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[Pathik Sahoo](https://orcid.org/0000-0002-5102-9482), [Pushpendra Singh](https://orcid.org/0000-0002-7274-6683), Komal Saxena, Subrata Ghosh, R P Singh, Ryad Benosman, [Jonathan P Hill](https://orcid.org/0000-0002-4229-5842), [Tomonobu Nakayama](https://orcid.org/0000-0001-9696-475X), [Anirban Bandyopadhyay](https://orcid.org/0000-0002-8823-4914)

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[A general-purpose organic gel computer that learns by itself](https://mdr.nims.go.jp/datasets/16f280ee-aa69-4fcb-b651-db93af85a27b)

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A general-purpose organic gel computer that learns by itselfNeuromorphic Computing andEngineering     PAPER • OPEN ACCESSA general-purpose organic gel computer thatlearns by itselfTo cite this article: Pathik Sahoo et al 2023 Neuromorph. Comput. Eng. 3 044007 View the article online for updates and enhancements.You may also likeMagnetic field directed assembly ofmagnetic non-spherical microparticlesIvna Kavre Piltaver, Andrej Vilfan, GregorKostevc et al.-Triboelectric ‘electrostatic tweezers’ formanipulating droplets on lubricatedslippery surfaces prepared byfemtosecond laser processingJiale Yong, Xinlei Li, Youdi Hu et al.-The bioengineering of perfusableendocrine tissue with anastomosableblood vesselsHiroki Yago, Jun Homma, HidekazuSekine et al.-This content was downloaded from IP address 144.213.253.16 on 02/09/2024 at 02:17https://doi.org/10.1088/2634-4386/ad0fechttps://iopscience.iop.org/article/10.1088/1361-648X/ad2bd9https://iopscience.iop.org/article/10.1088/1361-648X/ad2bd9https://iopscience.iop.org/article/10.1088/2631-7990/ad2cdfhttps://iopscience.iop.org/article/10.1088/2631-7990/ad2cdfhttps://iopscience.iop.org/article/10.1088/2631-7990/ad2cdfhttps://iopscience.iop.org/article/10.1088/2631-7990/ad2cdfhttps://iopscience.iop.org/article/10.1088/1758-5090/ace9fchttps://iopscience.iop.org/article/10.1088/1758-5090/ace9fchttps://iopscience.iop.org/article/10.1088/1758-5090/ace9fcNeuromorph. Comput. Eng. 3 (2023) 044007 https://doi.org/10.1088/2634-4386/ad0fecOPEN ACCESSRECEIVED27 September 2022REVISED14 April 2023ACCEPTED FOR PUBLICATION27 November 2023PUBLISHED6 December 2023Original content fromthis work may be usedunder the terms of theCreative CommonsAttribution 4.0 licence.Any further distributionof this work mustmaintain attribution tothe author(s) and the titleof the work, journalcitation and DOI.PAPERA general-purpose organic gel computer that learns by itselfPathik Sahoo1, Pushpendra Singh1, Komal Saxena1, Subrata Ghosh2,3, R P Singh4, Ryad Benosman5,6,7,Jonathan P Hill1, Tomonobu Nakayama1 and Anirban Bandyopadhyay1,∗1 International Center for Materials and Nanoarchitectronics (MANA), Research Center for Advanced Measurement andCharacterization (RCAMC), National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki 3050047, Japan2 Chemical Science and Technology Division, CSIR-North East Institute of Science and Technology, NEIST, Jorhat, Assam 785006, India3 Academy of Scientific and Innovative Research (AcSIR), CSIR-NEIST Campus, Jorhat, Assam 785006, India4 Quantum Science & Technology Laboratory, Physical Research Laboratory, Navrangpura, Ahmedabad, Gujarat 380009, India5 University of Pittsburgh School of Medicine; Biomedical Science Tower 3, Pittsburgh, PA 15260, United States of America6 CNRS, UMRS 968, UMR 7210, INSERM UMRI S 968, Institut de la Vision, Sorbonne Université (UPMC University Paris 06), 75012Paris, France7 Robotics Institute, Carnegie Mellon University, Pittsburgh, PA 15213, United States of America∗ Author to whom any correspondence should be addressed.E-mail: anirban.bandyo@gmail.comKeywords: organic computer, Optical vortex, Non-algorithmic computing, helical nanowire, Single shot learning,Deep learning network, Clique problemSupplementary material for this article is available onlineAbstractTo build energy minimized superstructures, self-assembling molecules explore astronomicaloptions, colliding∼109 molecules s−1. Thus far, no computer has used it fully to optimize choicesand execute advanced computational theories only by synthesizing supramolecules. To realize it,first, we remotely re-wrote the problem in a language that supramolecular synthesis comprehends.Then, all-chemical neural network synthesizes one helical nanowire for one periodic event. Thesenanowires self-assemble into gel fibers mapping intricate relations between periodic events inany-data-type, the output is read instantly from optical hologram. Problem-wise, self-assemblinglayers or neural network depth is optimized to chemically simulate theories discovering invariantsfor learning. Subsequently, synthesis alone solves classification, feature learning problems instantlywith single shot training. Reusable gel begins general-purpose computing that would chemicallyinvent suitable models for problem-specific unsupervised learning. Irrespective of complexity,keeping fixed computing time and power, gel promises a toxic-hardware-free world.One sentence summary: fractally coupled deep learning networks revisits Rosenblatt’s 1950stheorem on deep learning network.1. Main textDeep learning computers are revolutionizing human civilization optimizing user-conceived solution pathsaccurately, figuring out the shortest path by extensive training to reach the expected solution [1]. Hallmarksare switches and circuits. Demands are increasing speed and resources by compromising nature withenormous toxic waste. The next revolution would bring computers that synthesize new deep networks,invent learning