Direct-Shadow Completeness counterexample — primitive q=3 triple cover

Status: exact counterexample to universal DSC-0 and DSC-P

Date: 2026-08-15

Claim boundary: this falsifies the universal Direct-Shadow Completeness implication and the proposed universal Class-C local-escape route. It does not falsify the Erdős-Straus conjecture. In fact the candidate progression below is covered by earlier Type A/B layers, so it supplies no ES counterexample.

Read with:

• DIRECT-SHADOW-COMPLETENESS.md
• Q3-ABSORPTION.md
• Q3-WEAK-REDUNDANCY.md
• Q3-POINTWISE-ABSORPTION.md
• Q3-FIBER-INJECTIVITY.md
• Q3-NEXT-DIGIT-NORMAL-FORM.md

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1. Result

There exists an admissible Mordell-hard Type A/B target candidate that is not directly shadowed by any single earlier layer, but is nevertheless covered by the union of three earlier q=3 layers.

Hence

Therefore both proposed universal implications

are false.

The counterexample sits inside the active fixed-negative/Class-C geometry: the three covering layers are fixed-only, Jacobi-negative, active, and have q=3.

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2. Constructed target

Take

Then

Let

and

Choose the target Type A trap

Because

the target trap is compatible with every Mordell-hard class

For the simplest representative choose

The candidate progression is

where CRT gives

and

Thus

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3. The three covering q=3 layers

Use the earlier layers

Their moduli are

For the target progression:

Hence all three pullback moduli are exactly

Layer 25

Modulo 99,

Therefore the three parameter classes give

Since

and the other two lifts are not in T_25,

Layer 70

Modulo 279,

The three lifts are

Since

we get

Layer 187

Modulo 747,

The three lifts are

Since

we get

Therefore

Every integer parameter s hits at least one earlier Type A/B layer.

So this target candidate is union-shadowed.

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4. The cover is pointwise primitive

The three trap atoms used above are exactly the kind left after the strong/weak/pointwise reductions:

• j=25: u=94=-5 mod 99;
• j=70: u=259=-4*5 mod 279;
• j=187: u=730=-17 mod 747.

Each is pointwise primitive relative to every earlier layer whose modulus divides m_j/3.

Their residues modulo 9 are

Thus they occupy the three distinct next 3-adic digits predicted by Q3-NEXT-DIGIT-NORMAL-FORM.md.

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5. The three rows are active fixed-negative rows

Their squarefree signatures are

because the factor 3^2 is square.

All three primes 11,31,83 divide the target modulus M, so all three rows are fixed-only in the character-shield sense.

For the chosen target residue,

Since each has q_j=3>1, all three belong to the active fixed-negative core

Thus this is not merely a non-character residual accident. It is a genuine shared Class-C active-core cover.

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6. Direct novelty — exhaustive exact verification

The remaining question is whether some single earlier layer directly shadows the target. It does not.

A direct shadow at layer j requires

Since

only layers with

can possibly directly shadow.

Two independent exhaustive implementations were used.

Verifier A — full j scan + exact divisor-count sieve

For every

a linear divisor-count sieve found

(attained at j=14,414,400).

Hence any direct shadow must satisfy

The implementation scanned all 15,290,695 earlier layers, retained only those meeting the exact cardinality condition, reconstructed every trap pullback, and found:

possible after q <= 2*tau(j): 297
actual direct shadows:           0

The same result holds for each of the six hard classes.

Verifier B — independent square-root envelope

A separate Python control flow does not use the maximum-tau sieve.

It uses only the elementary bound

so a direct shadow must satisfy

It scans all earlier j for this coarse necessary condition, obtaining 111,057 candidates; independently factors those j, applies the exact q_j<=2*tau(j) test, leaving the same

possible layers; and reconstructs their exact pullbacks.

Result:

Therefore the candidate is directly novel but union-shadowed.

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7. How the counterexample was constructed

This was not obtained by blindly scanning fifteen million depths.

The primitive q=3 atoms impose congruences on the target divisor d.

Choose:

• row 25, atom -5, giving d=5 mod 11;
• row 70, atom -4*5, giving d=20 mod 31;
• row 187, atom -17, giving d=17 mod 83.

CRT gives

with least positive solution d=764.

The complementary target factor must satisfy the inverse factor-pair congruences

whose least positive solution is

Then

and automatically

The q=3 triple cover is therefore a CRT/factor-pair construction arising directly from the primitive next-digit theory.

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8. Consequences for the research architecture

Falsified

The following universal targets must no longer be used:

and

In particular:

• universal DSC-0 is false;
• universal DSC-P is false;
• universal shared-CN local escape is false;
• the proposed universal Class-C implication direct novelty -> local escape is false.

Not falsified

The following remain valid within their stated boundaries:

• all frozen finite DSC certificates through k<=1500;
• C1 single-layer escape;
• C2-coprime and CN-coprime CRT results;
• strong q=3 absorption;
• weak q=3 redundancy;
• pointwise q=3 absorption;
• q=3 fiber injectivity;
• q=3 next-digit normal form;
• ancestry rigidity theorems;
• character-shield completeness.

The counterexample appears far beyond the previous finite frontier and is fully consistent with those bounded certificates.

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9. Consequence for Erdős-Straus

This counterexample is not bad news for the Erdős-Straus conjecture itself.

The target progression is not an uncovered family. It is the opposite: every parameter is caught by one of the earlier Type A/B layers 25,70,187.

So the lesson is architectural:

collective Type A/B shadowing is real, and it can be constructive rather than pathological.

The research program should pivot from trying to prove that collective shadows never occur to classifying and exploiting collective shadows as an additional coverage mechanism.

That may strengthen the route toward López-all-primes, because a target candidate need not possess an exact-depth realization in order for its primes to be covered by Type A/B; it may instead collapse collectively into earlier covered layers.

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10. New frontier

The immediate replacement for DSC is a collective-shadow classification program:

1. classify primitive q=3 triple covers by CRT/factor-pair data;
2. determine when such covers are themselves forced for remaining Type A/B target candidates;
3. combine direct shadows, collective shadows, character shields, and explicit realizations into a complete prime-coverage theorem;
4. attack the López remainder directly rather than requiring every novel candidate to realize an exact-depth prime family.

The universal ES wall remains exactly where it should: