Extensions on "Reflections on phenomenic descriptions of the real part 1"

 I was thinking on probably showing mathematical symbolic proves to my previous post on this,but i still dont seem to think in my mind to be all that necessary,so i may do or may not,really im here again in order to dissert on the synthetic values of the previous post.

the first part of the previous post was to show how from the epistemological ubiquous complete cronic credible system idea one could derive the mathematics of curved diffeological spatial chart principles,curvature,connection,forms,etc,as explained all quantities are also qualitatively defined,they measure synomic patterns,or conscient ones(im not being a Husserl follower,or any of this,but here im recognizing that the double cover of science is the science itself nor it as in itself,science are the finitely cimented infinity,or the compact infinity intersection centers not an incognizable idealized pure thing as kant was thinking).All of them defined in phenomenic ideas,that organizing in limited amount of logical axioms would mean the following(maybe they arent inreductive,but Philosophical reductionism is somewhat meaningless here):

1. complexity is the pattern finitely intricated formed infinity local differentiation.
2. pattern formation lies in the linked object formation from the spatial-time-float conducted unit(here float also referes to conservation,but you dont need to ensure it for now).
3. float means a conservation standing point,universe category is self-sulficient,also finite digression from the origin point.
4. complexity forms from  the deep selfdeterministic patterns(where self is the  mean for genealogical positive singletons,so even quantum patterns are predictable and deterministic to some degree,from what we know,this ensures metric compatibility with quantum phenomena and ensures metric too as a degree of freedom)

2 means morphism composition(an subtle non category of a category,'composition closed non category' or wtv similar shit ) and folded decidability(provability completeness or constructability), that is a kind of choice to work on,to resign reality concepture as we are doing ,3 means dagger compact and folded decidability,1 means lawvere algebra indefinite recursion(limit/colimit closed categorys(the one doenst need to be specific)),4 not stricly speaking is the causal set confinability(irreversibility,an formed arrow consisting of causal connections,an kind of universal direction)

see that im not been much strict here,if i was this would form a physical system with all contradictions "excluded",we are picking minimal reasons by which the principles hold in a minimal way,this ensures a minimal foundation to mathematics and a strong one,2 means composition and the constructive logic,3 ensures identity morphism and 3 also ensures non externalities in regard to the constructability,dagger operation is regarded to make part on it,we so assume to be a category derived from metaphysical presupositions,also we are suposing some limit closed principle stated on lawvere algebras,the category so is complete,4 means an morphism from which all derive,this could mean an adjunction extension to which one only morphism is more preferable to give arise to other ones(an preordered monad),all can be derived from a only functor in such way that the functor make the algebra itself a definite place,we can focuses on the algebra behind this functor to restrict our foundations,it means an preorder in the category,this can be set to equalize continuity idealization,so an topological spatialization and then causal orders,an way to have a almost topological vectorial space defining a vector space parallel to the functional work,so we just had defined an quasicategory,we can make something more strong by having 4 making less efect,so thereby defining an topological category or just an topos,or a net if we didn't consider continuity,an sequential object,an graph,a network, a class structured thing(object/morphism property definition,a convergence structure or an collection target,an limit tower),or better,an thin category based on uniqueness relations to which has an underlying more depth dependence as every logic has,this is a dagger complete thin category(the polymorphic form derived easily from isomorphic sameness relations),if we consider no 4, then we just have a dagger complete category,that has monoidal cartesian closed category as subcategory(who knows,knows),having just the first three so would be sulficiently strong and sulficiently non abstract to form most known categories,and some unknown ones.Since we are working it as a logical category,an internal logic category,it is so an higher category or just a really direct internal logical category(an category in its non higher medium instantiation or seemly an higher medium instantiation when we consider it to be an higher category;making the alocation of both being the same,every category is a higher category,every higher category an lower one)if we made 3 more weak,we could have finalized with just an complete category,so:

•  topological vectorial space
• quasicategory
• topos space
• dagger complete thin category(to which has equivalence with some important non thin derived categories)
• irreversible monadic space(thin Eilenberg moore category)
• derived irreversible monadic spaces (irreversible monad adjunction)
• dagger complete categorys(ones that are also cocomplete)
• complete categorys
• some intermediators...
• derived monadic space (monad adjunction)
• monadic space(Eilenberg moore category)
• category 

being sure,this goes forever in the infinity  big,the infinitely little,the infinitely centered and also the signal variations that for some reason i didn't phrased in the original post for the sack of arthmetic comprehension, above gets more physical,middle more computational,below more logical,and thats all possibly covered  in Curry Howard Lambek triality,note that this forms an hierarchy of categories,a limit tower,since this is self-referential/self-framing there's a consense that this tower is itself derived from its idealizers(as before by ideal i meant and mean mathematical one,a sieve so),its not an hierarchy,its an heterarchy,a hyperdoctrine,that's philosophical ideals being derived from mathematical ones(also observe that the proper class CAT is not a really category of all categorys,this would rise contradictions,its just sulficiently large),all categories also interact in all possible manners,having adjacent properties,equal ones,almost equal ones,having a related subject,all of them as each and all are able to describe others of them as each and as all of them,mutating organized organisms,having to share this common sensibility to consent its conditional proposals(dynamis),this can be formalized as an really large amount of theorems and proof-arguments,for me centered at Yoneda lemma,kan extensions, grothendieck constructions(something i may say about later since its the principle to which an category can be seemly related to another by its indexing and well relates to time),adjunctions,initiality,terminality,generalizations,dualities,liminality,explicit indexing categorys, some generalized indexing theorems(that i dont know about the generalizations but im sure i can prove some with a amount of time,i just have to manage time) and representability variations of the ones above and higher or void variations of the ones above,all coalizing a marginal dynamical system lying beneath the directed logical/reason inferences,claimed to form semantical means to which the being that constitutes the very real circunstance of singleton/empty differentiation take as its attachment bond,the very constituent of the so called instantiated inequilibrated logical singularity,the diaresis of the infinite idea that perpertuates to claim a positive/negative directed subdefinition of the same infinite idea,we are working beings used to give a non realized vectorial orientation, being that amounts to be a double cover of the reversed imediated self;the cosmic preponderance and its multiple causal directed homotropies(homotropy here being a parallel with entropy and neguentropy,meaning also homotopy(a relation btw information and topology)) are a human bioproduct of its erratic deliriant behaviors to being one self,the unique/uno self,the supreme,the absolute,the numen,the daemon,the dümen,the dimen,the timus,the nomus/the nous,the nomic,the noetic,the god/the holy trinity,quatrinity,unity,ninity,elevinity,the tao,the enlightned,the bright light,the zohar,the sacred,the symbolical,the indefinite continuation of its prosperous life circunstances,a stupid made scattered/afronesic/hysteric/agnostic/chasmatic thought harding,grinding,struggling, burbling trough its existence, the tower order comes essentialy to its ecological one,its bio-homorarchys,life filaments that we create by assigning memetic float unitys,taking the middle infinitely long path(the infinitely centered infinity florishing as an ontoeidogenesis or an reverse of it,a co/dual of this).

