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

=== Assistant: Proceeding. ===
Proceeding. Below is METHODS BLOCK 11 — INTEGRATION CONTRACT (ARTIFACT #3), expanded to maximum algorithmic depth with machine-learning–style rigor (deterministic sampling, convergence certification, numerical contracts, and audit hooks). This block specifies how integrals are computed, not what is inferred. Any deviation invalidates evidence.

METHODS BLOCK 11 — INTEGRATION CONTRACT (ARTIFACT #3)

OBJECTIVE
Provide a deterministic, auditable, architecture-stable numerical integration contract for evaluating the Bayesian evidence integral defined in Block 10. This block specifies the only admissible integration algorithm and its invariants. No alternative integrators permitted.

INTEGRATOR IDENTITY
Integrator = NestedSampling
Variant = Deterministic Nested Sampling (DNS) with fixed pseudo-random sequence
All randomness is seeded and replayable.

Formal type:
Integrator : (ℒ, π, Θ_id, settings) → ⟨Z, diagnostics⟩

PRECONDITIONS
• Likelihood kernel ℒ normalized and frozen
• Prior π proper and frozen
• Identifiable parameter space Θ_id compact or compactified
• No collapse already triggered except attribution-only mode

If any precondition fails ⇒ INVALID.

PARAMETER SPACE PREPARATION
Define Θ_id ⊂ ℝ^{k_min}.
If Θ_id unbounded, enforce deterministic compactification via monotone bijection (e.g., tanh transform) declared pre-run and hashed.

Transform Jacobian included exactly in ℒ. No approximations.

LIVE POINT INITIALIZATION
Define N_live ∈ ℕ⁺, fixed pre-run.

Initialize live point set L₀ = {θ_j}ⱼ=1…N_live by deterministic low-discrepancy sequence (e.g., Sobol), seeded with s.

Constraints:
• Coverage uniform under π
• No rejection sampling with adaptive criteria
• Initialization reproducible bitwise

ML analogy: fixed exploration grid, not stochastic minibatches.

ITERATIVE UPDATE
At iteration t:
# Identify worst live point θ_w with minimum likelihood ℒ_w
# Record weight w_t corresponding to prior volume shrinkage
# Replace θ_w with new θ′ sampled from π subject to ℒ(θ′) > ℒ_w

Sampling constraint:
• Constraint satisfaction only
• No gradient use
• No learned proposal
• Deterministic accept/reject using seeded sequence

PRIOR VOLUME EVOLUTION
Prior volume X_t evolves deterministically:

X_t = exp(−t / N_live)

Weights computed exactly:

Z ≈ ∑t ℒ_t · (X{t−1} − X_t)

No stochastic estimators. No bootstrapping.

CONVERGENCE CRITERION
Define absolute evidence tolerance δ_Z > 0, fixed pre-run.

Terminate when:

Remaining evidence mass ≤ δ_Z

Formally:
max(ℒ_live) · X_t ≤ δ_Z

Relative tolerance forbidden.

ML analogy: absolute error stopping, not relative validation loss.

DIAGNOSTICS
Compute and emit:

• Effective sample size
• Evidence accumulation trace
• Maximum remaining contribution
• Iteration count
• Deterministic acceptance rate

Diagnostics are informational only. They do not alter execution.

DETERMINISM REQUIREMENTS
• Seed s fixed and emitted
• Floating-point order fixed
• Parallelism either forbidden or strictly ordered
• Any nondeterminism ⇒ INVALID

Two runs with same inputs MUST yield identical Z bitwise (within floating-point determinism guarantees).

FAILURE CONDITIONS
• Failure to find θ′ satisfying ℒ > ℒ_w within bounded attempts ⇒ INVALID
• Numerical underflow/overflow ⇒ INVALID
• Violation of volume monotonicity ⇒ INVALID
• Early termination without convergence ⇒ INVALID

NO ADAPTATION
Forbidden:
• Adaptive N_live
• Adaptive proposals
• Dynamic tolerance
• Heuristic stopping

Integrator behavior is fully specified and frozen.

ATTRIBUTION MODE
If upstream collapse already triggered:
• Integration may still run
• Z computed but flagged non-authoritative
• Used only for failure attribution graph

ARTIFACT EMISSION
Emit immutable IntegrationRecord:

⟨
IntegratorID,
N_live,
Seed,
δ_Z,
IterationCount,
Z,
ConvergenceFlag,
DiagnosticsHash
⟩

Hashed and audit-logged.

PIPELINE ROLE
Integration output feeds exclusively into:
• Bayesian Evidence decision (Block 10)
• Monte Carlo robustness checks (Block 13)
• Failure Attribution (Block 15)

ML INTERPRETATION
This block implements exact model scoring under a frozen hypothesis class:

• No learning
• No tuning
• No stochastic training dynamics
• Pure numerical marginalization

Comparable to evaluating a fixed probabilistic program, not training an ML model.

END METHODS BLOCK 11.

If you confirm, the next block will be METHODS BLOCK 12 — GLOBAL COLLAPSE, expanded into irreversible decision logic, short-circuit semantics, and certification interaction.

© Robert R. Frost
2026-01-03