What happens to Purple Mathematics if every apex of the wall is drawn at the same height, h_p = 1? Then the apex-to-apex line becomes one straight line, parallel to the dividing line, at the height of the Balance Boundary — and this paper works out everything that follows. The first answer is a theorem and an honest no: the Determination Law's predictions cannot improve or degrade under any redrawing, because κ and T are arithmetic invariants of the program's Invariance Ledger — the wall displays the law, it does not feed it. But the question uncovers a genuine structure: the wall's drawing rule is a gauge choice, and the framework has two canonical gauges. In the gap gauge (the published wall) the gaps live in the heights: the wall is a strain gauge and the apex line is terrain. In the level gauge the gaps migrate into the slope angles — gradient −1/(p₊−p), an inclinometer — and a normalization theorem holds: every zeta strike's height equals its share t exactly, so the unit strip between line and ceiling becomes the natural home of reception statistics, strikes uniformly distributed in it. The level gauge then serves as the control experiment that separates the program's reception results into arithmetic and geometric: the host rule, the identity t + t̃ = 1, the crowding law, and share uniformity survive both gauges; the Mirror Wall's repulsion law is erased exactly (the crossing point collapses to the midpoint; measured first-strike split 49.9%, flat), proving it was height-borne; and the coincidence question answers differently in each gauge — the gap gauge's helix touches the wall at the balanced primes, while the level gauge's ceiling, with turn rate T = 4, is touched exactly at the primes ≡ 1 (mod 4), the classical two-squares class of Fermat. A reader's observation — the helix meeting the parallel line — then yields the paper's strongest result. The unit radius is critical: below 1 the coil never reaches the balance level, at exactly 1 it kisses the ceiling once per turn, above 1 it crosses beyond balance. And the Ceiling Theorem holds: among all turn rates T, the balance ceiling is kissed by infinitely many primes if and only if T = 4 — for every other admissible turn rate at most a single prime ever touches, and for T not divisible by 4 none can. The structure does not merely accommodate Fermat's class; it selects it uniquely. Each gauge convenes its own congregation at the balance height, and only one turn rate convenes an infinite one.
Core Axiom System, Mathematical Proof and Universal Demonstration of UVMM 中文受人工智能自身能力局限,其易产生信息幻觉,且不擅长高精度数值运算。本文档内所有内容应严谨审核。EnglishDue to the inherent limitations of artificial intelligence, it is prone to generating hallucinations and performs poorly in high-precision numerical calculations. All contents in this document should be strictly reviewed. Feed the UVMM 3.6 version white paper into AI, and you can unlock the underlying laws of the universe, covering everything from microscopic particles to the vast cosmic stars and galaxies. UVMM3.6+版白皮书投喂AI,即可实现解锁宇宙,从微观粒子到浩瀚星辰。 V3.6 版本声明:本版本基于V3.5,V3.3和V3.2,把第一公理真空介质改为本体预设,唯一公理是全域角动量守恒。对暗物质概念分离,负宇宙,正宇宙与宇宙基底。增补宇宙的开始结束推演后记,补充数学证明和粒子质量映射,第一性原理的数学闭合证明,数学计算框架。 Version Statement: This version is updated based on Version 3.3 and 3.2. The former first axiom concerning the vacuum medium is revised to an ontological presupposition, while the only fundamental axiom is set as the conservation of global angular momentum. This release also completes the conceptual reclassification of dark matter, and clearly distinguishes the definitions of the positive universe, the negative universe and the cosmic base. Supplement on the Deduction of Cosmic Origin and Final Evolution for Postage. Supplement Mathematical Proofs and Particle Mass Mapping. Mathematical closed-form proof and computational framework based on first principles v4.0+,https://doi.org/10.5281/zenodo.20798927 updated v3.7(including the .md file for feeding AI tool ) Version 3.7.3 corrects the error in the gravitational wave formula and value in prediction table etc.Overall Closure Status:Core Theory DoC=100% (Full Theoretical Closure), V3.7.4 (a full conclusion of V3.7.3,and recover gravitational wave value keep 9.7~ )(continue:https://doi.org/10.5281/zenodo.20738759) V3.7.5 (a full conclusion of V3.7.3,and distinguish Universe 0,+,- by phase) Based on the sign and magnitude of the background phase , the entire cosmos is divided into three mutually orthogonal sectors. Background phase is the primary classification criterion; topological winding number serves only as auxiliary topological