Conditional Set Theory
A proposal to formulate Space-time as a conditional set. A compellingly logical ontological evolution on Causal Set Theory.
Introduction: The Need for a Conditional Framework
Conditional Set Theory, is a new, speculative physical theoretical approach to defining values that presents its use as Theory of Everything (ToE) by inviting us to consider the ontological context of any value, and integrate the known context into the calculation for enhanced prediction of real-world decay pathways. To do so, this model converts the heuristics of existing theories (String, LQG, CST, etc) into observable topologies of any mathematical model. These can then be re-classified as conditional posets with the condition being that there is metadata available for bias-correction and coherence (for real-world observation data). As such, CoST unifies QM's non-locality with observer-relativity, addressing anthropic gaps without losing CST's discreteness and is an ontological evolution on Causal Set Theory (CST).
Dot Theory, fundamentally explores the concept of reality as a participatory projection. In Dot theory’s version of reality, observers are individually attributed as co-creating meaning through subjective data/sentiment-shaped "dots" or information units filtered by relative differences and bias-corrected for predictive coherence. This led me to Causal Set Theory (CST), pioneered by Rafael Sorkin (1987), as a promising discrete foundation for quantum gravity. CST models spacetime as a locally finite poset of events, with causality as the primitive order, from which continuum geometry emerges (Sorkin, 1993). However, CST's observer-independence leaves gaps in anthropic effects and consciousness. These are the challenges Dot Theory addresses via conditional, fractal refinements.
In this paper, knowledge of CST is assumed, and in it, CST is refined into Conditional Set Theory (CoST) as a variant optional calculation where sets are "conditioned" by observer metadata, i.e. making causality dynamic and participatory. This monograph outlines CoST's foundations, derivations, predictions, and implications. This positions it as a ToE for evolving human experience, testable in physics and beyond.
Theoretical Foundations
CST posits spacetime as a causet: A poset (C, ≺), where C is finite events, and ≺ denotes causal precedence (transitive, irreflexive, locally finite) [Bombelli et al., 1987]. Volume approximations yield manifold dimensions, preserving Lorentz invariance without preferred frames [Dowker, 2005].
Dot Theory complements this by treating reality as recursive fractals: Dots as discrete units, connected via sentiment-conditioned relations (qualia). Consciousness emerges from coherence (R_coh) in the energy bath (a pre-geometric substrate [Vossen, 2024]). CoST merges these: Causets become conditional, modulated by observer state ψ (Hilbert space vector), yielding fractal orders in an infinte Hilbert space.
Key alignment: Both reject continuum primacy. CST via discreteness, Dot via fractal projections.
Divergence: CST is causal-objective; CoST is conditional-subjective, with causality emergent from observer-biased metadata.
Key Modifications: From Causal to Conditional Sets
CoST refines CST's poset with four tenets, incorporating Dot's observer corrections:
Spinor Adjustment: Modify QM spinors to treat observer data as local metadata: ψ → ψ + δ(ψ), where δ(ψ) = ∑ w_i · cos(θ_i) · e^{-β i^2}, with w_i Bayesian weights from historical data. This conditions QM trajectories, testable in simulations (e.g., decay pathways in fusion, improving accuracy vs. standard QM) [Vossen, 2024].
GR Modification: Introduce dynamic constant in e = (m ⊙ c³)/(k T), with ⊙ = ∑ Δθ_i (cumulative lensing biases from metadata). This aligns with CST's emergent geometry but adds fractal scaling ln(s/s_0), yielding D ≈ 1.25 bath to D ≈ 3 observed [Sorkin, 1993; Vossen, 2025].
Game Theory Integration: "Name-and-claim" as Bayesian optimisation: Priors are refined by metadata for posterior predictions. Testable in particle collisions (e.g., metadata inclusion boosts trajectory accuracy) (von Neumann, 1944).
Mother Matrix: M_{μν}(ψ) as bias-encoding tensor, analogous to GR's stress-energy but conditional on F_{μν}(ψ). Fractal spectrum (non-integer eigenvalues) validates in lensing/biometric data (Girard, 1972 for mimetic ties).
These create conditional causets: (C, ≺_ψ), where ≺ is observer-modulated, resolving CST's observer-gap.
Mathematical Formulations:
Core meta-equation: e = (m ⊙ c³)/(k T),
with ⊙ = 1 + k · R_coh · ln(s/s_0) · Φ(ψ) · S_info.
R_coh = (∫ |ψ_coh(x)|² d³x)/(∫ |ψ_dec(x)|² d³x) ≈ N · e^{-β · ln(s/s_0)}, β ≈ 0.1.
Φ(ψ) = ∑ w_i · cos(θ_i) · e^{-β i^2}, collapsing to 10^{-10} post-observation.
S_info = -∑ p_i ln p_i ≈ 1–10.
k = 1/(4π).
Gravity: G_{μν} = (8π G)/c^4 T_{μν} + k · R_coh · S_info · ρ_t · g_{μν}.
Lensing: Δθ = (4GM)/(r c²) · (1 + k · R_coh · S_info).
These derive from energy bath (infinite Hilbert space) projections, unifying forces via conditional orders.
Predictions and Testability
Computing: Enhanced meshing vs. traditional (to be tested via benchmarks).
Cosmology: Fractal peaks at k = 1/(4π) in EHT residuals (2026 data).
Healthcare: EEG mappings with δ(ψ) ≈ 0.05 for outcomes.
Physics: Modified spinors for trajectories in quantum simulations (e.g., fusion).
These leverage existing tools for falsifiability, extending CST's CMB predictions into practical applications for the prediction of observations.
Implications: A framework for understanding the Human Experience of reality
CoST unifies physics with consciousness where applicable: Wet substrates (humans) enable creative chaos (the unique individual perspective creating true mathematical randomness) whereas silicone/dry (AI) enable efficiency. This offers cheap, accessible and effective human welfare ethically and fosters unification.
As such, CoST refines CST into a participatory ToE*, where conditional sets bridge observer-reality. Like any evolution, it evolves physics for human needs and invites testing and adoption.
*here we acknowledge that the foundational ontology of reality is framed in terms consistent with the effective theories underpinning our current understanding. More specifically, those operating as low-energy approximations or phenomenological models that emerge from deeper discrete structures.
References
Sorkin, R. (1987). Causal Sets.
Bombelli, L. et al. (1987). "Space-time as a Causal Set."
Dowker, F. (2005). Causal Sets and the Deep Structure of Spacetime.
Vossen, S. (2024). Dot Theory. www.dottheory.co.uk.
von Neumann, J. (1944). Theory of Games and Economic Behavior.
Girard, R. (1972). Violence and the Sacred.
