If decompwlj became a standard mathematical lens

1How to read this counterfactual

This report imagines a future in which the decomposition is not merely known, but is independently studied, implemented, criticized, extended, and taught. It is therefore a conditional impact analysis, not a prediction of fame and not a retrospective verdict on the present project.

The public record is concrete. Eismann’s paper was submitted in 2007 and last revised in 2010; its abstract states the Euclidean decomposition, the generalized-sieve interpretation, the prime classification, and the numerical conjectures.^[1] The OEIS records the prime weight as A117078, the level as A117563, and the two prime classes as A162174 and A162175.^[2–5] The project atlas currently states that it contains 1,000 decomposed sequences and offers CSV, SQL, image, and 3D-graph downloads; it also explicitly says that its data have not been verified.^[6] That last sentence is not a condemnation. It is the starting specification for the imagined world: recognition would turn a remarkable personal atlas into a curated scientific corpus.

2The mathematical core and why it is unusual

2.1 Definition

Let be a strictly increasing sequence of positive integers. For each adjacent pair define the jump

Then set

and, when , define the weight and level by

Thus

The decomposition exists exactly when

If , the term is level-classified; otherwise it is weight-classified. The definition is local, unique, and order-sensitive: it uses the next term to set a threshold and then chooses the first divisor of beyond that threshold.

2.2 The exact additive–multiplicative bridge

The bridge is not simply the identity ; every Euclidean division has that form. The distinctive step is that the divisor is selected by the additive gap. Equivalently, the class boundary can be expressed as a divisor-window event:

This is the mathematically fertile formulation. The additive object does not merely sit beside a factorization; it determines the moving interval in which divisors are allowed to count. For primes, , , and . The transform therefore couples a consecutive prime gap to the divisor geometry of a reflected integer.

2.3 Why the natural numbers are the base case

For , the jump is constantly 1 and . The weight is the smallest prime factor of , while the level is the complementary largest proper divisor. Level 1 corresponds exactly to prime. The familiar Eratosthenes columns appear as weight fibers: weight 2 catches even ; weight 3 catches multiples of 3 not already caught by 2; weight 5 catches multiples of 5 not already caught; and so on.

2.4 A scale-free coordinate that widespread study would probably introduce

The traditional plots use . Because , a more portable statistic is the factor-balance coordinate

Then the class boundary is simply . Pairing this with a normalized gap coordinate such as

would separate size effects from arithmetic effects. A mature theory would likely use both representations: raw log coordinates for visual fibers, and normalized coordinates for cross-sequence statistics.

3What “wildly recognized” would mean in practice

Recognition in mathematics is not a single event. For this framework it would mean that a chain of institutions and practices had formed around it.

The closest institutional analogy would not be a new grand conjecture but a small mathematical database that gradually becomes semantic. The OEIS currently contains about 398,000 sequences and provides stable identifiers and curated metadata.^[7] The LMFDB provides another model: mathematical objects have pages, invariants, provenance, reliability information, links, and machine-accessible collections.^[8] In the imagined world, a “WLJ Observatory” would sit between these models: it would not replace the OEIS, but would attach a reproducible derived signature to eligible increasing integer sequences.

4Possible impacts on number theory

4.1 A new organizing language for local arithmetic

Number theory often separates additive questions—gaps, differences, sums, short intervals—from multiplicative questions—factors, divisors, smoothness, roughness, and residue constraints. Decompwlj would not erase that division, but it could provide a repeatable way to place both types of data in the same local object:

That tuple creates families of number-theoretic questions that are natural in the new coordinates:

• How frequently does have a divisor just beyond ?
• What is the distribution of the least admissible divisor conditioned on the gap?
• When does the complementary factor remain bounded, grow slowly, or occupy special residue classes?
• Which fixed weight columns or fixed level rows coincide with classical arithmetic families?
• Which phenomena survive after the gaps are permuted, or after the sequence is replaced by a density-matched random model?

4.2 Prime gaps and shifted divisors would meet in one object

For primes, the level-class condition is a moving divisor-desert condition on . This places the framework near two established bodies of theory: the distribution of integers with a divisor in a prescribed interval, studied in depth by Ford,^[9] and divisors of shifted primes, studied by Koukoulopoulos.^[10] The novelty of the WLJ coupling is that the shift and the divisor threshold are both determined by the next prime. That dependence is precisely what makes the object interesting and technically difficult.

