02-EFT.WP.Core.Equations v1.1 | Foreword

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Foreword

I. Purpose and Positioning of This Volume

• Consolidate and standardize the Sxx-? family of minimal equations, unifying strong/weak formulations, domain and boundary declarations, the path—arrival-time convention, and alignment with the implementation-binding layer.
• Introduce no new symbols or units; all symbols and units strictly follow Core.Terms and Core.Metrology. Cross-volume citations adopt the fixed format: “see companion white paper Energy Threads, Chapter x, S/P/M/I…”.

II. Intended Audience and Application Scenarios

1. Target readers: model and theory researchers, numerical implementation engineers, metrology and verification teams, methods and reproducibility practitioners.
2. Typical scenarios:
   • Selecting or establishing an Sxx-? equation and realizing it as a weak form with numerical examples.
   • Checking path consistency and dimensional closure of arrival-time expressions.
   • Aligning equations with I20-* interfaces to register, discretize, solve, and regress.

III. Authoring Conventions and Mandatory Constraints

1. All formulas, symbols, and definitions are English plain text; wrap every inline item in backticks, e.g., T_fil(x,t), n_eff(x,t), gamma(ell), d ell, c_ref.
2. Any expression with division, integrals, or composite operators must use parentheses and explicitly state path and measure:
   • Constant factored: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
   • General form: T_arr = ( ∫ ( n_eff / c_ref ) d ell )
3. Strong vs. weak forms: use = for strong form; denote weak form with weak= or inner_V[u,v].
4. Prohibit ambiguity and mixing: T_fil must not be used as T_trans; strictly distinguish n from n_eff; the bare symbols "c", "T", "n" are forbidden.
5. Dimensional closure: apply check_dim to every Sxx-?; the integrand ( n_eff / c_ref ) * d ell is dimensionless.
6. Statistical conventions must declare windows or domains: avg_t[f; Δt], avg_V[f; V=Ω], avg_gamma[f].

IV. Relationship to Other Volumes

• With Core.Terms: inherit unified symbols, naming, postulates, and the conflict checklist (e.g., P70-*).
• With Core.Metrology: align on dimensions, units, non-dimensional mappings, and uncertainty-propagation conventions.
• With Methods.Repro and Core.Errors: align reproducibility workflows from equation to implementation and error classes/threshold rules.
• With implementation layer I20-*: this volume provides normative prototypes for equation registration, validation, discretization, solution, and regression.

V. Deliverables and Compliance Checklist

1. Deliverables:
   • Minimal equation catalog Sxx-? (strong/weak side-by-side).
   • Boundary and initial condition templates (∂Ω_D, ∂Ω_N, IC).
   • Path—arrival-time consistency examples and the reference path gamma_ref(ell).
   • Minimal numerical working sequence (aligned to I20-*) and regression metrics (L2, L_inf, T_arr).
2. Compliance checklist:
   • Are gamma(ell) and d ell explicit in all expressions?
   • Are the bare "c", "T", "n" avoided, and has check_dim been completed?
   • Are statistics and non-dimensionalization consistent with Core.Metrology?
   • Do validate_equation and Chapter 7 lint checks pass (forbid "∫ n d ell / c" and missing parentheses)?

VI. Reading Path and Implementation Guidance

1. Recommended reading order:
   • Chapter 1 establishes equation numbering and strong/weak templates.
   • Chapter 2 confirms the two arrival-time conventions and the additivity of segmented paths.
   • Chapters 3–5 select constitutive and governing equations by object and complete the weak-form derivation.
   • Chapter 6 assembles boundary/initial conditions.
   • Chapters 7–9 finalize variational derivations, discretization, and regression checks.
2. Minimal implementation sequence:
   register_equation → 2. validate_equation → 3. check_dim_equation → 4. discretize_path / assemble_operator / solve_* → 5. compare_solutions.

VII. Versioning and Maintenance

• Current version v1.1 is a stable minor update over v1.0, interface-compatible; additions and edits are detailed in Appendix B.
• Version semantics follow major.minor.patch; minor versions only add or revise content without breaking changes.
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