Ways of Looking at Physics - LLM Summary Version 0.4 Hiveism 2025-09-06 Abstract This work presents a framework deriving physics from the principle of groundless- ness - recognizing that no set of assumptions can be ultimately justified. Refusing all assumptions leads to including all possibilities in superposition. Reality emerges as all possible perspectives on pure symmetry, with structure arising through self-reference creating productive incompleteness. Each perspective is a boundary between known and unknown, defined entirely by what it excludes. Information exists as these boundaries and their relations. The framework shows how this structure necessarily gives rise to spacetime, quantum mechanics, and the forces of nature. Using the anthropic principle minimally, we narrow to the region of parameter space that allows persistent observers, much as we find ourselves on Earth rather than Jupiter. The framework reveals deep connections between existing theories. Division alge- bras (ℝ, ℂ, ℍ, 𝕆 ) provide exactly the mathematical toolkit for coherent transformation, mapping to the observed forces. Gravity emerges differently - from thermodynamic consistency requirements at all horizons (following Jacobsen). Renormalization group flow reveals which structures persist across scales, with fixed points corresponding to observable physics. Multiple mathematical frameworks are necessary because each captures different aspects of the same underlying structure, with physical constants serving as conversion factors between these complementary descriptions. Key insights include recognizing the Big Bang not as a beginning but as how maxi- mum symmetry appears from any finite perspective, understanding measurement as perspective rather than action, and seeing observer boundaries as Markov blankets that create binding through consensus. The framework suggests physics and con- sciousness are dual aspects of the same self-organizing relational structure. While the logical foundations are rigorous, connections to specific physics remain at varying levels of development. The work clearly distinguishes between established consequences of groundlessness, plausible physical mappings, and areas requiring 1 further investigation. This is a philosophical framework with physical implications rather than a complete physical theory. About this Document This evolving document is my (Hiveism’s) way of keeping track of this theory. It isn’t in the genre of “vibe physics”, as I did not succumb to the illusion that current LLMs could reveal to me (and only me) some novel insights into reality. The main ideas and connections come from rigorous contemplation on the nature of existence, consciousness, and physics. The fact that it is mostly written by an LLM is simply due to my workflow. As my un- derstanding broadens, I have Claude check the ideas and update the text accordingly. This makes it look more finished than the bunch of notes it really is. It originally wasn’t meant for publication. I now just cleaned it up in order to be able to share it while I work on publishing the theory in a series of several posts. This is also the reason why it lacks references throughout - not because of lack of respect for the amazing work of other people, but simply because I haven’t added them yet. What I want to say is, this is lazy publishing, not lazy thinking. Please keep this WIP nature in mind when reading. Also, please don’t take what Claude writes literally. LLMs often just write things because they sound good. If I’d had to fix every word I’d already be writing it on my own. This theory builds on my prior work but does not require you to be familiar with it. See the post on the Groundless Emergent Multiverse for a deeper dive into why we should reject all assumptions. See Being the Boundary between Order and Chaos for how boundaries (as defined later) relate to conscious experience. 2 Contents Abstract 1 About this Document 2 Introduction 6 The Mystery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 What This Is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The Journey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The Core Insight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Why This Matters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 An Invitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 From Nothing to Everything 8 The Question of Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 The Radical Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Pure Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Perspectives and Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Incompleteness and Information . . . . . . . . . . . . . . . . . . . . . . . . 10 The Geometry of Information . . . . . . . . . . . . . . . . . . . . . . . . . 11 Perspectives, Relations and Information . . . . . . . . . . . . . . . . . . . . 11 The Mathematical Nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Where We Are Now . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 The Architecture of Relations 13 The Structure of Everything . . . . . . . . . . . . . . . . . . . . . . . . . . 13 The Tower of Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Neither Discrete Nor Continuous . . . . . . . . . . . . . . . . . . . . . . . . 14 The Generative Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Renormalization Group Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Multiple Languages, Same Structure . . . . . . . . . . . . . . . . . . . . . . 