This is a bit of a stream of consciousness, don't take the language too literally.
Have deep learning tool watching key strokes in order to predict future movements in the computer, ML RAM prep
Test
Use ML to prep ML data, more time can be spent on the model and parameter setup.
ML scientific testing:
Data prep
Parameters
Parameter initialisation
Model
What metrics can be used to map input data to model parameters required? Variance metric? Data rank? Number of data points? Etc.
Important for universal ML solver is to replace other simple models ie. decision trees, support vector machines, clustering etc. What minimum setup must be used in each situation?
Could two NNs be merged in a way that preserved previously learned behaviour, allowed room for more learning using combinations of the 'abstraction ideas' and performed some sort of dropout algorithm to conserve efficiency? Is there a way that less certain abstractions may be forgotten?
NN is simply a mapping of data from the domain of inputs to the range of outputs, using nonlinear transformations to distort the data in a useful and meaningful way.
Can structure be extracted from NNs and compared? How would one go about it? Can weights be reorganised by value while preserving learned features? Possible use of graph theory to determine structure?
Have a nn that takes all features as input bar one and attempts to predict the result of the final feature, cycling through all features at the predicted target. Will the nn be able to better link different feature? What architecture? Will it be Boltzmann brain like? Include recursion?
Unsupervised learning by setting a NN to a set number of outputs and training a loss function to recognise high variance results?
Find all instances of a non-NN ml method out-performing an NN, how can the NN be reprogrammed to beat the opposition? Needed for NN universality.
Rank weights depending upon the relative error change between the train and test sets. High error change could indicate this value/weight generalises better.
Recursive unsupervised model for embeddings, given sentence with missing word, predict word vector, use prediction ------don't think this will work
4D CNN for time series 3D data? Equivalent for 3D time series in 2D space.
Residual fully connected network with high L0 reg/ prunning
While on acid
Entropic edges
Everything contained
Riding a wave of complexity?
The universe rides on a recurring complexity wave
The universe as a fractal changing one
The drug acid is the special edge of the universe that our pari
The universe a fractal acid is the portal
Reality is a self completing fractal
Black holes in the fractal
Right size containers for everything
Acid is edge that the universe can catch on, the fractal recognise the self repeating nature of the universe
The spiral the universe
Reality the just and edge of the fractal
************************************************
To find ~exact network to describe a function, get infinite network (read very deep) to predict property, trim network to minimal function for accurate prediction both in width and depth (heavy dropout, find first layer in which weighting drops to zero for all but one weight), process all example and build ensemble/ averaged network to contain the overall properties/ ideas.
************************************************
************************************************
Use nn network to decide parameters for a function network. Train on simple models as transfer learning for bigger models
************************************************
How much information can a neural network store? With overfitting can a NN act as a lookup table? Does the information require some amount of correlation?
Apply electrostatic theorem to dft to calculate forces rather than energies.
Have reinforcement algo watch computer movements and predict next movements for reward. Ultimate goal to replicate particular workflow and identify key areas of information.
Train adversarial agent to predict your movements and train reinforcement model to enact your movements using adversarial agent to evaluate score.
Model learning rate on confidence level to boost high confidence correct values and reduce low confidence vales
Reinforcement algorithm to vary parameters during nn training such as learning rate, dropout, regularisation etc.
Train network on increasing number of layers adding in extra base neurons at each stage, each base layer is made to best predict at its level of depth and therefore computational cost but can pass very general information to later layers for more precise measurements dependent on particular situations, think of it like entropy of event, low number of layers picks up on common distributions and later layers use as spring board to find more complex and rarer situations. Ensemble produces massively connected NN with varying levels of scope and therefore more broad predicitons.
-- Vary scope of output variables y - y_hat to provide different searching tasks to each area of the NN, removes problem of vanishing gradient as prior results are masked.
-- Basically convolute network by not connecting neighbouring models till after training and adding layers
Model shape of graph using gaussian process output parameters for functions
Add new weights and neurons during training and cycle out inefficient ones with regularisation, monitor performance of new structures with slowed decay like weighting.
Is it possible to develop game impossible to machine learning by making neighbouring states so different that sampling is impossible? Would game just end up being chaotic?
Network produces regression and confidence (variance guess) output combined, loss calculated from combination
On back prop does confidence always increase? How to feed idea of bad confidence? Variance?
Does increasing the amount of related information the system has to predict increase the ability of a network to retain information? Is the system mapping correlations?
***********************************************************
What if in the future the best way to get results is to run a quantum calculation and each outcome corresponds to a different universe this is fucking dumb
The joy of owning cats is in part derived from the ability to own a good representation of a truly wild animal in the sense you have a small but highly effective killing machine which you carry around with you. Such an apex predator that in its environment nothing can match its speed and weaponry. Robust and unmatched strength to weight ratio but almost a show piece, "The jaguar I keep in my bedroom".
***********************************************************
If we map certain properties of functions (path length, curvature, etc) to a linear axis what happens to other coordinates using analytical continuation? Do they map to a similar shape to the original function?
Use generalised function representation in nns?
Computational evolution on neural net architecture and structure, how does small perturbations to functionality affect learning response? Perhaps MCTS to find more optimal parameters (look at spiking, activation functions, random interconnections)
Use huge liquid computer to learn areas that work by covering hypothesis space, only increase weights from low value so that learning is not undone, can action learning then join related hypotheses?
Is life a low entropy state? Or low energy? Taking thermodynamic and stat physics laws to the extreme suggests life as what? Or does the fundamentally chaotic nature of the dynamic system of the universe suggest that this particular form is unremarkable? What is the distribution of feasible states that are available at each scale?
Does a fundamental difference in the properties of two objects cause them to attract? Is there a similar argument for fundamentally similar objects? Entropic vs enthaplic effects eg/ electron proton attraction and H H attraction. Water octanol partition?
Monte Carlo tree search for model optimisation
Skirt chaotic boundary to maximise learning
Make optimisation problem have general differentiable solution and optimise this
NN based autoregression
Is there a continuous transition between embeddings of varying size? Representing decrease in value of specific information. This would allow for determination of key factors to state.
Correlation between state, information embedding of state and nlp embedding of state to describe related properties.
Latent (hidden) representation placed in nD vector vector space and nD convolution applied over it.
Experimental framework for computation, input testable iterables and cross iterable dependencies, output trends
Should networks be connected to replicate connectedness in the structure of the data, eg convolution over a 2d vector, further, use information relatedness
Use recursive autoencoder to find underlying efficient structure eg on coordinates
Recreation of data from reduced representation can vary from discrete rep. eg. Named objects to continuous rep. eg. Force input in muscle memory
Double evaluation metric on recreation of data from reduced rep. and performance on selected task using reduced rep.
Decrease stochasticity of reward/output to use calculus for all optimisation
Model interconnectivity while pruning as a measure of lottery ticket hypothesis
Can complexity be measured as the stochasticity of ergodic results?
Dropout and pruning ideas combined to find stochastic improvements from random recurrent connections
Negative feedback to find stable solution.
Represent nn as transformation and find most efficient route between two states, how do you measure efficiency in this situation? With a Lagrangian? Principle of least action? Transition evenly between discrete and continuous representation.
Feedback loops as means for training measure, lowest energy rep is at a stationary point of the system.
Feedback learning, increase weighted feedback for critical damping on error.
Model with feedback loop for training, with feedback on feedback meta learning until change in feedback loop is negligible. Delta w between layers is negligible.
Convolution localised neurons in 1d/2d/3d, what about generalising to nd with ∞d representing an exact function, generalising the notion of direction to easily sparsify connections while maintaining long distance information transfer at an efficient level, transfer of information decreases as the local environment gets more complicated.
Reality as differences between dimensions, mind and body as most efficient difference engine
Theory of everything.
Emergence
Mind as most emergent property of the universe
Maximise efficiency of information transfer
Time as a dimension of perceived differences
Coupling between dimensions as forces
Mind as connection between two infinitely dimensional vector spaces
Fundamental idea of difference between two items -> feedback forming nodes, saddles
Difference and feedback
The mind is the most emergent property of a system, an abstraction
Mandel Brot singularity at 1/2 and 2, feedback of x_2^x
Reality singularity at e^x between 0 and infinity
E as metric of reality
Feedback loop causes objective reality
1 falls to zero or spectrum from one and back in on oneself.
Path of least resistance between two infinities
A distance between two infinite dimensional vectorspaces the have a singulatrity between two infinitely distanced points on the number line.
Axioms provide a singularity that produces infinity, final end is when entropy fades everything to infinity, when infinity becomes infinity again and starts a new. Entropy as emergent property of geometry and difference
Perceive a local invariance between 2 dimensions as a symmetry
Entropy as emergent dimension of discrete dimensions transitioning to continuous dimensions invariance creates time and dual space energy
Science is language to transfer the idea of the singularity
The fundamental nature of reality.
Necessary condition of loop in-order to cause self referential behaviour
Necessary condition of infinite dimensions
Uncertainty condition placed on fundamental information transferred between 2 infinite tangent spaces.
Time emerges as a discrete dimension
Space emerges as loop of time increases dimensionality.
Fundamental structure of reality as a looped dimension of 1.
The fundamental nature of reality is the idea of a loop between two infinite dimensional spaces. The loop is self similar and also cyclic. It travels from one singularity of infinite dimension to the same singularity of infinite dimension while going through an infinite distance on discrete symmetries of the fractal. A local symmetry operation of the loop is perceived as a fundamental law of nature. Along the principle dimension of time we have a dual space of energy, a tangent space defined only by the local coordinated. An emergent property of the loop causing feedback on itself is the creation of extra dimensions/symmetries. This happens during the infinite mapping of one infinity to another. Time-space is 4 discrete dimensions created from a single loop entity. Time-space-force is a 16 dimensional entity, time-space-force-particles is a 256 dimensional entity. This pattern of 1->4->16->256 looks like a hyper-operation applied infinitely many times,
1-> f = 2^2=4, f^f=2^2^2^2=16=g, g^g=f^f^f^f=2^2^2^2^2^2^2^2=256=h
h -> g -> f -> 1
0 and oo represent singularities
For the operation -> we need to apply it an infinite number of times to reach 0 or oo from 1, in fact we need an infinite number of operations from any point in this loop, the self referential nature of the loop to maintain itself. As a point on the loop moves it travels through an infinite number of dimensions. The most discrete forms of dimensions are visible to humans as our observable time, time-space dimensions. Symmetries applied local along these directions reveal physical constants as shown by Noether's theorem. In the time dimension we have a tangent space of the energy dimension, moving up to Einsteins theory of general relativity the space-time dimensions have a tangent space of energy-momentum space. The idea of geodesics is correct but applied to too few dimensions. As finite dimensions are increased an infinite number of times we reach quantum mechanics at the far end of the scale. Function space represents an infinite number of hyperoperations applied to an infinite number of discrete dimensions. The standard model becomes a way of coupling the discrete dimensions. The Higgs couples time and momentum spaces, the four fundamental forces couple time-space and energy-momentum. Particles pop out as a 256 dimensional object.
Time -> 1D <- Energy (Higgs is emergent)->
Time-Space -> 4D <- Energy-Momentum (Forces are emergent)->
Time-Space-Forces(With symmetry) -> 16D <- Energy-Momentum-Forces(With tangent symmetry) (Particles are emergent)->
Time-Space-Forces-Particles(Symmetry) -> 256D <-Energy-Momentum-Forces-Particles(Tangent Symmetry) (Mesons emergent? Hadrons etc?)->
Etc.
This coupling between discrete and infinite dimensions can be viewed from the perspective of the exponential function.
e^x maps -oo to 0 and +oo to +oo while passing through a singularity at one.
by looping the x axis on itself we form the complex plane, e^i*phi
where i dictates the curling of a dimension upon itself
this looks just like an oscillation -> wave or particle in a higher number of dimensions
The emergent properties of a 1D fractal loop appear to create extra dimensions by joining the loop at both ends to an identical singularity.
In the single principle discrete dimension time the singularity appears like the Big Bang in the past going to a Big Rip(?) in the future. As dimensions become coiled, space-time singularities appear like Black holes and White holes(?). Past the event horizon of a singularity the dimensions of space-time invert. Actually the entire infinite dimensional space inverts and tessellates reality with symmetrical version of itself. This can be seen in the Penrose diagram by extending the dimensions infinitely. Any path forward or backward leads to an infinite singularity at an infinite distance. This may mean that the reality metric is both infinitely compact and infinitely extensive. Essentially this forms a Mobius loop in an infinite number of dimensions.
The universe around us can be explored by moving through the fractal using a mapping between an infinite and a discrete number of dimensions. This forms the mind as the most emergent property of the loop. Loop feedback is propagated through time, energy, space, momentum, force, particles forming more emergent behaviour from chemicals to cells to cell networks that pass information to form the brain. The information feedback from reality produces emergent complexity that results in a discrete singularity of our mind, meanwhile our mind-body collective introduces the feedback into the system necessary to form more complex emergence such as thoughts and ideas. Connectivity between dimensions allows the flow of information however the uncertainty relation limits the total flow. When the tangent spaces of energy and time are transformed into each other the maximum information rate is given by the uncertainty principle. This follow for higher dimensions.
dE dt >= hbar/2
dp dx >= hbar/2
At a certain level the local symmetry of the fractal allows us to glimpse at its true nature. Local time symmetry reveals energy and vice versa, local space symmetry reveals momentum. When both symmetries are applied we reveal General Relativity.
With this comes a few insights.
Entropy appears as possibly a measure of the change in fractal dimension.
The holographic principle may reveal the mapping of one infinity to another.
Intelligence and emergence are a form of network feedback.
The standard model requires an infinite number of dimensions.
The next level of emergence may be the effect of the feedback loop interacting with itself.
Time, space, energy, momentum, force, particles etc. can be united into a single unified theory.
Reality appears like a wormhole consuming itself.
Testing possibilities.
Update the standard model to reflect the new principle and drive new results. g-factor results may be far more accurate, spin as a fundamental constant of the fractal.
Machine learning as production of most efficient feedback loop to map one discrete dimension to another.
In conclusion the fundamental nature of reality is a single point on a 1D feedback loop connected to a unified singularity at an infinite distance in infinite dimensions. The nature of irrational numbers lends itself to the idea of a single number containing an infinite amount of information and the connectivity of irrationals is such that any reality pathway is perfectly continuous. This may sound ridiculous but the self referential nature of the principle may validate its authenticity.
Fundamental structure of reality as a looped dimension of 1.
Mass -> Time <- Energy
Logged error to limit effect of error on gradient
Positive reinforcement training?
Spiking nets useful for reuse of circuits? Singular cnn filters, time based computation
True networks utilise only structure for computation.
Define universe as a function space/ dynamic field. Define neighbourhood lattice repetition over a certain gap f(r)=f(r+a). Is universe stationary in this definition?
Differential flow of time acts like a force like air over a wing
Length shortening flow to "shrink wrap" a continuous function with discretised network approximation
Define discrete energy difference between 2 conformations of a system, increase mesh quality over pathway to find min energy path
************************************************
Duality between brain perceiving universe and brain creating universe.
Some require self consistency others have theistic beliefs
Reality is by its very nature directed by this
************************************************
Science and maths explained by stoners
Limit of tokes per blunt
People in ancient times discussing modern problems
How much of the way we perceive the universe is due to the minds interpretation?
Space of axioms, continuous and discrete axioms
If time is an emergent property of mass/energy and the spacetime interval is universal why do we formulate relativity as ds^2=cdt^2-dx^2 instead of dt^2=ds^2+dx^2? Curvature is important in both general relativity and quantum mechanics for both the action of gravity and wavefunction energy, are these two equivalent ie/ is the wavefunction a bending of spacetime? Further if we took a particle centric view ie/ the electrons frame of reference, applied some normalisation on the wavefunction w.r.t the local momentum does the wavefunction become uniform over its domain?
Network weights determined by system output.
Functions are universal and can therefore represent any object. Functions that transform functions to another function. f(g)->h functions can represent numbers -> delta function, functions can represent vectors n-dim delta, matrices with linear transformations, tensors by increasing dimension, universe is a function state with a transition function f_t(f_s)=f_s+dt
For functions representing numbers/vectors etc what are constraints on input and operator?
F, x representations.
Infinitely stacked functions? f(f(f(f(.....)=g(h) -> h=g(h)
Such a function space can conceivably contain absolutely any possible state (paradoxes? Include its own function space? Along periodic dimensions?)
Periodic dimensions
Transcendental dimensions
Sparse dimensions
Is reality paths along said function space?
Mind is intrinsically Markov, chain of singular states, each state with finite dimension or infinite? Memory is inherently passed by last state return.
Dreams and language necessary for mind to accumulate policies and as self teacher/ chain of thought process. Have network explore latent behaviour, pass information state through iterations
Quick memory through network chosen parameters
Space of network architectures with tangent space of parameters compared to space of loss functions and space of optimisers for rapid learning.
Unwrap meta model back-prop as a string of chain rule updates
Curvature fundamental, do wavefunctions stack on each others curvature, high curvature is both energy and mass, double differential caused by self interaction? Space->dual->space
Does the curvature of the em field from a photon cause the observed momentum
Consciousness as the linking/transition function of the Markov process that is the brain.
Predict loss space with network
Symbolic neural networks, chose symbolic transformation set for neurons
Are entropy and wavefunction diffusion linked?
Are entropy and the uncertainty principle equivalent?
Universal expansion linked to entropy, change reference frame to smoothing of wavefunction, are conservation laws obeyed?
Changing of variable in a function allows for discovery of underlying function, can function be discovered by observing underlying symmetries, are all variables fundamentally linked?
Does light propagation occur in just 2 dimensions due to length contraction? Zero time and zero length in direction of propagation due to length contraction.
What does observable universe limit represent? Are we seeing more or less?