protocols in a single shot or without training. Hallmarks would be new data structure,software free, circuit free, fully analog, reusable hardware adaptive to changing environment [2]. Demandswould be fixing the computing speed and resources irrespective of complexity while compromising the user’scontrol. Realizing all, we present organic nested deep learning network, ON2.Thus far, analog computers emulate either processor parts or mathematical operations [3, 4]. In contrast,gel based general-purpose analog computer ON2 executes an entire self-learning theory step-by-step.Precisely we required inventing a deep learning network operating in a chemical beaker to converge amathematical simulation using astronomical optimization power of a supramolecular synthesis. It isimpossible to read rapidly changing big data, instantly invent a suitable model. But, when self-assembling© 2023 The Author(s). Published by IOP Publishing Ltdhttps://doi.org/10.1088/2634-4386/ad0fechttps://crossmark.crossref.org/dialog/?doi=10.1088/2634-4386/ad0fec&domain=pdf&date_stamp=2023-12-6https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://orcid.org/0000-0002-7274-6683https://orcid.org/0000-0003-0243-944Xhttps://orcid.org/0000-0001-9696-475Xhttps://orcid.org/0000-0002-8823-4914mailto:anirban.bandyo@gmail.comhttp://doi.org/10.1088/2634-4386/ad0fecNeuromorph. Comput. Eng. 3 (2023) 044007 P Sahoo et almolecules are encoded with computational choices, it collides to explore 109 options s−1. Optimizationhappens almost instantly, irrespective of complexity. This ability has remained unused because we do notknow how to encode & decode computational choices to reactant molecules reversibly. Inspired by protein &neuron’s clock-assemblies [5], we invented a language to encode supramolecule [6]. We found no need toread the whole data bit-by-bit, but only periodic events or loops as clocks such that the whole data couldlargely be regenerated by assembling discrete clocks suitably in a 3D space. Clock is a variable, 3D clockassembly CA maps all possible relations between all parameters (movie S1), it is a theoretical model inventedin a chemical beaker. It represents geometric manifolds whose corners are resonance frequencies [7]. Thenew data structure (R3 : (x,z)→ (ϕ 1,ϕ 2,ϕ 3)) enables encoding a problem as geometric shape to a chemicalreaction using the surface potential of molecular assembly.Furthermore, extensively training a learning protocol is one of the greatest problems of automation.Learning requires finding invariants. An invariant is a geometric or mathematical correlation between similarevents or objects that remain unchanged within limiting variations. We found no need to train output.Instead, let supramolecular synthesis learn invariants to regenerate only one input using its internal clocks.While doing so, the spontaneous self-assembly takes control, finds the number of nodes and hidden layersrequired to frame deep neural network, maps the network of invariants for fully unsupervised learning. Forfaintly-related problems, invariants construct solution paths wirelessly with no training. Answers find thequestioner, just opposite to what current computers do [1]. With fully remote operation via microwave inputand instant optical read-out, no need to add chemicals or post-analysis of chemical structures (figure 1(A)).For better unsupervised learning, layers are often engineered in deep networks [8]. Gel computing is apush–pull loop that runs between two clock-structures, CAf made of field and CAm made of matter, togetherthey make network’s neuron [5]. Both CAs try to match each other minimizing resonance peak difference orbias B, synthesizing at least four layers of hierarchical architectures of field and matter in a chemical beaker.In four layers, while field structure simplifies to a single clock from a 3D clock-assembly, matter grows from asingle molecule to a supramolecule. While each layer ends by optimizing field-matter dual structures,network ∅sN sets optimized pairs as input to the next layer. 3D clock assembly is a complex tensor, whennanowires of two layers resonantly couple, it is a dot product of two tensors, we get weightW, that’s also aninvariant. Coupled resonance band runs second neural net ∅si to synthesize nanowire cluster for invariant.Four metastable states, CA, 4D, 5D and 6D run third convergence loop ∅sb to configure hidden layers of theneural network, as shown in ‘figure 1(B)’. Three nested deep networks, ∅sN −∅sb −∅si assembled within andabove continuously convolute each other’s clock-assembly or neuron-node while optimizing weight-tree orinvariant tree [9], see theory in figure captions. A common activation function AF, defines a neuron’s state,we replace complex tensors with higher-dimensional multinions Asijk...s [10].Thus, three deep learning networks governed by a common HamiltonianH synthesizes distinct fiberclusters that emit multiple resonance frequencies with correlated phases as clock-assemblies forming point,line, plane, and 3D shapes (figure 1(C) and 2). Together, its invariant tree. Invariants are four kinds ofgeometric shapes or clock assemblies resonantly match with the unknown input at a time, create a tree madeof beat frequencies, beat tree or classification tree. Then gel puts internally stored fixed clocks on thetree-edges in a 3D space such that real-world-like events are regenerated including further analysis. Networktransforms (1) to (2).Output= activation_function(dot_product(weights, inputs)+ bias) (1)Output=activation_function(dot_product(invariant_tree, fixed_clock) + bias_tree) . (2)Our computer’s operation