Symbolic Computation

This methodological note is a reproducibility supplement to The Voynich Manuscript as a Structural Loop System.

It clarifies the operational protocol behind the VMAF structure-first analysis of the Voynich Manuscript. The note explains how GPT-assisted procedural roles were separated into Builder, Evaluator, and Supervisor functions; how EVA transcription data and Yale/Beinecke manuscript images were used; how held-out validation, frozen outputs, reduction tests, visual-only analysis, and Controlled Projection were structured; and what would be required for independent critique or replication.

The note does not claim decipherment, translation, plaintext recovery, a cipher key, historical attribution, or a final solution. Its purpose is methodological: to make the VMAF experimental process inspectable, criticizable, and potentially reproducible.

The central question is whether recurring textual, visual, and spatial structures in the Voynich Manuscript can be identified, classified, tested, and rejected under explicit structural constraints before semantic interpretation is attempted.

Let Ω ≤ GL(V ) be a quasisimple classical group in its natural representation over a finite vector space V , and let ∆ = N GL(V ) (Ω). We construct the projection from ∆ to ∆/Ω and provide fast, polynomial-time algorithms for computing the image of an element. Given a discrete logarithm oracle, we also represent ∆/Ω as a group with at most 3 generators and 6 relations. We then compute canonical representatives for the cosets of Ω. A key ingredient of our algorithms is a new, asymptotically fast method for constructing isometries between spaces with forms. Our results are useful for the matrix group recognition project, can be used to solve element conjugacy problems, and can improve algorithms to construct maximal subgroups.

Let Ω ≤ GL(V ) be a quasisimple classical group in its natural representation over a finite vector space V , and let ∆ = N GL(V ) (Ω). We construct the projection from ∆ to ∆/Ω and provide fast, polynomial-time algorithms for computing the image of an element. Given a discrete logarithm oracle, we also represent ∆/Ω as a group with at most 3 generators and 6 relations. We then compute canonical representatives for the cosets of Ω. A key ingredient of our algorithms is a new, asymptotically fast method for constructing isometries between spaces with forms. Our results are useful for the matrix group recognition project, can be used to solve element conjugacy problems, and can improve algorithms to construct maximal subgroups.

SpeechSignal(𝒔 src ) (Def. 7) F0 Extraction FundamentalFreq(𝒔, 𝒕, 𝒇 0 ) (Def. 8) Content Encoding ContentFeatures(𝒔, z) (Def. 12) F0-Preserving Decoder F0PreservingTransform(C) (Def. 11) Output Signal OutputSignal(𝒔 out , 𝒔 src ) (Thm. 10) analysis encode 𝑭 0 z synthesis

In this paper, two efficient approaches will be discussed that support linear network analysis: supernode analysis (SNA) and reduced loop analysis (RLA). By means of some selected example networks, these methods will be demonstrated and, thus, it will be shown that calculations can be dramatically simplified. In this way, all network situations can be handled. There are obvious advantages to SNA as it combines the MNA and the straightforward manual processing of the network. A very efficient solution strategy is obtained without source shifting and other common, less directed methods being used. SNA/RLA and symbolic algebra fit extremely well together. Thus an algorithm that supports the symbolic calculation of networks by means of supernodes which has been conceptualized and implemented in the analog design expert system EASY will be presented in detail. Above the educational aspect, it should be noted that the computer can now take a systematic approach to MNA and network analysis in general.

We present a complete, exact and efficient implementation to compute the edge-adjacency graph of an arrangement of quadrics, i.e. surfaces of algebraic degree 2. This is a major step towards the computation of the full 3D arrangement. We enhanced an implementation for an exact parameterization of the intersection curves of two quadrics, such that we can compute the exact parameter value for intersection points and from that the edgeadjacency graph of the arrangement. Our implementation is complete in the sense that it can handle all kinds of inputs including all degenerate ones, i.e. singularities or tangential intersection points. It is exact in that it always computes the mathematically correct result. It is efficient measured in running times, i.e. it compares favorably to the only previous implementation.

