Pre.S. : This post extends the Generative Tree of fitness/EuCeleration measures framework I am trying to develop here.
If you want to read about its previous extended version or short version please read the most recent ones on the ResearchGate. Also, there is a poster on Fitness of Randomness in realistic natural systems therein. The abstract below has everything you need to prove (or believe!) and establish the framework. Thanks for your interest!
All systems are proved to be evolving from origin to enlightenment to reversal of their GenTrees as defined in the framework above. This is the most General Intermediate Maxima Phenomenon which includes the Strongest Universal Selection Law i.e. the maximality in the Intermediate Maxima theorem. The framework in gist has now three components-
1. The Universal Plot (UPlot) that is an arrangement of possible GenTrees on the Input Vs Output variables plot. This is the scenario before GenTree expressional activity. The UPlot for any system-environment configuration and infinite sub-units is a tetra-dome on the four quadrants given unsaturability of output variable(s) like fitness/EuCeleration measures over the input variable(s) due to internal aversion among prior and higher-later derivatives/differences, and the exhaustion of good, neutral and finally the negative output/fitness/EuC dynamical resources and instances to peak and infinitely dilute the unsaturated output/fitness/EuC measures and exhaust the experiences set to close the trajectories.
2. Dynamics Condition over the plot is given by the Connectivity Limit Theorem requiring an axiomatizable match in correspondence variables between evolutionarily active agents and the evolving system.
3. The Dynamics, I feel, essentially occurs given the pre-dynamical and thus unvaried/dynamic unified universally inevitably initiated natural singular trans-cycle/initiation(big-bang-evolution-dissolution of universe/an entity)-unique-universal origin-evolvability and its kinetor can nothing but express completely and irreversibly in terms of extent of expression and not necessarily the content of expression. This is because all seemingly trans-initiation-unique-universal infinite kind-type-instances, including the seemingly infinite unique-universal kind-type-instances active and self/trans and pan-activating singularities forming the final active singularity, are axiomatizably initiated or borne while the active ones of these express inevitably and the inactive ones join the expression as their self and/or trans-activated evolutionary cycle promotes them to activity phase. The expression of this origin-evolvability caused by the kinetor is on the origin axis and then on the evolution axis, with the Evolutionary Cycle (see below) of them together as the Singularity, as deriving and implied by the cycling of their output dynamics. This dynamics is here proved to be infinite for any within dynamics entity as once the entities are initiated by the evolvability and kinetors, they last axiomatizably for all the comprehensive and ultimate evolutionary dynamical axis and its sub-axes cycling as potential, expressive and experiencing entities pre-incarnation, in-incarnation and post-expressive mortem respectively. Note the origin-evolvability and the kinetors then seem to have dynamical activity. The axes and experiences-instances for qualitative-and/or-quantitative exploration thus seem to be infinite in lengths like all entities due to infinite initiations by the primordial and active seemingly infinite kind-type-instances infinite-entitizing-n-entity-pan-definition/profile-transitivizing origin-evolvability-kinetor-content instances perceiving and conceiving them. The entities once initiated axiomatizably last up to the end of the dynamics i.e. the extinction monopoly can only occur at the end of the respective experiences-instances set exploration i.e. there is no premature extinction.
The Evolutionary Cycle
These three combine to yield the Cyclic Evolution of the essentially born infinite evolving systems with definitional and definition-transitive infinite exploration of a unique universal non-vectorical and ultimate set of maximal positive monopoly/majority and as-above-positivity-weakened neutral and negative monopoly/majority experiences traversing the tetra-domes, for some-degree-dynamical entities including the origin-evolvability, kinetor and the outputs. The ultimate set seems to be unique-universal as all-realistic-transitionable-definitions set giving it is unique-universal and equates it as all entities or their components are infinitely transitive within it as except the ultimate dynamical sequence, each lower order sequence is infinitely definition-transitive to explore it completely.
The only phase structure of a dynamics at each hierarchical level (derivative/differences or causal unit) is 0ve to +ve to -ve out of the permutations and combinations of these as all the initiations and dynamics can only begin at 0, continue with +ve as there are active +ve gen singularities and there can’t be -ve fitness systems (-ve cannot follow 0ve phase) and then dilution and exhaustion leave only -ve phase to ensue and finish the cycle. Moreover, as suggested below, the General-to-Special-to-General cycles necessitate dynamics to be system originating-expansive and propagating Generality i.e. optimizing +ve phase, stable neutrality and then dissolutional -vity as this is the only sequence fitting in. The stacking up of such simultaneously initiating 0ve-+ve- -ve dynamics cycles at each hierarchical level and system axial location and components yields the ultimate three phase cycling.