If the framework became standard, it could motivate theorems in which prime-gap estimates and divisor-in-interval estimates are designed to communicate. Recent work of Gafni and Tao on rough numbers inside consecutive prime gaps shows that modern sieve methods can successfully control a different but structurally related coupling of consecutive gaps with least-prime-factor conditions, producing an unconditional density-zero result and a conditional singular-series asymptotic.^[11] This does not prove a WLJ claim, but it demonstrates that “consecutive gap + moving multiplicative threshold” is a legitimate modern research architecture.

4.3 A hierarchy of prime subpopulations

Wide adoption would turn weight columns and level rows into routinely studied fibers. Some have immediate classical meanings. The prime weight sequence A117078 is already formalized on the OEIS, where the decomposition and the relation to the gap and level sequences are recorded.^[2] In particular, for decomposable primes greater than 3, weight 3 corresponds to the smaller member of a twin-prime pair. Thus

is not a new easier conjecture; it is the twin-prime conjecture in new coordinates. This is still useful as a dictionary entry. A standard coordinate system can reveal that several apparently different questions are the same fiber viewed from different fields.

The more promising native questions are aggregate and comparative:

• the density or rarefaction of the whole level class;
• the decomposition of that class into and species;
• the distribution of prime versus composite weights within the level class;
• exact residue-lock identities for small columns;
• uniform bounds in the two parameters and ;
• singular-series constants attached to fixed fibers or averaged families of fibers.

4.4 A “PNT layer” for sequences

The prime number theorem describes the first-order density of the primes. For an arbitrary increasing sequence , one first has its ordinary counting function

Decompwlj would add secondary counting functions:

A mature “PNT for sequences” programme would not claim one universal formula for every sequence. It would seek asymptotics or universality classes for the vector . Dense polynomial sequences, prime-like sparse sequences, recurrence sequences, and exponentially growing sequences should not be forced into one model.

4.5 New proof decompositions

Even when the coordinates do not change a conjecture’s difficulty, they may change how a proof is organized. A WLJ proof strategy would typically split a population by:

1. large versus typical gaps;
2. weight versus level class;
3. prime versus composite selected weight;
4. small fixed levels versus growing levels;
5. ordinary versus exceptional divisor geometry;
6. local residue obstructions and global distribution.

This is a useful form of intellectual compression if different problems repeatedly admit the same architecture. The transform would then act as a proof coordinate system, not merely a data label.

4.6 Algorithmic number theory

Exact WLJ computation asks for the least divisor of above a moving threshold . On small dense ranges, sieving or smallest-prime-factor tables are effective. On large sparse values, factoring may dominate the cost. Widespread use could therefore stimulate algorithms for:

• batch factorization of shifted sequence values;
• finding the first divisor in an interval without enumerating all divisors;
• certifying that an interval contains no divisor;
• streaming adjacent-pair transforms over b-files;
• probabilistic triage followed by exact certificates;
• parallel scheduling by bit length, gap ratio, and factorization difficulty.

The computational status must allow three outcomes: decomposable, not decomposable by definition, and not yet resolved computationally. Treating “unknown” as zero would contaminate every large-scale conclusion.

4.7 Famous conjectures: realistic forms of impact

5A new laboratory for statistical number theory

5.1 The transform creates a family of random variables

Statistical number theory studies deterministic arithmetic objects through distributions, moments, local factors, and probabilistic models. Decompwlj naturally produces a multivariate process:

For primes, this process couples the consecutive-gap process to a thresholded divisor process. That makes several questions available:

• marginal laws for , , and ;
• conditional laws given , a residue class, or a normalized gap;
• correlations across adjacent indices;
• large deviations for unusually level-like or weight-like blocks;
• extreme-value laws for the first admissible divisor;
• finite-size corrections and slowly converging singular-series averages.

5.2 A hierarchy of null models

The greatest statistical contribution might be methodological: WLJ pictures force researchers to distinguish geometry implied by the definition from arithmetic carried by the sequence. A recognized field would rarely publish a raw cloud without several controls.

5.3 Generic geometry versus arithmetic fine structure

The arrowheads and two-sheet plots are visually powerful, but some of their shape is forced:

• forces log-coordinate tradeoffs;
• forces the diagonal class boundary;
• the cutoff in forces a triangular envelope;
• the existence condition filters sequences by relative growth;
• sparsity alone can produce rarefaction-like behavior.

