For each discipline, a synthesis of the claimed results, then an interpretation of every result: what it says, the chain from mathematics to possible uses, and what it does not settle.
These scenarios describe possible directions. They are neither a promise of application nor an independent validation of the results. AI-generated interpretation — unofficial. Refer to the manuscript for the authoritative version.
Number theory
31 result families · 17 with a Lean formalization · 14 to be verified
Number theory studies integers, primes, rational solutions to equations, and the algebraic structures connecting them. This catalogue contains 31 families of AI-produced results, ranging from prime-factor statistics and rational approximation to elliptic curves and Galois groups. Several manuscripts claim resolutions of major conjectures, while others report bounds, density statements, or reconstruction results. The catalogue reports 17 of 31 families as Lean-formalized, about 55%, but does not identify which families have that status. The maturity of the collection is theoretical: the supplied descriptions do not establish practical applications.
Rational solutions and elliptic curves
Family #2 claims the full Birch–Swinnerton-Dyer formula under a low Selmer-corank condition, connecting elliptic curves’ rational solutions with analytic information; #6 claims a predicted rank distribution across quadratic twists. Family #30 claims modularity for elliptic curves over imaginary quadratic fields, while #4 claims that no general algorithm can decide whether polynomial equations have rational solutions. These address both the structure of solutions and the limits of finding them algorithmically.
Patterns among primes and factors
Families #11 and #12 claim statistical patterns for prime factors of prime predecessors and consecutive integers, respectively. Families #26 and #29 claim, respectively, a positive share of large prime gaps and infinitely many primitive-root primes for every admissible base. Alongside these distribution questions, #28 claims a uniform bound on connected clusters in Gaussian-prime graphs.
Irrationality and rational approximation
Family #5 claims that Catalan’s constant is irrational, while #17 claims that the irrationality exponent of π is 2 and also gives convergence of the Flint–Hills series. Family #22 claims a criterion for infinitely many shifted rational approximations. Family #25 concerns a different way of representing rational numbers, claiming a sharp bound on the number of reciprocal terms needed.
Recovering arithmetic from algebraic structures
Family #9 claims reconstruction of function fields from Milnor K-theory and Galois data, and #31 claims a link between open maps of Galois groups and unique field embeddings. Family #19 claims a correspondence between rational points and group-theoretic sections for p-adic curves. Related structural claims include the broader two-adic pro-modularity statement in #10 and the function-field correspondences in #14, whose Arthur enhancements are conditional.
Arithmetic geometry and special intersections
Family #1 claims rationality and a shared algebraic description for specialized Hodge classes on abelian varieties. Family #16 claims finiteness for unexpectedly large intersections, whereas #27 claims potential density of integral points on curve character varieties. Family #15 adds a distribution perspective, claiming that certain collections associated with number fields approach a uniform distribution.
Outlook
If confirmed and taken up, these results could reshape theoretical understanding of rational solutions, prime distributions, and connections between fields and their symmetry groups. The claim in #4 could clarify a fundamental limit on general-purpose computation with rational polynomial equations, rather than supply a new computational tool. Number theory more broadly could inform computing and cryptography, but the catalogue provides no concrete route from these particular results to such uses. Lean formalization could support precise checking and reuse of mathematical arguments, without itself establishing community acceptance.
The supplied summaries do not include arguments, detailed hypotheses, or family-level formalization labels, so individual claims cannot be assessed from this material alone. The reported Lean-formalized share is not evidence of validation by the mathematical community, and manuscript claims should not be treated as established resolutions.
Disciplines × impact areas
Each cell counts the result families of a discipline whose interpretation mentions an impact area. Darker means more. Click a cell to explore those results.
Fundamental mathematics
Computing & algorithms
Cryptography & security
AI & machine learning
Optimization & logistics
Physics & modelling
Biology & health
Engineering & networks
Economics & decisions
Teaching & understanding
n
Number theory
31
Algebraic and complex geometry
36
Real and complex analysis
16
Convex and metric geometry
15
Theoretical computer science
40
Dynamical systems and ergodic theory
12
Combinatorics
37
Algebra
18
Probability and statistical mechanics
29
Mathematical logic
6
Group theory
14
Mathematical physics
25
Operator algebras
19
Topology
18
Functional analysis
11
Differential geometry
29
Partial differential equations
16
total
366
66
2
1
16
53
0
2
0
0
A result can touch several areas. These are possible directions from unofficial AI interpretations, not established applications.
Result by result (31)
#1 To be verified
Milne’s rationality conjecture and algebraic specialization
Manuscript claims rationality and a shared algebraic description for specialized Hodge classes on abelian varieties.