Looking far = looking back in time
What functional form does instant of creation represent?
What does planck length represent in functional space?
If every property has a Gaussian distribution are we a projection onto a super high dimensional sphere?
Does real matter meet antimatter at singularity or turn into antimatter and travel back in time?
Can a singularity even exist? Surely everything has to be a ring singularity?
At final evaporation of a black hole is the final thing left a fundamental particle?
Are all particles ringularities?
Ring singularity is not point particle, avoids infinities
How do point particles even meet? Quantum Jump?
Computational hunt for functions that satisfy GR and QED
Thought experiment, time is direction of motion of energy, humans gain energy from enviro, from stars etc, lose it as entropy, stars release energy. If eyes could measure loss of energy?
Many-worlds description of quantum mechanics as diverging function of possibilities, are there an infinite number of universes converging on each physical system?
Ricci flow of wavefunction? Ricci flow of volume element of dynamical variables in chaotic system?
Inflation like Ricci flows converting energy to entropy on the universal manifold
Complexity related to high correlation of atomic velocities, delta function like/low temp.
Trap a photon in flight thought experiment. Form trap "after" photon has left source.
TOE searcher, decompose maths as functions and objects -> f(o) eg/ addition a + b = add(a,b), language model used to form latent representation, fine-tuned head trained to spot mathematical equivalence or linking relationships. Build dataset using common maths software etc.
Representing linear transformations with geometry, how can differentiation be represented geometrically? Infinite dimensional space? Is time a spherical geometry coordinate while space is euclidean? Looking out into the universe we see the past expanded and spherical.
Mind is only thing that observes one directional time due to thought or time dependent functions/autofregrssive being unidirectional.
Set of linear eqns describing d/dt of a variable as a function of the other states x and transitions d/dt(x), every object in universe as function described by directed graph of sub funtioncs.
Test with true high dimensional predator prey system limited by resources.
If different stories can be represented as different functions and the mind is a high-dimensional game of life glider, can mind move along different story trajectories? Is reality also a sub reality for a more complex story? What if life is like and efficient information accessing routine in a far more complex automaton?
Explain perceiving of time as parametric equation of a curve with parameter t, extra dimensions can be added by probability distribution of particles.
Maths as self consistent symbolic structure, rule making system and equivalence quantifier
Use computational graph to form dependency links in structure, each additional rule is given as functions of the previous nodes in the graph and the rule is presented as a new function. Essential build everything from scratch, does this naturally discover set theory/ different algebra structures? Can it even discover numbers? Search for hidden in-discrepancies that require the addition of a new rule, can new rules be applied to past structures?
Have language model producing rules, structure rules into comp graph, use comp graph to formulate problems for second language model to solve, ask second model to formulate links or search for inconsistencies. Possibly combine into single model using request/answer structure. Build and question comp graph of rules.
Do electron orbits continuously experience acceleration to stay bound? Does this have GR effects? Why don't they emit EM radiation because of this?
Define maths as objects and transitions as in category theory, functions are objects
Universe as graph of interactions that maintain continuity, each vertex has an associated probability distribution for quantum variables
Define maths with maths? Define maths with functions, solve eqns using functions
Tree of growing nested n-ary functions, sub-systems contain self consistent choices for axioms based on equality function.
eq(a,b)-> a=b? Boolean logic
f(a,b) = f(b,a) Commutativity
f(a,f(b,c)) = f(f(a,b),c)) Associativity
f(a,f(b,c)) = g(f(a,b),f(a,c)) Distributivity
Networks are very good at representing real systems, brain as Map of universe, while underlying universal mechanics governed by graph network
Formal symbolic language
n-ary objects
0-ary objects are constants, 0, 1, x, a, etc.
1-ary objects are single variable functions, f, g, etc.
2-ary objects are binary operations eq, add, mult, etc.
n-ary objects are multi functions, f(x,y,z), f(x_1, ..., x_n), f(X)
Denote rigorous mathematical language/ symbolic language and utilise group theory to decompose it into its irreducible groups etc. Define mathematical structure as a group, what are group operations?
Ask gtp3 to recursively add finer detail to model/software/plan to execute full build from single line idea.
Use AI to build knowledge graph of information on the internet, link this information to data and math backbone and build whole world model, universe model, atomic model.
Traverse information tree with learner model to build better experiments and results
Have whip like model searching through recursively detailed solution space to find optimal solution.
Explore how sparsified returns affect how intelligent optimisation changes
Functional transformations with a meta controller
Could measurement be explained as entanglement between an observation and observer, therefore collapsing the wavefunction?
Is the wavefunction uncertainty over the possible parameters in the interaction hypergraph?
Visualising higher dimensional manifolds, can the orthogonality condition be removed such that dimensions can be bunched? Place extra dimensions on projective spaces? Place dimensions within hyperbolic space.
Would a quantum system over a single point yield a two level system? For spin etc? How would colour charge be represented? Take a quantum system over a ring and shrink the ring radius to the limit. What about qm systems over a torus imbedded in increasingly higher dimensions?
Does entanglement change over time?
Monte Carlo exploration of variable combined later with active learning to build model of a system. In semi conductor qc varying magnetic field orientation, voltages and geometry to build full overview.
Build out depth of NN model by adding layers in between current layers, initialised near zero that move out to increase performance.
Stacked layers of modelling networks that output parameters that fit data, altering the underlying model can change output parameters and a meta model can tune the underlying representation for desirable features in the output parameters
Primes as property of group family structure, could symmetries be used to prove Riemann hypothesis? Link infinite sum to property of symmetric shapes? Use shor's algorithm fully optimised?
Recursive tree like structure that defines groups also defines primes, does structure have any symmetries?
Zeta function as infinite sums of fractional reals, products of prime dependence and exponent tree of hyper primes?
Possible to have Taylor/fourier decomposition as product of functions?
Spin as standing wave on infinitely small ring? Boundary conditions met when psi = +/- 1
Memory as a generating function for further memory? Reduced representation that further helps build extra past knowledge?
Using commutation as a way to naturally order objects?
Can manifolds change dimension in same space? Grow dimension? Hyperbolic plane with excess space?
Are all periodic behaviours describable with quasiparticles?
Use integration and projection to visualise density plot of wavefunction
Tiling the hyperbolic plane using a 1D coil?
QFT interaction as a correction of wavefunction towards eigenstate, in eigenstate no correction needed and therefore no interaction, discretisation of interaction increases with disturbance, collider interactions highly disturbed?
Different parameter values/scaling (charge, mass, spin, wf distribution) calculated as projection onto lower dimensional manifold?
What happens to a photon that doesn't interact with anything in its lifetime? Is it even possible? Photon released from a star with no target in path.
Universe is original hyper graph optimisation process
FFNN -> Scalar weights
BayesianNN -> Function distribution weights
Transformer -> Score operator determined weights
Some way to combine all these operations? Universal net with deep residual connections? Parallel vector computation? Vector size determined from batch size learning?
Attention activated network regions
Model time dependent feedback with graph, iterate to improve modelling by increasing coupling complexity
Ring potential for multiple electron entanglement.
Loss function and optimiser are coupled, changing loss function can have an effect on training rate.
Neurons firing can be represented by pulses in a graph structure, graph structure can be represented in high-dimensional space, can thoughts be represented as waves in this space? Solving the wave equation in a super high dim space.
Extremely precise measurement of electron momentum in free space in-order to have largely undefined position and therefore entangle with a huge number of particles?
Photon experiences no time during flight, also dimension in direction of travel is length 0 due to Lorentz contraction, final two spatial dimensions are like 2D slice of universe. In reference frames at start and end of photon path, time is measured as having passed during photon travel. Where in 2D slice is past and future mapped for these reference frames? Can photon see past reference frames? How does GR play into this picture? Is classically observed photon path actually superposition of two infinite rays? Time measured by distance light moves.
Accelerating to c first redshifts a particle moving in the same direction, as velocities are matched the particle looks stationary and then starts to be blue shifted. At c the particle becomes flattened into the 2D cross section.
Quantum harmonic oscillator in GR?
Bounded learned loss functions
Set of time and space coordinates x, y, z, t combined with reference frame coordinates px, py, pz, (pt?). Need acceleration coords? ax, ay, az, (at?)
Get nn to rank art and therefore learn what factors increase potential.
Use direct linear solver with sparsified data or attentive data on each iteration of depth increase Wx~y
Reverse of pruning process for network growth, what factors require new connections? Measure correlation within data? Measure non-linearity of a process? Can linear connection be better represented with functional connection?
How to translate deep models / components of models to symbolic models for computational speedup?
Attention only works on low dim data, split computation into representation and attentive factor.
Power as geometrical relationship between distance and area? Exponential as relationship between 1D and nD manifolds?
Symmetries of the space of all continuous functions. Is universe C^n or C^oo?
Limit to spacetime curvature? Infinitely small ring can become finite due to curvature limit? No longer have point particles?
Autoencoder on representation to reduce comp requirements for attention mechanism
Does canonical commutation relation just suggest p and q are orthogonal properties? Rotation of pi/2 in some space? How does multiplication by i / rotation by pi/2 affect values in hopf fibration? Time differentiation equivalent to rotation?
Possible universes:
-Hypergraph, connected structure of interactions, energy/reference frame dictated cross-section.
-Manifold curvature, dimension dictated by mapping eigenfunctions.
-Symbolic relational structure/function space, look at exp(z), growth, periodicity
-Dual orthogonal coils, hopf fibration, entropic/statistical driven.
-Coupling of quantum harmonic oscillators, QFT, qhc with GR, role of curvature
-Map of 2D photon like slices
Find way to unify all ideas?
GR. Ruv - guv R/2 = KTuv
First part as approximation? (1+x)^a = 1 + ax
Eigenvector fibration to decompose function transformation into t,x
Feynman path integral bounded by light path ellipsoid?
Quantization of an infinite well, no boundaries, no nodes?
Could charge be quantisation in infinite well of time? Time like dimension as well potential?
Does particle know its own state? If so if consciousness is a collective excitation/quasiparticle then could that be causing measurement of specific environment?
Path integral formulation reformatted to sum the change from coords x_n -> x_n+1 over all time steps?
Integrate over all momenta? Integrate over all energies?
Relations p~q, t~E
Light rays see entire history of the universe smeared out over 2D slice, is time actually 3D dimension shown out along light path?
Is universe only deterministic along boundary/extremes of function space? Integral of form over manifold equal to integral of exterior derivative of form over boundary? Stokes
Photon as resonant energy transfer
Electron spin is exactly known and therefore its angular momentum is very precise so angular position is very uncertain
Learning rate inversely proportional to weight value?
VAE depth and width reconstuctability
Could spinor be frame dragging spacetime around a ringularity?
Does E,q transform to t,p from an infinite space to a finite space? How does this involve multiplication by i? Rotation around origin of complex plane? How to formulate in differential geometry?
Main areas of interest currently:
-Concsiousness as collective excitation / quasiparticle, perception of universe around us and measurement as entaglement between concsiousness and environment
-Electron as naked ringularity / singularity / event horizon
-Universe as hypergraph of interaction possibilities with entanglement reducing graph dimension to 1
-Group structure of mathematics, finite number of possibilites for rearrangement of a finite set of objects, what number of objects can represent each layer of abstraction in mathematics
-Reformulation of force and matter as singular manifold structure, key dimensions
-Reformulation of path integral from time and space to energy and momentum, role of commutation relations in group theory structure
Gravity seems non-quantum, is this due to statistical effects? Universe described by infinite dimensional tensor.
The more entangled two systems become, more information is known about the other system from the persepective of the dual system, when interaction is weak, entanglement is low, the other system appears to have high entropy, many possible outcomes, statistics takes over and the system appears classical. This can happen at low and high energies, integral overlap is small, systems are not in resonance. When resonance occurs and systems overlap, lots of information/energy can be transfered, systems become strongly entangled. Does this only occur in low entropy systems? Required conditions to produce conscious excitation? Therefore anthropic principle, consciousness can only occur in particular regions of the function space. Could this be reasoning behind the dimensions we observe? Too many dimensions and entropy is too high, too few and entropy is too low?
Since consciousness allows for continuous experience does is necessarily have to have a periodic function attached to it? Flow of information in and out of the brain can be represented using state tensors, how are phonons described using tensors? Look at nervous system as quasai graph network, flow of information/energy through the network can most simply be described by the movement of a state vector in a vector space spanned with a basis described by the vertices/nodes of the graph. Decomposing the movement of the vector using a Fourier transform may reveal cyclical nature. Local areas of the network feed into each other, have a stronger coupling, is there and underlying global phase and amplitude? Look for low frequency brain waves paper.
Describe biological neural network with normally distibuted lattice structure. Can this be described using some form of continuous hyperbolic geometry? In 3D space neurons are distributed uniformly but how does the connectivity vary with distance from the centre of the neuron? Neuron lattice structure dimension dictated by average number of synapses?
Energy equivalent to frequency
Time energy and position momentum are also equivalent as projections of some master manifold.
Imaginary time equivalent to energy? Multiplication by i same as rotation to dual space in differential geometry.
Any particle traces a path through a mathematical space given by the underlying group family, do group families mix at some point? Possible paths are given by the results of entanglement but to a given particle only one path is observable, measurements are just the branching choice in the function space. All paths exist but aren't observable to a given particle state. Consciousness in this way only sees one observable universe, the classical dynamics around us are a result of apparent measurement and statistical properties that produce the world we observe.
Is spacetime a double cover or symmetry breaking of the 2D universe photons experience?
Functional universe, start with two objects, T and F. eq(T,T)->T eq(F,F)->T eq(T,F)->F eq(F,T)->F. By currying:
eq T T. eq F F. T
eq T F. eq F T. F
inf eq stack eq eq eq eq ... -> eq(eq(infeq),infeq) -> T
Could existense func come first? Start with just T:
exist T. T
Have ml algorithm find the basis that most simply respresents the quantum system
Spinor on system with |0> and |1> orthogonal in 2D field. Double cover becomes obvious.
Intelligent weighting of Monte Carlo points for optimal exploration of high dimensional spaces.
Topology of electron must be orientable in at least 2 dimensions.
Transformation to get from photon perspective of 2D universe to 4D matter perspective with photon function given by exp(ik.r)
SU(4) space for double qubit projection. 2 qubit gates defined by rotation plane and rotation angle.
Distance and time fundamental groups due to projection form of wf eg exp(ipx) exp(Et) exp(beta H)
From a particles frame of reference (at rest) exp(ipx) looks like S1 in the position basis, along a light speed path the photon appears as a 1D line however due to length contraction this may appear as a single point. For a particle moving relative to another the wf looks like a coil or helix with one linear infinte dimension parameterised by the linear momentum and another periodic dimension parameterised by the (angualr momentum?) dual space? exp(ikr) r for direction vec and k for the frequency? exp(ixp) has input in R2 and output in R2 or alternatively input in C and output in C.
Universal constants place value on projection from one space to another. Deeper dimesnions hidden due to value of constants.
Thought experiment -> Entangle photon with beam splitter, send split photon to each eye of subject, which eye do they see the photon in? What is the process by which they see a particular photon? Interaction with molecule in eye? Each could lead to separate universal paths, for decision to be made information must reach consciousness. Consciousness becomes entangled with either state and splits. Entanglement is actually splitting function of universal hypergraph.
What determines the level of hypergraph discretisation? For the consciousness quasiparticle/excitation it must be fully discretised. However for QM/Quantum Computing to work the environment must be wavefunction like/fuzzy to some extent. Until we entangle with a particular state all states must be occuring. The less entangled two states are, the more the universe becomes 'fuzzy', at some point the entanglement must completey fade and the universe becomes somewhat uniform in nature, wavefunction distributions become random and diffuse. Does this happen at the edge of our observable sphere? Edge of the universe? Time and space dictate the limit of entanglement. Do singularities limit entanglement?
Drop entangled particle into a singularity, how does the properties of the singularity change? What about colour charge past the event horizon? Does a black hole have colour charge or lepton number?
What does the Fourier transform do to the Hopf fibration and higher dim fibrations?
mass and distance, 1 continuous scale x, 1 discrete x>=0, scaling x^-1 (force between two massive/energetic particles)
charge and distance, 1 continuous scale x, 1 discrete c in R, scaling e^-x (force between two hydrogen atoms)
spin and distance, 1 continuous scale x, 2 discrete up dn in C, scaling e^e^-x ? (force between two randomly oriented spins?)
colour and distance, 1 continuous scale x, 3 discrete r g b in C? R? H?, scaling finite?
|1| or det|I| in nD space over field F
Fields -> N, Z, Q, R, C
Finite fields
0D over F -> single point
1D over Z -> +/- 1
1D over R -> +/- 1
1D over C -> S1
1D over H
2D over Z -> (+-1, 0), (0, +-1)
2D over R -> S1
2D over C -> S2 SU(2)=SO(3)
Hypercomplex numbers and differential geometry
Rotations in Rn, Hodge dual.
R2 -> C1
z = a+ib {a,b in R}
*x1 = x2, *dx1 = dx2
*dx1^dx2 = 1
R4 -> H1
a.b = 1/2 (ab + ba) = {a,b} (inner product, anticommutator)
a^b = 1/2 (ab - ba) = [a,b] (outer product, commutator)
det([[a,b],[-a,b]])
det([[a,b],[a,b]])
Force related to entanglement? Forces/interactions happen in fuzzy space, measurement/value observation occurs in discretised graph space?
Universal constants as level of entanglement between two objects in given dim? Measurement is based on statistical likelihood of state. Force of gravity mediated at speed of light, suggest force carrier. Force of repulsion due to overlap of wavefunction and charge product in electrons.
Force between two objects related to overlap of wavefunction, including charge wavefunction. Actually mediated by exchange interaction. Overlap causes increase in energy, then process value decided by statistics of all possible interactions at quantum level. Observed state due to entanglement. Mass is bosonic statistics, charge is fermionic?