in equation (2) is a version of classic Turing machine, TM presented inequation (1), which entails demonstrating that the machine (1) encodes data, (2) memorizes data, (3)encodes an instruction, (4) executes an instruction, and (5) reads the output. Table T1 displays the five stepsfor each distinct problem, a single nanowire is a TM and nanowire assembly is a universal Turing machine,UTM (see Appendix for details).A review of computational complexity: Our computing protocol does not necessitate processing the entirepixel-by-pixel dataset. Rather, we only require access to recurring events or clocks, as clocks run, the 3D clockassembly becomes an automaton (movie S1). While for a conventional computer complexity means, insteadof 1001 use a longer bit 110001101000001, gel processor can generate a cube with triangles in its corners, andcontinue to add new structures in the corners of geometric shapes as the resolution or volume of informationcontent increases, we patented it as Geometric musical language or GML [6]. Complexity for gel computermeans many layers of geometric shapes grown within and above. The concept is orthogonal to existingcomputer’s increment of complexity, GML’s advantage is noted below.2Neuromorph. Comput. Eng. 3 (2023) 044007 P Sahoo et alFigure 1. Organic all chemical nested deep learning Network ONA2. (A) Three phase gel computer explained in three columns.Microwave input to 3D printer by four pre-processing steps (left column) (movies S1), four-layered organic synthesis,clock-assembly CA, 4D, 5D, 6D constitute ON2 (middle) and optical read-out by four post-processing steps (right column).Input Cheetah image converted to X-Y-F (F = frequency–pixel RGB) is fed to gel precursor solution via antenna array. Pixels ofleg edge forms a loop. Below, 3D clock assembly Aijk holds six primary body parts. Each clock in 3D clock assemblyCA::(R3 : (x,y,z)→ (ϕ 1,ϕ 2,ϕ 3), a positive-definite tensor Aijk = Uef, f = ln(PT); P is the deformation of input A/ijk frommemorized Aijk). ForON2 (middle) helical nanowire synthesis selects number of node classes (rows) and number of hidden layers(columns) as Aijk requires to attain convergent structure. Below ON2 (middle column) four neurons shown in sequence. Dotproduct of multinion tensors of weight functions or invariants form 4D (SV (∅1)), 5D (SP(∅2)) and 6D datasets (SL(∅3)). Weightfunction of neuron is a network too. Post processing (column right), sorts angular momentum of photons find (t) or relativephase of clocks. Since gel clocks are fixed, resonant oscillations of tree deliver input-like-output using transformation function (t)x= cost;y= sin tsinm(t2);m= 1−7. B. ON2 is shown using nested spheres or clocks, each layer of the primary deep learningnetwork holds a new deep learning network. Three coupled deep neural networks run together, ϕ, AF,W, B are 3D clockassembly, activation function, weight and bias respectively, in a deep neural network, i, b, S, N, C depict, invariant, beating, seed,nested loop, condensed loop respectively. Here, s is layer, or depth of the deep learning network. ∅s+1i = AF((Ws+1i ∗ ∅sb)+ Bsi)synthesizes invariants as materials. ∅s+1b = AF((Ws+1b ∗ ∅sS)+ Bsb) delivers 4D, 5D and 6D. ∅s+1S = AF((Ws+1S ∗ ∅sN)+ BsC)deliversCA, ∗ is dot product. C. Organic gel stores structures of 4D, 5D and 6D invariants as tree. A red line denotes a spontaneouslychosen active classification route or user-defined route. Reproduced with permission from DepositPhotos. © bronsonlil90(Nicholas Flowers).For example, if a cube has embedded triangles in all of its 8 corners, then the geometric structureprocesses 83 compositions of variables consuming 8× 3= 24 clocks. One nanowire writes 12 clocks(Dodecahedron, geometric shape of light [11]), so we need 2 nanowires. Adding more complexity, if eachcorner of a triangle embeds a pentagon, its N 835 ∼ 107 (7962 624) variations using 24× 5= 120 clocks, weneed 10 nanowires. If each pentagon has a pentagon, then we get 8355 ∼ 3× 1034 for 120× 5= 600 clocks,we need 50 nanowires. In general, Nrp (variations tooptimise)→ N× r× p (nanowireneeded), p nanowires areused first, resultant structures integrate r elements and finally, they integrate to N elements. By countingloops or spheres in table T1, one can observe that gel computer identifies one of 107–1034 possible choices ina chemical beaker [11] in a four-layer deep learning network for almost all the problems. Consequently,irrespective of complexity, movie S1 showed that the pre-processing time (∼2 ms), processing time(∼10–15 min) and post-processing time (∼2 ms), are constant for all problems noted in table T1.For digital processing, the ‘computing complexity’ is related to ‘lines of codes’, programs in C++ orPython are made of loops (e.g. for-next loop) and their complexity varies On (Online text E). However,within-and-above growth for gel processing logarithmically reduces the resource demand, with increasingcomplexity of On, n number of resources is required. So, we have taken intractable open challenge problem(confirming water is dropped in a glass or not; figures S15 and S16) as benchmark.3Neuromorph. Comput. Eng. 3 (2023) 044007 P Sahoo et alFigure 2. Live holographic visuals of supramolecular synthesis as nested deep learning network ON2: (A). Infra-red, IR tunableoptical signals erase informations or selective clock assembly CA by melting nanowires of particular dimensions. Spectrumanalyzer, SA1 and SA2 are beat signals produced by reflected and transmitted He–Ne laser signals by nanowire to electromagneticfield pattern in chemical beaker, sensed by Fabry Perot photo-detectors, which