Safe-prime shifts q = 23 mod 24 carry a forced nine-class Type-II obstruction

Corridor

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Status: proved uniform finite-group theorem

Date: 2026-08-15

Depends on: STRONG-ES-PRIME-SHIFT-KNESER-DICHOTOMY.md, STRONG-ES-Q23-EXACT-FILTER.md

Claim boundary: applies to any safe prime q=2ell+1 in the residue class 23 mod24. Infinitude of such safe primes is not known and is not assumed. The theorem gives a uniform local obstruction for every one that exists.

1. Safe-prime setup

Let

\boxed{q=2\ell+1}

with both q and ell prime, and assume

\boxed{q\equiv23\pmod{24}.}

Let p be Mordell-hard and put

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

Hard primes satisfy

p\equiv1\pmod{24}.

Hence

p+q\equiv24\equiv0\pmod{24}

and therefore

\boxed{6\mid C.}

So 2 and 3 are forced prime factors of the shifted integer.

2. Both forced factors generate the QR subgroup

The quadratic-residue subgroup

Q=(\mathbb Z/q\mathbb Z)^{\times 2}

has prime order

\boxed{|Q|=\ell.}

Because

q\equiv7\pmod8,

quadratic reciprocity gives

\left(\frac2q\right)=+1.

Because

q\equiv-1\pmod{12},

one has

\left(\frac3q\right)=+1.

Neither 2 nor 3 is 1 mod q.

Since Q has prime order, every nonidentity element generates it. Therefore

\boxed{ \operatorname{ord}_q(2) = \operatorname{ord}_q(3) =\ell.}

Use 2 as a generator of Q and write

\boxed{3\equiv2^c\pmod q}

for a unique nonzero

c\in\mathbb Z/\ell\mathbb Z.

3. The forced two-factor QR box

Using only one copy of each forced factor, their signed exponent choices give the subset

\boxed{ F_q = \{x+cy\pmod\ell: x,y\in\{-1,0,1\}\}.}

There are at most nine elements.

We prove there are exactly nine.

4. Any collision forces one of six tiny ratios

Suppose

x_1+cy_1 \equiv x_2+cy_2 \pmod\ell

for two distinct pairs in {-1,0,1}^2.

Then

\Delta x+c\Delta y\equiv0

with

\Delta x,\Delta y\in\{-2,-1,0,1,2\}.

If Delta y=0, then Delta x=0 because ell>=11, contradicting distinctness.

Therefore

\boxed{ c=-\Delta x/\Delta y.}

The only nonzero ratios obtainable from the difference set are

\boxed{ \pm1, \quad \pm2, \quad \pm\frac12.}

Thus a collision forces c into this six-element set.

5. Every exceptional ratio would force a small excluded prime q

Recall

3\equiv2^c\pmod q.

c = 1

Would give

3\equiv2\pmod q,

impossible.

c = -1

Would give

3\equiv2^{-1},

6\equiv1\pmod q,

forcing q=5.

c = 2

Would give

3\equiv4\pmod q,

impossible.

c = -2

Would give

3\equiv4^{-1},

12\equiv1\pmod q,

forcing q=11.

c = 1/2

Squaring gives

9\equiv2\pmod q,

forcing q=7.

c = -1/2

Squaring gives

9\equiv2^{-1}\pmod q,

18\equiv1\pmod q,

forcing q=17.

None is compatible with

q\equiv23\pmod{24}, \qquad q\ge23.

Therefore no collision occurs.

Theorem — forced QR nine-set

\boxed{|F_q|=9.}

6. Translate the nine QR classes into forbidden NR factors

The Type-II target is

-1.

Every quadratic nonresidue can be written uniquely as

\boxed{x=-h}

with

h\in Q.

If the shifted integer C has a prime factor in the NR class x=-h, then the signed local set of that factor contains both x and x^{-1}.

The target -1 is obtained from x exactly when the existing QR box contains

h^{-1},

and from x^{-1} exactly when it contains h.

The forced set F_q is symmetric under inversion in additive coordinates.

Therefore every NR class

\boxed{-h, \qquad h\in F_q}

forces an immediate Type-II hit using only the forced factors 2,3 and that one NR factor.

Since |F_q|=9, there are nine such forbidden NR prime residue classes.

7. Safe-prime miss dichotomy with a uniform exceptional obstruction

The safe-prime Kneser theorem says every miss is either:

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

In the second branch, the nine NR classes above are still impossible as prime factors of C.

Thus:

Theorem — forced-nine-class safe-prime obstruction

For every safe prime

q\equiv23\pmod{24},

a Mordell-hard Type-II miss satisfies one of:

Branch A

Every prime factor of (p+q)/4 is a quadratic residue modulo q.

Branch B

The signed box is aperiodic, and at minimum the nine quadratic-nonresidue residue classes

\boxed{-F_q}

are forbidden as prime divisors of (p+q)/4.

The branch may have additional restrictions; nine is a uniform guaranteed minimum.

8. Sieve contribution

Among the

q-1=2\ell

reduced prime residue classes modulo q:

Branch A forbids all ell NR classes, density 1/2;
Branch B forbids at least 9 classes, density

\boxed{9/(q-1).}

Therefore every miss branch contributes at least

\boxed{ \delta_q = \min\left(\frac12,\frac9{q-1}\right) = \frac9{q-1} }

for q>=23, before using any additional exact branch structure.

For q=23, the dedicated exact theorem is much stronger than this generic bound.

For q=47, the generic bound already contributes

\boxed{9/46}

of a sieve dimension.

9. Important logical boundary

No claim is made that infinitely many safe primes

q\equiv23\pmod{24}

exist.

That is a Sophie-Germain/safe-prime type problem and is open.

The theorem is conditional only in the harmless sense:

every safe prime in this progression that exists carries the stated nine-class obstruction.

Finite known shifts such as 23 and 47 can be used unconditionally in fixed-shift sieve arguments.