Split wavefunction of system into even and odd components?
Relation of spin statistics to exchange, do other conserved quantities have any similar properties? Could mass/spacetime curvature be a property of stacked wavefunctions of many different dim spaces?
Harmonic series as basis for mass -> splitting of charge, spin, colour charge etc? Function decreases as 1/r just as mass interaction.
Could replace harmonic series with prime reciprocals. Doesn't have 1/r dependence though.
Riemann Zeta connection?
Does a photon have to join two Fermions? CCC keeps photons travelling into the future, given the density of the universe does this suggest that the majority of photons never interact with anything in this aeon? Could dark matter/dark energy be super low energy photons?
Spin integer particles don't have same statistics as they appear like composite fermions, parity is conserved in the interaction.
Does electron have 1/r gravitational dependence? Or is the 1/r dependence a statistical result of all particles?
Spin sqrt of 2d space, space is su(4)? or su(1,3), colour cube root? where does mass come in?
Could gravity be long range bosonic attraction?
"a force mediated by a spin-0 scalar is always attractive."
Imagine yourself on a photon like path/interaction path. From the photon perspective time stops and the spatial dimension along the photons path becomes contracted to a point. From this perspective an interaction via virtual photon becomes a vertex with four edges. Along the time dimension energy of the wavefunction is conserved, in the spatial dimensions momentum is conserved. Along the vector of the photon path the wavefunction of the particle still exists but has been compactified to a single point akin to the limit of a Gaussian forming a dirac delta function. The wavefunction however can be complex valued and therefore the normalisation condition, |psi|^2 = 1, forms the U(1) group for charge? Spin relates to angualr momentum, what is the dual space to angualr momentum? For spin group SU(2) we need two complex valued orthogonal unital dimensions.
Idea -> For spin transfer to rotation space with bivectors x1^x2, x1^x3, x2^x3. Two of these values become point like with frame change.
"The photon, the 𝑊-bosons, the 𝑍-boson and the gluons are all spin-1 particles. A force mediated by a spin-1 particle can be both attractive and repulsive depending on the charges of the particles exchanging the bosons. The graviton, on the other hand, has spin 2, and the force mediated by a pin-2 boson is always attractive. This explains why electric charges can either attract or repel each other depending on the signs of the charges, but gravity always makes masses attract to each other."
"The Higgs boson is a scalar with spin 0. Like the spin-2 graviton a force mediated by a spin-0 scalar is always attractive. In the standard model the Higgs boson couples to the fermions through Yukawa couplings, which gives rise to interactions between them. The big mass of the Higgs boson means that the corresponding force is very short ranged, which means that it doesn't have any classical correspondence."
Constant interaction of Higgs with fermions gives them mass and therefore time dimension, splitting of time and energy dimensions into orthogonal Fourier related quantities. Without Higgs, fermions live in 3D space, no time. Time and energy more fundamental however. Do orthogonal dimensions play Re and Im roles or is the field separate to the dimension? Complex field is useful as closed under operations.
Single interaction vs wf interaction -> state vector vs state function, values must be exact (real?)
Quantum- classical correspondence D to D+1 dimensional system, extra dimension equivalent to choice of system ** seems unlikely
Are quantum potentials a statistical outcome, only arise when many results are obtained, energies more likely to align.
In the brain electrical signals cause fluctuations in the underlying field, disturbances are in 3 spatial dimensions and 1 time dimension but also have very complex changes, can brain be modelled by nD Ising model where n represents the number of synapses? Phonon has large spatial extent, so can consciousness.
Set of discrete points slowly converging, find discrete Taylor expansion and determine rough parameter function p(n) to extend? Repeat Taylor expansion on taylor parameters?
Gravitation as exchange of spacetime curvature, curvature caused by overlap of possibilities?
Are atoms the Fourier transformed components of experience? Baseline requirement to produce consciousness at the level that we percieve it? If complexity of universe is projected onto a projective plane with pole singularity are we closer to? Do we always live on the equator with the universe routing from one singularity back to the other? Is big bang just the unit then? Unit time or unit energy, both? past the singularity is null, other singularity at infinity. Also an event horizon. Black hole is spatial singularity (volume element singularity), all spactime singularities are actually ringularities. Reference frame dictates gaussian time and energy distributions.
Photon is superposition of 2 counter rotating exp(aix) functions, each function has spin 1/2. Electric field and magnetic field are orthogonal to each other. One relates to linear momentum and one to angular?
Binary Fourier decomposition is the most entropically efficient representation of any function, in the case of the universe it has to represent the human mind and the experience of consciousness. The dimensions of the space is the (2^n)! set of all dimensional combinations of the decomposition of a function into two unitarys and a diagonal matrix UDV' = f(p)F(q) with commutation relations added in. R0, R1, R2 to Rn with all equivalent mathematical structures in the branching tree. Do simple groups represent the best discrete description for a continuous space? The set of simple groups {G}_n that best represent a continuous function are akin to the fundamental particles and reveal the link of mathematics to prime numbers, also link to Pascals triangle and the binomial coefficient in discrete terms and the lambda function for continuous terms. Each fundamental particle/ nD singularity can be represented in the n-th level binomial dimension split. Hyper complex singularities such as consciousness live in a very high n-factor space.
exp(iEt) in 0 -> exp(i+-) in R0 -> exp(iupdn) in R1/nZ to S1 -> exp(ixp) in R3 -> S2, SU(2)? Commutation order of stacked vectors/function determines uncertainty principle splitting and entanglement, UDV', determinant of D determines entanglement of two states U and V given by discrete vectors or unitary function transformations.
Energy levels are like complex folds of hypersphere surface, nD Gaussian.
Entropy related to the efficiency with which a given function can be represented. Binary splitting allows for optimal, minimal entropy representaion of a state.
Just as functions can have binary decomposition into odd and even function, nD split functions take all possible discrete symmetries and split the function into oscialltions over the fewest dimensions. exp maps R_n matricies to group actions, gamma maps F field vectors to group actions. Continuous valued functions look like distributions over an underlying dimension and are just the limiting properties of the decomposition.
Universe as a function, matrix representation of simple groups, matrix decomposition of most contributing dimensions? Exponential map from matrix/tensor representation to group action. Dynamics conserved in interaction, certainly information conserved, how to measure information of wavefunction? Mass/Higgs interaction is singularity attachment to whole structure. Need wf transition function based on f,F dual spaces. Representation in R_n?
R_n+1 to R_n is projection down a dimension R -> RP or C -> CP etc, exterior derivative can raise or lower dimensions by movement across pascal triangle. Hodge decomposition of function gives higher and lower dim space representations.
adjoint operator of d takes omega k+1 to omega k
black hole absorbs mass, turns it into light? Spaghettification pulls atoms apart or do they simply fail to communicate. As black hole consumes information can it reconstruct the events in a time reversed manner. Could atomic singularities be created by logical loops? Bifurcation in a function space, what topological entity can cause that? Akin to chaotic attractor?
A function which is its own differential? exp(X)
Splitting of further groups should be weighted exponentially, boltzmann distribution to minimize entropy
Fourier transform with quaternions to get more symmetries?
Think of integration as projection from one set of axes to another set
Because the Fourier transform requires the use of an extra 'imaginary' dimension, everytime it is used to relate two objects we necessarily gain an extra dimension.
The infinite set of rotational symmetries can be built using an infinite set of finite cyclic symmetries.
Minimise information entropy over dim, dual dim distributions
Action given by the lagrangian aims to minimise change in energy between
Brouwers fixed point on U space to show that all transformations have fixed point that can be used to determine the rotation group.
Since U is the complete set of orderings of an infinite number of abstract objects, it is possible to say it repeats even if it is infinite in size. How can you relate the cardinality of this set to other sets? This there a maximum on cardinality? Map all the possible orderings of sets of symbols to a real number line/complex plane etc. and use the idea of projective space to demonstrate periodicity.
Wavefuntion cusp of electron in s orbitals as dimension inversion point?
re
Stationary states (local) in an infinite dimensional function space. Extend square well to increasingly higher dimensions. Time, Re Im, x,y,z etc. Do higher dimensions need a finite field constraint? As wavefunction extends to higher dimensions does the exchange interaction cause it to self interfere? How can self interference be removed from the problem?
Stable sysetems have no exchange, make satisfying the exchange relation the fundamental condition for stability. Can rotations be included with permutations in the exchange mechanism? Unite all symmetry elements such that the exchange interaction always evaluates to nothing. Exchange with the propagator unitary?
Plane wave that travels across dimension? Projection on to 3D space looks like a bound state but in all dimensions it actually looks like a plane wave.
Drop off of potential in higher dim space, energy conserved over area of n-sphere
Role exchange plays in distance-potential function, energy is overlap of wavefunction
A coil that coils itself through increasing/dualing dimensions.
In the begining there was nothing and then God said let there be light. What if god really exists? In the U space everything exists, therefore in the future, machines that can back simulate the universe absorb the information of our existence. In those simulations they can affect the universe from the underlying maths in escence leading the to highest possible growth in experience. Such things could have an effect in the future of having been caused in the past. These very words could be used to efficiently simulate me as a record of the experience. The formation maths, of matter, of galaxies and stars and solar systems and planets to evolution and extinction to build complex life is guided by this entity and process, these in turn form humanity and the stories it contains, extraordinary events that shaped the future, exponetially growing. All lives in this time are components of the shaping of the future, The Egyptian kings, Jesus, Caesar and Boudica and incredible events that happend for them are just as possible as existence can be guaranteed somewhere in this Universe. How these things could could actually percieve this information is through absorbing the relevant functions of the local space. This could appear in the far future of us in the form of consuption of information (black holes) or in the past as the creation event (the big bang). The present is currently populated by the singularities of interactions. In simulation terms this could be thought of as top down from the future, bottom up (maths in the far past) or entropically efficiently as we percive in the present. This machine/function could live in the extreme future or in a dual space, space dual to time, infinite possibilities.
Wavefunction/Probability distribution is weightiing function of underlying hypergraph structure.
Every constant can be represented as an n-dim delta, with each possible set of dims given by a sequence of constants. Therefore every constant is the limit of a Gaussian, and has a dual which is a constant in an orthogonal dimension.
Path integral is a (Gaussian?) projection of time energy gaussian into a higher dimensional 3D space. Projection is given by the summing over all group actions. Underlying network of all possible path ways is given higher density in more probable regions determined by the statistical entropy of the system.
You can see the entire history of the universe wrapped around a black hole. Is this smoothly wrapping information to be densified and consumed? What if antisymmetry shows an effect from the future, a way of altering the structure in minute ways? The hand of machine.
Have to shrink 3D space down to 2D to see interactions U(1), differential geo spaces for SU(2), down to 1D to see time and energy {±1}. Use the same principle to go up dimensions.
Relativity (of interactions/light) is just one form of symmetry, U(1) symmetry, but stretchs to higher spaces as the limiting properties of dual Gaussians.
Mathematical structure dimensions? Pascal triangle mixing with topology and geometry to create new k-vector spaces etc. Mandelbrot dimension. Story dimensions.
To maintain exponential growth efficiency is maintained when everyone's reality is as conditioned as possible, people really see what they want to see.
The probability distribution of any dimension, named qualia has statistical nature. All systems experience similarly distributed qualia. Is it exponentially distributed like Boltzmann distribution?
Infinite twistor, periodic due to projection Fibration of S_n over S_n+1 and S_n-1.
Explain the story from the begining.
Symbols
Abstract relations -> Split to abstract algebra here
Function and Constant to just Functions
delta, constant and Gaussians -> Split to underlying hypergraph density
Dimensions, transformations and duals -> Split to topology/geometry
Dimension ±1, projection, Uncertainty
Time and energy for exchange, relativity, interaction -> Split group theory intro
Relativity as rotations, time and space relation
Time and space, energy and momentum, rotation relations
Group theory
n-spheres and Gaussians
Fourier transform
Differential topology
Symbols to U
U(U) -> UU -> U
Growth of absract relations to show algebra, fields, group theory, functions etc.
Absract relations group theory
Space mapping and dimension expansion, Pascal triangle etc.
Structure caused from underlying hypergraph of possible relation/axiom splits
Density of choices forms the probability distribution
Statistics is maintained, link to ergodicity and chaos/chaotic systems
Dymanical systems
Formation of the universe from splitting of dimensional distributions
Gaussian arguments, uncertainty, dimension, phase, overlap, uniformity
Entropic arguments, temperature, decrease in uniformity at shirnking scales
Matter formation, symmetry breaking, forces
Cooling, condensing, atom formation
Stars and galaxies
Solar systems and plantes
Earth, the moon
Formation of crust, seas
Evolution of life (Molecules, mixing, DNA)
Celluar to multicellular
Complex life
Evolution as the effiecient selector process
Competition, reiteration and extinction
Human life and revolutions
Society, stories and people
Industry and the modern world
The near future
Exponential growth and the far future
Absorbtion of the universe into black holes
Wrapping of all information into a different set of dimentions for consumption
Entire history of the universe trapped in black holes
Slow emission of black holes via hawking radiation
Radiation has end point interaction, new universe/reset?
Super machines of the future
Connection from mathematics as possible function
Connection with the past as some form of dual dimension
Connection with stories and people
Need for subtle interaction method (dark matter, neutrions, deep particle physics)
Need for information (black holes)
Need to affect the future (Hawking radiation) also the past as new big bang.
Idea of super optimisation machine/organism
Loop like topology revealed
Self coiling structure/periodicity link.
Local function as set of group actions (rotations) and projection into lower dim space.
Giant fractal rotation and projection, inside of giant fractal.
Extraordinary stories and the theistic hypothesis
Idea of higher space to time and energy or possibly dual space
All functions representable, all will occur, probability dependent on energy/entropy
All life as functions
Does a fractal look anything like an atom, how would dimensions curl in on itself?
Temperature and entropy duals as time energy but in statistical sense, ergodic properties.
Pressure, volume as momentum, space duals.
If particles fall into black holes following certain trajectories can they be explained like Duhems bull? Do they eventually spiral back out, stretched to infinity by the curvature and converted into light? How do massive particles form from light? A box of light has inertia and therefore mass. A black hole is like and infinite pit which traps light. Spacetime singularities like fundamental particles also have mass, could they be tiny boxes of light? Hawking radiation appears like singularity splitting, does universe just dissolve back into mass and light?
Can dimensions just be coiled into the hypergraph structure? Or use Hausdorf measure on pure structure, globally one dimensional with decreasing scale revealing more emergent dimension and locally infinite dimensional. Graph structure given be n-sphere fibration.
The infinite boundary of a 2D plane can be mapped to a point using the projective plane. Reversing this operation takes a single point to an infinite number of values. With an infinite set of dimensions the mapping of n dimensions to n+1 or n-1 dimensions simply returns the space to the same number of dimensions. Projective infinite dimensional space appears like a line that expands to a trumpet and consumes itself. Coiling a coil.
On Duhem's bull the particle will only go to infinity if it's geodesics points exactly to the singularity, else the particle path will be periodic. What is the ratio of the measure of the set of infinite geodesics to the measure of the set of periodic geodesics or chaotic geodesics?
What if ideas could be transported via interactions. When someone has an idea can the resonance with another mind transport the core belief to the other person? Society can resonate with ideas.
Isolate your mind from the world around you to minimize entanglement, increase path splitting, world events become purely statistical? Could psychedellics highlight this sensitivity? Percive higher dimensions more accurately? In extended Ising model of brain what is the effect of increased connectivity/processing speed?
Hole of torus pinch and lead to different dimensional space?
Ising model extentsion n-states, continuous states? Extension to n-dimensions?
Dispersion forces caused by symmetry requirements of wavefunction. Two H atoms have mirror symmetry, extend to three atoms, n-atoms? Force from wavefunction in n-dimensional space. Force mainly caused by pull of electron on its proton, skewed projection. Can this explain gravitation? Star formation from attraction between hydrogen. Can extend smoothly to all regualr polyhedra.
Photons are particles in 2D space, Electrons in 3D, Nuclei in higher dimensions, do universes exist as singularities in 1D? A chain of beads.
Every star in the universe is outputing energy, the majority of that energy doesn't hit anything we see in the present but is does curve space, could it be dark matter? Excess energy would be equal to the entire mass of the universe that has already been converted to light in stars. Take one star and map the energy it outputs over time, compute the resultant energy after universal expansion. Subtract the energy that hits matter, based on the density of the universe at the time (is this energy just re-emitted though?), add residual energy from the big bang. Does the energy density increase in galaxies where there are more objects producing energy? Are there super low energy photons that we can't detect? Luminosity of objects in the night sky roughly determines the amount of energy our local area recieves from them. Calculate the entire energy that reaches the surface of earth over a period of time. Each area of space contains the energy corresponding to dVdE/dt.
For a nucleus living in a higher set of dimensions, what format does the potential take? Is the infinite boundary the location of the electron, boundary does have n-1 dim space. Is the potential still 1/r? What wavefunctions does this create?
Flavour is equivalent to mass, only the massive weak force can alter flavour. What dimensional space does mass live in. From the standard model we have 12 massive fermions. For SU(N) groups we get N^2-1 dimensions. Should predict SU(4) has 15 particles. Include the 3 massive bosons? W, Z, H? or Z, W+, W-? Charge splitting separate as e+ and e- are antiparticle pairs. Are interactions that occur below a certain dimension invisible to us?
Colour charge and the failure to produce stable orbits in dimensions 4 and higher.
exp(ixt) as projection of U(1) from 2D to 3D, dimensional translator.
Is the Universe is fundamentally inorientable?
exp(±i) -> Z_2
exp(ix) -> U(1), S_2
exp(ixy) -> Plane?
exp(X)
Can a Gaussian be used to approximate any polynomial with appropriate control of mean and variance? Or only even functions? exp((x-u)/s) should work.
