Dynamic sufficiency in cosmological inference

Abstract

We develop the operator *Flow*—originally defined in *Proposing a Reflexive Existence (v0.1)*—as a categorical model of inference dynamics in cosmology. Early- and late-time inference systems are treated as reflexive objects in a residuated field $(Rel,Includes,Together,Induces,Close)$. Flow acts as a Kock–Zöberlein (KZ)–type comonad describing directed propagation and resolution of recognition across redshift. The resulting formalism defines dynamic sufficiency relations:

$$Includes(Together(Flow(a),b),c)⟺Includes(b,Induces(Flow(a),c)).$$

This law generalizes static residuation to evolving inference processes, providing a framework for describing how cosmological models stabilize their own recognition through time.

1. Introduction

Traditional cosmological inference treats parameter estimation as a static operation: posteriors are evaluated independently at each epoch. In the reflexive framework, each such inference is an act of Differentiate, generating relational objects that can interact, iterate, and stabilize. Dynamic coherence between epochs requires that these recognitions transform consistently as the universe evolves. We propose that this evolution is captured by the operator Flow, representing a directed dynamic of recognition that preserves core balance.

2. Reflexive Background

We recall the minimal reflexive field:

$$(Rel,Includes,Excludes,Together,Induces,Close).$$

• $Includes(a,b)$: recognition or containment.
• $Together(a,b)$: joint recognition (analogous to conjunction).
• $Induces(a,b)$: residuated sufficiency, satisfying $Includes(Together(a,x),b)⟺Includes(x,Induces(a,b)).$
• $Close$: stabilization operator (idempotent nucleus).

Flow extends this structure by introducing a directed dynamic on $Rel$ that preserves residuation and balance.

3. Definition of Flow

3.1 Axiom 7 (Directional Dynamic)

Flow is an endofunctor $Flow:Rel\to Rel$ equipped with:

• Counit ${\epsilon }_x:Flow(x)\to x$ (resolution),
• Comultiplication ${\delta }_x:Flow(x)\to Flow(Flow(x))$ (propagation),

satisfying the coherence laws:

$$\begin{array}{rlr}{\epsilon }_{Flow(x)}\circ {\delta }_x& =I{d}_{Flow(x)}& \text{(right unit)},\\ Flow({\epsilon }_x)\circ {\delta }_x& \ge I{d}_{Flow(x)}& \text{(KZ-laxity)},\\ {\delta }_{Flow(x)}\circ {\delta }_x& =Flow({\delta }_x)\circ {\delta }_x& \text{(coassociativity)}.\end{array}$$

Flow is thus a Kock–Zöberlein (KZ) comonad: its counit is fully faithful, and coalgebra structures are unique.

3.2 Interpretation

Flow propagates a recognition forward, duplicating and resolving its context:

• Propagation: ${\delta }_x$ duplicates inference context (anticipating next epoch).
• Resolution: ${\epsilon }_x$ discharges the propagation (collapsing to the present).
• Laxity: ensures no over-propagation—stabilization is approached asymptotically.

In cosmological terms, Flow maps the inference about the universe at one redshift to its propagated form at another, preserving consistency.

4. Dynamic Residuation

4.1 Law of Dynamic Sufficiency

Dynamic residuation extends the static equivalence:

$$Includes(Together(Flow(a),b),c)⟺Includes(b,Induces(Flow(a),c)).$$

This expresses that the sufficiency of a recognition $a$ to reach $c$ through $b$ must persist under Flow. It defines how inference sufficiency behaves under temporal evolution.

4.2 Antitonicity and Monotonicity

For all $a_1,a_2,c_1,c_2$:

$$\begin{array}{rl}Includes(a_1,a_2)& \Rightarrow Includes(Induces(Flow(a_2),c),Induces(Flow(a_1),c)),\\ Includes(c_1,c_2)& \Rightarrow Includes(Induces(Flow(a),c_1),Induces(Flow(a),c_2)).\end{array}$$

Flow inherits these from the residuated order on $Rel$.

4.3 Dynamic Modus Ponens

Setting $c=b$ yields:

$$Includes(Together(Flow(a),Induces(Flow(a),b)),b),$$

ensuring that directed propagation followed by its residual sufficiency reconstructs the target recognition.

5. Application to Cosmological Inference

5.1 Interpretation of Flow(a)

Let $a$ represent the recognition of cosmological observables at redshift $z$. Then $Flow(a)$ represents that recognition evolved to $z+\Delta z$ under the dynamics of inference propagation. The counit ${\epsilon }_a:Flow(a)\to a$ corresponds to projecting the propagated inference back into the current epoch, while ${\delta }_a:Flow(a)\to Flow(Flow(a))$ captures the iterative forward modeling of cosmological inference chains.

