16-EFT.WP.Methods.Cleaning v1.0 | Chapter 4 Units, Dimensions, and Metrological Harmonization

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Chapter 4 Units, Dimensions, and Metrological Harmonization

One-Sentence Goal
Unify unit(x) and dim(x), perform affine unit conversions and dimension-conservation checks, and ensure every expression destined for the release plane satisfies check_dim(expr) = true, forming an auditable metrology loop.

I. Scope & Objects

1. Applicable targets
   • All numeric fields in D_std produced by Chapter 3, together with their declared unit(x) and dim(x).
   • All derived and intermediate quantities related to arrival time, path, time mapping, and density normalization.
2. Target artifacts
   • Produce D_metric (data after metrological harmonization) and report.units (conversion and validation report).
   • Record, in the manifest, the conversion chain, the failure list, and a summary of uncertainty propagation.

II. Terms & Variables (Memory Anchors)

• Units & dimensions: unit(x), dim(x), check_dim(expr).
• Uncertainty: u(x), U = k * u_c.
• Time & path: tau_mono, ts, offset, skew, J, Delta_t, gamma(ell), d ell, L_gamma.
• Arrival-time two forms:
  T_arr_form1 = ( 1 / c_ref ) * ( ∫ n_eff d ell ),
  T_arr_form2 = ( ∫ ( n_eff / c_ref ) d ell ),
  delta_form = | T_arr_form1 - T_arr_form2 |.
• Reserved conflicts: T_fil ≠ T_trans, n ≠ n_eff.

III. Axioms (P104-*)

• P104-01 SI consistency
  Adopt SI and compatible derived units throughout; use rad for angles, K for temperature, and treat probability density as dimensionless.
• P104-02 Affine conversion
  Any unit conversion is equivalent to an affine map x_SI = a * x + b, where b ≠ 0 is used only for zero-offset units (e.g., Celsius).
• P104-03 Dimension conservation
  Before publication, ensure check_dim( y - f(x) ) = true; any dimension mismatch is a contract failure.
• P104-04 Explicit units
  Every numeric field must declare unit(x) and dim(x); implicit units in names or comments are forbidden.
• P104-05 Uncertainty accompaniment
  Any numeric conversion or functional mapping must update u(x) and record, in the manifest, the propagation convention and coverage factor k.

IV. Minimal Equations (S104-*)

• S104-01 Affine unit conversion
  x_SI = a * x + b
  u(x_SI) = |a| * u(x)
• S104-02 Dimension algebra
  dim(x * y) = dim(x) + dim(y)
  dim(x / y) = dim(x) - dim(y)
  dim(x ^ p) = p * dim(x)
  dim( x + y ) is defined iff dim(x) = dim(y)
• S104-03 Time & path dimension checks
  dim(tau_mono) = [T], dim(ts) = [T], dim(ell) = [L], dim(L_gamma) = [L]
  dim(c_ref) = [L][T]^-1, dim(n_eff) = [1], dim(T_arr_form1) = dim(T_arr_form2) = [T]
• S104-04 Checking operator
  check_dim( y - f(x) ) = true ⇔ dim(y) = dim(f(x))
• S104-05 Uncertainty propagation (linear approximation)
  u^2(y) = sum_j ( ( ∂f / ∂x_j )^2 * u^2(x_j) )
  If correlations exist, add 2 * sum_{i<j} ( ( ∂f / ∂x_i )( ∂f / ∂x_j ) * cov(x_i, x_j) )
• S104-06 Density and normalization dimensions
  Probability density: ( ∫_Omega p(x) dx ) = 1, dim(p) = [1]
  Example physical density: dim(rho) = [M][L]^-3, with conservation check sum_i( rho_i * DeltaV_i ) = const ± tol_mass

V. Canonical Units Table (Default Conventions)

1. Base: [L]=m, [T]=s, [M]=kg, [Θ]=K, [I]=A, [N]=mol, [J]=cd
2. Common derived:
   • Velocity m/s, acceleration m/s^2, pressure Pa, energy J, power W, frequency Hz, angular velocity rad/s
   • Path arc length ell: m, arrival time T_arr: s, reference light speed c_ref: m/s, n_eff: 1
   • Timestamps: internal tau_mono: s; released ts: s or ms, which must be explicit in the manifest

VI. Cleaning Process (M10-4, Metrological Harmonization)

1. Field scan and completion
   Traverse numeric fields, fill any missing unit(x) and dim(x); quarantine missing entries and propose candidates.
2. Unit legality and whitelist check
   Verify unit(x) against the policy whitelist; reject or map to the nearest compatible unit otherwise.
3. Affine conversion to canonical units
   Compute x_SI = a * x + b and update u(x_SI) = |a| * u(x); standardize ts and ell (e.g., ms → s, mm → m).
4. Dimension-conservation checks
   Run check_dim( y - f(x) ) on key relations, including:
   • ts = map_to_pub( tau_mono ; offset, skew, J )
   • Consistent dimensions between T_arr_form1 and T_arr_form2
   • Consistency of L_gamma = ( ∫_gamma 1 d ell )
5. Record conversion chains and uncertainties
   In manifest.units, write from_unit → to_unit, a,b, u(x) → u(x_SI), coverage factor k, and correlation assumptions.
6. Reporting and anomaly quarantine
   Emit report.units containing unit_conv_applied, dim_failures, u_propagated; tag failing branches for manual or policy remediation.

