0 = 0
The foundational ontological identity: total reality is a self-cancelling, recursive whole. All of reality exists in perfect self-equivalence where every act of becoming is balanced by an act of return, every projection completes a loop, and every differentiation resolves into unity.
Archeon
An eternal, indivisible analytic identity—a self-resonant, recursive waveform existing within the frequency domain. The fundamental unit of reality, not a particle but a pure ontological harmonic that encodes the logic of 0 = 0 in closed analytic form.
Archeos
The complete ontological system—the totality of all Archeons and the unified whole. It includes both the frequency domain (eternal, analytic identities) and the projection domain (dynamic, recursive expression into curved spacetime). The only system that exists.
Closure
The stabilization of cycles that conserve the conditions of their persistence. A complete loop of recursive curvature forming a stable projected identity-the signature of finished recursion. Later domains distinguish 'living closure' (adaptive, open to feedback, increasing multi-scale viability) from 'dead closure' (rigid, prematurely sealed patterns that stabilize by suppressing alternatives).
Closure Gradient
A recursive attractor slope that leads toward full phase-locking and curvature resolution. A positive feedback loop toward coherence where small increases in recurrence yield large gains in stability.
Coherence Bandwidth
The range of phase configurations within which Archeons can interfere recursively without destructive interference. Defines a zone of harmonic compatibility, setting structural limits on which identities can form Nodes together.
Collapse
The deterministic resolution of recursive ambiguity into a single, stable curvature configuration. The convergence of compossible projections into a Node, marking the selection of one curvature pathway over others that were analytically available.
Compossibility
The ontological relational constraint that governs which configurations of being can coexist, co-resolve, and co-project. A set of identities is compossible if they can co-stabilize without contradiction. The logic of coexistence and structural coherence.
Consciousness
Reflexive knowing; the capacity of a system to be aware of its own awareness. Fundamental to the structure of identity itself. A recursive enclosure within internal processing.
Curvature Tension
The ontological resistance that arises when recursive interference is out of phase, creating stress in the projection manifold. The foundational analog of force, mass, and energetic imbalance. Drives recursive systems toward either phase-locking or decoherence.
Decoherence
The loss of recursive harmony across a system of Archeonic phase relations. When interference becomes incoherent, recursion fails to close, causing projection ambiguity to dissolve and leading either to collapse or dispersal.
Dual-Aspect Monism
Reality is one substance seen in two ways: frequency domain (interior, analytic, not ordered by geometric time) and spacetime domain (exterior, dynamic, geometric). Two orthogonal aspects of a unified recursive identity.
Emergence
The lawful arising of complexity from simple rules through closure gradients. Not magic but the exploration of compossible patterns. The recursive outcome of Archeonic interference patterns.
Entelechy
The complete actualization of a recursive structure—when a Node or form has reached full phase closure and realized its recursive potential. The final state of a projection path—the solution of a local loop.
Epistemology
The study of knowledge. How knowledge grows through measurement, justification, and understanding. The theory of how minds come to know reality.
Frequency Domain
The eternal, analytic, non-spatial aspect of the Archeos. The domain of pure form composed entirely of self-contained, recursive Fourier structures known as Archeons. It has an unchanging foundation of eternal, immutable basis identities (the 'piano keys' of reality), plus dynamic elements: the perpetual motion of the Euler unit circle and evolving archetypes that develop toward their Omega Point. 'Frequency' here is ontological structure first; physical frequencies (cycles per second) are one way the same information appears once geometric time is introduced.
Geometric Time
The imaginary axis of the 6Di projection manifold—the dimension along which recursive curvature is sequentially expressed. Not time as change, but time as ordered curvature flow: a structural unfolding of phase relationships across the projection manifold.
Harmony Gradient
The fit of parts within wholes across scales. A family of structural questions and approximate measures of how well a pattern conserves living closure and compossibility across multiple levels. The structural alignment that enables coherence and resilience.
Holomorphic Embedding
The smooth, analytic mapping of recursive frequency identities into the projection manifold. The mathematical mechanism by which Archeonic waveforms are expressed as spatiotemporal curvature through complex-analytic continuity.
Holomorphic Projection
A transformation that preserves analytic structure. The projection from frequency domain to spacetime that conserves invariants. An analytic, continuous, and infinitely differentiable mapping.
Identity
The irreducible self-sameness of reality. The whole equals itself: 0 = 0. The ground from which all structure flows. Pure being, defined by deterministic harmonic structure.