acts as SA. (B). Used gelator is(S)-Phenyl-tetradecanoylamino-acetic acid methyl ester in Hexane solution, sliced rectangular frame of cheetah fed in 12 rowsand 22 columns and fed every 12 pixels in a column at a time using 12 Yagi antennas arranged all around the GT (figure S1). Ascheetah runs, helices add new pitches, as gelator fill cage of fields. (C). One-to-one correspondence between pixel intensity ofinput data, resonance bands of superstructures and the optical vortices when no information is fed. This is background data. (D)When em loops condense to create first seed supramolecule, orthogonality between SA1 and SA2 signals is checked byOscilloscope, Os. N intertwined loops made of iso-frequency paths in the chemical beaker shifts periodicity xi due todipole–dipole interactions between helical nanowires, builds N vibrating modes, ∅sN = Nexp[−a∑Ni=1 xi2 + 2b∑Ni,j xixj + . . .]written as Asijk...s, a dodecanion tensor. When nanowire made loops condense, they hold the relative orientation of loops, theresonant oscillations governing conformal transition holds the relative phase θi between pair of loops Ki and Kj, building adifferential clock with a period ti, N vibrating modes transform to ∅sS =∑Ni (Ki −Kj)/2[cos2θi − t2i sin2θi]1/2written as As ′ijk..s, adodecanion tensor. SA1 and SA2 phases make transformation function f(t), the phase plot is shown to the right. (E). Three rowsshow live hologram, as monochromatic plane-polarized laser 633.5 nm (1 mW, He–Ne) is shined (figure S3) semiconductorcamera captures live hologram (movie S2). Four levels share elements of periodicity shifts in hologram followingWsS = α(x1x2 + x3x4)+β (x1x3 + x2x4)+ γ (x1x4 + x2x3); α, β,γ are coupling coefficients. During four condensations,energy emitsΨ n =∑ni gi following a phase space with 12 singularity domains, in which n holes are open, followingBsC = Lh̄gi = Lh̄(3∑cosxi + 4∏cosxi), where L= l1 + il2 + . . . .+ sl12, l is orbital angular momentum, xi = Cosθi + eiliϕ iSinθi,where ϕ i =∑ns=1∂xi∂KiAsijk..s is azimuthal angular momentum, θi relative phase between resonant oscillations of a pair of loops.Ψ n is seen an optical vortex assembly, rotating photon-condensate. The Hamiltonian driving the system isH= ∅sN + ∅sS +WsS + BsC. Equivalent CAs and corresponding structures SEMs are shown.Recent 3D printers use 3D electromagnetic field distribution in a cavity to print an entire object at once[12]. We built a similar 3D printer where we feed input static or dynamic data pixel by pixel aselectromagnetic frequencies of 10–15 nW power by an antenna array into the chemical beaker (figure 2(A);movie S2). Cavity geometry and the frequencies are so tuned that interference by reflection from cavity wallsbuilds intertwined isofrequency loops similarly to periodic events or loops in the input data (figure 2(B)). Itis a large structure. Along the loop path, identical helical nanowires couple due to dipole–dipole interactionform weak bonds. Finally, all weakly bonded intertwined loops condense into a gel superstructure or seedsupramolecule (online text A). Helical nanowire and all seeds made from it emits phase correlatedelectromagnetic signals that could be written as a 3D clock assembly. It is our data structure. Self-assemblinggelator molecules absorb it as an energy packet. Use it to optimize helical nanowire’s and seed structureslength, pitch, and diameter (figure S1), so that when we shine monochromatic light to read data, time periodand phase gap of recurring events in unknown data are written as two angular momentums and polarity onthe rotating photons or optical vortices. For complex data with many clocks, vortices condense into anintegrated hologram, by sorting rings [13] input video or static data could be retrieved, analyzed (figures S2and S3). A Cheetah video is fed-and-retrieved (figures 1(A), S4 and S5; movies S3, S4; online text B) todemonstrate all kinds of invariances (movie S5), position, rotation, size, count, environment as part ofgeneral-purpose computing (movie S6; online text C). Thermal energy∼13 Cal used to melt organic geldrives computation, but it needs 10–15 nW continuous em signals carrying the input data.We have been developing a programmable fractal synthesis for a decade where morphology, speed, andreaction kinetics for self-assembly are tuned remotely from a single molecular precursor to visible scale [14].We run a similar supramolecular synthesis so that gelator solution can build clocks roughly from4Neuromorph. Comput. Eng. 3 (2023) 044007 P Sahoo et alnanoseconds to hours long. 1012 orders of temporal ranges are encoded as loops in 106 orders of spatialvariation simply by tuning the basic geometry of nanowire and seeds (figure 2(C)). Spatio–temporal densityof clocks or resolution and time-range determine computing strength of a gel, not number of fibers. Thus,the concepts of scalability, resources, speed are redundant.Periodic events hold intelligence in random instances [15], but no efforts were made to organize clocksinto an integrated time-structure [16] for advanced computing. The gel emulates intricate relations ofperiodic events in space and time far from physical reality. The coupling factor (α,β,γ) between periodicevents or loops determines physical separation between clocks in the 3D clock assembly, not actualseparation. Fixed coupling offers a protein-folding like transfer function [17] delivering input-like output(figure 2(D)). Fractal growth shrunks real world’s space and time by power law for gel processing. Powerindices (P) are determined by the coupling strength of periodic events or clocks.Irrespective