The arithmetic signal lives in what remains after those facts are controlled: fiber occupancies, missing fibers, congruence locks, exact constants, exception sets, conditional biases, and stable cross-scale departures from null ensembles. This distinction would sharpen statistical number theory well beyond the project itself: it is an example of how every new visualization should carry its own generative baseline.

5.4 Singular-series observatories

Fixed gaps and fixed weights impose simultaneous congruence conditions. In prime problems, those conditions are naturally summarized by Hardy–Littlewood singular series. The counterfactual field could build a “singular-series observatory” in which each WLJ fiber has:

• a local admissibility calculation;
• a predicted first-order frequency;
• an observed frequency by scale and gap;
• a finite-size correction model;
• a residual plot and uncertainty band;
• a proof status for any claimed identity.

This would turn the graph’s vertical columns and horizontal rows into quantitatively testable objects. It would also connect to a broad modern pattern: averages of singular series convert families of local prime constraints into global constants. Gafni–Tao’s consecutive-gap work explicitly uses such averages for a related roughness problem.^[11]

5.5 Universality classes of sequences

Across many sequences, the statistical goal would not be one universal curve. It would be a phase diagram.

This classification exposes an important fact: the existence condition itself is a growth statistic. Polynomial sequences are eventually decomposable because their relative gaps tend to zero. A geometric sequence with ratio at least is eventually or always excluded. Fibonacci growth approaches the golden ratio, greater than , so its native tail fails the gate. A big-data atlas must not compare these regimes as if missing decompositions were random missing values.

6Sequence analysis at big-data scale

6.1 From 1,000 portraits to hundreds of thousands of signatures

The current atlas reports 1,000 decomposed sequences.^[6] The OEIS homepage reported 398,208 sequences on 11 August 2026.^[7] Not every OEIS entry is eligible: many are not strictly increasing, contain negative values, represent tables, have too few terms, or grow too fast. Even so, applying a standardized transform to the eligible fraction would be a substantial mathematical data project.

If 398,000 sequences each contributed only 1,000 usable adjacent pairs, the raw ceiling would already be roughly 398 million pair records; at 10,000 pairs it would be about four billion. Those are scenario calculations, not claims about current b-file coverage. The point is that the conceptual bottleneck would move from plotting to data engineering, factorization scheduling, and statistical validation.

6.2 The WLJ signature

A useful atlas entry would be more than an image. For a sequence and cutoff , one could store a signature such as

where:

• D_N is the decomposability fraction;
• F_lev is the level-class share among resolved decompositions;
• μθ and μρ are normalized factor-balance and gap distributions;
• Hk and HL summarize weight-column and level-row entropy or concentration;
• C stores small-modulus congruence-lock diagnostics;
• Z stores deviations from matched null models;
• S stores stability across cutoffs, implementations, and data versions.

This is exactly the kind of fixed-length mathematical fingerprint that prior work has explored for OEIS classification using Benford- and Taylor-law features.^[12] A 2026 sequence model, IntSeqBERT, goes further by combining log-scale magnitude with residue embeddings for 100 moduli, reporting substantial gains on masked and next-term prediction over a tokenized baseline.^[13] WLJ features would be complementary: they encode adjacent gaps and thresholded multiplicative structure rather than residue spectra alone.

6.3 What machine learning could do

Machine-learning-assisted mathematics has already demonstrated a productive pattern: use models to detect relationships, interpret the learned dependence, and return to human conjecture and proof.^[14] Program-search systems similarly pair generative exploration with a systematic evaluator that rejects invalid candidates.^[15] A WLJ corpus would be unusually suitable for this evaluator-first style because every row satisfies exact identities and inequality checks.

6.4 The data-leakage trap

This single protocol rule would prevent an enormous class of misleading high-accuracy results. WLJ is naturally retrospective at the current index; it is predictive only through its lagged dynamics.

6.5 A reproducible data schema

OEIS b-files already provide a strict, machine-readable index–value format and encourage provenance comments.^[16] A recognized WLJ pipeline could ingest those files while retaining the original sequence identity. Database-driven mathematics benefits when objects are searchable by stored mathematical properties rather than by text alone; this is precisely the case for derived WLJ signatures.^[17]

6.6 Compute architecture

A realistic production system would separate specification from acceleration:

1. Reference semantics. A concise PARI/GP implementation defines the exact output and supplies canonical test vectors.
2. Eligibility pass. Detect strict increase, positive values, pair count, bit lengths, relative-growth regime, and non-decomposable terms.
3. Fast path. Dense bounded -ranges use sieves; sparse moderate values use batch factorization and ordered divisor enumeration.
4. Hard path. Large integers are queued for partial factorization, interval-divisor search, or certified “unresolved” status.
5. Cross-check. Independent implementations compare every row on small ranges and random or adversarial samples on large ranges.
6. Publish. Data, code version, checksums, certificates, and plot parameters are released together.

