The Ring as Machine : A Mixed-Radix, LUT-Mediated, Self-Organizing Computational Architecture : Topological Computation Through Mixed-Radix Phase Lattices

The Ring as Machine : A Mixed-Radix, LUT-Mediated, Self-Organizing Computational Architecture : Topological Computation Through Mixed-Radix Phase Lattices : 10.5281/zenodo.18188896 Vening, E. J.-P. (2026). The Ring as Machine : A Mixed-Radix, LUT-Mediated, Self-Organizing Computational Architecture : Topological Computation Through Mixed-Radix Phase Lattices. Zenodo. https://doi.org/10.5281/zenodo.18188896… @chargen shares a preprint authored by Edwin Jean-Paul Vening, introducing the RING architecture—a deterministic, drift-free computational system based on a 720-point mixed-radix phase lattice that evolves symbolic states via layered lookup tables (LUTs) without branching or floating-point arithmetic. The architecture leverages topological invariants and contradiction metrics for self-organizing interactions among parallel "lanes," enabling constant-time symbolic transformations suitable for cyclic domains like harmonics and resonance. Key innovations include a physiological field layer modeling multi-rate memory (flow, afterglow, wear) and coherent projections across radices (e.g., 60, 360), positioning RING as a hardware-native alternative for photonic processors and spaceborne autonomy, validated through simulations and prototypes. Mixed-radix number systems (also called mixed-base or variable-base systems) are positional numeral systems where the base (radix) differs from one digit position to the next, unlike fixed-base systems (binary base-2, decimal base-10, hexadecimal base-16) where the radix is constant everywhere.Core Definition & Value CalculationIn a mixed-radix system with radices b₀, b₁, b₂, …, bₖ (from least to most significant position), a number with digits dₖ dₖ₋₁ … d₁ d₀ (where 0 ≤ dᵢ < bᵢ) has the value:value = dₖ × (bₖ₋₁ × bₖ₋₂ × … × b₁ × b₀) + dₖ₋₁ × (bₖ₋₂ × … × b₁ × b₀) + … + d₁ × b₀ + d₀The place values are the cumulative products of the radices below each position. The most significant digit usually has no upper bound on its own radix (effectively ∞), as there is no "next larger unit."Everyday & Historical Examples Sexagesimal time & angles — bases roughly [60, 60, 24, 7] (seconds : minutes : hours : days : weeks)Example: 3 days, 14 hours, 42 minutes, 15 seconds → digits [3, 14, 42, 15] with bases [7, 24, 60, 60] Mayan Long Count — mostly base-20, but the second position is base-18 so that 18×20 = 360 ≈ days in a year Pre-decimal currencies (e.g., old British) — pounds : shillings : pence → bases [20, 12] Factorial number system (factoradic) — bases [2, 3, 4, 5, 6, …] (increasing)Every natural number has a unique representation (no leading zeros).Example: 23₁₀ = 3 2 1 0! → 3×3! + 2×2! + 1×1! + 0×0! = 18 + 4 + 1 + 0 = 23Widely used to index permutations (0 to n!−1 permutations of n items in lex order via Lehmer code / inversion table). Advantages & Disadvantages Property Advantages Disadvantages / Trade-offs Natural fit Perfect for composite, hierarchical units (time, angles, calendars, astronomy) Requires conversion to/from fixed-base for most computers Arithmetic Generalized long addition/multiplication works (carry rules adapt per position) More complex carry propagation than fixed-radix; usually slower in software Radix economy Can theoretically outperform fixed-base-3 in average digits per number (debated; e-approximating sequences proposed on MathOverflow) Factoradic and similar systems are optimal for certain combinatorial tasks, not general encoding Precision in cycles Exact representation of periodic/cyclic quantities (no drift in angles, harmonics) Not natively supported in mainstream hardware Hardware/computing Useful in Residue Number Systems (RNS) conversions, parallel modular arithmetic, DSP, certain path-enumeration algorithms Rare direct hardware support; mostly emulated or used in niche FPGA/ASIC designs Computing & Specialized Applications Residue Number System (RNS) conversions — many RNS-to-weighted (mixed-radix) algorithms exist for efficient parallel modular arithmetic in signal processing, cryptography, and error-correcting codes. Combinatorial enumeration — factoradic maps integers → permutations bijectively; very useful in ranking/unranking, generating all permutations without recursion. Graph/tree traversal & path listing — represent all paths through nested lists/trees by treating lengths of sub-lists as radices (common in Python/Java libraries for systematic enumeration). Astronomical & calendrical software — Gregorian/Mayan/Tibetan calendars rely heavily on mixed-radix conversions for date day-number calculations. Experimental hardware — occasional FPGA/ASIC prototypes explore mixed- or high-radix arithmetic for FFT butterflies, modular multipliers, or optical/analog-friendly formats, though these remain niche compared to binary/fixed-radix