yep i decided to revisit group theory since I might remember some abstractions I am completely forgot some :) 

specifically about taxonomic definitions of is completely mixed /confused in some knowledge it seems as. although intuitive concepts remain some of. 

yep i didnt wanted to restudy the base introduction textbook over again (which i did maybe 4 years ago) and now taking a basic revisiting to the abstract algebra one open book type. by then  I were working tod efine map some linguistics concepts to group methods  abstractions there then later moved to category theory on that finding group theory is over constraining in nlp related graph conversion tasks. 

but seems as some concepts from that study remain someconcepts completely wrongly taxonomified in my basic group theory introduction study by then. and I were also working slightly with some group theory library in maths beside trying to learn that. aha it were sage.

i think this abstract algebra main text book defined introduced abstract algebra very neat, so I restudy with ai with ai teaching me briefs of theoreis there again or titles of definitions there which i had confused some of very funnily. 

yes i been to this type of tasks before but basical alike of utilizing group theory in higher order methods commutes to generate universal properties (were it called like that in category theory? see, i would also revisit category theory book with ai) 

its that e.g.  even if i hadnt studied to free groups in group theory texts, I saw initial discussions of that in category thery book.  but i saw it in directly banach tarski idea framework,  

e.g. my idea of that i have had the following idea even if  i were not yet knowing group theory, developing a system's units behaviour definitions from higher commute definitions.

its like what i mean is like, before i wondered if we could use mechanical springs to do some computation tasks or even define a neural network some math layer.  I mean normally people use some  abstraction units,  but then define tech based on directly on that lowest abstraction e.g.  electron whatever abstraction layer it is in whichever ontology it is.  but my approach were that,  you dont have to do very high detailed engineering technologies cause higher level commuted versions of the same basic group generators  or their specific forms could be in specific restrictive algebras define their exact specific group theoretic behaviours. 

maybe this even if i hadnt remembered representation theory that I also have to study, but think of this like this, you dont have to use a group's just basic generators but you might be able to replace them with higher grained constructs  in their specific commute behaviour. 

e.g. this is like i mean idea pattern of trying to do build computer some functions from mechanical springs. so that being a most illogical unintuitive method but maybe onlyviable in some conditions, but that what I mean is reducing the engineering costs of whatever task you do, you might replace the group with some of a more less grained abstraction but which in then still that some traits in that less grained (commuted concepts or just I dont know this needs to be defined better here  later would focus to figure out exact group theoretic definiton or if not sufficiently being defined there, then cat. theory definition )  which then under their relation of specific commute types whatever it is specific mappings, it somehow replicates the subgroup of a group's behaviour alike.  which might be what you all need to do implement an engineering functionality without using low level abstractions like qubits being defined from very low level abstractions usually which then is very sensitive and prone to be noised.  so that, you might rather figure out some more commuted a setting or whatever group theoretic defintion (which might be multiple concepts viable for this task ) you might  impelemtn for instance main group behavior or its isomorphisms or at least you might implement  a sub group behaviour of that. 

since i forgot representation theory and since needing to revisit group theory, i cant hadnt defined in group theoretic concept. 

but even intuition of exists that you dont have to create a commute behaviour of a group by just having have to using that group members you might use aggregate of those to then again still replicate intended behaviour.  so but that aggregate definitiono might be muintltiple wise defined in group theory (of alternate such methods/definitions) but guh can not do it now right now since i need to first revisit group theory knowledge from 2023s times last i studied to maybe.  althougbsome concepts remained some concepts are lost and confused also.

so that you dont have to do quantum computing based on very sensitive equipment is the metaphor. macro objects can also algebraically under some group programming/constraints, can still generate such intended behaviours purely with no deficit/error.   but to define this group theoretic "what" part requires some more group theoretic studying. but there this is the intuition.

