Spiral Mathematics
The Generative Face of Chaos Theory
A Note on the Non-Euclidean Nature of Spiral Mathematics
Throughout this and other applications of Spiral Mathematics, you will find little, if any, standard mathematical or physics notation. Standard notation generally requires normalized data to establish relevance and comparability between computational units. Spiral Mathematics is, in part, the science of normalization itself. Reduction to Cartesian units is highly application-specific and may not be possible when all relevant factors have not yet been exposed.
The absence of conventional notation should therefore not be interpreted as an absence of mathematical structure. Spiral Mathematics operates on relationships, transformations, trajectories, and energetic constraints that may precede the point at which a system can be reduced to conventional numerical coordinates.
Spiral Mathematics is the logical process through which Cartesian-style measurement systems are derived, enabling Euclidean calculations to operate.
Because those calculations must remain application-specific, premature imposition of Euclidean notation on still-unfolding relational structures is often spurious and, when the goal is to understand the shape of the calculus itself, actively distracting.
If we take Cartesian mapping of three-dimensional space as an example, the coordinates x, y, and z have no absolute relevance. They are externalizations of how humans experience and interpret reality relative to our own sensory and motor organization: forward/backward along the primary axis of locomotion and gaze, lateral to that axis, and vertical relative to our upright posture.
From any perspective that does not share those innate referents, the three axes are meaningless. Establishing a plane of translation therefore requires setting aside the already-normalized, perspectival coordinate system long enough to identify the common relational structure that can be re-expressed in an alternate frame.
Why Chaos Theory is an Incomplete Model of Reality
A silent truth of mathematics is that for any reductionist construct, there exists an equally powerful generative perspective.
When modern physics adopted the entropic viewpoint of reality constructs, where any energetic expression not directly connected to a defined goal as waste or chaos, the failure to establish a defined goal as a boundary and an agreement of direct force application direction meant that in the largest scope, there is no category for any energy but chaotic.
In short, if you set out on a drive with no goal in mind, then no goal can be reached and from that perspective, the trip can only be a waste, no matter what happens.
If, on the other hand, your goal is to enjoy the drive and get home safely, then little if any of the energy involved is wasted. If you wished, you could calculate how much each event contributed to the goals of enjoyment and safety.
The first example is chaos theory. Because the universe has no defined work to do, evolution in any direction is inherently chaotic because a trajectory without a goal can never be defined as approaching it.
The second example is the root of Spiral Mathematics, the science of combinatoric exposure and control that makes arbitrarily diverse, massive datasets calculable, navigable, and tractable for understanding.
Both are methodologies for normalizing data, but where chaos theory treats massive combinatoric environments as inherently chaotic and ultimately not understandable because their directly observable consequences are highly variable. Spiral Theory accepts that although the individual factors are too granular for distinct calculation, the shape and flow trajectories of a calculation through combinatoric space pass through salient points that can determine and adjust state. By mapping the absolute vector regions, those control points can be exposed and aligned.
Spiral Theory — The Generative Boundary of Chaos Theory
Spiral Theory is not new, although its distinct formalization appears to have been lost. I first encountered the idea through my older brother, who discussed it after taking his first university mathematics course in the mid-1980s. I suspect, however, that it has since been subsumed into other disciplines and obscured behind the rise of chaos theory.
If you have designed a pivot table, planned for an event, managed a process, or even figured out your menu for the week, you have practiced Spiral Theory, whether you knew it or not.
To show how spiral theory works in practice, we’re going to talk about Big Pie Co., a worldwide pie-making company with branches in every part of the world.
Big Pie Co — A Predictive Spiral Design
Big Pie Co. makes every kind of pie you could imagine. They own supply farms of every kind, and their factories make pies for local consumption while exporting their products to other company stores in other regions.
If you asked the head corporate officers to explain the company in one sentence, they might say something like, “We produce and sell pies at enough of a difference to be profitable.”
From their perspective, that is a completely valid and thoroughly explanatory statement. How a pie gets produced and sold isn’t directly their concern. To manage the company as a whole, any pie sold at a sufficient profit is a success.
If, on the other hand, you asked them how to make a pie, they would struggle to give you an answer that made sense, if they could attempt it at all. They control a company that makes fruit pies, berry pies, meat pies, dairy pies, dessert and savoury pies, with countless fillings and regional recipes manufactured in dozens of distinct ways.