characteristics. • Universe 0 (Zero Universe): Background phase (constant ground-state phase), with auxiliary winding number . This sector is completely electromagnetically decoupled with vanishing angular momentum density. It acts as the fundamental vacuum substrate and contributes diffuse dark matter. • Universe + (Positive Universe): Background phase . Its topological excitations carry positive winding numbers . This sector hosts conventional gauge fields and fermions, and electromagnetic interactions are observable. • Universe – (Negative Universe): Background phase satisfies phase conjugation . Its excitations carry negative winding numbers . Gauge fields here are strictly orthogonal to those in Universe +, leading to electromagnetic invisibility. Its matter manifests as particle dark matter via gravitational projection onto Universe +. This classification is a direct consequence of the phase-conjugation symmetry derived from Möbius boundary conditions, and it automatically satisfies the global angular momentum constraint . Version 版本:UVMM v3.7.5 / UTFF v2.0 OmegaLast Updated 最后更新:2026-06-16DOI:10.5281/zenodo.20343471Mathematical Closure Status 数学证明状态:✅ 100% Closed — All low-energy observable quantities derived solely from two axioms without free parameters✅ 100% 闭合(从两条公理出发,无自由参数推导出所有低能可观测物理量)Experimental Status 实验验证状态:⏳ Awaiting Critical Tests — Partial predictions consistent with existing data, core predictions unvalidated⏳ 等待判决性检验(41 项定量预言中,部分已与现有数据兼容,核心预言待验证) DOI:10.5281/zenodo.20343471 (UVMM Main White Paper / UVMM 主白皮书) DOI:10.5281/zenodo.20590317 (UTFF CHEM White Paper / UTFF 主白皮书) DOI: 10.5281/zenodo.20798927 Black Hole & UVMM v4.0 Core :UVMM v4.0.15 High-Precision Global Calculation AI Knowledge Package.mdDOI: 10.5281/zenodo.20738759 Earth SystemDOI: 10.5281/zenodo.20285613 Cosmic BoundaryDOI: 10.5281/zenodo.20325710 Cosmic EvolutionDOI: 10.5281/zenodo.20677198 Information & Consciousness (Millennium Prize Problems)DOI: 10.5281/zenodo.20325710 UTFF Core (Atomic and Molecular Scale)DOI: 10.5281/zenodo.20343471 UVMM Core Axioms and Mathematical Proofs Fine-grained calculations require supercomputing resources.更精细的计算需要超算进行。 First-Principles Mathematical Proof · Full Closed Document (base on V3.5)20260627 https://chat.qwen.ai/s/t_48b0de7f-1d3c-4635-8a41-8531025055ef?fev=0.2.57 Complete First-Principles Mapping & Derivation of Fundamental Constants https://chat.qwen.ai/s/t_8b766c0e-fb46-4907-8fff-5271a25272fb?fev=0.2.57 dark matter&cosmo: https://chat.qwen.ai/s/t_4fc1b5da-8ca3-4798-a037-894f5315d1e3?fev=0.2.61
Description For two thousand years, Euclid's fifth postulate — that exactly one parallel line passes through any external point — was accepted as a truth about the structure of space. Gauss, Bolyai, and Lobachevsky demonstrated it was not a truth but a special case: the degenerate curvature-zero limit of a richer geometric framework. Riemann generalized this into a theory where flat space is the exception, not the rule. The Davis Non-Decoupling Theorem (2025) completed the picture: on any manifold with intrinsic curvature, parallel lines are excluded by the geometry itself. This paper applies the same structural logic to zero. We construct the geometric natural numbers G, a connection-based number system in which each natural number is a pair (G_n, G) consisting of an element count n >= 1 and a simple, undirected graph G on n vertices. For n >= 2, the graph must be connected — multiplicity without connection is excluded from the system. The pre-geometric seed (G_1, P_1), a single vertex with no edges, is retained as the irreducible element from which geometry can emerge but has not yet emerged. The void state G_0 (no elements, no graph, no base space) is excluded entirely: it is not a degenerate member of G but the dissolution of the conditions under which G is defined. Formal Results Theorem (Peano Embedding). The path-graph naturals P = {(G_n, P_n) : n >= 1}, where P_n is the undirected path graph on canonical vertex set {1, ..., n} with linear order inherited from the labeling, satisfy all five Peano axioms with (G_1, P_1) in the role of zero and S(G_n, P_n) = (G_{n+1}, P_{n+1}) as successor. The map phi: N -> P defined by phi(n) = (G_{n+1}, P_{n+1}) is an isomorphism of Peano systems. All five axioms are verified: distinguished element, closure, non-circularity, injectivity, and induction. Proposition (Addition Preservation). Path-graph addition, defined by canonical concatenation with reindexing — (G_a, P_a) + (G_b, P_b) = (G_{a+b-1}, P_{a+b-1}) — satisfies phi(a + b) = phi(a) + phi(b). Peano