Key Points
- To formalize the Unified Carlo Functional mathematical specification and provide an interactive runtime visualization engine for modeling dynamic systems and collapse mechanics.
- Formulated the Unified Carlo Functional integral combining contradiction density, loop intensity, spatial energy gradients, and temporal energy derivatives.
- Developed a 12-step execution protocol and benchmark simulation with parameters T=10, N=1000, dt=0.02, lambda=0.3, mu=0.2, and critical threshold F_crit=5.0.
- Engineered a zero-dependency HTML5 canvas visualizer enabling real-time parameter manipulation and dynamic manifold deformation.
- Established standard mathematical equations governing pseudo-time evolution, topography classification (wells, peaks, saddles), and path action minimization.
- Implemented automated system reset logic that triggers state stabilization whenever the functional value exceeds the critical threshold (F_Carlo > F_crit).
- Demonstrated real-time rendering of manifold warping and system state classifications directly in web environments.
Abstract
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 E₂₀ₑ₋₎ = ₀^2 (u) C (u) \, du established in Version 1. 0, this release formalizes the Unified Carlo Functional (F₂₀ₑ₋₎): F₂₀ₑ₋₎ = ₀^2 (u) C (u) + E (u) + (dEdt) du The release incorporates system dynamics across pseudo-time, topography classification (wells, peaks, saddles), path action minimization S, and automated reset threshold logic (F₂₀ₑ₋₎ > F₂ₑ₈ₓ). Primary Release Deliverables This Zenodo archive provides researchers with two primary core components: Structured PDF Specification (CarloFrameworkᵥ2UnifiedSpecification₂. pdf) Executive Architecture & Mathematical Formulation: Formal definition of loop intensity C (t), contradiction density K (t), asymmetry index A₂₀ₑ₋₎, and charge Cₓ₎ₓ₀₋. Operational Pseudocode Engine Manual (v3. 0): Step-by-step algorithms detailing core energy integrals, dynamic step evolution (C/, K/), 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, =0. 3, =0. 2, F₂ₑ₈ₓ=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 (u), loop intensity C (u), and curvature weight. Live Threshold Dynamics: Real-time computation of F₂₀ₑ₋₎ 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