Hey folks,
Figured I’d share some stuff on my “FF Bootstrap Time Spiral.” It’s basically about tying together United Bootstrap 2 (closed-loop causality), Fractal Flux (continuous fractal drivers), and Time Spiral geometry (radial + angular expansions). Maybe that sounds like a mouthful, but the short version is:
We ditch the idea of a single “first cause” (like a Big Bang or someone flipping the on-switch). Instead, everything loops in time. Future states actually shape the present—like a retro time push—and fractal signals keep the system from ever getting stale or predictable. Then we model time as a spiral, so each cycle can repeat and expand or contract.
I wrote up the theory, did discrete and continuous equations, and tossed in an example or two (like how to do it in a simulation). In discrete time, you basically set the start and end to be identical—x(0) = x(N)—which closes the loop. In continuous time, you handle that with advanced-delay equations (the system depends on X(t+tau)) plus boundary conditions that ensure everything meets back up at the end. Kinda tricky to solve but the outcome can be neat: a self-consistent spiral in time.
FF Bootstrap Time Spiral
Uniting Bootstrap 2, Fractal Flux, and Spiral Time for a Self-Sustaining Universe
By Mitchell D. McPhetridge
(Independent Researcher and Creative Systems Architect)
Abstract
This short paper proposes a visionary framework—called the FF Bootstrap Time Spiral—that merges three powerful ideas:
Bootstrap 2 (Closed-Loop Causality): Eliminates the need for a “first cause,” positing a complete loop where the end shapes the beginning.
Fractal Flux (FF): Injects fractal/chaotic drivers into the system to sustain ongoing complexity.
Time Spiral Geometry: Replaces linear or merely cyclical time with a spiral structure, capturing both repetition and continuous change.
By enforcing loop closure (i.e., start = end) and allowing future states to influence the present (retrocausality), we obtain a paradox-free model of time that can maintain infinite novelty. Applications range from AI and narrative design to cosmology, offering a new lens on multi-scale, multi-directional causality and the genesis of complexity without a singular spark.
- Introduction and Motivation
Classical models of the universe—or any causal system—often hinge on a “first cause” (e.g., a Big Bang, primordial spark, or initial condition). Such assumptions create philosophical and scientific paradoxes, especially when one contemplates time loops or backward-in-time influence. The FF Bootstrap Time Spiral aims to resolve these issues by:
Closing the chain of cause-and-effect into a loop (no external start).
Sustaining ongoing complexity via fractals or chaotic drivers.
Representing time in a spiral geometry, uniting cyclical repetition with radial expansion or contraction.
In doing so, we meet the challenges of paradoxes (like the “grandfather paradox” or “bootstrap paradox”) head-on, revealing a self-consistent loop where “no first cause” is required. Simultaneously, fractal/chaotic forces keep the loop from degenerating into static repetition, giving rise to perpetual novelty.
- Conceptual Foundations
2.1 Bootstrap 2 (Closed-Loop Causality)
Bootstrap 2 suggests that time can close onto itself—every event emerges from prior (and future) states, without an external “spark.” Such loops might seem paradoxical if we only accept linear causality. However, once we acknowledge that the start and end of the timeline are the same, internal consistency becomes the only requirement—no infinite regress or original “trigger.”
2.2 Fractal Flux (FF)
Fractals and chaotic processes produce structure across many scales. In a looped system, a fractal driver (e.g., sums of sine waves or correlated noise) can perpetually introduce subtle variations. Over repeated cycles, these accumulate into ever-new patterns, preventing stagnation and keeping the loop “alive.”
2.3 Time Spiral Geometry
A time spiral captures two behaviors:
Angular Repetition: A cyclical pass over similar “angles” or phases.
Radial Expansion/Contraction: Gradual outward (or inward) drift that accumulates differences each loop.
This spiral perspective naturally suits generational models, cosmic cycles, and even game narrative loops, illustrating how each pass can revisit the same angles but shift its radius (i.e., the “scale” or “level” of evolution).
- Mathematical Sketch
3.1 Discrete-Time Model
Let
X
n
X
n
be the state at step
n
n, and let
D
n
D
n
be a fractal dimension or “complexity measure.” Define:
X
n
+
1
=
f
(
X
n
,
X
n
+
τ
,
α
sin
(
β
n
)
,
D
n
)
,
D
n
+
1
=
g
(
D
n
,
D
n
+
τ
)
.