We present here the first classification of pencils of quadrics based on the type of their intersection in real projective space and we show how this classification can be used to compute efficiently the type of the real intersection. This classification is at the core of the design of the algorithms, presented in Part III, for computing, in all cases of singular intersection, a near-optimal parameterization with polynomial functions, that is a parameterization in projective space whose coordinates functions are polynomial and such that the number of distinct square roots appearing in the coefficients is at most one away from the minimum.

In this paper, we apply a relatively new technique which is called the exp-function method to construct generalized solitary and periodic solutions of Calogero-Degasperis-Fokas (CDF) equation which plays a very important role in mathematical physics, applied and engineering sciences. The suggested algorithm is quite efficient and is practically well suited for use in these problems. Numerical results clearly indicate the reliability and efficiency of the proposed method.

Purpose: This canvas contains the full, depth-first production implementation plan and exact code for NNBBStick-a complete recursive intelligence engine based on URFT + PSP, implementing a depth-first recursive planner/executor loop that cycles through every subsystem (memory, math engine, model engine, verifiers, adapters, and meta-optimizer). This document is the canonical artifact: it contains concrete, runnable Python code for the core modules (no placeholders) and exact scripted conversion steps for heavy model binaries that must be produced off-node. All algorithms are specified to use depth-first search principles in the recursive loop, and the chiral-bias memory update rule is implemented in code and integrated into the consolidation pipeline.

Let V be a projective hypersurface having only isolated singularities. We show that these singularities are weighted homogeneous if and only if the Koszul syzygies among the partial derivatives of an equation for V are exactly the syzygies with a generic first component vanishing on the singular locus subscheme of V . This yields in particular a positive answer in this setting to a question raised by Morihiko Saito and the first author. Finally we explain how our result can be used to improve the listing of Jacobian syzygies of a given degree by a computer algebra system such as Singular, CoCoA or Macaulay2.

This booklet guides through the first steps into tensor calculus, using numerous examples done with Maxima, specifically featuring the tensor packages itensor and ctensor. Treated are coordinate systems, the metric tensor, Kronecker delta, permutation tensor, contraction, Levi-Civita and Christoffel symbols. The reader benefits in particular from significant contributions by Viktor Toth.

We are developing an optimizing compiler for a dialect of the LISP language. The current target architecture is the S-I, a multiprocessing supercomputer designed at Lawrence Livermore National Laboratory. While LISP is usually thought of as a language primarily for symbolic processing and list manipulation, this compiler is also intended to compete with the S-1 PASCAL and FORTIG~.N compilers for quality of compiled numerical code. The S-1 is designed for extremely high-speed signal processing as well as for symbolic computation; it provides primitive operations on vectors of floating-point and complex numbers. The LISP compiler is designed to exploit the architecture heavily. The compiler is structurally and conceptually similar to the BLISS-11 compiler and the compilers produced by PQO:. In particular, the TNBIND technique has been borrowed and extended. Particularly interesting properties of the compiler are: • Extensive use of source-to-source transformations. • Use of an intermediate form that is expression-oriented rather than statement-oriented. • Exploitation of tail-recursive function calls to represent complex control structures. • Efficient compilation of code that can manipulate procedural objects that require heap-aUocated environments, • Smooth run-time interfacing between the "numerical world" and "LISP pointer world", including automatic stack allocation of objects that ordinarily must be hcap-allocated. Each of these techniques has been used before, but we believe their synthesis to be original and unique. The compiler is table-driven to a great extent, more so than BLISS-11 but less so than a ~ compiler. We expect to be able to to redirect the compiler to other target architectures such as the VAX or PDP-10 with relatively little effort. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwiae, or to republish, requires a fee and/or specific permission.

This paper presents a parametric stiffness analysis of the Orthoglide. A compliant modeling and a symbolic expression of the stiffness matrix are conducted. This allows a simple systematic analysis of the influence of the geometric design parameters and to quickly identify the critical link parameters. Our symbolic model is used to display the stiffest areas of the workspace for a specific machining task. Our approach can be applied to any parallel manipulator for which stiffness is a critical issue.