P(i) is property or component of ith generality level, E(i) is ith entity in the GenLine i.e. a stream in the evolutionary cycle.
P(lesser-to-All(i)) = E(i)
P (lesser-to-All(i)) + P(lesser-to-All(i+1)) = E(i+1) (completer)
Thus, P(i) can exist without P(I’>i) in the GenLine and Line(i) without L(i’>i). And thus, the GenLine can have simultaneous or successive General to Special Origin-Evolution.
P(i+1) is <– (given by) P(i), this giving could be a combination of succesival, causal, relational-functional, essential and/or causal: the true net direction of a realtion is charged by its givitivity direction.
Thus, for example, if y=f(x) n x=f-1(y) n x causes y, then x –>y for the set of these two relations, not reverse.
We may thus calculate Generality or Specialization as some givitivity/unificativity intensified/weighted combination of values of all <– (lesser-to-many) and »(many-to-lesser) relations. There may be heterogenous and pure versions of the set of these relations producing net Generality/Specialization. The weights are aligned with the definitional givitivity/unificativity so there can’t exist contradiction in the real and calculated values of Generality/Specializations.
Similar to (<==) lesser-to-many relations, many-to-lesser (») relations have similar property:
P(many-to-lesser(i)+many-to-lesser(i+1))= E(i) (completer)
P(many-to-lesser(i+1)) = E(i+1)
The true many-to-lesser (») are charged by their real unificativity direction. Thus, if set of y succeeds into x and x has caused those y, then y » x and not the reverse.
And, E(i+1)< or=(Ei) i.e. Dissolution can be simultaneous of both general and special or from special to general.
Thus, in the origin-evolution, it being an essentially expansive succession of an evolver or its components, only lesser-to-many phenomena can occur, while in the dissolution, it being essentially a contractive succession, only many-to-lesser can fit in. This happens for each transition or point in the successions with their experiences-instances set given finite or infinite pace/graduality and irreversible (no partial expression) certain (all phases expressed) expression of creating and dissolving activity, giving exact General-to-Specialized-to-General origin-evolution-dissolution dynamics accompanying the Intermediate Maxima Evolutionary Cycle above. If we believe in the axiom that nascent singular primordial-post-mordial essence-causes, with their own Generality/Specialization, exist and are required for any and here the cyclic dynamics fixing their cause-effect sequence, the only way these very general, intermediate/lesser general and specialized outputs can fit in the tetradomes is most general spirit singularity to its development to outputs and back via dissolution to initiate the next cycle.
System variables like Complexity, Order/Disorder, Stochasticity, Robustness, Modularity, Information Content/Size/Activity and Novelty , Internality(inheritance)/Externality(sensitivity) all participate in their own versions of the framework with Intermediate Maximum and Connectivity Limit theorem and their Evolutionary Cycles, combining and forming the ultimate FrameWork for fitness/EuC measures.
Infinitely evolving systems reach their characteristic infinite fitness as basally teleological positively fit components stack up on the evolutionary axes including time and space, at their own pace and after their characteristic evolutionary duration. This is the Strong Law of Universal Selection.
The fitness is taken as the natural fitness combination of basal and advanced adaptability and evolvability which both equate being aptitude of solving required, bonus and recreational querries by the systems with descending weights precluding nullity or negativity of the fitness (leading to basal teleology and thus, Strong Universal Selection from the basal teleology).
The system evolution is increment in system-matter i.e. system activity. The system evolutionary activity axis involves layer of innovation-evaluation-output units for system evolver units and further layers acting on this and successive meta-evolutionary axes giving rise to the master-evolutionary activity axis. Now all of these layers have physical time axis immediate/self fitness and evolutionary-time/dynamical axis implied/outputs fitness basal and and stong universal selection, requiring the modulations of the evolvers and evolutionary activity of the evolvers (like innovation, complexification, specialization, co-operativity) positive i.e. the master-evolutionary activity axis has a strong universal selection in addition to the basal teleological net positivity. Thus, infinite evolution systems implying system-size-infinitizing eventually and ultimately self-fixative by nature positive activity on the master evolutionary activity axis post-evolutionary-non-positivity-and-above-the-Strong-Universal-Selection inter alia and corresponding to each and every activity of the system, suggests progressive evolutionary performance increasing the characteristic maximal infinite fitness of an evolutionary system above that due to plain steady stacking fitness of basally teleological system components. This is the Stronger Law of Universal Selection.
