Editing Openai/69596681-3f7c-8006-b9fd-01d9d8191763 (section)

=== Assistant: Proceeding. ===
Proceeding. Below is METHODS BLOCK 4 — RESIDUAL OPERATOR, expanded into a fully specified residual-field construction with metric structure, invariance properties, and enforcement logic. This block is the sole interface between model predictions and empirical data.

METHODS BLOCK 4 — RESIDUAL OPERATOR

OBJECTIVE
Define the unique, admissible mapping from empirical data and model predictions to residuals. Residuals are the only objects passed to statistical evaluation. No other comparison channel is permitted.

RESIDUAL DEFINITION
For dataset D={⟨yᵢ,Σᵢ,metaᵢ⟩}ᵢ and parameters θ∈M_IR, define residual vector:

rᵢ(θ)=φ(yᵢ)−φ(Pᵢ(θ,b))

where φ is the canonical measurement embedding mapping observables to ℝ^{dᵢ}.

Constraints on φ:
• Deterministic
• Invertible on its image
• Dimension-preserving
• Unit-consistent
• Frozen pre-run

No learned embeddings. No adaptive scaling.

RESIDUAL FIELD
Define residual field R(θ)={rᵢ(θ)}ᵢ=1…N.

The field is ordered, indexed, and aligned with dataset ordering. Reordering forbidden.

METRIC STRUCTURE
Each residual rᵢ endowed with metric induced by covariance Σᵢ:

‖rᵢ‖²_{Σᵢ}=rᵢᵀΣᵢ⁻¹rᵢ.

Σᵢ must satisfy:
• SPD
• Dimension match with rᵢ
• Calibration hash consistency with metaᵢ

Metric properties:
• Positive-definite
• Coordinate-invariant under φ
• Additive across independent data points

GLOBAL RESIDUAL FUNCTIONAL
Define total residual functional:

ℛ(θ)=∑ᵢ rᵢᵀΣᵢ⁻¹rᵢ.

ℛ used only for diagnostics and structural checks, not optimization.

INVARIANCE REQUIREMENTS
Residuals invariant under:
• Reparameterization of θ within M_IR
• Equivalent unit representations
• Permissible coordinate transforms declared in φ

Residuals must change only under changes to data or θ.

NO DATA MASSAGING
Forbidden operations:
• Smoothing
• Filtering
• Rebinning
• Outlier removal
• Winsorization
• Adaptive weighting

Any modification of yᵢ or Σᵢ beyond φ ⇒ INVALID.

FAILURE CONDITIONS
• Undefined φ(yᵢ) or φ(Pᵢ) ⇒ INVALID
• Σᵢ non-SPD ⇒ INVALID
• Dimension mismatch ⇒ INVALID
• NaN or Inf in rᵢ ⇒ INVALID

AUDIT EMISSION
For each execution emit:
• Residual hashes H(rᵢ)
• Metric values ‖rᵢ‖²
• Aggregate ℛ

Residuals are immutable artifacts.

ROLE IN PIPELINE
Residuals feed exclusively into:
• Structural Violation (Block 5)
• Likelihood Kernel (Block 8)

No other consumer allowed.

END METHODS BLOCK 4.

If you confirm, the next response will be METHODS BLOCK 5 — STRUCTURAL VIOLATION, expanded into tolerance geometry, hard constraint logic, and collapse triggering.

© Robert R. Frost
2026-01-03