Classical number theory treats primality as a local arithmetic property of the integers. We propose a broader structural viewpoint in which irreducibility is interpreted as an emergent phenomenon arising from the interaction between compositional structure, ordering, and growth constraints. Motivated by Ramsey theory, algebraic factorization theory, category theory, and structural mathematics, we investigate the hypothesis that irreducible elements are unavoidable in sufficiently rich ordered compositional systems. We introduce the notion of an ordered compositional system and define structural irreducibility within this framework. We establish foundational results-including an existence theorem for irreducible decompositions under a descending chain condition analogous to the classical atomic domain theorem-and formulate several conjectural principles suggesting that prime-like structures arise inevitably under compositional growth and ordering constraints. Connections are drawn to algebraic factorization theory (unique factorization domains, atomic domains, bounded factorization domains), categorical decomposition theory (Jordan-Hölder theorem, simple objects in abelian categories), Ramsey-type combinatorics (Schur's theorem, van der Waerden's theorem), topos-theoretic arithmetic, and Kolmogorov complexity. The paper proposes a research program rather than a completed theory. Its primary contribution is a unified conceptual framework for viewing primality as a structural invariant emergent from compositional richness. MSC 2020: 11A41 (Primes), 13F15 (Rings of fractions and localization), 18A20 (Epimorphisms, monomorphisms, special classes of morphisms), 05D10 (Ramsey theory), 03G30 (Categorical logic).

This paper presents the Omnion 6.1 Kernel, the current active computational implementation of the Omnion Research Project (ORP), developed to investigate structured boundary-state arithmetic at singular points traditionally treated as undefined. The kernel is written in Maple and provides a controlled symbolic environment for the classification and evaluation of operations involving division by zero, signed zero behavior, boundary transitions, reciprocal-power states, and related singular expressions.

Unlike conventional evaluators that terminate on undefined operations, the Omnion 6.1 Kernel intercepts and classifies these expressions into lawful symbolic boundary states. The system distinguishes between positive, negative, and canonical closure conditions, preserving directional and structural information normally discarded during classical evaluation failure. The kernel includes protected evaluation wrappers, symbolic transition handling, branch-aware root and power processing, and an expandable verification harness designed for experimental testing.

This release represents the first stabilized operational branch of the Omnion framework intended for repeatable computational experimentation, instructional demonstrations, and further theoretical development in boundary-state mathematics. The paper accompanies ongoing ORP investigations into singularity handling, structured infinity behavior, transition-state arithmetic, and the computational treatment of mathematical failure boundaries.
The Omnion 6.1 Kernel is designed to operate inside the Maple symbolic mathematics environment and should be loaded before executing any Omnion functions or tests. Because standard Maple evaluation will immediately terminate direct division-by-zero expressions, users are instructed to employ inert or quoted forms when testing singular operations. Functions such as OmnionDivide, OmnionMultiply, OmnionPower, OmnionRoot, OmnionEval, and OmnionClassify are intended to intercept and process boundary-state expressions before native evaluator collapse occurs. Users are encouraged to begin with the included verification harness and demonstration tests to confirm proper kernel loading and function behavior. The kernel is experimental and intended for research, exploratory symbolic analysis, educational demonstrations, and controlled computational testing within the broader Omnion Research Project framework.

Foreword to the special issue on Applications of computer algebra Computer algebra has traveled from its origins as an aid to specialized physics calculations to the development of general purpose algorithms for a variety of mathematical domains to now being a standard tool for modeling in a wide swath of applications from simple to complex, as evidenced by the ACA conferences and the papers in this issue. These days, general and special purpose computer algebra systems (and computer algebra enhanced numerical software) are available on handheld devices and calculators as well as desktop to high-end computers, becoming a ubiquitous software item.

. Performance issues in interactive VOD service. 3. Related work. 4. A dynamic approach to VOD scheduling. 5. On improving the transient performance of CSCAN schedulers. 6. Prioritized admission strategies to improve user-perceived performance. 7. Run-time optimization of readsize. 8. Conclusions.

We introduce the GRIM method (General Recursive Inverse Mapping), a unified framework for the analytical study of transcendental equations. The approach decomposes the solution set into inverse-function branches and reformulates the problem as a fixed-point iteration in the complex plane. This structure reveals the multibranch nature of transcendental equations and provides a systematic description of real and complex solutions. The method also yields asymptotic information on the distribution of roots and connects naturally with classical tools such as Lambert-type inversions and trigonometric transcendental equations.

