Constructive counterexample to universal Direct-Shadow Completeness

Shadow

Verifier:

Status: exact constructive counterexample; independently replayed and frozen in GitHub Actions

Date: 2026-08-15

Claim boundary: this falsifies the universal conjectures DSC-0 and DSC-P as stated in the research program. It does not falsify the Erdős-Straus conjecture, and it does not falsify López Type A/B coverage. The target candidate is collectively redundant rather than directly redundant.

Verifier:

verify_dsc_counterexample.py
DIRECT-SHADOW-SMOOTHNESS.md

Hosted certificate:

workflow run: 31862644146
head SHA:     ee08dd722164843da6cfa417fc6439d5ac09447d
artifact id:  9241048158
artifact digest:
sha256:7a8ee6c11d77b1d637238de10fae033be64b3099388c43e17942748b38b354df
verdict:      SUCCESS

1. The statements falsified

The program defined:

DSC-0

\text{not directly shadowed} \Longrightarrow \text{not union-shadowed}.

DSC-P

\text{not directly shadowed} \Longrightarrow \text{there exists a reduced avoiding parameter class}.

The candidate below is not directly shadowed, but three earlier q=3 rows cover every integer parameter. Therefore it has no integer avoiding parameter at all, hence certainly no reduced avoiding parameter.

Thus:

\boxed{\text{DSC-0 is false}}

and

\boxed{\text{DSC-P is false}.}

2. Constructed target

Set

D=2,218,779,486, \qquad N=1,900,986,818,

and

\boxed{ k=DN=4,217,870,554,934,815,548.}

Then

\boxed{ M=4k-1 =16,871,482,219,739,262,191. }

Its exact factorization is

\boxed{ M =19\cdot229\cdot433\cdot487\cdot3823\cdot4,809,977. }

All displayed factors are prime.

Because D|k, the residue

\boxed{ t\equiv-4D\pmod M}

is a Type A/B target trap. In least nonnegative form,

\boxed{ t=16,871,482,210,864,144,247.}

Also

\gcd(M,840)=1,

\boxed{ L=\operatorname{lcm}(840,M) =840M =14,172,045,064,580,980,240,440. }

Two Mordell-hard siblings work:

h=361 \qquad\text{and}\qquad h=529.

Their CRT bases are respectively

\boxed{ r_{361}=13,750,258,009,078,623,567,721}

and

\boxed{ r_{529}=2,412,621,957,413,839,375,369.}

Each satisfies

r_h\equiv h\pmod{840}, \qquad r_h\equiv t\pmod M, \qquad \gcd(r_h,L)=1.

Either sibling by itself falsifies DSC-0 and DSC-P.

3. Three earlier q=3 rows cover every parameter

Use the following earlier layers:

`j`	`m_j=4j-1`	trap divisor `e`	branch	trap `u`
6,820	27,279	20	`-4e`	27,199
8,602	34,407	506	`-e`	33,901
9,790	39,159	89	`-4e`	38,803

The trap divisibility is exact:

20  | 6820
506 | 8602
89  | 9790

and the moduli factor as

27279=3^2\cdot7\cdot433,

34407=3^2\cdot3823,

39159=3^2\cdot19\cdot229.

For the target L, their gcds are

9093, \qquad 11469, \qquad 13053,

so all three have

\boxed{q_j=3.}

Hard sibling h=361

The exact pullback classes are

j=6820  -> s = 1 mod 3
j=8602  -> s = 2 mod 3
j=9790  -> s = 0 mod 3

Hard sibling h=529

They are the cyclic shift

j=6820  -> s = 0 mod 3
j=8602  -> s = 1 mod 3
j=9790  -> s = 2 mod 3

Therefore in either candidate

\boxed{ R_{6820}\cup R_{8602}\cup R_{9790} =\mathbb Z/3\mathbb Z. }

By Q3-SINGLETON-PULLBACK.md, each row contributes only its displayed singleton class, but together the three singletons cover all classes.