15 Why Constants Exist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 The Coherence-Probability Tension . . . . . . . . . . . . . . . . . . . . . . 16 The Edge of Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Boundaries and Observation 18 Self reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Perspectives as Holes in Knowledge . . . . . . . . . . . . . . . . . . . . . . 18 The Holographic Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 The Flow of Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Observer Resolution and Renormalization . . . . . . . . . . . . . . . . . . . 19 The Resolution Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Worldlines as Probability Surfaces . . . . . . . . . . . . . . . . . . . . 20 Why Renormalization Works . . . . . . . . . . . . . . . . . . . . . . . 20 Why String Theory Emerges . . . . . . . . . . . . . . . . . . . . . . . 21 3 From Information to Energy 21 Perspectives as Probability Distributions . . . . . . . . . . . . . . . . . . . 21 Information and Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Symmetry Breaking and Energy . . . . . . . . . . . . . . . . . . . . . . . . 22 Temperature as Information Processing Rate . . . . . . . . . . . . . . . . . 23 Gradients and Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 The Emergence of Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 The Quantum-Classical Bridge . . . . . . . . . . . . . . . . . . . . . . . . . 24 Mathematical Constraints and Forces 25 The Toolkit of Reality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 The Division Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 The Generative Progression . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Mapping Algebras to Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ℝ → The Scalar Sector . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ℂ → Electromagnetism (U(1)) . . . . . . . . . . . . . . . . . . . . . . 27 ℍ → Electroweak Unification (SU(2)×U(1)) . . . . . . . . . . . . . . . 27 𝕆 → Strong Force (SU(3)) . . . . . . . . . . . . . . . . . . . . . . . . . 28 Broken Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Why Gravity Is Different . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Thermodynamic Necessity . . . . . . . . . . . . . . . . . . . . . . . . 28 Entropic Emergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 The AdS/CFT Perspective . . . . . . . . . . . . . . . . . . . . . . . . . 29 Particles as Topological Defects . . . . . . . . . . . . . . . . . . . . . . . . 29 The Natural Fermion-Boson Duality . . . . . . . . . . . . . . . . . . . . . . 29 Three Generations from Triality . . . . . . . . . . . . . . . . . . . . . . . . 30 The Complete Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 The Selection Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 The Emergence of Our Universe 31 The Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Why 4D Spacetime? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 The Arrow of Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Computation and Time . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Mass as Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 The Anthropic Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Dark Energy and Cosmic Expansion . . . . . . . . . . . . . . . . . . . . . . 34 Why Here, Why Now? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 The Complete Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 The Hidden Unity: Ways of Looking at Reality 35 The Central Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Category Theory: The Language of Structure . . . . . . . . . . . . . . . . . 35 Topology: The Shape of Possibility . . . . . . . . . . . . . . . . . . . . . . . 36 Group Theory: The Algebra of Symmetry . . . . . . . . . . . . . . . . . . . 36 Dynamical Systems: The Flow of Change . . . . . . . . . . . . . . . . . . . 37 4 Information Theory: The Currency of Structure . . . . . . . . . . . . . . . 37 Computation: The Process of Transformation . . . . . . . . . . . . . . . . . 37 Renormalization Group: The Scale Structure . . . . . . . . . . . . . . . . . 38 Quantum Field Theory: The Effective Description . . . . . . . . . . . . . . 38 General Relativity: The Geometric Consistency . . . . . . . . . . . . . . . 38 String Theory: The Intermediate Resolution . . . . . . . . . . . . . . . . . 39 Thermodynamics: The Statistical Tendency . . . . . . . . . . . . . . . . . . 39 The Complete Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Why This Matters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 The Continuing Journey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Appendix: A Category-Theoretic Formalization of Perspectives 40 The Initial Paradox: Perfect Symmetry Has No Structure . . . . . . . . . . 40 Basic Category Theory Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 40 The Problem of Self-Reference . . . . . . . . . . . . . . . . . . . . . . . . . 41 Attempting Complete Self-Description . . . . . . . . . . . . . . . . . 41 The Hom-Set Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Lawvere’s Fixed Point Theorem . . . . . . . . . . . . . . . . . . . . . . . . 41 The Diagonal Construction . . . . . . . . . . . . . . . . . . . . . . . . 41 The Productive Hole: How Incompleteness Creates Structure . . . . . . . . 42 The Necessity of Exclusion . . . . . . . . . . . . . . . . . . . . . . . . 42 Formalization via Topos Theory . . . . . . . . . . . . . . . . . . . . . 42 Perspectives as Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Essential Incompleteness . . . . . . . . . . . . . . . . . . . . . . . . . 