6.7 New forms of sequence search

The OEIS is often used as a fingerprint database for theorems: a short sequence can connect a researcher to an unexpected literature.^[18] WLJ could add a second search mode. Instead of matching term values, a researcher could match structural profiles. Two sequences with no common initial terms might share:

• the same rarefaction exponent;
• the same small-weight spectrum;
• the same normalized factor-balance law;
• the same congruence-lock residuals;
• the same response to thinning or subsequence extraction.

Such matches would be hypotheses, not equivalences. Their value would come from whether they lead to a common theorem or a meaningful counterexample.

7What the supplied figures could become

The supplied images already contain the seed of a visual language. In the imagined world, they would be treated as exploratory maps whose components are formally indexed, counted, modeled, and compared. The square frames below preserve the plotted aspect ratio; images are contained rather than stretched.

7.1 Standard visual protocol

Wide use would likely produce a plotting standard:

• square canvases and equal / scale;
• explicit logarithm base and cutoff;
• non-decomposable terms reported separately, never hidden at ;
• class colors accessible to color-blind readers;
• raw counts plus density-normalized panels;
• a matched null-model panel;
• uncertainty or sampling information for aggregated curves;
• links from every visible fiber to data and definitions.

7.2 From pictures to topology of arithmetic data

Once millions of points and thousands of sequences are involved, images alone cease to be adequate. Researchers could study persistent components, occupied-fiber complexes, transport across cutoffs, and distances between normalized distributions. These tools would not make the arithmetic topological in a theorem-proving sense by default; they would provide robust summaries of how discrete fibers appear, merge, or vanish across scale.

8Effects on mathematics as a whole

8.1 Experimental mathematics would gain a model case

The project could become a case study in how an individual visual idea matures into mathematics:

1. define an exact transform;
2. build examples and an atlas;
3. identify visually stable strata;
4. separate tautology, reformulation, heuristic, and theorem;
5. design controls and independent computation;
6. extract proof-sized statements;
7. publish both positive and deflationary findings.

This process is broadly useful. Many modern mathematical discoveries begin with large calculable objects whose patterns are difficult to see directly. Machine-learning work in pure mathematics explicitly describes a loop from data to interpretable pattern to conjecture and proof.^[14] Decompwlj could offer a relatively elementary but genuinely nontrivial training ground for that loop.

8.2 Mathematical databases would become more semantic

A sequence database usually indexes values, formulas, comments, and references. A WLJ layer would index derived behavior. This points toward a wider future in which mathematical databases store:

• canonical invariants and transforms;
• known relations between objects;
• completeness and reliability metadata;
• proof and computation certificates;
• searchable experimental signatures;
• negative results and failed conjectures.

The LMFDB already illustrates object pages, invariants, related objects, data sources, and programmatic access at very large collection sizes.^[8] A WLJ observatory would extend that database mentality to cross-sequence experimental arithmetic.

8.3 Education could connect four topics in one picture

For students, the natural-number case connects Euclidean division, prime factorization, Eratosthenes’ sieve, and logarithmic geometry. The prime case then introduces gaps, shifted values, conjectures, asymptotic density, and the difference between a reformulation and a proof. Few elementary-looking constructions naturally open so many doors.

A course module could ask students to:

• prove the existence criterion;
• derive the natural-number smallest-prime-factor rule;
• compute several sequences in PARI/GP;
• identify what features of an arrowhead are forced;
• build a random control;
• formulate one conjecture and try to destroy it;
• classify every claim as definition, equivalence, finite verification, heuristic, conditional theorem, or theorem.

8.4 A bridge between professional and citizen mathematics

If a framework originating outside a conventional academic path became widely studied, one cultural effect could be a healthier interface between amateur exploration and professional proof standards. The lesson would not be “large computation is enough” or “institutions do not matter.” It would be that curiosity, visualization, and sustained data collection can originate anywhere, while durable mathematical knowledge emerges through precise definitions, open code, reproducibility, criticism, and proof.

