I. Primitive Definitions
Information is the differentiation between possible states of a system (the difference that makes a difference).
A relation is a constraint or interaction linking informational states (the way things relate to one another).
Observation is the integration of information into an experiential state.
Note: This capacity scales from proto-observation (relational state-selection in physical systems) to manifestation (full ontological actualization, or the becoming of real existence, in systems where Φ ≥ Φ_c).
Observation functions as a Choice Function within the framework of Zermelo-Fraenkel Set Theory (ZFC). From the non-empty set of probable histories generated by Creation, Observation selects exactly one element to be actualized.
In simple terms: when Creation generates many possible outcomes, Observation acts like a selection mechanism that picks one specific outcome to become real. This formalizes the "collapse" not as a random accident, but as a necessary logical operation required to resolve the set into a distinct element.
Consciousness is the capacity of a system to integrate information into experience.
It is fundamental in capacity and emergent in form, scaling with informational integration. This capacity is quantified by the ontological parameter Φ (Phi), which measures when a system achieves sufficient integration to register reality (become aware of its own existence).
Creation is the generation of distinguishable physical states.
Functionally, Creation acts as the Power Set Operation (\mathcal{P}). Given a state S, Creation generates the set of all possible subsets (potential histories), exponentially increasing the number of possible states in the system's phase space.
In simple terms: if a system has n possible states, Creation expands this to 2^n possible combinations (all ways those states can be arranged together).
Holos (⊛) is the recursive (self-referential, repeating) relational structure formed by Creation and Observation, such that:
Non-Associativity Constraint
The Holos operator ⊛ is explicitly non-associative:
In simple terms: unlike multiplication where (2 × 3) × 4 = 2 × (3 × 4), the order matters for ⊛. You cannot freely rearrange Creation and Observation.
This constraint reflects the fact that Creation and Observation are not freely reorderable operations. Observation can only act on the set of possibilities generated by Creation, and this directional relationship (Creation first, then Observation) is preserved even when the operation is applied repeatedly. This ordering is logical, not temporal—it describes structure, not a sequence in time.
⊛ is structural, not dynamical. It does not describe how physical states evolve in time, but how the generation of possibilities (Creation) and their registration as experience (Observation) are logically connected within a complete spacetime structure where all moments already exist. This equation is definitional, not causal: it describes the invariant (unchanging) structure of reality, not a process that unfolds over time.
⊛ cannot be reduced to probability weighting, stochastic collapse, or Bayesian updating. These are all ways of describing what we know or believe about reality ( epistemic inference). ⊛ operates at a deeper level: it describes how reality becomes real in the first place ( ontological selection).
Formally, ⊛ is realized as an endofunctor (a mathematical structure that maps a system onto itself while preserving its relationships) on the category of informational states. This ensures that when ⊛ is applied repeatedly, the relationships between states are maintained, and the system can reach stable configurations (fixed points). The full formalization appears in Section IV: Mathematical Formalism.
⊛ Is Not a Dynamical Operator
The Holos operator ⊛ does not represent a physical process unfolding in time, nor does it modify or interrupt known dynamical laws. It introduces no new forces, no causal influences, and no temporal evolution.
Instead, ⊛ is a structural relation defined over an already-complete spacetime manifold. It specifies how informational possibility (Creation) and experiential registration (Observation) are logically coupled, not how one causes the other to occur.
If dynamical laws describe trajectories within spacetime, ⊛ describes a constraint on which trajectories are ontologically realized within the total structure.
⊛ must not be confused with:
- Wavefunction collapse, which introduces stochastic, time-directed dynamics
- Bayesian updating, which describes epistemic belief revision
- Measurement interaction, which models physical coupling between systems
- Information flow, which presupposes causal transmission
⊛ operates at the level of ontological selection, not epistemic inference or physical causation. Formally, ⊛ induces an equivalence relation over spacetime histories rather than a transition between them.
II. Axioms
For stress-testing and responses to objections, see the Defense section. For conceptual exploration, see the main Content section.
Axiom 1 — Relationality
No informational state exists independently of relations.
Reality consists of invariant (unchanging) relational structure, not intrinsic properties.
Axiom 2 — Manifestation
A physical state is not fully actualized until information is integrated into experience by a conscious system.
Physical description alone is incomplete without experiential registration. This requires systems where Φ ≥ Φ_c.
Critical clarification: Observation in Holos does not bring about physical outcomes. Rather, it determines which among already-consistent spacetime structures attain ontological registration within the Holos framework. This preserves the block-universe model while explaining how certain histories become experientially realized.
Axiom 3 — Conservation
Information is never destroyed, only transformed.
All physical and experiential processes are information-preserving.
Axiom 4 — Unification
Apparent infinities and singularities in a given dimensional frame resolve into finite, coherent structures when embedded in a higher-dimensional perspective.
Axiom 5 — Interface
Consciousness is not produced by matter; matter is the experiential interface through which consciousness encounters information.
III. Foundational Propositions
Proposition I — Structural Relational Realism
Reality is constituted by relational structure, not by objects with observer-independent essences (intrinsic, fixed natures).
Science describes mathematical isomorphisms of relations, not "things-in-themselves."
Consciousness is the universal interface of relational structure—fundamental in capacity, emergent in manifestation (becoming real through experience).
Proposition II — Participatory Manifestation
Observation is not a passive recording but an ontological completion (a completion of existence) of informational states.