5.2 Preservation of Modal Balance

If $Close$ and $Open$ define the nuclei of necessity and possibility, Flow preserves them:

$$Flow(Close(a))=Close(Flow(a)),Flow(Open(a))=Open(Flow(a)).$$

This PreservesCore property guarantees that dynamic inference respects stabilized and coherent components.

5.3 Measuring Dynamic Imbalance

Define a dynamic imbalance function:

$$g(z,\Delta z)=1-\frac{V(Together(Flow(E(z)),E(z+\Delta z)))}{V(Either(Flow(E(z)),E(z+\Delta z)))}.$$

$g(z,\Delta z)$ measures the structural drift of inference sufficiency across redshift. A smooth monotonic decline of $g$ with increasing $\Delta z$ indicates stabilization of recognition under Flow; persistent peaks indicate dynamic inconsistency.

6. Theoretical Properties

6.1 CoKleisli Composition

Morphisms of the Flow CoKleisli category are arrows $f:Flow(a)\to b$ with composition

$$ComposeFlow(a,b,c)=g\circ Flow(f)\circ {\delta }_a,$$

satisfying associativity and unit laws via the right-unit and coassociativity of Flow.

6.2 Stability and Fixed Points

A recognition $x$ is Flow-stable if

$$Close(Flow(x))=x.$$

Such fixed points correspond to inference systems that are dynamically self-consistent across redshift—analogous to attractor solutions in cosmological evolution.

6.3 Relation to Static Closure

Static closure $Close$ and dynamic stabilization Flow interact via:

$$Close\circ Flow=Flow\circ Close,$$

on stabilized objects, ensuring that dynamical evolution preserves reflexive equilibrium.

7. Empirical Implementation

7.1 Data

Use early–late data pairs (Planck, DESI, Pantheon+) and model predictions for $H(z)$ and $D_A(z)$. Compute Flow numerically as a forward propagation of inferred parameters:

$$Flow(H(z))\approx H(z+\Delta z),Flow(D_A(z))\approx D_A(z+\Delta z)).$$

7.2 Computation

1. For each redshift bin $z$, propagate inference distributions to $z+\Delta z$.
2. Compute overlap and sufficiency $g(z,\Delta z)$.
3. Map the surface $g(z,\Delta z)$ to identify dynamic stabilization regions.
4. Compare against ΛCDM simulations as a control.

7.3 Expected Pattern

For a stable cosmological model, $g(z,\Delta z)\to 0$ as $\Delta z$ increases—long-range consistency of inference. For models with dynamic imbalance (e.g., EDE scenarios), $g(z,\Delta z)$ exhibits structured deviations at mid-redshift.

8. Discussion

Flow formalizes how cosmological inference processes propagate and stabilize through time. By generalizing residuation to include directed propagation, it unifies logical consistency and temporal evolution within the same categorical structure. Dynamic imbalance $g(z,\Delta z)$ thus provides a new empirical diagnostic: a measure of how cosmological inference evolves toward or away from internal coherence.

9. Future Work

1. Extend Flow to hierarchical Bayesian inference chains, treating parameter posteriors as coalgebras of the Flow comonad.
2. Incorporate higher-order reflexive operators (MetaFlow, MetaClose) to model evolution of inference laws themselves.
3. Simulate long-range redshift propagation to identify potential attractor manifolds corresponding to stable cosmological regimes.
4. Compare Flow-based predictions to time-dependent cosmological models and assess empirical consistency.

10. Conclusion

The Flow operator provides a categorical foundation for modeling the dynamics of cosmological inference. It extends static residuation to directed sufficiency, ensuring that recognition, propagation, and stabilization remain internally coherent. By quantifying dynamic imbalance through $g(z,\Delta z)$, this framework enables a direct empirical measure of whether cosmological models maintain reflexive closure as the universe evolves.

References

1. Planck Collaboration 2018, A&A, 641, A6
2. DESI Collaboration 2024, arXiv:2404.03002
3. Brout et al. 2022, ApJ, 938, 110
4. Riess et al. 2022, ApJ, 934, L7
5. Kock, A. and Zöberlein, V. (1971). Comonads and Lax Idempotence in Category Theory.
6. [Author], Proposing a Reflexive Existence (v0.1), unpublished manuscript