25-EFT.WP.STG.Dynamics v1.0 | Chapter 1 — STG Domain Definition and Scope

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Chapter 1 — STG Domain Definition and Scope

One-Sentence Goal
Define the objects, I/O, and metrological boundaries of Spacetime Graph (STG) dynamics, and fix the traceability and compliance template for the entire volume.

I. Scope and Objects

1. Objects
   • Graphs and their time-varying forms: G(t) = ( V(t), E(t) ), supporting static and dynamic graphs.
   • Node states and edge flows: x_v(t), f_e(t); global quantities: Q(t).
   • Exogenous inputs and observations: u(t), y(t); noise and disturbances: η(t), w(t).
2. Inputs
   • Graph data and metadata: A, D, B, coordinates/partitions/constraints; time base & sampling: T_s, Δt.
   • Runtime & environment: RefCond (load, temperature, policy version, ops window, etc.), event streams and topology-change streams.
   • Priors & model families: forms of F/Φ, physical conservation, steady-state assumptions, control/observation structures.
3. Outputs
   • Trajectories & forecasts: x(t) or x_k; control laws and alerts: K_ctrl, alerts.*.
   • Metrological products: Inv / metrics (e.g., spectral_gap, energy dissipation) and uncertainties { u(x), U, nu_eff }.
   • Manifests: manifest.stg.* (Appendix C).
4. Boundaries & assumptions
   • Explicit time semantics: continuous t ∈ R_+ and discrete k ∈ N; enforce non_decreasing(time).
   • Mandatory units/dimensions: unit(field), dim(field), with check_dim.
   • When topology shocks or strong nonlinearity break differentiability, switch to MC/Bootstrap routes (Ch. 13, App. E).

II. Terms and Variables

• Graphs & operators: G = (V, E), |V| = N, A, D, L = D − A, L_norm, incidence B.
• State / input / observation: x(t) ∈ R^N, u(t) ∈ R^m, y(t) ∈ R^p.
• Noise: w(t), η(t); stochastic drivers may be written with dB_t.
• Kernels & semigroups: K_t = exp( − t L ), spectral decomposition L = U Λ U^T.
• Numerics & clocks: Δt, T_s, ρ(M).
• Dual-form gap: delta_form (continuous vs discrete), tolerance tol_Tarr (name reused cross-volume).
• Environment & corrections: RefCond, corr_env(x; RefCond).

III. Axioms P701-*

• P701-1 (Time & window monotonicity) — non_decreasing(time); event and window boundaries are closed.
• P701-2 (Parallel dual forms) — compute continuous/discrete forms in parallel and persist delta_form.
• P701-3 (Units & dimensions) — any field entering equations must provide unit() and dim() and pass check_dim.
• P701-4 (Explicit graph domain) — any sum/integral over the graph declares the domain & measure: ( Σ_{v∈V} • ), ( Σ_{e∈E} • ).
• P701-5 (Naming & conflicts) — forbid mixing T_fil and T_trans; strictly separate n and n_eff; formulas/symbols/definitions must not use Chinese.
• P701-6 (Traceability) — persist RefCond, method, solver, seed, and data provenance.
• P701-7 (Environmental corrections) — changes in external conditions enter as corr_env(x; RefCond).

IV. Minimal Equations S701-*

• S701-1 (Continuous-time state): dx/dt = F(x, u, t; θ) + G_w(x, t) w(t).
• S701-2 (Discrete-time state): x_{k+1} = Φ_{Δt}(x_k, u_k; θ) + G_ξ(x_k) ξ_k.
• S701-3 (Observation): y(t) = H(x, t) + η(t) or y_k = H_k x_k + η_k.
• S701-4 (Diffusion/conservative example): dx/dt = ( − L ) x + B u, energy E(x) = (1/2) x^T L x, with dE/dt ≤ 0.
• S701-5 (Graph heat-kernel solution): x(t) = exp( − t L ) x(0).
• S701-6 (Edge–node conservation): div = B f, dρ/dt + div = s.
• S701-7 (Dual-form gap): delta_form = || x(t+Δt) − x_{k+1} ||, assert delta_form ≤ tol_Tarr.
• S701-8 (Numerical consistency example): Euler x_{k+1} ≈ ( I − Δt L ) x_k; stability ρ( I − Δt L ) < 1.

V. Metrology Workflow M7-1 (Ready → Model → Check → Persist)

1. Ready
   • build_graph: clean/standardize G; compute L / L_norm / U / Λ; lock RefCond, units, schema.
   • Validate check_dim(all), graph connectivity, and anomalous weights.
2. Model/Estimate
   Choose F / Φ with constraints (diffusion/wave/consensus/conservative); estimate θ; configure integrator and Δt.
3. Checks
   Run continuous/discrete dual forms; record delta_form; enforce energy/conservation contracts; verify stability & spectral bounds.
4. Persist/Publish
   Produce manifest.stg.case, manifest.stg.runtime, manifest.stg.uncertainty; sign with traceable hashes.

VI. Contracts & Assertions C70-* (suggested gates)

• C70-011 (Dual-form gap): p95(delta_form) ≤ tol_Tarr (suggest tol_Tarr = 1e-3 * ||x||).
• C70-012 (Spectral stability): for diffusion, Δt ≤ 1 / λ_max(L); if Euler, require ρ( I − Δt L ) < 1.
• C70-013 (Energy dissipation): diffusion systems must satisfy dE/dt ≤ 0; violations trigger fallback.
• C70-014 (Data freshness): max_age(obs) ≤ 2 * T_s; coverage ≥ 0.95.
• C70-015 (Unit consistency): check_dim( y − f(x) ) = "[0]" must pass.
• C70-016 (RefCond completeness): missing critical RefCond.* fields blocks publication.

VII. Implementation Bindings I70-* (domain-initialization core)

• I70-11 build_graph(data, schema) -> G
• I70-12 compute_laplacian(G, mode) -> { L, L_norm, U, Λ }
• I70-13 annotate_refcond(meta) -> RefCond
• I70-14 simulate_stg(F_or_Φ, x0, u, G, tgrid, solver) -> traj
• I70-15 validate_units(ds) -> report
• I70-16 check_stg_contracts(ds, rules) -> report
• I70-17 emit_stg_manifest(results, policy) -> manifest.stg

Invariants: non_decreasing(time); delta_form ≤ tol_Tarr; check_dim(all); RefCond / method / solver are traceable.

VIII. Cross-References

• Time semantics & sync: companion Energy Filaments white paper Chs. 2/3 (S/P/M/I).
• Path/medium corrections & dual-form practice: EFT.WP.Metrology.PathCorrection v1.0, Chs. 2/10/11.
• Topological events & complex mapping: EFT.WP.Particle.TopologyAtlas v1.0, Chs. 6/7/9.

IX. Quality & Risk Control

• Suggested SLO/SLIs: delta_form_p95, spectral_gap, lag_ms_p95, coverage, energy_dissipation_rate.
• Audit: version/deps/seed/data provenance with hash checks; dual-environment recomputation with difference ≤ ε_dual.
• Fallbacks: switch integrators (Euler → RK/BDF), shorten Δt, enable graph coarsening or freeze topology windows; failing that, enter read-only observation mode.

Summary
This chapter fixes the domain boundaries, terminology, minimal equations, workflows, and contracts for STG dynamics, and provides I70-* initialization interfaces and invariants. All subsequent chapters must retain dual-form publication, unit/dimension checks, and RefCond traceability; all releases are grounded in manifest.stg.*.