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

=== Assistant: Proceeding. ===
Proceeding. Below is METHODS BLOCK 6 — FEASIBILITY / PARAMETER FORCING, expanded into explicit admissibility logic, minimal-dimension forcing, rank conditions, and irreversible collapse criteria. This block governs whether the frozen model can even be instantiated against data without violating axioms.

METHODS BLOCK 6 — FEASIBILITY / PARAMETER FORCING

OBJECTIVE
Determine whether there exists any admissible parameterization of the frozen model that satisfies all structural constraints. This block precedes likelihood and Bayesian evidence. It answers: can the theory even be instantiated without breaking itself?

FEASIBILITY DEFINITION
Define feasibility predicate:

Feasible(D) = 1 iff ∃ θ ∈ M_IR such that
• V_struct(θ)=0
• P(θ,b) defined for all required observables
• θ respects all manifold, envelope, and type constraints

Else Feasible(D)=0.

Existence only. No optimality. No preference over θ.

SEARCH SPACE
θ is searched only within M_IR.
No extension of parameter space allowed.
No relaxation of constraints allowed.

If M_IR empty ⇒ Feasible=0 immediately.

PARAMETER COUNT
Let k = dim(M_IR).
Let n_eff = effective number of independent constraints induced by D after identifiability projection (see Block 7).

Define minimal admissible dimension:

k_min = rank(Fisher) evaluated on admissible θ (formalized next block).

FEASIBILITY OUTCOMES
Three mutually exclusive outcomes:
# OK Condition: Feasible=1 AND k = k_min. Interpretation: model admits a non-degenerate instantiation without redundancy.
# FORCED Condition: Feasible=1 AND k > k_min. Interpretation: data force a reduction in effective parameter dimension; redundant directions exist. Action: parameters in nullspace(Fisher) MUST be projected out before likelihood evaluation. No new parameters may be introduced. FORCED is not collapse but is recorded.
# COLLAPSE Condition: Feasible=0. Interpretation: no admissible θ exists that satisfies structural constraints. Verdict forced to COLLAPSE independent of evidence.

NO OPTIMIZATION RULE
Forbidden:
• argminθ ℛ(θ)
• argmaxθ ℒ(θ)
• gradient-based searches
• adaptive proposal schemes

Permitted:
• Existence checks
• Constraint satisfaction checks
• Boundary evaluations if analytically required

Feasibility is a logical predicate, not a numerical minimum.

PARAMETER FORCING OPERATOR
Define forcing operator:

F_force: Θ → Θ_reduced
Θ_reduced = Θ / Null(Fisher)

Application of F_force is mandatory if outcome=FORCED.
Failure to apply ⇒ INVALID.

IRREVERSIBILITY
If outcome=COLLAPSE for dataset D, then for any D′ ⊃ D:
Outcome(D′)=COLLAPSE necessarily.

No later data may restore feasibility.

FAILURE MODES
• Attempt to expand M_IR ⇒ INVALID
• Attempt to introduce nuisance parameters ⇒ INVALID
• Attempt to relax τ_struct ⇒ INVALID
• Attempt to bypass feasibility and proceed to likelihood ⇒ INVALID

ARTIFACT EMISSION
Emit immutable record:
• Feasible flag
• Outcome ∈{OK,FORCED,COLLAPSE}
• k, k_min
• Projection operator if FORCED

Feeds directly into:
• Block 7 (Identifiability)
• Block 12 (Global Collapse)
• Block 15 (Failure Attribution)

END METHODS BLOCK 6.

If you confirm, the next response will be METHODS BLOCK 7 — PARAMETER IDENTIFIABILITY (ARTIFACT #2), expanded into Fisher geometry, nullspace projection, and basis invariance guarantees.

© Robert R. Frost
2026-01-03