Abstract
QMR-014 systematically examines direct and compound arithmetic equalities involving surah number (S), verse number (V), word count (W), Arabic-letter count (L), and traditional Abjad value (A).
The research covers three successively specified formula families: F1, F2-A and F2-B. The results are descriptive or post-discovery exploratory, not confirmatory evidence of intentional numerical design.
Quran 25:52 and 74:20 provide detailed examples of F2-A and F2-B equalities. The QMR-014 atlas presents all archived matches and evaluated formula/configuration entries, including those without a match.
Case study: Quran 25:52
The canonical DATA-001 text is:
فلا تطع الكافرين وجاهدهم به جهادا كبيرا
Its recorded measurements are identical in RAW and BODY:
| Measurement | Value |
|---|---|
| Surah number S | 25 |
| Verse number V | 52 |
| Whitespace-delimited words W | 7 |
| Arabic letters L | 33 |
| Traditional Abjad A_INDEX | 1,300 |
| Traditional Abjad A_LITERAL | 1,300 |
The exact F2-A equality is:
A = S × V = 25 × 52 = 1,300
The formula is nontrivial under the stated F2-A arithmetic classification. It occurs in all four inventories: RAW/A_INDEX, RAW/A_LITERAL, BODY/A_INDEX and BODY/A_LITERAL. These four records describe the same verse and must not be counted as independent cases.
Targeted word-by-word recount
| Canonical word | Arabic letters | Traditional Abjad |
|---|---|---|
| فلا | 3 | 111 |
| تطع | 3 | 479 |
| الكافرين | 8 | 392 |
| وجاهدهم | 7 | 64 |
| به | 2 | 7 |
| جهادا | 5 | 14 |
| كبيرا | 5 | 233 |
| Total | 33 | 1,300 |
The targeted recount used the literal traditional 28-letter Abjad table. Both QMR-014 Abjad conventions yield the same value for this particular text; this does not imply that the conventions are interchangeable elsewhere.
Case study: Quran 74:20
The canonical DATA-001 text is:
ثم قتل كيف قدر
The recorded measurements are identical in RAW and BODY:
| Measurement | Value |
|---|---|
| Surah number S | 74 |
| Verse number V | 20 |
| Whitespace-delimited words W | 4 |
| Arabic letters L | 11 |
| Traditional Abjad A_INDEX | 1,484 |
| Traditional Abjad A_LITERAL | 1,484 |
The exact F2-B equality is:
A = S × V + W = 74 × 20 + 4 = 1,484
The match is classified as nonreducible to F1 or F2-A under the specified F2-B arithmetic criteria. It occurs in all four RAW/BODY and Abjad-convention inventories. These four records describe one verse, not four independent observations.
Targeted word-by-word recount for 74:20
| Canonical word | Arabic letters | Traditional Abjad |
|---|---|---|
| ثم | 2 | 540 |
| قتل | 3 | 530 |
| كيف | 3 | 110 |
| قدر | 3 | 304 |
| Total | 11 | 1,484 |
This targeted recount uses the literal traditional 28-letter Abjad table. For this verse, A_INDEX and A_LITERAL give the same total.
The F2-B T_ABJAD exploratory calibration concerns the aggregate count of eligible verse references, not an individual probability for Quran 74:20. Under the specified null model, this aggregate does not show an unusually large excess.
Scope of the wider QMR-014 inventory
F1 examines 26 direct equality relations per inventory. The unique S = V = L case at 20:20 agrees with the separate QMR-011 analysis; QMR-011 retains its own statistical interpretation.
F2-A examines 60 addition or multiplication formulas per inventory, using three distinct variables. Thirteen distinct verses have an Abjad-related match across its four inventories. Ten are eligible for its V-and-A null-calibration statistic.
F2-B examines 60 formulas of the form X = Y × Z + T per inventory, using four distinct variables. Its archived results include 16 distinct verses with an Abjad-related formula. Reducible expressions are distinguished from nonreducible ones.
Exploratory null-model results
The F2-A and F2-B calibrations jointly circular-shift verse measurements within each surah, with opening verses fixed. Each uses 2,048 simulations over 6,122 eligible non-opening verse identifiers and counts distinct eligible verse references once across the four inventories.