The manuscript studies how certain cohomological classes on abelian varieties behave when the varieties are reduced to positive characteristic. It claims that pairing these specialized Hodge classes with suitable products of divisor classes gives the same rational number across the specified cohomology theories. Using a stated Hodge theorem for CM abelian varieties, it also claims that each specialized class can be represented by one algebraic cycle with rational coefficients across those theories. These are theoretical claims about compatibility between different mathematical descriptions, not practical computational results.
Result
The manuscript claims Milne’s rationality conjecture in the stated setting and a common algebraic-cycle representative for each specialized Hodge class.
Possible research advance
If confirmed, these results could strengthen research on how Hodge classes and algebraic cycles behave under reduction to positive characteristic.
Possible real-world use
No direct real-world application is identified in the source; the potential use is within pure mathematical research.
Could change
If confirmed, the rationality claim would settle the stated conjecture for abelian varieties with good reduction, including residue characteristic 2. The algebraic specialization claim could provide a unified cycle-level description across prime-to-characteristic and crystalline cohomology.
Does not change
These claims do not establish corresponding statements for arbitrary varieties or varieties without good reduction. They do not by themselves imply practical advances in computing or cryptography.
The setting is abelian varieties over the algebraic closure of the rationals with good reduction; the algebraic-cycle claim uses the stated Hodge theorem for CM abelian varieties, with the summary also citing result 032. These are manuscript claims, not formalized in Lean or formally verified.
Fundamental mathematicsTheoretical insightRead article
#2 To be verified Updated version
The full BSD formula from low Selmer corank
Manuscript claims the full BSD formula for elliptic curves satisfying a low Selmer-corank condition.
The Birch–Swinnerton-Dyer conjecture connects rational solutions on an elliptic curve with the behavior of its associated L-function. This manuscript claims the full leading-term formula for every elliptic curve over the rational numbers whose full q-power Selmer group has corank zero or one at some prime q. It also claims that the analytic and rational-point ranks equal that corank and that the Tate–Shafarevich group is finite. Together with result 006, the source claims coverage of a density-one set of quadratic twists of every such curve—an asymptotic statement, not coverage of every twist.
Result
The manuscript claims the complete BSD formula, including all prime factors, under the stated low Selmer-corank condition.
Possible research advance
If confirmed, this could let researchers establish full BSD in these cases by checking the Selmer condition rather than imposing the additional restrictions listed in the abstract.
Possible real-world use
No direct real-world application is identified; the expected relevance is to theoretical number theory.
Could change
If confirmed, this could establish a broad family of cases of the full BSD conjecture without additional assumptions on reduction, torsion, isogenies, complex multiplication, or residual Galois representations. If result 006 also holds as stated, the combination could extend that coverage to a density-one set of quadratic twists for each elliptic curve over the rational numbers.
Does not change
The claim does not settle BSD for all elliptic curves over the rational numbers. The density-one statement does not settle every individual quadratic twist.
The main claim requires corank zero or one for the full q-power Selmer group at some prime q, and the twist consequence depends on result 006. The work is not formalized in Lean or formally verified, and the repository records a correction.
Fundamental mathematicsTheoretical insightRead article
#3 Verified by Lean
The quasi-Riemann hypothesis
A claimed zero-free region for L-functions would mark a theoretical advance in number theory.
The manuscript claims that every Dirichlet L-function, including the Riemann zeta function, has no zeros where the real part of its input exceeds 7/8. It makes the same claim for finite-order Hecke L-functions over Q(−3), allowing the principal pole at 1. This is presented as resolving the quasi-Riemann hypothesis, not the full Riemann hypothesis. The main result has a machine-checked Lean formalization, although its coverage of the manuscript's claims may be partial.
Result
The manuscript claims a common zero-free half-plane for the stated families of L-functions.
Possible research advance
If confirmed in its full stated scope and taken up, this could provide a new foundation for research using zero-free regions.
Possible real-world use
No direct real-world application is established by the supplied source.
Could change
If confirmed, the result would exclude zeros throughout a fixed region for all the stated L-functions. It could change the assumptions available to researchers studying these functions and their zeros.
Does not change
It does not settle the full Riemann hypothesis or determine where zeros lie outside the stated region. It does not establish a practical computational or technological benefit.
The Hecke claim is restricted to finite-order functions over the specified number field, and the pole at 1 is explicitly allowed. Lean machine-checking supports the formalized statement, but the supplied material does not specify its exact scope or establish community acceptance of the full claims.