8.5 Generalizations—carefully delimited

The transform needs more structure than the slogan suggests: an ordered sequence, subtraction to form a gap and a reflected value, a divisibility relation, and a rule selecting the least admissible divisor. Generalization is therefore plausible, but not automatic.

The most valuable generalization theorem would identify the minimal axioms under which an analogue exists and state exactly which properties of the integer case survive. Failed generalizations would be informative: they reveal which ingredients do the real mathematical work.

9Boundaries, failure modes, and safeguards

An open-minded report should specify not only how the framework could succeed, but how its apparent success could be illusory.

9.1 Sensitivity is not automatically a defect

The transform is not invariant under translation: replacing by preserves gaps but replaces by , changing its divisors. Scaling also changes the admissible divisor structure in a nontrivial way, and taking a subsequence changes the gaps. This means WLJ is a descriptor of a presented ordered sequence, not of an abstract set of integers.

That sensitivity may be exactly what one wants when presentation and local adjacency are meaningful, as for consecutive primes. It becomes a problem only when researchers compare sequences without specifying which transformations should preserve identity.

9.2 Four tests of genuine value

A framework that passed only the first test could still be a valuable exposition tool. Passing all four would justify the much stronger claim that it had changed mathematical practice.

10A plausible recognition pathway

10.1 What success might look like at different levels

These levels are not mutually exclusive, and the first is not a failure. Mathematics is often changed by modest tools that become reliable and ubiquitous.

11Thirty research questions for that world

Foundations and invariance

1. What is the minimal algebraic and order structure needed to define a WLJ analogue?
2. Which transformations of a sequence preserve decomposability, class, or normalized signature?
3. Can the transform be characterized categorically or as an operator on adjacent-pair processes?
4. Is there a canonical tie convention that optimizes theoretical symmetry and compatibility with the historical data?
5. Which geometric features of the log plots are theorems of the definition alone?

Analytic and probabilistic number theory

6. What is the true asymptotic order of the prime level class and its and components?
7. Can divisor-in-interval estimates be made uniform when the interval endpoint is a consecutive prime gap?
8. What singular-series averages govern fixed-level and fixed-weight fibers?
9. Which residue-lock laws are exact, which are first-order, and which require a consecutiveness correction?
10. Is there a central limit theorem for the normalized factor-balance coordinate in any natural sequence class?
11. What are the extreme-value laws for unusually large selected weights or unusually small levels?
12. How much of aggregate rarefaction is predicted by one-point density alone?
13. What statistics optimally distinguish the primes from parity- and singular-series-aware random models?
14. Does permuting the prime gaps destroy all nontrivial WLJ correlations, or do one-point residue effects remain?
15. Can modern consecutive-gap sieve methods be adapted directly to reflected values ?

Sequence big data

16. What fraction of OEIS sequences are eligible under transparent, reproducible rules?
17. How stable are sequence signatures under longer b-files and corrected entries?
18. Can normalized WLJ features retrieve mathematically related sequences that term matching misses?
19. Do WLJ features improve held-out classification after controlling for growth, residues, and ordinary gaps?
20. Which clustering results remain stable across algorithms and random seeds?
21. How should unresolved factorizations be modeled statistically?
22. Can minimal-divisor certificates be compressed enough for a billion-row corpus?
23. What is the best sequence-specific null model, and can it be selected automatically without leaking labels?
24. Can an anomaly-ranking system prioritize conjectures that later admit short proofs?
25. How many distinct visual morphologies remain after scale normalization and definition-forced geometry are removed?

Generalization and mathematical practice

26. Is there a function-field model in which the analogue of prime rarefaction has an exact asymptotic?
27. Can ideal factorization supply a natural analogue in number fields without arbitrary choices among equal norms?
28. Which failures in nonunique-factorization monoids reveal genuinely essential axioms?
29. Can a formal proof assistant verify the reference transform, class identities, and finite certificates at scale?
30. What would count as the first result that the mathematical community agrees was discovered because of WLJ coordinates rather than merely expressible in them?

12Conclusion

In the imagined world, decompwlj would matter because it made local arithmetic data comparable, queryable, and mathematically productive—not because a new vocabulary was declared important.

