Exact two-target filter at the composite shift `k=15`

Status: proved exact group-theoretic filter

Date: 2026-08-15

Depends on: ES-TWO-TARGET-SIGNED-BOX-EQUIVALENCE.md, FAB-HARD-FIRST-FILTERS.md

Claim boundary: classifies the next corridor position after 3,7,11. It does not prove that a 3,7,11 survivor must hit at 15, and therefore does not prove Erdős--Straus.

1. Setup

Let p be Mordell-hard and put

Then C is an integer. Because p≡1\pmod8,

so

Hard residues modulo 5 are 1 or 4, so 5\nmid C. Also C\equiv1\pmod3 whenever A is supported on primes 1\bmod3, and in any case 3 need not divide C. Thus every odd prime factor of C is a unit modulo 15.

2. The two-primary subgroup

The unit group

has order eight. The residue 2 generates a cyclic subgroup of order four:

The two exact targets live outside H:

The six hard classes occupy only two residues modulo 15. Each is compatible with p≡1\pmod3 and p\equiv1 or 4\pmod5:

The Type-I targets are then

Neither 14 nor 11 lies in H.

3. The signed box stays in H exactly on H-supported factorizations

Theorem — H-trap

The signed divisor box of C modulo 15 is contained in H if and only if every prime factor of C is congruent to 1,2,4, or 8 modulo 15.

In that case the box equals H as soon as v_2(C)\ge2, and equals {1,2,4,8} or the three-element set {1,2,8} according as 4 is or is not independently generated; in all such cases

where H_{\mathrm{thin}}=\{1,2,8\} is the simple local set of the forced factor 2.

Proof

Every prime r\equiv1,2,4,8\pmod{15} is already an element of H, so the signed box they generate stays in H. Conversely a prime outside H contributes a residue in {7,11,13,14} and the box meets the complement. QED.

4. Combined miss on the H-trap

Theorem — H-trap misses both targets

If every prime factor of C lies in {1,2,4,8}\bmod{15}, then

Proof

The box lies in H, while both hard-class Type-I targets and the Type-II target lie in {11,14}=(\mathbb Z/15\mathbb Z)^\times\setminus H. QED.

Thus on the H-trap, Type I contributes no extra coverage.

5. Prime factors outside H are almost always immediate hits

Let r be an odd prime divisor of C.

r\equiv14\pmod{15}

Then -1 itself lies in the signed box. Type II hits.

r\equiv7\pmod{15}

One has 7^{-1}\equiv13, so the local set is {1,7,13}. The forced factor 2 multiplies 7 to 14. Type II hits.

r\equiv13\pmod{15}

The local set is again {1,7,13}, and 2\cdot7\equiv14. Type II hits.

r\equiv11\pmod{15}

Here 11^{-1}\equiv11, so the local set is {1,11}.

• If v_2(C)\ge2, then 4 lies in the box and 4\cdot11\equiv14. Type II hits.
• If v_2(C)=1 and every other prime factor lies in H, the box contains

and misses 14. Type II therefore misses. Type I hits if and only if the target is 11, i.e. if and only if <div class="math" role="math">\boxed{p\equiv4\pmod{15}.}</div>

For p\equiv1\pmod{15} both targets miss.

The last bullet is the only combined-miss geometry that uses a prime outside H.

6. Exact combined miss theorem

Theorem

For a Mordell-hard prime p, both exact targets miss at k=15 if and only if one of the following holds:

1. H-trap: every prime factor of (p+15)/4 is 1,2,4, or 8\bmod{15};
2. thin 11-packet: v_2(C)=1, p\equiv1\pmod{15}, every nonresidue prime factor of C is 11\bmod{15}, and there is no prime factor 7,13,14\bmod{15}.

In every other case at least one of -1 and -p^{-1} lies in the signed box, and p satisfies Erdős--Straus.

7. Corridor position

The integer (p+15)/4 is the consecutive neighbour

of the three already-classified forms A, A+1, A+2. A hypothetical counterexample that has escaped q=3,7,11 must therefore place four consecutive integers in prescribed multiplicative semigroups, of which the fourth is the H-trap or the thin 11-packet above.

8. Finite signal

Through 500{,}000, every Mordell-hard 3,7,11 residual that misses k=15 does so by the H-trap; the thin 11-packet did not occur in that range. Through 2{,}000{,}000 the first two-target hit after a combined 3,7,11 miss is supported on

with 15, 19, and 23 accounting for the great majority. No residual in that range was unresolved by shift 59.

This is finite evidence only. The next exact target is a combined classification of the prime shift k=19.