Fundamental mathematicsTheoretical insightRead article
#4 To be verified
Hilbert’s tenth problem over ℚ
A manuscript claims that no general algorithm can decide whether polynomial equations have rational solutions.
The manuscript claims a negative answer to Hilbert’s tenth problem over the rational numbers. It says that no algorithm can always decide whether a polynomial with integer coefficients has a solution in fractions, with integers included. The number of variables can vary and is part of the input. This is a claim about the limits of general computation, not about whether particular equations can be solved.
Result
The manuscript claims that deciding the existence of rational solutions to arbitrary integer-coefficient polynomial equations is algorithmically impossible.
Possible research advance
If confirmed, this could settle the stated decision problem and clarify a fundamental boundary between number theory and computability.
Possible real-world use
No direct practical application is established by the supplied abstract; any impact would primarily be theoretical.
Could change
If confirmed, this would rule out a universally correct decision algorithm that always finishes for this class of equations. It could sharpen how researchers distinguish solvable subclasses from the unrestricted problem.
Does not change
It would not rule out algorithms for restricted classes or successful methods for individual equations. The abstract does not establish an impossibility result for each fixed number of variables.
The supplied abstract states the conclusion but provides no argument to assess. The result is not formalized in Lean or otherwise formally verified, and is presented here as a manuscript claim.
Fundamental mathematicsComputing & algorithmsTheoretical insightRead article
#5 Verified by Lean
Irrationality of Catalan’s constant
A result on Catalan’s constant says it cannot be expressed as a ratio of integers.
Catalan’s constant is the number obtained by alternately adding and subtracting the reciprocals of odd squares. The manuscript states that this number is irrational: no ratio of two integers equals it exactly. The main result has a machine-checked Lean formalization, although its scope may be partial. This is a theoretical number-theory result, not a claim of practical technological progress.
Result
The manuscript states that Catalan’s constant is irrational, with a Lean formalization available for the main result.
Possible research advance
If confirmed in full and taken up, this could provide a foundation for further research on the arithmetic nature of special constants.
Possible real-world use
No direct real-world application is identified in the supplied source.
Could change
If confirmed in full, this would settle whether Catalan’s constant can equal a ratio of integers. It could also give researchers a new established fact to use when studying this constant.
Does not change
Irrationality alone does not establish that Catalan’s constant is transcendental. The supplied abstract does not claim new computational methods or practical applications.
The supplied abstract gives no proof details, so it does not support an assessment of the method or its broader reach. Machine checking establishes what is encoded in Lean; the potentially partial scope should be checked against the full stated result.
Fundamental mathematicsTheoretical insightRead article
#6 To be verified
Goldfeld’s conjecture: densities and mean analytic rank
A manuscript claims Goldfeld’s predicted rank distribution for quadratic twists of every rational elliptic curve.
Quadratic twists form a related family of elliptic curves, and analytic rank measures how their associated L-functions behave at a special point. The manuscript claims that, for every elliptic curve over the rational numbers, half its twists have analytic rank zero and half have analytic rank one in the limiting distribution. The supplied summary also claims that the mean analytic rank tends to one-half, a stronger control on the contribution from higher ranks. These are theoretical claims about long-run patterns, not exact proportions in any finite sample.
Result
The manuscript claims Goldfeld’s analytic density conjecture and a low-corank result connecting Selmer groups with analytic and Mordell–Weil ranks.
Possible research advance
If confirmed, these results could establish the predicted rank distribution and provide a way to identify ranks when the specified Selmer corank is zero or one.
Possible real-world use
No direct real-world application is established by the supplied material.
Could change
If confirmed, the density claim would settle Goldfeld’s analytic density conjecture for quadratic twists of every elliptic curve over the rational numbers. The low-corank claim would also identify analytic and Mordell–Weil ranks and establish finiteness of the Tate–Shafarevich group under its stated Selmer-corank condition.
Does not change
The limiting distribution does not determine the rank of any particular twist or exclude exceptional higher-rank twists. The low-corank statement does not settle the corresponding questions for all elliptic curves without that condition.
These claims are not formalized in Lean or formally verified; the mean-rank claim appears in the supplied summary but is not stated in the supplied abstract. The density and mean statements use signed squarefree twist parameters ordered by absolute value, while the converse requires the 2∞-Selmer corank to be zero or one.