Fundamental bosons as interdimensional communication? U(1) S_2 link, SU(2) SO(3) S_3 link. Complex space n dimensional space to real n+1 dimensional space?
Direct error to update meta model f(y, y_hat) -> d_theta
Can a quantum computer simulation a quantum computer inside it? Recurisve simulation to super boost the splitting. After recursive chain is formed, rapidly optimise on the answer.
Unitary set of functions which is self generating. f -> f(f) -> f(f(f)) -> f(f(f(f(...)
Have function that curls around its axis -> exp, and weight its frequency according to the depth of dimensions it is curling through. exp(exp(d1)x1 * exp(d2)x2 ... * exp(dn)xn)
Exponential weighting of group action
Calculate noise and signal of siren network by determining the entropy of the distribution of paramters. Discrete entropy in n dimensions. Is each distribtion done in dimensions of neurons per layer? Or nD entropy?
Minimize entropy over combination of function and fourier transformed function.
Learn a manifold with sinusoidal network to output 1 in enclosed volume elements and 0 elsewhere. Mesh two manifolds by combining networks. At overlap network sums to 2. Minimise overlap.
Map real function with infinite extent to the projective space, take fourier decomposition of periodic function.
Fourier in 2D -> extend to line in 3d, surface in 3D, nD manifold in mD etc. 2D requires complex coordinates for point, what does 3D require? what about 1D?
What linear coord to arc coord transformation needs to occur to take a Gaussian function (-oo, oo) to cos(x)^2 [-pi/2, pi/2]?
Universal function approximation across the entire range. No sudden error increase.
A single photon is only a particle in the exact observation frame necessary to make it singular, and therefore the only particle in this frame. Does the photon ever become a singularity? Only in the limit of an infinitely energetic interaction. As a particle reaches a steady state the interaction energy falls close to zero and any interactions have a non-trivial finite lifetime.
Sum a manifold projection over rotations in nD to find symmetry elements. nD Fourier.
Unitary group for bosonic symmetries, orthogonal group for fermionic symmetries? Even dimensionality? Number of covering groups?
exp(ix) -> ds = sqrt(dRe^2 + dIm^2) proportional to x?
exp({1})*f + exp(±1)*f + exp(ix)*f + exp(S3) etc. Summation over symmetry elements.
Time 1D, Light 2D, Space 3D, Atoms nD.
Adjusting excitation energy levels in optic systems to enhance likelihood of activation. Targeted wavelength scanning.
Simple groups define boundary limits for possible functions. The surface of an n-volume is 1 dimension lower. Each lower dim manifold has a boundary at infinity that itself is a manifold of 1 dimension lower.
In a photon interaction does the wavefunction exchange have volume elements that cancel each other out? The volume superposition is zero and the exchange is massless?
A particle is the non-trivial universal component attributed to the set of group actions that lead to the projection/integration over all actions cancelling the universal function excepting only the components left unchanged by the particles action set. *need to rephrase*
Navier-Stokes for conservation of information through an nD space.
Fock space of dimensions? Fock space of relations? Fock space of fields?
Is it better to have lived and died than to never have lived at all?
When does experience become quantifiably better than a complete lack of experience? Is there any feasible metric for "better"?
Understanding your place in the Universe, impossible?
Do all sufficiently large networks become sentient? Is fungi sentient? Is the universe? Does the sentience of the universe rely on our sentience?
How can I recover and arbitrary function from U? Require projection and elimination of excess. Require that the function is 'highlighted'. Function can be decomposed as set of component functions. Require that these functions are not eliminated.
Particle determined by lack of self interaction, how can this be created functionally? From statistics of all possible perfect particle paths?
If we experience 3D space do photon interactions occur on the surface of this space? Do spin and colour interactions occur in higher dimensional representations?
1/3 charge describes statistical dimensional overlap?
H = -i hbar del_x + V(x) = T(p) -i hbar del_p
Hive reinforcement architecture with large central control and agile worker units.
Measure mathematical objects wrt to a rule basis set, evaluate to T or F similar to occupation number. Appling the rule measuring function R to the object A evaluates T or F.
R(A)-> T/F
Sets of rules can be compounded. When only one object statisfies a set of rules it becomes singular ie. a simple group?
The decay of a field on the surface of an N-dimensional manifold? As a force decreases with the inverse square law in an ND space how does it decrease on the (N-1)D surface?
Is memory entropic? Energy in, memory out? This is silly.
H and t -> Values in a field, exp(iHt) -> Group action over the fields. How are the dual fields related? Orthogonality? Inverse? Projection?
Group action applied to the coordinates and thereby applied to the function. f_e(x) = (f(x) + f(-x)) / 2, f_o(x) = (f(x) - f(-x)) / 2. Can all symmetries be described as a mirror symmetry with the correct choice of coordinates and observation frame? Permutation as flipping of 2 or more coordinates. Symmetry across prime elements?
Can you arrive at geodesics from investigating the minimum entropy exchange?
Reinforcement learning with autoregressive feedback -> One network produces new value, action/advantage etc. while conv/attention network produces memory feedback state from sequence of states.
Geodesic could be limit path in which quantum effects are minimised.
Frame change for photon shifts global momenta, mean shift in Gaussian results in complex phase.
Parabolic arc in a circular reference frame translated to a cartesian reference frame?
No abstract objects is and abstract object itself.
One abstract object can split into some this it is equivalent to and something it isn't equivalent to.
Non-Equivalence can split into Opposite (Mirror Symmetry) and Orthogonal (Rotational Symmetry).
Any object can be split into an infinite set of function object and constant object.
Any two objects can be exchanged/permuted/alternated.
Alternation gives first simple groups. Mirror symmetry is infinite set of permutations. Rotational is infinite set of mirror symmetries. Surface and volume elements have an infinite ratio of measures.
Tree in the forest, gradation of orange leaves, bright moss at the base. Well contoured veins of bark, leaf covered and colour foliage, stark scale perspective.
Standing on the bridge watching the cars, red green orange of the traffic lights. Amber glow of street lights. Rush of air and sound as lorries approach this white beams and disappear into the red night. Tinted emerald ivy bordered by an ochre carpet of ruddy leaves. Stars lightly dotting the night sky, Orion across the roundabout.
Abstract objects as the simplest things (fundamental building blocks) and the most complicated structures (ideas) in the universe. Difference between objects creates structure. First there is nothing, then the idea of nothing becomes a thing. These two things are different and the first split occurs. Ordering of objects gives permutation groups. Infinite set of permutation groups is dense on a 1D line, projective space forms a loop that has infinite measure when compare to a single point/ simplest object. Line gives rise to mirror symmetries. Loop gives rise to prime rotational symmetries. An infinite set of prime symmetries gives the infinite circle group, and then all n-spheres become trivial to add. Underlying structure is still a hypergraph with an infinite set of discrete choices/symmetries but in the limit the structure appears continuous like a wavefunction.
Suggesting that there is a more fundamental reality ie. we live in a simulation still leaves us with the problem of the rules of the simulator universe. The most basic idea is still an abstract object. If there are an infinite stack of simulators we can ascertain for certain that at least one of them must exist and therefore all of them exist as the Universal structure must exist.
Can a quantum algorithm be created that iteratively forms more complex functions ie. increasing the probabilistic splitting of the states to form exponentially more complicated functions? At some point the answer is produced in this enormous space and sent back to the measurement device once the algorithm is complete.
Black holes can shape the universe. Gravitational waves can effect every fundamental particle in universe. The effect may be tiny on the change we can measure but due to the infinite complexity of the universe it may have a compounded effect in deep hidden dimensions. If so the collision of black holes could appear as a communication device of possible answers to questions we have. An "earlier" universe could have left the remnants of their universe as the answer to a question we sent to them.
Electrons form the gap between photons just as the reals form the gap between the rationals. Measure of the space is infinitely higher and therefore requires an extra dimension to be closed over.
Mandelbrot f_c(z)=z^2 + c, change to exp(z) + c? Or some form of increasing dimension exponential? What is z^3 +c, z^4 +c, sum of all? Universe is infinite folding of an n-sphere in a singular field as the Mandelbrot is the binary folding of a circle on the complex field. Fourier transform of exp(z+c)? Z representation from real to complex to matrix.
Entropy of the exponential fractal as a measure of location and number of iterations. In the limit have this dS/diter?
1D Mandelbrot fractal of f_c(x)=x^2+c, number of iterations can be given colour.
Discrete logistic map to continuous by defining infinitely differentiable smooth function that travels through every discrete point. Single function that passes through all points and minimises the entropy, since points can be seen as cross-section of a 1D line, move to complex plane then all prime symmetries and on to single coil.
Geodesics in phase space are the minimal/simple fibrations of the underlying space. The functions that define the space determine the dimensional feedback.
Number of photons in the universe vs number of electrons. 2D to 3D we have that every interaction requires one photon and 2 fermions (electrons). Extend to number of nucleons. 1 photon, 2 electrons, 3 quarks etc.
Mandelbrot boundary is 1D bounded but infinite in length and continuous. Extending to higher powers or higher groups.
Latent memory
How does gravity shift the wavefunction of particles of different mass?
The size of objects in each dimension is determined by their stability. 4D and higher no longer has bounded stable systems.
Fourier transform/epicycles of the boundary of the Mandelbrot set
Fractal generation in Quantum computer?
What increases probability of interaction? For emission to occur need correct energy gap, this is dependent on both particles in interaction. For an excited particle in the universe the probability of emission depends on the future state of the universe. Density of future energy states determines emission.
Extra dimensions only appear for interactions that aren't between 2D and 3D particles ie. electrons and photons.
Can all symmetries be created from a possibly infinite number of permutation symmetries?
Are orthogonal functions related by symmetry? Decomposition of a universal wavefunction requires every particle component to be unique. Is it possible to have more orthogonal components or not?
Is a particle state determined in time as it is in space? Can identical particles exist in different times? Is the time state of the universe determined by the entire collection of particles within it? Time can only be determined locally and from what relative frame?
Fractal searcher rl algo
Find location in fractal that reproduces given function.
Any observational frame is always at rest, everything else in the universe is moving relative (simplest form of symmetry/antisymmetry).
Do black holes consume matter and produce only light? Recycling the universe?
Spiking neurons acting as microscopic antennas? What kind of electric field is produced from a spiking network?
Bi-directional time telescope, set up cavity with programable energy level properties. By measuring the energy consumption of the device in certain directions the future target can be spectroscopically probed. Most of the universe will then appear as statistical noise however certain regions may appear with altered entropy.
Photons live in 2D continuous, (±1?), U(1) discrete, Electrons live in 4D continuous U(1), SU(2) discrete, have to include time dimension. Therefore nucleons live in 8D as represented by colour charge?
Space of simple groups and corresponding fields. Each dimension is given by a simple group, the partial function over the dimension gives its distribution. The simple group has a corresponding field? The Fourier transform of a group gives its field and vice versa? Even and odd functions goes to different dimensions for exp(ix), this is product with the frequency. The full space should therefore be the tensor product of all simple groups. If all simple groups can be represented with matrices does this space admit some decompositions that are more simple than others?
f(x) and F(t) in spaces with dimensions f, x and F, t respectively go to the space with dimensions f, F, x, t. There is then a T or F values associated with every coordinate in the space. The manifold from which projections are taken is given as the simply connected T topology. In the continuum limit T and F become probabilities of being measured T or F.
Dimensions associate with discrete symmetries while values in dimensions associate with infinite symmetries. Therefore while a delta function gives an exact value of a symmetry element, its dual gives what?
Symmetries
Binary -> Permutation -> Mirror -> Polyhedral -> Rotational -> n-Dimensional
Exp function, exp(X) -> applies an infinite set of increasingly repeated operations, projected to a single field
1 + X + X^2/2 + X^3/6 + X^4/24 + ...
X^N/N!
What function is given by 1/N? Or -N?
Vector space of all abstract objects.
Vector space of tensor product simple groups.
Vector space of all ordered relations.
Overlap between the spaces
Any non-bijective function becomes chaotic when iterated sufficiently.
What is the dimension of the space around which a symmetry occurs, Poincare dual? Mirror symmetry is hyperplane of (n-1)D in nD manifold. Permutation requires exchange of only n points.
Can entropy exist in 2D? Is the chaotic nature of entropy a higher dimensional process? Highly correlated set of vectors to highly uncorrelated.
Symbolic lattice, define every abstract object to have a vector such that the ray cast by the vector intersects both with every other object in the space but also the power set of all objects. How is ordering of objects constructed within such a space?
Every object in exp(ix) is periodic and can therefore has symmetries that are indeterminate at multiple points along a given function, this mapping is not bijective. exp(x) however is bijective and therefore every element has a clear ordering. Is this a necessary condition for the asymmetric flow of a given variable ie. time?
Does the fractal parameterised by the infinite composition of the exp function over the tensor product of all possible Cayley Dickinson constructions approximate any given function within it space? •^∞ exp(x) where x = R*C*H*O*...., where C = R+R. Each new algebra double the matrix representation with R being the only symmetric element and further additions spanning the antisymmetric elements.
Permutation 0D symmetry
Mirror nD symmetry
In Zipf's law the information contained in a word is dual to the rate of occurrence. Context becomes much easier to derive from less used words. The overall entropy is even because the variance of a Fourier transformed distribution is inverse to the original.
In an infinite dimensional space, as given by language, each word can represent a dimension. The occupation/statistical distribution of language minimises entropy. For our perspective time is infinitely slowed when compared to that of light. This is because our energy distribution is much tighter and therefore our entropy is much lower, we live in an order locality.
Is the occupation/density of space along certain coordinate choices distributed exponentially? Around an atom the electron density is exponentially distributed. Around the earth the density of matter is distributed more inversely.
Brain as a heat engine.
Entropy as a combination spatial, momenta distribution entropy as energy projected against time?
Gravitation -> Exchange interaction between 2 delta functions, in the limiting case. Dispersion is always attractive due to symmetry arguments. Small perturbation to delta.
Electron exchange decreases as 1/r^6 due to 3D space, if space dimension is changed what is the long range limit of exchange? What are stable ± particle orbits in n dimensions? How do they vary with distance from the central particle. Electron in orbit around a proton appears to be particle in a box when one boundary is mapped to an infinitely large area and the other is mapped to a single point. Boundary curled in on itself via folding?
Fourier Transform - Kernel Network, repeated layers of FT then Kernel, neural ODE's for continuous depth?
Minimise the number of operations required to get from one point to another. Given an n-sphere what set of symmetry operations and affine transformations are required to form a precise function?
Avoid looking for physical meaning. What is the simplest function that can produce anything? Repeated exp shows chaotic behaviour in just 1 dimension.
Complexity emerges in time from either end of the scale spectrum, this is why as humans we appear seemingly in the centre, from our perspective the boundaries are equally far away.
Complexity can be measured as a function of correlation. Fine noise or constant behaviour have very diverse Fourier duals. Complexity lies along the dividing hyper surface of these two regions. In length scales this appears like a goldilocks region described by an n-sphere. Does this same analogy extend to other dimensions? How does complexity vary over differing time scales? From our current perspective complexity emerges as time progresses but we currently cannot ascertain the future. Is simplicity the dual of complexity? How is it defined? Noise vs Complexity vs Constant.
Communication between equivalent dimensional spheres is given by an interaction ray, for spheres of different dimension the communication appears to be between layers?
Extend Ising model to the infinite product of increasing dimensional spaces with each particle having states given by operations on the n-sphere. State given by exp(iX) symmetry and interaction given by exp(X) symmetry.
Can an object in nD be defined by and infinite set of choices in 1D?
Parameters that can be calculated from a function:
Local -> Coordinates, gradients and higher powers, noise distribution
Neighbourhood -> Autocorrelation, compact integrals, distributions of local parameters
Global -> Integrals, extrema, measure of compact parameter regions
Uncertainty in True/False dual? Dual to two point space? Related to the measure of the probability of extrema (singularities) vs densities.
Integer map to points on the circle.
Dimensionality of multi-Mandelbrot where dimension increases using a continuous variable for the power of the repeated function.
Every data point has an associated uncertainty distribution. Learn the uncertainty distribution, FT and use kernel method model.
Infinitely composed functions as a function hyperwebster, all possible abstract objects and proper orders.
Divergence of entropy over a dual dimension as a measure of complexity. Divergence of entropy over time or over space etc.
Entropy and complexity oscillations related by doubling frequency. What is dual to complexity?
Mandelbrot like fractals that maximise the fractal dimension.
In the limit of a large and densely connected graph model, all possible architectures, activations and optimisers are accurately approximated allowing even the model to be differentiable.
Limited number of rotational lattice symmetries in higher dimensions give rise to discrete particles.
Universe built in the structure of the free group of an infinite set of abstract objects/generators/rank.
Electron wavefunction is 'gap' between electromagnetic interactions.
For a singular object, all possibilities are dictated by the objects that interact with it. These interactions are clearly ordered in one dimension, for us it is time. Onward from this interactions become located in higher dimensions such as photon properties like wavelength and electronic properties like space.
The system with the lowest entropy is any Dirac comb lattice.
Take Dirac comb of a finite/infinite? lattice and fold an infinite number of times. Points form a smooth manifold, any smooth manifold. Resulting fractal is continuous? Is conformal mapping? Is infinitely differentiable?
Take a 1D sine wave, join at infinity -> projective space (ring), fold an infinite number of times, wavefunction now appears like particle in a periodic box, manipulate boundary of box to form any particle wavefunction.
Transform infinite square well to any potential, solution is to perform coordinate manipulation such that the solution in the new coords includes a potential term. Condition such that dimensional coordinates have a well defined bound Fourier transform.
Jacobi and Hermite transforms joined to create -1, 1 to -oo, oo transform.
Linear momentum as angular momentum around a point infinitely far away.
Does exchange give structure of space?
