The Franquero Threshold Function: A Unified State Function for Systemic Operational Cost in KF Units

Abstract

Formal Academic Abstract I propose the Franquero Threshold Function (FTF) as a unified governing equation for systemic operational cost. The FTF introduces KF Units as a fundamental dimension of systemic cost, independent of energy, action, or entropy. The function sums a baseline constant, linear contributions from thermodynamic, temporal, mechanical, and quantum variables, and pairwise coupling terms mediated by Franquero Coupling Coefficients. A threshold criterion determines whether a system remains stable or undergoes a critical phase shift. I present empirical validation through a series of five water boiling experiments conducted by myself, Keny Wayne Franquero, and my co investigator, Tori Tianna Weiker. The experiments varied vessel geometry (glass Erlenmeyer flask versus stainless steel pan), heat setting (5, 7, and 10 out of 10), and boundary conditions (open versus sealed). The results demonstrate that the FTF can accurately model, adapt to, and predict thermodynamic phase transitions based on changes in boundary conditions. The framework successfully identified three distinct physical regimes: uniform bulk heating, evaporative limited heating, and stratified localized nucleation. In Test 5, the FTF predicted a boiling time of 300 to 340 seconds based on the reduction of heat loss from sealing the pan. The actual boiling event occurred at 270 seconds with localized nucleation at 90.8 degrees Celsius, demonstrating the predictive power of the framework. This manuscript is a theoretical proposal supported by empirical data, intended for further mathematical refinement and cross domain investigation. The KF Unit is a proposed base dimension. Its physical reality and usefulness are matters for future mathematical and experimental investigation. Plain English Summary I created a new mathematical framework called the Franquero Threshold Function. It measures something I call systemic operational cost, which is the total cost a system pays to maintain a given state. I invented a new unit of measurement called the KF Unit to track this cost. My wife, Tori Tianna Weiker, and I tested this framework by boiling water. We used two different vessels, a glass flask and a stainless steel pan, and we tried different heat settings. We also tested what happens when you cover the pan with a lid. The framework worked. It accurately described how the water heated up. It explained why the glass flask lost energy when the flame licked up its sides. It explained why the open pan hit a wall at 84.3 degrees Celsius and could not boil. And it predicted that sealing the pan with a lid would allow the water to boil. The actual boiling happened faster than predicted, which taught us that the water at the bottom of the pan was boiling before the water at the top, a phenomenon called localized nucleation. This framework is not yet a complete theory. It is a starting point. It proposes a new way to look at the total cost of any physical system. I believe it could eventually be applied to chemical reactions, mechanical systems, and quantum computers. I am publishing this preprint to establish priority and invite other researchers to test and refine the framework.