T3-3: Stochastic extensions

[rororowyourboat]

Problem

Extend the canonical form to support stochastic transition maps for epidemiological, financial, or agent-based models:

h: X × Ω → X    (random variable interpretation)
h: X → ΔX       (map to distributions over X)

Type

[MATH] [CODE]

Prioritization

• Criteria: C2 (Completeness)
• Tier: 3 — Research Frontier
• Phase: 5 — Triggered, not scheduled
• Dependencies: T1-1, T2-2
• Triggered by: Concrete model requiring stochastic semantics

Pre-condition

T1-1 (execution semantics) and T2-2 (behavioral verification) must be stable. Stochastic semantics make behavioral verification significantly harder; do not introduce until the deterministic case is well-characterized.

Scope (when triggered)

• StochasticMechanism role
• Distribution-valued Space
• Monte Carlo execution harness in gds-sim
• Behavioral checks extended to probabilistic assertions (satisfaction probability, expected convergence time)

---

Part of the GDS-Core Improvement Roadmap

Scientific Context

Evidence level: Level 3 (Behavioral) extension — stochastic semantics make behavioral verification significantly harder. BV-001 (universal invariant) becomes a probabilistic assertion (satisfaction probability ≥ p). BV-002 (fixed-point) becomes expected convergence time.

Verification Strategy

Known-distribution tests. Systems with analytically solvable stationary distributions (e.g., birth-death chains, Ornstein-Uhlenbeck processes). Verify Monte Carlo results converge to known distributions. This validates both the stochastic mechanism and the probabilistic behavioral checks.

Do not pursue until the deterministic behavioral verification (T2-2) is well-characterized — stochastic extensions layer on top of it.

[rororowyourboat]

Jamsheed's Math Review Feedback on T3-3

Jamsheed considers stochastic extensions the most important Tier 3 item and recommends earlier placement:

Of the three tasks here, stochastic extensions is likely the most crucial and could be considered a Tier in its own right, or if preferred as a sub-task of a previous Tier.

Combined with his recommendation to move continuous-time to Tier 1, this would create a reordered schedule:

• Tier 1: ExecutionContract + continuous-time formalization (all time modeling together)
• Tier 2: Stochastic extensions + behavioral verification

This "operationalizes gds-core earlier for those model paradigms which require continuous time models with uncertainty input rails."

Tracked as Decision 2 in #172. See improvement-plans/review_synthesis.md for full context.