of the actual space and time, by choosing pixel to antenna frequency conversion table, thelimiting times of input events are densely packed into the fastest time domain of gel and smallest spatial scaleso that beating-driven self-assembly fills slower time domains. By reducing beating clocks to one,self-assembly converges computation.Resonating nanowires in the seed generated by user encoded 3D clock assembly try to synchronize.Closely spaced, nearly similar frequency clocks generate beating signals, forms a 3D clock assemblyinternally, in 103 orders slower time domains than external input. So, two variants of 3D clock structurescoexist, from external input and thenceforth internal beating (figures 1(A) and 2(D)). Once beating 3D clockassembly becomes a target, surface potentials of seed hold choices. They reorient and collide astronomicalways to condense into optimized supramolecule by growing and shrinking like a feed-forward neuralnetwork. Thus, beating assists a supramolecular growth by filtering differential clocks from the inner layeremitting complex wave [18]. Therein, seed structures in a plane synchronize at resonance and vibrate at asingular beat frequency (figure 2(D)). Several such planes form a 3D clock assembly with a few clocks. It is anew target that triggers self-assembly. The resultant seed structure’s clocks arrange along discrete lines inspace. For each line, we get one effective beat frequency and a new 3D clock assembly form surface of atopology. When it drives self-assembly, the final structure does not beat. Three transitions, filter andmemorize point, line, plane, and 3D shapes made of clocks. These abstract geometric manifolds (figures 1(C)and 2(E)) of resonance frequencies are invariants, since fractal growth is equivalent to orthogonaltransformations of two tensors representing resonant clocks of gel fibers, for all problems (movies S7–S12;figures S6–S16; table T1). ON2 is a type of geometric deep network [19], where extracting differential clocksin the orthogonal space is like extracting a common sphere between overlapping clocks. The number oflayers of self-assembly required to complete three transitions depends on the composition of symmetries inthe geometry of input clock-assembly. Thus, steps needed to reduce symmetries is the number of hiddenlayers in its neural network.Since multiple layers of nanowires grow one above another, the interlayer beating adds azimuthal ororthogonal angular momentum in addition to orbital angular momentum (figure 2(C)). In ‘figure 3’, weperformed mathematical operations on the gel using microwave input and optical read-out. Since dilutinggel solution, the effective separation between nanowires could be increased deriving holograms for fewnanowires; we map step by step synthesis of fractal growth (figure 2(E)). Since nanowires build asuperstructure that acts as an elementary unit for the next layer, we get hierarchical structures. Thehierarchical assembly and the corresponding hologram in ‘figures 3(A) and (B)’. Normally we do not see thedifferential clocks or holographic parts exclusive to invariants. However, by applying suitable ac signals usingan antenna from outside, we can amplify electromagnetic beating signals in the fractal boundaries ofdifferent layers, as shown in ‘figure 3(C)’. Then we orthogonally project two holograms in pairs; 4D and 5Dshow only superposed invariants post projection. Similar orthogonal superposition of 5D and 6D hologramsdiminishes the holographic part due to 3D clock assembly. Only the invariant parts survive as a staticstructure in ‘figure 3(D)’.The use of antennas to read invariants live prompts us to isolate key invariant parts of the opticalhologram when Cheetah video was fed to gel (see deconvolution 5). Thus far, explainable AIs and deeplearning protocols select classifiers from a given database. ‘Figures 2(D) and 1(C)’ shows that gelsuperstructures resonantly activate geometrically similar peaks. Thus, creating new invariants and classifiersto analyze the unknown problem.‘figure 4(A)’ shows a chart of invariants for Cheetah where from 3D clock assemblies, higher-levelinvariants were found as new slower clocks in a network of clock-assemblies (following figure 1(C); tableT2). Once we get a gel trained by a single shot Cheetah video, we fed different four-legged animal videos tothe gel (movie S6). Since invariant parts of holograms are generated by beating or interference of boundaryoscillations of two sub-superstructures of gel, they could synthesis new invariants or clock assemblies usingmemorized ones as outlined in ‘figure 4(B)’. Each invariant forms an isolated cluster with a distinct5Neuromorph. Comput. Eng. 3 (2023) 044007 P Sahoo et alFigure 3. Experimental evidence for mathematical proof of invariants for Cheetah: (A). Elementary mathematical operationscarried out by feeding Cheetah video through gel precursors as the structures grow from singlet helical nanowire to doublet to apair of doublets or namely a multiplate (right to the left). Four such multiplates build a mesh. Five meshes could be investigated atmax by adjusting solvent precursor ratio, taking a sample from solution, carrying out SEM (first row, right to left). (B) Below eachSEM image, the corresponding optical vortex assembly Z is shown. (C) In the third row, the wireless antenna is switched ON toamplify a few bands by pumping additional electromagnetic signals that are orthogonal transformed vortices SV (∅1) SP(∅2), andSL(∅3). (D) After orthogonal transformation, by using mirrors and feeding two new electromagnetic signals to the gel, the dotproducts of