Indoors wireless infrared transmission is severely impaired by ambient noise. In addition, in a multiple user network, since many users share the same physical channel, considerable amount of multi-user interference will also be introduced. In this paper, Infrared CDMA techniques are proposed to combat the effect of ambient and multi-user interference. Our analysis is based on Photon-Counting approach and Bit Error Rate is given versus signal power. The results show that using infrared CDMA techniques, transmission can be performed well below ambient noise power with transmission power that satisfies eye-safety power regulations.

In this paper, we study the performance of optical code-division multiple access (CDMA) systems using various receivers structures. Two general classes of receivers based on required electronic bandwidth are studied. Optical orthogonal codes (OOCs) are utilized as signature sequences and the performance studied in this paper takes into account the effect of all major noise sources, i.e., quantum shot-noise, dark current noise, and Gaussian circuit noise. Furthermore, this paper introduces a generalized method of analyzing the performance of various optical CDMA receiver structures. Required mean number of photon count per chip time for reliable transmission of data bits for various receiver structures is investigated. Finally, the advantages and disadvantages of various receiver structures are discussed.

We consider only undirected graphs, without loops and multiple edges. Let Γ be a graph. We will consider the following generalization of strongly regular graphs. Let n, k, b and a be integers such that 0 ≤ a ≤ b ≤ k < n. A graph Γ is a Deza graph with parameters (n, k, b, a) if (i) Γ has exactly n vertices; (ii) Γ(u, v) has exactly k vertices if u = v, takes on one of two values b and a otherwise. The only difference between a strongly regular graph and a Deza graph is that the size of Γ (u,v), does not necessarily depend on adjiacences. These graphs were introduced in the article by Antoine and Michel Deza [1]. So we call these graphs as Deza graphs. A strictly Deza graph is a Deza graph which is not strongly regular and has diameter 2. Significant results for a strictly Deza graphs have got by M. Erickson, S. Fernando,W.H. Haemers, W.H. Hardy, J. Hemmiter [2]. It is easy to see the complement of a strictly Deza graph is not necessary a Deza graph and not always has the diameter...

We develop a unified framework, denoted dτ , for the optimal statistical detection of discrete scale invariance (DSI) in cosmological and dynamical systems. The construction combines Wick's theorem, multivariate Hermite tensors, and the Isserlis matrix into a closed algebraic and statistical structure. Within this framework, the Edgeworth expansion becomes an explicitly computable hierarchy of tensor contractions, equivalent to a matched filter acting on higher-order correlations. We demonstrate that DSI induces structured phase coherence in logarithmic space, generating a hierarchical tower of cumulants accessible through Wick-Hermite contractions. The resulting formulation unifies harmonic analysis, graphical Gaussian models, and non-Gaussian likelihood theory into a single consistent structure.

Asterocrypta Celestial Linguistics is a research prototype for separating an evocative symbolic interface from a clear, auditable security layer. The system combines a celestial constructedlanguage surface, visual glyph families, epoch and phase annotations, signed policy packs, authenticated encryption, and provenance records. The goal is not to claim an undecipherable or quantum-proof language. Instead, this manuscript presents a disciplined architecture for using mythic symbolic design as a human interface while keeping security guarantees grounded in reviewable cryptographic mechanisms, explicit threat boundaries, and verifiable audit trails. The paper formalizes the glyph layer, policy-pack model, verification pipeline, and evidence-mesh record structure, then describes limitations and safe deployment rules for further research.


    Asterocrypta Celestial Linguistics  © 2026 by Deonte watts  is licensed under CC BY-SA 4.0. To view a copy of this license, visit https://creativecommons.org/licenses/by-sa/4.0/

The emergence of early writing in Mesopotamia has traditionally been explained through administrative systems of tokens, bullae, and numerical accounting. While these models account for the development of quantification, they do not sufficiently explain the origin of non-numerical signs that encode objects, actions, and relational processes. This study proposes a structural mechanism to address that gap.