Hence for every integer s, at least one earlier Type A/B row is hit:

\boxed{ \forall s\in\mathbb Z, \quad r_h+Ls \text{ lies in an earlier Type A/B layer.} }

This is an exact union shadow.

4. Why the direct-shadow check is finite

A direct shadow cannot contain any prime outside the support of L.

DIRECT-SHADOW-SMOOTHNESS.md proves:

\boxed{ \text{direct shadow by modulus }m \Longrightarrow \operatorname{rad}(m)\mid\operatorname{rad}(L). }

Reason: if p|m but p∤L, the attained fibre contains a value divisible by p, while every Type A/B trap is a unit modulo m.

Therefore every possible earlier direct-shadow modulus is smooth over the odd prime support

\boxed{ \{3,5,7,19,229,433,487,3823,4,809,977\}. }

There are exactly

\boxed{270,836}

positive smooth integers below M, of which exactly

\boxed{135,402}

are nontrivial and congruent to 3 mod 4, hence are possible earlier Type A/B moduli.

The verifier enumerates all of them.

For each modulus m=4i-1, it computes

g=\gcd(L,m), \qquad q=m/g,

and checks every one of the q attained fibre points directly against the exact Type A/B trap criterion.

Result for h=361:

possible smooth Type A/B direct-shadow moduli checked: 135,402
direct shadows: 0

Result for h=529:

possible smooth Type A/B direct-shadow moduli checked: 135,402
direct shadows: 0

Thus both candidates are directly novel.

5. Independent hosted replay

The GitHub Actions verifier reconstructs the target factorization, candidate CRT data, all three q=3 pullbacks, the complete smooth direct-shadow search space, and every attained fibre without consulting a stored verdict.

The first hosted run completed successfully:

run id:      31862644146
head:        ee08dd722164843da6cfa417fc6439d5ac09447d
artifact:    9241048158
archive sha: 7a8ee6c11d77b1d637238de10fae033be64b3099388c43e17942748b38b354df

Thus the counterexample is now frozen as a replayable theorem-certificate, not merely a local search observation.

6. Exact conclusion

For each of the two hard siblings:

the target is a valid admissible Type A/B candidate;
no single earlier layer directly shadows it;
three earlier q=3 layers jointly cover every integer parameter.

Therefore

\boxed{ \text{directly novel} \centernot\Longrightarrow \text{not union-shadowed}. }

and

\boxed{ \text{directly novel} \centernot\Longrightarrow \text{reduced prime-realizable}. }

The finite k<=1500 and related DSC certificates remain correct finite statements. What fails is their universal extrapolation.

7. What this means for the research program

This result removes the proposed DSC shortcut. The shadow graph is not a complete obstruction theory when only direct edges are retained.

The replacement object must preserve collective local covers. At minimum, the correct structure needs hyperedges / covering cores rather than only single-layer ancestry edges.

For the Erdős-Straus goal, however, this is not a negative result: a union-shadowed target candidate is already covered by earlier Type A/B decompositions. Exact-depth prime realizability was a stronger auxiliary objective than pointwise Type A/B coverage requires.

The immediate program should therefore split cleanly:

Depth-spectrum track: classify minimal union-shadow cores and replace DSC with a hypergraph/covering-core theory.
Erdős-Straus track: attack pointwise Type A/B coverage directly, especially the zero-density prime-modulus survivor/composite-rescue core.
Retain Strong Absorption, Weak Redundancy, Pointwise Absorption, singleton q=3 pullbacks, and direct-shadow smoothness as exact reduction tools.
Keep López-all-primes separate: this counterexample concerns minimal-depth realizability of one target candidate, not existence of some Type A/B decomposition for a prime.
Keep the Erdős-Straus wall honest until the all-prime/composite remainder is actually closed.

The project has learned something stronger than another finite survival bound: the universal bridge itself has been falsified constructively, and the ES route can now discard that unnecessary burden.