09-EFT.WP.Core.Density v1.0 | Chapter 1 — Foundations of Density and Measure

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Chapter 1 — Foundations of Density and Measure

I. Scope and Terminology

• This chapter establishes a common calculus for densities and measures, fixing the domain Omega, base measure mu, and differential elements dV / dx / dS / d ell, and distinguishing three density classes: physical density rho(x,t), probability density p(x), and spatial/spatio-temporal intensity lambda(x,t). Every integral explicitly declares its measure and domain, e.g., ( ∫_Omega p(x) dx ) = 1, ( ∫_V rho(x,t) dV ) = M(t), and path integrals as ( ∫_{gamma(ell)} a(ell) d ell ).
• The postulates P91-* introduced here underpin later chapters; minimal equations S92-* begin in Chapter 2 (e.g., the continuity equation S92-1 is defined there).

II. Domains, σ-Algebras, and Measures

• Domains and base measures. Let Omega be the domain (spatial, spatio-temporal, or parametric), 𝓕 a σ-algebra, and mu the base measure. Common cases: Lebesgue dx, volume dV, surface dS, and path d ell.
• Product measures. On a Cartesian product Omega = X × T use ( dV × dt ); in non-orthogonal coordinates, the same principle applies with dV (including the Jacobian factor; see §V).
• P91-1 (measure explicitness). Every integral must declare its measure and domain; expressions omitting either are not eligible for publication.
  Examples: ( ∫_Omega p(x) dx ) = 1, ( ∫_V rho(x,t) dV ) = M(t), ( ∫_{gamma(ell)} n_eff d ell ).

III. Density Classes and Their Relations

• Probability density p(x). Dimensionless; must satisfy ( ∫_Omega p(x) dx ) = 1. Under a change of variables x → y,
  p_Y(y) = p_X(x(y)) * | det( ∂x/∂y ) |.
• Physical density rho(x,t). Dimensional (e.g., mass density rho_m, charge density rho_q, number density n(x,t)); total quantity M(t) = ( ∫_V rho(x,t) dV ). Together with flux J(x,t) it satisfies continuity with sources/sinks (see Chapter 2, S92-1).
• Spatial/spatio-temporal intensity lambda(x,t). Point-process intensity with ( ∫_A lambda(x,t) dV ) ≈ E[ N(A,t) ]; time-integrals yield arrival rates (see Chapter 5).
• P91-2 (unit/dimension closure). unit(x) and dim(x) must pass check_dim(expr); never mix dimensionless p(x) with dimensional rho(x,t).

IV. From Fields to Density and Total

• From total to density. Given a time-varying total M(t), define rho(x,t) such that ( ∫_V rho(x,t) dV ) = M(t); on discrete voxels {V_i}, approximate M(t) ≈ ( ∑_i rho_i(t) * V_i ).
• From counts to probability density. With samples {x_i}_{i=1..N} and bin width Delta, the histogram density is p_hat = count / ( N * Delta ) (see Chapter 7, S92-10).
• Conservation check (placeholder; see Chapter 2). Define the mass-conservation residual
  eps_mass = | dM/dt - ( ∫_V s(x,t) dV ) - ( ∫_{∂V} J•n dS ) |; it must meet a release threshold (see S92-2 and workflow Mx-91).

V. Coordinate Transforms and Jacobians

• Variable transforms.
  Probability: p_Y(y) = p_X(x(y)) * | det( ∂x/∂y ) |.
  Physical: rho_Y(y,t) = rho_X(x(y),t) * | det( ∂x/∂y ) |^{-1} to preserve ( ∫ rho dV ).
• Curvilinear coordinates and measures. In general coordinates q = (q1,q2,q3), dV = | det( ∂x/∂q ) | dq1 dq2 dq3; path measure d ell = || ∂x/∂s || ds.
• P91-3 (normalization discipline). Probability densities must pass a normalization error check before publication:
  eps_norm = | ( ∫_Omega p(x) dx ) - 1 | <= eps_tol, with eps_tol recorded in the manifest.

VI. Support, Boundaries, and Source/Sink Metadata

• Supports and measurable sets. supp(p) ⊆ Omega for probability, supp(rho) ⊆ V for physical densities. Manifests must state the domain closure and boundary class (Dirichlet/Neumann/Robin).
• Sources/sinks and boundary flux. The source term s(x,t) has the same dimension as ∂_t rho; boundary flux ( ∫_{∂V} J•n dS ) participates in the conservation relation (see Chapter 2).

VII. Path and Arrival-Time Anchors (Cross-Volume Consistency)

1. Path integrals and arrival time. Always cite both forms:
   • Constant pulled out: T_arr = ( 1 / c_ref ) * ( ∫_{gamma(ell)} n_eff d ell )
   • General form: T_arr = ( ∫_{gamma(ell)} ( n_eff / c_ref ) d ell )
   • Discrepancy: delta_form = | ( 1 / c_ref ) * ( ∫_{gamma(ell)} n_eff d ell ) - ( ∫_{gamma(ell)} ( n_eff / c_ref ) d ell ) |.
2. Use cases. When the time axis of rho(x,t) or lambda(x,t) depends on arrival-time alignment, you must report gamma(ell), d ell, c_ref, n_eff, and delta_form (as in Core.Sea Chapter 8).

VIII. Quality-Control Checklist (This Chapter)

• Normalization and dimensions.
  Probability: eps_norm <= eps_tol.
  Physical: publish unit(rho), dim(rho), and M(t).
• Measures and domains. Provide (Omega, mu) or (V, dV), boundary ∂V and its class; when paths appear, declare gamma(ell) and its parameter range.
• Minimal metadata fields.
  measure = {"space": "...", "base": "lebegue|count|surface|line", "coords": "...", "jacobian": "explicit|implicit"}
  support = {"domain": "...", "boundary": "D|N|R", "closure": "..."}
  norm = {"eps_norm": value, "eps_tol": value}
  units = {"rho": "...", "J": "...", "s": "..."}

IX. Reference Implementation Hooks (I90 excerpts for this chapter)

• Define measures and densities.
  define_measure(space:str, metric:str|None=None, base:str="lebegue") -> MeasRef
  density_from_field(field:any, units:str) -> DensRef (bind a field and units as rho(x,t) or n(x,t)).
• Normalization and binding.
  renormalize(pdf:PdfRef, domain:any) -> PdfRef (ensure ( ∫ p(x) dx ) = 1);
  bind_to_equations(eqn_refs:list[str]) -> bool (declare binding to P91-*, S92-*).

X. Example: Minimal Workflow from Counts to Density

• Measure & domain. Create MeasRef = define_measure(space="R^1", base="lebegue"), declaring Omega and dx.
• Histogram density. Build p_hat from samples and bin width Delta, then verify ( ∫_Omega p_hat(x) dx ) and record eps_norm in the manifest.
• Physical (number) density. Convert counts to n(x,t) on voxels {V_i} with n_i = count_i / V_i, and publish M(t) = ( ∑ n_i * V_i ).
• Metadata. Ensure the four blocks—measure/units/support/norm—are complete; if the time axis depends on arrival time, report both forms and delta_form per §VII.

XI. Outputs and Cross-Volume Alignment

1. Postulates released: P91-1, P91-2, P91-3.
2. Artifacts: density/measure metadata templates, normalization and dimensionality checklists.
3. Alignment:
   • Arrival-time dual forms and spectral notation consistent with Core.Sea;
   • Units and uncertainty (u(x), U = k * u_c) consistent with Core.Metrology (developed in later chapters);
   • Dataset and publication field names consistent with Core.Threads.