The Carlo Unified Functional v3 (With Visualiser)

This repository release contains the complete specification, formal documentation, and interactive visualization suite for the Carlo Framework (Version 3.0 Runtime / Version 2.0 Architectural Specification). Building directly upon the foundational circular energy functional ECarlo = ∫₀2π κ(u) C(u) \, du established in Version 1.0, this release formalizes the Unified Carlo Functional (FCarlo): FCarlo = ∫₀2π [ κ(u)C(u) + λ ∇ E(u) + μ (dE/dt) ] du The release incorporates system dynamics across pseudo-time τ, topography classification (wells, peaks, saddles), path action minimization S, and automated reset threshold logic (FCarlo > Fcrit). Primary Release Deliverables This Zenodo archive provides researchers with two primary core components: Structured PDF Specification (Carlo_Framework_v2_Unified_Specification_2.pdf) Executive Architecture & Mathematical Formulation: Formal definition of loop intensity $C(t)$, contradiction density $K(t)$, asymmetry index ACarlo, and charge Cₜₒₜₐₗ. Operational Pseudocode Engine Manual (v3.0): Step-by-step algorithms detailing core energy integrals, dynamic step evolution (∂ C/∂τ, ∂ K/∂τ), energy gradient mapping, path action evaluation, and system threshold enforcement. 12-Step Execution Protocol & Numerical Worked Example: A complete simulation benchmark (T=10, N=1000, dt=0.02, λ=0.3, μ=0.2, Fcrit=5.0) providing standardized verification values for third-party implementations. Interactive HTML5/Canvas Visualizer (Unified Function Visualiser.html) Real-time Geometric Deformation: Interactive HTML5 engine visualizer demonstrating manifold warping and boundary behavior driven by user-controlled parameters. Parameter Sliders: Real-time manipulation of contradiction density κ(u), loop intensity $C(u)$, and curvature weight λ. Live Threshold Dynamics: Real-time computation of FCarlo rendering dynamic status updates (STABLE SYSTEM vs. RESET (THRESHOLD EXCEEDED)). Single-File Zero-Dependency Architecture: Self-contained, High-DPI screen-optimized canvas visualizer playable directly in any browser without external compilation or library dependencies. KEYWORDS unified functional • behavioural mathematics • Carlo Framework • variational calculus • energy–curvature–dynamics • loop modelling • contradiction dynamics • temporal evolution • curvature wells • curvature peaks • curvature saddles • least‑action histories • threshold logic • multi‑layer system • dynamic systems • collapse mechanics • identity‑tension structure • full‑cycle execution • operator‑level interface • Quick Commands module • recursive systems • field formulations • multi‑loop interactions • stochastic dynamics • networked Carlo systems • structural cognition • developmental arc compression • evolution of C and K over τ • simulation template • operational pseudocode • researcher onboarding • contradiction accumulation • curvature structure • temporal behaviour • collapse‑based reasoning • Behavioural Mathematics • Dynamic Systems Theory • Variational Calculus • Cognitive Architecture • Collapse Theory • Contradiction Mechanics • Structural Reasoning Frameworks • Nonlinear Dynamics • Complex Systems • Identity‑Tension Modelling • Recursive Computational Models • Mathematical Psychology • Systems Engineering (Theoretical) • Computational Cognition • Operator‑Level System Design • Unified Functional Modelling • Curvature and Energy Dynamics • Temporal Evolution Models • Field‑Based Cognitive Systems • Behavioural Mathematics / Dynamic Systems • Unified Functional Modelling / Collapse Dynamics • Carlo Framework / Variational Architecture • Structural Cognition / Nonlinear Behaviour • Multi‑Layer Cognitive Systems / Temporal Evolution • Carlo Framework V2.0 • unified functional • behavioural maths engine • collapse dynamics • curvature manual • dynamics manual • action manual • contradiction calculus • full‑cycle Carlo execution • Quick Commands module • recursive cognitive systems • variational cognition Contact: For enquiries or research questions related to this work, email matthewcarlo.research@gmail.com