Cubic-log sieve bound from q=3,7,11,23 Type-II filters

Corridor

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Status: proved application of a classical upper-bound sieve

Date: 2026-08-15

Depends on: STRONG-ES-Q3-Q7-Q11-SIEVE.md, STRONG-ES-Q23-EXACT-FILTER.md

Imported classical tools: Selberg/Brun upper-bound sieve, prime number theorem in arithmetic progressions

Claim boundary: this is a specific quantitative consequence of four exact fixed Type-II filters. Classical full Erdős--Straus exceptional-set estimates are stronger. This does not prove universal strong/Type-II coverage.

1. Existing three-shift branches

Let

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

The exact q=3,7,11 analysis splits the survivor set into two q=11 branches.

q=11 Branch A

The sieve dimension is

\boxed{5/2.}

q=11 Branch B

After harmlessly enlarging the exact thin defect branch, the sieve dimension is at least

\boxed{27/10.}

Therefore every three-shift survivor belongs to a branch of dimension at least

\boxed{5/2.}

2. q=23 acts at a new shifted residue

The q=23 shifted integer is

\boxed{A+5.}

Thus every prime-factor exclusion at this position forbids the residue

\boxed{A\equiv-5\pmod\ell}

for the relevant sieve primes ell.

For all sufficiently large ell, this residue is distinct from:

the primality residue for 4A-3;
0, used by the q=3 filter;
-1, used by q=7;
-2, used by q=11.

Hence the local sieve dimensions add directly. The finitely many collision primes do not affect the logarithmic exponent.

3. q=23 Branch A adds one-half

The main q=23 miss branch requires every prime factor of A+5 to be a quadratic residue modulo 23.

Quadratic nonresidue primes modulo 23 have density

\boxed{1/2.}

among primes.

Therefore Branch A adds exactly one-half unit of sieve dimension.

Applied to the weaker existing q=11 Branch A:

\boxed{ \frac52+\frac12=3.}

This will be the dominant combined branch.

4. q=23 Branch B is substantially thinner

The thin q=23 branch allows ordinary prime divisors of A+5 only in the three residue classes

\boxed{1,5,14\pmod{23},}

apart from the fixed forced primes 2,3.

Thus it forbids

\boxed{19/22}

of the reduced prime residue classes modulo 23.

So any branch using the thin q=23 geometry gains at least

\boxed{19/22>1/2}

of sieve dimension.

The exact valuation cap on the two allowed nonresidue classes makes the true branch even smaller.

5. Four branch combinations

Combine the two q=11 branches with the two q=23 branches.

The resulting lower bounds for sieve dimension are:

\boxed{ \begin{array}{c|c|c} q=11 & q=23 & \text{dimension lower bound}\\ \hline A & A & 5/2+1/2=3\\ A & B & 5/2+19/22>3\\ B & A & 27/10+1/2=16/5>3\\ B & B & 27/10+19/22>3. \end{array}}

Thus every simultaneous four-shift survivor belongs to a sieve problem of dimension at least three.

6. Main theorem

Applying the classical upper-bound sieve branchwise and summing the four bounds gives:

Theorem — four fixed Type-II shifts leave a dimension-three prime set

\boxed{ \#\{p\le X:\ p\text{ prime and }q=3,7,11,23\text{ all miss}\} \ll \frac{X}{(\log X)^3}.}

The same estimate holds after restricting to Mordell-hard primes.

7. Relative prime density

Since

\pi(X)\sim X/\log X,

the relative density of four-shift survivors among primes satisfies

\boxed{ O\left((\log X)^{-2}\right).}

Thus four explicit small Type-II shifts already remove all but a doubly-logarithmically thin relative subset of primes.

8. Interpretation

The logarithmic exponent has increased as follows:

\boxed{ \begin{array}{c|c} \text{fixed shifts used} & \text{upper-bound sieve dimension}\\ \hline 3,7 & 2\\ 3,7,11 & 5/2\\ 3,7,11,23 & 3. \end{array}}

Each useful corridor position contributes an explicit positive-density prime-factor exclusion on a new shifted integer A+h.

The exceptional finite-group defect branches do not reduce the current dominant exponent because they are more restrictive than the main splitting branches.

9. Prior-art boundary

The method remains classical sieve theory, and the best known full-ES exceptional-set estimates are much stronger.

The point of the present calculation is structural:

the exact fixed-shift Type-II classifications produce a transparent local density ledger;
the shifted residues 0,-1,-2,-5 are distinct;
the sieve dimensions add visibly.

This creates a bridge between the finite-group signed-box analysis and analytic exceptional-set estimates.

10. Next target

Search for additional corridor primes q for which hard congruences force enough quadratic-residue generators into (p+q)/4 that every miss branch excludes a uniformly positive density of prime residue classes.

If an infinite sequence of such shifts can be controlled uniformly, the fixed-shift sieve dimension would grow without bound. Turning that growth into a universal theorem would require constants and uniformity far beyond the present fixed-family argument, but it is now a precise analytic direction.