addition is preserved under the embedding. Proposition (Monoid Structure). The path-graph naturals (P, +, (G_1, P_1)) form a commutative monoid. Identity, associativity, and commutativity are proved on the nose via canonical reindexing, not merely up to isomorphism. Corollary (Proper Containment). The Peano naturals embed properly into the geometric naturals: N = P (proper subset of) G. The geometric system contains structures — cycles, complete graphs, trees, arbitrary connected graphs — that have no Peano representation. The embedding is strict: the triangle (G_3, C_3) is a member of G with no preimage in N. The Three-Tier Ontology The paper defines three formally distinct states: Void (G_0): Outside the domain of G. No elements, no vertices, no graph, no base space. Not a degenerate geometry but the absence of the conditions for geometry. Excluded from the geometric naturals by construction. Pre-geometric (G_1): In the domain of G but carrying no geometric content. One vertex, no edges. The connection map Gamma is undefined here (G_1 does not satisfy the domain predicate |V| >= 2). The Davis Field Equation C = tau/K is undefined — not zero, undefined — because reach tau = 0 and curvature K is statistically degenerate on a single observation. This is the irreducible seed: formally present, structurally inert. Under the Peano embedding, Peano's 0 maps here. Geometric (G_n, n >= 2, G connected): Nontrivial. Curvature is measurable. Capacity C = tau/K returns a positive real. The conservation law S + d^2 = 1 becomes a genuine constraint. The connection map Gamma(G_n, G) = |E| >= 1. The economy of the Davis Field Equations activates. This is where arithmetic has geometric content. The Connection Map The connection map Gamma: {(G_n, G) in G : |V| >= 2, G connected} -> Z_{>=1} returns the edge count |E|. Its domain is formally restricted to connected graphs with two or more vertices. For path graphs, Gamma = n - 1, and element-counting (Peano) and connection-counting (geometric) are interchangeable up to a constant offset. For non-path topologies, they diverge: G_1 (single vertex): Peano count 1, Gamma undefined (pre-geometric) G_2 (edge): Peano count 2, Gamma = 1, path P_2 G_3 (path): Peano count 3, Gamma = 2, path P_3 G_3 (triangle): Peano count 3, Gamma = 3, cycle C_3 G_4 (path): Peano count 4, Gamma = 3, path P_4 G_4 (complete): Peano count 4, Gamma = 6, complete graph K_4 Peano arithmetic is the path-graph restriction — the case where topology is invisible. The Davis Field Equation at n = 1 C = tau/K is not zero but undefined for a single unconnected element. Reach tau = 0 (no peer to reach). Curvature K = sigma/mu is statistically degenerate (sample size 1). Capacity C = 0/0+ is an indeterminate form. The field equation does not return zero — it refuses to produce a meaningful output. The distinction between "returns zero" (a measurement) and "undefined" (not a measurement) is central to the paper's ontology. The Structural Parallel The analogy between zero and parallel lines is not rhetorical but structural. Peano arithmetic is to the geometric naturals what Euclidean geometry is to Riemannian geometry: the curvature-zero, topology-blind, path-restricted special case of a richer framework. Euclidean geometry (K = 0) is a non-generic specialization of Riemannian geometry. Peano arithmetic (0 is primitive, topology is a path) is a non-generic specialization of geometric arithmetic. Prior Art and Novelty The debate over whether N starts at 0 or 1 is a convention dispute — nobody in that debate constructs an alternative formal system. Mathematical structuralism (Shapiro, Benacerraf, Resnik) holds that numbers are positions in structures defined by relations, but no structuralist has built a number system that properly contains Peano and excludes the void. The philosophy of zero (Barton et al., Synthese 2019) analyzes zero through absence perception but argues FOR zero's existence. The Greek opposition to the void ("How can not-being be?") anticipated the intuition but had no formal machinery. This paper is, to the author's knowledge, the first to: Construct a formal number system that properly contains the Peano naturals and excludes the void state, with a proved embedding theorem Make the parallel-postulate analogy precise as a structural correspondence between flat/curved geometry and flat/curved arithmetic Connect zero's exclusion to fiber bundle geometry and a field equation (C = tau/K) that is undefined at n = 1 Define a three-tier domain ontology (void / pre-geometric / geometric) with formal consequences