QMR-014 — Numerical equality atlas

How to read this atlas

This is an exhaustive exploratory inventory of numerical equalities under declared counting rules, not a list of independent discoveries or a test of intentionality.

Corpus and measurements

All records use DATA-001: Tanzil Simple Clean, version 1.1, with 6,236 verse identifiers. The variables are: S (surah number), V (verse number), W (whitespace-separated words), L (Unicode Arabic letters), and A (traditional Abjad value).

RAW retains the canonical text as supplied. BODY follows the DATA-001 convention for removing the opening Basmala from the first verse of 112 surahs; surahs 1 and 9 are exceptions. Verse identifiers are unchanged.

A_INDEX applies the declared INDEX-001 handling of Arabic letter variants. A_LITERAL applies the literal traditional Abjad convention used in QMR-009. These are separate orthographic measurements: an equality under one may be absent under the other.

RAW/BODY and A_INDEX/A_LITERAL produce four configurations. Repeated records across configurations describe the same source verse and must not be counted as independent discoveries.

Formula families and classifications

F1 records direct equalities between measured variables, including higher-order intersections. F2-A tests X = Y + Z and X = Y * Z, with three distinct variables and unordered operands. A F2-A match marked "nontrivial" does not meet its recorded arithmetic-triviality criterion; this says nothing about rarity.

F2-B tests X = Y * Z + T using four distinct variables. "Reducible" marks an expression that reduces to an earlier equality family under the recorded criteria. "Nonreducible" means only that those reduction criteria do not apply. It does not establish statistical significance.

Classification belongs to each recorded configuration. The catalog retains the original technical flags so that these labels can be checked against the archived data.

What the totals count

An occurrence is one formula/verse/configuration match. A grouped result combines matching occurrences with the same family, formula, verse reference and numerical values, while preserving its individual configurations. A formula/configuration entry describes an evaluated formula, whether or not it matched any verse. These three totals are not interchangeable.

Download the archived results

Download the five derived research archives as JSON files. These files contain archived measurements and simulation results, not the canonical Quran text.

• f1-exploratory-inventory.json
• f2a-exploratory-inventory.json
• f2b-exploratory-inventory.json
• f2a-null-monte-carlo-exploratory.json
• f2b-null-monte-carlo-exploratory.json

Recalculate the archived results

Download the QMR-014 reproduction package (ZIP). This package contains the five archived JSON files, the analysis modules, two verification scripts, the research protocol draft and instructions for obtaining DATA-001. It does not contain the canonical Quran text.

The verification scripts recalculate the inventory reports and the core Monte Carlo results using the existing analysis code. They do not establish an independent replication or reproduce every documentary field of the JSON archives. Readers must obtain the exact DATA-001 source separately and verify its SHA-256 before running the scripts.

Tracing a result to its archived inventory

The source inventories are repository files:artifacts/qmr-014/f1-exploratory-inventory.json,artifacts/qmr-014/f2a-exploratory-inventory.json, and artifacts/qmr-014/f2b-exploratory-inventory.json. These repository paths identify the source artifacts; they are not themselves download URLs. Once QMR-014 is published, the three inventory JSON files can be downloaded from the links above. No archive download is provided while it is a draft.

The five downloadable JSON files allow readers to check the atlas against archived inventories and inspect the recorded simulation results. They do not include the canonical corpus, analysis scripts, independent source-audit scripts or full research protocol. Re-running the research independently requires those additional resources and the declared counting conventions; downloading the five JSON files alone is not an independent reproduction.

To trace a catalog row, select its family and each listed counting configuration, then locate the displayed formula inreports[configuration].relations for F1 orreports[configuration].formulas for F2-A and F2-B. The verse reference identifies the corresponding archived match.

For F1, find the verse reference in the formula entry'srefs array. Its displayed equality value comes from the matching group in that report'smaximalClasses, whose variables include every variable in the formula. For F2-A and F2-B, find the object with the same verse reference in the formula entry'smatches array, then compare its numerical values and configuration-specific classification flags.

To check a formula/configuration entry without a match, locate that formula under the selected report and inspect its archived count, which is zero. The atlas groups identical displayed results across configurations, but provenance must be checked in each configuration's original entry. This lookup verifies correspondence with the archives; it is not an independent source-text recount.

Statistical scope and interpretation

The F2-A and F2-B Monte Carlo calibrations are post-discovery exploratory analyses. Each uses 2,048 joint within-surah circular shifts of verse measurements, keeping opening verses fixed. The model examines positional alignment conditional on the observed measurements; it does not model the generation of the text.

Each calibration uses 6,122 eligible non-opening verse identifiers, excluding the first verse of each of the 114 surahs. The corpus itself retains all 6,236 identifiers. These are counts of distinct eligible verses, not counts of matching formulas or configurations.

For F2-A, T_ALL counts eligible verses with at least one nontrivial equality containing V; T_ABJAD counts the subset also containing A. For F2-B, the same two statistics count nonreducible matches instead. Each eligible verse is counted once across the four configurations.

The calibrations concern their respective eligible formula families containing V; they do not assess F1 or establish a probability for any individual verse. Formulas without V are invariant under this shift and require a different model for statistical assessment.

An exact equality is reproducible under the stated conventions, but does not by itself establish statistical rarity, novelty, intentionality or a mathematical miracle. See theQMR-014 research reportand the research methodologyfor the analysis and its limits.

Complete matching-record catalog

One row per grouped result. Every matching counting configuration remains listed within its row. Equations are checked against archived numerical values; independent source-text recounts are documented separately for the cases that underwent them.

959 grouped results displayed