Direct-Shadow Completeness attack through k = 1500

Status: exact finite theorem-certificate result; universal theorem remains open

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this record does not prove universal Direct-Shadow Completeness, López Type A/B coverage, or the Erdős-Straus conjecture.

This record supersedes DIRECT-SHADOW-K1200.md as the latest fully frozen all-stage candidatewise certificate frontier.

1. Completed workflow provenance

GitHub Actions run:

run id:       31849103304
head commit:  c508994fb48e6f701f15577352f275df5646cd78
artifact id:  9238241616

Artifact ZIP digest:

sha256:e181a66bec9a8e0d68b4b6b46892b6c71c50ebe8ab64c45944c8c17408c983dd

Configured range:

k_limit:      1500
search_limit: 3000000

Every workflow stage completed successfully: candidate attack, independent verifier, coordinate-core mining, coarse peeling, fiber peeling, bounded residual selector, quadratic-character analysis, CENTL certification, SHA-256 freezing, and artifact upload.

2. Exact candidatewise result

admissible candidates:             73,814
directly shadowed candidates:      20,574
directly novel candidates:         53,240
integer avoiding witnesses:        53,240
reduced avoiding witnesses:        53,240
unresolved integer candidates:          0
unresolved reduced candidates:          0

Therefore

directly novel hard-compatible candidates through k<=1500 are explicitly not covered by the union of all earlier Type A/B layers.

More strongly, every one has a reduced avoiding progression, and hence an infinite exact-depth prime progression by Dirichlet.

This is an exact finite-range certificate statement only.

3. Independent verification

The independent verifier returned:

{
  "direct_novel_candidates_checked": 53240,
  "integer_witnesses_verified": 53240,
  "k_limit": 1500,
  "reduced_witnesses_verified": 53240,
  "unresolved_integer_candidates": 0,
  "unresolved_reduced_candidates": 0,
  "verdict": "VERIFIED"
}

No candidatewise union-shadow counterexample appeared through depth 1500.

4. Hardest first reduced witnesses

The largest first reduced parameters include:

k	h	t	first reduced s	x = r + Ls
1488	121	5858	2,664,772	13,320,769,304,761
1443	121	5768	2,548,375	12,353,606,223,961
1338	289	5349	2,277,454	10,236,795,302,449
1488	169	5920	2,275,978	11,377,254,388,249
1285	121	4111	2,208,004	3,177,141,479,521
1380	529	5507	2,158,986	10,008,973,062,169

The finite survival is therefore not an artifact of tiny parameter choices.

5. Prime-power coordinate locality

For the 53,240 directly novel candidates:

canonical unary-safe assignment solves: 17,776 / 53,240 = 33.388%
maximum guided repair count:             10 prime-power coordinates

Cumulative guided upper bounds:

0 changes: 33.388%
<=1:       66.405%
<=2:       85.804%
<=3:       94.701%
<=4:       98.131%
<=5:       99.435%
<=6:       99.853%
<=7:       99.964%
<=8:       99.992%
<=9:       99.996%
<=10:     100.000%

These guided counts are proof-mining upper bounds, not proven minima.

6. Exact coarse and fiber peeling

The exact prime-power load criterion alone fully resolves

of candidates. Its largest residual kernel has 28 prime coordinates, largest residual prime 109, and the conservative universal first-bound leaves only primes at most 139 potentially non-peelable before the true candidate geometry is used.

The sharper exact augmented fiber peeling resolves

with an empty residual kernel.

Across all candidates, the largest residual fiber kernel has 9 prime coordinates and the largest residual prime observed is

The kernel-size distribution is:

size 0: 26,532
size 2:     28
size 3:  3,996
size 4:      6
size 5:    384
size 6:  1,582
size 7: 20,274
size 9:    438

Thus the theorem-driven reduction continues to collapse a global system with hundreds of earlier layers to a tiny small-prime interior.

7. Bounded-selector result

The workflow then tests the fixed residual selector menu

All nonempty fiber kernels were solved:

Therefore the independent two-stage construction

resolves

directly novel candidates through k<=1500, without consulting the stored sequential witness to decide either stage.

Unresolved bounded-selector kernels:

The largest selector radius actually required remains

the same maximum observed in the earlier k<=1200 replay.

8. Quadratic-character layer

The exact trap-signature checker performed

explicit Jacobi trap checks and all passed.

The scalar character shield independently certifies

of the directly novel candidates.

The remaining character residual is

The stronger theorem in CHARACTER-SHIELD-COMPLETENESS.md explains why collective scalar-character inconsistency cannot arise except from a fixed-negative earlier layer. These residuals therefore mark where exact trap geometry, higher local signatures, multiplicative quotients, square-lift structure, and p-adic refinement are actually needed.

9. Relation to the theorem program

The exact finite frontier now reads:

k <= 600:   19,016 / 19,016 directly novel candidates reduced-realizable
k <= 1000:  33,644 / 33,644
k <= 1200:  41,470 / 41,470
k <= 1500:  53,240 / 53,240

No candidatewise collective-shadow counterexample has appeared.

More importantly, the latest range is not supported only by sequential witness search. Every directly novel candidate is also independently resolved by the theorem-driven fiber-peeling plus bounded-selector construction.

The immediate universal target remains

for all admissible Type A/B candidates.

The active proof route is now to explain why the residual fiber kernels remain confined to small prime coordinates and why the selector radius remains bounded, using the quadratic-signature, multiplicative-quotient, reciprocity, square-lift, and exact two-box trap structure already recorded elsewhere in this directory.