The problem is not whether they know the answer. It is that, from their perspective, the combinatoric space of possible answers is unreasonably large.
They could articulate the flow of elements that must be coordinated to produce and sell a pie. They could specify how the price of a pie is established and what determines profitability as a concept, but how any specific pie is made and sold is something they would neither know nor care about unless there were a specific reason to examine it.
Big Pie Co. is large and successful because its data systems have undergone spiral compression. At each level of the company, every position has access to the data relevant to its responsibilities, scoped to what it can control, along with the authority to make the changes required.
When a condition the company needs to navigate, whether a positive or negative effect on profitability, manifests, the right person can assess it, address it, and report on it effectively both upward and downward along the data path.
At any given time, 20% of the effort of Big Pie Co. is focused on making and selling today’s pies. The other 80% is concerned with controlling the combinatoric trajectory of the company to ensure the conditions allow tomorrow’s pies to be made on time.
When a gardener in an apple orchard noticed an insect nest of a known, highly prolific pest that would impact the quality of apples for the next two years but not harm the trees or decrease the size of the yield, Big Pie Co didn’t ignore the problem and have a critical shortage of apple pies.
It was already too late to eliminate the pests, once they begin nesting control is impossible, but Big Pie Co could reduce the demand for apple pies, bring them in from other regions, and find a profitable use for the apple yield that couldn’t be made into pies the normal way.
Marketing instantly begins planning a series of Limited-Edition International Pies for the affected region.
R&D accelerates its research into Jelly Pies. Apples that were no longer suitable for conventional pies were excellent for cider, while research into an entirely new line of pies was already underway.
Production and sales begin assessing which pies from which regions can be imported, which should be imported, and which the underused apple-pie facility can produce locally as a version “inspired by” an international favourite.
This all happens near instantly because, as any element of the network changes, Spiral visibility allows each control point to see the flow it needs to control long before it arrives.
The gardener’s report changes the local yield manager’s projections, which causes the expected usable and substandard apple projections to be revised. Those revised projections then flow to the other departments that depend on them for planning, and the process continues throughout the network.
Since Big Pie Co. has invested heavily in ensuring that it can always produce and sell a profitable line of pies, regardless of what that might be in any particular instance, it has outlasted and outgrown competitors who reacted to the effects of events as they became problems rather than controlling the probability space in which they were problems.
How the Spiral Compresses Complexity
This is possible because the data flow is thoroughly mapped and the specific processes arranged in loops based on the specific data frequency, prediction and control planes that provide the longest horizon of prediction to each node.
The yield manager has sufficient information to reasonably predict the effect of the insect infestation on the overall local crop and provide guidance to the gardener (don’t bother to spray, it’s too late) and provide an accurate combinatoric range of future action to the next levels of the company.
At each level, the node can count on the combinatoric trajectory established at the previous node to carry through to their own node and allow them to proactively prepare to refine and redirect the combinatoric trajectory of the next phase of the flow.
We’re going to sidestep for a moment and look at the classic Chaos Theory example, the flow of a drop of water dropped onto an irregular surface.
My instinct has always been that any practitioner of fluid dynamics has a prepared speech for exactly why that situation is not in any way chaotic and to use it for chaos theory is counter intuitive.
Where Chaos Theory says that no two drops of water experience exactly the same conditions and therefore their specific trajectories are inherently unpredictable, Spiral Theory, as applied through fluid dynamics, says that although the exact trajectory of each drop is combinatorically prohibitive to predict, there are multiple points along that trajectory where the flow can be actively controlled, reducing the range of possible end states.
For example, the Spiral view would be that the source and destination planes can be fixed to ensure every drop had minimal variation in contact point. It would also show what factors of the surface being dropped on to were most likely to influence the flow of the droplet and suggest adjustments to the parameters around those points.
Applying the Methodology Predictively
Unlike most formal methods, which prescribe a fixed sequence of questions, Spiral Mathematics is recursive and multi-perspectival by nature. Every element is examined both through its own causal chain and through its transitive effects on secondary chains. The aim is not to complete a checklist, but to expose the control points that govern the overall trajectory.
When examining any one facet of the system the questions asked are:
Which factors are expressions of existing trajectories?
Which factors are adjustable and directable at this point?
Which factors are predetermined by later trajectory needs (ie: mandated by downstream organizational needs)?