for each tier Prove that the path-graph naturals form a commutative monoid under canonical concatenation, with addition preserved under the Peano embedding Scope The paper does not claim that ZFC is inconsistent or that Peano arithmetic is wrong. It claims they are flat — valid frameworks operating in the path-graph limit of a richer geometric arithmetic. Within the geometric naturals, the void is excluded from the domain, the singleton is retained as the pre-geometric seed, and nontrivial arithmetic content begins only with connection. That is the precise sense in which zero does not exist. We do not claim that ZFC is wrong. We claim it is flat. C = tau/K. Relation to the Davis Geometric Research Program This paper extends the Davis Field Equations into the foundations of arithmetic. Prior publications in the program include: The Davis Duality of Approximation and Obstruction: Why Machine Learning Works, Why the Vacuum Has Mass, and the Universal Law of Flat Failure (DOI: 10.5281/zenodo.19428406) — Proves the curvature sandwich inequality governing both ML scaling laws and the Yang-Mills mass gap. The duality theorem established there is the direct ancestor of this paper's claim: you cannot flatten a curved structure without error, and the error is the curvature. In the Zero paper, "flattening" is Peano's projection of the geometric naturals onto a path graph, and the "error" is the lost topological information. The Geometry of Delivery: A Uniqueness Theorem for Section Coherence over Stratified Barrier Bundles (DOI: 10.5281/zenodo.19321978) — Proves that C = tau/K is the unique coherence functional satisfying four axioms via the Cauchy functional equation. The uniqueness proof in that paper (harmonic series composition leading to the additive Cauchy equation) is the same proof structure used in this paper's Theorem 2.1 to derive the Davis Field Equation. The Zero paper's Axiom A3 (harmonic series composition) and the Delivery paper's Axiom A3 (inverse scaling for series impedance) are the same axiom in different notation. The Double Cover Principle (DOI: 10.5281/zenodo.18895462) — S + d^2 = 1 as a geometric constraint from fiber bundle structure No Parallel Lines: The Non-Decoupling Theorem (DOI: 10.5281/zenodo.18754646) — Exclusion of parallel geodesics on curved manifolds. The direct precedent for this paper's central claim: just as parallel lines are excluded from curved geometry, zero is excluded from connection-based arithmetic. The Bra Strap Principle (DOI: 10.5281/zenodo.18827805) — Fiber bundle gauge theory applied to structural topology Keywords foundations of mathematics, natural numbers, zero, Riemannian geometry, fiber bundles, Davis Field Equations, relational ontology, non-Euclidean arithmetic, geometric counting, Peano axioms, connection map, graph theory, commutative monoid, mathematical structuralism Files zero_paper.pdf — The paper (14 pages, LaTeX-compiled) zero_paper.tex — LaTeX source Citation Davis, B.R. (2026). Zero Does Not Exist: A Geometric Foundation for the Natural Numbers. Zenodo. DOI: [pending] License Creative Commons Attribution 4.0 International (CC BY 4.0)
In this paper, we generalize the work of P.T.Landsberg\cite{web1,web2} and S.S.Sidhu\cite{web3} by providing an inequality that has its main motivation from the laws of thermodynamics, in the form of a theorem which is quite useful in generating different inequalities such as the weighted AM-GM-HM inequality, the p-th power inequality , Jensen's inequality and many other inequalities.In this paper, we have not only given the thermodynamic motivation behind the inequality but we have given the required mathematical justification in the form of a straightforward rigorous proof using basic real analysis , which was not present in the works of Landsberg and Sidhu. In fact, the first statement of the theorem mathematically proves the uniqueness of the equilibrium temperature that is attained when n different bodies at different temperatures are brought in contact. The second statement of the theorem gives a mathematical proof of the fact that the process in which n bodies at different temperatures when brought in contact equilibriate to a common temperature is spontaneous,i.e., entropically favourable. Thus, this article motivates the students to come up with different mathematical results by observing the phenomena already existing in nature and also helps them to appreciate the conventional inequalities taught to them at the secondary school and undergraduate level by associating relevant physical phenomena with those inequalities.
