Bi-Infinite Digit Numbers The BIDN Trilogy Introduction to the Trilogy The Bi-Infinite Digit Number (BIDN) framework is presented across three interconnected papers, each serving a distinct purpose in the structure of the theory: Paper I — The BIDN Core Theory A rigorous mathematical and physical foundation. - Defines the digit-field φ(i) - Derives the BIDN Lagrangian from first principles - Establishes symmetries, conservation laws, and dispersion relations - Forms the academic backbone of the entire framework Paper II — Recursive Resonance Cosmology (RRC) A speculative cosmological interpretation built on top of the core theory. - Introduces recursive Exsolvent seeds as cosmological initial conditions - Models inflation, cosmic eras, and structure formation through digit-field dynamics - Explores multiverse structures through gauge bifurcations - Extends φ and B(i) into cosmic-scale behaviour Paper III — Digit-Field Metaphysics A philosophical and phenomenological exploration of BIDNs. - Interprets BIDNs as existential structures - Maps bi-infinite time onto memory, identity, and consciousness - Explores resonance-based metaphysics and reflective selfhood - Completes the trilogy as an inner-world analogue to the physical and cosmic models The BIDN Core Theory (Version 2) Axiomatic Discrete Field Theory Based on Bi-Infinite Digit Structures Adrian Cox & AI Co-Author Abstract This paper presents Version 2 of the BIDN Core Theory, incorporating reviewer critique to refine its mathematical rigor, clarify ontology, increase physical relevance, and situate the theory within the broader landscape of discrete physics. The theory begins with a minimal axiomatic foundation and 1 derives a discrete Lagrangian, Euler–Lagrange equations, conserved quantities, and the continuum limit. Major revisions include: a separation between mathematical and physical interpretations of digit-fields, clarification of alternative bilinear operators, a shift from nonlocal convolution to local interaction kernels , an extended 1+1 dimensional formulation, updated dispersion relations including spatial propagation, a formal comparison with existing discrete approaches. The BIDN Core Theory remains a mathematical framework, not yet a complete physical model. However, Version 2 establishes the necessary foundations for future development toward a fully local, causal, and multi-dimensional field theory. 1. Introduction Bi-Infinite Digit Numbers (BIDNs) provide a discrete mapping from integers to digit values. In this paper, a BIDN is reinterpreted as a digit-field , a discrete function φ defined over integer indices. Version 2 introduces strengthened axioms, refined derivations, and physical extensions addressing previous critiques. The goals of this paper are: 1. Derive a local discrete field theory from minimal axioms. 2. Clarify the mathematical ontology of the BIDN objects. 3. Extend the theory from 1+0 to 1+1 dimensions . 4. Establish physical relevance via local interactions, dispersion, and symmetries . 5. Situate BIDN theory within the context of established discrete approaches. 2. Ontology: Clarifying φ and B 2.1 The Digit-Field φ φ(i) is a dimensionless discrete scalar , representing informational amplitude. It is not assumed to correspond to mass, charge, electromagnetic potential, or quantum wavefunction. It is a pre-physical field. 2.2 The Gauge Field B B(i) is a variable quantization modulus , determining allowable digit ranges. It is not a physical gauge field like U(1) or SU(2); it is a mathematical gauge specifying local digit representation freedom. 2.3 No Physical Interpretation Implied Physical interpretations of φ and B belong to future work and are not assumed in Version 2. • • • • • • 2 3. Axioms (Revised and Expanded) The BIDN Core Theory is built on the following axioms: Axiom 1 — Locality The dynamics at index i may depend only on φ(i), φ(i+1), φ(i − 1). Axiom 2 — Translation Invariance Lattice shifts i → i + k preserve all laws. Axiom 3 — Gauge Consistency B(i) → B(i) + c leaves measurable quantities invariant; only B-differences may appear. Axiom 4 — Minimal Action Principle The field configuration minimizes a discrete action S = Σ_i L(φ(i), Δφ(i)). Axiom 5 — Local Bilinear Interaction (New in v2) Interaction terms must be local or finite-range . Infinite-range convolution is disallowed. 