The second set refines the first groupings, with a primary focus on the operational band:
Of the operational factors at this level, what information is required to provide the strongest redirection or reinforcement of the incoming trajectories of information through the structure?
What is the earliest point at which that can be derived?
What information can this node provide downstream to ensure the operating constraints of this node and its antecedent nodes are correct?
The third set focuses directly on the node’s control plane: which factors it is actually in the best position to manipulate, redirect, or reinforce.
What is the optimal frequency and granularity of review for each factor to provide the optimal leverage over the process?
Do any of the controlled factors have significant combinatorial effects on any other factors and, if so, is that a fixed or variable relationship? For example, is there a variable compounding factor based on ordering?
Is there information that this node requires but does not have that would improve its combinatorial leverage?
Are there missing or excess parent or child nodes obscuring the correct point at which that information should be exposed?
Are any parent or child nodes making out of scope decisions that arbitrarily decrease the combinatoric leverage of this node?
Are any disconnected nodes making decisions that materially affect the combinatorial leverage of this node, and if so, why? Regulatory bodies are a working example of this.
Last and possibly most important:
Is there information being discarded at this node that would materially affect the combinatorial leverage of other nodes in the network?
As this process is repeated recursively and laterally throughout the data structure, following the paths of optimization it reveals, those optimizations create further optimization paths of their own, much as a harmonic system can self-tune toward increasingly refined harmonic structures when left to evolve within its constraints.
Why it’s Called Spiral
A visualization of this process shows why spiral theory is the chosen representation.
If you imagine the information flow of an organization to be like an orb weaver’s web, the actual flow of information is along the sticky part of the spider web that winds in a single line from the innermost point to the outer edge.
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If that line were stretched out in a single thread, it would be incredibly inefficient to support, monitor and travel along. The spider would have the same amount of sticky web, but no way to make it relevant or self-supporting.
By wrapping the sticky web over a set of non-sticky radial lines, the lines of control through a data structure, the spider is able to combine the power of the line itself because an insect encounters not one, but many lines, each of which makes it exponentially harder to escape. Additinoally, via the vibrations through the structure, a transitive information flow, a single contact point can convey immense amounts of sensory data about the entire web.
As it is used, the web is damaged and the spider must constantly assess whether the web is still viable, how to best patch if it is, and whether to replace it today or rest and wait for tomorrow. In the same way, an information system must be able to detect and address when its structure has begun to become unpredictable and when that unpredictability must be normalized.
The Forensic Application
When we apply Spiral Theory forensically, the intent remains the same, but the questions become causative rather than predictive.
Where in predictive spiral theory, the goal is to determine what factors create preferred trajectories as the lowest energy path, in forensic spiral application, the goal is to determine what factors must have been applied in what order to make the end state or spectrum of end states the most energetically favourable result.
To fully understand the forensic side of Spiral Theory requires accepting an expansion of Friston’s Free Energy principles as a guiding precept for all interactions in any system. At the most fundamental levels, any element that can be defined as an event is a series of energetic exchanges and an energetic exchange will always occur in the most energetically favourable path over any chosen span.
Classic crime scene recreation and other forensic sciences are all applications of Spiral Theory, but the scope, intent, and reliability of data restrict it to its least informative mode, determination of proximal cause for a specific end state resolution.
For most forensic needs, the precursor reasons that made the proximal causes possible in the first place are, at best, moderating factors on any moral or ethical judgment of the end results. Why is only relevant as it relates to anything that might be termed as exculpatory.
When applied to a larger, more complex system, the more diverse the end point spectrum incorporated in the causal determination and the more distinct data points along the combinatoric trajectory of motion that can be definitively connected, the greater the power of the exercise in determining energetically probable causal chains required to create the combined trajectory through all exposed points.
The Forensic Process
For full explanatory power, a Spiral reconstruction is recursive and laterally demanding, as is a prediction, because an initial application, working from an incomplete causal chain, is refined by a series of targeted induction and confirmation stages.
When examining any two points in the causal chain, the first question is:
Is there a direct and obvious, energetically favourable, causal chain between these two points, and does it provide a sufficient trajectory for all points that must follow it?
If the answer is yes, then the points can be provisionally considered as directly connected with no significant causal points between. This must be provisional as it cannot rule out non-obvious intermediary causal connections with strictly transitive effects on future trajectories.
