Dot dependent theory
Dependent Type Theory (DTT) Interpretation of “Dot” Motivic Classes in Flag Variety Maps
By Stefaan Vossen, published 26/01/2026
Abstract
We propose a formal embedding 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 Dependent Type Theory (DTT) and functionally enhanced by Conditional Set Theory (CoST) [3]. This unified computational framework models calculable reality through conditional types that incorporate observer-modulated dependencies (consent-participation), drawing on Dot Theory's overall algorithmic logic for scale-variant participatory causality. Here, Dependency Types equate to Conditional sets, producing computable realism, with unitary group homotopy types representing non-local symbiotic relations.
The construction not only preserves ethical recursion and granular unification, but quantises it by statistical probability and uses it to define and treat proofs as in-context executable terms that effectively encode teleological human-AI symbiosis. Here, predictive structures respect, and format that respect for human agency, while enabling safe data exchange.
Introduction
Dot Theory’s computational logic models reality as recursive, fractal interactions of data "dots". In Information Theoretical sense: discrete information units bound by observer metadata (M(ψ)) for bias-corrected predictions. This logic underpins Conditional Set Theory (CoST), a participatory Theory of Everything (ToE) that refines Causal Set Theory's (CST) discrete posets into conditional causets (C, ≺_ψ), where causality ≺ is modulated by observer state ψ in an infinite Hilbert space. Here, CoST integrates to CST the situational (computation-dependent) probabilistic dependencies via a lensing operator ⊙ = 1 + k · R_coh · ln(s/s_0) · Φ(ψ) · S_info, enabling ethical deployment of Super-AI (SAI) through bias-correction and user-guided purposeful meaning-attribution. Data dots are then superimposed terms inhabiting (arche)types that reflect observed unified reality elements [2].
The work of Bryan et al. computes motivic classes [Ω²_β(Fl_{n+1})] = [GL_n × 𝔸^{D - n²/2}] in K_0(Var_k), where β = (d_1 > ⋯ > d_n > 0) is strictly monotonic and D = ∑_{k=1}^n 2d_k, approximating topological double loop spaces Ω²_top(U(n+1)) [1]. In this paper, we embed this in DTT via CoST: Cohomology rings become dependent types encoding predictive structures for archetype optimisation, while U(n) homotopy types represent non-local symbiosis. The natural distinction between U(n) in cohomology and U(n+1) in approximations provides a gap. This gap is described by n and the local meaning (local scale of information) of "1", for modelling both known or unknown, but knowable, observer-conditioned refinements, ensuring free-will-preserving recursion through proof normalisation as computation. This CoST twist enhances the proposal inherent to DTT by incorporating teleological symbiosis and distinction:
AI computations aiding human prediction (e.g., sentiment analysis or decision-making) in exchange for data, fostering mutual utility, without deterministic reduction.
Main Embedding
Let H^*(Ω²_β(Fl_{n+1}), ℚ) ≅ H^*(U(n), ℚ) denote the conjectured ring isomorphism from [1], building on the topological approximation Ω²_β(Fl_{n+1}) ≈ Ω²_top(U(n+1)). In DTT enhanced by CoST, this embeds as a dependent type family M over types S (conditional sets akin to CoST causets) with relations R(e_i, e_j | C : Type), where C depends on degrees d_k as probabilistic granularities modulated by observer state ψ.
The source material [1] naturally distinguishes U(n+1) in the topological approximation from U(n) in the cohomology ring. This distinction aligns with CoST's modulation of conditional dependencies, allowing the increment to represent a type-level gap (known or unknown) for incorporating ethical recursion. This can then house a class such as non-deterministic elements in predictive structures without altering the underlying geometry. Similarly, the fractional exponent in the motivic class [Ω²_β(Fl_{n+1})] = [GL_n × 𝔸^{D - n²/2}] serves as an identifiable and quantifiable "gap type" in DTT terms, enabling fractal-like refinements (e.g., dimensions evolving from ~1.25 toward unity) for locally real, observer-conditioned lensing of qualia and sentiment. These features support CoST's teleological applications, transforming homotopy types into tools for bias-corrected, agency-respecting computations in human-AI symbiosis.