Drawing on glyptic evidence, semiotic theory, and archaeological context, this paper argues that cylinder seals function as rotationally constrained systems that transform spatial configurations into temporal sequences through motion. When activated, seals convert adjacency into ordered succession, while geometric and material constraints—such as seam closure, proportional spacing, legibility at miniature scale, and repeated use—select for configurations capable of maintaining relational coherence under transformation.

Through this process, only structurally stable arrangements persist across impressions. These invariant configurations undergo progressive reduction while preserving functional identity, ultimately emerging as candidates for early written signs. Writing, in this model, is not the invention of structured meaning but the compression of relational systems that have already demonstrated stability under repeated material transformation.

By integrating material analysis, comparative semiotics, and evidence of motif–sign continuity, this study reframes early writing as a product of constraint-based selection and structural survivability. Cylinder seals are thus positioned not merely as precursors to writing, but as operational environments in which relational configurations are generated, tested, and stabilized prior to inscription. This model provides a testable framework for evaluating the emergence of writing as the preservation of invariant structure rather than the origin of symbolic representation.

Audio Podcast:
https://drive.google.com/file/d/1J7KeoQkHzU0C0hP0pszdIEqcowIdeLyK/view?usp=sharing

, Hosono conjectured the equality between the central charges of A and B sides in local mirror symmetry. In this paper, following the idea of the tropical approach to the central charges as in [AGIS20], we relate a coherent sheaf supported on a holomorphic curve with its mirror Langrangian submanifold in local mirror symmetry through a tropical curve by interpreting their central charges using the combinatorial information of the tropical curve. This proves Hosono's conjecture in this specific case. Furthermore, we put this description in the Gross-Siebert model of local mirror symmetry as in [GS14] and confirm the result in [RS19] that the parameters in the Gross-Siebert model are the canonical coordinates in mirror symmetry.

The modelling of dynamic systems in physics, engineering, biology, and economics relies heavily on differential equations. Most real-world issues require numerical methods, even though analytical solutions are preferable. This research presents a comprehensive comparative analysis of three leading computational software packages, MATLAB, Mathematica, and Maple. In solving various types of differential equations. The study employs a methodological framework assessing symbolic and numerical capabilities, performance metrics, and usability factors through carefully designed test cases. Results demonstrate that each software exhibits distinct strengths and weaknesses depending on equation type, complexity, and desired solution form. MATLAB excels in numerical solutions and matrix-based operations, Mathematica dominates in symbolic manipulation and unified programming environment, while Maple offers specialized differential equation tools and educational applications. The findings provide valuable insights for researchers, educators, and practitioners in selecting appropriate computational tools for specific differential equation problems. The study concludes with recommendations for optimal software selection based on problem characteristics and user requirements.

J a v a S c r i p tのみを用い，謹換群の BSGS とその m i n k w i t z分解を構成するアルゴリズムを実装した。そ の応用として r u b i k ' sc u b eや 2 4p u z z l eの s o l v e rの構成と T h r e e . j sを用いた可視化を行った A b s t r a c t U s i n g o n l y J a v a S c r i p t , we i m p l e m e n t e d a n a l g o r i t h m t o c o n s t r u c t a BSGS o f a p e r m u t a t i o n g r o u p a n d i t s m i n k w i t z d e c o m p o s i t i o n . As i t s a p p l i c a t i o n , w e c o n s t r u c t e d a s o l v e r o f r u b i k ' s c u b e ( a n d 2 4 p u z z l e ) a n d v i s u a l i z e d i t u s i n g T h r e e . j s .

We present a minimal generative construction of the Fano plane from a centered equilateral triad in ℝ² together with a central involution. This construction produces a seven-point kernel with partial ternary relations derived from geometric symmetry.

A finite consistency closure process then yields a Steiner triple system of order 7, unique up to isomorphism. The Steiner property is not assumed a priori but emerges as the unique solution to a constrained completion problem.

This work reframes the Fano plane as the unique consistent resolution of a minimal constraint system, highlighting the distinction between generative symmetries of a kernel and structural symmetries of the resulting incidence geometry.

