Five-halves sieve bound from the exact q=3, q=7, and q=11 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-SIEVE.md, STRONG-ES-Q11-EXACT-FILTER.md

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

Claim boundary: classical Erdős--Straus exceptional-set results are stronger when larger parametric families are used. This note records the specific quantitative consequence of the three exact fixed Type-II shifts 3,7,11; it does not prove universal strong/Type-II coverage.

1. Consecutive corridor variables

Let

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

The first three prime shifts in the Type-II corridor are

\boxed{ \begin{array}{c|c} q & C=(p+q)/4\\ \hline 3 & A\\ 7 & A+1\\ 11 & A+2. \end{array}}

For a simultaneous survivor:

A has only prime factors 1 mod3;
A+1 has only quadratic-residue prime factors mod 7;
A+2 lies in one of the two exact q=11 miss branches.

We sieve each q=11 branch separately.

2. Baseline dimension from primality, q=3, and q=7

For every large sieve prime ell, the first two-filter argument forbids:

one residue making p=4A-3 divisible by ell;
residue A=0 for the half of primes with ell=2 mod3;
residue A=-1 for the half with (ell/7)=-1.

These residues are distinct away from finitely many small primes.

Therefore the baseline sieve dimension is

\boxed{2.}

and the baseline survivor count is

\ll X/(\log X)^2.

3. q=11 Branch A adds half a sieve dimension

Branch A requires every prime factor of

A+2

to be a quadratic residue modulo 11.

Thus for every sieve prime ell with

\left(\frac\ell{11}\right)=-1,

the residue

\boxed{A\equiv-2\pmod\ell}

is forbidden.

Quadratic nonresidue primes modulo 11 have relative density

\boxed{1/2.}

among primes away from 11.

The residue -2 is distinct from the three baseline forbidden residues for every sufficiently large ell; collisions occur only at finitely many primes and do not affect sieve dimension.

Hence Branch A has dimension

\boxed{ 2+\frac12=\frac52.}

The upper-bound sieve gives

\boxed{ N_A(X) \ll \frac{X}{(\log X)^{5/2}}.}

4. q=11 Branch B is even thinner

The exact thin Branch B requires:

v_3(A+2)=1;
every other QR factor is 1 mod11;
no prime factors 7,8,10 mod11;
the only allowed nonresidue classes are 2,6 mod11, with total valuation at most two.

For an upper bound we may discard the valuation restrictions and enlarge the set.

After ignoring the special fixed prime 3, every prime divisor of A+2 is then allowed only in the three classes

\boxed{1,2,6\pmod{11}.}

Therefore primes in the seven reduced classes

\boxed{3,4,5,7,8,9,10\pmod{11}}

are forbidden as divisors of A+2.

These classes have prime density

\boxed{7/10.}

by the prime number theorem in arithmetic progressions.

Thus Branch B has sieve dimension at least

\boxed{ 2+\frac7{10}=\frac{27}{10}.}

Consequently

\boxed{ N_B(X) \ll \frac{X}{(\log X)^{27/10}}.}

The actual exact Branch-B set is smaller because the ignored primitive valuation mass is capped by two.

5. Combined three-shift theorem

Every simultaneous q=3,7,11 miss belongs to Branch A or Branch B at q=11.

Therefore

\begin{aligned} N_{3,7,11}(X) &\le N_A(X)+N_B(X)\\ &\ll \frac{X}{(\log X)^{5/2}} + \frac{X}{(\log X)^{27/10}}. \end{aligned}

The first term dominates.

Hence:

Theorem — three fixed Type-II shifts leave a five-halves-dimensional sifted set

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

The same bound holds after restriction to the six Mordell-hard classes.

6. Relative prime density

Since

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

the relative density of triple survivors among primes is

\boxed{ O\left((\log X)^{-3/2}\right).}

Thus the first three prime Type-II shifts alone capture all but a very thin relative subset of primes.

Again, this does not approach the strength of the best classical full-ES exceptional-set bounds. Its value is that it arises from three explicit exact strong/Type-II corridor positions.

7. Local-density interpretation

The dominant Branch-A sieve dimension decomposes as

\boxed{ 1 +\frac12 +\frac12 +\frac12 = \frac52.}

The four pieces are:

primality of 4A-3;
inert-prime exclusion from A modulo 3;
quadratic-NR exclusion from A+1 modulo 7;
quadratic-NR exclusion from A+2 modulo 11.

The three shifted factor restrictions occur at distinct residues

0,-1,-2\pmod\ell

for almost every sieve prime, so their local dimensions add cleanly.

8. Prior-art boundary

Vaughan, Elsholtz, and later work obtain substantially stronger exceptional-set estimates for Erdős--Straus using broader parametric solution families and deeper sieve arguments.

No novelty claim is made for:

the upper-bound sieve;
density-zero exceptional sets;
the general use of several parametric families to increase sieve dimension.

The specific contribution here is the transparent translation of the exact q=3,7,11 Type-II factorization filters into an additive sieve-dimension ledger.

9. Next target

The corridor now suggests a systematic program.

For each small prime shift

q\equiv3\pmod4,

classify the exact miss into:

a main splitting branch excluding a positive-density set of prime divisors of (p+q)/4;
finitely many low-entropy defect branches.

If the main splitting density and every defect-branch density can be quantified uniformly, each new corridor position can add positive sieve dimension.

The immediate algebraic targets are q=19 and q=23.