25-EFT.WP.STG.Dynamics v1.0 | Chapter 10 — Causality and Intervention (Multi-Environment Invariance)

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Chapter 10 — Causality and Intervention (Multi-Environment Invariance)

One-Sentence Goal
Define a unified program on G = (V, E) for invariant mechanisms across environments and intervention simulation, running structural causal models (SCMs) in parallel with dynamical equations to enable computable do(•) inference, counterfactual simulation, and contract-governed publication.

I. Scope and Objects

1. Objects
   • Parallel SCM–dynamics:
     Continuous: d x / d t = f(x, u; G, θ_e) + w_e(t); Discrete: x_{k+1} = Φ_{Δt}(x_k, u_k; G, θ_e) + w_{e,k}.
     Structural layer: x_i = g_i( x_{pa(i)}, u_i, ζ_i ; θ_i ), where invariant mechanisms mean each g_i preserves form and parameters across e ∈ 𝔈.
   • Interventions: hard node do(x_S := a); soft interventions (replace conditional mechanism) p(x_S | x_{pa(S)}) → q(•); edge interventions (cut/reweight).
2. Inputs
   Multi-environment data { D_e }, environment labels e, optional prior graph G_0, candidate intervention sets S, time window [ t0, t1 ], units / RefCond.
3. Outputs
   Estimated causal parents pa_c(i), invariance evidence & scores, do-effects E[Y | do(X = a)], counterfactual trajectories, and manifest.stg.causal.
4. Constraints & boundaries
   • unit(x), unit(u), unit(Y) are mandatory; do(•) is an operator with dim( do(•) ) = "[1]" (dimensionless).
   • Identifiability assumptions (transferable perturbations, sufficient cross-environment heterogeneity, mechanistic separability) must be recorded.

II. Terms and Variables

• Environments & mechanisms: environment index e ∈ 𝔈; stable mechanisms g_i; perturbations w_e(t), ζ_i(e).
• Graphs & sets: causal graph G_c, parent set pa_c(i), intervention set S ⊆ V.
• Risk & invariance: environment risk R_e(θ), invariance constraint Inv(•); scores from IRM / ICP / Anchor.
• Effects & contrasts: τ = E[Y | do(X = a)] − E[Y | do(X = a')]; counterfactual x^{cf}(t).
• Dual forms: predictive (observational conditioning) vs causal (do operator) gap delta_form_do.

III. Axioms P710-*

• P710-1 (Parallel dual forms) — For any X → Y, compute both E[Y | X = x] and E[Y | do(X = x)]; record
  delta_form_do = | E[Y | X = x] − E[Y | do(X = x)] |.
• P710-2 (Invariant mechanisms) — For stable nodes i, g_i and parameters θ_i do not vary with e; environments enter through ζ_i(e) or exogenous distributions.
• P710-3 (Explicit measures/domains) — Expectations/integrals are explicit: ( ∫ p_e(x) • dx ), ( Σ_k • ); dynamical do-trajectories use ( ∫_t • dt ) with γ(time) declared.
• P710-4 (Unit consistency) — check_dim( E[Y | do(X = x)] − Y ) = "[Y]"; all I/O fields must declare unit(•).
• P710-5 (Identifiability) — Declare assumptions that break confounding (multi-env diversity, intervention reachability, regularity 0 < p(X = a) < 1 or simulable do).
• P710-6 (Topology traceability) — If G_c is inherited from Ch. 7 or a prior in Ch. 2, persist hashes and versions.

IV. Minimal Equations S710-*

• S710-1 (SCM and do-operator):
  x_i = g_i( x_{pa(i)}, ζ_i );
  do(X_S = a) sets x_S := a and cuts inbound edges pa(S) → S, inducing p^{do}(x).
• S710-2 (IRM invariance risk):
  With representation Φ and linear head w,
  min_{Φ,w} Σ_{e∈𝔈} R_e( w ∘ Φ ) subject to ∀ e, ∇_w R_e( w ∘ Φ ) |_{w=1} = 0.
  Penalized form: R(Φ,w) + λ Σ_e || ∇_w R_e ||^2.
• S710-3 (ICP):
  For a candidate parent set S, test Y ⟂ E | X_S (cross-environment conditional independence). If true, accept S as invariant parents; control FDR via multiple testing.
• S710-4 (Dynamical do-effect propagation):
  For continuous systems, do(X_S(t0) = a) yields expected effect
  ΔY(t) = ( ∫ ( ∂Y(t) / ∂x_S(t0) ) da ); linear approximation ΔY(t) ≈ C e^{A (t − t0)} B_S a.
  For discrete: ΔY_k ≈ C A^{k−k0} B_S a.
• S710-5 (Counterfactual trajectories):
  Given posterior noise ζ̂, build x^{cf}(t) under the same ζ̂ and replaced intervention:
  x^{cf}(t) = Solve( f, x(t0), do(X_S = a), ζ̂ ); contrast δ_cf(t) = x^{cf}(t) − x^{obs}(t).
• S710-6 (Dual-form gaps):
  delta_form_do = | E[Y | X = x] − E[Y | do(X = x)] | (point or window averages).
  Trajectory gap: delta_form_traj = ( ∫ || x̂_do(t) − x̂_pred(t) ||_2 dt ).
• S710-7 (Soft/edge interventions):
  Soft: p(x_i | x_{pa(i)}) → q(x_i | x_{pa(i)});
  Edge: modify A ← A + ΔA (cut j → i by setting A_{ij} = 0) or edit inputs of g_i.