For number theory, it could organize a real family of gap-conditioned divisor problems and create a shared proof architecture linking prime gaps, shifted values, divisor deserts, residue obstructions, and singular-series averages. For statistical number theory, it could supply a multivariate process whose generic geometry can be cleanly separated from arithmetic fine structure through disciplined null models. For big-data sequence analysis, it could turn an atlas of 1,000 images into a corpus of reproducible signatures, capable of clustering, anomaly detection, semantic retrieval, and AI-assisted conjecture formation. For mathematics more broadly, it could become an instructive model of how visual experimentation, community databases, computation, and proof reinforce one another.

The most important conceptual impact would be a change of question. Instead of asking only, “What is the next term?” or “How fast does the sequence grow?”, researchers could also ask:

That is a legitimate and surprisingly rich question. Whether it becomes a major branch of mathematics would depend on the answers it generates: theorems, constants, algorithms, counterexamples, cross-sequence laws, and unexpected connections. The counterfactual world is therefore open rather than celebratory. It invites the framework to become testable at the same scale as its ambition.

ATechnical appendix

A.1 Exact inequalities behind the two classes

Assume . Since :

Minimality of shows that the weight class occurs exactly when a divisor is present in . In the level class, every divisor at or below is at most . This divisor-window formulation should be the default bridge to standard analytic terminology.

A.2 Reference PARI/GP kernels from the supplied source pack

decomp(a,b) = {
  my(d=b-a,l);
  if(a<=2*d, return([0,0,d]));
  l=a-d;
  fordiv(l,k, if(k>d, return([k,l/k,d])))
}

decomp_fact(a,b) = {
  my(d=b-a, l=a-d);
  if(a<=2*d, return([d,0,0]));
  my(k=vecmin([e | e<-divisors(l), e>d]));
  [k,l/k,d]
}

The two supplied files use different output orders for the non-decomposable case. A standardized library should choose one schema and preserve it across every implementation.

A.3 Suggested claim labels

A.4 Source and evidence notes

1. [1] Rémi Eismann, “Decomposition into weight * level + jump and application to a new classification of primes”, arXiv:0711.0865v4 (2010). Primary definition and founding results.
2. [2] OEIS, A117078 — weight of the nth prime.
3. [3] OEIS, A117563 — level of the nth prime.
4. [4] OEIS, A162174 — primes classified by level.
5. [5] OEIS, A162175 — primes classified by weight.
6. [6] Rémi Eismann, decompwlj.com atlas: 1,000 sequences, downloadable data and images, and the site’s own verification caveat.
7. [7] The On-Line Encyclopedia of Integer Sequences, live sequence count observed 11 August 2026.
8. [8] LMFDB overview and database API, used here as a model for object pages, invariants, provenance, and programmatic mathematical data.
9. [9] Kevin Ford, “The distribution of integers with a divisor in a given interval”, Annals of Mathematics 168 (2008), 367–433.
10. [10] Dimitris Koukoulopoulos, “Divisors of shifted primes”, IMRN 2010(24), 4585–4627.
11. [11] Ayla Gafni and Terence Tao, “Rough numbers between consecutive primes” (2025). Used only as a related methodological comparison, not as evidence for a WLJ theorem.
12. [12] Chai Wah Wu, “Can machine learning identify interesting mathematics?” (2018), on empirical-law fingerprints for OEIS sequences.
13. [13] Kazuhisa Nakasho, “IntSeqBERT: Learning Arithmetic Structure in OEIS via Modulo-Spectrum Embeddings” (2026).
14. [14] Alex Davies et al., “Advancing mathematics by guiding human intuition with AI”, Nature 600 (2021), 70–74.
15. [15] Bernardino Romera-Paredes et al., “Mathematical discoveries from program search with large language models”, Nature 625 (2024), 468–475.
16. [16] OEIS Wiki, B-files specification.
17. [17] Steven Clontz, “Database-Driven Mathematical Inquiry” (2024).
18. [18] Sara C. Billey and Bridget E. Tenner, “Fingerprint databases for theorems”, Notices of the AMS 60 (2013), 1034–1039.

A.5 Project materials inspected

The report also used the attached 12-page PDF 0711.0865v4.pdf, links.txt, the supplied PARI/GP algorithm files, seven project figures, the seventh and eighth Fable5 reports, and the modular-arithmetic report. The supplied 2026 reports were used to identify current project questions and status distinctions; public-source claims in this report were independently checked against the paper, OEIS, project site, and primary literature listed above.