Automated WS-CAND-003 results — 2026-08-14

This note freezes the first successful automated run of the CENTL Erdős-Straus research harness. It records finite computational results only and must be read with the claim boundaries in PRIOR-ART.md and THEORY.md.

Run configuration

• hard-prime limit: 10,000,000
• Type A/B layer limit: k=3000
• hard residue classes modulo 840: 1, 121, 169, 289, 361, 529
• selected hard primes: 20,513
• selected hard primes resolved: 20,513
• unresolved: 0

Frontier

The independently rechecked record frontier remained:

1009       -> C_AB 3
1201       -> C_AB 8
2521       -> C_AB 22
3361       -> C_AB 25
9601       -> C_AB 28
33289      -> C_AB 45
76441      -> C_AB 70
83449      -> C_AB 170
1095481    -> C_AB 245
1423321    -> C_AB 1050
2031121    -> C_AB 1403
4728649    -> C_AB 1435
9658489    -> C_AB 2622

Shadow geometry through k=3000

The automated direct-shadow map found:

• 460 fully directly shadowed layers;
• 1,108 completely direct-fresh layers;
• 875 partially directly shadowed layers;
• 3,544 unique direct-shadow layer edges;
• 1,808 shadow edges for which the earlier modulus divides the later modulus;
• 1,736 shadow edges not explained by simple modulus divisibility alone.

This shows that modulus ancestry is a major part of the graph but not the entire shadow mechanism.

Global non-union-shadow witnesses

The first-hit computation produced 1,164 distinct hard-class/current-trap CRT classes containing at least one first-hit prime. Every such prime is an explicit witness that its current class is not covered by the union of all earlier Type A/B layers.

The independent verifier rechecked all 20,513 first-hit primes against all smaller layers required by their claimed depth.

The current record class is

k = 2622
m_k = 10487
h = 169 mod 840
t = 10449 mod 10487
witness prime = 9658489

and is therefore globally non-union-shadowed by explicit witness.

Independent-verifier totals

The first successful workflow run independently checked:

• 7,113 rejected lower frontier levels;
• 3,000 trap-cardinality layers;
• 10,710 direct-shadow certificates;
• 1,164 witnessed globally non-union-shadowed classes;
• 20,513 first-hit prime witnesses;
• 3,544 ancestry-edge arithmetic records.

Final verifier verdict: VERIFIED.

CENTL certification

The workflow built CENTL from the checked-out repository commit and used it to certify:

• 13 closed exact Egyptian-fraction identities, one for every frontier record;
• 13 fixed-parameter polynomial Type A/B families, one for every frontier witness;
• 214 distinct observed modulus-ancestry quotient identities;
• the dedicated polynomial family for the current record p=9,658,489, C_AB=2622.

Final CENTL certification verdict: CENTL exact contracts verified.

Modulus-ancestry families

For an ancestry quotient q=4s+1, the arithmetic relation is

The automation grouped fully shadowed layers having a common modulus-dividing ancestor into finite candidate families. The largest groups in the first run were:

quotient q	s=(q-1)/4	observed uniform full-shadow points
5	1	130
9	2	38
17	4	31
25	6	31
13	3	29
21	5	22
33	8	14
41	10	13
65	16	11
37	9	10
73	18	10
81	20	10

The strongest first theorem target is therefore the q=5 ancestry family

The 130 observed points are evidence for studying this family, not a claim that all j satisfy full shadowing. The run contains examples of full, partial, and absent immediate-ancestor shadowing, so the mathematical problem is to derive the exact condition on j.

Important interpretation

direct-fresh does not mean globally irredundant. It only means no single earlier layer covers the complete class. The stronger non-union conclusion is currently made only when an explicit first-hit prime supplies a counterexample to collective earlier coverage.

The finite ancestry groups are theorem candidates, not infinite-family theorems. A future proof must supply necessary-and-sufficient arithmetic conditions rather than extrapolate from these counts.

Reproducibility

The successful workflow uploaded a complete research artifact bundle containing generated JSON, direct-shadow certificates, independent-verifier output, CENTL receipts, the generated .centl contract, build information, and SHA-256 manifests.

GitHub workflow: .github/workflows/erdos-straus-research.yml

Research harness: research/erdos-straus/