This improvement in master evolutionary activity for infinite evolving systems implies last-state optimality for all evolutionary systems.
Basal Teleology and Anti-Extinction Basal Teleology: Survival teleology for infinitely evolving systems as their survival <==> basal ‘fitness’/EuCeleration (positivity of dynamics).
Transition Non-Monopoly: The non-monopoly or majority of either +ve, -ve or neutral derivatives and that of restrictive properties of the infinite evolvers and their components to permit transitions among their non-extremes.
Internal Aversion Non-Saturability: Internal Aversion among higher-later derivatives together and the prior derivatives causing the derivative triangle of fitness dynamics to not saturate with all high derivatives limiting the total positivity of the fitness dynamics i.e. EuCeleration to be limited.
These phenomena suggest Stronger Evolutionary Dynamics among infinite evolvers i.e. ultimate fitness and EuCeleration to be some higher infinite than the basal teleological Strong Evolutionary Dynamics.
The Strongest Law of Universal Selection:
The Infinite Evolvers essentially have quadrangularized limit derivative triangle such that all points on the Dynamical Axes produce infinite independent derivatives rows, columns and higher units, to explore unique and universal all-variables/properties/patterns-maximal covering infinite +ve monopolical and intermediate dynamical experiences set, mapping bijectively to sub-units over their infinite lengths or sizes so that they add up to highly infinite unique universal maximum of net fitness/EuCeleration measure.
I feel and have assumed that there are infinite possibilities to make the above experiences set infinite. The set of all realizable instances of an evolver is partitioned into the infinite sub-classes of system-states in the state space over the Dynamical Axes variables and the corresponding experiences. The partitions are bound to get crossed to cause experience transitions in the unending time and other abstractly formed dynamical/variable axes that the instance set partitions. The transitions are universal because all entities are infinite evolvers in evolutionary time including physical time, as they potentially/conceived-ly/instanced-ly exist before their physical/expressive existence and also exist post-physical/expressive mortem as abstract evolvers (e.g. history-context-outputs feelers and grabbers in the abstract and ultimate evolutionary axis). There may be and are restrictions on the explorability and realizability of candidate experiences elements, but the final transition-able set seems to be infinite for both highly experience transitive and lesser transitive entities, and all the evolving entities infinite evolvers, making them be essentially and exhaustively Strongest Universally Selected.
Thus, all entities obey the Strongest Law of Universal Selection.
They big or small, are all surely infinite, so they surely break the non-monopoly- for the good and the bad!
For the good-
Due to basal teleology, the only monopoly the infinite evolvers can reach is positive monopoly over the derivative triangle/quadrangle and they definitely reach it and explore fully its variants and their instances latest after they, being infinite by definition, exhaust the above non-monopolous experiences set and its instances set. This happens for all infinite units and sub-units of all higher or lesser infinite evolvers and is fixed within and above them once the respective ultimate derivative +ve monopoly is instituted to produce all entity wide essential fitness/EuCeleration maximization for each order of the derivative triangle/quandrangle and thus all units and sub-units.
For the bad-
All the possible experiences could mean all possible entity profiles including definitions up to their instance variants. All entities being infinite evolvers and/or such(sub)-units can reach higher infinities only by traversing this set completely whether the experiences are monopolous or not. Thus sooner or later, all entities traverse the unique universal experiences set including full positive, and inevitable-basal-teleology-damped neutral and negative monopoly/majority in the derivative triangle/quadrangle tops and finish at their first negative monopoly (as the start of fitness/EuCeleration is at 0 and negative monopoly in the triangle/quadrangle implies sub-0 fitness/EuCeleration measure i.e. extinctional -ve dynamics of the system. At this point, the comprehensive and total evolutionary dynamical axis and resources vanish as its inhabitants, due to negative monopoly acting on them all as the reversal of the Generative Tree of qualitative-to-quantitative system variables where outputs of die out with their causal line or are ceased by it with the causal line dying with or after it, respectively. This precludes further potential, expressive, non-expressive and post-mortal existence of all entities to close the cycle.) Whenever they finish- earlier or later, they being infinite evolvers/(sub)-units, definitely experience the instances of the positive monopoly/majority of all extents as they traverse the infinite experience and its instances set indefinitely.
Once they finish, there has to be the next big-bang to start a new cycle of evolvers!