If no, then there must exist some factor that is not yet exposed along the causal chain that accounts for the changes necessary to make the next point energetically favourable.
From the moment the induction begins, a second line of consideration must also be maintained. What probable secondary effects or influences would the required energetic transformations leave; is there evidence that those secondary transformations did occur; and if not, why?
Is there a way to narrow the potential search vectors for evidentiary support to confirm or narrow potential causal pathways.
The next step is consideration of unfactored constraints. Are there any conditions, existing or nascent to the gap, that would provide an external transformation to the energetic landscape?
If neither an existing factor, nor one introduced by the environment is a sufficient proximal cause for the change, then the inquiry must ask if the cause is non-proximal or non-obvious.
The first step in determining that is searching for similar causal patterns in the dataset, arranging them in temporal and transformational development lines and determining whether any overlapping non-obvious factors lead to similar outcome spectrums.
The search is not a pure Boolean construct. Matching of causal structures must be accomplished across multiple planes of causal relationships as force effects are not monolithic and the same essential processes can create highly divergent surface presentations through secondary compounding factors on a common combinatoric base.
When examining any similar patterns, the next step is then analysis of what the commonality is and whether it indicates a direct relationship, a non-proximal common unresolved force or a repeatedly emergent energetic pattern, generated by the inherent nature of the exchanges.
As with any Spiral analysis, this must be done across as many concurrent planes of commonality as possible to establish a causality grid and expose transitive effects through common secondary transformation points.
The Energetic Nature
In social analysis, potential planes of commonality include demographics, external pressures, internal pressures, evolving social structures blurring the external/internal pressure gradients, etc.
The point of this aspect is to determine if there is a common energetic pattern that is independent of the specific presentation of it. Any factor that retains similar energetic properties throughout a transformation sequence to another factor in another sequence is a strong indicator of either common non-proximal causal root, or an energetic pattern that will inherently be present under similar energetic interactions.
This allows provisional classification of co-occurrence patterns as causally related or natural co-emergence under completely different causal patterns.
If, within the scope of definable traits, a common but differentiated energetic property provides sufficient transformational potential to account for the variation in surface presentation, and that primary factor has a probability distribution that matches the observed end state spectrum, then the relationship is almost certainly an energetic flow pattern independent of direct causal relationship.
To provide an example. If a demographically similar group across different temporal, technology or geographic regions respond in similarly energetic ways expressed in their scope, then there is an underlying thermodynamic flow related to that demographic energy structure that controls the relationship, not the specific surface forms that it presents through.
Energetic Favourability — A Temporally Scoped Construct
Before we continue, we must address the concept of energetically favourable paths, as it is distinctly different from energetically minimizing actions.
In any energetic landscape, the primary presentation of any transformation will follow the path of predominantly lower energy actions. The universe tends towards laziness. In reality it is not a tendency, but a rule, but it can only be handled as a tendency because specific micro-state calculation creates landscape perturbations where short-term expenditure is weighted with sufficient variation to provide exploration of paths that are not statistically favourable.
As an example, we can look at a shortcut that develops over time. For most people, before the shortcut has been defined, the energy required to forge a new path through unfamiliar terrain is assessed as an energetic waste. It requires less effort to follow the established path, even if it is objectively longer.
A certain percentage of people will assess the energetic landscape differently based on their own actual state, and choose to take a more linear, but as yet unproven path. In their calculation, the effort of forging a new path is less than the wasted effort of a less efficient path that has been established.
As others notice the effect on the efficiency of the ones who do take the shortcut, and as the shortcut itself requires less effort to use, the energetic landscape shifts from the most apparently energetically minimizing action to the most persistently energetically favourable path. The shortcut becomes more popular and in time becomes the default route.
From an objective stance, the flow can appear chaotic, but that is a micro-calculation aggregation error, not a mathematical stance.
For a determinable percentage of the subject units through the flow, a sufficient number will have differing constraint patterns to explore adjacent, seemingly higher energy paths.
This is where a system can make a counterintuitive transformation because the downstream effects of the apparently lower energy choices have an aggregate effect that exceeds the primary energy savings.
Aggregation can be both subtle and slow, and depending on the nature of the forces aggregated can create both incremental and sudden changes to an energetic structure retroactively as fluidly as proactively.
Temporal Asymmetry and Reconciliation of Effects
At first glance this seems like a moment to throw up your hands in confusion, but it actually provides the clearest framework to examine large societal schisms of established institutions.
