{"id":31195626,"url":"https://github.com/gatanegro/logos","last_synced_at":"2026-02-16T04:02:52.510Z","repository":{"id":314377465,"uuid":"1055267885","full_name":"gatanegro/LOGOS","owner":"gatanegro","description":"Logos Theory from 3D Collatz Framework -UOFT","archived":false,"fork":false,"pushed_at":"2025-09-12T03:39:48.000Z","size":5,"stargazers_count":0,"open_issues_count":0,"forks_count":0,"subscribers_count":0,"default_branch":"main","last_synced_at":"2025-09-20T03:48:10.491Z","etag":null,"topics":["app-for-calculation-dist-py","collatz-conjecture","dynamical-systems","fine-structure-constant","mathematical-physics","modulo-arithmetic","nonlinear","number-theory","recursive-functions-"],"latest_commit_sha":null,"homepage":"","language":null,"has_issues":true,"has_wiki":null,"has_pages":null,"mirror_url":null,"source_name":null,"license":"mit","status":null,"scm":"git","pull_requests_enabled":true,"icon_url":"https://github.com/gatanegro.png","metadata":{"files":{"readme":"README.md","changelog":null,"contributing":null,"funding":null,"license":"LICENSE","code_of_conduct":null,"threat_model":null,"audit":null,"citation":null,"codeowners":null,"security":null,"support":null,"governance":null,"roadmap":null,"authors":null,"dei":null,"publiccode":null,"codemeta":null,"zenodo":null,"notice":null,"maintainers":null,"copyright":null,"agents":null,"dco":null,"cla":null}},"created_at":"2025-09-12T02:55:05.000Z","updated_at":"2025-09-12T03:54:16.000Z","dependencies_parsed_at":"2025-09-12T05:47:52.670Z","dependency_job_id":"61f624dc-ed5c-4e26-96e0-0293d8b7df29","html_url":"https://github.com/gatanegro/LOGOS","commit_stats":null,"previous_names":["gatanegro/logos"],"tags_count":1,"template":false,"template_full_name":null,"purl":"pkg:github/gatanegro/LOGOS","repository_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/gatanegro%2FLOGOS","tags_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/gatanegro%2FLOGOS/tags","releases_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/gatanegro%2FLOGOS/releases","manifests_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/gatanegro%2FLOGOS/manifests","owner_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners/gatanegro","download_url":"https://codeload.github.com/gatanegro/LOGOS/tar.gz/refs/heads/main","sbom_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/gatanegro%2FLOGOS/sbom","scorecard":null,"host":{"name":"GitHub","url":"https://github.com","kind":"github","repositories_count":286080680,"owners_count":29499807,"icon_url":"https://github.com/github.png","version":null,"created_at":"2022-05-30T11:31:42.601Z","updated_at":"2026-02-16T03:57:51.541Z","status":"ssl_error","status_checked_at":"2026-02-16T03:55:59.854Z","response_time":115,"last_error":"SSL_connect returned=1 errno=0 peeraddr=140.82.121.6:443 state=error: unexpected eof while reading","robots_txt_status":"success","robots_txt_updated_at":"2025-07-24T06:49:26.215Z","robots_txt_url":"https://github.com/robots.txt","online":false,"can_crawl_api":true,"host_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub","repositories_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories","repository_names_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repository_names","owners_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners"}},"keywords":["app-for-calculation-dist-py","collatz-conjecture","dynamical-systems","fine-structure-constant","mathematical-physics","modulo-arithmetic","nonlinear","number-theory","recursive-functions-"],"created_at":"2025-09-20T03:45:31.607Z","updated_at":"2026-02-16T04:02:52.478Z","avatar_url":"https://github.com/gatanegro.png","language":null,"funding_links":[],"categories":[],"sub_categories":[],"readme":"\n\n---\n\n### **The Logos Theory: Axiomatic Mathematical Framework**\n\n#### **Axiom 1: The Primordial Substrate**\nThe fundamental substrate of reality is a countably infinite, discrete lattice of pure information states.