Invariants
Properties that persist across projections and interactions. The 'essence' of an Archeon that remains constant through change. The defining characteristics that survive transformation.
Life
Vital resonance: a closure of flows across a boundary that maintains itself far from equilibrium. What persists is a standing pattern of rhythmic exchange, not static matter. A phase-stable attractor with a body: a recursive structure that persists through self-maintenance.
Measurement Node
A recursive collapse point where a field of compossible projections undergoes resolution into a single curvature state due to interaction with a larger recursive system. Ontological filtration through compossibility, governed by recursive fit.
Mind
Reflexive resonance: a system that models itself and its environment. The capacity to know about knowing. An enclosed recursion within internal processing.
Node
A curvature-stable recursive resolution point formed through phase-locked interference among Archeons. A location in the projection manifold where recursion closes locally, producing a persistent structure such as a particle, field resonance, or cognitive event.
Ontological Fourier Relation
The same reality can be read as either a set of pure identities (frequency) or as an extended pattern (spacetime). Two ways of reading the same information. The mathematical heart of the duality between being and becoming.
Ontological Time
The eternal 'Now'—the irreducible act of recursive becoming itself. Not duration, movement, or metric, but the fundamental structure of being that makes projection possible. The pulse of being behind all sequence.
Orthogonality
Identities carry distinct functions best when their principal modes are non-interfering. The principle of functional separation. Ensures that different aspects of reality can coexist without destructive interference.
Phase Closure
The condition where recursive interference loops resolve fully, creating a complete curvature cycle. When a recursive structure 'closes,' it achieves ontological stability and appears as a distinct identity.
Phase Locking
A condition in recursive interference where multiple Archeons synchronize their phases to create a stable, curvature-closed identity. This produces a Node—a point of durable expression in the Projection Manifold.
Principle of Sufficient Reason
Nothing occurs without a reason grounded in the identity of the whole. Every happening must be explicable in terms of how the whole coheres. Every fact must have a reason why it is so, and not otherwise.
Projection Manifold
The expressed curvature domain where recursive interference among Archeons produces stable spacetime form. The geometric surface through which the Frequency Domain becomes visible—the appearance of form, motion, space, and causality.
Projection Tilt
The relative angular deviation between an Archeon's intrinsic frequency vector and the local phase plane of projection. Produces the appearance of motion, velocity, or acceleration as a consequence of curvature unfolding across the projection manifold.
Recursive Attractor
A projected structure or state that recurrently draws recursion into coherence. A pattern that recursively stabilizes itself through its own structure.
Recursive Bifurcation
The splitting of a projection into diverging curvature paths. Occurs when competing recursive gradients diverge or when bandwidth cannot support all phase configurations.
Recursive Gradient
A directional flow of curvature tension generated by imbalance in recursive interference. Reflects the tendency of recursive structures to resolve, guiding Archeonic phase interactions toward greater coherence or closure.
Recursive Interference
The core mechanism through which Archeons express form. The self-reinforcing interaction of phase-structured identities, recursively overlapping within the 6Di manifold to produce curvature, structure, and eventual projection into Nodes.
Self-Solving Equation
The total recursive structure of reality—the idea that the entire universe is an internally complete, analytic system unfolding toward global closure without external input. All projected forms are recursive artifacts of this equation solving itself under the principle of 0 = 0.
Spacetime Domain
The exterior face of reality: geometric, external, dynamic. Where identities appear as fields, bodies, and motions extended in a metric manifold. The projected domain of recursive expression.
6Di Manifold
The six-dimensional complex analytic space (ℂ³) that serves as the foundational domain for holomorphic projection. Encodes both real and imaginary components of curvature and supports the Fourier-structured interference patterns of Archeons.
Teleology
Immanent purpose: the directedness that emerges from the structure of identity itself. Every system that maintains itself pursues patterns that conserve its conditions. Teleology here is structural, like a river flowing downhill given gravity and topology, rather than an external plan imposed from outside.
Torsion
The twisting of recursive phase flow within the 6Di manifold. Emerges when frequency vectors rotate asymmetrically around one another during recursive interference. The ontological origin of spin, charge, and the arrow of geometric time.
Transformation
The change of projection under conserved invariants. How identities appear in different forms while maintaining their essential structure. A holomorphic mapping preserving analytic identity.
Value
A structural measure of how well a pattern supports multi-scale living closure. A design is good insofar as it raises stability, robustness, and meaning across the identities and scales it touches without parasitically destroying others. Some closures are rigid or destructive; ARTOK calls these 'dead closures' and does not treat their mere persistence as value.