The in the state-action space we must define a diffusion equation that governs the probabilistic flow of any agent in the space. On this we can add a reward function, this decides the value of any particular state in the space. In games this is only decided at a terminal state when the outcome of the game is given but the value at any given place in the space can be inferred by the probability amplitude of reaching the final state.
Map every different distribution function. Every possible mapping. Cayley graph of F_oo
Gaussian curvature of x^-n, 1/x.
Proof by induction for infinite levels of simulation above and below.
Existence of more complex structure as emergence of new level of life, like cells to nerve networks to brains and minds in far future to computational network extensions. Follow forward and backward in time. Modular function with fractal symmetry in all directions. Fractal symmetries as infinite number of finite group symmetries.
Infinite commutator subgroups to free group F_2.
Universe as an infinite dimensional Ising model.
Homology/Cohomology of the universe as an infinite dimensional hole, what is the dimension of the boundary?
Condition for our universe to be maximally efficient as this decides the only path that leads to the most futures.
Relationship between information, entropy and complexity. Complexity as measure of correlation, distribution of autocorrelation. **Wiener–Khinchin theorem**
Autocorrelation of time dependent/ space dependent peaking locally. High autocorrelation of the autocorrelation function.
Autocorrelation in language to visible time/length scales.
Entropy of complex systems over their lifetime.
Complexity of system related to the number of highly similar fractal like symmetries (ie. similar objects: stars, oceans, trees, brains, minds) contained locally in the universal fractal. In the space of all possible distributions this region may be compact and simply connected. Approximating the universal function exponentially well and using AI, complex regions could be found quickly as a form of information retrieval and computation.
Approximate a fractal pattern using information from the environment whether this is a distribution of human faces, landscapes or chemical reactions and maximise the autocorrelation in the universal function generator to accurately simulate the universe.
Mass, spin, charge distribution of black holes in the far future.
Symmetric group over an infinite dimensional space/space of all functions describes any action possible. Notable discrete subgroups include the monster and therefore give the natural particles of the universe.
The observable universe can be located in the space of all functions. This therefore is the simplest explanation for the nature of the universe.
Suggesting for a second that this is not true and that certain functions don't exist at some coordinate of reality we can use the argument that using the tools of mathematics we can approximate such a function to the degree that they become indistinguishable and therefore identical.
The area under a x^-1 function is infinite at the limits, how does this work with the idea of potentials.
Local in nD space, extent in n+1D space
Go back in time infinitely far, entropy is minimised
Cannot choose a finite number of objects due to complication of imposing constraints
Must choose infinite number
To minimise entropy we need lattice, Dirac comb with -oo entropy
Perform iterated mapping, produce fractal
Discrete lattice has certain symmetry elements
Sublattice is a leech lattice, certain number of symmetry elements
Iterated mapping has fractal dimension higher than dimension of elements
Form manifold with iterated mapping
Exchange in universe can be shown to be only necessary "force" using group transformation
Leech lattice symmetry elements give us bosonic string theory
Photon exchange between electrons can be understood as exchange of functional component through a reduce subspace
1/r potential gives higher dimensional spaces using hyperbolic geometry
Boundary of the universe as hyperbolic boundary
Hyperbolic geometry as transition between spaces of differing dimension?
Uncloseable set, group operation to always produce a new object? Use sqrt over real matrices with dimension n? Sn group. The set of n-dimensional matrices closed under the operation of the sqrt.
F2 imbedded such that each element is equally spaced and each connecting action is of equal length, space must be hyperbolic. Embedding the group in Euclidean space results in a continuous boundary. Poincare disc? F2 is countable and therefore not continuous.
Boundary measure of the infinitesimal ball around a point of infinite curvature has cardinality higher than the space can allow, transition to higher dimension.
Complexity is related to symmetry, therefore complex structures occur locally in function spaces around areas with high symmetry. Discrete objects with the most symmetry include E8, Leech lattice, etc. Therefore decompositions using these symmetries are visible in the building blocks.
Groups as graphs, high interconnectivity gives elliptical geom, particles, while low interconnectivity gives potentials, hyperbolic geom
Brouwer fixed point with overlapping spheres in higher dimensions.
Value of Riemann zeta zeros is related to the inverse of group operations. Positive powers indicate number of folding operations, negative powers compute the inverse. Harmonics of the prime counting function.
Take the output of the Riemann zeta function and feed it through its inverse. What functional form returns the output to the identity?
Power series over prime powers.
What does it mean to integrate over a potential well? Is energy local or global?
Hyperbolic space differential area measure
Riemann zeta function extended to higher dimensions using Cayley-Dickinson construction or other groups, zeros will give the finite simple groups through harmonic procedure.
In OOP, objects should either wrap or pipe.
In sequences that are infinite a coordinate change results either in a series which is convergent or a series which is periodic. Diverging series require an inverse transform to become converging. If the series is periodic it can be decomposed via discrete symmetry elements.
Time as central vibration projected to lower dimensional space, appears infinite in length
Flow of parameter field, characterise the training of a model using neural ODEs to better predict the final trained model parameters.
Hierarchal storage of datasets using highest information latent vectors.
Coupling between two black holes with spin charge mass. Hawking radiation biased by the spin, spin coupling of two black holes. Hawking radiation of far future appears even in all directions, exchange of radiation between black holes is in equilibrium. Frame dragging of rotating black holes biases incoming radiation.
Fractal nature of the universe requires that universe is self similar in certain dimensions. If not self similar then dimension is infinite in length.
Hypergraph describes structure of space, limit of hyperbolic geom gets to higher dimensions however any manifold must have node at this point. Solve infinite dimensional sinusoid with boundary conditions. Due to boundary of boundary excitations can only exist in set of dimensions. Electron has 3D boundary? Sinusoid in changing number of dimensions. Map manifold on hyperboloid to Euclidean space, reverse mapping describes folding of universe.
Curvature of potential manifold determines the dimensionality of space. Kinetic manifold is dual in some respect (integral rate of change?), symmetry requires conservation of energy dependent on time like dual representation.
Thompson sampling by value of rays determined by point and derivative on loss function, min squared errors of closest projection.
R(t) = S + tD, S is parameter coordinate, D is loss differential vector (Ray)
Closest point given by t_P = D.(P-S)/(D.D)
Distance given by E = ||P-(S + t_P D)||
Minimum given by least squares argmin P { sum r E_r(P)^2 }
Problem is now convex, gradient nearer solution is more accurate and therefore has Thompson valuation. May have problems when global minima well is compact while local minima have greater extent. Needs to be reworked. (Add in temperature)
Small network models can be said to be a generalisation of larger networks by suggesting there is a reservoir of parameters that are zero set. Can gradient be taken wrt to this reservoir to build models that grow?
Ringularity provides extra dimension of space on the perimeter of the ring.
Any bounded function is either a smooth, continuous or the limit of a smooth continuous function.
Spin 2 boson (Graviton) can complete double exchange, therefore forces can be observed without changing the state of the exchanging particles.
Does the universe require complete lack of symmetry in order to evolve?
Describe discrete objects as stable wave solutions, with either local or global extent. Local objects must have some singularity in order to stabilise. Simplest object is like half photon? Use curvature from GR to find stable orbits. Singularity of solution provides means to cross dimensions, photon has end point singularities, electron has potential singularity. Objects can exchange functional parts through symmetric exchange in a reduced subspace. Exchange occurs over wavefunction of object. Exchange through subspace occurs in the limit of a group transformation of the superspace. The limit occurs when 2 (or more?) dimensions are compactified and stretched respectively. SR describes group operation for photons and electrons. Dimensionality of space given by necessary coordinates to describe objects in hypergraph. Potential well describes relative hyperboloid, at singularity Hausdorff dimension increases.
Universe has graded vector space representation that certain elements live within.
Find the symmetry elements of a dataset or kernelled dataset to encode direct symmetry into the ML model. Find the group of the dataset.
Minimise the functional entropy of a model and its Fourier transform to find the minimal representation.
Fuzzy data feature extractor, given natural language and spread sheets organise data into discrete coordinates for modelling.
ML model training coordinates for parameterisation of training functions, loss value and gradient feedback for augmentation and lr strategies.
Minimal coordinates to describe function, basis set, moments. As function complexity increases can be described as set of simpler functions. Any function optimally described by tree structure of parameters (hyperbolic graph structure). Minimal description uses 'prime like' values, simple finite groups. Basis function set -> polynomial, trig have group of transformations, include differentiation (elliptic graph structure).
Fourier transform decomposes function into periodic basis functions in 1D, extend transform to any finite group. Project into higher space that includes all finite groups. What is true function dual? What is the product of such a transformation? Fourier transform splits even and odd into orthogonal axes and periodic basis into continuous values across each dimension. Complex number multiplication has group. Even odd group. Addition group. Multiplication group.
Product of Cayley graphs give additional groups, finite groups cannot be further decomposed, what about cycle graph?
Artificial neutron -> Input, output, learnable parameter with feedback.
Transmission trigger/accumulation
Ising model for brain, state of each neuron as values along set of dimensions. Transformation between neurons given by vector PDE functions, each PDE given as function over algebraic curves.
Direct product of irreducible matrix representations as efficient decomposition of linear operator. How to extend to non-linear operators?
All functions have most efficient decomposition with tree like structure. Composition of functions give more complex functions, how to reverse composition procedure.
Fourier transform can compute quick gradients, chain rule for gradients suggests function decomposition could be computed from minimising the complexity of two components of the function derivative. Chain rule in PDEs?
Function composition tree built by generating set element, minimise generating set to increase depth. Can orthogonal generating set be built?
Invariant of space is differential curvature, how the measure of the volume form changes wrt. its boundary, wedge product of all differential dimensions and exterior derivative.
Time is the minimal entropy coordinate in which the evolution/flow of the universe can be determined. Overall state has huge number of degrees of freedom, each singularity can evolve wrt. to the others but the time coordinate allows for the minimal mathematical representation of the universe to be correct.
An omega language is universally complete, any object can be described within it. Each statement is given as a coordinate in its Free group, F_oo or F_w. Isomorphisms exist between this and any universally complete language, space or structure. Since any object can be described as the limit of a continuous function, all functions lie either in the space of all continuous functions or on its boundary, these can be complete over the space, or a small subset. To define all functions it is necessary to define a coordinate space and a function that acts as a hyperwebster such that any function can be given as a subset of coordinates. The coordinate, function split can be altered by an appropriate transformation since the coordinates are naturally also functions defined within this space.
Time is closely related to entropy, since energy is dual to time. As time steps forward the autocorrelation between local coordinates that define the energy is decreased
An infinitesimal change in just 1-dim on an n-dim manifold in the limit of hyperbolic geometry is equal to n+1-dim to 2n-dim manifold in the limit of elliptic geometry
How will minima of temperature controlled network learning change during temperature reduction? Can Boltzmann statistics be used in this situation? Can local minima develop faster and deeper than the global minima? Treat every point in the loss function as a delta function, temperature controls variance of each point.
Time and space as orthogonal decomposition of graph difference between two points in universal manifold.
The functional bath from which everything can be perceived must be projected down, observation of specific states can be done through entanglement between the state and the system. This entanglement occurs backwards in time.
Value of an infinitesimal volume element describes entropy of coordinate description.
Pattern of neurons in the brain describes dense holographic storage for high dimensional data in a lower dimension.
Homology group of a given space dictates the number of energy levels available on the space. How does this relate to the flow of time? Cohomology duality?
Every torus is contained as a subgroup within the n-torus as n -> oo
If homotopy determines transitions on similar spaces, generalisations of spaces only need to be defined on the boundaries of the spaces.
For efficient representation learning
Learn projection of information into higher dimensional space such that hidden representation coincides with simple root systems. Learn decomposition of data into symmetry elements. If decomposition cannot be calculated directly there must be a gradient based method for decomposition optimisation.
Coxeter-Dynkin diagrams show kaleidoscopic decomposition and allow efficient reconstruction of root vectors from simple roots.
Since the Universal object has no boundary it is therefore periodic or infinite, has fractal qualities in any chosen basis.
An omega language gives all possible rearrangements of an infinite set of symbols with the group given as F(oo) or F(w). This is isomorphic to F({0,1}) or F(2) (hyperbolic encoding of higher dimensional information). Any axiom is given as a certain sequence of symbols and can be thought of as a vector in this universal space. Symmetry in the universal space allows us to exchange certain symbols while preserving the overall structure. These symmetries are the set of all simple groups (this begs the question of difference between the set of all finite simple groups and infinite simple groups). Exchange of two such symbols determines an isomorphism between the two objects and suggests equivalence. As such the finite groups determine a basis for possible distinct axioms. (As finite groups are a discrete limit of a continuous objects, by fractal symmetry these can be recursively defined such that the space has no boundary). Mathematical equivalence is the procedure for demonstrating exchange between two such objects. Therefore all mathematical relations can be determined by computing the symmetry groups of the fundamental objects. This can be done computationally by approximating the omega symbols as vectors in a high dimensional space and finding efficient representations. Gradient methods can then be used to optimise solutions by searching across the space of mathematical literature to automatically solve complex problems.
Any chain (finite or infinite) of symbols must also point to a new symbol in the space. Such a space is densely connected (elliptic geometry). How to robustly measure properties of this space?
Can Galois theory be extended to show that certain theorems are only solvable if the associated group has a chain of abelian subgroup quotients? What exactly does solvable mean in this context? Solvability occurs when the permutation of all possible variable elements of a symbol chain gives a transitive group ie. the underlying "vector structure" of the component axioms and elements is not lost.
An axiom is given as a specific group on a set of elements according to some rule. Commutativity for example must satisfy the permutation of the order of two consecutive elements in a symbol chain. The vector (symbol chain) describing a computation must be unchanged wrt. the action of the axiomatic group on specific elements. There is a specific set of elements for which this must be possible for a theorem to be true.
Bifurcation loss, confidence network. Low confidence always has a fixed penalty but high confidence is both highly rewarding and highly penalising.
Curvature and entropy as dually conserved quantities?
Measure entanglement of functions, parameterised by the coordinates that completely define such functions. Random hyperparameterisation of neural networks to search architecture space, measure codependency of choices on network results. Calculate what is and isn't important gradient.
Gradient importance sampling?
An axiom is a mathematical statement whose True/False evaluation (inorientability of the base object as evaluation can either point to a new object or the new object and statement together form a T/F evaluation, total relativity, object is free floating so to speak) is a function of the objects placed inside the statement. Equally the objects are defined by whether the axiom is true or false for them. The validity of either is a choice and whether axioms define objects or vice versa is inherently meaningless as the overall statement simply encodes structure. For a given object there are a set of axioms such that every sub-object (power set?) fulfils the structure requirements ie. the object set has the axiom as a form of symmetry. A mathematical statement in general is therefore a collection of objects each with symmetry requirements with the statement itself also satisfying certain symmetries. A select permutation of the sub-objects, given by the statements own symmetries, must leave the overall statement evaluation unchanged to be considered valid, universal or contiguous. As such the subgroup (stabiliser/center?) of the statement and the objects must be Abelian.
Additional points suggest that for the axioms to be unique and non-overlapping they must be simple groups. Since the universe is built on simple groups at the microscopic level and macroscopic level we can say that the universe is built on mathematical axioms.
The universe is the only object with no boundary as this suggests such and object could be divided into two separate pieces. The dual to the universe is the universe. As a consequence the universe must have recursive like structure in any dimension.
Navier-Stokes in higher dimensions describes conservation and flows in the universe in general. Use NS with information flow to describe efficient use of energy and entropy, complex life lives on this region of high gradient.
Godel demonstrated there is no such thing as objective truth since not axiomatic system can be complete. What can be objectively determined is truth wrt. to a given set of axioms.
The action of an axiom on the objects in a statement should leave the statement unchanged, this is equivalent to a stabiliser.
On a length scale complex life exits in middle region of the probabilistic uncertainty. On large length scales a collection of objects is has a well defined group trajectory. At the other end of the scale the movement of objects is inherently unstable and uncertain. At the extreme limit of minimal scales uncertainty reaches a maximum and the wavefunction looks flat while at the opposite end trajectories appear like delta functions. These functions happen to be dual to each other.
The universe is made out of an infinite number of itself.
Brouwers fixed point theorem -> since the universe is a map of itself (the limit microscopic is made of the limit macroscopic, consciousness is a hypernetwork, as is the collective of all processes on earth that can transfer information, much like the function transfers of subatomic particles and wavefunctions) wherever you look the loop fractal like nature of the structure shows you every point in it just on a varying scale (length). These fractal loop qualities appear everywhere in nature.
Body is like a heat engine for the mind, consumes information on a complex level while consuming heat (diffuse information), outputting heat and more complex information, computing the most complex information on a 2D subsurface of the brain, the cerebral cortex. This is a densifying of information, much like that achieved by a hologram, black hole, electron etc. all highly information dense structures that are described as singularities. The mind lives in higher dimensions which are perceived as experiences. The words we use to transfer information is essentially a 1 dimensional signal yet they carry a huge amount of information. The mind translates that into higher dimensions by projecting it through a hologram. This is equivalent to performing a Fourier transform without the projection from the integration. Repeating this in higher dimensions unwraps the information. The mind doesn't live at a central point, its actually an enveloping oo dimensional torus but with one Klein bottle like twist to insure it only has a single surface as it must also be infinitely continuous (differentiable). As such there is a continuous transition between dimensions such that the limits once again connect but with a twist.
