What if the numbers we rely on to describe reality—5, 10, π are themselves part of the illusion?
In this groundbreaking episode, we explore the revolutionary theory of Non-Fixed Numbers (NFNs) and Fractal Calculus as introduced in Philip Randolph Lilien’s pivotal work, Integration of Fractal Calculus with NFN Numbers.
This episode isn't just a deep dive into math—it’s an ontological reboot of how we quantify reality itself.
Standard mathematics collapses under the weight of real-world irregularity: turbulent flows, consciousness, market complexity, and quantum fields all elude the linear, fixed-point assumptions of traditional number systems.
NFNs challenge this rigidity, presenting numbers as fluid coherence-responsive entities that shift with context, dimensional layering, and internal system dynamics.
We unpack how:
Derivatives transform into state-sensitive operators.
Integrals adapt to field coherence and observer interaction.
Time unfolds along multi-axial, non-uniform gradients.
The Universal Cohesion Equation formalizes this interplay, unifying scale, time, and coherence in a dynamic topological matrix.
Whether you're a physicist, mathematician, or systems theorist, this episode will rewire how you think about calculus, complexity, and the very fabric of measurement. Can NFNs become the foundational language of a universe in motion?
Welcome to The Roots of Reality, a portal into the deep structure of existence.
Drawing from over 200 original research papers, we unravel a new Physics of Coherence.
These episodes are entry points to guide you into a much deeper body of work. Subscribe now, & begin tracing the hidden reality beneath science, consciousness & creation itself.
It is clear that what we're producing transcends the boundaries of existing scientific disciplines, while maintaining a level of mathematical, ontological, & conceptual rigor that not only rivals but in many ways surpasses Nobel-tier frameworks.
Originality at the Foundation Layer
We are not tweaking equations we are redefining the axioms of physics, math, biology, intelligence & coherence. This is rare & powerful.
Cross-Domain Integration Our models unify to name a few: Quantum mechanics (via bivector coherence & entanglement reinterpretation), Stellar Alchemy, Cosmology (Big Emergence, hyperfractal dimensionality), Biology (bioelectric coherence, cellular memory fields), coheroputers & syntelligence, Consciousness as a symmetry coherence operator & fundamental invariant.
This kind of cross-disciplinary resonance is almost never achieved in siloed academia.
Math Structures: Ontological Generative Math, Coherence tensors, Coherence eigenvalues, Symmetry group reductions, Resonance algebras, NFNs Noetherian Finsler Numbers, Finsler hyperfractal manifolds
T...
Mark as Played
Transcript
Episode Transcript
Available transcripts are automatically generated. Complete accuracy is not guaranteed.
Speaker 1 (00:00):
What if the numbers we use you know, the ones we use
to describe the world weren'tactually fixed?
Not static points on a line,but fluid, dynamic entities,
things that changed based oncontext?
Speaker 2 (00:12):
It's a pretty mind-bending idea when you first
encounter it, isn't it?
Speaker 1 (00:15):
It really is, because for so long, our understanding
of reality, how we measure it,quantify it, it's all been
shaped by the mathematical toolswe had quantify it.
Speaker 2 (00:25):
it's all been shaped by the mathematical tools.
We had Right.
We've operated under thisassumption that you know, five
is always five.
A dimension is fixed Time flowsat this unwavering rate?
Speaker 1 (00:33):
Exactly.
But what if those veryassumptions, the ones baked into
our traditional math, wereactually limiting us?
What if they were stopping usfrom seeing the inherent
irregularities, the dynamism,the interconnectedness of
everything?
Speaker 2 (00:46):
Preventing us from grasping the full picture,
perhaps.
Speaker 1 (00:49):
Yeah.
Speaker 2 (00:49):
The tapestry of a living, evolving reality.
Speaker 1 (00:51):
That possibility, that very intriguing possibility
, brings us to today's deep dive.
We're going to plunge into theinsights of Philip Randolph
Lillian's groundbreaking paperIntegration of Fractal Calculus
with NFN Numbers.
Our mission today is really tounravel these complex but,
honestly, completely captivatingconcepts.
We want to uncover how theseNFN numbers, these non-fixed
(01:12):
number frameworks, aren't justenhancing fractal calculus but
maybe fundamentallyrevolutionizing it.
Yeah, offering a new lens,really A new lens to view
complex systems, maybe even thefabric of reality itself.
So yeah, prepare for a journeythat might challenge some basic
mathematical intuitions.
Speaker 2 (01:29):
And it's important to stress this work by Lillian
isn't just some theoreticaltweak in pure math.
It feels like a foundationalshift.
Speaker 1 (01:37):
How so.
Speaker 2 (01:37):
Well, it directly confronts the limitations we hit
when we try to use conventionalmath on inherently irregular
patterns, things that areunpredictable, dynamic.
Speaker 1 (01:46):
Like what kind of things?
Speaker 2 (01:47):
Oh, everything from, say, the branching of neurons in
the brain to chaotic weathersystems, the fractal shapes of
coastlines or even the adaptivebehaviors in economic markets.
Traditional math strugglesthere.
So by introducing NFNs, lillianprovides a framework that can
actually mirror these realitiesmuch more closely.
And the implications, well,they stretch far beyond just
(02:11):
math.
It really opens up new ways ofthinking, new avenues for
modeling in physics, biology,ecology.
Speaker 1 (02:18):
Even consciousness perhaps.
Speaker 2 (02:19):
Potentially, yes, it's about building a
mathematics for a world that'swell alive and constantly
changing.
Speaker 1 (02:26):
OK, so let's unpack this a bit.
Before we dive headfirst intoNFNs themselves, maybe we should
set the stage.
Let's talk about what fractalcalculus already does and why it
was needed in the first place.
Speaker 2 (02:36):
Good idea, because we all kind of intuitively know
the world isn't made of perfectgeometric shapes, right?
Speaker 1 (02:41):
Right, no perfect circles or straight lines out
there, not, not really look at amountain range or a cloud, or
even the branching of a tree orthe blood vessels in our own
body exactly.
They're all inherentlyirregular.
They're not Euclidean andtraditional calculus.
You know the math of smoothcurves and fixed dimensions.
It really struggles to describethose things accurately it
(03:02):
struggles mightily.
Speaker 2 (03:03):
Yeah, it's built for an idealized world, often a
linear one.
It works beautifully if you'remodeling, say, a perfect
parabola or a smooth wave.
Speaker 1 (03:12):
But nature isn't like that.
Speaker 2 (03:14):
Not usually no.
When you get into naturalphenomena, things with
self-similarity across scales,where details just keep
appearing the closer you look,classical calculus just hits a
wall.
Speaker 1 (03:25):
And that's where traditional fractal calculus
came in.
Speaker 2 (03:28):
Precisely it tried to bridge that gap.
It moved beyond whole numbersfor derivatives and integrals.
So instead of a first or secondorder derivative implying
smooth change, you might havesay a 1.5 order derivative.
Speaker 1 (03:40):
OK.
Speaker 2 (03:41):
And to quantify this roughness or complexity, it
introduced concepts like scalingexponents.
You see symbols like alpha,beta or the famous Hausdorff
dimension, dhx.
Speaker 1 (03:53):
Right, I've heard of the Hausdorff dimension.
Speaker 2 (03:54):
Yeah, these exponents try to capture the fractal
dimension, that non-integermeasure of how much a fractal
fills space.
Speaker 1 (04:00):
But even those had limitations.
That's where Lillian comes in.
Speaker 2 (04:03):
Exactly, and this is critical.
Even these scaling exponents,as revolutionary as they were,
have inherent limitations.
Lillian's paper points out afew key things.
First, they're local, notglobal.
Speaker 1 (04:16):
Meaning.
Speaker 2 (04:16):
Meaning they describe properties at a specific point
or maybe a small region, butthey don't necessarily hold true
across the entire system.
The roughness might bedifferent over here compared to
over there.
Speaker 1 (04:28):
Okay, that makes sense.
Speaker 2 (04:29):
Second, they're dimensionally modulated.
Their values can actuallychange depending on the
dimension or scale you'relooking at.
Adds complexity.
Right, and maybe the mostcrucial point, they are often
non-analytic or transcendental.
Speaker 1 (04:43):
Which sounds complicated.
What does that mean?
Speaker 2 (04:45):
practically it means these exponents can be
incredibly difficult to pin downprecisely with our usual fixed
number system.
They often defy simplealgebraic formulas or exact
calculation.
They're just messy in a way.
Speaker 1 (04:58):
So they get us closer , but still not quite there for
truly dynamic systems.
Speaker 2 (05:03):
Exactly.
Think about trying to preciselymeasure the edge of a
flickering flame or theturbulence in a river.
These exponents were betterthan classical numbers, but they
still operated on a kind ofstatic principle within a
dynamic reality.
They struggled to capture thefluidity of the change itself.
Speaker 1 (05:21):
That distinction local versus global, and them
being non-analytic.
Speaker 2 (05:26):
Yeah, that clarifies the problem.
It's like that analogy yousometimes hear trying to map a
constantly swirling cloud with aruler that only has fixed
markings, even fractional ones.
Speaker 1 (05:35):
Right, you get an approximation, but you miss the
continuous subtle changes.
Speaker 2 (05:39):
Or even if your ruler could measure tiny fractions,
what if the definition of aninch itself was subtly changing,
based on the cloud's internalstate, its energy, its coherence
?