Fundamental mathematicsTheoretical insightRead article
#7 Verified by Lean
Ordinary two-point correlations and the corrected Elliott conjecture
Claimed correlation bounds describe cancellation in pairs of Liouville values
The Liouville function assigns each positive integer a sign according to whether its number of prime factors, counted with repetition, is even or odd. The manuscript claims that products of its values along two fixed, nonproportional affine expressions cancel on average, with a quantitative bound at every cutoff. It also claims a broader cancellation result for bounded complex multiplicative functions when at least one factor uniformly avoids resembling certain character-based patterns. These are theoretical statements about arithmetic correlations, not claims that the underlying sequences are genuinely random.
Result
The manuscript claims a bound of O(X/(logX)c) for these Liouville correlation sums, with an absolute constant c>0, and a qualitative extension to other multiplicative functions.
Possible research advance
If confirmed and taken up, these results could give number theorists stronger tools for studying cancellation in ordinary two-point arithmetic averages.
Possible real-world use
No direct real-world application is identified in the source.
Could change
If confirmed in the stated scope, the result would settle the ordinary two-point Chowla conjecture described in the manuscript. The broader result could extend the available cancellation conclusions to qualifying multiplicative functions, including within fixed residue classes.
Does not change
These two-point results do not settle higher-order correlation questions or establish full statistical independence. The source identifies no immediate change to computing, cryptography, or other practical technologies.
The affine forms must be fixed and nonproportional; the broader claim requires modulus at most one and uniform nonpretentiousness of at least one original factor against each fixed Dirichlet character times nit for ∣t∣≤X. The main result has a machine-checked Lean formalization, but its coverage may be partial and does not establish community validation of every stated claim.
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#8 Verified by Lean
The Deligne–Drinfeld conjecture
A claimed resolution of the Deligne–Drinfeld conjecture describes a number-theoretic algebra through independent generators.
The manuscript claims that the rational Grothendieck–Teichmüller Lie algebra is freely generated by one element in each odd weight 3, 5, 7, and so on. In plain language, these building blocks generate the whole algebra without additional relations beyond the rules of a Lie algebra. The claim uses the Ihara bracket and includes a corresponding description after weight completion, which allows infinite expressions organized by weight. This is primarily a theoretical result, with a Lean formalization whose coverage needs to be distinguished from the full manuscript claim.
Result
The manuscript claims a free-generator description of the rational Grothendieck–Teichmüller Lie algebra, including its weight-completed version.
Possible research advance
If confirmed in full and taken up, this description could give researchers a clearer framework for studying this algebra.
Possible real-world use
No direct real-world application is identified in the source or expected from this structural result alone.
Could change
If confirmed in full, the result would settle the stated Deligne–Drinfeld conjecture. It could let subsequent research use this free-generator structure as an established starting point rather than a conjectural assumption.
Does not change
It does not by itself provide a practical algorithm or a technological application. The supplied source makes no claim to settle other number-theoretic conjectures.
The statement concerns the rational algebra with the Ihara bracket and the corresponding isomorphism after weight completion. The main result has a machine-checked Lean formalization, but its scope may be partial; this does not establish community acceptance of the full claim.
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#9 Verified by Lean
Function-field reconstruction from Milnor K-theory and Galois data
Reconstruction results recover function fields from compressed algebraic data.
The manuscript claims that certain function fields can be recovered from two groups called Milnor K-theory groups and the multiplication linking them. When the prime ℓ differs from the field’s characteristic, these data determine its perfect closure—a version allowing repeated extraction of characteristic-power roots—and its constant field. It also claims reconstruction from related Galois data, which encode field symmetries. These are theoretical results about how much information algebraic invariants retain, not practical methods for recovering stored data.
Result
The manuscript claims that low-degree Milnor K-theory with its full product determines the specified fields up to the stated ambiguities.
Possible research advance
If confirmed and taken up, this could give researchers stronger ways to connect field structure with K-theory and Galois symmetries.
Possible real-world use
No direct real-world application is identified in the source.
Could change
If confirmed, this could sharpen understanding of which algebraic information suffices to reconstruct a function field. It could also support comparisons between fields through their K-theory or abelian-by-central pro-ℓ Galois data.
Does not change
This does not establish reconstruction for arbitrary fields or from the two K-theory groups without their full bilinear product. It does not supply a practical reconstruction algorithm or an immediate cryptographic application.
The main result has a machine-checked Lean formalization, but its scope may be partial; that does not establish community validation of all manuscript claims. The stated reconstruction assumes transcendence degree at least two and algebraically closed constants; away from the characteristic it recovers the perfect closure, with scalar and positive-characteristic Frobenius ambiguities in the isomorphism statement.