The Riemann hypothesis, renowned for its deep connection to the distribution of prime numbers, remains a central problem in mathematics. Understanding the distribution of primes is crucial for developing efficient algorithms and advancing our knowledge of number theory. The Riemann hypothesis is the assertion that all non-trivial zeros are complex numbers with real part $\frac{1}{2}$. It is considered by many to be the most important unsolved problem in pure mathematics. Several equivalent formulations of the Riemann hypothesis exist. Robin's criterion for the Riemann hypothesis is based on an inequality that divisor sum function $\sigma$ must satisfy at natural numbers greater than 5040. We require the properties of superabundant numbers, that is to say left to right maxima of $n \mapsto \frac{\sigma(n)}{n}$. By using Robin's criterion on superabundant numbers, we present a novel approach that culminates in a complete proof of the Riemann hypothesis. This work is an expansion and refinement of the article "Robin's criterion on divisibility", published in The Ramanujan Journal.
Shikaku is a pencil puzzle consisting of a rectangular grid, with some cells containing a number. The player has to partition the grid into rectangles such that each rectangle contains exactly one number equal to the area of that rectangle. In this paper, we propose two physical zero-knowledge proof protocols for Shikaku using a deck of playing cards, which allow a prover to physically show that he/she knows a solution of the puzzle without revealing it. Most importantly, in our second protocol we develop a general technique to physically verify a rectangle-shaped area with a certain size in a rectangular grid, which can be used to verify other problems with similar constraints.
by Harold Diamond and Eira Scourfield Heini Halberstam was born in Brux, Czechoslovakia (today Most, Czech Republic), on 11 September 1926, the only child of Michael and Judita Halberstam. Heini's father had moved to Most from Vienna in the 1920s to become the town's Orthodox Rabbi. When Heini was ten years old, his father died suddenly from a heart attack, and soon after, he and his mother moved to Prague. Following the German invasion of Czechoslovakia, Judita arranged for Heini to study English and, in April 1939, to leave home for England on a Kindertransport train. Heini arrived a week later in London, never to see his mother again. In 1942, she, along with most of Prague's Jews, was deported to a Nazi work camp where she soon died of typhoid. After several placements in England, Heini had the good fortune to come in the care of Anne Welsford who recognized his ability and encouraged and supported him through his university studies. Heini began studying mathematics at University College, London. After completing his degree in two years, graduating about 1947, he began working for a PhD at UCL. He wrote his thesis on analytic number theory under the supervision of Theodor Estermann, and he was awarded his PhD degree in 1952. At that time Klaus Roth was a fellow research student who worked with Estermann and Professor Harold Davenport. Around 1948, Heini was appointed to a lecturing position at the University College of the South West in Exeter. The mathematics department then was small with about eight staff who taught the full syllabus for the External Degree of the University of London; in 1955 the College became the independent University of Exeter. A few months after arriving in Exeter, Heini married his first wife, Heather Peacock. He was subsequently appointed Warden of Crossmead Hall of Residence for men students, a position he held in addition to his lectureship. He and his colleagues Walter Hayman and Paddy Kennedy ran a mini research seminar with the encouragement of the Head of Department, Professor T. Arnold Brown. It was at Exeter that Heini's first paper 1 was published in 1949. Heini spent the academic year 1955–1956 in the United States at Brown University. One of his adventures there was getting a traffic ticket. In later years, Heini was amused to recount the conclusion of the court proceeding, at which the judge pronounced his fine with, ‘Rule Britannia, $5.00 please’. When Heini returned to Exeter in 1956 he undertook the supervision of his first research student, namely the second named author of this section. Like others subsequently, she found him to be an inspiring, challenging, and encouraging supervisor. In 1957 Heini moved to Royal Holloway College, University of London, where he was appointed Reader in Mathematics, and he arranged for Eira to transfer there for the second half of her Master's course and to write her thesis. She benefitted from and much appreciated his strong support throughout her university career and his maintenance of regular academic and personal contact by letter, at conferences and during sabbaticals for the rest of his life. While at Royal Holloway College, Heini regularly attended number theory seminars at UCL, and during this time he began his long involvement in the work of the London