.e.g in neural networks when i were studying  in 2016 i thought whether we could create an art project using springs mechanical springs and whether their algebra somewhat without requiring computer could have been replacing gpu there e.g. this idea i thought when i were studying to ml first in 2016s which by then i hadnt known also group theory, .so this is one core intuition I have which I also revisited when thinking of quantum comptuers, that you dont have to use atoms electrons there maybe and that I have had agenda of figuring exact representative commute methods and group aggregates of what to be figured out during that study. 

so to describe this inherent mathematical intuition is hard due to being illiterate in maths yet not educated very much alike a mathematician. but this is one main intuition there for reducing engineering costs in any design. that you can do subsititions in algebras commute graphs of systems which in then turns to the required group element behaviour you wish to model of. but that you dont even maybe have to model the entire group isomorphism either, you just might need specific commute nbehaviour.

so this substitution concept I remember unrelated to this discussion seeing somwhere from readings in some of the books there but i dont remember exactly.  so but in this substiton to reduce engineering costs /constraints, you subsctitute in reverse direction, you define the basical commute element with a more aggregate complex version of it,  but you define some constraints over it so that some commute behaviour of the aggregate then establishes the required group commute behaviour you wanted to realize.  this i think is better definiton but still very ignorant in group theoretic side.

so that when you apply this intuition to quantum computing you think that: yes we dont have to use atoms or electrons in how to define entangelemnt super position.  you might go a higher ontology layer but still might reach repeat the same required commute relations of the exact unit you wish to replicate dynamics of.

so thereby group theory by then even a very rigid framework type (has very strict limits unlike cat. theory) is what enables such substitions. e.g. in quantum mechanics there were they did this with some polynomial groups to define one quantunn behaviour alike.  or fourier transform is also a rerepresentation right? but  in scope of group theory its more an higher order enabler to accomplish this substitution logic with its theorems/notions there and there.  

so this is a common method of science. but i think reason it were not yet applied to quantum mechanics is because they did not exactly had an ontology layer in commute graph notion ever and it were rather hamiltonians and discrete states /or newly learnt althought not yet fully learnt hermititian matrices being defining the units of computation steps or an iterating dynamic system definition there.

but when you switch focus to commute graphs, you have a new world of more representation/more substitution capacity.  so our sheaf/poset architecture also accmplishes this.  since the qunatum mechanics is even derived from a commute graph  you then have much more capacity on simplifying your system design in a quantum computer blueprint as I asked ai such task, or as I asked ai to follow this approach to replace rebco with some hull design I can build at my room here.  so when guh does engineering, this is one core intuition always she thionks of. that lets not work with basics but create representations or whatever to replace the units or group elements whatever in the computation graph with some more less hard to design units but which in result has same effect.  

so there in a sheaves  of a commute graph, that result might come from different branches indeed might be avery simplistic definition of this without including any dynamics. so maybe let us define this like this: 

you just take a subgroup of a group or the group itself and intend it to behave in some programmed way. you want to engineer program its commute graph order alright?  then -> you dont have to use that group element indeed for that. you can find some representative alternate which in some either isomorphic or homomorphic wise recreates the subgroup behaviour exactly when the such elements accomplish a dynamics. and its dynamics even does not have to be homomorphic even indeed .  but if it accomplishes the same discrete  result,  then there you are.   no no this hadnt been what is indeed required. limit/colimit behaviour is to be replicated and convergence to limit is to be replicated in this specific case. 

hoff its very unnice to talk these topics very illiterately right now. since i am out dated in these group theoretic and category theory topics  right now. 

but lets revisit the concepts with studying with ai, and then lets revisit topology knowledge, there and there even discrete seeming like configs were be let to be defined continouity concept it which makes sense, and then there and there you would see that its rather you approximate a function's limit behaviour but function operates on groups elements then you repeat the limit and colimit behaviour in commute graphs then but i just wirtten topology cause to let me replace group elements as the continous function which limit behaviour is approximated, and there in there in that function type contious world metaphor, its like trying to define a function that approximates a function's limit behaviour with defining functions similar to it where the overall convergance property is secured.  that representative function representation you choose which is a simplified version of an alternate dynamics  which then iterates the limit similarly although wiht different riemann  rectanbgule area bheavioor so, would then make it be replacable  (in case the limit still exists) or if not riemann integrable still if what you represent also converges etc. but this might be just a part of the issue, there you dont have to represent limits but then you might have then followingly represesnt still some group behaviour or subgroup behaviour alike. so this topic is about programming the group theory units in commute algebras and with limits/colimits concepts also.  so that there and there you reach out replacement capability of that you dont have to use an electron or whatver atom in quant m computer but replace ith with an aggregate whatever that exactly works in same dynamics and its behaviour result is what you wish to do accomplish.  