SV (∅1) & SP(∅2), and SP(∅2), & SL(∅3) are derived. Frequencies are tuned to amplify SV+ SP and SL+ SP, We findsignificant disappearance of clocks other than the invariant elements of the invariant tree. To the left, an image of the cheetah’sinvariant tree is shown. Two 3D clock assemblies, one memorized (Aijk) and the other unknown input (A/ijk) resonating withmemory. Deformation η (η = (A/ijk −Aijk)/Aijk) in the 3D clock assembly for train and test datasets are taken as differential∂η/∂∅ signal along three orthogonal axes ∅1, ∅2, ∅3, in general ∅S. The plot is R3 : (x,y,z)→ (ϕ 1,ϕ 2,ϕ 3), The invariantcondition: partial derivatives of S, with respect to η,(∂Sϕ 1∂η):(∂Sϕ 2∂η)vanishes when ϕ 1 ̸= ϕ 2 (A : B= tr(ABT)). T denotestranspose, tr trace.resonance signature; hence they get wirelessly connected by electromagnetic resonance. No circuit is needed.A new animal fed to a single shot trained gel tries to activate its own invariant triplet. The differencesbetween two trees in beating activate matching invariants, they form a new invariant-tree (figures S9 andS13). On derived tree-edges, matching clocks stored inside are put to create a new 3D clock assembly, oroutput (figures 1(C) and 4(C)). Thus, gel composes a problem-specific invariant network, adds new clocksanalyzing it without training.Thus far, automatons failed to discover generic invariants because variables governing complex events innature are interconnected in a tree-like dynamic network [20]. A deep learning tool has to discover thatdynamic invariant network grown within and above. An algorithm has to explore combinations as the powerof clocks self-assemble to optimize these fractally connected astronomical variations. In a chemical beaker,geometric shapes made of resonance peaks are stored. They find way to arrange emulating the compositionof complex shapes in a problem by resonance, its inverse of computing. By that, gel solves an intractableClique problem [21]. Fractal growth initiates fractal resonance chain, but, clique matching is assisted bydifferential clocks. In the 1960s, deformation along three axes was filtered along three orthogonal axes to findinvariant [22], the organic gel does it here (convolution for belief, 9), but with a difference. It finds invariantsof differential clocks along one axis for the 4D database. Then its orthogonal 5D database is made of 4Dinvariants. Thus, clique error in 4D is minimized in 5D.To prove that extracting three orthogonal invariants from the spatial assembly of clocks isgeneral-purpose computing, nine open challenge problems in AI were selected. Gel detected invariant treesfor nine different static and dynamic inputs for problems varying from genetic data for diabetes, coronavirus,swarm intelligence, the complex composition of classical songs, the open classification challenge of pouringwater, and detecting the face of a Japanese lady (figures S6–S16; movies S7–S12). In all cases, after a singletraining, the gel classified unknown events naturally, or the user could choose an exclusive classifier. The6Neuromorph. Comput. Eng. 3 (2023) 044007 P Sahoo et alFigure 4. Invariant-tree driven synthesis of new classifiers as a dedicated analysis protocol. (A) Four columns show four phases ofclassifications to analyze events without training. The first column shows single shot learning of a running Cheetah dynamics by agel. The second column shows the determination of spontaneous classifiers by gel in its vortex assembly. The third column showscombining different classifiers to synthesize new sets of classifiers. The fourth column shows how synthesized classifiers are usedto analyze events for which the gel was not trained (less than one shot or LO shot learning; movie S7). Identify the classifiersspontaneously and sort test sets by solving the Clique problem: train & test structures are matched. The classification efficiencyscore is the combination of three parameters (online tables T2–T8). The ratio of maximum overlapping area between the inputand output, the ratio of differences in the clocks used relative to the number of clocks used, and weighted average angulardifferences between different geometric planes. 3D geometric shape mismatch is estimated in the detection score. (B) The livegrowth in the gel superstructure, synthesis of combinatorial classes adopts distinct structural symmetries which couple byelectromagnetic resonance of frequency triplets (υ1, υ2, υ3). (C). Invariant matching is shown by superposing train and the test3D clock assemblies. It starts from SL, transcends to SP, and finally reaches SL. Then the nested loops run. Differential ∂η/∂∅signals along three orthogonal axes ∅1, ∅2, ∅3, in general ∅S.tables outlining high classification scores are shown online (tables T2–T8). Gel naturally labels objects,classes, invariants using distinct time ranges of neurons; stores for decades without refreshing.The same gel was melted and reused for all problems. Computation times for all problems appearedidentical,∼10 min. To enhance optical resolution, we use well-established methods to break the diffractionbarrier [23, 24]. Please find supporting online Video 2 of our recent report [9] for evanescent wave-inducedamplification of a nanowire generated 3D vortex structure. One could solve multiple problems in one gel at atime, in a single shot. So, multiple labeling or co-synthesizing varied datatypes are feasible. Thus far,parallelism meant sequential hardware arranged parallely, here, its fractal deep learning, layers of neural netsgrow within and above, not side by side.Data availability statementAll data that support