V. Metrology Workflow M7-10 (Ready → Model/Estimate → Validate → Persist)

1. Ready
   • Aggregate { D_e } with metadata (sampling, RefCond, units).
   • Choose the assumption layer { IRM, ICP, Anchor, IV } and prior G_0.
   • Define the intervention library: node/edge/mechanism-replacement and physically attainable bounds.
2. Model / Estimate
   • Structural: estimate pa_c(i) via ICP/Anchor, or sparsify on G_0 with cross-env stability selection.
   • Dynamics: from Ch. 7’s A, B, C, g, split stable/unstable parameters; learn Φ minimizing Σ_e R_e plus invariance penalties.
   • Produce candidate G_c and mechanism parameters θ̂.
3. Intervene / Simulate
   • Build intervention models: do by edge cuts / conditional replacements.
   • Counterfactuals: condition on ζ̂ from observations and advance x^{cf}(t).
   • Estimate effects τ̂ and uncertainty (bootstrap / wild bootstrap or Delta method).
4. Validate
   • Invariance tests: Y ⟂ E | X_{pa_c} with FDR control;
   • Dual-form gaps: delta_form_do / traj p50 / p95;
   • Effect robustness: leave-one-environment-out variance of τ̂;
   • Physical consistency: check_dim, conservation / nonnegativity (if applicable).
5. Persist / Publish
   manifest.stg.causal = { G_c.hash, methods: { ICP / IRM / … }, envs, do_lib, effects: { pairs: [ X → Y ], τ̂, CI }, deltas: { do, traj }, tests: { pvals, FDR }, RefCond, units, method.hash }.

VI. Contracts & Assertions C70-10xx

• C70-1001 (Invariance pass): for each published Y, some S = pa_c(Y) satisfies Y ⟂ E | X_S with q ≤ q_max (suggest q_max = 0.1).
• C70-1002 (Dual-form bound): delta_form_do_p95 ≤ tol_do (default tol_do = 3 • ( atol + rtol • || Y || )).
• C70-1003 (Effect interpretability): report 95% CI for τ̂ and check_dim(τ̂) = unit(Y).
• C70-1004 (Reachability / regularity): for each intervention variable X, reachability or simulability holds (coverage cov_e(X) ≥ ρ_min or actual interventions exist).
• C70-1005 (Topology consistency): G_c.hash matches versions of runtime A/B/g; otherwise reject publication.
• C70-1006 (Counterfactual completeness): counterfactuals use the same ζ̂ as observations; persist inference method and seeds.
• C70-1007 (Unit consistency): check_dim( E[Y|do] − E[Y|obs] ) = "[Y]" passes.

VII. Implementation Bindings I70-10*

• I70-101 infer_invariant_parents(D_env, G0, method) -> { G_c, pa_c, scores } (method ∈ { ICP, Anchor, IRM })
• I70-102 fit_invariant_repr(D_env, loss_cfg) -> { Φ̂, ŵ, invariance_metrics }
• I70-103 simulate_do(model, do_spec, horizon, ζ_mode) -> { traj_do, stats } (ζ_mode ∈ { match_posterior, resample })
• I70-104 estimate_effect(pair: X→Y, do_values, model) -> { τ̂, CI, delta_form_do }
• I70-105 counterfactual_trajectory(model, obs_traj, do_spec) -> { traj_cf, δ_cf }
• I70-106 invariance_tests(D_env, pa_sets) -> { pvals, FDR }
• I70-107 edge_intervention(A, edges, op) -> A' (op ∈ { cut, reweight, replace })
• I70-108 emit_causal_manifest(results, policy) -> manifest.stg.causal

Invariants: non_decreasing(time); traceable RefCond / method / hash; delta_form_do ≤ tol_do; all check_dim pass.

VIII. Cross-References

• Spectral operators (stability & propagators): Chs. 2 & 4.
• Dynamics identification & controllability/observability: Chs. 7 & 8 (reachability & feasible driver sets).
• Numerical integration & events: Ch. 9 (stable advancement during do simulation with atomic events).
• Runtime/panels & manifests: Ch. 14 and Appendix C.
• Noise/cleaning: see the TopologyAtlas and PathCorrection volumes for cleaning policies (cross-volume reuse).

IX. Quality & Risk Control

1. SLIs/SLOs: invariance_pass_rate, delta_form_do_p95, effect_CI_width, transport_risk (cross-env gen error), fdr_icp, do_sim_stability.
2. Fallbacks
   • Invariance failure: restrict target subgraph, diversify environments, or add anchors/instruments.
   • Excessive dual-form gap: inspect confounding/missing parents, or switch to soft-intervention approximations.
   • Unstable counterfactuals: choose stiff-stable integrators (Ch. 9), shorten horizon, or regularize A.
   • Unit/contract failures: block publication and revert to the last compliant model.
3. Audit: persist G_c changes, test details (p-values/FDR), intervention scripts, seeds, delta_form_do traces, and panel snapshots.

Summary
This chapter aligns SCM invariance with network dynamics, delivering a closed loop from parent identification → do-intervention → counterfactual trajectories → compliant publication. With C70-10xx contracts, dual-form gaps delta_form_do, and manifest.stg.causal.*, causal inference becomes reusable and auditable across environments and at runtime.