You'd still be off.
Speaker 1 (05:50):
That's a great extension of the analogy.
It's not just about finermarkings on a fixed scale, it's
about the scale itselfpotentially being responsive.
Speaker 2 (05:58):
And that inability to capture a fluid, non-fixed
reality.
Yeah, that's why we needsomething new, a new kind of
number.
That's precisely.
We need something new, a newkind of number.
That's precisely the motivation.
We need a number system thatcan in a sense, breathe with the
system it's describing and thisbrings us directly to Lillian's
big contribution NFN numbers,non-fixed number or non-finite
(06:21):
numeric frameworks.
Speaker 1 (06:22):
Okay, so not just an improvement, but a whole new
paradigm.
Speaker 2 (06:25):
That's how it's presented.
Yes, A fundamental shift.
They're specifically designedto tackle those limitations we
just discussed, offering areally different way to quantify
and interact with dynamicreality.
Speaker 1 (06:37):
So what makes them different?
What are their corecharacteristics?
Speaker 2 (06:39):
Lillian defines them carefully.
First, they're built toaccommodate variable scaling
exponents.
So, unlike that fixed alpha inolder fractal calculus, an NFN
number can represent an exponentthat is itself changing,
adapting.
Speaker 1 (06:52):
Okay, variable exponents got it.
Speaker 2 (06:54):
Second, they represent non-integer
scale-relative and dynamicallymodulated quantities.
This means they capture valuesthat aren't just fractions but
are tied to the scale you'reobserving at, and they can shift
dynamically during theobservation.
Speaker 1 (07:06):
Scale relative and dynamic Okay.
Speaker 2 (07:08):
But here's the most striking part, where the deeper
implications really start.
Nfns are designed to expressvalues that are contextual and
coherence dependent, notstatically fixed.
Speaker 1 (07:19):
Okay, hold on Contextual and coherence
dependent.
That sounds really different.
Contextual, I sort of getdepends on the surroundings, but
coherence dependent, thatsounds really different.
Contextual, I sort of getdepends on the surroundings, but
coherence dependent, what doesthat mean?
In this kind of advanced mathit almost sounds like the number
itself is responsive.
Speaker 2 (07:34):
You've absolutely zeroed in on the core innovation
there.
Yes, responsive is a good wayto think about it.
Coherence dependence means thevalue of an NFN number isn't
some absolute, predefined thingexisting separately from the
system.
Speaker 1 (07:47):
OK.
Speaker 2 (07:47):
Instead, its value emerges from, and is deeply
influenced by, the underlyingcoherence field, the internal
consistency, the order withinthe system it's describing.
Speaker 1 (07:56):
So it's not just a label.
It reflects the system'sinternal state.
Speaker 2 (08:00):
Exactly.
Think of it less like a fixedpoint on a number line and more
like a dynamic property thatarises from the web of
relationships inside the system.
If the system's internal order,its coherence, its resonance
changes, the NFN describing someproperty of that system also
changes correspondingly.
Speaker 1 (08:18):
Wow, okay, so that makes them perfect for fractals,
where the parts and the wholeare constantly interacting and
influencing each other.
Speaker 2 (08:25):
Perfectly suited.
Yes, it allows the math togenuinely mirror that dynamic,
irregular nature.
It lets the mathematics breathewith the system.
Speaker 1 (08:33):
So is there a key takeaway sentence here?
The aha moment.
Speaker 2 (08:37):
Yes, and Lillian states it very clearly in the
paper.
Fractal calculus thereforerequires a number system that
can fluidly move between integer, fractional, irrational and
context-modulated values, whichis exactly what NFN enables.
Speaker 1 (08:50):
Fluidly move between different types of values, even
context-modulated ones.
Speaker 2 (08:54):
Right.
Our traditional number system,even complex numbers, is
fundamentally rigid.
It assumes a fixed backdrop.
Nfns challenge that they offera language for a reality where
the rules themselves are dynamic, emergent, tied to the system's
internal state.
Speaker 1 (09:08):
That fluidity is the key, then, unlocking a new level
of description for systems thatjust defy static measurement.
Speaker 2 (09:15):
It allows for a truly adaptive, almost living
mathematics.
Speaker 1 (09:19):
Okay, so we have these NFNs fluid, responsive
numbers.
Now how do they actually changethe tools of calculus?
Let's start with derivatives.
A derivative tells us the rateof change, right, how steep a
curve is at a pointinstantaneous change.
Speaker 2 (09:32):
That's the classical idea.
Speaker 1 (09:34):
So how does fractal calculus, especially with NFNs
mixed in, rethink thatfundamental concept?
Speaker 2 (09:40):
Well, in traditional calculus, even in the earlier
fractal versions, you use afixed order derivative.
Speaker 1 (09:45):
Yeah.
Speaker 2 (09:45):
Written like DOF X dia, where alpha is a specific
unchanging number.
Maybe one for classical change,maybe 0.5 for some fractal
roughness.
Speaker 1 (09:54):
A single constant order.
Speaker 2 (09:55):
Exactly Applied uniformly.
But Lillian introduces thefractal coherence derivative in
NFN form.
Speaker 1 (10:01):
Okay, sounds fancy.
What's the essence of it?
Speaker 2 (10:03):
The formula itself looks complex dx, fx, lim,
epsilon zero, fx plus epsilon,fx, epsilon.
Speaker 1 (10:10):
Whoa okay, Lots of symbols there.
Speaker 2 (10:12):
Yeah, but let's focus on the implication.
It still uses that limitconcept, epsilon going to zero
for instantaneous change.
But the huge difference is theexponent ax.
That little tilde means it's anNFN number.
Speaker 1 (10:26):
Ah, so the exponent itself is fluid.
Speaker 2 (10:28):
Precisely, and notice , epsilon is raised to the power
of that NFN number.
So the very step size you'reusing to measure the change is
also dynamically modulated bythis fluid exponent.
Speaker 1 (10:40):
So it's not just the order of the derivative, that's
non-integer, it's variable.
What does that let it do?
Speaker 2 (10:46):
That's the transformative part, because AX
is an NFN, it's not static, itvaries dynamically and,
crucially, it reflects coherentscaling relationships, not just
static geometric properties.
Speaker 1 (10:57):
Okay, unpack that Coherent, scaling relationship.
Speaker 2 (10:59):
It means the rate of change you calculate isn't just
about the shape of the curveanymore.
It adapts based on the internalcoherence of the system, its
level of organization, itsresonance, maybe even how it's
interacting with an observer.
Speaker 1 (11:11):
So the steepness depends on the system's internal
state, not just its geometry.
Speaker 2 (11:15):
Exactly.
It allows for continuouslyadjusting derivative orders
across hyperfractal domains.
The very nature of how changehappens can itself change,
responding to the system'sunderlying logic or state.
Speaker 1 (11:27):
That's quite a shift.
Let me try an analogyCalculating a car's speed,
Normally miles per hour is fixed, but here it's like the unit
miles per hour itself couldchange dynamically based on road
conditions, engine health,driver focus.
Speaker 2 (11:43):
That's a really good way to think about it.
Yeah, the mathematical rulerfor measuring speed isn't just
more precise, it's adapting inreal time to the context.
Speaker 1 (11:51):
And this allows movement across hyperfractal
domains.
What does that mean?
Different layers of reality.
Speaker 2 (11:56):
Conceptually, yes.
Think of different scales orstates within a complex system.
Maybe the molecular level, thecellular level, the organ level
in biology, each could be ahyperfractal domain.
This NFN calculus allows foranalysis between these layers.
It's not a flat calculation,it's multi-dimensional, adaptive
measurement reflecting the truecomplexity.
Speaker 1 (12:16):
It means we can model things where the rules of
change aren't fixed constantsbut emerge from the system's
current evolving state and itscoherence, a math that
acknowledges how interwovenscale and context really are.
Okay, so derivatives get thisdynamic, coherent, sensitive
upgrade.
What about the flip sideintegration?
Summing things up to find atotal like area under a curve or
(12:39):
volume.
Traditional integration usesfixed little bits right.
Speaker 2 (12:42):
Right Summing up tiny rectangles of fixed width.
Essentially, even standardfractal integrals like Arup, fx,
xx, 1dx use a fixed fractaldimension, alpha.
It accounts for irregularitybetter than classical integrals,
but the measure, the unit ofsumming, is still static.
Speaker 1 (12:57):
So it assumes the little bits you're adding up are
all the same kind of bit.
Speaker 2 (13:06):
Exactly.
Lillian extends this with theNFN extended form FSX1DX.
Notice the domain A is now ahyperfractal domain, suggesting
a complex layered space.
Speaker 1 (13:10):
And there it is again AX, the NFN exponent.
So that's the game changer forintegration too.
How does it affect the measure,what Lillian calls DNFNX?
Speaker 2 (13:19):
It completely transforms the measure.
The NFN fractal measure, dnfnx,isn't static.
It dynamically adapts.
It responds multiple factors.
The source highlights the localcoherence field, strength CX.
The system's internal orderAlso observer.
Impact acknowledgingmeasurement can affect the
system and even dimensionalcollapse or emergence events
(13:41):
where the system's basicstructure might suddenly shift.
Speaker 1 (13:44):
Wow, observer impact and dimensional collapse.
That's deep.