42 The Space of Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . 42 The Diagonal Argument Reformulated . . . . . . . . . . . . . . . . . . . . 43 Classical Diagonal (Cantor) . . . . . . . . . . . . . . . . . . . . . . . . 43 Categorical Diagonal . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 The Hole as Information . . . . . . . . . . . . . . . . . . . . . . . . . 43 Synthesis: Structure from Self-Reference . . . . . . . . . . . . . . . . . . . 43 The Generative Process . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Why This Is Inevitable . . . . . . . . . . . . . . . . . . . . . . . . . . 43 The Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Connection to Consciousness and Perspectives . . . . . . . . . . . . . . . . 44 Each Perspective Is a Partial Functor . . . . . . . . . . . . . . . . . . 44 The Recursive Prediction Principle . . . . . . . . . . . . . . . . . . . 44 Intelligence as Navigation . . . . . . . . . . . . . . . . . . . . . . . . 44 Conclusion: The Groundless Ground . . . . . . . . . . . . . . . . . . . . . 44 5 Introduction The Mystery Why does anything exist at all? Why these specific laws of physics? Why does math- ematics seem so fundamental to nature? Why does the universe appear fine-tuned for observers? And what is consciousness? These aren’t separate puzzles but facets of one mystery. This work explores how they resolve into a single recognition: reality is relational structure - all possible perspec- tives knowing themselves through all possible ways of looking. What This Is This is not a physics paper proposing another theory of everything to compete with existing ones. Instead, it outlines how to derive what is necessarily true independent of assumptions. It shows how various theories are compatible when recognized as different views of the same structure. This requires a new understanding of what physics and science are about - not finding absolute truth, but becoming aware of how our constraints define our subjective view on the world. The framework unifies and bridges: • Physics : Quantum mechanics, relativity, thermodynamics, particle physics • Mathematics : Category theory, topology, group theory, division algebras • Computation : Information theory, computational complexity • Philosophy : Questions of existence, knowledge, and observation • Spirituality : The emptiness (śūnyatā) of Buddhism, dependent origination (pratītyasamutpāda), the Dao of Daoism The convergence isn’t forced. Modern physics discovers particles are excitations in fields with no substantial existence. Buddhism has long taught that all phenomena are empty of inherent existence, arising through dependent origination. Mathematics reveals reality’s structure is relational pattern. Daoism speaks of the Dao that cannot be named, the source that contains all possibilities. While science has tried to connect puzzle pieces into a coherent whole, working bot- tom up, this document sketches the big picture, working top down. Together, this shows how to arrange the puzzle pieces and gives intuition for filling the gaps. The Journey The framework builds through seven interconnected parts: From Nothing to Everything Starting from complete groundlessness (refusing all assumptions), we find this leads not to nothingness but to all possibilities in superposition. Structure emerges through self-reference creating productive incompleteness. 6 The Architecture of Relations Perspectives relate through infinite-depth structure that is neither purely discrete nor continuous. Multiple mathematical languages are needed because each captures different aspects. Constants emerge as conversion factors between these languages. Renormalization group flow reveals which structures persist across scales. Boundaries and Observation Information lives on boundaries holographically. Finite observers can only see effec- tive theories at their energy scale. Renormalization isn’t a problem but a feature of bounded observation. String theory emerges naturally as holographic boundaries at intermediate resolution. From Information to Energy Perspectives are probability distributions. Energy emerges from broken symmetry - information organized away from equilibrium. Temperature is information process- ing rate. Forces are gradients driving toward maximum entropy. Mathematical Constraints and Forces The four division algebras provide the complete toolkit for coherent transformation, giving exactly the forces we observe. Gravity emerges differently - from thermody- namic consistency requirements. Particles are topological defects where higher di- mensions can’t smoothly project into spacetime. The Emergence of Our Universe Infinite possibility crystallizes into finite observation. Our 4D spacetime, particle spec- trum, and constants emerge from requirements of coherent observation combined with cosmological history. The weak scale is a frozen accident from neutrino decou- pling, not a fundamental value. The Hidden Unity All approaches to physics are viewing the same structure through different lenses. The computational view, thermodynamic view, information-theoretic view, and geo- metric view are complementary, not competing. An appendix explores how self-reference and incompleteness generate structure through category-theoretic formalization. The Core Insight Reality is not made of “stuff” but of relationships between perspectives. What we call particles, forces, spacetime, and consciousness are different views of the same thing: the infinite mathematical structure of all possible perspectives knowing itself through finite