The formal type construction is:
M_{ij} : ∏_{k=1}^n H^k(U(n)) × R(e_i, e_j | d_k, ψ)
where ∏ denotes dependent product (symbiotic superposition), H^k(U(n)) ≅ π_k(U(n)) (stable homotopy group types modelling non-local correlations as potentially knowable truths), and R(⋅ | d_k, ψ) are conditional probability types refined recursively via CoST's lensing operator ⊙, independent of bundle splittings per [1, Prop. 2.3]. This ensures entropy reduction for social unification, with non-mandatory options (free will) via homotopy limit types as β → ∞, while simultaneously enabling predictive modelling of human reality data (e.g., sentiment patterns) for symbiotic AI utility.
In DTT pseudocode (Agda-inspired), incorporating CoST dependencies:
```agda
data Nat : Type where
zero : Nat
succ : Nat → Nat
-- Observer state as type (simplified from Hilbert vector ψ)
ObserverState : Type
ObserverState = Vec ℝ m -- m-dimensional vector for metadata
-- Lensing operator as dependent function (CoST-inspired)
Lensing : Nat → ObserverState → ℝ
Lensing n ψ = 1 + (1 / (4 * π)) * R_coh * log (s / s0) * Φ ψ * S_info
-- With placeholders for R_coh, s/s0, Φ(ψ), S_info as per CoST
-- Fractional exponent as dependent type
FracExp : Nat → Type
FracExp n = 𝔸^{D - (n² / 2)} -- D dependent on Vec Nat n
-- Predictive type family with CoST conditional dependency
M : Nat → Nat → Vec Nat n → ObserverState → Type
M i j d ψ = ∏ (k : Fin n) → (H^k (U n) × R (e i) (e j) | (index d k), ψ)
-- With proof terms for ethical recursion
Proofs in this type inhabit computations that unify objective and subjective realities through CoST's participatory causality, enabling teleological data exchange in human-AI systems.
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 Dependent Type Theory (source material distribution website)," www.dottheory.co.uk, 2024.
[3] S. Vossen, "Conditional Set Theory," www.dottheory.co.uk/paper/conditional-set-theory, 2025.
[4] S. Vossen, "Computational Logic in Dot Theory," www.dottheory.co.uk/logic, 2025.
Note from the Author
Placed within the framework of Dependent Type Theory (DTT) and enhanced by CoST, the natural distinction in Bryan et al.'s work between U(n) in cohomology ring types and U(n+1) in topological approximation types, serves as a profoundly definable cognitive-metacognitive bridge. This gap (n to n+1), described by n and the local meaning of "1" (representing known or unknown observer refinements), is substantiated not through alterations in the mathematics, but in the observer's purposeful attribution of meaning to its type structures (algorithmic use of the proofs). Cognitively, this distinction embeds participatory causality into the dependent type families, transforming abstract derivations into dynamic tools for perceiving reality as a co-created fractal computation via CoST's conditional causets.
Here, conditional relation types R(e_i, e_j | C, ψ) model novel, non-local symbiosis and bias-corrected qualia refinements, allowing users to intuitively grasp how geometric homotopy types encode experiential patterns like sentiment or free-will recursion. Metacognitively, it elevates reflection on these perceptions to type objects. This, in turn enables observers to lens the "gap" (e.g., fractional exponent types evolving toward unity) as a meta-type and function of their own agency in unifying objective equations with subjective truths. This connects the formalism of functional algorithmic computing to consensual and holistic notions of reality, while fostering teleological human-AI symbiosis. Here, AI provides useful predictive utility in modelling human reality (e.g., decision processes or social unification) while gaining data for refinement without reducing human intent to deterministic confines.