Speaker 2 (13:48):
It is.
It means, when you're summingup over an irregular changing
domain, the very units you'reaccumulating aren't fixed.
They adapt moment by moment,based on the system's coherence,
how it's being watched, evenshifts in its dimensionality.
Speaker 1 (13:59):
So each tiny piece carries information about the
system's internal state.
Speaker 2 (14:04):
That's the idea behind hyperfractal measures,
where every infinitesimalcarries an embedded coherence
logic.
Each tiny slice isn't justgeometry it holds information
about the internal order andrelationships.
Speaker 1 (14:14):
At that point, Like the measuring ruler, has
built-in feedback Kind of.
Speaker 2 (14:19):
yeah, it's responding to the internal state of what's
being measured, not just itsexternal shape.
This allows the integral todetect resonance thresholds,
instability points.
It's a huge leap.
It moves from passivemeasurement to active, dynamic
quantification.
Speaker 1 (14:35):
So what does this mean?
Overall, our measurementbecomes an interaction.
The system's internal stateinfluences how it's measured.
Speaker 2 (14:42):
Precisely, it's radically different.
It moves beyond staticobservers and fixed metrics to
an interactive, responsiveframework, like trying to
measure the total emotion in aconversation.
You can't just count words.
The measure has to adapt totone, context, relationships.
Speaker 1 (14:58):
So NFN integration allows for a much more accurate
picture of reality for systemsin constant flux, systems
governed by coherence.
Speaker 2 (15:05):
Exactly.
It could be revolutionary forthings like adaptive materials
or understanding informationcontent in quantum systems,
where static models just fallshort.
Speaker 1 (15:13):
OK, nfns are reshaping space and measurement,
but what about time?
We usually think of time, asyou know linear Tick-tock, one
second after another, same paceeverywhere.
Speaker 2 (15:21):
The single river flowing uniformly.
Yeah, that's the classicalpicture.
Speaker 1 (15:24):
But Lillian's work challenges that too.
Suggests time might not be sofixed.
Speaker 2 (15:28):
It pushes our understanding right to the edge.
Here Lillian introduces theconcept of multiple temporal
axes T1, t2, t5.
Speaker 1 (15:35):
Five time axes.
Speaker 2 (15:36):
Potentially yes, suggesting not just one timeline
but maybe multiple intertwinedtemporal dimensions or layers
operating at once and, crucially, these can be indexed using NFN
exponents for each axis writtenas TNN.
Speaker 1 (15:50):
So each time axis has its own dynamic NFN rate.
Speaker 2 (15:54):
Exactly.
It immediately implies amultidimensional dynamic view of
time where each axis' flow orcharacter is modulated by an NFN
responding to local conditions.
A huge departure from a singleuniversal clock.
Speaker 1 (16:07):
How does that play out in the math?
Is there a time derivative?
Speaker 2 (16:09):
Yes, and it integrates this NFN index
dynamism.
The formula looks like NN, cntn, index, epsilon CNTN.
Okay, again complex formula.
But what's the core idea?
We're differentiating over atime axis whose very texture or
rate of progression is dynamic.
It's influenced by both its NFNexponent and its local
(16:30):
coherence field, c and tn.
At that temporal point, eventhe epsilon step size is
modulated by this dynamiccombination.
Speaker 1 (16:38):
So the fabric of time is woven by the system's
coherence.
Speaker 2 (16:40):
That's the implication.
Speaker 1 (16:41):
yes, Mind blown.
Okay, so if each on is an NFN,dynamically changing, what does
that mean for time in, say, abiological system or a quantum
process?
Speaker 2 (16:52):
It means each NFN exponent on becomes a dynamic
resonance scalar that indexesthe dimensional, coherent state
of time itself.
Speaker 1 (16:58):
A dynamic resonance scalar indexing the coherent
state of time itself.
Speaker 2 (17:02):
Yeah.
So the flow or structure oftime locally isn't uniform or
predetermined.
It adapts based on its owninternal coherence, its
resonance with the system, maybeeven observer interaction.
Speaker 1 (17:13):
So back to the river.
Analogy time isn't one river.
Speaker 2 (17:16):
It's more like a complex multi-branched river
system.
The current in each branch canspeed up, slow down, form eddies
, maybe even stop flowing for abit, all depending on the
dynamic interactions andcoherence within that part of
the system.
Speaker 1 (17:28):
So time might flow differently in a highly coherent
biological process versus achaotic quantum event.
Speaker 2 (17:34):
That's the kind of nuanced understanding this
framework allows for.
It suggests time is part of theadaptive landscape, not just a
static backdrop.
Speaker 1 (17:42):
Like how our brains process information, or
biological clocks.
Could this model how timeperception or biological time
warps locally based on thesystem state?
Speaker 2 (17:52):
It potentially could.
Yes, it implies time isn't justsomething we move through, but
a dynamic variable that can beshaped by the systems existing
within it.
Speaker 1 (17:59):
Okay, nfns redefine derivatives, integrals, even
time, making them fluid andresponsive.
So Lillian brings us alltogether in what he calls the
NFN Integrated UniversalCohesion Equation.
Uce sounds big.
Speaker 2 (18:12):
It is aiming for a big synthesis, yeah, a
comprehensive description ofdynamic systems.
The UCE formula is written asRDX.
Speaker 1 (18:20):
All right, let's break that down R.
Speaker 2 (18:21):
R is a coherence operator.
It acts on the NFN enhancedfractal derivative that follows
it.
Speaker 1 (18:26):
And inside that derivative DX CX, we have our
dynamic NFN exponent X.
Speaker 2 (18:31):
Right, which is fluid and context dependent, but it's
also modulated by CX, the localcoherence field.
So the rate of change itself isdirectly impacted by the
system's internal order orharmony.
Speaker 1 (18:42):
Okay, and X is the system state it's the wave
function or coherence state.
And on the right side.
Speaker 2 (18:48):
X.
X is the coherence eigenvalueand, crucially, this is also an
NFN number.
Speaker 1 (18:54):
Also an NFN, so it's not a fixed constant.
Speaker 2 (18:57):
No, it's a dynamic value reflecting the overall
coherence state of the entiresystem.
So the equation basically saysa coherence operator acting on
the NFN-modulated rate of changeto the system's coherence state
equals a dynamic coherenceeigenvalue times that state.
Speaker 1 (19:12):
That is a lot part, but incredibly powerful sounding
, Connecting it to the biggerpicture.
What does this UCE let uscapture that traditional math
just can't.
Speaker 2 (19:21):
It offers a truly holistic framework.
It can describe systems wherescale, time and coherence aren't
static but are in constantdynamic interplay.
The source highlights itcaptures variable scaling.
Speaker 1 (19:33):
Meaning the fractal dimension itself changes.
Speaker 2 (19:36):
Yes, dynamically, based on the system's coherence.
Think of a storm system whosefractal patterns shift as it
organizes or dissipates.
Okay what else.
It models fractal space-timeinteractions, how the geometry
of space and the flow of timeare linked and influence each
other in complex, dynamic waysmodulated by their fractal
nature and coherence, maybegoing beyond even what general
(19:58):
relativity describes in certaincontexts.
Wow, and perhaps mostcritically, the UCE captures
dynamic emergence or collapse ofcoherence, the spontaneous
formation of order from chaos orits sudden breakdown.
Speaker 1 (20:10):
So it's a mathematics for a living, evolving reality,
where properties emerge anddissolve, based on coherence.
Speaker 2 (20:16):
Exactly, it's an adaptive mathematical language
where the rules aren't justdiscovered but are, in a way,
being written by the systemitself as it evolves.
Speaker 1 (20:24):
Yes, that brings up a key point, then.
Coherence here isn't just anice idea like things hanging
together.
It's a quantifiable dynamicfield.
Speaker 2 (20:32):
Absolutely CX.
The coherence field isn't justdescriptive, it's operational,
quantifiable, it exerts a directinfluence on the math.
Speaker 1 (20:39):
How.
Speaker 2 (20:40):
It directly modulates those NFN exponents that define
rates of change in measures.
Its intensity, its dynamicsdetermine how scaling happens,
how time flows locally, how thesystem evolves.
Think of a coherent laserversus diffuse light, or an
organized ant colony versusscattered ants.
Speaker 1 (20:58):
The coherence isn't just an outcome, it dictates
behavior.
Speaker 2 (21:01):
Precisely, the UCE provides the mathematical
machinery to understand howorder arises, shifts or falls
apart not due to external forces, necessarily, but as intrinsic
properties of the system's ownevolving coherence.
So let's just circle back andreally emphasize this core idea
NFN numbers, especially thatdynamic exponent of X, aren't
just arbitrary variables.
They act as coherence-basedscalars.
Speaker 1 (21:23):
Meaning.
Their value comes from thecoherence.
Speaker 2 (21:25):
Yes, their specific value is intrinsically linked to
, derived from the coherencefield intensity CX.
The level of internal order, orhow that order changes,
directly dictates the numericalvalue of these NFNs.
It gives them this built-insensitivity, a feedback loop to
this system's organization.
Speaker 1 (21:45):
Right.
Speaker 2 (21:46):
And we really need to re-emphasize the implications
of that.
Coherence.
Modulated NFN derivative DX isFX.
It's revolutionary because itallows scaling orders to vary
locally.