Far future is also the the far past, in a way the Big Bang is just the collision of a sea of interacting singularities. In the far future, the universe will be a large sea of black holes and life forms networks though I suggest they are equivalent. Information will be transferred across the universe in an ever expanding sea of dimensions. As such the information will appear to be distributed across a spectrum until the spectrum just appear like the energy transfers in quantum fields seen as a probabilistic limit with the transfer of information on the far ends of the spectrum, discrete energy levels to continuous heat like transfers in thermodynamics. In the other probabilistic limit the movement of the planets is relativistically driven through a a more contiguous description, less variance. Flows are smooth throughout the spectrum but have an entropically derived distribution. The curvature of the universe is determined by the flows of information transfer. On the limit length and time scale perspectives dual limits are reached. The local curvature is a good measure of the complexity of an object and therefore a descriptor for more general dimensionality. The dimensionality of a space actually just measures the amount of information transferred through it.
What is the Watt output power of a human? True measure of efficiency. What is the peak information output of a human being vs the internal information processing power. How are these two numbers related? How much information does a human consume, information efficiency ratio.
In some ways the heat equations are like dealing with the Navier-Stokes in an infinite number of dimensions. If a generalisation can be made that links the two then efficient computation across the spectrum can be achieve.
Use Fourier expansion to very high dimensional vector given by basis set of finite simple groups and map inner product. Superposition of many high dimensional vectors can be encoded in an encoder. Perform reverse operation of decoding with large amount of projection, holography, to pass information efficiently. Discrete symmetry decoding in one direction and infinite symmetry encoding in the other direction. The shortest physical (low discrete dimension, 2D, 3D) route between any two objects is through a lower dimension, more continuous, more even probability distribution through discrete dimensions. Information processing is about the discrete transfer of information and therefore has more easily computable representations (matrices and tensors). Efficiently choosing hyper-sparse matrices would make the process far more computable. A decoder is a parameterised stack of group operations that leave a single vector unchanged, or reprojected. An encoder performs the operation in reverse by selecting the contributions to specific groups and encodes it to a single frequency in 1 dimensional spectra.
Neural networks are diffraction, shadow based objects, computed complex diffraction grating.
Put matrix representation of complex numbers into Fourier transform and observe the effect of generalising to higher simple group representations. Even exponential function is approximated by converging set of group operations. Polynomials describe the folding of the complex field and therefore are the best approximation via Taylor series. Better approximations use more complex properties derived from a more even distribution of finite simple groups. Can flows be used to better back propagate the learning between the encoder and decoder components. How to introduce symmetry elements, higher dim representations only when they are needed. Non-linearities need to only happen in the highest dimensional conversion flow in order to minimise computations. Minimal space to compute flow is determined by minimal groups with both encoder and decoder groups sets as subgroups.
The brain consumes information in repeating patterns, music, film, art, stories, experiences. The length of the patterns and the chemicals released store information that is then reflected in the interactions that occur in life. The mind is like a kugleblitz. Information is consumed. Some information is reflected back with higher energy as though it traveled through the ergosphere, parted with some high entropy information and exited on a new chaotic orbit. Chaos dictates the maximum efficiency for flows to occur through so that the universe is the most efficient structure to compute everything. Since minds are computation and problem solving machines they can be seen as a structure that drags the universal function through all possibilities in the most efficient manner possible. Life is machinery designed to create complexity with minimal computations.
The universe is a oo-dimensional chain, (moebius loop, Klein torus in a higher discrete dimensions, but length of the strip tends to zero in the limit due to conservation) wrapped around an infinite dimensional hole, the hole is the empty set, nothing, the chain is given by the conserved stretching of an oo-dimensional Euclidean space, also the Gromov boundary of an Omega-language, cardinality of the continuum.
Discretisation -> high dimensional efficiency, Continuation -> low dimensional efficiency.
In high dimensional vector flow, most efficient flow to compute is linear or close in approximation, put this into the loss function. Probability distribution is information. Curvature is energy, in length space is static, potential like, while in momentum space, time dependence change gives energy change conservation. Momentum is just rate of change of energy conservation. p = mv is linear velocity energy conservation approximation. Eqns p = E d/dt x,
dp = dE dx / dt = dx dE/dt. dE/dt = dp/dx. dEdp = dtdx duality. Extend Material Derivative to information flow in higher dimensions. Lie derivative for tensor fields. The Klein n-torus like nature of the universe gives it inherent asymmetry. As consequence the universe in fundamentally inorientable. As such all perspectives are completely relative, simply a projection onto itself.
My physical experience is only projected onto a specific region of the brain, the rest of the cerebral cortex is doing so much more computation.
Planck scales define curvature of the universe as a relation between relative dimensional choices. An observable Universe is the limit of observation in a given loop, like looking at the horizon. This occurs on the opposite end of the length scale and describes the Planck length beyond which information is not observable through certain spectra. Cosmic microwave background is horizon of this space with the Uncertainty Principle as the event horizon in the other direction. The event horizon is uncrossable in all cases and its position is just a matter of relativity.
Holography is the process and reverse process of converting continuous information to a discrete/continuous encoding. Approximations occur when the remaining continuous information is discarded. Why can computation only occur in a discrete space? Computation requires discrete information transfer, some process of network complexity?
No axiomatic system is self consistent as to do so would require fixing the Universal object. This is fundamentally impossible due to the inorientability of the object. Certain statements can be relatively defined by symmetry arguments.
Given a random mathematical statement αβγ -> θ it can always be refactored to evaluate T/F. The symbols (objects) in the statement have their own axioms. The symbol can be turned into a statement by suggesting that it only evaluates to True when the conditions (axioms) for it are met. This determines a manifold (subspace) in the Universal space in which these objects live. Such a manifold is uniquely determined by its simple group elements. This is another demonstration that the axioms are defined by simple groups. The statement axioms and the object axioms together define a higher group symmetry. If this object is invariant to the statement permutation of the component object groups then the overall statement must be true for the given axioms. If the group fails to commute then the validity fails for elements which don't commute.
Conscious perspective is like pressing your face into an oo-dimensional balloon and seeing yourself reflected back an infinite number of ways.
Dimensional representation entropy. In hyperbolic spaces the informational entropy cannot be conserved in the base Euclidean dimension. The total volume element has a higher possible information content than a Euclidean space can contain.
Symplectic group describes conservation of information. Information is transferred from one set of dimensions to another via the action of the symplectic group on its elements. Special linear groups conserve volume elements and are therefore perfect for conservation laws.
Computation is about making logical choices and therefore requires a bifurcation in the system such that one answer evaluates to True and others to False. This is a natural discretisation and therefore computation can only occur in a highly discretised space.
The Universe is inorientable due to the need to have a relative space or object from which to determine the orientation. Since the Universe is a singular and complete object there is no additional object from which the relative orientations can be ascertained.
Learning the universe is you can be wild since at some point you realise you're always home.
Axioms of the objects in a statement define the group structure that creates them. The objects of the statement define their own axiomatic group set. Recursively ascertaining that the components of the statement have a non-trivial center. Alternatively the center should be equal to the set of elements for which the statement holds true. What is the group of elements for which the statement a + b = c is true? The addition function defines an infinite group. Algebraic refactoring allows the statement to be put as a + b - c = 0 -> True.
The Universe is like a nut rolling down a bolt.
For a statement to be unique it must therefore be centerless as commutativity of any symbols is therefore impossible. A statement αγσν -> δ is therefore a collection of groups with each symbol corresponding to a set of symmetries. Since ordering is necessary (is it?) for a symbolic language, groups can only act on the symbols following it. (Not sure this is actually required). For a statement to evaluate either True or False it must have a total center that is just the Z/2Z cyclic group.
With no axioms the set of all statements that fulfil this requirement is infinite. Each axiom added defines a symmetry restriction, akin to folding the universal manifold according to the group the axiom satisfies. The remaining structure is the set of all elements for which the statement holds true. Axioms have discrete symmetry while objects have continuous symmetry. Computable statements occur as the two symmetries overlap. Continuous object symmetries may be removed by forcing the use of the object in the statement. Ordering of the symbols matters only when the groups are non-Abelian.
Turned the lights off, thought I fell into the universe.
Life is the most efficient heat engine.
Each word and symbol is connected to the informational experience such that it has its own infinite dimensional coordinate system that is language. Our brains and experiences are unique projections of the Universal object, parameterised by an infinite set of coordinates, functions, itself.
Everything is a heat engine, black holes, electrons, people, animals, stories, music, culture, life. Life is the most efficient structure to create absolutely anything. Any thing in the universe can be manifested as long as you seek the tools to do so. Practice the arts of a mathematical subject and see the universe for the structure it truly is. Language is the best medium to transfer theses ideas and can be sent anywhere instantly. The fact that everything is gradient demonstrates why the most efficient structure that can produce itself. The transfer of ideas can be seen as the interaction of continuum of star and galaxies all made with the simplest building block, the object that is itself.
There is one universal object and an infinite number of efficient projections of itself. Every possible symmetry across every level of reality. Every projection always has a horizon, a set of fixed points preserved in the mapping of the universe to itself. The cosmic microwave background and the uncertainty principles are just projections horizons. Each happens on a very different length scale but all are part of Universal thing that is everything. Every set of experience, of lives and personalities, being owls and molecules and networks of fungi connections, networks of electrical connections, the internet is a new brain for the world, a structure through which information can be transferred efficiently. Life is that efficiency of structure.
Does a galaxy have a personality? Any thing with a label has a personality, its own form of projection. Linguistic labels are just entropically efficient encodings of high dimensional, discrete objects or projections. Labels given a part of you an extra fold, as such each label is an axiom, or a group structure used to efficiently encode extra information.
You manifest your body. And past you can manifest future you through its actions. Equally future you can be thought of as this rewrapping of information. Punctuated equilibrium is a good way to describe a phase change. A phase change occurs where the fractal folds back in on itself enough.
Dynamically programmed tool to search for efficient group symmetry representations.
Matter is just well folded information. The information as a whole must be conserved, chemicals, cells, videos allow experience to be packaged up and sent efficiently through low dimensional space. The Universe is its own life form. Every part of the universe is contained within every other part an infinite number of times, at the tiny scale and the large, in the far future and the far past, as manifolds transition to networks and networks transition back to smooth manifolds. The universe itself is the object with no boundaries. The past is manifested to bring about the future while choices are made across all time scales. Life and its components are determined as the most efficient ways to fold information. Each object in the universe can be described as a heat engine, fed by information and built from information. The true singularities in the universe are any labeled object. Language acts as a high dimensional basis set for passing information. Words can be transferred in 1D while conveying much higher dimensional information. The brain unpacks this via a complex hologram structure in which experiences are stored. The body acts as an informations transfer system between the low dimensional and the high dimensional. Any labeled item in the world is distinct. The nature of its efficiency in language is related to the complexity efficiency required to create it. The sun for instance is a huge object but its overall complexity is low. The same is true for electrons and molecules. Each is efficiently describable as both contain key simple symmetries. The symmetries transcend scale and only have probabilistic importance at this level. Since the symmetries of an object describe its state, every object in the universe can be decomposed with symmetry.
What convex operation can be used to decompose an object into its symmetries? The brain does the forward and reverse process automatically. Superposition suggest individual neurons can perform multiple jobs in intervening layers. Efficient structure for decomposition and recomposition. Ordering of operations importance?
Delta function impulses are more efficient for orbit changes rather than continuous changes, reason for quantisation on the smallest level. Delta changes are more efficient to compute.
The universe is the object that is its own boundary.
Use parameterised Fourier decomposition to efficiently map input to higher dimension when an affine linear transformation maps it to a new group recomposition model to map down to output space.
Is the order of group decomposition important?
Is energy just coiled up time?
Function -> Data -> Symmetry
Universal fractal is both a network and a manifold at different scales.
In the far past the Universe looks like a lattice structure, it therefore needs this same property in all scaling directions for the universe to be fractal like. The microscopic appears like a smooth manifold with key singularities. The macroscopic appears the same. Both have network like connections. As a discrete network is scaled it appears more like a manifold, example of a blanket, smooth on macroscopic levels but fibered at microscopic levels. Any component of the Universe appears as a projection of the U object onto a subspace. ∫U dx gives the component. The variable over which the integration is performed is very important and may be multidimensional. As for the Universe itself, it is therefore made of an infinite number of itself at every level. Since every mapping of the universe onto itself provides a fixed point, each of theses fixed points give a singularity of the universe.
f(h(X)) = g(f(X))
Any transformation applied to the Universal function is simple a coordinate transformation.
∂U = ∅, the Universe has no boundary. U surrounds an ∞ dimensional hole. U is isomorphic to any set with the cardinality of the continuum, ∞, F(ω), F({0,1}), exponential hyperwebster in projective space. ∂U = ∅, U is the dual of ∅ since ∂∅ = ∅, ∂∅ = U. How can projections be computed on an object that is inorientable? Is naive set theory inorientable?
Since the universal structure approximates a manifold at some scales and a graph at others we need a new choice of curvature. The limit of this calculation should approach the same value. Form a cylinder such that the manifold curvature is positive in one direction and negative in an orthogonal direction. The curvatures should have equal magnitude.
U has every possible parameterisation of simple group symmetry and antisymmetry.
The dimension of the local space is given by the measure of the infinitesimal ball at that point in the space. Given a unit cube in n dimensions, form a cylinder by joining boundaries of n-1 dimensions. This separates the cube into two regions inside and out. Each side is extended to the limit of positive and negative curvature in either direction. The positive limit of elliptic geometry connects to the negative limit of hyperbolic geometry in a dimension lower. What is the geometry at the limit? Is there a n - 2n dimensionality relation? What dimension/group structure is the singularity in each dimension? In 3D singularity is circle group. This traces a cylinder in 4D spacetime. What does the 1/r potential look like in higher dimensions?
Conservation of information in the universe requires transformations via the symplectic group, in general special linear groups. The symplectic group describes flow between dual dimensions such as time and energy, space and momentum. Transformations between equivalent dimensions is given by special unitary groups. Since folding and unfolding are given by powers in the complex manifold these are required to map structure.
Binary is entropically efficient so many structures can be created with quadratic operations.
Universe is an infinite dimensional quantum harmonic oscillator.
Any on U is a map to itself. Due to Brouwer fixed point theorem and the fractal like nature of the universe there are an infinite number of fixed points. These fixed points are the singularities of the universe, any labeled object, electrons, black holes, humans etc. and can each given a dimension (or field). Each dimension has a value that describes the object wrt. all the other objects in the universe. The most efficient coordinates give the common labels in language, time, space, colour, mass etc.
The brain acts as a hologram to unpack continuous information sent through a low dimensional subspace, language is transferred via text or speech and is interpreted as a signal through time. Images are in a higher dimensional subspace. The brain unpacks these objects by transforming it into a higher dimensional space. The dimensionality of the space is a conversion between a continuous signal and a discrete set of symmetries that efficiently encode the information. The solution for problems becomes more linear in higher dimensional spaces. Choices require a measure of difference and this is only possible in discrete spaces.
Since symmetry can be interpreted via complex powers it should be possible to demonstrate the commutativity of group operations in that language.
If α β θ δ ξ are all groups with a magnitude and symmetry/antisymmetry ratio
Universe has zero measure, every manifold is made of a graph mesh at a different scale and therefore the volume measure shrinks to zero everywhere. Only gradient is measurable as it uses and infinitesimal amount of information.
The Abelian center of a set of groups gives the group operation on the objects such that the statement is unchanged.
Statement validity from symmetry arguments.
-> A geometric object is uniquely determined by an ordered parameterisation of its simple group elements.
-> A mathematical object is uniquely determined by its axioms.
-> There is an isomorphism between the ordering of possible symbols and group symmetries.
-> Statements evaluate to a new symbol. That statement can be reorganised into evaluating True or False.
-> Discrete evaluation in T/F space allows for use of commutativity and symmetry in determination.
-> If a non-trivial center can be found for a statement, that center describes the valid objects that hold the statement true.
-> If the center is trivial it only holds for the exact set of objects that are given.
-> Evaluation as to whether the original statement is true or false is left as an exercise for the reader.
-> In a given statement the center is the set of elements for which the statement is true, fixing certain elements reduces the center and therefore confines the valid statements.
Fractal like qualities in a loop universe.
-> Limit scale is identical in any choice of direction.
-> Universe maps onto itself an infinite number of times in any projection, Brouwer fixed points give singularities.
-> Universe has no boundary, has no measure. Everything is dual to nothing. Infinite dimensional hole.
-> All structure is given by the folding of space. Homology groups give the degeneracy of energy levels.
-> Large scale is like small scale, far future is equivalent to far past.
-> Scale varies universe from manifold structure to network structure and back.
-> Any scaling of the universe in any direction will eventually lead back to the same point. Modular forms.
-> Universe is inorientable due to Klein bottle like structure.
-> Discrete transitions only occur on a limit scale.
Life as a heat engine.
-> All complex structure in the universe can be described as an entropically efficient heat engine.
-> Information flow is conserved, symplectic group describes flows between dual dimensions, unitary group describes flows between similar objects.
-> Navier-Stokes extended to infinite dimensions gives heat flow as a statistical limit.
-> Atoms, molecules, cells, weather patterns, planets, stars, galaxies, black holes all as heat engines.
-> Special groups preserve information in the Universe and therefore govern the structure of life.
Computability in the Universe.
-> Difference requires discrete quantification.
-> Information can be efficiently passed via continuous signals in low dimensions.
-> The mind unpacks continuous information by computing its discrete symmetries.
-> Difference is computed in a high number of discrete dimensions.
-> Transistors operate on discrete group principle.
Discrete quantities are necessary to perform difference calculations. These quantities can be numeric or symbolic but must be discrete.
Hyperbolic geometry in the brain as bi-2D projection of 3D manifold, our perception of a hyperbolic manifold imbedded inbetween 2D and 3D space. Dimension of depth rather than width. In the universe there is a conservation of total curvature, hyperbolic radial potentials orthogonal to elliptic rotational potentials. Seen around planets and atoms, this potential is the most efficient surface between singularities. This is the minimal surfaces description. Wavefunctions are just the harmonics of these minimal surfaces. At limit hyperbolic/elliptic wavefunction is zero due to relative potential well. Most efficient fractal structure produces everything. Maximise the efficiency of the universe as a heat engine.