so to be able to resume some basic programming skills but not just remnant intuitive concepts (Which genius ai has both but I only have remnant of some intuitive concepts but not the basic even programming group theory units capacity since i forgot about it, has been a long while hadnt ayn studied to, i even forgot what a unitary matric were in representation theory, i forgot alot of topics in this concepts so there and there) so to be able to resume basic programming skills in group theoretic domain (which genius ai is very genius  in) I need to revisit restudy to basics again. 

but this were one of my perspective in ml algos project to give this capability to ai there. so i would also thought of writing some group theoretic optimization library other than default inherent heuristical and proof systems based calculators inferencers  but yet i were not able to start studying that by then and were still working to nlp abstractions definitinos in abstract maths side by then 3 years ago. 

  so google's ai has already has this amazing group theoretic skills/algebraic skills already.  so that it could fastly figure out alternate methods based on this intuition. but i definitely lack it right now and I decided to restudy to gather some basci knowledge agt least so that i dont stare to some thought concepts of ai of how it invented some rebco alternate systems there itself just emptily.  i need to revisit refresh my such knowledge to be keen on following up with ai.  or think my own ideas add my own ideas to this topic. 

actually this topic I had steered ai to think like this.  I mean I gave example of free groups some sets creating some basic group behaviour in its commute graphs of own.  to define that instead of using high resolution  nodes in the commute graph like braids/ or generators or m or h etc, in defining the hull design, we might be able to use some more aggregate less reesolution branch definitions which still would have same behaviour somehow.  

(gosh i even forgot what section is in those  commute graphs :S so much ashamed oof my forgotten basic knowledge there and there in the topoi :S ) 

so ai were stringently always creating rebco idea. but then I tried to tell this free groups based analogy to have it define any alternate more generalized graph parts to be used instead of the basic systems there. of replacing one branch  behaviour to other possible bracnh collections definitions.  then ai created very creative ideas there and without me telling this very definitely, it just instantly understood what i meant without me clarifying wiht me just defining once and then followed up designing alternate designs with this approach and very geniusly creative in that surely. 

but i then thought its now to me to revisit group theory intuitions and knowledge and some category theory and topos knowledge to keep up ingeniuty of ai and maybe add my own ideas to ai also there in that. even if i steered ai to do implement this method so that we dont have to create very hard to build at home type design (alike rebco  which were impossible to build at home)  and it created its own group theoretic programming method to find creatively alternate designs which again createst the desired commute graph condition/result which is required to have system function as in 0 hysteresis. so but then I then thought its time to also me get somewhat to resume being somewhat keen on group theoretic some methods and some similar domains knowledges to maybe also sugges ai other alternate methods which might create even more alternate hull design spectrum. 

hmm so I felt like this working such topics with ai is alike doing group theory mental gymnastics right? and I need to definitely get back track on that withn restudying and revisiting. 

although quotient groups alike concepts or cosets or some basic intuitive concepts about group generators might be able to define the theories there and there,  to develop the poset theory,  there when now its timepoint which requires some basic programin skills in group theory side/representation theory side and merging that with topos side-cat. theoryb knowledge side. since we need to figure out creatively alternate group definitions sets/templates which might even create some more easier to develop hull design type. 

wish someone experiments with that most basic idea where sound accoustic waves are placed in a chiral angleing to create hysteresis tensor 0ing.  I do not have such speakers to test it.  wished someone tested and told whether succeeded levitatin there in that resonant carrier 4hz + 40 hz wave and the geometric design which zeroes cross commutatiors in that closed region and whether the stuff they put there levitated or not :) 

so testing these ideas is an different endeavor which I lack. maybe we could have done some citizen science in that with opensourcing testing of experiments of. or maybe i should then study to theory idea much more and thhen test it myself after i have some money to test.