the findings of this study are included within the article (and any supplementary files).All data are available in the manuscript or the supplementary materials.7Neuromorph. Comput. Eng. 3 (2023) 044007 P Sahoo et alAcknowledgmentsAuthors acknowledge the Asian office of Aerospace R&D (AOARD), a part of the United States Air Force(USAF), for the Grant No. FA2386-16-1-0003 (2016–2019) on the electromagnetic resonance-basedcommunication and intelligence of biomaterials.Author contributionsA B Conceptualized the research; P Sa Did the experiment and data analysis; P S, K S and S G assisted inbackground data development, P Sa analyzed the result and plotted the data; A B and P Sa wrote the paper, RP S, R B, J P H and T N reviewed the optical vortex studies.Conflict of interestThe authors declare no competing interests.Appendix. Turing completeness analysis from a computer science perspectiveIn six classes of problems summarized in table T1 online, the input video or static dataset is first convertedinto a 3D clock assembly written as an electromagnetic spectrum. Although they look different from bits,input 3D clock assembly is the first TM (see the tape in figure 2(E)), and the clock/vortex parameters definethe cell states of a Turing tape that is processed by a nanowire. Similar to a TM, a nanowire can read andwrite data to an arbitrarily long sequence of clocks or tape by changing its geometric parameters(figure 2(C)), which act as its memory. Output can be read using refracted and transmitted optical vorticesfrom a helical nanowire (One clock= one vortex= one variable [11]).The output of a nanowire TM is a 3D assembly of optical vortices sent to the other members of thenanowire assembly as input [11]. As multiple nanowire TMs reorient in a 3D arrangement following Hasse’slaw, the necessary Turing tape manipulations are made via essential state transitions to derive an invariant foran accurate output. The nanowire assembly can perform a sequence of instructions a certain number oftimes or until a certain condition is met, satisfying the criteria to be a UTM. The loop counter for the UTM isdetermined by the ratio of coupled vortices or the number of phase singularity points in the optical vortexassembly. Its grammar is listed in figure S13 and musical problems like figure S6 is a perfect example, becausewithout high precision counter, there is no music, all that one would get is noise.However, a nanowire assembly can process long stretches of single or smaller nanowire assemblygenerated vortex assemblies without the need for external input, much like a UTM that can computefollowing Hasse’s law without external control. Thus, the nanowire assembly is a TM and runs singlenanowire made TMs similar to ribosomes running m-RNA, satisfying the criterion for a UTM ([25, 26]).Running part of algorithm is shown in figures S15 and S16, where a benchmark open challenge was solved.Gel ran multiple TMs made of isolated clusters in a global TM, yet addressed all specific features of localissues.Furthermore, all problems in table T1 online have two phases, solving one part of a problem and usingthe derived 3D nanowire assembly or processing circuit to solve a faintly associated but new problem forwhich it was not trained (figures S9, S10, S11 and S12). Thus, the nanowire assembly is a Turing completegeneric algorithm processor and can compute any computable function as listed in figure S13.Finally, to be a true UTM, the nanowire assembly must perform Boolean algorithms. While reading asingle nanowire TM generated vortex assembly, the nanowire assembly measures polarization of all clocks,density, and sum of phase singularity points, 3D coordinates of vortices, and adjusts its own configuration toinclude suitable symmetries. Polarization of vortices or clockwise (1) and anticlockwise (0) rotation of anoptical vortex is essentially used here to encode Boolean codes. Instead of (0,1) it is a rotating circle orpolarized optical vortex. The phase singularity points on a vector vortex beam or perimeter of the ring oflight counts the number of loops to run for executing a program. So, UTM has a counter for an instructionalor conditional halt (see falcon attacking birds figure S12). Polarization of vortices leads to constructive anddestructive interference that ensures a logic gate-like Boolean operation, execute conditional stop, run,branching out, and wait for a value to arrive (figure S13).Previous reports have described how to write various existing computer’s data formats into a 3D clockassembly (Online text D); Chapter 4 [27]). Consequently, any algorithm can be written using a 3D clockassembly, and its distinctive features are processed as a separate engine in the nanowire assembly or UTM inthe chemical beaker. However, a reader should note that a true UTM cannot exist in principle, all so calledUTMs partially satisfy the criterion.8Neuromorph. Comput. Eng. 3 (2023) 044007 P Sahoo et alORCID iDsPushpendra Singh https://orcid.org/0000-0002-7274-6683Ryad Benosman https://orcid.org/0000-0003-0243-944XTomonobu Nakayama https://orcid.org/0000-0001-9696-475XAnirban Bandyopadhyay https://orcid.org/0000-0002-8823-4914References[1] Rosenblatt F 1961 Perceptrons and the theory of brain mechanics (Cornell Aeronautical Lab Inc.) VG-1196-G 621[2] Navlakha S and Bar-Joseph Z 2011 Algorithms in nature: the convergence of systems biology and computational thinkingMol. Syst.Biol. 7 546[3] Lin X, Lin X, Rivenson Y, Veli M, Luo Y, Jarrahi M and Ozcan A 2018 All-optical machine learning using diffractive deep neuralnetworks Science 361 1004–8[4] Bandyopadhyay A, Pati R, Sahu S, Peper F and Fujita D 2010 A massively parallel computing on an organic molecular layer Nat.Phys. 6 369[5] Singh P, Sahoo P, Saxena K, Manna J S, Ray K, Ghosh S and Bandyopadhyay A 2021 Cytoskeletal filaments deep inside a neuron arenot silent: they regulate the precise timing of nerve spikes using a pair of vortices Symmetry 13 821[6] Bandyopadhyay A, Ghosh S and Fujita D 2020 Universal geometric-musical language for big data processing in an assembly ofclocking resonators US Patent App. 16/635,900 JP-2017-150171, 8/2/2017: World patent received February 2019, WO 2019/026983[7] Cohen U, Chung S, Lee D D and Sompolinsky H 2020 Separability and geometry of object manifolds in deep neural networks Nat.Commun. 