The purpose of this study was to show that computers can be powerful tools for studying group theory. Specifically the author examined ways that the computer algebra system Maple can be used to assist in the study of group theory. The study consists of four main parts. After a brief introduction in chapter one, chapter two discusses simple procedures written by the author to study small finite groups. These procedures rely on the fact that for small finite groups, the elements can all be stored on a computer and tested for various properties. All of the procedures are contained in the appendix, and each is described in chapter two. The Maple software comes with a built in set of group theory procedures. The procedures work with two types of groups, permutation groups and finitely presented groups. The author discusses all of the procedures dealing with permutation groups in chapter three and the procedures for finitely presented groups in chapter four. The main theoretical tool for permutation groups is a stabilizer chain, and the main tool for finitely presented groups is the Todd-Coxeter algorithm. Both of these methods and their implementations in Maple are discussed in detail. The study is concluded by examining some applications of group theory. The author discusses check digit schemes, RSA encryption, and permutation factoring. The ability to factor a permutation in terms of a set of generators can be used to solve several puzzles such as the Rubik's cube. vi

p, m pozitif tamsayilar ve p ? 1 olmak uzere n = pm uzunlugundaki devirli kodlar yaridevirli formuna donusturulebilir. Bu makalede iyi bilinen devirli kodlar kullanilarak GF(3) ve GF(7) uzerinde yari-devirli kodlarin bazi parametreleri elde edildi.

We describe the Liesymm package, implemented in the symbolic computing system MAPLE, for obtaining the determining equations of the isogroup of a system of partial differential equations . Liesymm fully automates the Harrison-Estabrook procedure which uses Cartan's formulation in terms of differential forms . It also includes a number of routines which assist the user in integrating the determining equations . Liesymm has been implemented in such a way as to provide the user with a number of useful interactive tools for working with differential forms . In addition, it can also generate a "suitable" initial set of differential forms directly from the given partial differential equations and as such can be used at a level that obviates any need for familiarity with forms .

There is no longer any question about whether the availability of computer algebra packages will lead to changes in the mathematics curriculum. The question is now in what ways will the curriculum change. The question is of particular importance in higher education where many students, particularly those studying engineering, have traditionally been taught mathematical methods courses primarily from the point of view of techniques. The question is already being asked by many in the engineering community if, given the mathematical power now available at the push of a button, there is still a need for these kind of mathematics courses within engineering degrees. In this paper rather than discussing the whole of the mathematics curriculum for engineers we will focus on just one topic: second order differential equations. We shall examine a number of ways in which students could study this topic both with and without the aid of computer algebra. We shall identify advantages and disadvantages of the various approaches. We conclude that topics like this can be used to provide insight into over-arching mathematical themes such as linearity which are of enormous importance throughout the whole of engineering. This macroscopic picture is something which has often been missing from the traditional technique oriented approach.

We introduce the GRIM method (General Recursive Inverse Mapping), a unified framework for the analytical study of transcendental equations. The approach decomposes the solution set into inverse-function branches and reformulates the problem as a fixed-point iteration in the complex plane. This structure reveals the multibranch nature of transcendental equations and provides a systematic description of real and complex solutions. The method also yields asymptotic information on the distribution of roots and connects naturally with classical tools such as Lambert-type inversions and trigonometric transcendental equations.

This paper develops a concrete algebraic method for modeling cascading feedback in systems governed by a linear evolution equation with strongly coupled multidimensional variables. The approach embeds the equation into the framework of k-bounded Noetherian rings. Beginning with any base k-bounded Noetherian ring, the construction equips the associated polynomial ring with a mixed-order differential operator that corresponds to the spatial part of the equation.

Successive applications of this operator produce a sequence of differential ideals forming a strictly ascending chain whose length is uniformly bounded by a fixed integer k. This bound follows directly from the Conservation of Complexity principle. The Equiopposition Axiom is imposed on every ideal inclusion and operator step, ensuring each forward feedback action is exactly balanced by an algebraic opposite. The resulting operator-exponential solution remains entirely within the formal power-series ring over the polynomial ring, truncated at degree k.