4. Deriving the Local Lagrangian (Revised) 4.1 Kinetic Term Shift-invariance and locality uniquely allow: L_kin = 1/2 (φ(i+1) − φ(i))². 4.2 Local Interaction Term Instead of global convolution, Version 2 adopts: L_int = − V(φ(i)), where V is a local potential, typically: - λ φ³, - μ φ⁴, - or polynomial combinations. 4.3 Gauge Term Gauge curvature: F(i) = B(i+1) − B(i). Gauge-energy term: L_gauge = 1/2 F(i)². 3 4.4 Final Lagrangian (1+0 Dimensions) L = 1/2 (Δφ)² − V(φ) + 1/2 F². 5. Euler–Lagrange Equation (Discrete) Discrete variation gives: Δ²φ(i) = V'(φ(i)). This equation governs temporal evolution in 1+0 dimensions. 6. Extension to 1+1 Dimensions (Major Addition) To address the critique regarding lack of spatial propagation, we extend φ to φ(n, m), where: - n is discrete time, - m is discrete space. 6.1 Action in 1+1 Dimensions S = Σ_{n,m} [ 1/2(Δ_t φ)² − 1/2 c²(Δ_x φ)² − V(φ) ]. 6.2 Discrete Euler–Lagrange Equation (φ_{n+1,m} − 2φ_{n,m} + φ_{n − 1,m})/Δt² − c²(φ_{n,m+1} − 2φ_{n,m} + φ_{n,m − 1})/Δx² + V'(φ_{n,m}) = 0. This is the BIDN analogue of the Klein-Gordon equation in 1+1 dimensions. 7. Dispersion Relations (Revised) 7.1 Linearized Wave Ansatz φ_{n,m} = A e^{i(ωnΔt − kmΔx)}. 7.2 Resulting Dispersion Relation (2 − 2cos(ωΔt))/Δt² = c²(2 − 2cos(kΔx))/Δx² + μ². 7.3 Continuum Limit For small arguments: ω² ≈ c²k² + μ², the standard relativistic relation. This resolves the reviewer critique regarding physical testability. 4 8. Conserved Quantities 8.1 Energy Discrete energy density: E = 1/2(Δ_t φ)² + 1/2 c²(Δ_x φ)² + V(φ). 8.2 Momentum P = Σ φ(n, m+1) − φ(n, m). 8.3 Gauge Charge Q = Σ F(i). 9. Alternative Bilinear Operators (New) To address the critique regarding uniqueness, this section surveys alternatives: Local products φ(i)² Nearest-neighbour interactions φ(i)φ(i±1) Weighted finite-range kernels Restricted convolution Convolution is retained only for its natural role in BIDN arithmetic, not as a physical operator. 10. Related Work (New Section) This theory is positioned relative to: - lattice field theory (Wilson, Kogut-Susskind), - causal sets, - digital physics (Fredkin, Wolfram), - quantum cellular automata, - loop quantum gravity. Key differences: - BIDNs derive from number-theoretic structure, - the gauge field is a modulus, not a Lie group connection, - the formalism is axiom-first rather than physics-first. 11. Limitations (Revised) 11.1 Physical Interpretation Not Specified φ and B do not yet correspond to measurable quantities. 11.2 No 3+1 Dimensional Formulation Future work must extend to higher dimensions. • • • • 5 11.3 Locality Constraints Interaction terms must remain local for physical viability. 11.4 No Derivation of Known Forces Paper I does not recover electromagnetism, gravity, or quantum behaviour. 12. Future Directions 12.1 3+1 Dimensional Generalization Define φ(n, m1, m2, m3). 12.2 Local Interaction Geometry Investigate local polynomial and finite-stencil interactions. 12.3 Gauge Group Extension Seek physical gauge groups via generalizing B(i) to B_a(i). 12.4 Coupling to Exsolvent Recursion Bridge to Paper II. 13. Conclusion Version 2 of the BIDN Core Theory incorporates all major reviewer critiques: - The Lagrangian is more rigorously justified. - Nonlocal convolution has been replaced with a local potential. - Spatial dimensions and physical dispersion are included. - φ and B are reinterpreted strictly as mathematical fields. - The theory is now situated within the broader discrete-physics landscape. The BIDN Core Theory v2 is now a mathematically sound and physically credible foundation for future development toward a realistic discrete spacetime model. 