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

=== Assistant: Proceeding. ===
Proceeding. Below is METHODS BLOCK 5 — STRUCTURAL VIOLATION, expanded into explicit tolerance geometry, hard-constraint semantics, monotonic collapse logic, and machine-enforceable failure conditions. This block implements non-statistical falsification. Statistical evidence is irrelevant here.

METHODS BLOCK 5 — STRUCTURAL VIOLATION

OBJECTIVE
Detect violations of the frozen structural constraints of the theory. Structural violations are decisive, binary, and non-probabilistic. Any violation here is sufficient for collapse regardless of likelihood or Bayesian evidence.

TOLERANCE DEFINITION
Define global structural tolerance τ_struct > 0, fixed pre-run and frozen under A2.

τ_struct applies uniformly across all residual components unless explicitly block-indexed. No adaptive tolerances permitted.

Residual-wise tolerance:
For each datum i, define admissibility condition:

Sᵢ(θ) = [ rᵢ(θ)ᵀ Σᵢ⁻¹ rᵢ(θ) ≤ τ_struct ]

STRUCTURAL VIOLATION OPERATOR
Define structural violation indicator:

V_struct(θ) = 1 iff ∃ i such that Sᵢ(θ) = FALSE
V_struct(θ) = 0 otherwise

This operator is Boolean-valued and irreversible.

GEOMETRIC INTERPRETATION
Each Σᵢ defines an ellipsoid in residual space.
τ_struct defines a hard boundary on squared Mahalanobis distance.

Violations correspond to residuals exiting admissible ellipsoids.

No probabilistic interpretation. This is not a confidence region.

PARAMETER-INDEPENDENT CHECK
Structural violation may be evaluated at:
• Any θ ∈ M_IR
• The feasibility-check θ
• Boundary points of M_IR if required

If ∀ θ ∈ M_IR, V_struct(θ)=1 ⇒ infeasible ⇒ collapse.

No minimization over θ allowed. No “best-fit” escape.

NO AVERAGING
Forbidden:
• Averaging violations
• Counting fraction violated
• Weighting by Σᵢ
• Trading off violations across i

Single violation sufficient.

MONOTONICITY
If V_struct=1 for dataset D, then for any superset D′ ⊃ D:
V_struct(D′)=1 necessarily.

This enforces A4 monotonicity to falsification.

FAILURE MODES
• τ_struct undefined ⇒ INVALID
• τ_struct modified post-freeze ⇒ INVALID
• Any NaN/Inf in rᵢ or Σᵢ⁻¹ ⇒ INVALID
• Attempted softening of violation ⇒ INVALID

OUTPUT ARTIFACT
Emit:
• For each i: Sᵢ(θ)
• Index set I_fail = { i | Sᵢ(θ)=FALSE }
• StructuralViolationFlag = V_struct

These artifacts are immutable and feed directly into Block 12 (Global Collapse) and Block 15 (Failure Attribution).

ROLE IN PIPELINE
Structural Violation is evaluated before:
• Feasibility / Parameter Forcing
• Likelihood
• Bayesian Evidence
• Monte Carlo

If V_struct=1 ⇒ downstream blocks may execute only for attribution, not for rescue.

END METHODS BLOCK 5.

If you confirm, the next response will be METHODS BLOCK 6 — FEASIBILITY / PARAMETER FORCING, expanded into admissibility logic, minimal dimension forcing, and irreversible collapse criteria.

© Robert R. Frost
2026-01-03