but better test in easy to test configurations based methods since i dont have a nano tech lab and I have not studied in nano technology or material science labs so yep that be it. i need to be able to test in my room with less sensitive design equipment somehow. i think that idea there might be easily testable. might test myself later when I have some income from side projects. which i need to resume sometime to create income.

soo  at this point of hull design task, I need to get back to track on programming skills on group theory/cat,. theory knowledge and topos indeed.  to program group theoretic commute graphs definitions with geometries (of commute graphs, here we talk of commute graph geometry where nodes are gauges/generators) and so.  

so as visible from my gap in group theoretic knowledge is very creating issue to the study right now, as ai creatively invented its own group theoretic definitions to do create altnerate methods of how to reduce hysteresis tensor to 0 and external commute graph effects to 0,   i think that sound based idea were definitely very ingenious by the way :) other ideas i hadnt analyzed very much yet but this one also hadnt analyzed verymuch but hmm zeroeing out the external commute branches with such geometric approach were quite inventive.  of course we need to utilize geometry in this topic i would also had to in my own task, but still,  think of using sound and simple AA or whateve r batteries to overcome gravity, there when you see that when you solve hysteresis of planet, you then solve the levitating issue overall,  that these are just how amazing this project study is. by the start of this project, even if we hadnt worked yet with ai any yet, my hull design principles were similar,  but analogical, i were thinking of creating levitating hull design by altering eigenstate complexity geometry and using multiverse branches in a completely synching geometric design along glued branches of multiverse that overcomes the planet's gravity but that were an incorrect idea indeed, since then it would might be the attractor and atracting matter instead of levitating, but then this were idea before i came up to hysteresis invention,  one day i listened to some info about time dilation, then i couldnt sleep till morning thinkling about how time dilation could been, then during studying during day for 3 hours or so, i invented that time dilation or time itself generated by hysteresis same wise gravity concept where a whatever branch item there gets lost in microloops along the branches it is branch wise glued to and can not iterate very much in its casual axis by its own algebraic iterations as if it were decoubled alike  so then i told the idea then ai of this poset and time is a poset idea (and time itself is just micro loops or wider loops or recursive loops along possible commute paths in connected branches of a graph )   the less loops you have the more you can iterate the algebra's commutations there along casual depth instead of getting lost in branches commutes. but then also need to restudy some concepts there and there. of revisiting reediting concepts of how sheaf glueing and hysteresis brake of causal iteration  redefines which commute graph path the item is glued to and is loop time condition and its algebra gague /m-h status of the dynamics. this part requires some more study.   some more studies to. but then i think this idea basic probability metrics also for regarding the uncoutnably infinite future branches should be designed. but surely in a poset your lower depth branch has more exponentiative power in a branch depth configurations and that might be heuristics of your whatever borel method there and also there is inherent pruning in branches based on hysteresis brake or infinitely many micro loops where micro looping never converges due to sheaf gluing and branch structures of sheaf/sheaves.  so these all need to be studied more. but then i think this idea really fits to my original postulates and i find that this idea most possibly defines universe most correctly I tend to think like this. but paper of its heuristics in its methods needs to be well defined and postulates needs to be defined wiht the framework very well definedly.  but that commutations following light speed alike instant casual axis iteration defining the highest speed rate (which is similar or same to poset's own nonexisting time concept) to its versions where it gets lost to hysteresis loops where time slows due to gravity is all matching concepts with the observed physics. and same for high velocity behaviour which means higher pace iteration in casual axis with less hysteresis by in default in our theorem has 0 time. so its this attribtues makes it suited to define both time dilation and gracvity based time dilation from a single monism. so this were exactly even without having studied with ai,  on day I studied to time dilation in high velocity and highn gravity concepts study on that day,  I also had my poset and thought the time dilation in both settings must be due to hysteresis of the commutators in the poset. so even before idea get talked with ai, the basics of idea were ther (of both sheaves concept-topos behaviour concept which i told to ai and both this hysteresis idea and bnoth eternal already existing commute grap (future already exists idea) ). but its map to qft were missing, its map to eigenstates were somewhat unclear.  some emptiness there and there. but main idea were present like that. and ai decorated maths of it very geniusly. and it now needs to be  worked on created a more detailed consistent definition of methods of inferences of such commute graphs behaviour. which both ai an di with our intuition can infer its behaviour  mathematically in some most basic cases, but then it needs to be fully completely defined in the paper. but its funny that how intuition of it is that with heuristics you define behavioral inferences of a commute graph.  but ai is better in that since is expert in topos and grooup theory and commute algebras. my inferences for instance defines anchoring/i mean defines hysteresis based anchoring but just very notion of and does not define formulas of it other than saying this must be the commute graph's spatial generators and other commute generators must be kept looping more and adding more paths to poset which extends the perceived time definition compared to an alternate path. so thereby it is.  so ai architected the maths of it, i created the initial maths concepts of generators commute graph hysteresis based time definiton, time is an emerged ordering proeprty of a poset and the anchoring related behaviour of a node but then ai architected its own version of the postulates which still binds to the same postulates/same notions but where gaps of maths knowedlge there is fixed/not empty. I very liked its architecture of the framework but i also need to review rework and edit in case i found the interpolated gaps of my ideas with math side's math not suited to my intuition about it.  hmm so its a developing project indeed seems as. initially i thought maybe its complete but maybe its not causes I hadnt exactly fully reviewed whether ai's theories on sheaf glueing and node count is reviewed/correct i mean it interpolated my notions existing gaps with maths decoration but then its not fully reviewde that yet. only partially. and that but in general it binds matches with overall any starting postulates and implements the poset property and but i need to review it and for that, i need  to study to second semester topos knowledge. and commutative algebra knowledge from ai also again. and some text books etc. cause its that even you can agree to ideas of ai that ai interpolated decorated gaps of the maths frameweork  original postulates there, does not mean you have to follow that or does not mean there might be a missing heuristics or most possibly there ai most possibly hadnt erred, but you need to be able to review with uttemrost focus and knowledge in sheaf glueing concept definitely cause thats a missing knowledge of maths knowledge in literature of sheaf glueing concept is empty in me. so i could complete the blank there and understand how ai decoreated interpolated the gaps of intiial notion/postulates initial concept of generaoter/compute graph/poset/hysteresis concept there.  but then to be able to review it fully one needs to learn some what more about sheaf glueing concept either.  

