Erdős-Straus — Wall

Status: OPEN

Major route correction

Universal Direct-Shadow Completeness is no longer part of the wall because it is false.

DSC-COUNTEREXAMPLE.md gives a hosted-verified admissible hard candidate that is not directly shadowed but is collectively covered by three earlier q=3 rows. Thus DSC-0 and DSC-P are false as universal statements.

This does not move Erdős-Straus backward. A collectively shadowed target is already solved by an earlier Type A/B layer. Exact-depth realizability was stronger than pointwise ES coverage requires.

Closed / retained on this route

• Density-one Type A/B coverage of primes via the prime-modulus backbone
• Finite candidatewise DSC through k<=1500 as a finite theorem-certificate
• Exact reduced-parameter domain gcd(r+Ls,LQ)=1
• Strong q=3 absorption
• Weak q=3 redundancy
• Pointwise q=3 absorption
• q=3 singleton-pullback theorem
• Direct-shadow smoothness
• Explicit counterexample showing universal DSC-0 / DSC-P are false
• Ancestry rigidity and the existing Type A/B local structure
• Exact two-target corridor companions at q=3,7,11 and k=15,19, plus the linear form 2p+1
• {2,3,5,7}-smooth aligned Type-II covering is impossible (external prime ℓ≥11 is necessary)
• A public infinite hard-prime hunt with content-addressed letter numbers (ES-HUNT.md). Finite coverage, not a proof

Still required for the ES route

1. All-prime two-target coverage

Prove that every prime has some Type A/B decomposition, or more generally some two-target signed-box hit. The density-one theorem is not enough. López A/B would suffice but is stronger than original ES requires.

After the prime-modulus backbone, the pointwise burden is concentrated in the zero-density survivor core:

This is the principal current wall.

2. Composite n

Once every prime is solved, extend to arbitrary composite n. The standard divisor/scaling reduction means a solution for a divisor can be scaled to a solution for n; the repo should carry a clean self-contained proof when this stage is promoted into the final chain.

What is no longer required

The following are useful depth-spectrum questions but are not prerequisites for Erdős-Straus:

• universal DSC-0;
• universal DSC-P;
• proving every directly novel candidate has an exact-depth prime;
• eliminating every collective shadow.

These belong to the separate covering-core / exact-depth program.

Honest floor

A density-one set of primes is structurally captured. The remaining all-prime problem is a zero-density core: some later two-target shift must hit. Direct-shadow graphs alone cannot encode all redundancy, so future ES work should attack existence of some Type I or Type II hit, not universal exact-depth realizability. The next exact corridor target is the Type-I companion at q=23.