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

=== Assistant: Proceeding. ===
Proceeding. Below is METHODS BLOCK 7 — PARAMETER IDENTIFIABILITY (ARTIFACT #2), expanded into full Fisher-geometry formalism, nullspace projection mechanics, invariance guarantees, and enforcement logic. This block is the only place where parameter redundancy is diagnosed and resolved. No other block may alter parameter dimension.

METHODS BLOCK 7 — PARAMETER IDENTIFIABILITY (ARTIFACT #2)

OBJECTIVE
Determine which parameter directions are empirically identifiable given the frozen model, residual operator, and covariance structure. Remove all empirically redundant directions before any likelihood or evidence computation. Identifiability is structural-relative-to-data, not epistemic or subjective.

PRECONDITIONS
• Axioms satisfied
• Types validated
• Model Map frozen
• Residual Operator well-defined
• Feasibility outcome ∈ {OK,FORCED}
If outcome=COLLAPSE, this block executes only to emit attribution artifacts.

JACOBIAN DEFINITION
For admissible θ∈M_IR, define Jacobian:

J(θ) = ∂r(θ)/∂θ ∈ ℝ^{N_d × k}

where N_d = total residual dimension after embedding φ and stacking all rᵢ.

Constraints:
• J exists almost everywhere on M_IR
• J finite, bounded
• No numerical differentiation allowed unless explicitly frozen and audited
• Automatic differentiation permitted if deterministic and hash-checked

FISHER INFORMATION
Define Fisher information matrix:

F(θ) = J(θ)ᵀ Σ⁻¹ J(θ)

where Σ is the block-diagonal covariance formed from all Σᵢ.

Properties:
• F symmetric
• F positive semidefinite
• rank(F) ≤ k

F evaluated at any admissible θ that satisfies V_struct(θ)=0.
No averaging over θ. No expectation over likelihood.

IDENTIFIABILITY CRITERION
Define minimal identifiable dimension:

k_min = rank(F)

Identifiable basis exists iff k_min ≥ 1.

Nullspace:
Null(F) = { v ∈ ℝᵏ | Fv = 0 }

These directions are empirically unidentifiable under current data.

BASIS INVARIANCE
Identifiability is invariant under any smooth reparameterization of θ.
Formally, for θ′=g(θ) with invertible Jacobian G:

F′ = Gᵀ F G ⇒ rank(F′)=rank(F).

Thus identifiability is coordinate-free.

PROJECTION OPERATOR
Define orthogonal projection onto identifiable subspace:

Let U = eigenvectors of F corresponding to nonzero eigenvalues.
Define P_id = U Uᵀ.

Reduced parameter vector:

θ_id = P_id θ ∈ ℝ^{k_min}.

All downstream computations MUST use θ_id exclusively.

APPLICATION RULES
• If k = k_min ⇒ no projection required
• If k > k_min ⇒ projection mandatory (FORCED case)
• If k_min = 0 ⇒ Feasible=0 ⇒ COLLAPSE

No partial projection. No soft regularization.

NO PRIORS, NO BAYES HERE
• No priors used
• No likelihood weighting
• No Bayesian marginalization

This block is purely geometric and data-constrained.

FAILURE MODES
• rank(F) ill-defined ⇒ INVALID
• Numerical rank tolerance altered post-freeze ⇒ INVALID
• Attempt to retain nullspace parameters ⇒ INVALID
• Attempt to add constraints to lift degeneracy ⇒ INVALID

ARTIFACT EMISSION
Emit immutable artifacts:
• F matrix hash
• rank(F)=k_min
• Nullspace basis vectors
• Projection operator P_id
• Identifiability outcome {FULL,FORCED,COLLAPSE}

PIPELINE ROLE
Outputs feed exclusively into:
• Likelihood Kernel (Block 8)
• Feasibility record (Block 6)
• Failure Attribution (Block 15)

Identifiability resolved here is final and irreversible for this MethodHash.

END METHODS BLOCK 7.

If you confirm, the next response will be METHODS BLOCK 8 — LIKELIHOOD KERNEL (ARTIFACT #1), expanded into full probabilistic semantics, heavy-tail mixture formalism, normalization guarantees, and admissible covariance structure.

© Robert R. Frost
2026-01-03