For F2-A, T_ALL counts verses with at least one nontrivial equality containing V; T_ABJAD counts the subset with at least one nontrivial equality containing both V and A. For F2-B, the corresponding statistics require nonreducible equalities instead. Each verse is counted once even if multiple formulas or configurations match.
The 24 F2-A formulas and 12 F2-B formulas without V are invariant under this shift and are not statistically assessed by these calibrations.
| Family | Statistic | Observed | Simulated mean | Exploratory upper tail |
|---|---|---|---|---|
| F2-A | T_ALL | 384 | 356.769 | 0.072718 |
| F2-A | T_ABJAD | 10 | 4.070 | 0.009761 |
| F2-B | T_ALL | 39 | 48.226 | 0.932650 |
| F2-B | T_ABJAD | 12 | 11.192 | 0.446559 |
The F2-A run used seed 20260919; the F2-B run used seed 20260920. For F2-A, the upper-tail exceedance counts were 148/2,048 for T_ALL and 19/2,048 for T_ABJAD. For F2-B, they were 1,910/2,048 and 914/2,048, respectively. Each reported Monte Carlo tail estimate uses (k + 1)/(N + 1), with N = 2,048 and k the number of simulated counts at least as large as the observed count.
The archived Wilson intervals describe Monte Carlo sampling uncertainty for k/N; they do not quantify uncertainty from the choice of null model. The statistics within a family are correlated, and no independent replication of the full null runs or confirmatory multiplicity adjustment is claimed.
These are post-discovery exploratory results from different formula families. In particular, the F2-A T_ABJAD upper-tail estimate is not a probability assigned to the individual equality at Quran 25:52.
Under the specified model, F2-B does not exhibit an unusually large aggregate count. Exact arithmetic matches remain exact regardless of this aggregate calibration.
Interpretation and limitations
All results depend on the fixed corpus and declared tokenization, Arabic-letter and Abjad conventions. The null model concerns conditional positional alignment, not the generation of Quranic text. Formula selection, earlier exploratory inspection and model choice limit the strength of statistical inference.
This research does not establish novelty, intentionality or a mathematical miracle. Historical precedence for the individual 25:52 equality has not been established by this draft.
Reproducibility records
artifacts/qmr-014/f1-exploratory-inventory.jsonartifacts/qmr-014/f1-exploratory-summary.mdartifacts/qmr-014/f2a-exploratory-inventory.jsonanalysis/qmr-014/f2a-independent-2552-source-audit.mjsartifacts/qmr-014/f2a-null-calibration-note.mdartifacts/qmr-014/f2a-null-monte-carlo-exploratory.jsonartifacts/qmr-014/f2b-exploratory-inventory.jsonanalysis/qmr-014/f2b-independent-source-audit.mjsartifacts/qmr-014/f2b-null-calibration-note.mdartifacts/qmr-014/f2b-null-monte-carlo-exploratory.jsonartifacts/qmr-014/scientific-synthesis.mddocs/research/qmr-014-protocol-draft.md
Once QMR-014 is published, its atlas will offer five JSON downloads: the F1, F2-A and F2-B exploratory inventories and the F2-A and F2-B Monte Carlo archives. It will also offer a reproduction ZIP containing these five JSON files, the existing analysis modules, two verification scripts, a protocol draft, reproduction instructions, a DATA-001 acquisition notice and a file-integrity manifest. The ZIP does not contain the canonical Quran text.
Readers must obtain DATA-001 separately and verify its SHA-256 as specified in the acquisition notice. Using the existing analysis implementation, the scripts recalculate the archived inventory reports and check the core Monte Carlo results. They do not constitute an external independent replication or verify every documentary field of the JSON archives.
The other listed repository paths are not additional public download URLs. The targeted word-by-word audits check the two cited passages separately from the main inventory calculations; they do not independently replicate the complete research.
The exploratory analysis code, inventories, synthesis and
null-calibration records were archived at
ad8f9d45a9dfdd3a01f83122067e5161f9ae024a.
The exhaustive atlas implementation was added at
fa89602ac015ba076ee95fac9be5667e4dcb780e,
and the targeted independent source audit of Quran 25:52 at
5621220e902068d3fddcd4fa2b4244a297d65728.
This unpublished research draft was developed in subsequent commits.