Dimension-two sieve from the exact q=3 and q=7 strong/Type-II filters

Corridor

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Status: proved application of a classical upper-bound sieve

Date: 2026-08-15

Depends on: FAB-HARD-FIRST-FILTERS.md, STRONG-ES-Q7-EXACT-FILTER.md, STRONG-ES-FINITE-SHIFT-CORRIDOR.md

Imported classical tools: Selberg/Brun upper-bound sieve, prime number theorem in arithmetic progressions, bounded prime reciprocal sums for nonprincipal Dirichlet characters

Claim boundary: the sieve method and almost-all Erdős--Straus philosophy are classical; Vaughan and later Elsholtz prove much stronger exceptional-set estimates using larger parametric families. This note records the specific consequence of the exact q=3,7 Type-II filters. It does not prove universal strong/Type-II coverage.

1. The two exact miss conditions

Let p be Mordell-hard and put

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

Then

\frac{p+7}{4}=A+1.

The exact q=3 theorem gives

\boxed{ q=3\text{ misses} \iff \text{every prime divisor of }A\text{ is }1\pmod3.}

The exact q=7 theorem gives

\boxed{ q=7\text{ misses} \iff \text{every prime divisor of }A+1\text{ is a quadratic residue mod }7.}

Thus a simultaneous survivor must satisfy both splitting restrictions on two consecutive integers.

2. Convert the conditions into forbidden sieve residues

For every prime ell>7, define three possible forbidden residues for the variable A.

Primality of p = 4A-3

A\equiv 3\cdot4^{-1}\pmod\ell,

then

4A-3\equiv0\pmod\ell.

Except for the single possibility p=ell, such an A cannot correspond to a prime p.

Thus primality contributes one forbidden residue for every sieve prime.

q = 3 miss

\ell\equiv2\pmod3,

then a simultaneous survivor cannot satisfy

A\equiv0\pmod\ell,

because such an ell is forbidden as a prime divisor of A.

q = 7 miss

Let

\chi_7(\ell)=\left(\frac\ell7\right).

\chi_7(\ell)=-1,

then a simultaneous survivor cannot satisfy

A\equiv-1\pmod\ell,

because ell would be a quadratic-nonresidue prime divisor of A+1.

For every ell>7, the three candidate residues

3\cdot4^{-1},\qquad0,\qquad-1

are pairwise distinct whenever their corresponding conditions are active.

3. Local sieve dimension

For primes ell>7, let

\rho(\ell) = 1 + \mathbf1_{\ell\equiv2\ (3)} + \mathbf1_{(\ell/7)=-1}.

This is the number of distinct forbidden residue classes modulo ell.

Write chi_3 for the nonprincipal quadratic character modulo 3. Away from the finitely many ramified primes,

\mathbf1_{\ell\equiv2\ (3)} = \frac{1-\chi_3(\ell)}2,

and

\mathbf1_{(\ell/7)=-1} = \frac{1-\chi_7(\ell)}2.

Therefore

\boxed{ \rho(\ell) = 2-rac{\chi_3(\ell)+\chi_7(\ell)}2.}

Summing over primes and using the boundedness of prime reciprocal sums of nonprincipal Dirichlet characters gives

\boxed{ \sum_{\ell<z}\frac{\rho(\ell)}\ell = 2\log\log z+O(1).}

Thus the simultaneous primality-plus-two-filter problem has sieve dimension

\boxed{\kappa=2.}

4. Selberg upper bound

Apply the standard upper-bound Selberg sieve to the interval

1\le A\le Y

with the residue sets above.

The local density product satisfies

\prod_{\ell<z} \left(1-\frac{\rho(\ell)}\ell\right) \asymp \frac1{(\log z)^2}.

Taking a fixed positive power of Y as the sieve level gives

\boxed{ \#\left\{ A\le Y: \begin{array}{l} 4A-3\text{ prime},\\ q=3\text{ misses},\\ q=7\text{ misses} \end{array} \right\} \ll \frac{Y}{(\log Y)^2}.}

The finitely many small sieve primes and the exceptional equality 4A-3=ell contribute only lower-order terms.

5. Prime formulation

Since

p=4A-3,

the same estimate becomes

\boxed{ \#\{p\le X:\ p\text{ prime and both }q=3,7\text{ Type-II shifts miss}\} \ll \frac{X}{(\log X)^2}.}

Restricting further to the six Mordell-hard residue classes can only decrease the count.

Thus the exact q=3 and q=7 Type-II families alone leave at most a dimension-two sifted prime set.

6. Relative prime density

The prime number theorem gives

\pi(X)\sim\frac{X}{\log X}.

Hence

\frac{ \#\{p\le X:\ q=3,7\text{ both miss}\} }{\pi(X)} \ll \frac1{\log X} \longrightarrow0.

Therefore:

Theorem — two fixed Type-II shifts solve a relative density-one set of primes

\boxed{ \text{The union of the }q=3\text{ and }q=7\text{ Type-II families captures a relative density-one set of primes.}}

This is a statement about those two explicit strong/Type-II families, not about a universal proof.

7. Why the sieve dimension is exactly two

The local arithmetic has a useful interpretation.

Among large primes ell:

primality of 4A-3 forbids one residue universally;
half of the primes also forbid A=0 through the mod-3 splitting condition;
half also forbid A=-1 through the mod-7 quadratic condition;
one quarter satisfy both splitting obstructions and therefore forbid two extra residues.

The average number of forbidden classes is

1 +\frac12 +\frac12 =2.

Equivalently, sorting primes into the four independent character combinations gives

\frac14(1)+\frac14(2)+\frac14(2)+\frac14(3)=2.

8. Prior-art boundary

Almost-all results for Erdős--Straus are classical and substantially stronger than this specific bound.

Vaughan's 1970 work and later Elsholtz parametric-sieve results produce much thinner exceptional sets using richer families of solutions.

Therefore the responsible interpretation is:

The exact FCF q=3,7 factorization filters fit naturally into classical sieve theory and, by themselves, already have enough combined local dimension to capture a relative density-one set of primes.

No novelty claim is made for the sieve method or for density-one solvability in general.

9. Next analytic target

The q=11 exact filter adds a third consecutive shifted integer A+2.

Its main branch is another half-density quadratic splitting condition, while its exceptional branch contains at most two units of tightly prescribed nonresidue valuation.

A natural next target is to prove a three-position bound of the shape

\boxed{ \#\{p\le X:\ q=3,7,11\text{ all miss}\} \ll \frac{X(\log\log X)^{O(1)}}{(\log X)^{5/2}},}

or stronger, by treating the thin q=11 defect packet separately.