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Axiomatic Panbiogeography

offers an application of incidence geometry to historical biogeography by defining collection localities as points, tracks as lines and generalized tracks as planes.
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Incidence Geometry
Composite Construction
Quaternion Algebraic Geom
Primate Vicariances
Individual Track Construc
Generalized Tracks
Main Massings
Track Analysis and MetaCo
Martitrack Panbiogeograph
Replies to Criticism
Multimodel Selection
Search Encounter
Track Analysis beyond Pan

Propositions - Converting the plurivocal to the identity of a unified consensus in Axiomatic Panbiogeography

 Proposition 2.1 - If l and m are any two distinct generalized tracks that are not parallel, then l and m have a unique point in common.
        if this point is a node then there is a derivative at the node
        if this point is not a node then the continuousness (continuity) in the area is a plane boundary cutting collinear points.
Proposition 2.2 - For every individual track or geographic variation there is at least one node outside the area in the vicinity of which Catastrophe theory of (Thom 1975) can be applied.

Tracking  Communites in Time: Panbiogeographic threading from concurrent distributions back into the Jurassic.

Dice 1952 suggested the existence of natural communities of species assemblages and Panbiogeographyhas  become detailed enough that it may be possible to suggest from current  intrataxon vicariance comparisons -- a common community descent.  Heads has accomplished this in his past two books.

This is a regionalization in ecology(ies) - not of geography alone but is found from biogeography.

Thus the names to be given to these communities (which are subject to evolutionary ecosystem alteration are not the same thing giveable to areas of common history as proposed by comparative biogeographers since the depth in time will be limited by the interaction of the ecosystem of the entire planet of/to the given ecology-evolutionary lineages components summed within.   Panbiogeography unlike compartive biogeography can function for iertiages that might extend back in time off the Earth. Panbiogeography  operates for data off Earth.  Thus while projection back into the Jurassic based on these environmental conditionings may be possible, extension say beyond the Permian temporally required more certainty in form- making as suggested on the Taxogeny paige

The Data that are being looked at include salamander differentiation (bifurcation sequences) in the Central US with connection via Croizat tracks to Nuclear Central America and from there globally (China – vs. - France; Africa vs. South America; East vs. West, one vs. two world continents)  depending on the biogeographic synthesis available.  The analysis suggests that extension to the Permian may in principle be possible.

The analysis is composed of within Eurycean splitting compared to other phyla similar and a  biogeographic synthesis is explored geographically (and thus applicable and made available to natural communities) in space time and form within the urodelan distributions and their distal origins towards angles available to investigate temporal community possibilities further back in that time/temporality.

Properties of natural communities that share a common vicariance are proposed as either internal genetic or external physical force self-similarities.  This development shows that comparative biogeographic speculation that failing to separate geography from prior biogeography is the cause of some opinion that geography itself cannot be indicative beyond spurious probability indications of species’ presence(s). The eminently statistical nature of Croizat tracks suggested by Leon are thus confirmed with natural community properties tied to geography. Panbiogeography thus leads to predicting ecological niche relation homogeneities paraphyletically and this is correlated between edges in biogeographic databases representing in the active on-going analysis.

While it is possible to be critical of this means to interrogate how earth and life may have evolved together (there is the verisimilitude of life off Earth) the  statistical correlations of geography across communities can be used as input to compartmental models of ecosystem functionality and can be useful when making projections of effects of future proposed global changes on biogeographical phenomenological kinematics into suggested geographical place equilibria at certain proposed times of the future.  This is needed as man continues to alter the niche parameters of living things at increased paces.  Thus Panbiogeographic output (no matter how criticized) is helpful towards amelioration of human population increases on the rest of life (what species will survive with man before the Sun expands? And/or man migrates life off Earth with living things with it or moves with natural speed concussively alive from within the Solar System tangentially out).

Proposition 2.3 - For every collection there is at least one individual track or geographical variation not passing through it's area (principle of terminal taxons)

Proposition 2.4 - For every collection locality there exist two distinct tracks oriented from some baseline that pass through the geographic coordinates of the locality.
      Notice - let MrN be three points reconstructing the baseline, Prop. 2.4, there fore an individual baseline's exsitence is or co-exsitence was (at least)
 Proposition 2.5  There exist 3 distinct lines such that form Space + Time + Form; either the two distinct lines of Prop 2.4 are not time or time is correctly inferred and is Craw's (1983) reduction of the information content of a cladogram.


“The surprising fact is the amount of infinitesimal and infinitely large numbers which such a positional numeral system can represent.:”

On Sergeyev’s Grossone: How to compute effectively with Infinitesimal and Infinitey large Numbers.

The ability for the gross powers to be recursively defined in terms of prior gross one numbers allows for this large but not cardinally inaccessible size of representation.  How far it can be empirically utilized where other larger infinites are at play remains to be seen and how to populate a set of grosspowers and grossdigits  via path analysis for instance (where the two directional arrows represent undetermined divisions within the a posteriori instantiated empirics) needs to be materialized.

Consider any given set of distribution points as having an association with some number with the Natural Number set N as a grossnumber numeral.

This particular division would be suitable wherein the distributions shows two lineages and where the the  track^node^mass^baseline organizational difference between the two is simply a shift relative to the baseline.

One way to find this shift as the data for both lineages are populated into the outline and framework is by the constraint that N also presents grossone squared at the same time. One is thus left with the more less difficult task of describing the track^node^mass^baseline coefficients as the cross (vicariant) diagonal through ordered pairs of actual distribution collection localities.

To do this the lineages without shift (relative to sister) are demonstrated in the upper half of the x illustrated diagonal.  How high grossone minus one goes before it is subtracted (in data actually inputted) may depend on the masses actually present.  The lower end of the x may be those areas wherein the node antinodes are clearly directed to the baseline.