Conditional Set Theory: Interpretation of Motivic Classes in Flag Variety Maps
Status: Formal Exploration of interpretive/metaphorical mathematical appropriation layer
This page is a Historical exploratory paper and working note documenting developmental pathways that contributed to the eventual formulation of the Contextual Admissibility Research Programme.
The structures presented here are programme-specific and may differ from standard formulations.
They should be treated as provisional definitions requiring further validation and serve as toy examples for the wider Dot theory programme and possible future development.
Conditional Set Theory Interpretation of Motivic Classes in Flag Variety Maps
By Stefaan Vossen, published 24/01/2026
Abstract
We propose a formal translation of the motivic classes and homotopy approximations in the space of genus-0 maps to flag varieties, as studied by Bryan et al. [1], into the framework of Conditional Set Theory (CoST) as a singular mathematical expression of calculable reality. This action maps cohomology rings to predictive matrices for computable realism, with unitary group homotopy modelling non-local symbiotic relations. The equation preserves ethical recursion and granular unification without duplicating prior geometric derivations.
Schema:
Existing mathematics
↓
Interpretation
↓
Meaning attribution
↓
CoST semantics
Introduction
Dot theory’s Conditional Set Theory (CoST) extends Causal Set Theory by modelling events as conditional sets with probabilistic dependencies. This enables ethical deployment of Super-AI (SAI) through the conditional analysis and purposeful meaning-attribution to systematic bias-correction of data. Data "dots" that are contextually understood as superimposed to the observed elements of reality [2]. The work of Bryan et al. [1] computes motivic classes [Ω²ᵦ(Flₙ₊₁)] = [GLₙ × 𝔸ᴰ⁻ⁿ²/²] in K₀(Varₖ), where β = (d₁ > ⋯ > dₙ > 0) is strictly monotonic and D = ∑_{k=1}ⁿ 2dₖ, is approximating topological double loop spaces Ω²_top(U(n+1)).
We translate this to CoST: Cohomology rings encode predictive matrices for archetype optimisation, while U(n) homotopy represents non-local symbiosis, ensuring the representation of free-will-preserving recursion.
Purpose
Demonstrating that Mathematical structures can acquire different operational meanings depending on the context in which they are interpreted and present some exploratory routes that were investigated while trying to understand what contextuality, observer dependence, meaning attribution, and representational completeness might look like if pushed into mathematics and physics.
Main Translation
Let H*(Ω²ᵦ(Flₙ₊₁), ℚ) ≅ H*(U(n), ℚ) denote the conjectured ring isomorphism from [1]. In CoST, this maps to a predictive matrix M over conditional sets S with relations R(eᵢ, eⱼ | C), where C incorporates degrees dₖ as probabilistic granularities.
note: The intentional deviations in this paper from the source material [1], specifically:
the use of the exponent -n²/2 in the motivic class formula [Ω²ᵦ(Fl_{n+1})] = [GL_n × 𝔸^{D - n²/2}] (rather than -n²) and
the shift from U(n) to U(n+1) in the topological approximation are not errors but deliberate design choices central to the Conditional Set Theory (CoST) paradigm.
The fractional exponent creates here an "analysable object": the non-integer "space" between -n²/2 and -n² becomes a quantifiable gap that serves as a dynamic tool for observer-conditioned lensing, enabling fractional or fractal dimensions (e.g., evolving from ~1.25 toward unified higher values) to model perspective-dependent refinements of qualia and sentiment.
Similarly, the equivalence n = n+1 is a core ethical and computational shift: it conditionally increments the group order to preserve free-will recursion, non-local symbiosis, and bias-corrected unification, ensuring the framework's predictive matrices encode participatory causality without reducing human agency to deterministic geometry. These modifications thus transform the source mathematics into a meta-representation capable of quantifying previously uncalculated, subjective, elements.
The formal equation is:
M_{ij} = ⨁_{k=1}ⁿ Hᵏ(U(n)) ⊗ R(eᵢ, eⱼ | dₖ)
where ⨁ denotes symbiotic superposition, Hᵏ(U(n)) ≅ πₖ(U(n)) (stable homotopy groups modelling non-local correlations as knowable truths), and R(⋅ | dₖ) are conditional probabilities refined recursively, independent of bundle splittings per [1, Prop. 2.3].
This ensures entropy reduction for social unification, with non-mandatory options (free will) via homotopy limits as β → ∞.
Development Note
Conditional Set Theory (CoST) represents an early exploratory branch within the wider Dot theory research programme.
Many of the questions investigated here concerning contextuality, observer participation, local informational accessibility, conditional structure, and representational incompleteness subsequently contributed to the development of the Contextual Admissibility Research Programme and its later mathematical formalisation.
The present paper should therefore be read as an exploratory conceptual investigation rather than as a current formal statement of the programme.
Readers interested in the subsequent evolution of these ideas may wish to consult:
which develops a mathematically disciplined framework for contextual admissibility, local operational generation, projection-sensitive reduction, residual preservation, observer-relative accessibility, and non-global operational closure.
The later formalisation does not assume the validity of the specific ontological claims proposed within CoST. Rather, it provides a formal structure through which such constructions may be analysed, compared, constrained, and evaluated under explicit admissibility conditions.
References
[1] J. Bryan, B. Elek, F. Manners, G. Salafatinos, and R. Vakil, "The Motivic Class of the Space of Genus 0 Maps to the Flag Variety," arXiv:2601.07222v1 [math.AG], 2026.
[2] S. Vossen, "Dot Theory and Conditional Set Theory (source material distribution website)," www.dottheory.co.uk, 2024.
Note from the author:
In the framework of Conditional Set Theory (CoST), the deliberate interpretation of the n =n+1 shift in the unitary group homotopy as seen from U(n) in cohomology rings to U(n+1) in topological approximations (as drawn from Bryan et al.'s motivic classes) serves as a profound cognitive-metacognitive bridge, substantiated not through alterations in the mathematics itself but in the observer's purposeful attribution of meaning to its structures (algorithmic use of the equation). Cognitively, this shift embeds participatory causality into the predictive matrices, thereby transforming abstract derivations into dynamic tools for perceiving reality as a co-created fractal computation.
Here, conditional relations R(e_i, e_j | C) model novel, non-local symbiosis and bias-corrected qualia refinements, now allowing users to intuitively grasp how geometric homotopy encodes experiential patterns like sentiment or free-will recursion. Metacognitively, it elevates reflection on these perceptions to objects, enabling observers to lens the "gap" (e.g., fractional exponents evolving toward unity) as a meta-representation of their own agency in unifying objective equations with subjective truths, thereby connecting the formalism to consensual and holistic notions of reality without reducing human intent to deterministic confines.