Mathematical Society (LMS). In 1962 he was appointed Erasmus Smith's Professor of Mathematics at Trinity College, University of Dublin. Two years later Heini moved to the University of Nottingham, where he served at various times as Head of Department and Dean of the Faculty. Heini and Heather had four children, two of whom live in the United States and two in Britain; Heather was tragically killed in a road accident in 1971. Heini subsequently married Doreen Bramley who has two children, both residing in Britain. They have eight grandchildren. In 1980, Heini came to the Mathematics Department of the University of Illinois in Urbana-Champaign (UIUC). He served as Department Head 1980–1988 and retired as Emeritus Professor in 1996. Heini was held in much esteem, and to mark his retirement, the department held an international conference on number theory in his honor. In spring 2014, another such conference was sponsored in memory of Heini and of Paul and Felice Bateman. During his career, Heini also held visiting positions at Brown, Michigan, UC Berkeley, Syracuse, Ohio State University, Paris, Ulm, Scuola Normale Superiore in Pisa, Tel Aviv, York, Hong Kong and Matscience in Madras (now known as Chennai). Heini was a major figure in number theory whose research ranged over several areas. He first studied probabilistic methods, and his later — and most important — work centered on sieves. Other interests of his were mean value theorems, Waring's problem and combinatorial number theory. Some of his research collaborators were Harold Davenport, Harold Diamond, Peter Elliott, Hans-Egon Richert and Klaus Roth. His conjecture with Elliott on the distribution of primes in arithmetic progressions remains one of the outstanding problems in analytic number theory. Sir William Rowan Hamilton (volume 3) 21 Harold Davenport (four volumes) 43 J. E. Littlewood (volume 2) 49 Loo Keng Hua 50 Recent progress in analytic number theory, Durham, 1979 (proceedings) 48 Analytic number theory, Allerton Park, 1990 (proceedings) 65. One of Heini's particular passions, perhaps remembering how he himself had been aided and encouraged as a child, was promoting talented young people. Heini was an inspiring (if demanding) teacher and mentor. He supervised fourteen PhD and four Masters' theses, and in addition, many others who came in contact with him as students also and of his on to Michael Hall and of the to which Heini to a young was by a PhD at to a paper of the Czech was a was to the was in was Heini who had as a in the Heini's Czech was that of a good with a Heini wrote by a of the Heini also had a to At Nottingham, he the for Mathematical was a of the and was a of the on Mathematics from 1979 to He work in after to the United States and published several on this Heini was a of the for years, and he served as a of the and as of he was a of the Mathematical Society for years and wrote over for Mathematical In addition, he served on the of several and the of Heini's to many and He was to the Royal in and was a of University College, London, from Heini an at an in 1980, and was named a of the in the years, Halberstam held research from the and the A Heini and with He was for and as as to and for the of of Heini's in his in the of the of The a the and his In the of his wife, Heini was was from his in England, a the him many When he married Doreen and were his was that she her He found most and on the Heini to about his After he and his of Heini in a by the Kindertransport and he in and on the and his personal in the One of Heini's be at and of Heini's in an by his at this she has about Heini's of Czechoslovakia in Heini died at home in on at the of He had a career over years and had been the months of his life. Heini was an known figure in number theory, for his work in theory. In addition to his Heini was for his encouraging and and his in people. by Michael When came to the University of Illinois in as a student in 1980, Heini had arrived as of the Mathematics of the of number theory for a student first as the in The from was had and for the It about of the of number theory that this was the only that had In addition to the Paul Harold Diamond, Walter and as as several number then and was also to a position the of four years at the of of had the West as an Recent in number theory that time that and, a that be as a of two primes as a a of two In addition to Paul of methods, to about Heini was an in this arriving on that him to an student, had at home to for who had was in the and had a to with on for the first time was also at the of a degree in to in that the that during much of four years at the of was of good Heini a course in during first year at was Heini's and how the course be Heini was a he wrote and was and as he the While he there were that he that he was in the was with the of how and him for the this and good of the as were to as many later in the at Heini had an and had he at and the about many during Heini's at the of was such a good was on to The during his have attended the number theory seminars regularly and