Direct-Shadow Completeness attack through k = 1200

Shadow

This record extends the candidatewise falsification attack through k=1200 and freezes the first full automated prime-power coordinate-core diagnostic.

Source in the repository

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

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Research lineage: DIAMOND.md -> DIRECT-SHADOW-COMPLETENESS.md -> DIRECT-SHADOW-K1000.md -> this record

This record extends the candidatewise falsification attack through k=1200 and freezes the first full automated prime-power coordinate-core diagnostic.

It does not prove universal Direct-Shadow Completeness, López Type A/B coverage, or the Erdős-Straus conjecture.

1. Exact candidatewise result

The completed GitHub Actions run used:

all six Mordell-hard classes modulo 840;
every Type A/B layer through k=1200;
every prime-compatible admissible target residue;
witness search through s=2,000,000;
an independent modular verifier;
the prime-power coordinate proof-mining analyzer;
CENTL exact certification of selected hardest CRT progression identities;
SHA-256 freezing and artifact publication.

The exact result was:

admissible candidates:             57,367
directly shadowed candidates:      15,897
directly novel candidates:         41,470
integer avoiding witnesses:        41,470
reduced avoiding witnesses:        41,470
unresolved integer candidates:          0
unresolved reduced candidates:          0

Therefore

\boxed{ 41,470/41,470 \text{ directly novel candidates through }k=1200 \text{ are explicitly not union-shadowed.} }

And more strongly,

\boxed{ 41,470/41,470 \text{ have reduced avoiding progressions.} }

Every one therefore has an explicit Dirichlet progression containing infinitely many primes whose first Type A/B hit occurs at that candidate's depth.

This is an exact finite-range certificate statement, not an asymptotic or universal theorem.

2. Independent verification

The independent verifier returned:

{
  "direct_novel_candidates_checked": 41470,
  "integer_witnesses_verified": 41470,
  "k_limit": 1200,
  "reduced_witnesses_verified": 41470,
  "unresolved_integer_candidates": 0,
  "unresolved_reduced_candidates": 0,
  "verdict": "VERIFIED"
}

Thus the growth from 33,644 directly novel candidates at k<=1000 to 41,470 at k<=1200 introduced no collective-shadow counterexample and no failure of reduced prime realization.

3. Hardest first reduced witnesses

The largest first reduced parameters in this run included:

k	h	t	first reduced s	x = r + Ls
1146	169	4579	1,232,114	4,743,294,875,089
1170	529	3899	1,200,079	4,716,744,233,569
1160	289	4607	1,066,259	4,154,955,601,729
1116	169	4427	1,065,106	3,992,999,483,929
1148	529	4550	1,060,095	4,088,194,001,329
1200	529	3199	1,059,772	4,272,114,097,969
1176	361	4691	1,019,129	4,026,090,113,161
1165	529	4426	1,012,370	1,320,658,207,369
1200	529	4784	1,009,638	4,070,014,316,449
1148	529	4583	991,301	3,822,896,533,369

The appearance of first avoiding parameters above one million reinforces that the positive certificates are not arising merely from trivial tiny parameter choices.

4. Prime-power coordinate locality

The run also froze the first complete coordinate-core diagnostic for all 41,470 directly novel candidates.

The analyzer begins from a canonical assignment satisfying all single-prime-support constraints, preferring local residue 1 wherever possible. It then uses the already-certified reduced witness only to guide which prime-power coordinates to change. The repair count is therefore an explicit upper bound on local Hamming distance to a satisfying assignment; it is not asserted to be minimal and is not an independent proof of witness existence.

Results:

canonical unary-safe assignment solves: 15,715 / 41,470 = 37.895%
maximum guided repair count:             9 prime-power coordinates

Repair distribution:

changed prime-power coordinates	candidates	cumulative
0	15,715	37.895%
1	13,838	71.264%
2	7,292	88.847%
3	3,017	96.122%
4	1,080	98.727%
5	367	99.612%
6	123	99.908%
7	34	99.990%
8	3	99.998%
9	1	100.000%

Thus every one of the 41,470 certified candidates lies within at most nine prime-power coordinate changes of this simple unary-safe basepoint under the guided construction.

That is a proof-mining observation, not a universal bounded-repair theorem. Its importance is structural: a huge global congruence system is exhibiting very low local repair complexity.

5. CENTL certification and artifact

CENTL verified all generated symbolic progression identities for the selected hardest cases. The independent number-theoretic verifier remained responsible for earlier-layer avoidance and reducedness.

The completed artifact bundle was uploaded by workflow run 31846146909 with artifact ID 9236427053.

Artifact ZIP SHA-256 digest:

a2479a4113d693af2e647ffc2e007d3d7b1cf628ce7190f72c4ad6282a98ba14

The bundle contains the complete candidate certificate JSON, independent verifier result, coordinate-core result, CENTL receipt, build information, generated contracts, reports, and per-file hashes.

6. What changed conceptually

The candidatewise pattern has now survived three increasingly strong exact ranges:

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

No union-shadow counterexample has appeared.

At the same time, the coordinate-core result says the certified solution space is not merely nonempty. It appears close, in prime-power-coordinate distance, to a very simple local basepoint.

This is exactly the behavior motivating the stronger local elimination theory in SHADOW-KERNEL.md and FIBER-SHADOW-KERNEL.md.

7. New target

The research frontier is no longer just to push the number 1200 upward.

The main target is to prove that the Type A/B pullback system admits a local-to-global construction:

\boxed{ \text{directly novel} \Longrightarrow \text{small-prime fiber kernel} \Longrightarrow \text{reduced local solution} \Longrightarrow \text{infinite exact-depth prime family}. }

The first implication is now partially supported by exact peeling theorems rather than computation alone. The second is where the deepest remaining structure appears to live.