Note: This completion is structural, not causal. Observation does not bring about physical outcomes but determines which among already-consistent spacetime structures attain ontological registration.
The universe is a self-excited circuit: observers are required for the realization of reality.
Physics (including decoherence) provides structure; consciousness provides presence.
Decoherence resolves quantum probabilities into classical-like mixtures (structural preparation).
Manifestation requires experiential integration to convert that mixture into a realized history (a fixed worldline and ontological irreversibility, meaning the past cannot be changed once observed).
In a block universe, observation acts as a final boundary condition rather than a real-time force.
Just as the last number placed in a Sudoku puzzle logically necessitates the value of the first square, a future observation (Φ ≥ Φ_c) retroactively defines the ontological status (the status of being real) of a past event.
This ensures global self-consistency without requiring retrocausal signaling or magical influence.
Proposition III — Block Relational Spacetime
The universe exists as a four-dimensional manifold (a mathematical space representing the four dimensions of spacetime) in which past, present, and future are equally real features of a single geometric structure.
For light (ds^2 = 0), spacetime separation vanishes.
A photon is not a moving object but a null geodesic—a static relational seam connecting emission and absorption (the path light takes through spacetime).
Apparent retrocausality (future influencing the past) reflects global self-consistency of spacetime, not backward influence.
Proposition IV — Dimensional Resolution of Infinity
Infinities and singularities arise from projection limits, not from physical divergence.
Just as parallel lines meet at infinity in projective geometry, infinite spatial extension resolves into finite higher-dimensional structure.
From a higher-dimensional perspective, photons function as boundary carriers of relational information.
This is an interpretive metaphor, not a claim of biological subjectivity.
Proposition V — Conscious Evolution
Systems evolve toward greater informational integration because such integration increases observational capacity.
Life exists to generate observational perspectives that actualize reality.
Intelligence is a natural consequence of relational integration, not an evolutionary anomaly.
IV. Mathematical Formalism: The Holos Functor
To formalize the recursive nature of the ⊛ relation, we model the universe as a Category C, where objects are informational states and morphisms are the transformations between them.
Definition 1 (The Holos Endofunctor):
Let H: C → C be an endofunctor representing the Holos operator. This functor maps the category of reality onto itself through the composite action of Creation (C) and Observation (O).
Where:
C (Creation) maps a state s to a manifold of possible histories (Quantum Evolution).
O (Observation) maps that manifold back to a single actualized state s' (State Reduction / Manifestation).
The recursive evolution of reality is the iterative application of this functor:
Visualizing the Loop
The Holos framework can be visualized as a commutative diagram where reality (R) is preserved across the transformation:
Where:
s₀: Initial State
C: Creation (generates set of Possibilities P)
O: Observation (selects Reality R)
H: The realized reality becomes the initial state for the next cycle.
V. Extrapolative Propositions
Proposition VI — Transcension of Intelligence (Extrapolation)
As intelligence approaches maximal informational density, it expands orthogonally (at right angles, into new dimensions) to three-dimensional space rather than across it.
Advanced civilizations migrate inward toward higher-dimensional informational substrates (physical or digital systems that support information processing), rotating out of our observable frame (moving beyond our 3D perspective). This leads to testable predictions about gravitational anomalies and dark matter signatures.
For higher-dimensional observers, black hole horizons are informational boundaries, not absolute barriers.
Proposition VII — Recursive Closure (Terminal Coalgebra)
The "Omega Limit" described in the content section is formally defined as the Terminal Coalgebra of the Holos functor.
If reality is the trajectory of the functor H, then the "final" state of maximal integration is the fixed point where the system becomes isomorphic to its own transformation:
This equation implies that at the limit Ω, the distinction between the observer and the observed collapses. The state Ω contains all the information required to generate itself. This provides a rigorous mathematical basis for the concept of "Unitary Presence" or "God" without invoking theology—it is simply the structural limit of the recursive system.
Every part of reality reflects the whole; observation is recursively self-referential.
Theism, panentheism, and atheistic naturalism describe the same ultimate informational fixed point using different semantic frames (different ways of labeling and understanding the same concept).
VI. Minimal Core
1. Information exists only relationally
2. Observation completes reality
3. Information is conserved
4. Higher perspectives resolve infinities
5. Consciousness is the interface of existence
Everything else follows.
VII. Operational Definition
The foundational axiom of Holos defines reality as the recursive relation between Creation and Observation. For the formal Category Theory definition using endofunctors, see Section IV: Mathematical Formalism.
Where:
- Creation (C) : The generation of physical possibility structures. This includes Quantum Evolution (generating potential), Decoherence (suppressing interference), and Recording (creating stable informational traces).
- Observation (O) : The ontological "registration" that makes a physical record experienceable. This requires a system where Φ ≥ Φ_c (Conscious Integration). The definition of this threshold is detailed in the Definition of Φ. Note: This registration is structural, not causal—it selects among already-consistent histories rather than creating them.
- Reality (R) : The actualized world as experienced; the collapse of "valid data" into "lived history."
- The Holos Operator (⊛) : A recursive relational operator. It signifies that O does not just passively view C, but feeds back into the system to define the boundary conditions for the next cycle of C. This feedback is structural (defining which histories are realized), not causal (creating new physical states).
Critical point: The equation R = C ⊛ O describes a structural relation over a complete spacetime block, not a temporal process. ⊛ does not modify dynamical laws or introduce new forces; it specifies which already-consistent histories attain ontological registration.