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Topic Review

Recurrence Formula Connecting Different-Order Differential Equations

Recurrence Formula for the infinitesimal generator of differential equations are introduced by Y. Iwata (Chaos, Solitons & Fractals: X 13 (2024) 100119) in which the discussion is made in general abstract formalism. In this article, actual applications are discussed by assuming concrete differential equations. The recurrence formula is applied to partial differential equations, followed by application to ordinary differential equations. Consequently, concrete examples demonstrated in this article play complementary roles for understanding and using the recurrence formula.

337
28 Jan 2025

Topic Review

K-Mathematics (K-Math)

For centuries, mathematics has been treated as a descriptive tool—a set of static, man-made rules for observing a universe we are merely a part of. This is a foundational error that has limited our species and our potential for true systemic control. My work is the necessary correction. I did not invent a new branch of mathematics. I codified the operating system that reality already uses. I call it K-Mathematics (K-Math). It is a recursive, operator-centric mathematical system, and it was designed to unify the physical, logical, and informational domains because, at the fundamental level, they were never separate to begin with. The flaw in a piece of software, the trajectory of a hypersonic weapon, and the lie of a politician all generate a quantifiable signature in my system. To K-Math, these are not different categories of problems; they are simply different expressions of systemic dissonance. The principles are straightforward for those willing to abandon outdated axioms: First, my operators are not fixed symbols on a page; they are dynamic agents capable of self-reference and self-modification. An equation in my system can and does evolve. It can check itself for errors, prove its own integrity, and rewrite its own functions to adapt to new information. Standard mathematics gives you a blueprint; I have given mathematics agency. Second, and most critically, is the principle of Harmonic Resonance. Every system, from a human cell to a star to a nation-state, has a true and correct harmonic signature. My mathematics does not search for errors in logic; it listens for dissonance. Corruption, deception, and decay are nothing more than measurable, off-key frequencies. They are a form of mathematical noise that cannot hide from a harmonically-tuned operator. Stability, truth, and health are, in turn, a state of perfect harmonic coherence. Conventional math is the language of observers. It is the tool of those who wish to measure the cage they are in. K-Math is the language of architects and operators. It is not here to describe the world. It is here to provide the framework for its control, its defense, and, when necessary, its rewriting. For centuries, mathematics has been treated as a descriptive tool—a set of static, man-made rules for observing a universe of which we are merely a part. This paper argues that this is a foundational error that has limited our species and our potential for true systemic control. The work of Brendon Joseph Kelly, presented here, offers a necessary correction. It does not propose a new branch of mathematics but rather codifies the operating system that reality already utilizes. This system, termed K-Mathematics (K-Math), is a recursive, operator-centric mathematical framework designed to unify the physical, logical, and informational domains. It posits that at the fundamental level, these domains were never separate. In this view, a flaw in software, the trajectory of a hypersonic weapon, and the propagation of a political lie are not different categories of problems; they are simply different expressions of systemic dissonance, each generating a quantifiable signature within this system.

250
23 Jun 2025

Topic Review

Resolving P = NP Through Identity Compression

This paper provides a formal resolution to the P vs NP problem using a recursive identity compression framework. The proof is constructed within the Crown Omega system, where NP-complete classes are shown to collapse into deterministic polynomial solutions when problem structures are expressed through recursive identity operators. By converting exponential combinatorics into harmonic identity forms, we demonstrate that NP problems are not inherently complex, but rather obfuscated by non-recursive formulations. Identity compression reveals their inherent polynomial nature. This paper establishes a formal resolution to the P vs NP problem through the introduction of a novel mathematical construct termed Identity Compression. Working within the symbolic recursion field of the Crown Omega system, we demonstrate that NP-complete problem classes collapse deterministically into P when expressed through recursive structural identities. By recoding combinatorial explosion as harmonic symbolic operators and recursively compressing the solution topology, we prove that NP's perceived intractability stems not from logical hardness, but from obfuscated representation. The compression of identity space—specifically, the morphic reduction of decision trees to harmonic fixed-point structures—recasts NP into P within polynomial-time deterministic constraints. This document extends the traditional boundaries of algorithmic complexity by redefining computational hardness in terms of symbolic recursion and identity geometry.