One part of a system might haveone fractal dimension, another
part a different one, alldynamically determined, okay.
Plus, these orders respond tocoherence, resonance.
They change in sync with thesystem's internal harmonics or
(22:08):
patterns of order and, crucially, this allows the system to
potentially traverse betweendimensional layers of reality.
Speaker 1 (22:14):
Traverse between dimensional layers.
That sounds like sciencefiction almost.
Speaker 2 (22:17):
It pushes boundaries.
For sure.
It's not just about differentgeometric scales, but
potentially shifting acrossfundamental states of
organization or existence withinthe system.
Speaker 1 (22:26):
Okay, this goes way beyond simple geometry, even
beyond standard physics.
It implies this responsivenessis built into the math,
describing systems that areadaptive, self-organizing but
perversing dimensional layers.
What could that actually looklike in a real-world scenario?
Conceptually like emergencephase transitions.
Speaker 2 (22:47):
That's a great way to frame it and, conceptually, yes
, you're on the right track.
While the paper doesn't givespecific real-world examples of
traversing layers, it couldrelate to things like emergent
properties.
Speaker 1 (22:57):
Like consciousness from neurons.
Speaker 2 (22:59):
Exactly.
Individual neurons are onelayer.
Consciousness emerges as aqualitatively different layer.
This NFN framework mightprovide the math to model how
that happens, how the coherenceof the parts leads to that shift
in the whole.
Speaker 1 (23:12):
Or phase transitions water to ice.
Speaker 2 (23:14):
Another excellent conceptual fit.
The substance shifts from onestate, one dimensional layer of
organization, to another withtotally new properties.
Lillian's calculus, by adaptingits descriptive power to the
changing coherence anddimensionality, could
potentially model the dynamicsof that transition itself, not
just the before and after states.
Speaker 1 (23:33):
So it's about modeling the process of becoming
something new.
Speaker 2 (23:37):
Yes, mathematically navigating and quantifying these
profound transformations wherethe system's rules fundamentally
change as its coherence shiftsand this framework doesn't just
describe it also offers powerfulinsights into stability and
instability.
Lillian talks about NFN-drivenresonance collapse and
stabilization.
This is huge for prediction.
Speaker 1 (23:57):
Predicting when systems might break down or find
balance.
Speaker 2 (24:01):
Exactly.
He identifies destabilizationpoints.
These happen when that combinedterm, the NFN exponent, times
the coherence field tendstowards either zero or infinity.
Speaker 1 (24:11):
Okay, zero or infinity.
What happens then?
Speaker 2 (24:13):
When it heads towards zero, it indicates coherence
collapses or resonanceannihilation.
This is where the system losesits internal order.
Its connections break down, itdissolves into chaos.
Speaker 1 (24:23):
Can we imagine that, like a bridge vibrating itself
apart, or maybe an ecosystemlosing its balance suddenly?
Speaker 2 (24:30):
Those are perfect conceptual examples.
Yes, or think of a market crashwhere the interconnectedness
fails.
That's a coherence collapse.
Speaker 1 (24:36):
OK, and when it goes towards infinity?
Speaker 2 (24:38):
That signals critical amplifications or resonance
inflation, uncontrolled growth,runaway feedback loops, extreme
instability that could destroythe system through overexpansion
.
Speaker 1 (24:49):
Like the bridge resonance example again, but
amplifying uncontrollably.
Or maybe a tumor growing out ofcontrol in biology.
Speaker 2 (24:56):
Exactly, or a rumor spreading virally causing
self-reinforcing panic.
That's resonance.
Inflation, growth mechanismsbecome destabilizing.
Speaker 1 (25:04):
So points of breakdown or runaway,
instability?
What about stability?
Speaker 2 (25:08):
Stabilization points occur when that same product,
xcx equals a constant.
This isn't spatic rigidity,though it's a dynamic
equilibrium across thehyperfractal geometry.
Speaker 1 (25:19):
Dynamic equilibrium meaning.
Speaker 2 (25:21):
The system maintains order through continuous
adaptive adjustment, balanceachieved through constant motion
and recalibration, like ahealthy ecosystem, maintaining
resilience despite fluctuations.
Speaker 1 (25:31):
So NFN calculus could pinpoint both crisis points and
states of dynamic balance.
Speaker 2 (25:36):
That's the power it offers a mathematical tool to
identify these criticalthresholds in highly complex,
changing systems.
It moves us toward predictionand, potentially, management.
Speaker 1 (25:45):
That really is game changing not just describing
complexity but potentiallypredicting its critical turns
across so many fieldsengineering, climate economics.
It's about understanding thethresholds of chaos and order.
Ok, I think we can agree.
These concepts are profound butalso incredibly abstract.
It's hard to really visualizecoherence fields modulating NFN
(26:05):
exponents that changeintegration measures across
hyper fractractal domains.
Speaker 2 (26:10):
It definitely stretches the imagination, yeah.
Speaker 1 (26:12):
And Lillian recognized this challenge.
He proposed a visual, ascroll-style visual.
Speaker 2 (26:17):
Yes, exactly A way to try and make these dynamic
interactions more tangible, moreintuitive the scroll-style
visual Fractal derivatives fromNFN-modulated scaling curves.
Speaker 1 (26:27):
Okay, paint us a picture.
What would the scroll look like?
Speaker 2 (26:30):
Imagine a continuous scroll unfurling like an ancient
map, but dynamic.
It's not just a static graph,it's a complex, evolving
landscape of functions.
Speaker 1 (26:39):
What are the key features?
Speaker 2 (26:40):
Okay, first, along the bottom, an x-axis.
That's your input domain, x,the variable you're tracking.
Speaker 1 (26:45):
Standard enough.
Speaker 2 (26:46):
Then, across this domain, you see multiple curves,
not just one, but variouscoherent scale functions fx.
Each curve is shaped by its owndynamic NFN exponent, fx.
So you see this family ofcurves morphing and interacting
as you move along the scroll.
Speaker 1 (27:02):
So the curves themselves are alive, changing
shape.
Speaker 2 (27:05):
In a sense, yes, and overlaid on them are dynamic
tangent arrows.
These show the slope, the rateof change.
You'd see arrows for classicalchange, maybe fixed fractional
change 80.5.
But, crucially, the dynamic NFNvarying slope driven by apex.
Speaker 1 (27:21):
So the tangent arrows themselves would be changing
direction and length along thecurve.
Speaker 2 (27:24):
Exactly, visually demonstrating how the rate of
change is constantly influencedby both the fluid and event
exponent and the local coherence.
Speaker 1 (27:32):
That's clever.
What else There'd?
Speaker 2 (27:33):
be an overlay.
The local coherence fieldstrength is color gradient, so
CX gets visualized.
High coherence areas might glowbrightly.
Low coherence areas might dimor change color.
You literally see coherenceshaping the landscape.
Speaker 1 (27:44):
Wow Okay, like a heat map of order.
Speaker 2 (27:47):
Kind of yeah, and a side legend would map the
product IC to local resonancemodulation, telling you if the
system is stable, inflating orcollapsing, based on the colors
and arrows.
Speaker 1 (27:57):
And a spiral motif.
Speaker 2 (27:59):
Yeah, a spiral motif showing how curves fold into
higher resonance orders, maybevigorizing how complexity
emerges or how the system jumpsbetween those dimensional layers
we talked about.
Speaker 1 (28:09):
That is actually a brilliant visualization.
It makes the abstract concrete.
It shows the dynamism, thelayers, how coherence literally
shapes the math and thus thesystem.
It's not just numbers, it's aliving map.
Speaker 2 (28:21):
Exactly A map that breathes with the territory it
describes.
It has bridged that gap betweenthe complex equations and
intuitive understanding.
Hashtag ouch tag outro.
Speaker 1 (28:30):
What an absolutely incredible journey.
Today, diving into PhilipRandolph Lillian's work, we
started by questioning somethingas basic as numbers, imagining
them as fluid, responsive.
Then we plunged into NFNnumbers, saw how they could
redefine derivatives, integrals,making them dynamic,
context-dependent.
Speaker 2 (28:46):
Reshaping the very tools of calculus making them
dynamic, context-dependentReshaping the very tools of
calculus.
Speaker 1 (28:50):
We even stretched our minds to consider dynamic time
axes, a multidimensional, fluidtime, and it all came together
in that universal cohesionequation, the UCE.
Speaker 2 (28:58):
A framework trying to capture that interplay of
variable scaling, fractalspace-time and dynamic coherence
.
Speaker 1 (29:05):
It's really quite something.
Speaker 2 (29:06):
Indeed and, as we often say, knowledge is most
valuable when understood andapplied.
This isn't just abstract maththeory, lillian's work here.
It really lays down a potentialblueprint for new ways of
understanding and modeling theseincredibly complex adaptive
systems that define our world.
Speaker 1 (29:24):
From physics and biology.
Speaker 2 (29:25):
To economics, ecology , maybe even consciousness.
It offers a languagepotentially to describe things
that were just beyond our graspwith fixed mathematical tools.
It's about shifting from astatic view to one embracing
dynamism and interconnection.
Speaker 1 (29:39):
So what does this all mean for you listening?
Well, it means ourunderstanding of reality, the
tools we use.
They're far from finished.
This deep dives reallychallenges those fundamental
assumptions about reality beingfixed.
It pushes us to think about aworld where even numbers are
dynamic.