11 746[8] Goodfellow I et al 2009 Measuring invariances in deep networks Advances in Neural Information Processing Systems vol 22[9] Honglak L, Grosse R, Ranganath R and Ng A Y 2009 Convolutional deep belief networks for scalable unsupervised learning ofhierarchical representations Proc. 26th Int. Conf. on Machine Learning (Montreal, Canada)[10] Singh P et al 2020 Quaternion octonion to dodecanion manifold: stereographic projections from infinity lead to a self-operatingmathematical universe Advanced Intelligence Systems and Computing (Springer) p 1169[11] Sahoo P, Singh P, Manna J, Singh R P, Hill J P, Nakayama T, Ghosh S and Bandyopadhyay A A 2023 Third angular momentum ofphotons Symmetry 15 158[12] Loterie D, Delrot P and Moser C 2020 High-resolution tomographic volumetric additive manufacturing Nat. Commun. 11 852[13] Junliang J, Chang Z, Yang H, Liu Q, Wang F, Gao H, Li F and Zhang P 2019 Mode sorter designed for (de)multiplexing vectorvortex modes Appl. Opt. 58 7094–9[14] Ghosh S, Dutta M, Ray K, Fujita D and Bandyopadhyay A 2016 A simultaneous one pot synthesis of two fractal structures viaswapping two fractal reaction kinetic states Phys. Chem. Chem. Phys. 18 14772–5[15] Blake A 1942 Seismic periodicit Nature 149 308[16] Winfree A 1977 Biological Rhythm Research 8, 1; The Geometry of Biological Time (Springer) 2nd edn[17] Okeyoshi K, Kawamura R, Yoshida R and Osada Y 2015 Microtubule teardrop patterns Sci. Rep. 5 9581[18] Su H, Niu B, Li H, Liu F, Yuan T, Chen H-Y and Wang W 2020 Evanescent wave-guided growth of an organic supramolecularnanowire array Angew. Chem., Int. Ed. 59 19209–14[19] Monti F et al 2017 Geometric deep learning on graphs and manifolds using mixture model cnns Proc. IEEE Conf. on ComputerVision and Pattern Recognition pp 5115–24[20] Buhlmann P 2018 Invariance in heterogeneous, large-scale and high-dimensional data Proc. Int. Congress of Mathematicians (Rio deJaneiro, Brazil) pp 2785–800[21] Rhodes N, Willett P, Calvet A, Dunbar J B and Humblet C 2003 CLIP: similarity searching of 3D databases using clique detection J.Chem. Inf. Comput. Sci. 43 443–8[22] Criscione J C, Humphrey J D, Douglas A S and Hunter W C 2000 An invariant basis for natural strain which yields orthogonalstress response terms in isotropic hyperelasticity J. Mech. Phys. Sol. 48 2445–65[23] Hao X, Kuang C, Gu Z, Wang Y, Li S, Ku Y, Li Y, Ge J and Liu X 2013 Frommicroscopy to nanoscopy via visible light Light Sci. Appl.2 e108[24] Hao X, Kuang C, Li Y and Liu X 2013 Evanescent-wave-induced frequency shift for optical superresolution imaging Opt. Lett.38 2455–8[25] Caetano-Anollés G and Caetano-Anollés D 2015 Computing the origin and evolution of the ribosome from itsstructure—uncovering processes of macromolecular accretion benefiting synthetic biology Comput. Struct. Biotechnol. J. 13 427–47[26] Shapiro E 2012 A mechanical Turing machine: blueprint for a biomolecular computer Interface Focus. 2 497–503[27] Bandyopadhyay A 2020 Nanobrain: The Making of An Artificial Brain From a Time Crystal (Taylor & Francis Inc. Imprint CRCPress Inc.) p 3369https://orcid.org/0000-0002-7274-6683https://orcid.org/0000-0002-7274-6683https://orcid.org/0000-0003-0243-944Xhttps://orcid.org/0000-0003-0243-944Xhttps://orcid.org/0000-0001-9696-475Xhttps://orcid.org/0000-0001-9696-475Xhttps://orcid.org/0000-0002-8823-4914https://orcid.org/0000-0002-8823-4914https://doi.org/10.1038/msb.2011.78https://doi.org/10.1038/msb.2011.78https://doi.org/10.1126/science.aat8084https://doi.org/10.1126/science.aat8084https://doi.org/10.1038/nphys1636https://doi.org/10.1038/nphys1636https://doi.org/10.3390/sym13050821https://doi.org/10.3390/sym13050821https://doi.org/10.1038/s41467-020-14578-5https://doi.org/10.1038/s41467-020-14578-5https://doi.org/10.3390/sym15010158https://doi.org/10.3390/sym15010158https://doi.org/10.1038/s41467-020-14630-4https://doi.org/10.1038/s41467-020-14630-4https://doi.org/10.1364/AO.58.007094https://doi.org/10.1364/AO.58.007094https://doi.org/10.1039/C6CP00447Dhttps://doi.org/10.1039/C6CP00447Dhttps://doi.org/10.1038/149308a0https://doi.org/10.1038/149308a0https://doi.org/10.1038/srep09581https://doi.org/10.1038/srep09581https://doi.org/10.1002/anie.202007319https://doi.org/10.1002/anie.202007319https://doi.org/10.1021/ci025605ohttps://doi.org/10.1021/ci025605ohttps://doi.org/10.1016/S0022-5096(00)00023-5https://doi.org/10.1016/S0022-5096(00)00023-5https://doi.org/10.1038/lsa.2013.64https://doi.org/10.1038/lsa.2013.64https://doi.org/10.1364/OL.38.002455https://doi.org/10.1364/OL.38.002455https://doi.org/10.1016/j.csbj.2015.07.003https://doi.org/10.1016/j.csbj.2015.07.003https://doi.org/10.1098/rsfs.2011.0118https://doi.org/10.1098/rsfs.2011.0118https://doi.org/10.1201/9780429107771 A general-purpose organic gel computer that learns by itself 1. Main text Appendix. Turing completeness analysis from a computer science perspective References