In particular every prime divisor of C, which appears individually in the signed box, is a quadratic residue modulo q.

Conversely, if every prime divisor of C is a quadratic residue, then the entire signed box lies in Q, so -1 is missed.

The full stabilizer may in that direction be smaller than Q unless the box fills Q; but the Type-II miss itself is automatic.

Thus the exact n=2 normal form is pure quadratic splitting.

4. Any nonresidue factor forces index at least six

Assume the Type-II target is missed and some prime divisor

r\mid C

is a quadratic nonresidue modulo q.

Then the signed box contains r, so it is not contained in the quadratic-residue subgroup.

Therefore the full stabilizer cannot have index two.

The defect index is even by Section 2.

Also index four is impossible because

v_2(q-1)=1,

so 4 does not divide q-1 and G has no subgroup of index four.

Hence:

Corollary — splitting-or-high-defect dichotomy

A prime-shift Type-II miss satisfies either:

every prime divisor of C is a quadratic residue modulo q; or
the full stabilizer index satisfies

\boxed{n\ge6\text{ and }n\text{ is even}.}

This is the universal form behind the explicit q=7,11,23 filters.

5. One-target Kneser budget

Factor

C=\prod_i r_i^{e_i}.

For each local factor define

s_i = \min\left( 2e_i+1, \operatorname{ord}_{G/H}(r_iH) \right).

Kneser's theorem gives

|R| \ge

|H|\left(1+\sum_i(s_i-1)\right).</div>

A Type-II miss removes at least one H-coset, so

|R|\le(n-1)|H|.

Hence

\boxed{ \sum_i(s_i-1) \le n-2.}

Thus every higher defect is a low-expansion factorization in the stabilizer quotient.

6. Safe-prime specialization

Suppose

\boxed{q=2\ell+1}

with ell an odd prime.

Then

|G|=2\ell.

The subgroup indices are only

1,2,\ell,2\ell.

Index 1 means the box is all of G and therefore hits.

Index ell is odd, impossible for a miss by Section 2.

Therefore a Type-II miss has only two possibilities:

\boxed{n=2}

\boxed{n=2\ell=q-1.}

That is:

Theorem — safe-prime miss dichotomy

For a safe prime shift

q=2\ell+1\equiv3\pmod4,

every Type-II miss is either:

the pure quadratic-splitting branch; or
a full-stabilizer-trivial aperiodic defect.

No intermediate quotient geometry exists.

7. The small filters as specializations

The shifts

q=7,11,23

are all safe primes:

7=2\cdot3+1, \qquad 11=2\cdot5+1, \qquad 23=2\cdot11+1.

Therefore each exact filter has the same architecture:

main branch: all prime factors of C are quadratic residues;
exceptional branch: the full signed box has trivial stabilizer.

The hard-prime congruences force small QR factors into C, which makes the aperiodic branch extremely small:

at q=7, the forced factor 2 eliminates it completely;
at q=11, forced 3 leaves only a two-unit primitive packet;
at q=23, forced 2,3 leave only a two-unit primitive packet in two residue classes.

This explains the parallel exact classifications without treating them as unrelated coincidences.

8. Research consequence

To find additional useful fixed shifts, prioritize primes

q\equiv3\pmod4

for which:

the possible stabilizer indices are sparse, ideally safe primes;
hard congruences force one or more generators of the QR subgroup into (p+q)/4;
those forced local intervals consume most of the aperiodic Kneser budget.

Such shifts contribute a clean half-density main splitting branch and only low-entropy exceptional branches to the consecutive-corridor sieve.