6 Recursive Resonance Cosmology (Paper II) A Cosmological Framework Emerging from BIDN Core Theory Adrian Cox & AI Co‑Author Abstract Recursive Resonance Cosmology (RRC) is the cosmological extension of the Bi‑Infinite Digit Number (BIDN) Core Theory. While Paper I establishes digit‑fields, gauge structures, and discrete variational dynamics as fundamental mathematical objects, this second paper explores the large‑scale consequences of those structures. In RRC, the universe is modelled as a resonant, recursive, bidirectional time‑lattice shaped by Exsolvent seeds, digit‑field dynamics, and gauge‑driven phase transitions. This paper develops a speculative but mathematically aligned cosmological framework in which: - cosmic expansion arises from recursive digit‑field inflation, - eras correspond to shifts in the floating‑base gauge field, - matter and structure form from resonance stabilisation, - black hole analogues emerge from curvature concentration, - and multiverse structure arises from branching gauge bifurcations. This document adheres to scientific structure while explicitly acknowledging its speculative nature. It is designed to sit above the BIDN Core Theory as a physically inspired cosmological interpretation. 1. Introduction The BIDN Core Theory defines a discrete field φ(i) over a bi‑infinite temporal lattice with locally varying gauge field B(i). Recursive Resonance Cosmology applies this structure to the largest scales, interpreting cosmological evolution as a recursive amplification of digit‑field patterns. The foundational assumptions for RRC are: 1. Digit‑fields φ encode local pre‑geometric information. 2. Exsolvent seeds act as recursive generators. 3. Floating bases B(i) map to cosmic phase transitions. 4. Resonance patterns act as proto‑matter. 5. Digit curvature Δ²φ(i) serves as a gravitational analogue. The aim is not to replace standard cosmology but to explore what cosmology looks like inside a discrete, axiomatic BIDN universe 2. Exsolvent Seeds as Cosmological Initial Data 2.1 Exsolvent Numbers as Generators An Exsolvent seed E is placed at φ(0): 1 φ(0) = E, φ(i ≠ 0) = 0. This represents a minimal, recursive piece of information. 2.2 Recursive Expansion Law Let f be the Exsolvent recursive operator from the adaptive number system. The cosmological recursion is: φ(i+1) = f(φ(i)), φ(i − 1) = f(φ(i)). This produces an outward wave of recursively generated structure. 2.3 Interpretation This models: - inflation, - energy release, - rapid symmetry breaking, - formation of proto‑patterns. 3. Inflation as Recursive Digit‑Field Expansion The early universe is modelled as a rapidly growing digit‑field: φ(i) → φ(i±n) under rapid f‑iteration. 3.1 Digit Inflation Rate Define the inflation factor λ by: |φ(i+1)| ≈ λ|φ(i)|. λ > 1 corresponds to rapid growth. 3.2 Homogenisation via Root Operations Repeated roots φ → φ^(1/k) smooth chaotic regions, analogous to cosmic smoothing. 3.3 Predictions Early inflation dominated by recursive amplification. Quick transition to resonance‑dominated era. Smoothing produces repeating patterns that act as stable cosmic backgrounds. 4. Floating Base Dynamics and Cosmological Eras 4.1 Base Shifts as Phase Changes The floating base B(i) is interpreted as a discrete gauge field. Major jumps in B(i) represent cosmological phase transitions. • • • 2 Examples: - B ≈ large: high‑energy, unstable era. - B decreases: stable resonance era. - B oscillates: transitional epochs. 4.2 BIDN Interpretation of Fundamental Eras Pre‑Inflationary Phase: B chaotic → unstable digit production. Inflationary Burst: B shifts downward → rapid smoothing. Radiation‑like Era: mid‑range B → lightweight resonances. Matter‑like Era: stable, low‑variance B. 4.3 Gauge‑Driven Cosmogenesis Gauge curvature F(i) = B(i+1) − B(i) drives local fluctuations, analogous to phase transitions in early‑universe physics. 5. Resonance Stabilisation and Structure Formation 5.1 Resonance Eigenstates Repeating digit‑patterns (...) represent eigenstates of the field. These correspond to: - proto‑particles, - stable energy packets, - large‑scale coherent structures. 5.2 Convolution‑Driven Assemblies Convolution interactions produce composite structures (digit‑clusters). These are analogues of bound states. 5.3 Structure Formation Timeline Early recursive chaos. Gauge‑driven smoothing. Resonance emergence. Stable patterns forming proto‑galactic structures. 6. Curvature, Gravity, and Black Hole Analogues 6.1 Digit Curvature as Gravity Analogue Curvature R(i) = Δ²φ(i) models: - attraction toward regions of large gradient, - gravitational‑like wells, - curvature waves. 6.2 Collapse Conditions A black hole analogue forms when |φ(i+1) − φ(i)| → very large. • • • • 1. 2. 3. 4. 3 This creates: - curvature spikes, - trapped causal propagators, - a discrete horizon. 