Thats the secret that hegel kind realized (even that not directly); mathematics flourishs trough its science,all made on its philosophies and conjectured on its beliefs,the notions are philosophies as well as the reason behind it,the essential notion i mean(ontoeidogenesis context).mathematics goes deep to philosophy and philosophy transitates trough the realms by the guiding moonlight,the satori vaguely apearing lasting its numbered particles in math,or the ubiquitous shadow that covers us all as an umbraculum diving into mathematics,the dynamics as physics(computation)allows the computational(physical) that creates the ontological martingale,unfolding the math and assuming/transversing its barriers,so the demiurge comes to divague with god,divaguely equating both sectioned abstract imaterial  entitys,the imaterial enters,penetrates the material,to the scale of confusing the limits as its parts and the parts as its over-reaching whole, transdisciplinarity,so the finite is then infinity,the connections beared as neurological ressonant patterns sticks with the purely digital consonances,and the somatic the ontic.

Using grothendieck constructions one could build an categorical index that creates a dynamics in btw the heterarchy forming common rightly ascendend homorarchies,vector topology time indexing serving as time for categorical abstract constructions,guiding both in the right direction,featuring the middle path,and since the logic besides all is self referential so the content itself,never forgetting the rudimentar basic dialetical suprassume;the form-content duality.

We so can navigate trough all means.

The transitation btw linguistics/language/semiotics/semiology,literature,mathematics,computation,physics,philosophy,and all its body's/head's is more clear than ever was before so.About Yoneda lemma,i have some web images that i finded on Web relating it greatly to semiotics,also has some math(that i somewhat have dificults to put the logic as said before):

Remember the Introduction of the first post underlying the way universe presents itself is by the contingent means a mean to which he is in and for itself,thats the basis of all category theory almost i would say,it can be used to prove topological correlations(such as using presupositions to infer homology/cohomology groups ),symetric relations,representation and equivalence,adjoint,...,mereological proof-arguments,etc (all this among the most useful tools for number theory such as the dualities behind it and others pure/meta disciplines),here some theorems that can be proved:

• Representability Theorem
• Cayley's Theorem 
• Freyd's Adjont Functor theorem
• Grothendieck's Representability Theorem
• Lagrange's theorem
• Fermat little theorem
• Fermat last theorem(i dont know how,i never had seen the proof of it,but i know the ideas behind it and i can asure that is not directly(directly would be too dificult))
• theorems related to kan extensions and adjunctions
• list goes on,to say that probably(also there's the fact that all proves are in the same order of sameness in category theory)

Cayley's theorem having deep relation with Yoneda lemma,since groups are just one adjoint simple relation away from categories, there's a deep sense that the true nature of functorial semblants are deeply connect to all provability arguments,i would say the same to Yoneda Lemma(that i might one time or another call it a theorem bcs doenst make any sense stick with others people past common sense of provability,the theorem applies to so much things and is so important that the statement of it being an lemma kind that insult's me),other picked web image of the theorem realizing the symbolical math arguments:

this structure so includes the semiotic basal questions of why ,how,...,consisting basic prove relations by amounts of  duality's and poly-duality's,poly-relations/multi-relations,this happens to be the same for all the related structures that i said before.I could go further at Curry Horward Lambek triality and others categorys and proofs too but for this time its this,to be sure i may do another extension if necessary,but wtv.

probably from this it might be more obvious some of the anomaly cancelation limits that i used in the post,how i made the logic itself and how the physical relates to the mathematical,the physical being the deterministic open-closed domain that always runs trough its contradictions to be finite/discrete(as center numeration)/non contradictory/pattern made/continuos(space filled by the phenomena picturing),the physical accumulates in the mathematical and the mathematical turns physical,computable to say better,thats the principle,the borders are reasonable,reason is everywhere(hegel quote), because the essential of reason is the infinite-finite holy unity connection,following the idea that humans/sentient beings are essential when they are quaternary( catuskoti),since all medium is the connective,and the instant is the unit of reason seemly all frameworks corrodes and we are again talking about patterns,consistence of them,yeah thats the basis of the posts(previous ,this and the other ones),how physical arises literally from the mathematical(type theory point of view) and the mathematical erases the boundarys of the physical,both relying on type theory(the infinity comes finite and the processes overloads and reloads,changing and reversing, getting changing acumulation points at its corners,they are inseparable,the axioms can be retraced and traced,the argument of time and logic is a metaphysical one),bye.