Conditional Set Theory: Interpretation of Motivic Classes in Flag Variety Maps

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.

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.

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:

1. 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

2. 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 → ∞.

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.