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

=== Assistant: Proceeding. ===
Proceeding. Below is METHODS BLOCK 1 — AXIOMS (HARD), expanded to maximum formal density, machine-operational semantics, and enforcement logic. All axioms are non-derivable, non-negotiable, and globally binding. Violation of any axiom forces INVALID regardless of downstream behavior.

METHODS BLOCK 1 — AXIOMS (HARD)

Axiom Set A = {A0,…,A7}.
Each axiom is enforced as a Boolean invariant evaluated at runtime.
Let Violation(Ai)=1 indicate violation.
Global invariant: ∑ᵢ Violation(Ai)=0 required for validity.

A0 — EMPIRICAL AUTHORITY
Statement: Epistemic authority holds iff realized empirical data exist and are admitted.
Formalization:
∀ execution E, ∃ dataset D s.t. 𝒬(D)=VALID.
No operator, parameter, likelihood, or verdict may be evaluated in absence of VALID empirical data.

Enforcement:
• If D=∅ OR 𝒬(D)=INVALID ⇒ abort
• Synthetic, simulated, interpolated, or bootstrapped data forbidden at authority level
• Empirical data defined as externally realized, time-stamped, instrument-derived measurements

Failure mode:
Violation(A0)=1 ⇒ INVALID, Verdict undefined.

A1 — NO SYNTHETIC DATA INSTANTIATION
Statement: Synthetic data may not instantiate operators or satisfy axioms.
Formalization:
∀ operator O, inputs(O) ⊄ D_synth where D_synth ≡ data generated by model, MC, resampling, GAN, or prior predictive draw.

Permitted:
• Synthetic data ONLY for robustness Monte Carlo after verdict logic frozen
• Synthetic draws may not influence likelihood normalization, Bayes evidence, feasibility, or collapse decision

Enforcement:
• All datasets tagged with provenance hash
• Any provenance hash matching internal generator ⇒ INVALID

Violation(A1)=1 ⇒ INVALID.

A2 — STRUCTURAL FREEZE PRE-RUN
Statement: All structure frozen prior to execution.
Formalization:
Let S_struct = {operators, types, thresholds, tolerances, priors, integrator settings}.
∀ t>t₀, S_struct(t)=S_struct(t₀).

No dynamic model extension, hyperparameter tuning, threshold adaptation, or rule injection allowed.

Enforcement:
• Structural hash computed before run
• Any mutation detected ⇒ abort

Violation(A2)=1 ⇒ INVALID.

A3 — BINARY, IRREVERSIBLE DECISIONS
Statement: All terminal decisions are binary and irreversible.
Formalization:
Verdict ∈ {0,1}.
Once Verdict=1 (COLLAPSE), ∀ future executions with same MethodHash ⇒ Verdict=1.

No probabilistic verdicts. No confidence scores. No soft thresholds.

Enforcement:
• Verdict stored immutably
• Re-execution cannot revert collapse

Violation(A3)=1 ⇒ INVALID.

A4 — MONOTONICITY TO FALSIFICATION
Statement: Additional data may only preserve or worsen theory status.
Formalization:
Let D₁⊂D₂ ⇒ Verdict(D₂) ≥ Verdict(D₁).

No data may “rescue” a collapsed theory.

Enforcement:
• Historical verdicts cached
• If Verdict(D₁)=1 AND Verdict(D₂)=0 ⇒ INVALID

Violation(A4)=1 ⇒ INVALID.

A5 — NO OPTIMIZATION OBJECTIVES
Statement: The pipeline performs evaluation, not optimization.
Formalization:
∄ objective function J to be minimized or maximized.
∄ gradient descent, ascent, tuning, fitting loops.

Likelihood used solely for evidence comparison, not parameter improvement.

Enforcement:
• Static analysis forbids argmin/argmax loops
• No adaptive proposals except fixed integrator mechanics

Violation(A5)=1 ⇒ INVALID.

A6 — NO HIDDEN STATE
Statement: Execution is memoryless except for declared artifacts.
Formalization:
∀ executions E₁,E₂ with identical inputs ⇒ identical outputs.

No global caches, hidden priors, learned weights, or persistent mutable state.

Enforcement:
• All state serialized and hashed
• Any nondeterminism not seeded ⇒ INVALID

Violation(A6)=1 ⇒ INVALID.

A7 — REPRODUCIBILITY REQUIRED
Statement: Identical inputs yield identical outputs across architectures.
Formalization:
∀ platforms P₁,P₂:
Run(P₁,S,D,seed)=Run(P₂,S,D,seed).

Enforcement:
• Fixed seeds everywhere
• Deterministic integrator
• Floating-point nondeterminism checks

Violation(A7)=1 ⇒ INVALID.

GLOBAL AXIOM CHECK
Define AxiomValid = ∏ᵢ (1−Violation(Ai)).
If AxiomValid=0 ⇒ Abort before Block 2.

These axioms dominate all subsequent blocks. No block may override, weaken, or reinterpret them.

END METHODS BLOCK 1.

If you confirm, the next response will be METHODS BLOCK 2 — TYPES, expanded into strict type theory, dimensional constraints, covariance legality, and static type-checking rules for machine enforcement.

© Robert R. Frost
2026-01-03