VII. Contracts & Assertions (Chapter Must-Pass Items)

• Unit declarations complete: forall x ∈ numeric_fields: exists unit(x) ∧ exists dim(x)
• Whitelist compliance: forall x: unit(x) ∈ policy.allowed_units(field)
• Affine conversion success: forall x: convert_status(x) = ok
• Dimension conservation: check_dim( y - f(x) ) = true (cover at least both T_arr forms, L_gamma, and the time mapping)
• Uncertainty accompaniment: forall x: defined u(x) and, after conversion, u(x_SI) = |a| * u(x)
• Metrology report present: exists(report.units) and aligned with the manifest

VIII. Implementation Binding (I10-4)

1. Interface prototypes
   • repair_units(ds, policy) -> report
   • coerce_unit(field, to_unit) -> field'
   • check_dim_set(ds, eqset) -> results
   • propagate_uncertainty(ds, f, inputs) -> u(y)
2. Preconditions
   • policy provides the whitelist, affine parameters {a,b}, target units, and correlation assumptions.
   • ds already satisfies Chapter 3 key and index contracts.
3. Postconditions & invariants
   All numeric fields in ds' are in canonical units; check_dim_set passes; report.units is replayable; manifest.units matches the data version.
4. Failure semantics
   E_UNIT_UNKNOWN, E_DIM_MISMATCH, E_AFFINE_PARAM_MISSING, E_UNCERTAINTY_MISSING, each pinpointing fields and record ranges.

IX. Uncertainty & Common Operator Propagation

• Addition: y = x1 + x2 → u^2(y) = u^2(x1) + u^2(x2) (independence assumption)
• Multiplication: y = x1 * x2 → (u(y)/y)^2 = (u(x1)/x1)^2 + (u(x2)/x2)^2
• Ratio: y = x1 / x2 → (u(y)/y)^2 = (u(x1)/x1)^2 + (u(x2)/x2)^2
• Power law: y = x^p → u(y)/y = |p| * u(x)/x
• Affine: y = a * x + b → u(y) = |a| * u(x)

X. Unit Conventions for Common Fields (with Examples)

1. Time
   • tau_mono: s, ts: s (if the source is ms, then a = 1e-3, b = 0)
   • offset, skew, J: in s or as dimensionless rates, explained in the manifest
2. Path
   • ell: m (if the source is mm, then a = 1e-3)
   • L_gamma: m, derived from L_gamma = ( ∫_gamma 1 d ell )
3. Arrival time
   • c_ref: m/s, n_eff: 1, T_arr: s
   • The two forms must share the same dimension to pass check_dim( T_arr_form1 - T_arr_form2 )
4. Density & normalization
   • Probability density p(x): 1; physical density rho: kg/m^3.
   • Normalization error eps_norm and conservation residual res_mass are detailed in Chapter 13.

XI. Automated Check Set (Recommended Minimal eqset)

• check_dim( ts - map_to_pub( tau_mono ; offset, skew, J ) )
• check_dim( T_arr_form1 - T_arr_form2 )
• check_dim( L_gamma - ( ∫_gamma 1 d ell ) )
• check_dim( rho * dV ) and assertions related to normalization and conservation

XII. Audit & Panel Fields

• Minimal panel
  unit_dim_filled, unit_conv_applied, dim_failures, arr_dim_ok, u_propagated, eps_norm, res_mass
• Alert suggestions
  dim_failures = 0, arr_dim_ok = 1, u_propagated = 1, eps_norm ≤ tol_norm, res_mass ≤ tol_mass

XIII. Cross-References

• Standard inputs and schema: Chapter 3.
• Time axis and synchronization: Chapter 5.
• Path and arrival time: Chapter 6.
• Density and normalization: Chapter 13.
• Uncertainty propagation: Appendix E.
• Contracts and release freeze: Chapter 10 and Appendices B, C.

Summary
This chapter establishes strong constraints for metrological harmonization via P104-*, provides S104-* for affine conversion, dimension algebra, and uncertainty propagation, and, through process M10-4, transforms D_std into D_metric. After completion, the key relations (time mapping, path length, the two arrival-time forms) all pass check_dim, while the conversion chain and uncertainty trail are preserved in the manifest, supplying a solid metrology foundation for the consistency work in Chapters 5–6 and the release-freeze procedure in Chapter 10.