designs. Relation to RING Architecture (from prior context)The RING system deliberately chooses highly composite mixed radices (3, 12, 60, 360, 720, etc.) because they create a dense, divisor-rich phase lattice ideal for symbolic processing of angles, frequencies, harmonics, orbits, and resonance — domains where fixed-binary floating-point accumulates drift over long computations. By performing all operations via deterministic lookup tables on this lattice (no adders, no multipliers, no branches), RING turns the mixed-radix property into a hardware-native, drift-free computational primitive — far beyond conventional uses in timekeeping or combinatorics.In essence, mixed-radix systems are a flexible generalization of positional notation that excel whenever quantities have non-uniform scaling factors between layers — whether for human convenience, mathematical bijections, or (in rare cases like RING) radical re-thinking of symbolic machine architecture. Explaining RING architecture applications Mixed-radix number systems The RING architecture (as presented in Edwin Jean-Paul Vening's 2026 preprint "The Ring as Machine: A Mixed-Radix, LUT-Mediated, Self-Organizing Computational Architecture") is a proposed computing paradigm that departs from conventional von Neumann / floating-point / branch-heavy designs. It uses a fixed 720-point mixed-radix phase lattice (with radices such as 3, 12, 60, 360, 720 that have many divisors and connect naturally to cyclic/periodic phenomena), layered lookup tables (LUTs), purely deterministic symbolic state updates, contradiction-based self-organization among parallel lanes, and no runtime branching or accumulated rounding error.Because of these properties, the architecture is positioned for domains where traditional computers struggle with precision, predictability, energy, thermal/noise resilience, or real-time symbolic/cyclic processing.Key Application Areas (as described or strongly implied) Photonic / optical computing hardwareRING is explicitly proposed as a natural mapping target for photonic processors. Reason: symbol transformations are LUT-based → can be implemented as passive or low-energy optical interference / waveguide lookup structures Phase lattice concept aligns directly with optical phase accumulation and interference No floating-point → avoids precision loss that plagues analog optical compute→ Potential role = core logic fabric for future all-optical or hybrid photonic AI / signal processors Spaceborne & deep-space autonomous systemsOne of the strongest emphasized target domains. Drift-free / deterministic forever behavior (no accumulating numerical error) Very low or zero branching → predictable worst-case timing, simple formal verification Radiation-hard by design (symbolic, table-driven, no charge-pump floating point units that are SEU-sensitive) Self-organizing contradiction resolution can provide forms of fault-tolerance / graceful degradation→ Guidance, navigation & control (GNC), on-board science data reduction, long-duration autonomy in environments with no ground contact (interstellar probes, permanently shadowed lunar/planetary landers, high-radiation orbits) High-precision cyclic, harmonic, resonant & periodic computationThe underlying 60/360/720-radix system is natively tuned to circles, angles, frequencies, calendars, orbital mechanics, acoustics, RF/wave propagation.→ Direct symbolic processing of: Audio / musical harmonics & synthesis (phase-accurate without aliasing accumulation) Vibration analysis, structural health monitoring of rotating machinery RF / radar / sonar beamforming & direction finding Celestial navigation & ephemeris calculation Cryptography / coding that exploits finite fields or ring-like structures Real-time symbolic reasoning & non-numeric AIBecause transformations are table-driven and parallel lanes interact via topological invariants rather than arithmetic, RING can be viewed as a massively parallel symbolic rewriting engine.→ Possible uses: Theorem proving / formal verification engines that must run forever without drift Reactive / autonomic rule systems in safety-critical or mission-critical settings Bio-inspired or neuromorphic computing where multi-timescale memory ("flow / afterglow / wear") is modeled natively Energy- & thermally-constrained edge devices Constant-time execution for many operations No dynamic branch prediction or speculative execution hardware needed Very regular memory access patterns (LUTs) → Edge AI in satellites, underwater systems, high-altitude drones, or industrial IoT nodes that need predictability and low power variation Summary Table of Fit Domain Why RING fits particularly well Traditional computing pain point being addressed Photonic processors Phase & LUT optical interference & passive routing Analog precision loss, thermal tuning difficulty Deep-space autonomy Drift-free, deterministic, radiation-tolerant design