When a system like religion formally splits into sects or denominations, the differences are rarely rooted in interpretations of how the congregation should conduct themselves specifically, but in the underlying understanding of the ruleset that determines the bounds of acceptable conduct.
The transformation occurs at the temporal root where it began, regardless of alignment with that root with current time.
When applied across other domains, this also provides a gradient for evidentiary expectations.
Generally, evidentiary expectations can be set by a combination of temporal disconnect between inception and resolution of a divergence increasing the evidentiary expectations with the ‘violence’ of the transformation and relative vector of behaviour reducing evidentiary expectations.
The longer a divergent path was viable, the more evidence should have accumulated that it existed.
The more violent the state transition is, whether that be internally or externally forced, the more evidence can be expected to be destroyed during the transformation.
The more opposed the resultant vector is to the deprecated vector, the more destructive interference is experienced and the more evidence actively suppressed, destroyed or reinterpreted.
Where Missing Evidence is Evidence by Missing — A Spiral Look at Prehistory
An important corollary of this is that it allows commonalities to form across disparate evidentiary collections if the underlying evidentiary expectation supports the fragmentation patterns.
If the overall paths from A to B and C to D appear similar, but each is missing critical evidence that would directly establish their relationship, then the apparent discontinuities may still be corroborative rather than divergent if the evidence missing from each is appropriate to its full energetic path.
A theoretical but not speculative application of this can be made to early hominid semi-cannibalistic predation on emerging hominid species. Although there is scattered evidence that this occurred, the rarity and fragmentation of that evidence have contributed to a prevailing view that it did not happen systemically.
If you apply Spiral analysis to the question, the evidentiary expectation, based on later hominids largely subsuming and replacing advantageous early hominid populations, would be that any evidence that did remain would be actively destroyed or suppressed except in specific circumstances and there should be a predictable preponderance of evidence under those constraints.
Evidence should cluster in relatively isolated areas where attrition was the primary method of transition and the evidence was likely inaccessible or not socially relevant enough for the energy expenditure to destroy it.
The other predictable evidentiary pattern would suggest less clustering but more prominent overall presentation where the resultant vectors were less diametrically opposed to the cannibalistic trend. Regions where cannibalism was socially accepted or incorporated would have less existential need to remove evidence, but a stronger transformative need to claim the power implied in the history.
The other predictable evidentiary chain would be that behaviours like human sacrifice would reflect predictable moderation adaptations to continued predatory cycles, and the prevalence, extent and cycle of sacrificial elements in a region may provide evidentiary frameworks for where corroborative or divergent evidence is most likely to survive, and in what forms.
Although far from the final one possible, the last factor I will mention is constraints where no evidence was likely to have been generated in the first place, either because no co-occurrence happened over a significant period or where the environmental constraints did not provide the kind of caloric crisis required for cannibalistic predation.
Examples of places with low evidentiary baselines would be examples like the Indus valley or early North America, where the predominant trend of cultures rooted in that region is towards the abundant, non-meat-based or plentiful game staples available.
As you can see from this example, where traditional investigation concludes that a lack of direct evidence means the event did not occur, a spiral investigation asks whether the absence of direct evidence is circumstantial evidence with similar if not greater weight.
A crime scene investigator who does not ask why the murder weapon was not disposed of is missing a large part of the equation.
In Conclusion
This paper is presented both as application of and explanation of Spiral Mathematics. Spiral Mathematics is a methodology for establishing testable predictive or forensic controlling factors of an arbitrarily diverse, massive dataset to allow standard measurement, notation and calculation via Cartesian and Euclidean derived methodologies.
Although the mathematics of chaos theory can be directly applied within a generative structure to calculate the precise variances that must be accounted for in a fully formalized construct, for the purposes of understanding the process and for most applications of it, chaos derived parameters can be reasonably estimated without granular calculation.
An aspect of the process is exposing those points where that granularity provides distinct benefits.
In this way, Spiral Mathematics does not replace conventional mathematics, it reveals the conditions under which conventional mathematics becomes meaningful.
Patrick Handley-McCarthy
This work, like all pieces in this series, is published as research documentation under the Human Cognome Project and inherits all rights and obligations therein in perpetuity.
https://github.com/Human-Cognome-Project/human-cognome-project
https://www.linkedin.com/in/patrick-handley-mccarthy-92716b202/