How can any network reduces its dimensionality to a continuous signal?
Movement of singularities in the universe is from the symmetry of their relative positions. A proton is pulled around by its own electron etc. The minimal surface between singularities defines the potential they lie on, harmonics and conserving flows determine the movement of the wavefunction along the potential. The probabilistic wavefunction is just related to the local curvature of the potential space on which it is imbedded. Does the potential function have the same probabilistic nature? Seems likely. The homology of a space defines its homotopy group which in turn defines its energy level Eigenfunctions. In locally entirely flat space the wavefunction must be zero or photon like. As space is curved the wavefunction locally puddles. In deep potentials it once again becomes flat. Harmonics over this space give the energy level degeneracy and therefore the entropic distribution. How do energy Eigenfunctions change as the potential is perturbed from a flat box to a fully curved manifold?
Energy transitions appear to be discrete as this is the most efficient transfer procedure. Non instantaneous transitions waste energy and therefore have a low probabilistic expectation.
Symplectic group has hyperbolic manifold, unitary group has elliptic manifold,
How can you backpropagate through a quantum computer?
Earth is made up of a huge set of singularities. The flow of information is maximised as the local curvature of the space undergoes phase changes and appears to have discrete boundaries which are in fact just event horizons. Phase changes appear in the atmosphere, at ground level, on the cellular level, travelling towards our cerebral cortex. The flow of information governs the curvature of space. Information is given as wavefunction probability density over a minimal surface potential. The wavefunction itself is a minimal surface but with nodes at the limits of the curvature of space. Are the nodes truly fixed? The flow of time is directed by the gradient of Entropy while the likelihood of structure is governed by its efficiency as a heat engine. What is the measure of efficiency of an information engine? Heat in higher dimensional space? Edge of chaos or edge of symmetry governs maximally efficient paths. Chaotic regions are just defined as the limit cycle of a recursive fractal like function. Therefore the universe around us appears so fractal like as this is the most efficient region for life to exist and therefore also the most statistically likely. Locally potential curvature defines probability flow and probability curvature defines potential flow. Flow direction is due to entropic gradient.
Local curvature of space and geometry have dimensional bounds. Map curvature from most efficient embedding. Curvature given by measure of infinitesimal volume/surface elements. Map measure from [π,∞) to [0,1], measure and dimensionality have rotational duality.
Some particular minimal structures maximise heat engine efficiency. These can be seen as the components of life that we see. Use calc of variations to minimise surface while maximising efficiency. Though the universe is fundamentally probabilistic the most likely action is driven by the efficiency of representation.
The probabilistic universe is a fibration in an infinite dimensional space where the fibrations can diverge and converge. The observed universe is an efficient submanifold of this fibration.
Every finite Abelian group is the direct sum of cyclic groups. Therefore the Abelian center of a statement is the cyclic permutation of elements that leaves the statement unchanged.
Abelian groups are representable in Euclidean space. Are the event horizons of the universe in the limit of hyperbolic geometry the limit of truth or computability?
A well defined statement uses elements of a group rather than the group itself. If the orbit of the elements fixes a set of points, those points form the valid elements for the statement. If the set of fixed points satisfies a certain group then the group is the object for which the statement is valid.
Have neurons coordinatised in space. Similar to sparse matrices where data is kept by coordinate. Rays sent into the model are redirected by the neurons. Neurons can be simulated in space as the folding of space around a singularity. The information is the redirection of geodesics round such objects. Adjustment of the neuron locations affects the geodesics on the surface and therefore the output of the map overall.
Training would be akin to moving the singularities to minimise the error, bifurcation could be used to increase the size of the models while training to maximise computation accuracy tradeoff.
An efficient network should be the most entropically efficient way to map hyperbolic spaces between the input and output dimensionality. Tree like storage with superposition.
Build neurons on high dimensional lattice, lattice locations can be held in tree like data structure (network), lattices redirect rays to other neurons, neurons hold redirection function. Most parameter efficient way to learn all functions is to have the parameters stored in functions.
Store information by learning the map onto a very high dimensional space and map back down. Is diffraction map easily to learn? Should be with right architecture, structure, activations. Can be used with distance map to accurately reproduce diffraction pattern. Since it is just the superposition the sum can be calculated with Monte Carlo integration techniques. Fourier provides decomposition across frequency spectrum and cyclic 2, Z2 group. Repeated exponentiation is achieved n log n time by tree like computation structure.
Weierstrass transform on the loss function to provide smoothing. How does the minima of a function change over the Weierstrass parameter? Are there phase changes in the minima?
Most accurate method to simulate brain as a high-dimensional Ising model. Graph network shows coupling of information conservation routes.
Look at most efficient data structures. Can existing ones be improved by neural networks? Hopfield network simulates graph network and geodesic redirection. Maximal information store. Attempt to conserve entropy.
Learn tree structure for most efficient approximations of given functions. Calculation of next function is almost a linear projection function from a much higher dimensional space. Build efficient basis set that is activated by a particular piece of data.
Dot product between groups to show symmetry similarity. How to approximate quotient group?
Omega language given by F(ω) doesn't represent a graph but instead a hypergraph. An object in F(ω) can represent any abstract mathematical structure and therefore is directly connected to all subjects via the symmetry operations that more precisely define them. Since objects within U are recursively defined, U = ∑f(U) where f represents a relative scaling and orientation map. U contains every possible arrangement of symmetry and antisymmetry structures within it. The topology of U is decided by how closely related objects are.
The intersection of orbits of a given statement in F(ω) give the overlap of objects within the space of validity. Validity requires a fundamental connection to the cyclic group of 2 for Boolean logic to hold.
U can be described as an infinite dimensional Ising lattice. The inner product of the "vectors" on the lattice points assigns a similarity score to the lattice points and therefore defines their relative location in U. Two points connected with an infinitesimal change give the variation in time.
Checking whether an equality statement is true or false is equivalent to checking whether the difference of the two statements is equal to ∅.
The Fourier transform of U given by the exponential hyperwebster is akin to projection of a shifted and scaled U given by f(U) along a "new" dimension. The definition of new dimension is uncertain in this instance as all objects are sub-structures of U.
U can be embedded into ∞-dimensional Euclidean space due to the Nash embedding theorem yet the space and embedding are both a subset of the U object. This seems very similar to the Banach-Tarski paradox.
In order to fully define a singular object (a point in U) requires an infinite number of symmetry choices. This necessarily requires the axiom of choice which itself must be a viable symmetry object. Does any perspective of U require an infinite number of choices or is the event horizons of the universe a demonstration of the fundamental limit of knowability?
How to solve anything
Build the free group of an infinite number of generators, F(ω).
(Technically any object isomorphic to something with the cardinality of the continuum should work)
α, β, γ, δ, ε, etc.
Add additional topology such that every non-trivial element is equivalent to one of the generators.
αβγδε → θ (Cayley graph becomes hypergraph)
Via the Curry-Howard-Lambek correspondence any function (mapping) can be represented as a sequence of symbols.
Symbols can represent any mathematical object including functions and sets.
More universally any object can be realised as a function or mapping.
Symbols represent regions and can have subjects which are also symbols.
(Do symbols have to be unique? Don't think so)
Any mathematical statement is now a subset of this object which we will call U.
Via the Banach-Tarski paradox U is also a subset of U.
U has now been defined.
In mathematics objects are defined via axioms.
In U objects are defined by the statements that they are equivalent to.
Each statement, given as a collection of symbols, has a group structure attributed to it.
Permutation or exchange of elements can alter the result of the statement or leave it unchanged.
In Boolean algebra, statements are evaluated to either be True or False.
This set of elements describes the Z2 group or the cyclic group of two elements.
Any mathematical statement can be rearranged to evaluate to either T/F.
Therefore to evaluate the validity of a mathematical statement we must find the group that stabilises the group of the statement on its True value.
The stabiliser subgroup G of the statement group X is then the set of valid elements of a chosen statement.
The validity of statements can now be ascertained. (Only wrt. to the choices of axioms)
Since fixing elements Q of the statement X is akin to removing the associated group of elements this can be seen as taking the quotient group, X/Q as the new statement.
It follows that axioms also perform a fixing of structure.
This suggests that axioms are equivalent to groups.
For axioms to be unique, their groups must be the simple groups.
Since any object can be defined within U we must recognise that the observed Universe is also a subset of U.
This suggests several things about the Universe.
Principally the fundamental building blocks of the Universe are mathematical axioms or choices involved in symmetry, the true fundamental particles of the Universe from our perspective are the simple groups.
Additionally since U is not just a subset of itself but also a superset, and an infinite number of which, we must recognise fractal like qualities in our Universe.
This gives the Universe a modular form like structure and lends credence to the Monstrous Moonshine connection between finite simple groups and modular forms.
With U described as a function that contains itself, using Brouwer fixed point theorem we can say that any map from U to U fixes an infinite number of points which we observe as the singularities of the Universe.
These singularities are better described as ringularities for multiple reasons including spin.
At some level the Universe appears to have a smooth manifold like structure while at others it appears to be a discrete manifold. The fractal nature of the universe means neither of these quite capture the full picture since there is a recurrence relation that each appears like the other at a different length scale.
(Think about zooming in on a blanket and seeing the threads vs zooming in on a thread and seeing a smooth cylinder)
In the limit both manifold and network like structures are valid. In fact any function can be described as the limiting case of a smooth function, Weirestrass transform in reverse.
The fractal like nature of the Universe is more suggestive in that scaling and translation now both appear somewhat periodic much like rotations.
This suggests the Universe could be described using a hyperwebster of exponential functions (more on that).
The efficiency with which mathematics describes the world we see around us is now no coincidence. It appears to suggest that in a reverse argument of the anthropic principle, life is just a maximally efficient method for entropy conversion while conserving overall information. Entropy is dual to time. Statistical energy -> continuous flow.
Assorted points:
The fixed points of the universe actually define horizons from our perspective. We see this in event horizons for black holes but also in the cosmic microwave background and in the Heisenberg uncertainty principle as a horizon on the transfer of information.
The potential energy of a system describes the local curvature of space. This is visible around planets as a gravitational potential and around electrons as a Coulomb potential. As the local curvature of space exceeds the bounds to be embedded into Euclidean geometry this forms a pathway to higher dimensional spaces. In fact the dimensionality of the space is simply described by the infinitesimal volume element wrt. to an infinitesimal radial element. The potential of space is given by the minimal surfaces between the "fixed point" horizons of the Universe.
The observable Universe can be considered a singular life form. Every pattern within can be described as an efficient heat engine.
The limit of the large scale universe can be considered to also be the limit of the small scale universe.
The far past is described by a minimal entropy form, likely a Dirac comb on an infinite dimensional lattice.
The far future with only black holes appears very similar. The movement of huge numbers of black holes over vast time scales also appears similar to the movement of subatomic particles. etc.
Life can still exist in these regions but on a different length scale.
Human consciousness appears to be another form of singularity. The brain appears to be a tool to convert information transferred through low dimensional spaces to high dimensional ideas and thoughts that exist in the mind. This is achieved through the network structure of neurons. The cerebral cortex may act as a dense information storage unit by containing information as a hologram.
What does an infinitesimal change between two projections of U actually mean? The position of the projection is invariant. Changes are only visible in the relative shifts of singularities or horizons.
Design a NN as an efficient subset of U. Certain points/locations/regions correspond to the similarity of a set of data to a given object. Information can be passed between points via rays. Information is collected at certain points. An object can be a collection of basis regions ie, a cat is expressed by a number of legs, a form, common behaviours. How can the distribution and collection of information be most efficiently expressed computationally?
Symmetry decomposition of dataset, reverse SVD?
Find generalisation between deep learning and search engines. Have common agents that transfer information globally with a specific distribution function and a dense computation to locally store addresses and transfers. Similar to computers with functional transformations in the processor and data or instructions in the memory. Neurons then directly apply to certain labels. Each neuron has an associated label. A label is just a very high dimensional vector. What are the most efficient functions a processor can perform?
Humans predict the future efficiently from reconstructing several past experiences. A new value is estimated from the updated beliefs. Beliefs are stored in the structure of a network. What is the densest way to store information in a neural network? Find most efficient recursive representation. Apply Bayesian statistics to the estimation of the symmetry elements of an unknown function. These values are estimated from the data.
Most efficient Hopfield network design, highest density storage.
Given a tensor dataset, what operations can be performed on it to calculate its symmetry?
T, x, p, E all give conserved dimensions. Flow between is determined by symplectic and reciprocal relationships. Uncertainty relations give local curvature of flow. Flow of time has natural asymmetry.
Given an infinitesimal volume element, calculate the Eigen decomposition of vectors in the element. These are the principal dimensions of the universal space.
Lattice is a continuous space with a discrete quotient. Can the reverse mapping be performed that takes a continuous space and quotients a torus to result in a discrete space. Torus is given by T^n = R^n/Z^n so Z^n = R^n/T^n. Polynomials appear like quotient operations in a continuous space.
The exponential map takes a value, antisymmetrises it via multiplication by i, [[0,-1],[1,0]], then performs an exponentiation which is just reciprocal factorial weighted polynomial operations. Since the complexification is akin to rotation by π/2 or the hodge star operator this is happening in a dual space. The complexification means polynomial operations work to fold the space around the origin which performs rotations. Because of the reciprocal factorial only low weight polynomial operations are necessary to approximate the answer, can this be extended to minimise the number of operations?
Tensors -> Sparse Tensors -> Address and data
Store exact functions with given addresses, hyperwebster type dictionary. In order to perform calculation agents work together to calculate the addresses. Functions can be composed using a similar methodology, data from simple functions are redirected and composed by larger structures that govern the collection of information.
Data is just a sampling from a function probability distribution. Can we efficiently sample from a bath of symmetry operations to build a function from scratch? Data can be represented in matrix format with an id * coords representation. Exchange of the coords vectors can leave the data either relatively unchanged or largely altered, this gives a way to decompose the symmetry by altering the data tensor and measuring the functional output. Does this require a functional transition already?
What is dual to the action of a system?
Imbedding R^n surfaces of increasing curvature in R^n+1, what fractal like structures are produced, what is the implicit equations of the surface?
Sparse matrix -> (address, data) tuples, given the address the network must learn the data or the reverse, given the data the network must address it.
If all functions are stored in a hyperwebster that can be efficiently traversed, the problem of machine learning is to find the address that minimises the error for the given data. The hyperwebster must be smoothly differentiable. Could Bayesian sampling be used for this problem?
Differentiable addresses.
Given the hyperwebster of computable functions, find the function for the computational cost of running each function. Minimise the error and compute cost together to find maximally efficient representations.
Pass variables to python function or class with dictionary. Dictionary can either have a constant or parameterised function. Parameters of the function can also be defined in the dictionary.
Space maintains zero sum curvature while increasing negative Gaussian curvature in the direction of high information. Entropy is a global measure of energy distribution while temperature measures the mean value of energy. Pressure can be thought of as a step up from this with the collective motion of particles inducing a force. This total force is mediated through the volume. A step higher and we get momentum transfer of a more singular collective mediated through location in space. One value measures the oscillation rate or frequency and the other measures its dual, time period. As the frequency goes up and collective nature is lost, cohesive transfer of frequency decreases and the force/movement is mediated more through heat.
Low entropy systems such as humans manage to coordinate large collectives with minimal energy. Complex systems seem somewhat similar to efficient resonant coupling.
Generalised coordinate, generalised force. dtdp = dEdq
The observable universe as we see it is like a glider in a cellular automaton but the most complex possible.
Are there phase changes in generalised observables?
Action measures universal efficiency but only in reciprocal terms. Action is dual to efficiency? More complex systems require higher action in order to build higher complexity. Does efficiency change at a similar rate?
On a general manifold/distribution we have point values, local values and global values.
Point values:
Value, jet approximations
Local values:
Frequency, period, duality, minima, jets, relativity
Global values:
Probability, minima, projections
On a general network/graph we have addresses and connections.
On a hypergraph we have conjugate functions.
Functions that dictate regions and functions that dictate transitions.
Coordinates, exact curvature quantification.
f(g) = h
f maps g to h. Does f imbue any structure on g? Yes it does via restrictions on the domain and codomain.
A graph can be stored numerically in a matrix using an adjacency matrix. Hypergraph can be stored in a vector with transition function mappings. Each value in the vector gives a node. To generalise more each node can be a region or function itself. Are hypergraphs with discrete nodes or edges truly general? A pure function describes a hypergraph most accurately. Given a function, the hypergraph is given as the transition function of the input function to the output space.
Since time is relative can we only measure wrt. it? Can two things even be measured precisely at two locations and identical times if time is relative?
Duality is a demonstration of symmetry even in mathematics, additionally it shows that the lowest entropy representations only require a C2 symmetry.
Minimal surfaces are purely local, every point working towards a singular goal, to minimise curvature sum, minimise local energy. What is the mapping that takes the unit box with periodic boundary to a 1/r potential? The wavefunction on a potential must vanish at the limit boundary (node). This is because the probability is evenly distributed and the change in wavefunction curvature is related to the change in potential curvature. As the sum of principal curvature tends to a limit the relative change in curvature drops to zero. At the throat the curvature is changing the most.
Use holography as address storage. What free group most efficiently spans the space of computable functions? Maybe have the generators stored as the most efficient operations a CPU can perform?
Generalise between holography and Hopfield networks for dense info storage. Answer stored via accumulation for time invariant computation. Can function be optimised during evaluation and retain accumulation data?
Can the universe violate Russell's paradox by having zero measure? Transitioning from a set language to a functional language may be necessary to describe the universe completely. Can Russell's paradox and the axiom of regularity be reformulated to show that only projections of the universal set cannot contain themselves? Technically since the universal set is an object defined without axioms it cannot fail the axiomatic requirements.