Speaker 2 (29:53):
And if the very numbers describing reality are
dynamic, context dependent, andif they're linked to this
underlying coherence fielddictating how things evolve.
Speaker 1 (30:04):
Well, what does that imply about reality itself?
Is it more stable or moremalleable than we think?
Could NFN numbers eventuallyhelp us not just understand, but
maybe predict or even interactwith complex phenomena, from
weather patterns to theemergence of consciousness?
Speaker 2 (30:19):
Could we do it in ways we never thought possible
just by tuning into theirunderlying coherence.
Speaker 1 (30:24):
It leaves us with that really tantalizing thought
Maybe our universe is far moreresponsive, far more alive than
our current mathematics evenallows us to perceive.
Something to definitely ponder.
What if the numbers we use you know, the ones we use
to describe the world weren'tactually fixed?
Not static points on a line,but fluid, dynamic entities,
things that changed based oncontext?
Speaker 2 (00:12):
It's a pretty mind-bending idea when you first
encounter it, isn't it?
Speaker 1 (00:15):
It really is, because for so long, our understanding
of reality, how we measure it,quantify it, it's all been
shaped by the mathematical toolswe had quantify it.
Speaker 2 (00:25):
it's all been shaped by the mathematical tools.
We had Right.
We've operated under thisassumption that you know, five
is always five.
A dimension is fixed Time flowsat this unwavering rate?
Speaker 1 (00:33):
Exactly.
But what if those veryassumptions, the ones baked into
our traditional math, wereactually limiting us?
What if they were stopping usfrom seeing the inherent
irregularities, the dynamism,the interconnectedness of
everything?
Speaker 2 (00:46):
Preventing us from grasping the full picture,
perhaps.
Speaker 1 (00:49):
Yeah.
Speaker 2 (00:49):
The tapestry of a living, evolving reality.
Speaker 1 (00:51):
That possibility, that very intriguing possibility
, brings us to today's deep dive.
We're going to plunge into theinsights of Philip Randolph
Lillian's groundbreaking paperIntegration of Fractal Calculus
with NFN Numbers.
Our mission today is really tounravel these complex but,
honestly, completely captivatingconcepts.
We want to uncover how theseNFN numbers, these non-fixed
(01:12):
number frameworks, aren't justenhancing fractal calculus but
maybe fundamentallyrevolutionizing it.
Yeah, offering a new lens,really A new lens to view
complex systems, maybe even thefabric of reality itself.
So yeah, prepare for a journeythat might challenge some basic
mathematical intuitions.
Speaker 2 (01:29):
And it's important to stress this work by Lillian
isn't just some theoreticaltweak in pure math.
It feels like a foundationalshift.
Speaker 1 (01:37):
How so.
Speaker 2 (01:37):
Well, it directly confronts the limitations we hit
when we try to use conventionalmath on inherently irregular
patterns, things that areunpredictable, dynamic.
Speaker 1 (01:46):
Like what kind of things?
Speaker 2 (01:47):
Oh, everything from, say, the branching of neurons in
the brain to chaotic weathersystems, the fractal shapes of
coastlines or even the adaptivebehaviors in economic markets.
Traditional math strugglesthere.
So by introducing NFNs, lillianprovides a framework that can
actually mirror these realitiesmuch more closely.
And the implications, well,they stretch far beyond just
(02:11):
math.
It really opens up new ways ofthinking, new avenues for
modeling in physics, biology,ecology.
Speaker 1 (02:18):
Even consciousness perhaps.
Speaker 2 (02:19):
Potentially, yes, it's about building a
mathematics for a world that'swell alive and constantly
changing.
Speaker 1 (02:26):
OK, so let's unpack this a bit.
Before we dive headfirst intoNFNs themselves, maybe we should
set the stage.
Let's talk about what fractalcalculus already does and why it
was needed in the first place.
Speaker 2 (02:36):
Good idea, because we all kind of intuitively know
the world isn't made of perfectgeometric shapes, right?
Speaker 1 (02:41):
Right, no perfect circles or straight lines out
there, not, not really look at amountain range or a cloud, or
even the branching of a tree orthe blood vessels in our own
body exactly.
They're all inherentlyirregular.
They're not Euclidean andtraditional calculus.
You know the math of smoothcurves and fixed dimensions.
It really struggles to describethose things accurately it
(03:02):
struggles mightily.
Speaker 2 (03:03):
Yeah, it's built for an idealized world, often a
linear one.
It works beautifully if you'remodeling, say, a perfect
parabola or a smooth wave.
Speaker 1 (03:12):
But nature isn't like that.
Speaker 2 (03:14):
Not usually no.
When you get into naturalphenomena, things with
self-similarity across scales,where details just keep
appearing the closer you look,classical calculus just hits a
wall.
Speaker 1 (03:25):
And that's where traditional fractal calculus
came in.
Speaker 2 (03:28):
Precisely it tried to bridge that gap.
It moved beyond whole numbersfor derivatives and integrals.
So instead of a first or secondorder derivative implying
smooth change, you might havesay a 1.5 order derivative.
Speaker 1 (03:40):
OK.
Speaker 2 (03:41):
And to quantify this roughness or complexity, it
introduced concepts like scalingexponents.
You see symbols like alpha,beta or the famous Hausdorff
dimension, dhx.
Speaker 1 (03:53):
Right, I've heard of the Hausdorff dimension.
Speaker 2 (03:54):
Yeah, these exponents try to capture the fractal
dimension, that non-integermeasure of how much a fractal
fills space.
Speaker 1 (04:00):
But even those had limitations.
That's where Lillian comes in.
Speaker 2 (04:03):
Exactly, and this is critical.
Even these scaling exponents,as revolutionary as they were,
have inherent limitations.
Lillian's paper points out afew key things.
First, they're local, notglobal.
Speaker 1 (04:16):
Meaning.
Speaker 2 (04:16):
Meaning they describe properties at a specific point
or maybe a small region, butthey don't necessarily hold true
across the entire system.
The roughness might bedifferent over here compared to
over there.
Speaker 1 (04:28):
Okay, that makes sense.
Speaker 2 (04:29):
Second, they're dimensionally modulated.
Their values can actuallychange depending on the
dimension or scale you'relooking at.
Adds complexity.
Right, and maybe the mostcrucial point, they are often
non-analytic or transcendental.
Speaker 1 (04:43):
Which sounds complicated.
What does that mean?
Speaker 2 (04:45):
practically it means these exponents can be
incredibly difficult to pin downprecisely with our usual fixed
number system.
They often defy simplealgebraic formulas or exact
calculation.
They're just messy in a way.
Speaker 1 (04:58):
So they get us closer , but still not quite there for
truly dynamic systems.
Speaker 2 (05:03):
Exactly.
Think about trying to preciselymeasure the edge of a
flickering flame or theturbulence in a river.
These exponents were betterthan classical numbers, but they
still operated on a kind ofstatic principle within a
dynamic reality.
They struggled to capture thefluidity of the change itself.
Speaker 1 (05:21):
That distinction local versus global, and them
being non-analytic.
Speaker 2 (05:26):
Yeah, that clarifies the problem.
It's like that analogy yousometimes hear trying to map a
constantly swirling cloud with aruler that only has fixed
markings, even fractional ones.
Speaker 1 (05:35):
Right, you get an approximation, but you miss the
continuous subtle changes.
Speaker 2 (05:39):
Or even if your ruler could measure tiny fractions,
what if the definition of aninch itself was subtly changing,
based on the cloud's internalstate, its energy, its coherence
?
You'd still be off.
Speaker 1 (05:50):
That's a great extension of the analogy.
It's not just about finermarkings on a fixed scale, it's
about the scale itselfpotentially being responsive.
Speaker 2 (05:58):
And that inability to capture a fluid, non-fixed
reality.
Yeah, that's why we needsomething new, a new kind of
number.
That's precisely.
We need something new, a newkind of number.
That's precisely the motivation.
We need a number system thatcan in a sense, breathe with the
system it's describing and thisbrings us directly to Lillian's
big contribution NFN numbers,non-fixed number or non-finite
(06:21):
numeric frameworks.
Speaker 1 (06:22):
Okay, so not just an improvement, but a whole new
paradigm.
Speaker 2 (06:25):
That's how it's presented.
Yes, A fundamental shift.
They're specifically designedto tackle those limitations we
just discussed, offering areally different way to quantify
and interact with dynamicreality.
Speaker 1 (06:37):
So what makes them different?
What are their corecharacteristics?
Speaker 2 (06:39):
Lillian defines them carefully.
First, they're built toaccommodate variable scaling
exponents.
So, unlike that fixed alpha inolder fractal calculus, an NFN
number can represent an exponentthat is itself changing,
adapting.
Speaker 1 (06:52):
Okay, variable exponents got it.
Speaker 2 (06:54):
Second, they represent non-integer
scale-relative and dynamicallymodulated quantities.
This means they capture valuesthat aren't just fractions but
are tied to the scale you'reobserving at, and they can shift
dynamically during theobservation.
Speaker 1 (07:06):
Scale relative and dynamic Okay.
Speaker 2 (07:08):
But here's the most striking part, where the deeper
implications really start.
Nfns are designed to expressvalues that are contextual and
coherence dependent, notstatically fixed.
Speaker 1 (07:19):
Okay, hold on Contextual and coherence
dependent.