\n\n**Mathematical Representation:**\n- Let $\\mathcal{N} = \\mathbb{Z}^3$ be the set of all **nodes**. Each node is identified by an integer triple $\\vec{n} = (n_x, n_y, n_z)$.\n- Each node has a single intrinsic property: its **value** $v(\\vec{n}) \\in \\mathbb{Z}$.\n\n#### **Axiom 2: The Logos Operator ($\\hat{\\Lambda}$)**\nReality is generated by the application of a specific, discrete operator.\n\n**Mathematical Representation:**\n- The **Logos Operator** $\\hat{\\Lambda}$ is a nonlinear operator that maps the state of the lattice onto itself.\n- Its action on a node's value is defined by a **3D Collatz-type rule**:\n  $$\n  \\hat{\\Lambda} \\circ v(\\vec{n}) = \n  \\begin{cases} \n  \\frac{v(\\vec{n})}{2} \u0026 \\text{if } v(\\vec{n}) \\equiv 0 \\pmod{2} \\\\\n  3 \\cdot v(\\vec{n}) + \\Pi(\\vec{n}) \u0026 \\text{if } v(\\vec{n}) \\equiv 1 \\pmod{2}\n  \\end{cases}\n  $$\n- **Crucially,** $\\Pi(\\vec{n})$ is a **coupling function** that connects the node $\\vec{n}$ to its neighbors (e.g., $\\Pi(\\vec{n}) = v(\\vec{n}+\\vec{e}_x) - v(\\vec{n}-\\vec{e}_y) + ...$). This introduces non-locality and entanglement from the outset.\n\n#### **Axiom 3: Emergence of Spacetime**\nSpace and time are not fundamental but emerge from the state of the lattice.\n\n**Mathematical Representation:**\n\n- **Emergent Distance:** The perceived spatial distance between two nodes $\\vec{n}_i$ and $\\vec{n}_j$ is a function of the difference in their *wave amplitudes*.\n  $$\n  d_{ij} \\propto |a(v(\\vec{n}_i)) - a(v(\\vec{n}_j))|\n  $$\n  where $a: \\mathbb{Z} \\to \\mathbb{R}$ is a function mapping a node's value to an amplitude (e.g., $a(v) = \\log(v)$).\n  \n- **Emergent Time:** The perceived sequence of events (\"time\") is the count of iterative applications of $\\hat{\\Lambda}$.\n  $$\n  \\tau = k\n  $$\n  where $k$ is the step in the sequence $\\{\\mathcal{S}_k\\} = \\{\\mathcal{S}_0, \\hat{\\Lambda}\\mathcal{S}_0, \\hat{\\Lambda}^2\\mathcal{S}_0, ...\\}$. This is not a background time but an ordering parameter.\n\n#### **Axiom 4: The LZ Scale Constant ($\\Lambda_{LZ}$)**\nThere exists a fundamental scale that bounds the emergent universe.\n\n**Mathematical Representation:**\n- $\\Lambda_{LZ}$ is an **attractor** or **invariant** of the dynamical system defined by $\\hat{\\Lambda}$.\n- It could be defined as the asymptotic limit of the geometric mean of node values across the entire lattice:\n  $$\n  \\Lambda_{LZ} = \\lim_{k \\to \\infty} \\left( \\prod_{\\vec{n} \\in \\mathcal{N}} v_k(\\vec{n}) \\right)^{1/|\\mathcal{N}|}\n  $$\n \n- It defines the maximum extent of an emergent \"causal patch\" or field.\n\n#### **Axiom 5: Ontology of States**\nThe values and their dynamics define what we perceive as physical entities.\n\n**Mathematical Representation:**\n- **Matter/Energy (Stable Nodes):** Integer values that are **fixed points** or **limit cycles** under $\\hat{\\Lambda}$.\n  $$\n  \\exists k \\in \\mathbb{N} \\text{ such that } \\hat{\\Lambda}^k \\circ v(\\vec{n}) = v(\\vec{n})\n  $$\n  These stable, persistent patterns are identified as particles.\n- **Forces/Fields (Gradients):** The **local differences** or **gradients** between node values.\n  $$\n  \\vec{\\nabla}v \\approx (v(\\vec{n}+\\vec{e}_x) - v(\\vec{n}), v(\\vec{n}+\\vec{e}_y) - v(\\vec{n}), ...)\n  $$\n  These gradients are the source of interactions between stable nodes.