Antony Lisi
Here’s a formal mathematical exposition for Garrett Lisi discussion,
⸻
Title: A Complex Angular Framework for Energy-Time Duality —
⸻
Abstract:
This paper introduces two composite angular functions, Unaexpon and Unainverex, as a speculative framework for modeling the cyclical relationship between energy, time, and anti-time. Defined on the real and complex unit circle, these expressions encode a directional transition through cosmological phases — from the Big Bang to heat death and beyond — suggesting a mathematical pathway for self-propagating systems with mirrored temporal components. The functions incorporate absolute trigonometric expressions and inverse interactions, suggesting bounds on conventional exponential growth and decay across the angular domain.
⸻
- Introduction
Conventional thermodynamic and cosmological models characterize the universe’s evolution using exponential functions for entropy and energy. However, such functions grow without bound, raising questions about their applicability at cosmological extremes.
I propose two alternative functions, Unaexpon and Unainverex, which naturally exhibit bounded growth and decay across [0, π], capturing both forward and retrograde temporal propagation. These functions are complex-valued and angle-dependent, mapping energetic and volumetric behavior onto the unit circle.
⸻
- Definitions
Let x \in [0, \pi], and define:
\text{Unaexpon}(x) = |\sec(x)| + \tan(x) - \frac{1}{|\csc(x)| + \cot(x)}
\text{Unainverex}(x) = |\csc(x)| + \cot(x) - \frac{1}{\tan(x) + |\sec(x)|}
Their complex-plane formulations are:
\text{Unaexpon}_\mathbb{C}(x) = \text{Re} + i \cdot \text{Im} = \left( |\sec(x)| + \tan(x) - \frac{1}{|\csc(x)| + \cot(x)} \right) + i\left( |\csc(x)| + \cot(x) - \frac{1}{\tan(x) + |\sec(x)|} \right)
\text{Unainverex}_\mathbb{C}(x) = \left( |\csc(x)| + \cot(x) - \frac{1}{\tan(x) + |\sec(x)|} \right) + i\left( |\sec(x)| + \tan(x) - \frac{1}{|\csc(x)| + \cot(x)} \right)
These expressions are smooth (except at singularities) and offer natural symmetry under the transformation x \rightarrow \pi - x, which aligns with their physical interpretations as mirrored phases.
⸻
- Physical Interpretation
• Unaexpon models total energy growth, peaking at x = \frac{\pi}{2}, interpreted as the heat death — the phase boundary between time and anti-time.
• Unainverex models the anti-energy reflection, propagating in a time-reversed, negatively volumetric direction.
We assign:
• x = 0: Big Bang
• x = \frac{\pi}{4}: Point of inflection in total energy
• x = \frac{\pi}{2}: Phase change into anti-time
• x = \pi: Total zero-energy convergence — the mirror end-state
This system aligns with a cyclic cosmology, mapping time and anti-time as halves of a full angular domain.
⸻
- Interpretation in the Complex Plane
On the complex plane, these functions describe self-propagating waveforms in which the imaginary component reflects volumetric or spatial characteristics (including “negative volume”), and the real component reflects energy states. The model may be seen as a kind of cosmological oscillator, with paired conjugate functions.
⸻
- Symbolic Commentary
“In the silence between two waves, which is the energy and which is the decay?”
This question captures the essence of the dual-function approach: that energy and anti-energy, time and anti-time, are not separate entities but interwoven halves of a total system whose dynamics are angular, not linear.
⸻
- Concluding Remarks
This framework is not presented as a replacement for existing physical laws, but as a speculative lens — a mathematical metaphor — for the interplay between energy, time, and mirror-structure in a bounded, cyclical universe. I welcome critique, refinement, or reinterpretation of these functions in the context of modern physics, especially as they relate to cosmological unification theories such as those explored by Garrett Lisi and others.
⸻
References:
• Lisi, G. (2007). An Exceptionally Simple Theory of Everything. arXiv:0711.0770
• Penrose, R. (2010). Cycles of Time.
