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

=== Assistant: Proceeding. ===
Proceeding. Below is METHODS BLOCK 3 — MODEL MAP, expanded into a fully explicit, machine-enforceable operator specification with topological, geometric, and admissibility constraints. All predictive content of the theory is confined to this block. No downstream block may introduce new structure.

METHODS BLOCK 3 — MODEL MAP

OBJECTIVE
Define the complete, frozen mapping from parameters and baseline inputs to predicted observables. This block defines what the theory says, and nothing else may add to it.

PREDICTION OPERATOR
Define deterministic operator
P: Θ × B → ℝⁿ
where
Θ ⊂ ℝᵏ is the parameter space
B ≡ b ∈ ℝᵐ is the frozen baseline
Output dimension n fixed and declared pre-run.

Constraints:
• P is total on Θ×B
• P is deterministic
• P has no side effects
• P has no hidden state
• P is pure: identical inputs ⇒ identical outputs

No stochasticity allowed in P.

BASELINE
b is fixed, immutable, and globally visible.
b may include constants, calibration offsets, reference scales.
b may not depend on data D or parameters θ.

Any attempt to alter b post-initialization ⇒ INVALID.

IR MANIFOLD
Define admissible parameter manifold M_IR ⊂ Θ.

Formal properties:
• M_IR closed
• M_IR connected
• M_IR finite-dimensional
• Dim(M_IR)=k_fixed
• Boundary ∂M_IR explicitly defined or empty

All θ used in evaluation must satisfy θ ∈ M_IR.
If θ ∉ M_IR ⇒ infeasible.

No extension of M_IR allowed post-freeze.

IR ENVELOPE
Define prediction envelope E_IR ⊂ ℝⁿ.

E_IR constrains allowed predictions irrespective of data.
Formal constraint:
∀ θ ∈ M_IR: P(θ,b) ∈ E_IR.

E_IR may encode physical admissibility, causality bounds, positivity, monotonicity, or known symmetries.

Envelope violation is structural, not statistical.

GEOMETRIC CONSISTENCY
Define Jacobian J(θ)=∂P/∂θ.

Constraints:
• J exists almost everywhere on M_IR
• J finite and bounded
• Rank(J) ≤ k_fixed
• No singularities permitted unless explicitly declared and masked

Topology:
• No bifurcations allowed
• No branch switching
• Prediction surface is single-valued over M_IR

MODEL COMPLETENESS
All observables in dataset D must map to components of P.
No partial predictions allowed.

If ∃ datum yᵢ with no corresponding P-component ⇒ INVALID.

NO DATA DEPENDENCE
P, M_IR, and E_IR must not reference D, Σ, likelihood, or residuals.
Any data-conditioned prediction logic ⇒ INVALID.

STATIC ANALYSIS CHECKS
• Operator signature matches Types block
• Output dimension matches dataset dimension
• All constraints checkable without data access

FAILURE MODES
• Envelope violation ⇒ STRUCTURAL failure
• Undefined P(θ,b) ⇒ INVALID
• Dimension mismatch ⇒ INVALID

This block defines the entire hypothesis space. Nothing outside this block is part of the theory.

END METHODS BLOCK 3.

If you confirm, the next response will be METHODS BLOCK 4 — RESIDUAL OPERATOR, expanded into residual field construction, metric structure, and invariance properties.

© Robert R. Frost
2026-01-03