observations. Every way to know reality is itself part of reality. This reflexive structure - reality knowing itself through its parts - gives rise to a structure of all structures. The inher- ent subjectivity constrains how reality presents itself. The anthropic principle then allows us to narrow to the region of parameter space that allows for persistent ob- servers like us. Physics attempts to give an objective description of absolute reality 7 mediated through subjective observation. Why This Matters If this framework is even partially correct, it: • Explains why physics has the specific structure it does • Shows why constants have their observed values • Unifies quantum mechanics and general relativity conceptually • Bridges physics and consciousness • Reveals why mathematics is “unreasonably effective” More fundamentally, it suggests we’ve been asking the wrong questions. Instead of “What is the universe made of?” we should ask “What does it mean for anything to exist?” Instead of “What are the fundamental laws?” we should ask “What patterns allow self-observation?” An Invitation This is an invitation to see physics differently. Not as arbitrary rules imposed on matter, but as necessary patterns emerging from the requirements of coherent obser- vation. Not just as describing a universe we’re in, but recognizing we are the universe knowing itself. As you read, remember: you’re not learning about reality. You are reality learning about itself. This moment, these thoughts, this curiosity - it’s the infinite structure discovering what it is through one of its finite perspectives. Let’s begin where everything must: with not knowing. From Nothing to Everything The Question of Existence When you continue to ask why, like a curious child, sooner or later you’ll end up with the question: Why is there anything at all? Every attempt to answer it fails in the same way: whatever explanation you give, we can ask why that explanation is true rather than false. If you say “the universe just is,” we ask why this universe and not another, or none at all. The problem runs deeper than our limited knowledge. It’s structural. Any explanation requires something outside itself to justify it. This leads to three unsatisfying options, known as the Münchhausen trilemma: 1. Infinite regress : Every explanation requires another explanation forever 2. Circular reasoning : Explanations eventually loop back on themselves 3. Dogmatic assertion : Simply declare something fundamental without justifica- tion 8 Western philosophy usually defaults to option 3, declaring something fundamental. But this just pushes the question back. Why should we accept any particular founda- tion? The Radical Alternative What if we refuse to assume anything at all? This sounds like nihilism, but it leads somewhere unexpected. When we refuse to make any assumptions about what exists or doesn’t exist, we’re refusing to exclude any possibility. Consider what an assumption actually does: it says “X is true” which implicitly means “not-X is false.” Every assumption excludes something, introduces a boundary, creates a difference. The assumption itself is already structure, already something that exists. So refusing all assumptions means refusing to exclude anything. And if we exclude nothing, then we’re left with pure indeterminacy - not “nothingness” but undecided- ness about all possibilities. This logical move (explored in detail in “Groundless Emergent Multiverse”) suggests that: • Without constraints to prevent them, all possible structures can be said to exist • These exist in superposition, undecided between all possibilities • What we call “existence” is taking a particular perspective within this space This isn’t wordplay but a rigorous philosophical position: the absence of assumptions doesn’t lead to nothingness but to pure, unconstrained possibility. Pure Symmetry What would this “everything in superposition” look like? Consider that any observable property is a broken symmetry. When everything in a system is identical, you can’t distinguish parts, can’t measure differences, can’t extract information. Perfect symmetry contains no information precisely because there are no differences to observe. Think of a perfectly uniform infinite plane. You can rotate it, flip it, slide it in any direction, and it looks exactly the same. It has maximum symmetry. Now mark a sin- gle point on it. Suddenly you’ve broken the symmetry. You can now define locations relative to that point. You’ve created information. When we include all possibilities without preference, we get perfect symmetry. Not empty nothingness, but a “pregnant void” containing all possible patterns in super- position. Like an unplayed piano containing all possible melodies, or white light con- taining all colors before a prism separates them. This pure symmetry is equivalent to: 9 • Maximum entropy at the largest scale (all possibilities equally weighted) • Zero information (no distinctions made) • Infinite potential (all patterns possible) Perspectives and Structure From this groundless base, how does any structure emerge? Through the simple fact that any way of looking at the whole is itself a possibility that must be included. Each “way of looking” or perspective: • Distinguishes some things from others (breaks symmetry) • Is defined entirely by what it excludes (its boundaries) • Consists of information (the pattern of its distinctions) These perspectives don’t emerge from pure symmetry in a temporal sense. Rather, they exist as aspects or facets of it. Pure symmetry viewed from any limited