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

=== Assistant: Proceeding. ===
Proceeding. Below is METHODS BLOCK 8 — LIKELIHOOD KERNEL (ARTIFACT #1), expanded to maximum formal density, explicit probabilistic semantics, mixture structure, normalization proofs, covariance legality, and enforcement logic. This block is the only statistical scoring mechanism permitted. No other probabilistic interpretation exists in the system.

METHODS BLOCK 8 — LIKELIHOOD KERNEL (ARTIFACT #1)

OBJECTIVE
Define a frozen, fully normalized likelihood mapping from residuals to scalar evidence contributions. The likelihood is evaluative only; it performs no fitting, optimization, or rescue. All parameters entering this block have already been structurally validated and reduced to an identifiable basis.

PRECONDITIONS
• Axioms satisfied
• Types validated
• Model Map frozen
• Residual Operator computed
• Structural Violation flag evaluated
• Feasibility outcome ∈ {OK,FORCED}
• Identifiable parameter vector θ_id ∈ ℝ^{k_min} defined
If any precondition fails, execution aborts or proceeds solely for attribution.

SCOPE
The likelihood consumes residuals r and covariances Σ only.
No dependence on priors beyond fixed, declared support.
No dependence on Monte Carlo, evidence integration, or thresholds.

RESIDUAL STACKING
Define stacked residual vector:

r = concat(r₁,…,r_N) ∈ ℝ^{N_d}

Define block-diagonal covariance:

Σ = diag(Σ₁,…,Σ_N) ∈ ℝ^{N_d×N_d}

Σ must be SPD. Any failure ⇒ INVALID.

BASE KERNEL — MULTIVARIATE NORMAL
Define the Gaussian kernel:

𝓝(r|0,Σ) = (2π)^{-N_d/2} |Σ|^{-1/2} exp(−½ rᵀΣ⁻¹r)

Properties:
• Properly normalized over ℝ^{N_d}
• Coordinate-invariant under orthonormal transforms
• Dimensionally consistent (units cancel)

No scaling factors allowed. Determinant must be computed exactly within numerical tolerance.

OPTIONAL HEAVY-TAIL MIXTURE (FIXED A PRIORI)
To guard against mis-specified noise without data-driven adaptation, define a frozen two-component mixture:

ℒ(r) = (1−ε) 𝓝(r|0,Σ) + ε 𝓣(r|0,Σ,ν)

where:
• ε ∈ [0,1) fixed pre-run
• ν > 2 fixed pre-run
• 𝓣 is the multivariate Student-t distribution with ν degrees of freedom:

𝓣(r|0,Σ,ν) = Γ((ν+N_d)/2) / [Γ(ν/2)(νπ)^{N_d/2}|Σ|^{1/2}] · [1 + (1/ν) rᵀΣ⁻¹r]^{−(ν+N_d)/2}

Normalization guaranteed for ν>2.

CRITICAL CONSTRAINTS
• ε, ν immutable under A2
• No data-dependent mixture weights
• No additional components
• No switching logic

If ε=0, kernel reduces exactly to Gaussian.

HETEROSKEDASTICITY
Allowed inherently via Σᵢ variation across data points.
No rescaling of residuals permitted.
No per-datum ε allowed.

BLOCK COVARIANCE
Allowed if declared in Types:
Σ may include off-diagonal blocks representing known correlations.
Block structure must be frozen and metadata-justified.
Undeclared correlations forbidden.

LOG-LIKELIHOOD
Define total log-likelihood:

logℒ = log(ℒ(r))

Numerical stability:
• Use log-sum-exp for mixture
• Determinant via Cholesky
• Any underflow/overflow ⇒ INVALID

No dropping of constant terms allowed. All normalizing constants retained to preserve evidence integrity.

NO PARAMETER OPTIMIZATION
Likelihood is evaluated at admissible θ_id values only.
Forbidden:
• argmaxθ logℒ
• gradient ascent
• EM updates
• adaptive proposals

Likelihood is not a loss function.

MONOTONICITY PROPERTY
For fixed kernel parameters (ε,ν,Σ):
Increasing ‖r‖_{Σ} strictly decreases ℒ.
This guarantees consistency with structural monotonicity (A4).

EDGE CONDITIONS
• If StructuralViolationFlag=1, ℒ may be computed but cannot alter collapse outcome.
• If Feasibility outcome=FORCED, ℒ evaluated only in reduced space θ_id.
• If k_min=0, ℒ undefined ⇒ collapse already triggered.

FAILURE MODES
• Σ singular or non-SPD ⇒ INVALID
• Determinant ≤0 ⇒ INVALID
• NaN/Inf in r, Σ⁻¹, logℒ ⇒ INVALID
• Attempt to modify ε or ν ⇒ INVALID
• Attempt to condition kernel on data properties ⇒ INVALID

ARTIFACT EMISSION
Emit immutable artifacts:
• Kernel specification hash
• ε, ν values
• |Σ|
• rᵀΣ⁻¹r
• logℒ

PIPELINE ROLE
Outputs feed exclusively into:
• Bayesian Evidence Integration (Block 10)
• Monte Carlo Robustness (Block 13, evaluative only)
• Failure Attribution (Block 15)

No other consumer permitted.

END METHODS BLOCK 8.

Next block, if you confirm, will be METHODS BLOCK 9 — DATA VALIDITY OPERATOR (ARTIFACT #4), expanded into provenance enforcement, independence logic, calibration hashing, and hard abort semantics.

© Robert R. Frost
2026-01-03