Significance of the Elara-Cortex founder Kgomotso Lekola on the GIMPS: an efficacy analysis with eight machine-checked statements

Lekola, Kgomotso

Elara-Cortex analysis · data frozen 17 July 2026

Significance of the Elara-Cortex founder Kgomotso Lekola on the GIMPS

An efficacy analysis of a selection-driven contribution to the Great Internet Mersenne Prime Search

Kgomotso Lekola

ELARA·CORTEX Institute for Decision Systems and Number Theory · New Jersey · Johannesburg

17 July 2026

Subjects. Number theory · Distributed computing · Efficacy analysis · Machine-checked arithmetic

Original PDF · 17 July 2026 · Cite this paper · Share by email

Abstract. We analyse the trial-factoring record of the GIMPS account KgomotsoLekola using the public PrimeNet leaderboard [2], captured on 17 July 2026, and archived figures from May. The account recorded five attempts and one success. Eight arithmetic statements were checked with Z3 [3]; each negation returned UNSAT under the supplied assumptions. The analysis describes this small sample and does not establish statistical significance or future performance. GIMPS [1] has operated as a volunteer distributed-computing project since 1996. Historical figures require the archived captures because its live leaderboard changes.

Keywords. GIMPS; trial factoring; PrimeNet leaderboard; machine-checked arithmetic; Z3; pre-registration.

The result under analysis

Between May and July 2026, KgomotsoLekola logged five trial-factoring attempts and 368 GHz-days of credited work. One candidate yielded a factor; four did not at the tested depths. The archived May capture records a rise of 130 places in 24 hours, compared with 55 for the next member, after four attempts and 316.8 GHz-days. On 17 July 2026 the account ranked 432 of 500 by credited work, up a further 43 places with 16.2% more credited compute.

Fivetrial-factoring attempts, May to July 2026
Onecandidate factored; four without a factor at the tested depths
368GHz-days of credited work
+130places in 24 hours on the May capture, against +55 for the next member
432 of 500rank by raw compute on 17 July 2026
+43further places on 16.2% more compute between the two samples

Archived capture. The image is held with the paper's receipts and does not appear on this page.

The public PrimeNet trial-factoring leaderboard on the May 2026 participation day, with the account at rank 475: 316.8 GHz-days, four attempts, one success and a one-day rank change of +130 against +55 for the next member. Those participation-day figures come from archived captures of a board that updates daily; the current all-time standing is re-verifiable live at mersenne.org/report_top_500_tf.

Screenshot of the report interface. The image is held with the paper's receipts and does not appear on this page.

The official PrimeNet report interface, the Customized Producers page with the Stats type set to Trial Factoring, which is the public source of every figure in this analysis; any reader can reproduce the table there.

Interpreting credited work

GIMPS credits trial-factoring work in GHz-days. Deeper assignments search more factor space; finding a factor early ends an assignment before that work is complete. A success count alone therefore leaves out work spent excluding possible factors.

Candidate eliminated: finding a factor proves the candidate composite. No factor found: the test excludes factors within the searched range; it does not prove the candidate prime.

This analysis reports GHz-days per attempt alongside successful factor finds. Writing \(G\) for credited GHz-days and \(A\) for attempts, contribution density is \(\rho = G / A\). The board's attempt-weighted average is \(\bar{\rho} = \sum_m G_m \big/ \sum_m A_m\) over all members \(m\).

The machine-checked statements

Table 1 records comparisons drawn from the captured data. Z3 returned UNSAT for each negation under the supplied assumptions. This checks the calculations; the five-attempt record remains too small to establish the selection method's performance.

The eight statements, with the measured basis of each. Every negation returned UNSAT.

Statement	Measured basis
T0 · Participation-day double first place: the highest success-per-attempt among all visible members with a success (25% against at most 9.1%)¹ and the steepest rise, +130 places in 24 hours against +55.	May 2026 board, archived screenshots
T1 · In the captured all-time table: sixth of 309 members with a success, ranked by attempts per success.	frozen 500-row table
T2 · A constant success probability of 0.02216 gives 45.1 expected attempts per factor find. The account recorded one in five attempts.	frozen table
T3 · The account rose 43 places with 16.2% more credited compute between the two samples.	May capture and frozen table
T4 · Contribution density of 73.6 GHz-days per attempt, at least 3.8 times the board's attempt-weighted average of 19.18.	frozen table
T5 · 418,954 of 14,064,524 board attempts fall in the account's depth category. Under independent random draws from this distribution, four consecutive draws in that category have probability below one in a million.²	frozen table
T6 · The 259 deep-assignment specialists average one factor find per 85.8 attempts; the account recorded one in five. Two of the 500 members meet all three criteria: at most five attempts, density at least 70, and a factor find.	frozen table
T7 · The observed ratios are nine times the board's eliminations and 3.8 times its deepening per attempt. Their weighted sum is at least the board average for every non-negative weighting.	frozen table

The arithmetic behind T2, T4, T5 and T7

Under a model of independent attempts with constant success probability \(\hat{p}\), the expected attempts per factor find are the reciprocal of that probability. T2 substitutes the board's observed success rate:

\[ \mathbb{E}[\text{attempts per break}] \;=\; \frac{1}{\hat{p}} \;=\; \frac{1}{0.02216} \;=\; 45.1 . \]

The density in T4 follows from the definitions of Section 2, and its multiple over the attempt-weighted board average is what the solver certifies:

\[ \rho \;=\; \frac{368}{5} \;=\; 73.6 \ \text{GHz-days per attempt}, \qquad \frac{\rho}{\bar{\rho}} \;=\; \frac{73.6}{19.18} \;\ge\; 3.8 . \]

Let \(p\) be the share of all board attempts that sit at the account's day depth. Then

\[ p \;=\; \frac{418954}{14064524} \;=\; 0.030, \qquad p^{4} \;=\; 7.9 \times 10^{-7} \;<\; 10^{-6}, \]

so four independent draws at that depth under this random-assignment model have probability below one in a million. This calculation is not a significance test of the selected portfolio.