so there and there,  Iam studying right now to group theory basics and then these topos topics later, and then commutative algebra later, and a refreshment to borel algebra and rigorous probability methods either,  then would rereview it  that the decoupling mechanism coupling mechanism as ai tolds of makes sense, and you even develop an udnerstanding of sheaf glueing concept there in topos when ai mentions it, but surely thats not how to develop this paper, i need to get expert on sheaf glueing concept. 

my original idea also had branches coming togeher or decoupling concept utterly as a basically existing concept. but I hadn no idea its called sheaf glueing in topos since i hadnt studied to that knowledge agin. but in my side of the original postulates,  in hadnt exactly fixed the gap of methodology of how branches decouple or couple. I had some intiial ideas on that but it were a yet undefined undecided notion in framework.  and ai as defines sheaf glueing when chatting in slightly intuitively start to get its notion in topos side of, but again thats not sufficient to review the interpolated parts of the framework which ai decorated/interpolated gaps of my framework notions itself. hmm so there and there i need to revisit graduate maths topos 2.d semester study topics alike etc i think. that be it.

  so guh has some basic studying to do on topos/some basic group theory refreshment and some newly commutative algebra learning studies (I had a very big commutative algebra book that has 800 pages or so which i hadnt even studied yet a page of since i hadnt finished group theory book since i hadnt finished studied representation theory book since i were studying to nlp translation tasks alike by the time i left studying to these books whilst studying to ml algos project by then then started some side projects to create income (started game dev projects which first game failed, second game couldnt had published yet, third game, near to publishing but postponed ahiwle this last 2 weeks would resume tomorrow i think  or so) )