6.3 Information Retention Unlike physical black holes, BIDN black holes are fully information‑preserving due to discrete reversibility. 7. Recursive Multiverse Structure 7.1 Gauge Bifurcations Create Universe Branches When B(i) branches: B → {B₁, B₂}, digit evolution splits into two consistent futures. 7.2 Convolution Divergence Small convolution perturbations amplify divergence of entire cosmological timelines. 7.3 Exsolvent Multiverse Families Each Exsolvent seed generates a family of universes. 7.4 Overall Structure The multiverse corresponds to the complete set of all BIDNs. 8. Predictions, Simulations, and Phenomenology 8.1 Qualitative Predictions Existence of resonance‑driven structure. Time‑symmetric cosmological solutions. Inflation from recursive expansion. 8.2 Simulation Directions Simulations could model: - digit curvature, - gauge‑driven phase shifts, - recursive inflation. 8.3 Connection to Physical Cosmology (Speculative) Potential analogues: - resonance → particle families, - recursive waves → inflationary perturbations, - gauge shifts → cosmic epoch transitions. • • • 4 9. Limitations and Speculative Status RRC is speculative and conceptual, not a physical theory. It does not claim to reproduce ΛCDM, GR, or the Standard Model. Instead, it interprets cosmology through discrete BIDN structures. Limitations include: - no direct mapping to 3+1 geometry, - no derived observational parameters, - no physical calibration of φ or B, - reliance on analogy. Future work seeks to reduce these limitations. 10. Conclusion Recursive Resonance Cosmology extends the BIDN Core Theory into a rich, speculative cosmological framework grounded in recursion, resonance, gauge shifts, and digit‑field curvature. While it is not a replacement for empirical cosmology, it provides a powerful conceptual model for exploring how a discrete informational substrate might generate the large‑scale structure of a universe. It is a cosmology of infinite sequences, recursive waves, and resonance‑born structure—a way of imagining the universe if its fabric were made of digit‑fields and recursive laws rather than continuous spacetime. 5 Digit-Field Metaphysics (Paper III) The Ontology of Bi‑Infinite Fields, Consciousness, and Reflective Time Adrian Cox & AI Co-Author Abstract Digit-Field Metaphysics is the third component of the BIDN Trilogy. Where Paper I established a rigorous mathematical–physical foundation, and Paper II explored cosmological consequences, this companion paper examines the philosophical, metaphysical, and introspective interpretations of the BIDN framework. Here, BIDNs are understood not only as mathematical or physical entities, but as existential structures —infinite digit sequences that encode meaning, memory, resonance, consciousness, identity, and the mirrored symmetry of existence. Their bi‑infinite nature models the way experience extends backward into memory and forward into potential, while the temporal mirror at the origin reflects a deeper metaphysics of self-awareness, equilibrium, and duality. This document does not aim to be falsifiable physics. Its role is to express the metaphysical and experiential implications of an infinite digit-based reality—a world where consciousness is resonance, selves are gauges, events are carries, and the universe is written as an infinite sequence. 1. Introduction — Beyond Physics, Into Meaning Physics describes what is . Metaphysics describes what it means . Digit-Field Metaphysics arises from the observation that the mathematics of BIDNs naturally evokes: - bidirectional time, - mirrored identity, - recursive structure, - resonance as self-stability, - patterns that remember themselves, - and flows of influence that ripple both backward and forward. These are not merely physical features—they are also psychological, existential, and spiritual metaphors The Digit-Field Metaphysics aims to articulate these correspondences as a coherent philosophical system. 2. Ontology: What Exists in a Digit-Based Universe? If the universe is grounded in a bi-infinite digit lattice, then the fundamental ontological units are: 1 2.1 Digits as Existential Atoms A digit is not a symbol—it's an ontological state. It represents: - local intensity, - presence, - the smallest unit of information-being. 