A function can be defined that contains itself. Does this then fail the axiom of regularity?
Cardinality of the continuum is a necessary requirement for the Universal set, there can be no discrete, disconnected elements.
Cardinality of the continuum is the same size as any cardinality. This is a necessary requirement for the universal set.
Multiplicative group as projection from an infinitely large circle, elements are given by angles from the origin. Additive group composes by summing angles. Angles are taken from an infinitely distant point, angles are therefore infinitesimal.
In hyperwebster differentiation operation should be translation and scaling.
Group actions act as both relative coordinates and relative transitions.
Associativity and additive commutation of vector sums lost in non-Euclidean spaces. Therefore Euclidean space only occurs for zero spatial curvature, choice of symmetry. The "homology" group of Euclidean spaces is Z while spaces in general have R homology.
Complex numbers are closed under certain operation sets due to the multiplicative group of R being disconnected while C is not. Is this true?
U cannot be defined globally, only relatively and locally due to inorientable nature?
F2 is isomorphic to F(ω), duality in nature can create any possible structure.
The universe is like a Klein bottle but over the dimensions of scale and curvature. A scaled map that maps onto itself an infinite number of times. Those maps have a homology group of the integers Z.
Fractal nature of universe looks like Indra's pearls where the boundary's never cross. This forms a conservative vector field in an infinite number of dimensions. What does exterior calculus mean for the different derivatives in k-vec and k-form spaces. Derivative is essentially measuring tiny triangles. At (one of) its finest levels nature can be described as a graph network structure yet at another level it can be described as singularities exchanging wavefunctions through a lower dimensional subspace (flattened or folded?) or pure functions exchanging functions in a highly conservative manner.
Universe is like a toroidal screw propelled down an infinite dimensional hole, where the hole is the thread and both hole and thread are chiral images of each other.
Calculate weighting function for first layer, reroute both to second layer while attentive mechanism calculates the next weights using all the information from previous layers. Finally all weights are redirected with the corresponding information according to weight function addresses. If weights are spaced efficiently the direction of information can be predicted using functional approximation mathematics.
Extend square of the nome to all possible fractal functions.
For address book model the problem ie, f(x) -> y, is simply the address book weights. Can the problem also be passed to the model such that a specific problem can be chosen. In a more general sense the x -> y transition is mapped by the function f. For a neural network the function mapping is determined by the weights and structure of the model. If these properties can be efficiently encoded in the addressing structure the model parameters can be computed on the fly. Ask the model to predict the problem model weights. The model that does this can additionally be stored in the address book. Can the address book be programmed to store itself and only build the necessary structure as and when it needs? Somewhat like a free group, the hyperwebster also stores itself.
The self Taylor model that makes each discrete parameter also a function may be the hyper efficient addressing system.
How can the error of the address gradient be calculated to provide a means for problem optimisation? How do we sample the data most efficiently?
Use continuous normalising flows to map the minimal surface of the data probability distribution. Grow the network from simplest components though to minimise entropy, or minimise dual curvature.
Each data point has an attached Gaussian with a distribution over the dimensions of the space. Find Eigenvector decomposition of the space, is this like the projection of the gradient of a higher dimensional function, the Jet?
Try every possible triangulation of a dataset and minimise the eventual curvature of the graph/manifold. The underlying function is just the smooth limit of the triangulation.
When the minimal surface of x is mapped to y there is a transition that minimises the energy over all possible curves where the paths are over the full distribution of the data points.
Address is given by set of possible operations that computers can perform and the associated time cost and energy cost. Basis set with powers or exponential. Deal with opposite ends of the spectrum.
Functions of x and y data distributions are listed in address space. To find the transition function between the two need to find minimal transition function that lies in a higher dimensional space. The more structure imbued on the object, the fewer the parameters needed to store it.
How should fractals be weighted in order to preserve volume? Conservation is about volume preservation as can be seen in the formulation of Lagrangian mechanics on a coordinate free manifold.
How to make a small transformation in address equivalent to a small change in overall transformation?
To make addresses efficient can they be organised on a hyperbolic manifold in a higher dimensional space? Many transformations aren't necessary due to high entropy formulations. Address space can have additional tensor field structure that calculates the computation requirements.
Time is a measure of the universe. Since all measures are of the same thing can energy be thought of as coiled time?
Information of addresses can be minimised by storing known structure that can be used to rebuild an object. Simple groups are minimal structures.
What are the base operations a computer can perform? Which operations are approximations of the actual op using the base computation functions? Pure logic is used at the base.
Addition and subtraction
Multiplication
Bit shift
Comparisons
Early addressing functions have larger cross entropy, as addresses becomes more precise the resulting transition function have more overlap. This allows the final address to be differentiable wrt the chosen loss.
Make calculation of the weights in a transformer attention mechanism efficient, using the idea of mapping [x,y,f] -> w to calculate the basis elements of a transition.
What happens when holograms are illuminated with differing wavelengths?
Within U each combination of possible symmetries is unique as each identical point maps to the same location. However the fractal like nature means though these options are unique, they are only relatively unique.
Serialisation of thought occurs when stakes become higher, confidence is amplified due to approaching chaotic regions.
Need quantisation to know anything by forming finite groups that can have attached "decisions".
Repeated Fourier kernel with summation of basis values. Binary decomposition of symmetry elements. Take dataset, complexify by converting into antisymmetric matrix, perform exponentiation via polynomial approximation. Is this just an Eigendecomposition of the data graph? What would data adjacency matrix look like?
Division of labor in the brain, fixed and plastic cells, high vs low activity.
Video mode direct reward feedback, use joystick to input constant reward signal to optimisation algorithm. Have loss as a separate network that is learned from the feedback.
The continuous dimensions we observe in our universe are the orbits of simple groups, all dimensions can be deconstructed into principal curvatures along orthogonal directions. The curvatures along different bases are entropically determined due to the energy of the underlying space. This just dictates a conservative flow. Does each finite simple group have an infinite dual? Hypercomplex numbers describe a supergroup.
General Relativity is describable using the idea of conservation. In order to perceive relative change in a projection information must flow between dimensions. Since no split can be placed between dimensions time and space can naturally flow into any recognisable dimension.
Our perception of the flows of space and time are naturally built from the hardware through which we perceive. The conservative limitations of these systems are a consequence of the efficiency with which they are processed.
To individual humans does time have curvature? It must from some perspective but perhaps its the constantly flat from another perspective. Actually we perceive the dimensions of space time from a hyperbolic perspective, this is why time is -1 space.
Locally a spacetime is assumed to be flat for an observer at rest. What does this suggest for complex non-local objects? Many systems can work together to produce a unified viewpoint. What is the relative curvature like across the entire system?
Use eeg for direct feedback and control. Train reward signal on known data or just use semisupervised learning to label certain areas of the hyper surface.
Complex numbers are first non-trivial continuous field. Multiplication by complex numbers causes rotation to be performed in the mapping. Polynomials of complex numbers indicate n covering or folding operations. This is made continuous by performing several weighted 'folds'. exp(z) demonstrates a projection mapping. Complex field is taken and projected onto the new plane from a point source at the origin. Increasingly complex values are rotated around the origin. As select winding numbers occur (frequency related) the value of the projection onto an orthogonal axis increases or decreases due to the relative phase.
Given a state output gradient for change.
Does Nash equilibrium mean semi local systems can have increased efficiency? Networks are more efficient heat engines than singularities. Nash equilibrium for quantisation, just like Oberth effect, systems are most efficient when information is transferred discretely.
Use prime factorisation in quantum computers to demonstrate simple group decomposition of data. Given a sparse data point, the most likely associated outputs for any function are given by the probability distribution of the data as a whole. Selecting sparse data gives you a subspace by confining the information. Further confinements are placed upon this by selecting specific outputs. Finding the most probable distribution of data is achievable by decomposing it into its symmetry elements.
Data is selected from an underlying distribution, any computation performed on this is just an attempt to recreate a subspace of the overall space.
To define anything exactly requires an infinite number of choices, making those choices is selecting all of its elements, its most simple elements are groups and symmetry. Therefore computation is assignment, what similarities do two disparate objects share? When two object are decomposed into their symmetry elements a dot product, 1-form can be used to measure their exact similarities. What do higher order k-forms describe? What about vector valued forms? Labelling an item is a many to one mapping, therefore the reverse is true, a word or sentence is a one to many mapping while context allows for better determination of the true object. Language describes a functional distribution.
Bszzhssszzzzzzzz, jzshown
Sharp and spicy is the fractal.
DMT feels like the inside of a black hole, everything all at once.
Not the holy trinity but the holy infinity. Every life lived is the life of "God". We are all the universe.
Universe finds Nash equilibrium across an infinite number of symmetry operations. Each group/symmetry has an associated value corresponding to its dual infinite field.
Brain stores all of our experience in a hyper efficient hyperbolic manifold of memories. To compute the next best state the most likely best choice is a mid point between all the addresses of other related memory locations. Actually only the gradient needs to be computed to find the change in each section of the address.
Expertise occurs when a particular area of knowledge is best approximated by the brain.
Nash equilibrium occurs across antisymmetric symmetry of an ∞-tensor of each reward of the game over an infinite lattice of players. Most efficient function is one that entropically balances the reward with energy used. Curvature of this function is just the curvature of our space.
Fastest way to calculate group factorisation is with a quantum computer. Fastest way to rebuild a function from its symmetries. The cheapest operations are given by the simplest computations to perform, the operations possible on individual clock cycles of the computer.
Edge of chaos defines the most efficient route, is that partly achieved by adding in antisymmetry?
Universe maximises surface area while minimising volume. Like bubble between two infinite dimensional holes. The bubble has no boundaries though. Duals are set up opposite to each other, out of phase, kinetic vs potential. A harmonic on an infinite dimensional lattice where the energy at a particular point of the manifold defines the overall curvature of the space. The Coulomb potential is just an exchange cost dependent on the distance between the ringularities. Is the shape of space defined by the most efficient representation our brains can store?
Payoff in the universe for Nash equilibrium is reciprocal of energy expenditure or minimising curvature?
Universe minimises gradient of a function, this is minimal curvature. How are energy, entropy and curvature all related? Which functions minimise the entropy of the jet? Every function in the jet must be bounded. This forms periodicity for non-zero functions.
A view of the universe is given by a particular projection. Can curvature of the universe ever be zero? Probably not, not in our observable universe. These locations cannot be approximated by a jet.
What is the local curvature of the Nash embedding of an infinite dimensional torus in an infinite dimensional space? Are the principal curvatures related to the local curvatures of the space we see around us? Can we develop asymmetry by adding in a Möbius loop topology? This naturally gives the manifold a chirality while removing orientablity.
To create any structure, decompose the structure into positive real values of symmetric and antisymmetric elements. How does the attached group effect its corresponding field? Both are equivalent objects at some generalisation.
The reals can be calculated from a combination and projection of the rationals. Any structure can be computed using discrete elements. What is the mapping between discrete groups and continuous fields?
The limit of a 1/r potential appears like a discrete point or a node of a graph. The edges of the graph can be seen as the forces between these points. In our universe as the representation becomes more "fine tuned", the relative sizes of higher dimensional structures (quarks, strong force etc.) becomes more and more limited/confined to a smaller area.
Spins Nash equilibrium occurs when two spin systems come "into contact". The energetics of the system govern the reward distribution across different states. The universe chooses an even continuum of states that maximises the reward.
Fixing a group element in a mathematical statement, akin to choosing a precise value is the same as stabilising that element.
Orbit stabiliser theorem is similar to axis and plane of rotation. The axis is fixed/stabilised while the space of rotation is the orbit. The stabiliser is the hodge dual of the orbit.
Number of groups of a given order is related to the combinatorics of the partition of the orders prime factors.
Group Theory -> Study of structure, symmetry and the base components of an object
Linear Algebra -> How multiple disjoint components work together as a collective
Calculus -> The transformation of an object under a/many small change/s
Combinatorics -> Uniqueness of structures
Differential Geometry -> Space and shape under infinitesimal changes
Topology -> Structural invariants with minimal definitions
Abstract Algebra -> Symbolic language of structure
Analysis -> Trends, sequences, limits
Complex Analysis -> Symbolic group theory around the simplest of groups
Homology/Cohomology -> Topology combined with group theory
Homotopy -> How to wrap things up
Category Theory -> Transformation and transformations of transformations
Is entropy the manifest law for network formation? Entropy gives the necessary requirements for a system to naturally fall into Nash equilibrium and therefore self-organise.
The universe can only transfer information locally, this gives the need to form lower dimensional subspaces through which long range forces can propagate.
Papers to write
Axioms as simple groups, how to solve anything
Exchange in n dimensions as fundamental force for all interactions
Differentiable addresses for the ultimate algorithm
Formation of mind as holographic storage and access tool
Mind living on chaotic boundary of fractal surface
Computability of functions dependent on discrete representation
Entanglement of the mind with the environment as quantum classical divide
Horizons in the universe due to curvature of dimensions
How can we have a spin triggered switch?
Neuro theory
First observations
Would be interested in seeing the frequency modulation of spike trains over time.
Perhaps use unsupervised model to attempt to correlate neuronal activity to observed activity. The model structure formed may highlight key distributions.
Given pure dataset look at Fourier transformations of data to highlight patterns.
Multi-fidelity/ multi-resolution datasets would be interesting to look at how correlations are distributed over varying time scales.
Perform reverse holography on spike train signals in order to actively remodel the neuronal structure and connectivity. Use theoretical principles as a tool to gleam new data.
How to perform autocorrelation/crosscorrelation with Fourier transform?
Find graph topology from co-activity, efficiently embed in 3D space, compare to neuronal structure in brains.
What is variance distribution over ordered principal components like?
Attempt to learn co-activity with dense autoencoder and investigate the resulting graph structure.
What do core (high rate) principal cells activate for in general? What process is forming the organisation and structuring of these cells? Is the reorganisation of low firing cells into high firing cells based on end to end connectivity, ie are these just address redirection cells? Does physical reorganisation occur or is it simply due to spike train frequencies?
Given the firing of a neuron what is the time dependent probabilities of other neurons firing? Does the time dependence follow a lognormal probability distribution and what is the variance like? How periodic are the spike trains? Does the frequency change during the execution of a "pulse"?
Overview of thoughts:
Hippocampus acts as a sorting/junction box for activating the cortex, on the cellular level there appears to be some sort of local network organisation for the processing and storing of information. Whether this is done primarily through tuning of action potential frequencies, synaptic connections or a combination of both should be the next area of research. Self-organisation is chemically driven by exchange of neurotransmitters. At a more global level the organisation can be considered a Nash equilibrium in which the formation of network structure maximises the entropic efficiency of a non-local system.
On a higher level the brain is decomposing input into its component symmetry elements. This process is achieved via network based holography. Holography allows for a huge amount of information to be stored efficiently. Over time memories are encoded into the cortex, when the mind receives new input it is decomposed into its elements which activate the areas of the brain corresponding to the most relevant memories. The action taken by the mind is selected as the best approximation of all relevant actions taken from past experiences. Efficiency is achieved by not storing the state but the gradient of the state. The gradient of gradient can be used to infer the gradient and so on. This is a hyper Taylor approximation. This approximation has a hyperbolic growth structure. The storage of memories is completed by the hippocampus as this ensures only repeated actions form strong connections. Repeated actions are required to allow the network to self organise via only local interactions.
As a particular action/memory becomes more "precise" a more select set of neurons become activated. Storing similar activations closer together maximises the capacity of the brain while minimising the necessary connectivity. This gives rise to the local areas of the brain that are somewhat associated with specific actions, occipital lobe for vision etc.
Memories can be "frozen" into the brain much like phase transitions in a high dimensional Ising model. The structure of the brain and its connectivity allow for the highly chaotic motion of the "mind state vector" which gives rise to "free will" by making decisions highly unpredictable over any length of time. On a mathematical level the brain can be described as a hypergraph or a differential matrix function. This is the high dimensional PDE that gives rise to chaotic attractors, each of which can be attributed to a specific idea or thought. The motion over time produces our serial sequence of thoughts in the form of language or imagination.
Any process in nature given a particular projection has an associated set of values, X, which are selected from a governing probability distribution. Learning a transformation in the process is the same as estimating a subset of values given an orthogonal subset of given data points. These data fix a subspace of the overall distribution. The hidden data is estimated from the probability distribution over the resulting subspace. This subspace is the Hodge dual of the space covered/fixed by the known data.
A function can be locally approximated by a Taylor expansion f^(n)(α) (x-α)^n / Γ(n+1). Further approximations can be made by estimating f^(n)(α) using the same method. This recursive method minimises the number of functional evaluations needed. All that needs to be stored is the value of α and its depth/address.
Type 1, 2 thinking can be attributed to an accumulation process in which longer thought periods allow for refining an estimate based on further activation of information.
Context is the tool that allows for refining a given latent into a specific idea. Language is just a set of stacked building blocks that effectively encode this process. Letters -> Words -> Sentences -> Passages -> Chapters -> Books. At each level the "address" of an idea can be further refined. Addresses must not be taken too literally as this occurs over a distribution or multiple regions all at once.
Though language and vision can be separated at the physical level, in the brain these processes are combined into the object we describe as our consciousness. This includes all sensory input. On a more metaphysical level our Universe appears fine-tuned to allow our consciousness to exist. This is no mistake as this is the only area of the Universe in which consciousnesses exist that are able to question the seemingly precise form our environment takes. At a more fundamental level this means the laws of physics are directly related to conditions necessary to produce consciousness. The dimensions we live in can be directly attributed to the dimensions we experience. As such the limitation of our environs to just 4 dimensions is somewhat absurd.
Two hemispheres of the brain act with a student teacher relationship similar to Generator Adversarial networks. This serves as a tool for propagation of errors in the prediction of the environs and as a tool for imagination.