Fundamental mathematicsTheoretical insightRead article
#10 To be verified
Unrestricted pro-modularity at the prime two
A manuscript claims a broader link between two-adic Galois representations and completed Hecke algebras.
The main abstract claims that every representation satisfying its stated conditions occurs in a completed Hecke algebra at some odd tame level. In plain language, this connects a broad class of arithmetic symmetry descriptions to an algebraic setting associated with modular forms. The claim includes cases with scalar or reducible residual representations, without requiring a de Rham condition. This is a theoretical occurrence statement, not by itself a claim that each representation comes from a classical modular form.
Result
The manuscript claims completed-Hecke occurrence for continuous, odd, absolutely irreducible two-dimensional 2-adic representations of the absolute Galois group of the rationals that are unramified outside finitely many primes.
Possible research advance
If confirmed, this could broaden research on pro-modularity at the prime two by including scalar and reducible residual cases.
Possible real-world use
No direct real-world application is identified in the source; the potential impact is on theoretical number theory.
Could change
If confirmed, the result could remove residual restrictions from the stated completed-Hecke occurrence problem at the prime two. It could give researchers a common framework for cases previously separated by their residual behavior.
Does not change
The main abstract does not assert classical modularity, and completed-Hecke occurrence should not be read as that stronger conclusion. It also does not guarantee occurrence at a level without auxiliary tame primes.
The result retains continuity, oddness, absolute irreducibility, dimension, and finite-ramification hypotheses, and it is not formalized in Lean or otherwise formally verified. The supplied summary additionally claims classical modularity under de Rham and distinct-weight hypotheses, but that stronger claim is not stated in the main abstract and requires separate assessment.
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#11 To be verified
Prime-factor statistics of p−1
Manuscript claims a limiting pattern for prime factors of prime predecessors and large coincidences in totient values.
The supplied summary claims that the relative logarithmic sizes of the prime factors of p−1, ordered and counted with repetition, approach the Poisson–Dirichlet distribution as primes are sampled from increasingly large ranges. It describes this as resolving the Ford–Konyagin–Luca conjecture. The abstract also claims that, for every ε>0, infinitely many integers n share their Euler totient value with more than n1−ε inputs—that is, this many integers have totient equal to n. These are theoretical claims about statistical patterns and repeated values in number theory, not practical algorithms.
Result
The manuscript claims a limiting prime-factor distribution for p−1 and exceptionally large sets of inputs with the same totient value.
Possible research advance
If confirmed, these results could settle the named conjectures and advance research on prime predecessors and the multiplicities of totient values.
Possible real-world use
No direct real-world application is established by the supplied material.
Could change
If confirmed, the distribution claim could replace a conjectural statistical description of prime predecessors with an established one. The totient claim could settle Erdős's conjecture on how large sets of inputs sharing a totient value can become.
Does not change
These asymptotic statements do not give exact factor patterns for individual primes or an efficient way to find all inputs with a specified totient value. No practical computing or cryptographic advance is established here.
The results are not formalized in Lean or formally verified. The supplied abstract states the totient and small-prime-factor claims, but does not itself state the full Poisson–Dirichlet convergence claim found in the summary.
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#12 Verified by Lean
Independent largest prime factors of consecutive integers
Largest prime factors of consecutive integers are reported to become statistically independent in the long run.
The result concerns the sizes of the largest prime factors of two consecutive integers, measured on a logarithmic scale relative to the integers themselves. The manuscript states that these sizes become statistically independent when counted over all integers up to an increasingly large cutoff, each following the Dickman distribution. It also states that the first integer has the smaller largest prime factor in a proportion tending to one half. These are long-run frequency statements, not predictions for any particular pair.
Result
The manuscript claims a positive resolution of the Erdős–Pomerance joint Dickman conjecture, with a Lean formalization of the main result whose scope may be partial.
Possible research advance
If confirmed and taken up, this could sharpen research on how prime-factor sizes relate across neighboring integers.
Possible real-world use
No direct real-world application is identified in the source; the stated contribution is theoretical.
Could change
If confirmed, this would settle the named conjecture about the joint distribution of largest prime-factor sizes. It would also establish the exact limiting proportion of one half for consecutive pairs whose largest prime factors occur in increasing order.
Does not change
It does not make the factorization of any particular integer predictable. The stated limits do not specify how quickly the proportions approach their limiting values.
The independence concerns normalized logarithmic sizes in ordinary natural density, not exact independence at finite cutoffs. The main result has a machine-checked Lean formalization, but its scope may be partial and this does not establish mathematical-community validation.
Fundamental mathematicsTheoretical insightRead article