\n\n---\n\n### **Derivation of the Higher-Order Structure (HQS) and Recursive Energy**\n\nDeriving the curvature of the emergent spacetime from the computational process.\n\n#### **1: Recursive Wave Function ($\\Psi$)**\nThe wave function describes the \"computational effort\" or \"total activity\" of the lattice.\n\n**Mathematical Representation:**\n- Let $\\Psi(k)$ be the sum of all node values after $k$ applications of $\\hat{\\Lambda}$:\n  $$\n  \\Psi(k) = \\sum_{\\vec{n} \\in \\mathcal{N}} v_k(\\vec{n})\n  $$\n- This follows a recursive relationship based on the Logos rule. A simplified, mean-field version of this recursion:\n  $$\n  \\Psi(k) \\approx \\sin(\\Psi(k-1)) + e^{-\\Psi(k-1)}\n  $$\n  This equation captures the nonlinear saturation ($\\sin$) and convergence ($e^{-\\Psi}$) behavior.\n\n#### **2: Fixed Point ($\\Psi^*$)**\nThe system seeks equilibrium. The fixed point is found by solving:\n$$\n\\Psi^* = \\sin(\\Psi^*) + e^{-\\Psi^*}\n$$\nThe numerical analysis correctly found $\\Psi^* \\approx 1.23498228$.\n\n#### **3: HQS as Ricci Curvature ($R$)**\nThe fixed point $\\Psi^*$ is the equilibrium \"density of computation.\" We posit that this density *is* the source of curvature in the emergent spacetime.\n\n**Mathematical Bridge:**\n- In General Relativity, the **Ricci curvature scalar ($R$)** measures how much the volume of a small ball in a manifold deviates from a Euclidean equivalent. It is related to the energy density.\n- **Therefore, we define:**\n  $$\n  R \\propto \\Psi^*\n  $$\n  Specifically, the local Ricci curvature at a node is proportional to the fraction of the total \"computational energy\" $\\Psi^*$ that is \"used\" or \"bound\" at that node for recursion.\n- If a node has value $v(\\vec{n})$, its contribution to curvature is:\n  $$\n  R(\\vec{n}) \\propto \\frac{v(\\vec{n})}{\\Psi^*}\n  $$\n **\"HQS is the 23.5% of energy used per node\"**. The fraction of the fixed point value that is \"active\" is:\n  $$\n  \\frac{\\Psi^* - e^{-\\Psi^*}}{\\Psi^*} \\approx \\frac{1.23498 - 0.29049}{1.23498} \\approx \\frac{0.94449}{1.23498} \\approx 0.765\n  $$\n  This implies about **76.5%** of the value is \"active\" (the $\\sin$ component), leaving **23.5%** as the \"potential\" or \"binding\" energy ($e^{-\\Psi}$ component) that governs the curvature and stability of the node itself. This **23.5% is the HQS**—the energy cost of maintaining the node's structure in the emergent geometry.\n\n---\n\n### **Summary of the Mathematical Pipeline**\n\n1.  **Initial State:** $\\mathcal{S}_0 = \\\\{ v_0(\\vec{n}) \\in \\mathbb{Z} \\\\} \\text{ for } \\vec{n} \\in \\mathbb{Z}^3$\n2.  **Dynamics:** $\\mathcal{S}_{k+1} = \\hat{\\Lambda} \\mathcal{S}_k$\n3.  **Emergent Geometry:** $d_{ij} \\propto |a(v(\\vec{n}_i)) - a(v(\\vec{n}_j))|$\n4.  **Global Invariant:** $\\Lambda_{LZ} = \\text{Attractor of } \\\\{\\mathcal{S}_k\\\\}$\n5.  **Total Activity:** $\\Psi(k) = \\sum v_k(\\vec{n})$\n6.  **Fixed Point:** $\\Psi^* = \\lim_{k \\to \\infty} \\Psi(k)$\n7.  **Emergent Curvature (HQS):** $R(\\vec{n}) \\propto \\frac{v(\\vec{n})}{\\Psi^*}$ where the constant of proportionality is set by the **~23.5% binding energy factor** derived from the recursive equation.\n\nThis framework provides a complete, mathematically defined pipeline from a discrete, digital substrate to a curved, emergent spacetime with properties resembling our physical universe.\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fgatanegro%2Flogos","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fgatanegro%2Flogos","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fgatanegro%2Flogos/lists"}