Write \(E\) for eliminations per attempt and \(D\) for deepening per attempt, with subscript \(a\) for the account's policy and \(g\) for the measured global trajectory. For every non-negative weighting of the two goods,

\[ w_e E_a + w_d D_a \;\ge\; w_e E_g + w_d D_g \quad \forall\, w_e, w_d \ge 0, \qquad\text{with}\quad E_a = 9.0\, E_g,\;\; D_a = 3.8\, D_g . \]

The inequality follows from the observed ratios. It does not establish that those ratios will persist in future attempts.

Comparison cohorts

Table 2 lists all five members in the captured table with fewer than five attempts per success.

Members with a strictly better attempts-per-success ratio than five to one on the all-time table.

Member	Rank	GHz-days	Attempts	Successes	GHz-days per attempt	Attempts per success
TJAOI	147	61,268	90,225	88,850	0.7	1.0
ANONYMOUS	414	503	36	33	14.0	1.1
Kuyiyterea	242	11,154	286	99	39.0	2.9
Abdelrahman Mohamed	494	98	3	1	32.7	3.0
Kelvin From China	350	1,514	4	1	378.5	4.0

The members differ in attempt count and credited work per attempt, and the public table does not describe their methods. Table 3 narrows the comparison to members with at most seven attempts and at least one success.

The low-attempt breakers cohort: at most seven attempts and at least one success. The highlighted row is the account under analysis.

Member	Rank	GHz-days	Attempts	Successes	GHz-days per attempt	Attempts per success
Abdelrahman Mohamed	494	98	3	1	32.7	3.0
Kelvin From China	350	1,514	4	1	378.5	4.0
M8909387	395	708	7	1	101.1	7.0
KgomotsoLekola	432	368	5	1	73.6	5.0

The closest comparable: Kelvin From China

Kelvin From China has a similar low-attempt record: rank 350, 1,514 GHz-days from four attempts, one success and 378.5 GHz-days per attempt. The public data do not reveal that member's method or intent. For KgomotsoLekola, the report also records the May rank movement (T0), the four candidates with no factor found at the tested depths, and the proposed pre-registration in Section 7.

Limits of the analysis

The verification record lists the following limits on interpretation.

No all-time first place: Five members hold better all-time ratios.
No raw-density superlative: Deep-assignment specialists run 400 to 1,300 GHz-days per attempt. The certified claim is the 3.8 times multiple over the attempt-weighted board average, held jointly with T0 and T2.
Additional work: The rank increase was accompanied by additional credited compute, as T3 records.
Intent: Intent is a first-person account. The public record is consistent with it and cannot prove it.

Rank changes also depend on other members' activity. Neither rank movement nor the small number of successful attempts establishes a causal advantage for the selection method.

Proposed pre-registration

The proposal is to register the next five candidate identities, predicted outcomes and depth targets before computing begins. An evaluation would also need a fixed comparison model and scoring rule. Equation (3) concerns four independent draws in a depth category; it does not give the probability of repeating the portfolio's success rate. A prospective trial would provide new evidence, with its own sample-size limits.

Further research

This study forms part of the Institute's work on mathematical decision systems. GIMPS provides a public record of the reported trial-factoring activity; the results do not measure performance in routing, compression or chat. Readers can check the historical calculations against the frozen dataset and archived captures of the public leaderboard [2]. The Z3 scripts and verification records are available on request.

References

Great Internet Mersenne Prime Search. GIMPS, a volunteer distributed-computing project running since 1996. mersenne.org
PrimeNet. Top 500 trial-factoring producers, the Customized Producers report with the Stats type set to Trial Factoring; the frozen copy used here is dated 17 July 2026. mersenne.org/report_top_500_tf
de Moura, L., & Bjørner, N. (2008). Z3: An efficient SMT solver. In Tools and Algorithms for the Construction and Analysis of Systems (TACAS), LNCS 4963, 337–340.

25% is the archived May participation-day figure, one of four; the 17 July record is one of five, 20%. ↩
This calculation assumes independent random assignment from the stated distribution. It describes that model and is not a test of statistical significance for the deliberately selected portfolio. ↩

Cite this paper

Cite as. Lekola, K. (2026). Significance of the Elara-Cortex founder Kgomotso Lekola on the GIMPS: an efficacy analysis with eight machine-checked statements. Analysis, ELARA·CORTEX Institute for Decision Systems and Number Theory, New Jersey and Johannesburg. https://ai.elara-cortex.com/AI/research/gimps

@techreport{lekola2026gimps,
  author      = {Lekola, Kgomotso},
  title       = {Significance of the {Elara-Cortex} founder {Kgomotso Lekola} on the {GIMPS}: an efficacy analysis with eight machine-checked statements},
  institution = {ELARA-CORTEX Institute for Decision Systems and Number Theory},
  address     = {New Jersey and Johannesburg},
  type        = {Analysis},
  year        = {2026},
  month       = jul,
  url         = {https://ai.elara-cortex.com/AI/research/gimps}
}

Web edition. The original report is Version 1, dated 17 July 2026, with the leaderboard data frozen the same day. This web edition includes editorial revisions and clarifications of scope. The linked PDF retains the original text and is the version of record. No DOI has been issued; cite the address above.

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