X
n+1
=f(X
n
,X
n+τ
,αsin(βn),D
n
),D
n+1
=g(D
n
,D
n+τ
).
Retrocausality:
X
n
+
τ
X
n+τ
means the system references a future state.
Fractal Flux:
α
sin
(
β
n
)
αsin(βn) or more advanced fractal noise ensures unending novelty.
Loop Closure:
X
0
=
X
N
,
D
0
=
D
N
.
X
0
=X
N
,D
0
=D
N
.
We solve all states
{
X
n
}
{X
n
} and
{
D
n
}
{D
n
} simultaneously to ensure that the final state matches the initial.
3.2 Continuous-Time Model
Now consider
X
(
t
)
X(t) on
[
0
,
T
]
[0,T]. The advanced-delay (anticipatory) system:
d
X
d
t
(
t
)
=
F
(
X
(
t
)
,
X
(
t
+
τ
)
,
D
f
(
t
)
)
,
dt
dX
(t)=F(X(t),X(t+τ),D
f
(t)),
d
D
f
d
t
(
t
)
=
G
(
D
f
(
t
)
,
D
f
(
t
+
τ
)
,
X
(
t
)
)
.
dt
dD
f
(t)=G(D
f
(t),D
f
(t+τ),X(t)).
Boundary Conditions:
X
(
0
)
X
(
T
)
X(0)=X(T) and
D
f
(
0
)
D
f
(
T
)
D
f
(0)=D
f
(T). Numerically, one employs forward–backward sweeps or collocation to find a solution that meets at both ends.
- Visionary Aspects and Applications
4.1 AI and Machine Learning
A system that references its future (predicted) state can refine present learning in a novel way—like a network anticipating its own outcomes. Fractal flux ensures that no training pass is strictly repetitive, continually revealing new sub-patterns. This could yield more robust self-correcting loops, or “ethical AI” that simulates the impact of future states on present choices.
4.2 Narrative, Game, and Storytelling
Time loops fascinate creative storytelling. By embedding fractal flux, each “loop” can vary in subtle ways, ensuring no single runthrough is exactly the same. Characters’ future decisions might justify past motivations, producing a self-consistent story arc with no external first event needed.
4.3 Social and Behavioral Modeling
Generational feedback in societies often seems cyclical—economies boom and bust, cultural trends repeat with variations. Fractal drivers can capture how small-scale events echo at larger scales, while bootstrap closure models the sense of “history repeating itself but never exactly.”
4.4 Cosmological Theories
If the universe is indeed cyclical—like Penrose’s “Cycles of Time”—the FF Bootstrap Time Spiral offers a more explicit fractal forcing. Rather than a singular Big Bang, the cosmos could be an endless loop that re-injects complexity at each pass, forever avoiding heat death or trivial repetition.
- Conclusion: Toward a Self-Sustaining Time Spiral
The FF Bootstrap Time Spiral outlines a closed-loop view of time, augmented by fractal drivers, and geometrically interpreted as a spiral. It offers:
Elimination of First Cause: The loop explains itself.
Continual Novelty: Fractal flux keeps the system evolving.
Potential Retrocausality: Future states help shape the present without paradox.
Spiral Expansion: Each cycle revisits earlier “angles” but shifts the radial dimension.
From advanced AI frameworks to cosmic models that bypass the Big Bang, this paradigm invites us to rethink time as a self-sustaining, creative loop. By bridging mathematics, physics, and narrative design, the FF Bootstrap Time Spiral stands as both a rigorous and imaginative lens on how systems might perpetually unfold without needing an external spark—much like a cosmic ouroboros, forever generating its own tail to devour.
References
Lorenz, E.N. (1963). Deterministic Nonperiodic Flow. Journal of the Atmospheric Sciences.
Mandelbrot, B. (1983). The Fractal Geometry of Nature.
Penrose, R. (2010). Cycles of Time: An Extraordinary New View of the Universe.
Wheeler, J.A. (1978). The “Past” and the “Delayed-Choice” Double-Slit Experiment.
Barbour, J. (1999). The End of Time.
(Additional references and detailed equations are available in the extended manuscript.)