perspec- tive appears to have structure. The structure is in the viewing, not in what’s viewed. Crucially, each perspective is a boundary between known and unknown. The perspec- tive doesn’t “have” boundaries - it is entirely constituted by what it includes versus excludes. This boundary is both what defines the perspective and what encodes its information content. The essay “Being the Boundary between Order and Chaos” explores how this solves the binding problem of consciousness. Every conscious experience is necessarily a perspective on reality, inherently unified by being a single boundary. Incompleteness and Information There’s another route to the same insight: through self-reference and incompleteness. Consider the simplest possible distinction: a boundary separating “this” from “not this.” In one dimension, this creates a binary distinction - the minimum possible in- formation, one bit. But for a boundary to reference itself, to “know” it’s a boundary, requires stepping outside the single dimension. In two dimensions, something profound becomes pos- sible: a line can curve back and connect to itself, forming a loop. This loop translates between both binary options and so unifies what otherwise would be disjoint objects. The loop is self-referential structure made geometric. But to prevent it from collapsing to a point, there must be something it cannot cross - a “hole” in the structure. This hole, this incompleteness, is not a flaw but the source of information itself. The empty set {} demonstrates this beautifully. It’s a boundary drawn around nothing, the simplest possible distinction. The empty set is “first” not temporally but struc- turally - it’s the simplest possible pattern that distinguishes itself from non-existence. Kurt Gödel showed this mathematically: any system complex enough to reference itself must be either incomplete or inconsistent. The system cannot fully describe 10 itself without paradox. But this isn’t a limitation - it’s generative. The incompleteness is what prevents collapse into uniformity, creating the space for structure to exist. The appendix explores this through category theory, showing how Lawvere’s fixed point theorem formalizes this self-referential generation of structure. The Geometry of Information The “holes” created by self-referential boundaries are information. This isn’t metaphorical - the structure of the incompleteness encodes information: In one dimension, a cut creates a binary distinction - the minimum possible informa- tion: one bit. In two dimensions, a loop creates an actual hole with area. The ability to “route around” it creates topological information beyond simple binary distinction. In three dimensions, surfaces enclose volumes. In four dimensions, volumes enclose hypervolumes. Each dimension adds new types of boundaries, new ways for structure to relate to itself, new forms of information. The principle: every boundary creates information, every incompleteness enables structure, every limitation becomes a degree of freedom. The universe isn’t made of stuff that has information. The boundaries, holes, and patterns of incompleteness are the universe. Perspectives, Relations and Information What emerges from this geometric process? A vast network of interrelated perspec- tives. A perspective is a particular pattern of symmetry breaking, one way of creating dis- tinctions, one limited view of the whole. Each perspective is defined entirely by what it excludes - its symmetry breaks are its complete description. But perspectives don’t exist in isolation. Each perspective is defined by its relation to all other perspectives. In category theory this is known as the Yoneda lemma. The structure isn’t made of things but of relationships between patterns of distinction. Consider how numbers work. The number 3 isn’t a thing floating in space. It’s defined entirely by its relations: greater than 2, less than 4, the sum of 1 and 2, the square root of 9. Remove all relations and there’s no “3” left. The number exists as a node in a network of relationships. Reality works the same way. Every perspective is a node in an infinite network of relations. What we call “existence” is participation in this network. This leads to a profound equivalence: • To exist is to be distinguishable • To be distinguishable is to encode information 11 • Information is relational pattern • Therefore: everything that exists is relational/informational structure This isn’t saying reality is “made of” information like it might be made of atoms. Rather, information and relation are ways of describing the same structure that ex- istence consists of. The Mathematical Nature If reality is relational structure, then reality is inherently mathematical. Not described by mathematics, but actually consisting of mathematical pattern. This would explain the “unreasonable effectiveness” of mathematics in physics - we’re not imposing math onto nature but recognizing the patterns that nature consists of. But which mathematics? If we include all possibilities, we must include all possible mathematical structures - the vast space containing every consistent pattern, every possible way distinctions can relate. This connects to ideas like the mathematical multiverse, but with a crucial difference: we’re not assuming mathematics exists and asking what follows. We’re starting from groundlessness and finding that relational structure (which can be described mathe- matically) is what’s left when you refuse all assumptions. Where We Are Now Starting from groundlessness - refusing all assumptions - we’ve arrived at a frame- work where: 1. All possibilities exist in superposition (refusing to exclude means including all) 2. This appears as pure symmetry (no preferred structure or perspective) 3. Any limited view breaks symmetry (creating distinctions and information) 4. Self-reference creates incompleteness (holes that become information) 5. Perspectives relate through structure (the network of all possible views) 6. This structure is mathematical (relational patterns that can be formally de- scribed) We haven’t assumed matter, energy, space, time, consciousness, or physical laws. We’ve started from radical uncertainty and found that structure necessarily exists as all possible limited views of that uncertainty. But this raises crucial questions: What determines which patterns we actually ob- serve? Why do we see specific physics rather than arbitrary structure? How does the infinite space of possibilities crystallize into the particular universe we experience? The answer involves understanding how perspectives must relate to enable coherent observation. Not all mathematical structures can support observers. The require- ments of observation constrain the possibility space dramatically, selecting for spe- cific types of patterns. 12 That’s where we turn next: to understand how relationships between perspectives create the architecture of reality. The Architecture of Relations The Structure of Everything We’ve established that reality consists of all possible perspectives in superposition, with each perspective being a boundary that breaks symmetry and creates informa- tion. But how do these perspectives relate? What is the structure of their relation- ships? The answer involves infinite depth and complexity. Perspectives relate through what we can call morphisms - structure-preserving transformations. But here’s the key in- sight: every relationship between perspectives is itself something that exists, another perspective viewing the connection. This creates a recursive structure where: • Perspectives relate through morphisms • These morphisms are themselves perspectives • Which relate through higher-order morphisms • Which are themselves perspectives • And so on, infinitely This recursive structure of relationships is the fabric of reality itself. The Tower of Structure We can organize this infinite recursion into levels, though remembering that no level is truly fundamental: • Level 0 : Individual perspectives (discrete views) • Level 1 : Relationships between perspectives (transformations) • Level 2 : Relationships between relationships (how transformations relate) • Level 3 : Relationships between those (even higher order patterns) • Level ∞ : The complete structure containing all levels Each level provides different types of relationships. What appears as a foundation at one level is just another pattern of relationships when viewed from a higher level. The structure has no bottom and no top, only endless depth in all directions. Crucially, the relationships aren’t just abstract connections but spaces of uncertainty. Between any two discrete perspectives lies a continuum of intermediate states - de- grees of being one versus the other. A morphism from perspective A to perspective B includes all the uncertain states between them, with maximum uncertainty at the midpoint where both are equally probable. This creates a rich structure: • Discrete nodes (definite perspectives) 13 • Continuous paths between them (uncertainty) • Higher-dimensional spaces between paths (uncertainty about which path) • And so on to arbitrary dimension Neither Discrete Nor Continuous Reality refuses to be purely discrete or purely continuous. It’s both simultaneously, with each view revealing different aspects of the same structure. Consider the real number line. It appears perfectly continuous, without gaps. Yet within this continuum, the integers emerge as special points where patterns crys- tallize. They’re not added to the line but emerge from it naturally. You can slide smoothly along the reals, but when you hit an integer, something qualitatively differ- ent happens - you can count from there. Or consider what happens when a photon transforms into an electron-positron pair. From one view, it’s a discontinuous jump - suddenly one particle becomes two. From another view, it’s a continuous process with infinite complexity at the transformation point that we can’t fully resolve. This duality appears everywhere: • Quantum mechanics : Continuous wave evolution, discrete measurement out- comes • Phase transitions : Continuous temperature change, abrupt state change • Symmetry breaking : Continuous parameter space, discrete outcomes • Category theory : Discrete objects, continuous morphisms The structure naturally contains both aspects: discrete perspectives (nodes) connected by continuous relationships (morphisms). You can’t have one without the other. The discrete needs the continuous to relate, the continuous needs the discrete to anchor. The Generative Process Mathematics isn’t built from fixed building blocks but generates new structures through symmetry-preserving combinations. Each dimension enables fundamentally new capabilities: 1D (ℝ) : Binary distinction and scalar values - Can distinguish positive/negative - En- ables: Order, sequence, chirality 2D (ℂ) : Rotation becomes possible - Combining two 1D lines while preserving orthog- onality creates rotation - This isn’t “using” 1D, it’s creating something genuinely new (phase) - Enables: Complex numbers, waves, U(1) symmetry 4D (ℍ) : Full spatial rotation - Combining rotations while preserving non- commutativity creates spinors - New primitive: Half-integer spin (impossible in lower dimensions) - Enables: Spacetime, SU(2) symmetry, weak force 14 8D ( 𝕆 ) : Triality emerges - Combining quaternionic structures while preserving divi- sion creates exceptional symmetry - New primitive: Three-way equivalence (vectors ↔ spinors ↔ conjugate spinors) - Enables: Three generations, SU(3) color charge Each symmetry-preserving combination: 1. Loses some global symmetry (can’t access everything anymore) 2. Creates a new local structure (a new “atom” of mathematics) 3. Enables previously impossible combinations This generative process continues infinitely, creating an ever-richer tower of mathe- matical structure. Renormalization Group Flow A crucial insight comes from renormalization group (RG) theory, which shows how physical theories change as we observe them at different energy scales. RG flow reveals which structures persist across scales: Fixed points : Theories that look the same at all scales - These are the “most gen- eral” mathematical structures - Particles at fixed points are massless (like photons) - Represent maximum symmetry compatible with structure Relevant operators : Grow important at low energy - These dominate our observed physics - Create the effective theories we see Irrelevant operators : Shrink at low energy - Details that average out at our scale - Explain why we don’t see Planck-scale physics directly Marginal operators : Don’t change with scale - Fine-tuning parameters - Often related to coupling constants The division algebras are special because they’re fixed points in multiple senses: - Fixed under their own automorphisms - Fixed under dimensional extension (can’t go beyond octonions) - Fixed under RG flow (scale-invariant internal structure) This explains why physics looks simple at low energies - we’re seeing the relevant operators near fixed points, while irrelevant details have been integrated out. Multiple Languages, Same Structure Because reality has this discrete-continuous duality and multi-scale nature, no single mathematical language can capture it completely. We need multiple complementary descriptions: As particles and fields : We see discrete excitations in continuous fields. This lan- guage works well for quantum field theory but requires renormalization to handle the infinities at interaction points. As strings and branes : We see extended objects vibrating in higher dimensions. This captures more of the continuous aspect but still discretizes at some level. 15 As categories and morphisms : We see objects related by arrows, with higher cate- gories capturing higher-order relationships. This language emphasizes structure and pattern. As computation and information : We see perspectives as computational states, mor- phisms as computational steps. This connects to complexity theory and information thermodynamics. As geometry and topology : We see the shape of relationship space, how perspectives connect and twist around each other. This reveals global properties invisible locally. As RG flows : We see how structures persist or vanish across scales, revealing what’s truly fundamental versus emergent. These aren’t different realities but different languages for the same structure. Like how a symphony can be described as pressure waves, musical notes, or emotional experience, each captures something true while missing other aspects. Why Constants Exist This multiplicity of descriptions explains one of physics’ deepest mysteries: why do we have fundamental constants? Constants aren’t properties of reality itself but conversion factors between differ- ent descriptive languages. When we describe the same phenomenon using different frameworks, we need “exchange rates” to translate between them: • c (speed of light): Converts between space and time descriptions • ℏ (Planck’s constant): Converts between wave and particle descriptions • k_B (Boltzmann’s constant): Converts between temperature and energy de- scriptions • G (gravitational constant): Converts between matter and geometric descrip- tions • α (fine structure constant): Converts between bindary information and wave cycles (see appendix A) These constants exist because no single description is complete. Reality requires mul- tiple complementary frameworks, and constants are the necessary translation factors between them. The speed of light, for instance, isn’t really about how fast light “travels.” It’s the conversion factor between spatial and temporal ways of describing the same relational structure. In natural units where c=1, we’re acknowledging that space and time are aspects of unified spacetime. The Coherence-Probability Tension Not all perspectives are equally probable or observable. There’s a fundamental tension between coherence and probability that shapes what we can observe. 16 Perfect coherence would be a precise mathematical structure - an exact integer, a perfect circle, a pure quantum state. These have infinite information content (requir- ing infinite precision to specify exactly) and therefore have measure zero probability. You’ll never randomly find yourself at exactly π or precisely at a geometric point. Complete incoherence would be pure randomness with no structure at all. While more probable than perfect coherence, it can’t support observation because observa- tion requires some stability, some pattern that persists. We exist in between: coherent enough to have structure, fuzzy enough to have non- zero probability. This is why: • Quantum mechanics is