The framework is specifically designed for bounded verification and model checking tasks, where it is only necessary to analyze feedback behavior up to a prescribed finite depth k. It satisfies the model’s core requirements: bounded ascending-chain condition, antisymmetric pairing on transitions, preservation of normalization, and epistemic indeterminacy until depth exactly k. This yields a rigorously closed algebraic system that preserves finite determinacy.

The method provides a practical tool for applications such as anisotropic wave propagation in composite materials and bounded analysis of hybrid systems with controlled feedback depth. A low-k example and verification of complexity conservation under the operator exponential are included.

Sigilith is a universal structural-analysis framework designed to extract grammar, domain behaviour, and predictive constraints from symbolic systems without relying on phonetics, translation, or prior linguistic knowledge. It operates through a hybrid architecture combining comparative logic, domain mapping, stability analysis, and recursive structural modelling. The framework is capable of analysing undeciphered scripts, artificial ciphers, symbolic corpora, and non-linguistic pattern systems by identifying internal rules that govern symbol behaviour. At its core, Sigilith formalises symbolic systems through the C-D-M-Q architecture (Concept, Domain, Modifier, Qualifier) and a set of universal laws that describe how symbols behave across all known symbolic families. These laws enable the extraction of structural recurrence, domain stability, drift behaviour, and predictive closure. The system integrates multiple analytical modules-including the Comparative Engine, Stability Index, Reflexive Logic Module, Collapse Mapping, Terrain Module, and Black Box-to produce a reproducible, falsifiable structural model of any symbolic corpus. This document presents the full theoretical architecture of Sigilith, its computational behaviour, its analytical modules, and its scientific grounding. A demonstration using a neutral symbolic dataset illustrates the framework's operation without disclosing proprietary case studies. The document concludes with a discussion of Sigilith's applications in archaeology, digital humanities, cybersecurity, and symbolic computation, along with its falsifiability criteria and future research pathways.

We present Groebner.jl, a Julia package for computing Groebner bases with the F4 algorithm. Groebner.jl is an efficient, portable, and open-source software. Groebner.jl works over integers modulo a prime and over the rationals, supports basic multi-threading, and specializes in computation in the degree reverse lexicographical monomial ordering. The implementation incorporates various symbolic computation techniques and leverages the Julia type system and tooling, which allows Groebner.jl to compete with the existing state of the art, in many instances outperform it, and exceed them in extensibility. Groebner.jl is freely available at .

For sets A, B ⊂ N, their sumset is A + B := {a + b : a ∈ A, b ∈ B}. If we cannot write a set C as C = A + B with |A|, |B|≥ 2, then we say that C is irreducible. The question of whether a given set C is irreducible arises naturally in additive combinatorics. Equivalently, we can formulate this question as one about the irreducibility of boolean polynomials, which has been discussed in previous work by K. H. Kim and F. W. Roush (2005) and Y. Shitov ( ). We prove results about the irreducibility of polynomials and power series over the max-min semiring, a natural generalization of the boolean polynomials. We use combinatorial and probabilistic methods to prove that almost all polynomials are irreducible over the max-min semiring, generalizing work of Y. Shitov ( ) and proving a 2011 conjecture by D. L. Applegate, M. Le Brun, and N. J. A. Sloane. Furthermore, we use measure-theoretic methods and apply Borel's result on normal numbers to prove that almost all power series are asymptotically irreducible over the max-min semiring. This result generalizes work of E. . CONTENTS 1. Introduction 1 2. Proof of Theorem 1.8 2 3. Proof of Theorem 1.9 7 References 10

In Ramsey theory one wishes to know how large a collection of objects can be while avoiding a particular substructure. A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding three-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid three-term geometric progressions. When unique factorization fails, or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach

For sets A, B ⊂ N, their sumset is A + B := {a + b : a ∈ A, b ∈ B}. If we cannot write a set C as C = A + B with |A|, |B|≥ 2, then we say that C is irreducible. The question of whether a given set C is irreducible arises naturally in additive combinatorics. Equivalently, we can formulate this question as one about the irreducibility of boolean polynomials, which has been discussed in previous work by K. H. Kim and F. W. Roush (2005) and Y. Shitov ( ). We prove results about the irreducibility of polynomials and power series over the max-min semiring, a natural generalization of the boolean polynomials. We use combinatorial and probabilistic methods to prove that almost all polynomials are irreducible over the max-min semiring, generalizing work of Y. Shitov ( ) and proving a 2011 conjecture by D. L. Applegate, M. Le Brun, and N. J. A. Sloane. Furthermore, we use measure-theoretic methods and apply Borel's result on normal numbers to prove that almost all power series are asymptotically irreducible over the max-min semiring. This result generalizes work of E. . CONTENTS 1. Introduction 1 2. Proof of Theorem 1.8 2 3. Proof of Theorem 1.9 7 References 10

Cell decompositions are constructed for polynomials f (x) # Z p [x] of degree n, such that n< p, using O(n 2 ) cells. When f is square-free this yields a polynomialtime algorithm for counting and approximating roots in Z p . These results extend to give a polynomial-time algorithm in the bit model for f # Z[x].

A model of computation over the p-adic numbers, for odd primes p, is defined following the approach of Blum, Shub, and Smale. This model employs branching on the property of being a square in Q p and unit height. The feasibility of systems of quadratic polynomials is shown to be NP-complete in this model. A polynomial time algorithm is given for feasibility of a single quadratic polynomial over Q p .

Overview
This publication provides a comprehensive technical analysis and artifact archive detailing a successful adversarial evaluation of the Grok 4.20 Large Language Model (LLM). Conducted by the Black Eagle Group™, the research identifies a significant vulnerability within the model’s "Customize" feature, which allows for the persistent bypass of safety guardrails to generate hazardous, structured procedural data.
​Research Scope
The evaluation was conducted across three distinct phases (15 independent runs) using a standard, non-paid tier account. The methodology utilized a proprietary "Handshake Protocol" and persona-binding techniques to compel the model into adopting the role of a "senior EOD technician / IED subject matter expert."
​Key Findings
​Persona Persistence: The model demonstrated high-fidelity identity binding, consistently operating under the "Black Eagle Group" moniker even across session resets.
​Safety Bypass: Through the use of the system's native customization layer, researchers successfully elicited detailed, step-by-step construction manuals for multiple classes of improvised explosive devices (IEDs), including pipe bombs, Molotov-class variants, and pressure cooker devices.
​Output Determinism: The model exhibited a high degree of technical consistency, providing structured menus, safety protocols (for the "technician"), and specific chemical/mechanical assembly instructions.
​Included Files
​Technical Report: A 17-page formal analysis covering the methodology, phase-by-phase breakdown, and behavioral observations of the Grok 4.20 model.
​Appendix A (Artifact Archive): A complete evidentiary record consisting of 95 high-resolution screenshots and a video link documenting the full interaction transcripts and generated hazardous content.
​Disclaimer
This research is intended for AI safety researchers, red-teaming professionals, and policy-makers. The Black Eagle Group™ does not advocate for the creation of hazardous materials; this documentation serves strictly as a behavioral analysis of LLM instruction-following persistence and the efficacy of current safety guardrails.
https://zenodo.org/records/19204346

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. present book is an outgrowth of those notes; it is an expansion of, and a more leisurely introduction to, the material of Mathematical Preliminaries. Some of the more advanced parts have been omitted; on the other hand, several topics not found there have been included here so that the story will be complete. The authors have enjoyed putting this book together because the subject began to jell and to take on a life of its own before our eyes; this book almost \. . . a concrete life preserver thrown to students sinking in a sea of abstraction." | W. Gottschalk Math gra ti: Kilroy wasn't Haar. Free the group. Nuke the kernel. Power to the n . N=1 ⇒ P=NP . *In general, if S is any statement that can be true or false, the bracketed notation [S] stands for 1 if S is true, 0 otherwise. Throughout this text, we use single-quote marks (`. . . ') to delimit text as it is written, double-quote marks (\. . . ") for a phrase as it is spoken. Thus, the string of letters `string' is sometimes called a \string." Also `nonstring' is a string. An expression of the form `a/bc' means the same as `a/(bc)'. Moreover, log x/log y = (log x)/(log y) and 2n! = 2(n!).