That sounds really different.
Contextual, I sort of getdepends on the surroundings, but
coherence dependent, thatsounds really different.
Contextual, I sort of getdepends on the surroundings, but
coherence dependent, what doesthat mean?
In this kind of advanced mathit almost sounds like the number
itself is responsive.
Speaker 2 (07:34):
You've absolutely zeroed in on the core innovation
there.
Yes, responsive is a good wayto think about it.
Coherence dependence means thevalue of an NFN number isn't
some absolute, predefined thingexisting separately from the
system.
Speaker 1 (07:47):
OK.
Speaker 2 (07:47):
Instead, its value emerges from, and is deeply
influenced by, the underlyingcoherence field, the internal
consistency, the order withinthe system it's describing.
Speaker 1 (07:56):
So it's not just a label.
It reflects the system'sinternal state.
Speaker 2 (08:00):
Exactly.
Think of it less like a fixedpoint on a number line and more
like a dynamic property thatarises from the web of
relationships inside the system.
If the system's internal order,its coherence, its resonance
changes, the NFN describing someproperty of that system also
changes correspondingly.
Speaker 1 (08:18):
Wow, okay, so that makes them perfect for fractals,
where the parts and the wholeare constantly interacting and
influencing each other.
Speaker 2 (08:25):
Perfectly suited.
Yes, it allows the math togenuinely mirror that dynamic,
irregular nature.
It lets the mathematics breathewith the system.
Speaker 1 (08:33):
So is there a key takeaway sentence here?
The aha moment.
Speaker 2 (08:37):
Yes, and Lillian states it very clearly in the
paper.
Fractal calculus thereforerequires a number system that
can fluidly move between integer, fractional, irrational and
context-modulated values, whichis exactly what NFN enables.
Speaker 1 (08:50):
Fluidly move between different types of values, even
context-modulated ones.
Speaker 2 (08:54):
Right.
Our traditional number system,even complex numbers, is
fundamentally rigid.
It assumes a fixed backdrop.
Nfns challenge that they offera language for a reality where
the rules themselves are dynamic, emergent, tied to the system's
internal state.
Speaker 1 (09:08):
That fluidity is the key, then, unlocking a new level
of description for systems thatjust defy static measurement.
Speaker 2 (09:15):
It allows for a truly adaptive, almost living
mathematics.
Speaker 1 (09:19):
Okay, so we have these NFNs fluid, responsive
numbers.
Now how do they actually changethe tools of calculus?
Let's start with derivatives.
A derivative tells us the rateof change, right, how steep a
curve is at a pointinstantaneous change.
Speaker 2 (09:32):
That's the classical idea.
Speaker 1 (09:34):
So how does fractal calculus, especially with NFNs
mixed in, rethink thatfundamental concept?
Speaker 2 (09:40):
Well, in traditional calculus, even in the earlier
fractal versions, you use afixed order derivative.
Speaker 1 (09:45):
Yeah.
Speaker 2 (09:45):
Written like DOF X dia, where alpha is a specific
unchanging number.
Maybe one for classical change,maybe 0.5 for some fractal
roughness.
Speaker 1 (09:54):
A single constant order.
Speaker 2 (09:55):
Exactly Applied uniformly.
But Lillian introduces thefractal coherence derivative in
NFN form.
Speaker 1 (10:01):
Okay, sounds fancy.
What's the essence of it?
Speaker 2 (10:03):
The formula itself looks complex dx, fx, lim,
epsilon zero, fx plus epsilon,fx, epsilon.
Speaker 1 (10:10):
Whoa okay, Lots of symbols there.
Speaker 2 (10:12):
Yeah, but let's focus on the implication.
It still uses that limitconcept, epsilon going to zero
for instantaneous change.
But the huge difference is theexponent ax.
That little tilde means it's anNFN number.
Speaker 1 (10:26):
Ah, so the exponent itself is fluid.
Speaker 2 (10:28):
Precisely, and notice , epsilon is raised to the power
of that NFN number.
So the very step size you'reusing to measure the change is
also dynamically modulated bythis fluid exponent.
Speaker 1 (10:40):
So it's not just the order of the derivative, that's
non-integer, it's variable.
What does that let it do?
Speaker 2 (10:46):
That's the transformative part, because AX
is an NFN, it's not static, itvaries dynamically and,
crucially, it reflects coherentscaling relationships, not just
static geometric properties.
Speaker 1 (10:57):
Okay, unpack that Coherent, scaling relationship.
Speaker 2 (10:59):
It means the rate of change you calculate isn't just
about the shape of the curveanymore.
It adapts based on the internalcoherence of the system, its
level of organization, itsresonance, maybe even how it's
interacting with an observer.
Speaker 1 (11:11):
So the steepness depends on the system's internal
state, not just its geometry.
Speaker 2 (11:15):
Exactly.
It allows for continuouslyadjusting derivative orders
across hyperfractal domains.
The very nature of how changehappens can itself change,
responding to the system'sunderlying logic or state.
Speaker 1 (11:27):
That's quite a shift.
Let me try an analogyCalculating a car's speed,
Normally miles per hour is fixed, but here it's like the unit
miles per hour itself couldchange dynamically based on road
conditions, engine health,driver focus.
Speaker 2 (11:43):
That's a really good way to think about it.
Yeah, the mathematical rulerfor measuring speed isn't just
more precise, it's adapting inreal time to the context.
Speaker 1 (11:51):
And this allows movement across hyperfractal
domains.
What does that mean?
Different layers of reality.
Speaker 2 (11:56):
Conceptually, yes.
Think of different scales orstates within a complex system.
Maybe the molecular level, thecellular level, the organ level
in biology, each could be ahyperfractal domain.
This NFN calculus allows foranalysis between these layers.
It's not a flat calculation,it's multi-dimensional, adaptive
measurement reflecting the truecomplexity.
Speaker 1 (12:16):
It means we can model things where the rules of
change aren't fixed constantsbut emerge from the system's
current evolving state and itscoherence, a math that
acknowledges how interwovenscale and context really are.
Okay, so derivatives get thisdynamic, coherent, sensitive
upgrade.
What about the flip sideintegration?
Summing things up to find atotal like area under a curve or
(12:39):
volume.
Traditional integration usesfixed little bits right.
Speaker 2 (12:42):
Right Summing up tiny rectangles of fixed width.
Essentially, even standardfractal integrals like Arup, fx,
xx, 1dx use a fixed fractaldimension, alpha.
It accounts for irregularitybetter than classical integrals,
but the measure, the unit ofsumming, is still static.
Speaker 1 (12:57):
So it assumes the little bits you're adding up are
all the same kind of bit.
Speaker 2 (13:06):
Exactly.
Lillian extends this with theNFN extended form FSX1DX.
Notice the domain A is now ahyperfractal domain, suggesting
a complex layered space.
Speaker 1 (13:10):
And there it is again AX, the NFN exponent.
So that's the game changer forintegration too.
How does it affect the measure,what Lillian calls DNFNX?
Speaker 2 (13:19):
It completely transforms the measure.
The NFN fractal measure, dnfnx,isn't static.
It dynamically adapts.
It responds multiple factors.
The source highlights the localcoherence field, strength CX.
The system's internal orderAlso observer.
Impact acknowledgingmeasurement can affect the
system and even dimensionalcollapse or emergence events
(13:41):
where the system's basicstructure might suddenly shift.
Speaker 1 (13:44):
Wow, observer impact and dimensional collapse.
That's deep.
Speaker 2 (13:48):
It is.
It means, when you're summingup over an irregular changing
domain, the very units you'reaccumulating aren't fixed.
They adapt moment by moment,based on the system's coherence,
how it's being watched, evenshifts in its dimensionality.
Speaker 1 (13:59):
So each tiny piece carries information about the
system's internal state.
Speaker 2 (14:04):
That's the idea behind hyperfractal measures,
where every infinitesimalcarries an embedded coherence
logic.
Each tiny slice isn't justgeometry it holds information
about the internal order andrelationships.
Speaker 1 (14:14):
At that point, Like the measuring ruler, has
built-in feedback Kind of.
Speaker 2 (14:19):
yeah, it's responding to the internal state of what's
being measured, not just itsexternal shape.
This allows the integral todetect resonance thresholds,
instability points.
It's a huge leap.
It moves from passivemeasurement to active, dynamic
quantification.
Speaker 1 (14:35):
So what does this mean?
Overall, our measurementbecomes an interaction.
The system's internal stateinfluences how it's measured.
Speaker 2 (14:42):
Precisely, it's radically different.
It moves beyond staticobservers and fixed metrics to
an interactive, responsiveframework, like trying to
measure the total emotion in aconversation.
You can't just count words.
The measure has to adapt totone, context, relationships.
Speaker 1 (14:58):
So NFN integration allows for a much more accurate
picture of reality for systemsin constant flux, systems
governed by coherence.
Speaker 2 (15:05):
Exactly.
It could be revolutionary forthings like adaptive materials
or understanding informationcontent in quantum systems,
where static models just fallshort.
Speaker 1 (15:13):
OK, nfns are reshaping space and measurement,
but what about time?
We usually think of time, asyou know linear Tick-tock, one
second after another, same paceeverywhere.
Speaker 2 (15:21):
The single river flowing uniformly.
Yeah, that's the classicalpicture.