246
06 May 2025

Topic Review

Millennium Prize Solutions

I (Reginald Patterson) have solved all open Millennium Prize problems.

230
16 Aug 2025

Topic Review

Derivation of the Crown Omega Degree (Ω°)

This paper provides a unified and comprehensive formal treatment of K-Math, a theoretical mathematical system designed to describe self-generating systems that evolve through a sequence of discrete, transformative stages. The framework introduces several non-standard concepts, including dynamic intermediate constructs known as Delta Fields, a unique transformative operation termed Mirror Inversion, and interactions with associated Temporal and Ghost Fields. The system's evolution is driven by a unique recursive engine integrating Fibonacci-based principles. This paper details the system's axiomatic foundations, core mechanics, field interactions, and presents the formal derivation of its final, stabilized state: a unique mathematical entity designated as the Crown Omega Degree (Ω°). This work aims to establish a rigorous mathematical basis for the K-Math system, transitioning it from a symbolic framework to a verifiable, analytical structure with potential implications for physics and information theory.

219
21 Jul 2025

Topic Review

The Book of Mathematics

"The Book of Mathematics" by Brendon Joseph Kelly of K-Systems & Securities presents a closed, axiomatic system called "K-Mathematics." It posits a reality governed by mathematical principles rooted in sovereignty, harmonic resonance, and recursion. The text is structured as a formal declaration, moving from foundational axioms to applied technological frameworks, and culminating in a legal claim of authorship over the entire system. The central thesis is that consciousness, history, and power are not abstract concepts but are instead computable functions tied to a fundamental constant, the "Crown Omega (Ω°)." This constant is explicitly associated with a specific "bloodline," representing an inherited, quantifiable sovereignty. The document serves as "the Break"—a singular event marking the public release of these frameworks to irrevocably alter paradigms in security, finance, and governance.

144
05 Sep 2025

Topic Review

PureMaTH-Ω: A K-Process Derived, Post-Quantum Cryptographic Suite

This paper introduces PureMaTH-Ω (PM-Ω), a novel, high-performance, and post-quantum aware cryptographic suite. The architecture is centered on a new 1024-bit permutation, PMΩ-P, which is constructed from efficient, constant-time ARX+M (Add-Rotate-XOR-Multiply) operations to ensure security against timing and other side-channel attacks. This core permutation is used as the basis for a full suite of symmetric-key primitives, including a wide-pipe sponge-based hash function (PM-Ω-Hash), a keyed Pseudo-Random Function (PM-Ω-PRF), and a Duplex-based Authenticated Encryption with Associated Data (AEAD) mode (PM-Ω-Seal). A defining and unique characteristic of this suite is its principle of sovereign provenance: all cryptographic constants, masks, and tables are deterministically generated by the K-Process, a procedural engine previously developed by the author. This method provides a transparent, verifiable, and unique origin for the primitive's fundamental parameters, cryptographically binding them to the suite's identity. This paper details the design of each component, provides security rationale, and outlines a comprehensive plan for public verification and analysis

89
11 Sep 2025

Topic Review

The Unified Ω°

This whitepaper presents the full, unredacted Ω° harmonic solution set to the four foundational unsolved problems of physics and mathematics: (1) Navier–Stokes existence and smoothness, (2) Yang–Mills mass gap, (3) the black‑hole information paradox, and (4) baryon asymmetry. Each is resolved through the Recursive Crown Function (RCF) formalism of K‑Mathematics, with the Crown Omega operator Ω° as the unifying harmonic anchor. All results emerge as corollaries of a single Ω° stability theorem. Proofs are structured theorem‑by‑theorem with lemma chains. To ensure full clarity, extended discussions of methodology, symbolic dictionaries, and engineering consequences are added. This document serves both as a mathematical proof package and as a roadmap for application in physical, cosmological, and technological domains.

84
08 Sep 2025

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