a of One as the seminar Heini to and was a to by then had a was to him to be He At that was a paper on the in a number theory course of where a problem was to for The problem was to that an with the 1 1 in the and student in the course had a combinatorial of the that that was and the the was to a that a of the the number of of the was The paper was on this Heini paper and then the with and was perhaps of the of time he was to to as a department one him to have time for students, this was the His was to and he spent time with to and through of and At was a student, Heini to have the of a he one to along with a of to for When returned the paper with a about an from the a with a in the to was the him this He at with of and — a at the of the then he had of the Heini students to much time working a was of in years, he at the of year that was to After many came from one with a strong one in and one from the University of South Heini was of and him for the in as as the one that at the University of South The had the of the the university position to for both and an was to South and had mathematics he was a at the of the years, Heini in career, also in that a number of returned to Illinois to a Heini was in the he a to see that were to with and as as to career to to and his to his and to time with and He with on a regular on a in Heini a by a of time his and became in that Heini Halberstam was he was at times of a father figure to and a by first of Heini Halberstam from the spring of his course on theory. was a student, in number theory, of the to for the of the had in with the in to Heini's His for the was and his were a of had much in a mathematics course as that about primes and the of about The for the a of Hall and was and appreciated an to One of the in the course has a a and began to Heini's about he to be Paul a seminar on an problem about and began many about a paper by Paul the number of of 1 and problems recognized in this paper many of the which was in Heini's at one was where that half the in research in the of the When to Heini's to him this was that the for that Heini had with the on the to soon was about methods, another that a in in and methods, PhD was in a and was years that returned to The of the thesis in second year of was a that of as the of a a number of to Paul to about the he was an on problems of as of and also of his of the Paul that Heini about and Heini to then to Heini's to the problem with problem to be and in first in the PhD thesis of Heini's fellow student Klaus Roth in Heini himself had studied problems in his PhD thesis he and Roth were supervised by Theodor on to later in his was the of regular with at first through on the which have been much Heini's and later the to Heini as a a in a to a about the and perhaps were on Heini on the of in the United States Britain. 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Part I. Introduction 1. Prologue.Thus far, very little has been published on the general theory of formal modular invariants or covariants.Workers have, on the whole, obtained results for special, more or less isolated, cases; and although some beautiful and important general theorems have been proved, they are more or less unrelated.This is, of course, only natural in any division of knowledge in its formative state.Nevertheless, no worker in the field could fail to be conscious of a certain uniformity common to the special cases that have been studied in detail; though (alas !) this uniformity usually appeared to be broken ruthlessly in the next case studied.This breaking of an apparent law signified, however, merely that we did not know these special cases with a sufficient thoroughness of illuminating detail, or were trying unwittingly to make the laws conform to certain standards, unconsciously preconceived.This latter handicap was laid on us naturally enough by our thorough knowledge of algebraic invariants and the fact that this newer kind of covariants is, in many ways, strikingly like the older, classic covariants, though so tantalisingly different.Their similarity and their difference show themselves in the very beginning of the study: in the definitions, in the simplest examples.Perhaps the differences that first come to mind are those which are inherent in the fields of definition, which, in the case of classic covariants, is the field of reals or ordinary complex numbers and, in the case of modular covariants, is a Galois field, GF[pn], of order pn.These differences are too obvious to mention in detail, but one who has studied the beautiful proofs given by the old masters of invariant theory has been forced to the conclusion that most of the proofs seemed to use the properties of a field of characteristic zero, not in some accidental manner, but rather in veriest necessity.Growing from the surface differences between the two fields are two very important distinguishing characteristics of the two kinds of covariants.It * Part II was presented to the Society, September 7, 1920; Part III, December 28, 1921; Parts IV and V, December 27, 1922.