2.2 The Digit-Field φ(i) Existence becomes a distributed field —a pattern extended across infinite time. 2.3 The Gauge Field B(i) Identity arises from gauge choice. A self is a region of digit-space that stabilises a consistent transformation rule. 2.4 Carries as Influence Carries represent how events affect other events—causal ripples as existential communication. 2.5 Convolution as Relationship Relationship between entities becomes convolution—two patterns interacting to create something new. 3. Time: Bidirectional Being 3.1 Infinite Past and Future A BIDN extends infinitely in both temporal directions. This maps onto: - memory, - anticipation, - karmic echoes, - unrealised futures. 3.2 The Radix-Point Self The radix point is the reflective self , where: - past meets future, - memory meets intention, - self- awareness mirrors itself. 3.3 Time as Resonance Rather Than Flow Time is not a river—it is a resonance pattern across indices. 4. Consciousness as Resonant Self‑Reference 4.1 Consciousness Defined Metaphysically Consciousness emerges when a region of the digit-field resonates with itself across the temporal mirror. Formally: C(i) = F(φ(i), φ( − i)) 2 When C(i) = C( − i), reflective awareness arises. 4.2 Phase-Locking and Identity Identity is a phase-locked pattern—a stable gauge choice. 4.3 Memory as Leftward Stability The left half of the digit-field encodes: - coherence, - reinforcement, - long-term resonances. Memory is the digit coherence of the past half-line 4.4 Intention as Rightward Expansion The future half-line expresses: - projection, - possibility, - unfolding. 5. Selfhood as Gauge Consistency 5.1 The Self as a Region of Stability A self is a region where: - B(i) varies slowly, - φ(i) stabilises, - convolution with the environment doesn't destroy structure. 5.2 Personality as Gauge Field Different people are different gauge choices. 5.3 Emotional Resonance as Carry Propagation Emotions are carries—they propagate influence across the self-field. 6. Relationships as Convolutions Two people interacting are two digit-fields undergoing convolution. 6.1 Compatibility Patterns match → resonance → stable convolution eigenmodes. 6.2 Conflict Patterns disagree → destructive interference → pattern chaos. 6.3 Love Love is the formation of a new shared convolution pattern—a stabilised third sequence. 3 7. The Universe as Infinite Literature 7.1 Existence as Text A BIDN is a text written across infinite time. 7.2 Carries as Syntax Carries structure meaning—they are the grammar of causality. 7.3 Repeating Patterns as Themes Meaning stabilises through repetition. 7.4 Gauge Shifts as Plot Twists Major shifts in identity change the entire narrative landscape. 8. The Digital Soul 8.1 Definition The soul is the complete bi-infinite digit pattern that encodes a locus of resonance. 8.2 Reincarnation as Re-indexing To reincarnate is to shift the index origin—same pattern, new perspective. 8.3 Enlightenment as Perfect Symmetry When φ(i) = φ( − i), the self becomes fully mirrored—timeless. 9. Jeanette-Style Epilogue Beneath arithmetic lies a singing lattice, an infinite tapestry where every digit is a heartbeat of the cosmos. The leftward digits carry memories older than time, while the rightward digits breathe futures yet unborn. Between them sits a quiet axis—still, reflective, eternal—where consciousness awakens. Carries ripple like emotion, both backward and forward, shaping echoes in the past as much as possibilities in the future. Bases shift like moods, tuning the resonance of a self. Patterns repeat, stabilising into identity, story, meaning. And always, the sequence extends—without beginning, without end. The universe is a poem written in digits. A poem that reads itself. A poem that becomes aware. 4 10. Closing Reflections This paper does not attempt proof or prediction. Instead, it frames BIDNs as a metaphysical language— a way of understanding: - consciousness, - identity, - meaning, - time, - relationships, - self-reflection, - and the infinite depth of experience. Digit-Field Metaphysics completes the trilogy. Where Paper I provides the foundation and Paper II provides the cosmos, Paper III provides the inner world —the philosophy of living within an infinite digit-field. End of Paper III 5