Speaker 1 (15:24):
But Lillian's work challenges that too.
Suggests time might not be sofixed.
Speaker 2 (15:28):
It pushes our understanding right to the edge.
Here Lillian introduces theconcept of multiple temporal
axes T1, t2, t5.
Speaker 1 (15:35):
Five time axes.
Speaker 2 (15:36):
Potentially yes, suggesting not just one timeline
but maybe multiple intertwinedtemporal dimensions or layers
operating at once and, crucially, these can be indexed using NFN
exponents for each axis writtenas TNN.
Speaker 1 (15:50):
So each time axis has its own dynamic NFN rate.
Speaker 2 (15:54):
Exactly.
It immediately implies amultidimensional dynamic view of
time where each axis' flow orcharacter is modulated by an NFN
responding to local conditions.
A huge departure from a singleuniversal clock.
Speaker 1 (16:07):
How does that play out in the math?
Is there a time derivative?
Speaker 2 (16:09):
Yes, and it integrates this NFN index
dynamism.
The formula looks like NN, cntn, index, epsilon CNTN.
Okay, again complex formula.
But what's the core idea?
We're differentiating over atime axis whose very texture or
rate of progression is dynamic.
It's influenced by both its NFNexponent and its local
(16:30):
coherence field, c and tn.
At that temporal point, eventhe epsilon step size is
modulated by this dynamiccombination.
Speaker 1 (16:38):
So the fabric of time is woven by the system's
coherence.
Speaker 2 (16:40):
That's the implication.
Speaker 1 (16:41):
yes, Mind blown.
Okay, so if each on is an NFN,dynamically changing, what does
that mean for time in, say, abiological system or a quantum
process?
Speaker 2 (16:52):
It means each NFN exponent on becomes a dynamic
resonance scalar that indexesthe dimensional, coherent state
of time itself.
Speaker 1 (16:58):
A dynamic resonance scalar indexing the coherent
state of time itself.
Speaker 2 (17:02):
Yeah.
So the flow or structure oftime locally isn't uniform or
predetermined.
It adapts based on its owninternal coherence, its
resonance with the system, maybeeven observer interaction.
Speaker 1 (17:13):
So back to the river.
Analogy time isn't one river.
Speaker 2 (17:16):
It's more like a complex multi-branched river
system.
The current in each branch canspeed up, slow down, form eddies
, maybe even stop flowing for abit, all depending on the
dynamic interactions andcoherence within that part of
the system.
Speaker 1 (17:28):
So time might flow differently in a highly coherent
biological process versus achaotic quantum event.
Speaker 2 (17:34):
That's the kind of nuanced understanding this
framework allows for.
It suggests time is part of theadaptive landscape, not just a
static backdrop.
Speaker 1 (17:42):
Like how our brains process information, or
biological clocks.
Could this model how timeperception or biological time
warps locally based on thesystem state?
Speaker 2 (17:52):
It potentially could.
Yes, it implies time isn't justsomething we move through, but
a dynamic variable that can beshaped by the systems existing
within it.
Speaker 1 (17:59):
Okay, nfns redefine derivatives, integrals, even
time, making them fluid andresponsive.
So Lillian brings us alltogether in what he calls the
NFN Integrated UniversalCohesion Equation.
Uce sounds big.
Speaker 2 (18:12):
It is aiming for a big synthesis, yeah, a
comprehensive description ofdynamic systems.
The UCE formula is written asRDX.
Speaker 1 (18:20):
All right, let's break that down R.
Speaker 2 (18:21):
R is a coherence operator.
It acts on the NFN enhancedfractal derivative that follows
it.
Speaker 1 (18:26):
And inside that derivative DX CX, we have our
dynamic NFN exponent X.
Speaker 2 (18:31):
Right, which is fluid and context dependent, but it's
also modulated by CX, the localcoherence field.
So the rate of change itself isdirectly impacted by the
system's internal order orharmony.
Speaker 1 (18:42):
Okay, and X is the system state it's the wave
function or coherence state.
And on the right side.
Speaker 2 (18:48):
X.
X is the coherence eigenvalueand, crucially, this is also an
NFN number.
Speaker 1 (18:54):
Also an NFN, so it's not a fixed constant.
Speaker 2 (18:57):
No, it's a dynamic value reflecting the overall
coherence state of the entiresystem.
So the equation basically saysa coherence operator acting on
the NFN-modulated rate of changeto the system's coherence state
equals a dynamic coherenceeigenvalue times that state.
Speaker 1 (19:12):
That is a lot part, but incredibly powerful sounding
, Connecting it to the biggerpicture.
What does this UCE let uscapture that traditional math
just can't.
Speaker 2 (19:21):
It offers a truly holistic framework.
It can describe systems wherescale, time and coherence aren't
static but are in constantdynamic interplay.
The source highlights itcaptures variable scaling.
Speaker 1 (19:33):
Meaning the fractal dimension itself changes.
Speaker 2 (19:36):
Yes, dynamically, based on the system's coherence.
Think of a storm system whosefractal patterns shift as it
organizes or dissipates.
Okay what else.
It models fractal space-timeinteractions, how the geometry
of space and the flow of timeare linked and influence each
other in complex, dynamic waysmodulated by their fractal
nature and coherence, maybegoing beyond even what general
(19:58):
relativity describes in certaincontexts.
Wow, and perhaps mostcritically, the UCE captures
dynamic emergence or collapse ofcoherence, the spontaneous
formation of order from chaos orits sudden breakdown.
Speaker 1 (20:10):
So it's a mathematics for a living, evolving reality,
where properties emerge anddissolve, based on coherence.
Speaker 2 (20:16):
Exactly, it's an adaptive mathematical language
where the rules aren't justdiscovered but are, in a way,
being written by the systemitself as it evolves.
Speaker 1 (20:24):
Yes, that brings up a key point, then.
Coherence here isn't just anice idea like things hanging
together.
It's a quantifiable dynamicfield.
Speaker 2 (20:32):
Absolutely CX.
The coherence field isn't justdescriptive, it's operational,
quantifiable, it exerts a directinfluence on the math.
Speaker 1 (20:39):
How.
Speaker 2 (20:40):
It directly modulates those NFN exponents that define
rates of change in measures.
Its intensity, its dynamicsdetermine how scaling happens,
how time flows locally, how thesystem evolves.
Think of a coherent laserversus diffuse light, or an
organized ant colony versusscattered ants.
Speaker 1 (20:58):
The coherence isn't just an outcome, it dictates
behavior.
Speaker 2 (21:01):
Precisely, the UCE provides the mathematical
machinery to understand howorder arises, shifts or falls
apart not due to external forces, necessarily, but as intrinsic
properties of the system's ownevolving coherence.
So let's just circle back andreally emphasize this core idea
NFN numbers, especially thatdynamic exponent of X, aren't
just arbitrary variables.
They act as coherence-basedscalars.
Speaker 1 (21:23):
Meaning.
Their value comes from thecoherence.
Speaker 2 (21:25):
Yes, their specific value is intrinsically linked to
, derived from the coherencefield intensity CX.
The level of internal order, orhow that order changes,
directly dictates the numericalvalue of these NFNs.
It gives them this built-insensitivity, a feedback loop to
this system's organization.
Speaker 1 (21:45):
Right.
Speaker 2 (21:46):
And we really need to re-emphasize the implications
of that.
Coherence.
Modulated NFN derivative DX isFX.
It's revolutionary because itallows scaling orders to vary
locally.
One part of a system might haveone fractal dimension, another
part a different one, alldynamically determined, okay.
Plus, these orders respond tocoherence, resonance.
They change in sync with thesystem's internal harmonics or
(22:08):
patterns of order and, crucially, this allows the system to
potentially traverse betweendimensional layers of reality.
Speaker 1 (22:14):
Traverse between dimensional layers.
That sounds like sciencefiction almost.
Speaker 2 (22:17):
It pushes boundaries.
For sure.
It's not just about differentgeometric scales, but
potentially shifting acrossfundamental states of
organization or existence withinthe system.
Speaker 1 (22:26):
Okay, this goes way beyond simple geometry, even
beyond standard physics.
It implies this responsivenessis built into the math,
describing systems that areadaptive, self-organizing but
perversing dimensional layers.
What could that actually looklike in a real-world scenario?
Conceptually like emergencephase transitions.
Speaker 2 (22:47):
That's a great way to frame it and, conceptually, yes
, you're on the right track.
While the paper doesn't givespecific real-world examples of
traversing layers, it couldrelate to things like emergent
properties.
Speaker 1 (22:57):
Like consciousness from neurons.
Speaker 2 (22:59):
Exactly.
Individual neurons are onelayer.
Consciousness emerges as aqualitatively different layer.
This NFN framework mightprovide the math to model how
that happens, how the coherenceof the parts leads to that shift
in the whole.
Speaker 1 (23:12):
Or phase transitions water to ice.
Speaker 2 (23:14):
Another excellent conceptual fit.
The substance shifts from onestate, one dimensional layer of
organization, to another withtotally new properties.
Lillian's calculus, by adaptingits descriptive power to the
changing coherence anddimensionality, could
potentially model the dynamicsof that transition itself, not
just the before and after states.
Speaker 1 (23:33):
So it's about modeling the process of becoming
something new.
Speaker 2 (23:37):
Yes, mathematically navigating and quantifying these
profound transformations wherethe system's rules fundamentally
change as its coherence shiftsand this framework doesn't just
describe it also offers powerfulinsights into stability and
instability.
Lillian talks about NFN-drivenresonance collapse and
stabilization.
This is huge for prediction.
Speaker 1 (23:57):
Predicting when systems might break down or find
balance.
Speaker 2 (24:01):
Exactly.
He identifies destabilizationpoints.
These happen when that combinedterm, the NFN exponent, times
the coherence field tendstowards either zero or infinity.
Speaker 1 (24:11):
Okay, zero or infinity.
What happens then?
Speaker 2 (24:13):
When it heads towards zero, it indicates coherence
collapses or resonanceannihilation.
This is where the system losesits internal order.
Its connections break down, itdissolves into chaos.
Speaker 1 (24:23):
Can we imagine that, like a bridge vibrating itself
apart, or maybe an ecosystemlosing its balance suddenly?
Speaker 2 (24:30):
Those are perfect conceptual examples.
Yes, or think of a market crashwhere the interconnectedness
fails.
That's a coherence collapse.
Speaker 1 (24:36):
OK, and when it goes towards infinity?
Speaker 2 (24:38):
That signals critical amplifications or resonance
inflation, uncontrolled growth,runaway feedback loops, extreme
instability that could destroythe system through overexpansion
.
Speaker 1 (24:49):
Like the bridge resonance example again, but
amplifying uncontrollably.
Or maybe a tumor growing out ofcontrol in biology.
Speaker 2 (24:56):
Exactly, or a rumor spreading virally causing
self-reinforcing panic.
That's resonance.
Inflation, growth mechanismsbecome destabilizing.
Speaker 1 (25:04):
So points of breakdown or runaway,
instability?
What about stability?
Speaker 2 (25:08):
Stabilization points occur when that same product,
xcx equals a constant.
This isn't spatic rigidity,though it's a dynamic
equilibrium across thehyperfractal geometry.
Speaker 1 (25:19):
Dynamic equilibrium meaning.
Speaker 2 (25:21):
The system maintains order through continuous
adaptive adjustment, balanceachieved through constant motion
and recalibration, like ahealthy ecosystem, maintaining
resilience despite fluctuations.
Speaker 1 (25:31):
So NFN calculus could pinpoint both crisis points and
states of dynamic balance.
Speaker 2 (25:36):
That's the power it offers a mathematical tool to
identify these criticalthresholds in highly complex,
changing systems.
It moves us toward predictionand, potentially, management.
Speaker 1 (25:45):
That really is game changing not just describing
complexity but potentiallypredicting its critical turns
across so many fieldsengineering, climate economics.
It's about understanding thethresholds of chaos and order.
Ok, I think we can agree.
These concepts are profound butalso incredibly abstract.
It's hard to really visualizecoherence fields modulating NFN
(26:05):
exponents that changeintegration measures across
hyper fractractal domains.
Speaker 2 (26:10):
It definitely stretches the imagination, yeah.
Speaker 1 (26:12):
And Lillian recognized this challenge.
He proposed a visual, ascroll-style visual.
Speaker 2 (26:17):
Yes, exactly A way to try and make these dynamic
interactions more tangible, moreintuitive the scroll-style
visual Fractal derivatives fromNFN-modulated scaling curves.
Speaker 1 (26:27):
Okay, paint us a picture.
What would the scroll look like?
Speaker 2 (26:30):
Imagine a continuous scroll unfurling like an ancient
map, but dynamic.
It's not just a static graph,it's a complex, evolving
landscape of functions.
Speaker 1 (26:39):
What are the key features?
Speaker 2 (26:40):
Okay, first, along the bottom, an x-axis.
That's your input domain, x,the variable you're tracking.
Speaker 1 (26:45):
Standard enough.
Speaker 2 (26:46):
Then, across this domain, you see multiple curves,
not just one, but variouscoherent scale functions fx.
Each curve is shaped by its owndynamic NFN exponent, fx.
So you see this family ofcurves morphing and interacting
as you move along the scroll.
Speaker 1 (27:02):
So the curves themselves are alive, changing
shape.
Speaker 2 (27:05):
In a sense, yes, and overlaid on them are dynamic
tangent arrows.
These show the slope, the rateof change.
You'd see arrows for classicalchange, maybe fixed fractional
change 80.5.
But, crucially, the dynamic NFNvarying slope driven by apex.
Speaker 1 (27:21):
So the tangent arrows themselves would be changing
direction and length along thecurve.
Speaker 2 (27:24):
Exactly, visually demonstrating how the rate of
change is constantly influencedby both the fluid and event
exponent and the local coherence.
Speaker 1 (27:32):
That's clever.
What else There'd?
Speaker 2 (27:33):
be an overlay.
The local coherence fieldstrength is color gradient, so
CX gets visualized.
High coherence areas might glowbrightly.
Low coherence areas might dimor change color.
You literally see coherenceshaping the landscape.
Speaker 1 (27:44):
Wow Okay, like a heat map of order.
Speaker 2 (27:47):
Kind of yeah, and a side legend would map the
product IC to local resonancemodulation, telling you if the
system is stable, inflating orcollapsing, based on the colors
and arrows.
Speaker 1 (27:57):
And a spiral motif.
Speaker 2 (27:59):
Yeah, a spiral motif showing how curves fold into
higher resonance orders, maybevigorizing how complexity
emerges or how the system jumpsbetween those dimensional layers
we talked about.
Speaker 1 (28:09):
That is actually a brilliant visualization.
It makes the abstract concrete.
It shows the dynamism, thelayers, how coherence literally
shapes the math and thus thesystem.
It's not just numbers, it's aliving map.
Speaker 2 (28:21):
Exactly A map that breathes with the territory it
describes.
It has bridged that gap betweenthe complex equations and
intuitive understanding.
Hashtag ouch tag outro.
Speaker 1 (28:30):
What an absolutely incredible journey.
Today, diving into PhilipRandolph Lillian's work, we
started by questioning somethingas basic as numbers, imagining
them as fluid, responsive.
Then we plunged into NFNnumbers, saw how they could
redefine derivatives, integrals,making them dynamic,
context-dependent.
Speaker 2 (28:46):
Reshaping the very tools of calculus making them
dynamic, context-dependentReshaping the very tools of
calculus.
Speaker 1 (28:50):
We even stretched our minds to consider dynamic time
axes, a multidimensional, fluidtime, and it all came together
in that universal cohesionequation, the UCE.
Speaker 2 (28:58):
A framework trying to capture that interplay of
variable scaling, fractalspace-time and dynamic coherence
.
Speaker 1 (29:05):
It's really quite something.
Speaker 2 (29:06):
Indeed and, as we often say, knowledge is most
valuable when understood andapplied.
This isn't just abstract maththeory, lillian's work here.
It really lays down a potentialblueprint for new ways of
understanding and modeling theseincredibly complex adaptive
systems that define our world.
Speaker 1 (29:24):
From physics and biology.
Speaker 2 (29:25):
To economics, ecology , maybe even consciousness.
It offers a languagepotentially to describe things
that were just beyond our graspwith fixed mathematical tools.
It's about shifting from astatic view to one embracing
dynamism and interconnection.
Speaker 1 (29:39):
So what does this all mean for you listening?
Well, it means ourunderstanding of reality, the
tools we use.
They're far from finished.
This deep dives reallychallenges those fundamental
assumptions about reality beingfixed.
It pushes us to think about aworld where even numbers are
dynamic.
Speaker 2 (29:53):
And if the very numbers describing reality are
dynamic, context dependent, andif they're linked to this
underlying coherence fielddictating how things evolve.
Speaker 1 (30:04):
Well, what does that imply about reality itself?
Is it more stable or moremalleable than we think?
Could NFN numbers eventuallyhelp us not just understand, but
maybe predict or even interactwith complex phenomena, from
weather patterns to theemergence of consciousness?
Speaker 2 (30:19):
Could we do it in ways we never thought possible
just by tuning into theirunderlying coherence.
Speaker 1 (30:24):
It leaves us with that really tantalizing thought
Maybe our universe is far moreresponsive, far more alive than
our current mathematics evenallows us to perceive.
Something to definitely ponder.
Popular Podcasts
24/7 News: The Latest
The latest news in 4 minutes updated every hour, every day.
Crime Junkie
Does hearing about a true crime case always leave you scouring the internet for the truth behind the story? Dive into your next mystery with Crime Junkie. Every Monday, join your host Ashley Flowers as she unravels all the details of infamous and underreported true crime cases with her best friend Brit Prawat. From cold cases to missing persons and heroes in our community who seek justice, Crime Junkie is your destination for theories and stories you won’t hear anywhere else. Whether you're a seasoned true crime enthusiast or new to the genre, you'll find yourself on the edge of your seat awaiting a new episode every Monday. If you can never get enough true crime... Congratulations, you’ve found your people. Follow to join a community of Crime Junkies! Crime Junkie is presented by audiochuck Media Company.
The Clay Travis and Buck Sexton Show
The Clay Travis and Buck Sexton Show. Clay Travis and Buck Sexton tackle the biggest stories in news, politics and current events with intelligence and humor. From the border crisis, to the madness of cancel culture and far-left missteps, Clay and Buck guide listeners through the latest headlines and hot topics with fun and entertaining conversations and opinions.